waves, particles, and interactions in reduced dimensions
TRANSCRIPT
Waves, Particles, and Interactions in Reduced Dimensions
A dissertation presentedby
Yiming Zhang
toThe Department of Physics
in partial fulfillment of the requirementsfor the degree of
Doctor of Philosophyin the subject of
Physics
Harvard UniversityCambridge, Massachusetts
2009
c© 2009 by Yiming ZhangAll rights reserved.
Dissertation Advisor: Professor Charles M. Marcus Author: Yiming Zhang
Waves, Particles, and Interactions in Reduced Dimensions
Abstract
This thesis presents a set of experiments that study the interplay between the wave-
particle duality of electrons and the interaction effects in systems of reduced dimensions.
Both dc transport and measurements of current noise have been employed in the studies; in
particular, techniques for efficiently measuring current noise have been developed specifically
for these experiments.
The first four experiments study current noise auto- and cross correlations in various
mesoscopic devices, including quantum point contacts, single and double quantum dots,
and graphene devices.
In quantum point contacts, shot noise at zero magnetic field exhibits an asymmetry
related to the 0.7 structure in conductance. The asymmetry in noise evolves smoothly into
the symmetric signature of spin-resolved electron transmission at high field. Comparison to
a phenomenological model with density-dependent level splitting yields good quantitative
agreement. Additionally, a device-specific contribution to the finite-bias noise, particularly
visible on conductance plateaus where shot noise vanishes, agrees with a model of bias-
dependent electron heating.
In a three-lead single quantum dot and a capacitively coupled double quantum dot, sign
reversal of noise cross correlations have been observed in the Coulomb blockade regime, and
found to be tunable by gate voltages and source-drain bias. In the limit of weak output
tunneling, cross correlations in the three-lead dot are found to be proportional to the two-
lead noise in excess of the Poissonian value. These results can be reproduced with master
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equation calculations that include multi-level transport in the single dot, and inter-dot
charging energy in the double dot.
Shot noise measurements in single-layer graphene devices reveal a Fano factor indepen-
dent of carrier type and density, device geometry, and the presence of a p-n junction. This
result contrasts with theory for ballistic graphene sheets and junctions, suggesting that the
transport is disorder dominated.
The next two experiments study magnetoresistance oscillations in electronic Fabry-
Perot interferometers in the integer quantum Hall regime. Two types of resistance oscilla-
tions, as a function of perpendicular magnetic field and gate voltages, in two interferometers
of different sizes can be distinguished by three experimental signatures. The oscillations
observed in the small (2.0 µm2) device are understood to arise from Coulomb blockade, and
those observed in the big (18 µm2) device from Aharonov-Bohm interference. Nonlinear
transport in the big device reveals a checkerboard-like pattern of conductance oscillations as
a function of dc bias and magnetic field. Edge-state velocities extracted from the checker-
board data are compared to model calculations and found to be consistent with a crossover
from skipping orbits at low fields to ~E× ~B drift at high fields. Suppression of visibility as a
function of bias and magnetic field is accounted for by including energy- and field-dependent
dephasing of edge electrons.
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Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction 11.1 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Dc transport and current noise . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Dc transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Current noise auto- and cross correlations . . . . . . . . . . . . . . . 5
1.3 Material systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.1 GaAs/AlGaAs heterostructures . . . . . . . . . . . . . . . . . . . . . 9
1.4 Basic properties of mesoscopic devices . . . . . . . . . . . . . . . . . . . . . 121.4.1 Quantum point contacts . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.2 Quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.3 Quantum Hall effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 System for measuring auto- and cross correlation of current noise at lowtemperatures 232.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Overview of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 Design objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.2 Overview of the circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.3 Operating point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.4 Passive components . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.5 Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Digitization and FFT processing . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Measurement example: quantum point contact . . . . . . . . . . . . . . . . 32
2.5.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.2 Measuring dc transport . . . . . . . . . . . . . . . . . . . . . . . . . 332.5.3 Measuring noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5.4 System calibration using Johnson noise . . . . . . . . . . . . . . . . 35
2.6 System performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Current noise in quantum point contacts 41
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3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 QPC characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Current noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 0.7 structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3.2 Bias-dependent electron heating . . . . . . . . . . . . . . . . . . . . 51
3.4 Conclusion and acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 52
4 Tunable noise cross-correlations in a double quantum dot 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.4 Double-dot characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.5 Sign-reversal of noise cross correlation . . . . . . . . . . . . . . . . . . . . . 584.6 Master equation simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.7 Intuitive explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.8 Some additional checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.9 Conclusion and acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 63
5 Noise correlations in a Coulomb blockaded quantum dot 645.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2 Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 Noise in the two-lead configuration . . . . . . . . . . . . . . . . . . . . . . . 685.5 Noise in the three-lead configuration . . . . . . . . . . . . . . . . . . . . . . 725.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6 Shot noise in graphene 766.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.3 Shot noise in single-layer devices . . . . . . . . . . . . . . . . . . . . . . . . 796.4 Shot noise in a p-n junction . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.5 Shot noise in a multi-layer device . . . . . . . . . . . . . . . . . . . . . . . . 836.6 Summary and acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 85
7 Distinct Signatures For Coulomb Blockade and Aharonov-Bohm Interfer-ence in Electronic Fabry-Perot Interferometers 877.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.2 Device and measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.3 Resistance oscillations in the 2.0 µm2 device . . . . . . . . . . . . . . . . . . 907.4 Resistance oscillations in the 18 µm2 device . . . . . . . . . . . . . . . . . . 937.5 One more signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8 Edge-State Velocity and Coherence in a Quantum Hall Fabry-Perot In-terferometer 998.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008.2 Device and measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
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8.3 Checkerboard pattern and interpretation . . . . . . . . . . . . . . . . . . . . 1038.4 Edge-state velocity and energy-dependent dephasing . . . . . . . . . . . . . 1068.5 Nonlinear magnetoconductance in a 2 µm2 device . . . . . . . . . . . . . . . 1088.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
9 Unpublished results 1119.1 Current noise modulated by charge noise . . . . . . . . . . . . . . . . . . . . 1129.2 Quasi-particle tunneling between filling factor 2 and 3 in a constriction . . . 1189.3 The 3/2 quantized plateau in quantum point contacts . . . . . . . . . . . . 1249.4 Non-linear transport in N ≥ 2 Landau levels . . . . . . . . . . . . . . . . . 128
A Fridge Wiring: Thermal Anchoring and Filtering 131A.1 Simple RC filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132A.2 Sapphire heat sinks and circuit boards . . . . . . . . . . . . . . . . . . . . . 133A.3 Mini-circuit VLFX filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135A.4 Thermocoax cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
B Igor implementation of virtual DACs 140B.1 Igor implementation of virtual DACs . . . . . . . . . . . . . . . . . . . . . . 141
C Effects of external impedance on conductance and noise 152C.1 Effects of external impedance on conductance . . . . . . . . . . . . . . . . . 152C.2 Effects of external impedance on current noise . . . . . . . . . . . . . . . . . 153
D Conductance matrix measurement and multi-channel digital lock-in 156D.1 Simultaneous conductance matrix and current noise measurement . . . . . . 156D.2 Multi-channel digital lock-in . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
E The master equation calculation of current and noise in a multi-lead,multi-level quantum dot 161
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List of Figures
1.1 Different types of noise, in time and frequency domains . . . . . . . . . . . 71.2 GaAs/AlGaAs heterostructure and conduction band diagram . . . . . . . . 101.3 Conductance as a function of gate voltage in a quantum point contact . . . 131.4 Illustration of a source-drain voltage applied across a barrier in a 1d system 131.5 Conductance as a function of gate voltage in a quantum dot . . . . . . . . . 171.6 Differential conductance as a function of source-drain bias and gate voltage
in a quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.7 Simulations of differential conductance as a function of source-drain bias and
gate voltage of a quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . 191.8 Bulk 2DEG transport in the quantum Hall regime . . . . . . . . . . . . . . 21
2.1 Block diagram of the two-channel noise detection system . . . . . . . . . . . 262.2 Schematic diagram of the amplification lines . . . . . . . . . . . . . . . . . . 272.3 Equivalent circuits valid near dc and at low megahertz . . . . . . . . . . . . 282.4 Biasing the cryogenic transistor . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Setup for QPC noise measurement by cross-correlation technique . . . . . . 332.6 Power and cross-spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 Circuit model for QPC noise measurements extraction . . . . . . . . . . . . 362.8 Johnson-noise thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.9 Noise measurement resolution as a function of integration time . . . . . . . 38
3.1 QPC characterization by dc transport . . . . . . . . . . . . . . . . . . . . . 433.2 Noise measurement setup and micrograph of QPC . . . . . . . . . . . . . . 453.3 Demonstration measurements of bias-dependent QPC noise . . . . . . . . . 463.4 The experimental noise factor . . . . . . . . . . . . . . . . . . . . . . . . . . 483.5 Comparison of QPC noise data to the phenomenological Reilly model . . . 503.6 Bias-dependent electron heating in a second QPC . . . . . . . . . . . . . . . 51
4.1 Double-dot device and setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Measured and simulated cross-spectral density near a honeycomb vertex . . 584.3 Energy level diagrams in the vicinity of a honeycomb vertex . . . . . . . . . 614.4 Measured cross-spectral density at other bias configurations . . . . . . . . . 63
5.1 Micrograph of three-lead quantum dot, noise measurement setup, and mea-surements in the two-lead configuration . . . . . . . . . . . . . . . . . . . . 67
5.2 Excess Poissonian noise in the two-lead configuration . . . . . . . . . . . . . 715.3 Cross-spectral density in the three-lead configuration . . . . . . . . . . . . . 735.4 Relation between total excess Poissonian noise and cross-spectral density in
the three-lead configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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6.1 Characterization of graphene devices using dc transport at B⊥ = 0 and inquantum Hall regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2 Shot noise in single-layer devices . . . . . . . . . . . . . . . . . . . . . . . . 806.3 Shot noise in a p-n junction . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.4 Shot noise in a multi-layer device . . . . . . . . . . . . . . . . . . . . . . . . 84
7.1 Measurement setup and the electronic Fabry-Perot devices . . . . . . . . . . 907.2 Resistance oscillations as a function of magnetic field for the 2.0 µm2 device 917.3 Magnetic field and gate voltage periods for the 2.0 µm2 device . . . . . . . 927.4 Magnetic field and gate voltage periods for the 18 µm2 device . . . . . . . . 947.5 Resistance oscillations measured in a plane of magnetic field and gate voltage 96
8.1 Measurement setup and the electronic Fabry-Perot device . . . . . . . . . . 1028.2 Nonlinear magnetoconductance in an 18 µm2 interferometer . . . . . . . . . 1048.3 Magnetic field dependence of extracted velocity and damping factor . . . . 1068.4 Nonlinear magnetoconductance in a 2 µm2 device . . . . . . . . . . . . . . . 109
9.1 Device, noise measurement setup, and charge sensing in conductance . . . . 1149.2 Conductance and current noise near a honey-comb vertex . . . . . . . . . . 1159.3 Maximum current noise as a function of barrier transparency of a nearby dot 1179.4 Device and measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . 1199.5 Diagonal resistance as a function of dc current and magnetic field, at various
temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209.6 An example of the best fit to bias- and temperature-dependent tunneling
data with the weak tunneling formula . . . . . . . . . . . . . . . . . . . . . 1219.7 Fit error as a function of prefixed e∗ and g . . . . . . . . . . . . . . . . . . . 1229.8 Best-fit e∗ and g as a function of R0
D . . . . . . . . . . . . . . . . . . . . . . 1229.9 Bulk Hall resistance and diagonal resistance as a function of magnetic field,
showing 3/2 quantized plateaus in two different QPCs . . . . . . . . . . . . 1259.10 Temperature and bias dependence of the 3/2 plateaus . . . . . . . . . . . . 1269.11 Bulk Hall and longitudinal resistances as a function of magnetic field, at two
different temperatures and with two different crystal directions . . . . . . . 1299.12 Bulk longitudinal and Hall resistances as a function of dc current and mag-
netic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.1 Photograph of a bank of simple RC filters . . . . . . . . . . . . . . . . . . . 132A.2 Photographs of sapphire circuit boards and heat sinks . . . . . . . . . . . . 135A.3 18 Mini-circuits VLFX-80 filters assembled with the cold finger . . . . . . . 136A.4 Quantum Hall bulk transport . . . . . . . . . . . . . . . . . . . . . . . . . . 136A.5 Thermalcoax cables assembled on the Microsoft fridge . . . . . . . . . . . . 139
B.1 Channel definition table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141B.2 Virtual DACs parameters table . . . . . . . . . . . . . . . . . . . . . . . . . 143
C.1 Circuit schematics for calculating intrinsic conductance . . . . . . . . . . . 153C.2 Circuit schematics for calculating intrinsic current noise . . . . . . . . . . . 154
D.1 Circuit schematics for measuring conductance matrix and two-channel cur-rent noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
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D.2 Multi-channel digital lock-in control panel . . . . . . . . . . . . . . . . . . . 158
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Acknowledgements
Writing the acknowledgement section of my thesis signals the end of a long journey,
and gives me a rare chance to look back at my six-year-long Ph.D. life doing experimental
condensed matter physics research at the Marcus lab, Harvard University. There are so
many people who have contributed to make this thesis possible, and I am deeply grateful
to all.
I would like to first thank my advisor Charlie Marcus, for taking me into his research
group, for his superb guidance and support over the years, for his deep insight into experi-
mental physics and unending steam of creative ideas, for educating me the fine art of being
an experimentalist, for the encouragement to open discussions, for teaching me to avoid
using the word “problems” unless precedented by the the word “solved”, and to replace
the use of “I” by “we”, for providing such a well-organized environment unmatched among
physics labs, and finally for teaching me hand-in-hand how to make a perfect latte. I will
cherish all of them as treasuries for life.
I would like to next extend special thanks to Leo DiCarlo and Doug McClure. Leo
had been my day-to-day mentor since I joined the lab the first day, teaching me almost
everything such as making a purchase order, wire-bonding, cryogenic techniques and low-
noise electronics. Doug, joining the lab as an undergraduate a few month before I did,
has been my collaborator on every single project. I feel especially fortunate to be able
to form the “noise team” with Leo and Doug. Our perseverance finally led us out of the
1/f noise dominated regime. By uniting everyone’s strengths, we were able to develop a
noise measurement setup of the state-of-the-art performance, with which we had achieved
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fruitful results. After Leo’s departure to Yale as a post-doc for Rob Schoelkopf, Doug
and I continued our collaboration on the 5/2 project, and shared some initial success on
the quantum-Hall interferometry experiments. Over the years, I have been really glad to
observe that Doug has matured from a programming genius to a careful experimentalist
now taking the leading role in a very rewarding project.
I should thank all other lab mates, who have defined the unique culture of the Marcus
group with his or her talents and personalities. Reilly, as we call him, not only introduced to
us the “Reilly model” of the QPC 0.7 structure, but he has also been a great source of low-
temperature and high-frequency knowledge—in fact, he owns a vast breath of knowledge,
as he has been known as Reilli-pedia. Jeff Miller is both a fine scientist and a fine artist,
taking amazing pictures out of almost anything. In addition, he and Reilly both possess
the talent of mimicking lab mates, or professors, bringing unlimited supply of humor to the
lab. I am thankful to be able to work with Michi Yamamoto visiting from University of
Tokyo and Reinier Heeres visiting from Delft. Michi had been a very quick learner and very
helpful collaborator, helping me with the single-dot noise measurement, and after returning
to Japan, he was able to install and make improvements on the noise measurement setup in
the Tarucha group. Talented and hard-working, Reinier has helped me with both developing
the “Virtual DACs” routines, and getting the electrons cold (below 20 mK) in the Microsoft
fridge, after dozens of thermal cycles, fixing a newly-emerged problem during each one. I
must thank all other fellow students, post-docs, and visitors, with whom I have had the
privilege to work and have fun with, including Nadya Mason, Jason Petta, Slaven Garaj,
Ferdinand Kuemmeth, Max Lemme, Alex Johnson, Dominik Zumbuhl, Michael Biercuk,
Edward Laird, Jimmy Williams, Sang Chu (who taught me fab), Christian Barthel, Angela
Kou, Vivek Venkatachalam (from Amir Yacoby’s group), Eli Levenson-Falk, Hugh Churchill,
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Yongjie Hu (from Charlie Lieber’s group), Maja Cassidy, Patrick Herring, Nathaniel Craig,
Jennifer Harlow, Jacob Aptekar, Shu Nakaharai, Susan Watson, and more.
Many thanks to my collaborators on the various projects that I have been involved,
including David Reilly, Michi Yamamoto, Hansres Engel, Bernd Rosenow, Jimmy Williams
and Eli Levenson-Falk, and to the material providers: Micah Hanson, Art Gossard at
UCSB, and Loren Pfeiffer, Ken West at Bell labs, without whom these projects will never be
possible. I would like to thank many fellow students, post-docs and leading scientists in the
field for helpful discussions, including Iuliana Radu, Jeff Miller, Izhar Neder, Nissim Ofek,
Bert Halperin, Marc Kastner, Amir Yacoby, Mike Stopa, Yigal Meir, David Goldhaber-
Gordon, Leonid Levitov, Moty Heiblum, Ady Stern, Claudio Chamon, Jim Eisenstein, Mike
Freedman, Xiao-Gang Wen, Alexei Kitaev, and many more with whom I have communicated
with. I also acknowledge the funding agencies that have supported my research during these
years, including NSF, ARO/ARDA/DTO, Harvard NSEC, Harvard CNS, Microsoft Project
Q and Harvard University.
Additionally, I would like to express my thanks to the lab administrators: James Got-
fredson, Danielle Reuter, and Jess Martin, for keeping the lab running smoothly. I would
like to thank Jim MacArthur at the electronics shop for building amazing DACs for us,
Mark Jackson, Nick Wilson, Nick Dent from Oxford Instruments for their assistance in fix-
ing the fridge problems, and Yuan Lu, JD Deng, Steve Shepard, Noah Clay, Ed Macomber,
John Tsakirgis at CNS for their help with device fabrications.
Moreover, I am thankful that I could be admitted into the Harvard physics program
six years ago. I am thankful to my teachers at Harvard, including Charlie Marcus, Bert
Halperin, Donheen Ham, David Nelson, Eugene Demler, Jene Golovchenko, Mark Irwin
and Yoonjung Lee. I am thankful to be able to gain teaching experience working with
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Paul Horowitz, Tom Hayes, and the other TF Anne Goodsell on the demanding course
Physics 123. I am thankful to Sheila Ferguson and other staff of the physics department for
their administrative and personal assistance. I am thankful that Prof. Bert Halperin, Prof.
Donhee Ham, and my advisor Charlie Marcus have agreed to be on my Ph.D. committee,
evaluating my progress and giving me support in every possible way.
I thank Benjamin N. Levy for being my host at Harvard, picking me up when I first
landed in the U.S., sharing meals and watching movies together. I thank all the friends I
met ever since the English Language Program in the summer of 2003, and I really appreciate
your company and friendship in a country so far away from my family.
Last but not least, I am deeply indebted to my parents for their love, trust, encourage-
ment, and never-ending support from around the globe in Hangzhou, China.
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Chapter 1
Introduction
In recent years, the advent of semiconductor technology, including the ability to grow
extremely pure and crystalline semiconductor materials, engineering of band-structures,
and advanced lithography methods, has enabled researchers to study numerous intriguing
quantum mechanical phenomena in systems of reduced dimensions [1, 2, 3, 4]. These studies
have been developed into a new branch of physics called mesoscopic physics, which aims
at understanding the world at length scales between the macroscopic and the microscopic
worlds.
When temperatures are lowered to within a few degrees or less from the absolute zero,
electrons in the mesoscopic world can behave sometimes like waves [4, 5, 6, 7, 8, 9, 10, 11],
and sometimes like particles [12, 13]; sometimes both properties coexist [14, 15, 16, 17,
18, 19, 20] and sometimes they compete [21, 22]. The presence of many-body interactions
introduces a whole new level of complications, producing phenomena such as the now-well-
understood Coulomb blockade [12, 13, 3], cotunneling [14, 18] and Kondo [17] physics in
quantum dots, the two-decade-old open question of the “0.7 structure” [23] in quantum
wires, and fractional quantum Hall effects [24] in two-dimensional electron systems under
a strong perpendicular magnetic field, with the possible existence of exotic non-Abelian
quasiparticles [25, 26]. Indeed, the wave-particle duality of electrons and the many-body
1
interaction effects, in systems of reduced dimensions, have been the main themes of research
for mesoscopic physics.
1.1 Organization of this thesis
Along these lines, this thesis will present several experimental projects that I have accom-
plished during the six years of my Ph.D. life at Harvard, exploring the interplay between the
wave-particle duality of electrons and the interaction effects, in various mesoscopic devices.
In the rest of this introductory chapter, I will first introduce electronic transport mea-
surements, including dc transport and current noise auto- and cross correlations. Then I
will give a brief description of the material systems, based on which mesoscopic devices are
made, in particular GaAs/AlGaAs heterostructures. Finally in this chapter, I will describe
the basic properties of the various types of mesoscopic devices to be used in subsequent
chapters.
Chapters 2 through 8 form the main body of this thesis, presenting seven projects that
have been published or submitted for publication. Chapter 2 describes the detailed construc-
tion and operation of the two-channel current noise auto- and cross correlation detection
system. Chapter 3 studies shot-noise signatures of the 0.7 structure and spin in quantum
point contacts, as well as bias-dependent electron heating. Chapter 4 realizes sign reversal
of noise cross correlations in a capacitively-couple double quantum dot in a fully controlled
way. Chapter 5 reports the observation of super-Poissonian auto-correlation and positive
cross correlation in a multi-lead quantum dot, and establishes a proportionality between
auto- and cross correlations. Chapter 6 studies shot noise in graphene devices, revealing an
almost constant Fano factor in single-layer devices, and a density-dependent Fano factor in
a multi-layer device. Chapter 7 describes the observation of three distinct signatures for
2
Coulomb blockade and Aharonov-Bohm interference in electronic Fabry-Perot interferome-
ters in the integer quantum Hall regime. In the large interferometer that Aharonov-Bohm
interference dominates, Chapter 8 studies edge-state velocity and bias-dependent dephasing
using non-linear transport. Then in the last chapter, I will describe four interesting, yet for
one reason or another unpublished experimental results.
The five appendices that follow provide more technical details related to the experiments
and theoretical calculations. Appendix A describes several methods of filtering and thermal
anchoring for fridge wiring, which are essential for achieving low electron temperatures.
Appendix B provides detailed explanation and source codes for implementing in Igor Pro
the set of tools called “Virtual DACs”, which allows easy definition and simultaneous control
of a linear combination of multiple independent parameters, and are especially useful when
working with devices of many gates. Appendix C gives derivations for the expressions used
to extract the intrinsic conductance matrix and multi-terminal current noise of the device
in the presence of finite-impedance external circuits. Appendix D describes the circuit
for simultaneous measurements of conductance matrix and multi-terminal current noise,
and also describes the operations of a multi-channel digital lock-in developed in-house and
capable of measuring 16 conductance matrix elements simultaneously. Finally, Appendix E
provides the Igor routines for the master equation calculation of current and noise in a
multi-lead, multi-level quantum dot.
3
1.2 Dc transport and current noise
This section will describe electronic transport measurements, including the conventional di-
rect current (dc) transport, and measurements of current noise auto- and cross correlations.
All chapters will study dc transport, while studies of current noise will be the focus for
Chs. 3 through 6.
1.2.1 Dc transport
Since the inception of mesoscopic physics in the 1980’s, dc transport has been the ubiqui-
tous tool, and has played a key role in many discoveries of the field, including integer [5]
and fractional [27] quantum Hall effects, conductance quantization [6, 28] in quantum point
contacts, Coulomb blockade [12, 13] in closed quantum dots, universal conductance fluctua-
tion [4] in open quantum dots, and Aharonov-Bohm interference in ring structures [7], and
in Mach-Zehnder interferometers [11], etc.
Dc transport measures current I through the device in response to a source-drain volt-
age excitation V , or vice versa, extracting the device resistance R = V/I or conductance
G = I/V . Although termed dc transport, the resistances and conductances are often mea-
sured with lock-ins at near dc frequencies (3− 1000 Hz), because the measurement is often
much more quiet away from dc due to the ubiquitous 1/f noise present in devices and in
instruments.
To avoid smearing out features, the lock-in excitation should be kept below the tem-
perature or some intrinsic energy scale of the device. The lock-in excitation can also be
superimposed with another dc bias, to measure differential resistance r = δV/δI or differ-
ential conductance g = δI/δV in the non-linear regime.
There are two ways of biasing for dc transport: voltage bias, applying V and measuring
4
I, is more suitable for devices with higher resistances (26 kΩ or higher); current bias, apply-
ing I and measuring V , on the other hand, is more suitable for devices of lower resistances.
There are also two types of measurement configurations: two-wire measurements, where
the voltage is either applied or probed at the same contacts as the source and the drain,
are sensitive to dc wire or ohmic contact resistances; four-wire measurements, where the
voltage probes are different from the source and drain contacts, on the other hand, can
eliminate the resistances from dc wires and ohmic contacts.
1.2.2 Current noise auto- and cross correlations
Current noise, the temporal fluctuation of currents, can yield complimentary information
to dc transport, such as electron temperature, transmission, quantum statistics, and many-
body interaction effects [19, 29, 30]. Chapters 3 through 6 provide studies of current noise
in various mesoscopic devices, including quantum point contacts [31, 32], single [33] and
double [34] quantum dots, as well as graphene devices [35].
We first need to understand two important concepts about current noise: power and
cross spectral densities, which are Fourier transforms of the auto- and cross correlation
functions of current fluctuations. Also, they have the physical meanings of power per unit
frequency, therefore they have the units of A2/Hz. In addition, they can be calculated by
taking the product of Fourier transforms of current fluctuations. The derivations of these
results are provided as follows.
Define the current Iα(t) at terminal α, and its Fourier transform Iα(f) as:
Iα(fn) =1T
∫ T
0Iα(t)e−i2πfntdt,
Iα(t) =+∞∑
n=−∞Iα(fn)ei2πfnt,
where T is the measurement time and fn = n/T . The time-averaged current is 〈Iα〉 =
5
1T
∫ T0 Iα(t)dt = Iα(0), and the total cross-correlated power for δIα(t) and δIβ(t) is1:
〈δIαδIβ〉 = 〈IαIβ〉 − 〈Iα〉〈Iβ〉
=+∞∑
n,m=−∞〈Iα(fn)Iβ(fm)ei2πfn+mt〉 − Iα(0)Iβ(0)
=+∞∑
n=−∞Iα(fn)Iβ(−fn)− Iα(0)Iβ(0)
=+∞∑n=1
Iα(fn)I∗β(fn) + I∗α(fn)Iβ(fn)
=∫ +∞
0Sαβ(f)df, (1.1)
and Sαβ(f) = T · [Iα(f)I∗β(f) + I∗α(f)Iβ(f)], (1.2)
where Sαβ(f) is the power (for α = β) or cross (for α 6= β) spectral density.
We may also write:
Sαβ(f) = T · [Iα(f)I∗β(f) + I∗α(f)Iβ(f)]
=1T
∫ T
0dt1
∫ T
0dt2〈δIα(t1)δIβ(t2) + δIα(t2)δIβ(t1)〉e−i2πf(t1−t2)
=2T
∫ T
0dτ
∫ +∞
−∞dt ·Kαβ(t)e−i2πft
= 2Kαβ(f), (1.3)
where τ = (t1 + t2)/2, t = t1 − t2, Kαβ(t) = 12〈δIα(t)δIβ(0) + δIα(0)δIβ(t)〉 is the auto- or
cross correlation function, and Kαβ(f) =∫ +∞−∞ dt ·Kαβ(t)e−i2πft is its Fourier transform.
To summarize, Eq. (1.1) provides literal meanings to the power and cross spectral den-
sities, Eq. (1.2) gives the relationship between them and the Fourier transforms of currents,
and Eq. (1.3) gives the relationship between them and the Fourier transforms of auto- and
cross correlation functions of currents.
Current noise can be classified into different categories according to their frequency
1If α 6= β, it is the total cross-correlated power; otherwise, it is the total power of δIα(t).
6
I
Time
(a)
SI
Frequency
(b)
I
Time
(c)
Log
SI
Log Frequency
(d)
1 / fI
Time
(e)
Log
S I
Log Frequency
(f)
1 / f 2
I
Time
(g)
Log
S I
Log Frequency
(h)
1 / f 2
I
Time
(i)
SI
Frequency
(j)
Figure 1.1: Different types of noise: (a,b) white noise, (c,d) 1/f noise, (e,f) Brownianmotion noise, (g,h) random telegraph noise, and (i,j) pick-ups or continuous-wave signals.They are presented in the time domain (a,c,e,g,i), and in the frequency domain as theirpower spectral densities (b,d,f,h,j).
dependence. Figure 1.1 shows different types of noise as a function of time, as well as
their power spectral densities, SI , as a function of frequency. White noise, implying the
auto-correlation to be a delta function in time, has a flat frequency dependence of SI , as
shown in Figs. 1.1(a) and (b). Flicker noise, also called 1/f noise, is the least understood
type; its power spectral density, as its name implies and shown in Figs. 1.1(c) and (d), has
a 1/f dependence on frequency. Brownian motion can be viewed as white noise integrated
over time, and its SI , as shown in Figs. 1.1(e) and (f), has a frequency dependence of 1/f2.
7
Random telegraph noise occurs when the current jumps between two meta-stable states; as
shown in Figs. 1.1(g) and (h), its power spectral density is flat up to a corner frequency
given by the transition rates2, before rolling off as 1/f2. Pick-ups or continuous-wave signals
concentrate all the power at a certain frequency, and their power spectral densities are delta
functions in frequency, as shown in Figs. 1.1(i) and (j).
Among these types, white noise is what we are most interested in, because most of the
electron dynamics in mesoscopic systems happen at much shorter time scales than we can
measure, thus they appear white to the measurement setup. Indeed, no types of noise can
be strictly white; otherwise the total power would be infinite. Random telegraph noise,
for example, is white up to a corner frequency set by the rates of dynamics [36]. For this
reason, the noise we and most other people measure is often referred to as zero-frequency
noise in literatures [19, 29, 30].
In particular, two types of white noise are of interest to us. At equilibrium without bias,
only thermal noise is present, and has been used for gain calibration and measurement of
electron temperature, as will be described in detail in the next chapter. At a finite source-
drain bias greater than temperature, shot noise, arising from the quantization of charge
and partial transmission, dominates. It is the shot noise that makes noise measurement
interesting, allowing researches such as shot-noise thermometry [37], detection of fractional
charge [38, 39, 40, 41], observation of two-particle interference [42], and investigation of
interaction effects [43, 44, 45, 46, 34, 33], etc. Our studies of shot noise in different systems
will be presented in Chs. 3 to 6, following Ch. 2, which describes in detail how we measure
noise power and cross spectral densities.
2The power spectral density for random telegraph noise has a Lorentzian shape. Pleasesee Ref. [36] and Sec. 9.1 in Ch. 9 for the explicit expression.
8
1.3 Material systems
The material systems of choice for mesoscopic physics research, including Si inversion
layers, GaAs/AlGaAs heterostructures, graphene, carbon nanotubes, and semiconductor
nanowires, often have one or more dimensions restricted, forming one- or two-dimensional
systems. These reduced dimensional systems are realized by the so-called size quantiza-
tion [1]: when the electron motion is restricted to within a certain size in one direction,
the kinetic energy associated with the motion in that direction becomes quantized and sub-
bands are formed; if the density is low enough such that the Fermi energy lies between
the lowest and the first excited sub-bands, and the temperature is much lower than the
sub-band energy spacing, the electron motion in that direction is frozen, effectively reduc-
ing the dimension by one and creating a two-dimensional system. In carbon nanotubes or
nanowires, two of the three dimensions are restricted by size quantization, thus they form
strictly one-dimensional systems.
In addition to reduced dimensions, these material systems often have several other
desired properties. First, they need to be contactable by electrodes, therefore can be probed
by transport measurements. Second, their carrier densities are usually quite low, giving large
Fermi wavelength (∼ 50nm) and allowing easy control by electrostatic gates, which can
further reduce the device dimensions down to zero. Finally, the quality of these materials
are quite high and their mean-free path quite long; in particular, the highest achieved
mobility in GaAs/AlGaAs heterostructures has exceeded 30, 000, 000 cm2/Vs [47, 41].
1.3.1 GaAs/AlGaAs heterostructures
Among these material systems, the GaAs/AlGaAs heterostructures, grown by molecular
beam epitaxy [48] under ultra high vacuum, are the most mature, most flexible and clean-
9
Figure 1.2: Schematic of the GaAs/AlGaAs heterostructure for the 010219B wafer grown byMicah Hanson and Arthur Gossard at UCSB. On the right is its conduction band diagram.(Figure adapted from Ref. [49]).
est. They have been used throughout this thesis except Ch. 6. Figure 1.2 shows a schematic
of the GaAs/AlGaAs heterostructure used in Chs. 3 to 5 and grown by Micah Hanson and
Arthur Gossard at UCSB. This heterostructure starts with bulk GaAs substrate, then a
layer of AlxGa1−xAs (x ∼ 0.3), and finally a thin (∼ 10 nm) GaAs cap layer to prevent
oxidation. Inside the AlGaAs layer closer to the lower GaAs/AlGaAs interface than the
upper one, there is a thin δ-doping layer of Si. As shown on the right in Fig. 1.2, the Si
δ-doping layer lowers the conductance band edge towards the Fermi energy, but because Al-
GaAs has a much wider bandgap than GaAs, the global conduction band minimum is at the
GaAs/AlGaAs interface below the doping layer, crossing the Fermi energy. Mobile electrons
are trapped inside the triangular confining potential formed at this GaAs/AlGaAs interface.
Due to size quantization mentioned above, only the lowest sub-band of this confining poten-
tial is populated, creating a two-dimensional electron gas (2DEG). The separation between
the doping layer and the 2DEG is called modulation doping; together with a near perfect
match of lattice constants between GaAs and AlGaAs, it gives GaAs/AlGaAs 2DEG the
10
highest mobility among solid state systems.
The GaAs/AlGaAs heterostructures used in Chs. 2, 3, 7, and 8 are grown by Ken West
and Loren Pfeiffer at Bell Labs. The heterostructure used in Chs. 2 and 3 is similar to
the UCSB wafer I just described; the heterostructure used in Chs. 7 and 8, however, is
different in several ways. First, the 2DEG is 200 nm away from the chip surface, and
located in a 30 nm wide GaAs quantum well sandwiched between AlGaAs layers, instead
of at the interface between GaAs and AlGaAs. Second, the sample is doubly doped, with
Si δ-doping layers 100 nm below and above the quantum well. Finally, the two Si layers
are each placed in separate narrow AlGaAs/GaAs/AlGaAs doping wells. These features
have made this heterostructure the highest mobility wafer that we have measured, with a
mobility of ∼ 20, 000, 000 cm2/Vs, two orders of magnitude higher than the UCSB wafer.
The 2DEG can be electrically contacted by Au/Ge ohmic contacts (see Fig. 1.2), en-
abling transport measurements. These contacts are made by depositing Ni/Au/Ge or
Pt/Au/Ge, followed by annealing at around 500 C for tens of seconds. The optimal
annealing recipe will depend on the specific 2DEG material.
Devices of arbitrary shape and size are created by applying negative voltages on sur-
face gates, usually made with Cr/Au or Ti/Au using electron-beam lithography. Negative
voltages raise the conduction band relative to the Fermi energy underneath the gates, and
if the conduction band minimum becomes higher than the Fermi energy, all the electrons
under the gates are depleted. This approach is especially flexible since each gate voltage
can be precisely controlled with independent digital-to-analog converters (DACs), tailoring
the confining potential of the device.
11
1.4 Basic properties of mesoscopic devices
In this section, I will be introducing the basic properties of the various mesoscopic devices
to be studied in later chapters. Quantum point contacts (QPCs) are short one-dimensional
(1d) wires, and show quantized conductance as a function of gate voltage. Quantum dots
and double quantum dots are zero-dimensional (0d) systems; when their sizes are sufficiently
small, and their leads are sufficient opaque, Coulomb charging energy dominates, and they
exhibit single-electron tunneling behaviors. Subject to a strong perpendicular magnetic
field, two-dimensional (2d) electronic systems can exhibit integer and fractional quantum
Hall effects, when transport occurs along the edges, making the system effectively one-
dimensional. The subsequent three subsections will describe these properties in detail.
1.4.1 Quantum point contacts
Quantum point contacts, formed by depleting two facing gates as shown in the inset of
Fig. 1.3, are the simplest gated structure. The gates restrict the electron motion in one
direction, making the QPCs one-dimensional due to size quantization. As shown in Fig. 1.3,
the conductance g through a QPC, measured as a function of the gate voltage Vg, shows
steps of size 2e2/h. Indeed, the quantized conductance suggests the realization of an electron
waveguide, and is the hallmark of 1d ballistic transport [6, 1, 2].
The derivation of quantized conductance in 1d systems is pretty straightforward, as
given below. Consider that in a 1d system, the density of states with positive wave vector k
is ρ+(ε) = (dk/dε)/2π, and the velocity is v(ε) = (dε/dk)/~, where ε is the kinetic energy.
The product of these two quantities cancels the dispersion terms, giving a constant incident
rate per unit energy: ρ+(ε) · v(ε) = 1/h. When a source-drain voltage Vsd is applied across
a barrier in a 1d system, as shown in Fig. 1.4, the transmitted current is simply given by
12
12
10
8
6
4
2
0g
[ e2 /
h ]
-1600 -1200 -800 -400 0Vg [ mV ]
Quantum Point Contact
500 nm
Figure 1.3: Conductance as a function of gate voltage in a quantum point contact. Inset:Scanning electron micrograph of a device with identical design to the one measured.
μdμs
k(ε)τ(ε)
ds
eVsd
Figure 1.4: Illustration of a source-drain voltage applied across a barrier in a 1d system,with transmission probability τ , in the zero-temperature limit.
the total indicant rate eVsd/h multiplied by the transmission probability τ and the electron
charge e: I = (e2/h)τ · Vsd. Therefore, the conductance is just g = (e2/h)τ . Note that
even in the absence of any barriers, with τ = 1, the conductance e2/h is still finite and only
related to fundamental constants, regardless of the length and width of the 1d system.
A QPC is not a single-channel 1d system though. First, the size of quantized conduc-
tance has been doubled to 2e2/h at zero magnetic field due to spin degeneracy, which can
be lifted at high magnetic fields, as in Ch. 3. Second, size quantization in the constricted
direction leads to sub-bands with sub-band energy spacing given by the confining poten-
13
tial curvature. As a function of gate voltage, the sub-bands can be populated one by one,
leading to conductance steps, as observed in Fig. 1.3.
Between adjacent plateaus, the conductance risers are not infinitely sharp, but have
some width. A more realistic model for QPC is to assume the confining potential has the
form of a saddle point [50, 51]: V (x, y) = V0−mω2xx
2/2 +mω2yy
2/2, where x is the current
flowing direction, and y is the confinement direction. In this model, the sub-band energy
spacing is given by ~ωy. For the sub-band indexed n with sub-band edge at εn, the energy-
dependent transmission has the form τn(ε) = 1/(1 + e2π(εn−ε)/~ωx), therefore, the widths of
risers are given by ~ωx when kBT ~ωx.
Taking both spin and sub-bands into account, explicit expressions for transport, both
conductance and noise, at arbitrary bias can be calculated using the Landauer-Buttiker
formalism [52, 53, 54, 55]. The dc current is:
I =e
h
∫ ∑n,σ
τn,σ(ε)[fs(ε)− fd(ε)]dε, (1.4)
where σ denotes the spin, and fs (fd) is the Fermi distribution in the source (drain). The
differential conductance can be calculated by taking the derivative with respect to the
source-drain voltage, and assuming the bias the applied symmetrically:
g =e2
h
∫ ∑n,σ
τn,σ(ε)12
[−fs(ε)
ε− fd(ε)
ε
]dε
=e2
h
∫ ∑n,σ
τn,σ(ε)1
2kBTfs(ε)[1− fs(ε)] + fd(ε)[1− fd(ε)]dε. (1.5)
And finally, the current noise is given by:
SI =2e2
h
∫ ∑n,σ
τn,σ(ε)fs(ε)[1− fs(ε)] + fd(ε)[1− fd(ε)]dε
+2e2
h
∫ ∑n,σ
τn,σ(ε)[1− τn,σ(ε)][fs(ε)− fd(ε)]2dε. (1.6)
14
Note that the first term in SI is exactly 4kBTg, which can be viewed as thermal noise,
even in the nonlinear regime3. This is why we define the partition noise as SPI ≡ SI−4kBTg
in Ch. 3, and its expression given in Eq. (3.2) is just the second term in Eq. (1.6).
Understanding most features in a QPC, we need to note one more subtle feature that
has remained an open problem until now: the shoulder-like feature near 0.7 × 2e2/h, as
can be seen in Fig. 1.3. This is termed 0.7 structure [23], and found to be related to
spin and many-body interactions, yet its exact origin is still under debate. Since the 0.7
structure happens below the 2e2/h plateau, there are only two spin channels, and the
natural question to ask is whether the transmission coefficients are different for the two spin
channels. Conductance would give the sum of them; the additional information brought
by the shot-noise measurements [56, 31], which is related to∑
σ τσ(1 − τσ), can give the
second equation needed. Indeed, the shot-noise study of the 0.7 structure, presented in
Ch. 3 show that they are different, and the spin is almost fully-polarized for conductance
above 0.7× 2e2/h even at zero magnetic field.
Quantum point contacts, as simple as they are, can already demonstrate the wave-
particle duality of electrons and many-body interaction effects. On the quantized con-
ductance plateaus, the electrons flow in channels of reflectionless waveguides, and show
vanishing shot noise. On the risers, on the other hand, shot noise arises in the presence of
partial transmission, demonstrating the particle nature of electrons. Finally, the 0.7 struc-
ture in QPCs, a many-body interaction effect, makes the single-particle picture to break
down, and has remained to be understood.
3This result assumes symmetrically applied bias. Although in experiments, bias is oftenapplied only on one side, interaction effects tend to effectively symmetrize the bias.
15
1.4.2 Quantum dots
While the QPCs have left the electrons to flow freely in one direction, quantum dots confine
the electrons on 0d islands, which are connected to outside reservoirs by QPCs. Quantum
dots can be classified as open or closed dots, depending on the coupling between the dots
and the reservoirs. In open quantum dots, each QPC passes at least one channel, thus
the charge on the island is not quantized, and transport shows wave-like behaviors such as
universal conductance fluctuation and weak localization [4]. In closed quantum dots, when
the transmission through each QPC is less than one, Coulomb charging becomes dominant
and quantizes the charge on the island; transport occurs when electrons tunnel on and off
the island one by one, explicitly demonstrating the particle nature of electrons [3, 57]. Since
the open quantum dots are out of the scope of this thesis, I will only describe the closed dots
in the sequential tunneling limit below. Readers who wish to learn systematically about
quantum dots should refer to several excellent reviews [3, 4, 57, 58] available on these topics.
Shown in the inset of Fig. 1.5 is an example of a quantum dot design, which consists
of two plunger gates and two QPC leads. Conductance as a function of a plunger gate
voltage in the closed dot regime, as shown in Fig. 1.5, is zero almost everywhere, except
at some regularly spaced gate voltage points, drastically different from that in QPCs [see
Fig. 1.3]. This phenomenon, termed Coulomb blockade (CB) [3, 57], occurs because in
contrast to QPCs, where the non-interacting picture can explain most of the observations,
closed quantum dots are in the interaction-dominated limit, and electrons are quantized on
the island.
To understand the observed CB behavior, we first consider the simplest model of
16
0.04
0.03
0.02
0.01
0.00g
[ e2 /
h ]
-860 -840 -820Vg [ mV ]
Quantum Dot 500 nm
Figure 1.5: Conductance as a function of gate voltage in a quantum dot. Inset: Scanningelectron micrograph of a device with identical design to the one measured.
Coulomb energy with N electrons on the dot [57]:
E =1
2C(CgVg − eN)2, (1.7)
where C is the dot total capacitance, and Cg is the capacitance between the dot and the
gate Vg. This energy is purely classical, and the only quantum ingredient needed is that the
QPCs are in the tunneling regime, quantizing the number of electrons on the island. The
number of electrons N is chosen to minimize this Coulomb energy, but since N is restricted
to be an integer, transport is blocked most of the time, except when CgVg/e equals half
integers, in which case N can fluctuate between two values without energy cost, giving finite
conductance. This model explains the CB behavior observed in Fig. 1.5, and gives the gate
voltage period between the CB peaks as e/Cg, expected to be roughly constant.
Figure 1.6 shows the differential conductance g = dI/dVsd as a function of Vg and Vsd,
revealing much richer structures. First, the conductance is zero in diamond-shaped regions,
which is the well-known Coulomb diamonds. Inside these diamonds, the electron number
is well defined, and transport is blocked. The boundaries of the diamonds show sharp
conductance peaks, and they converge to a single point at zero bias, corresponding to a
zero-bias CB peak seen in Fig. 1.5. In addition, outside the diamonds, there are lines of
17
-1224 -1220 -1216Vg [ mV ]
2
1
0
-1
-2
Vsd
[ m
V ]
0.03
0.02
0.01
0.00
g [ e2 / h ]
Figure 1.6: Differential conductance as a function of source-drain bias and gate voltage ina quantum dot.
finite conductance parallel to the diamond boundaries, and ending at the boundaries with
the opposite slope. Moreover, the positive-slope lines generally have higher conductance
than the negative-slope lines.
These features can be understood from energy level diagrams and the electrostatic
considerations of the dot, and are captured by CB simulations in the sequential tunneling
limit4, as shown in Fig. 1.7. Near a diamond vertex at zero-bias, transport occurs when
electrons tunnel on and off the discrete energy levels of the dot, and the electron number
fluctuates between two integer values N and N + 1. Because differential conductance is
measured, the lines correspond to either the source (for positive-slope lines) or drain (for
negative-slope lines) chemical potential aligning with the dot levels. Therefore, the height
of the diamonds from zero-bias gives the charging energy e2/C (assuming charging energy is
much greater than level spacings), and the positive-slope lines have a slope of Cg/(C −Cs),
4The same simulations are also used in Ch. 5, and the source procedures are provided inAp. E.
18
-1.0
-0.5
0.0
0.5
1.0
Vsd
[ m
V ]
-10 -5 0 5 10Vg [ mV ]
0.060.040.020.00
-10 -5 0 5 10Vg [ mV ]
g [ e2 / h ]
(a) Single-level simulation (b) Multi-level simulation
s d s d s ds d
A
A
B
B
C
C
D
D
Figure 1.7: (a) Single-level and (b) multi-level simulations of differential conductance asa function of source-drain bias and gate voltage of a quantum dot. At the four pointsindicated by A, B, C, and D, the corresponding energy diagrams are shown at the bottom.The parameters of the simulation are chosen rather arbitrarily.
while the negative-slope ones have a slope of −Cg/Cs, where Cs denotes the capacitance
between the source and the dot. Furthermore, the conductance for positive-slope lines
is proportional to (C − Cs)/C, and that for negative-slope lines is proportional to Cs/C;
since Cs/C is usually less than 0.5 (equals 0.3 for the simulations shown in Fig. 1.7), the
positive-slope lines would have higher conductance than the negative-slope lines.
All the observed features can be captured by the single-level CB picture described above,
as its simulation shows in Fig. 1.7(a), except the presence of lines outside the diamonds.
Indeed, they are related to transport through multiple excited states [59, 60, 61], and can
be easily taken into account in a multi-level simulation, as shown in Fig. 1.7(b).
At zero-bias, transport occurs only through one level: the lowest unoccupied level in
the N -electron ground state, which is also the highest occupied level in the (N +1)-electron
19
ground state. We call the orbital levels above it electron excited levels, because they are
empty in the ground state and allow electrons to be excited on to them; similarly, we call
the orbital levels below hole excited levels, because they are full in the ground state, thus
allow holes to be excited on to them. The alignment of the chemical potential in the leads
to these excited levels can produce the lines outside the diamonds. The electron excited
levels can only be aligned with higher chemical potential of the two leads, while the hole
excited levels can only be aligned with the lower one. At negative bias, when the source
chemical potential is higher, an electron (hole) excited level produces a positive- (negative-)
slope line, as illustrated in the diagram C (D) of Fig. 1.7. At positive bias, on the other
hand, the drain chemical potential is higher, thus an electron (hole) excited level would
produce a negative- (positive-) slope line. As we have seen, transport through excited state
explains the presence of lines outside the diamonds; since there are ∼ 100 electrons in the
quantum dot studied in Fig. 1.6, there should be many orbital levels, producing many lines
observed outside the diamond regions.
1.4.3 Quantum Hall effects
A clean two-dimensional electron system, placed in a strong perpendicular magnetic field,
can exhibit integer [5] and fractional [27] quantum Hall (QH) effects, where the Hall conduc-
tance is precisely quantized at integer or fractional multiples of the conductance quantum
(e2/h). Quantum Hall effects are yet another example that wave and particle phenom-
ena as well as interaction effects show up in one way or another. While the integer QH
effects can be understood in a non-interacting picture, as will be described shortly, the
fractional QH effects will not be present without interactions. While the currents are
being carried in chiral waveguides with very little backscattering, measurements of shot
20
0.10
0.08
0.06
0.04
0.02
0.00
RX
X [
h / e
2 ]
1086420 B [ T ]
1
1/2
1/3
1/41/51/61/81/10
3/4
3/5
0
RX
Y [ h / e2 ]
18
16
14
12
10
8
6
4
2
0
GX
Y [ e2 / h ]
1612840Filling Factor
Figure 1.8: Bulk longitudinal resistance RXX (red) and Hall resistance RXY (black) as afunction of magnetic field. Inset: Hall conductance GXY ≡ 1/RXY as a function of fillingfactor, converted from magnetic field using a sheet density of 2.6× 1015 m−2.
noise [38, 39, 40] have revealed fractional quasi-particle charge in the fractional QH regime.
While the current-carrying edges can be brought together to interfere in a Fabry-Perot in-
terferometer, Coulomb charging may dominate the transport and quantize the charge.
An example of transport measurement in the QH regime is shown in Fig. 1.8: bulk longi-
tudinal resistance RXX and Hall resistance RXY are measured as a function of perpendicular
magnetic field B, showing both integer and fractional QH plateaus in RXY with vanishing
RXX. The quantized conductance, regardless of sample sizes, reminds us of transport in
quantum point contacts. Indeed, we can plot in the inset of Fig. 1.8 Hall conductance
GXY ≡ 1/RXY as a function of the filling factor ν ≡ n/nφ, showing remarkable resemblance
to the QPC conductance versus gate voltage data in Fig. 1.3. Here, n is the electron sheet
density, nφ = B/φ0 is the flux density, and φ0 = h/e is the flux quantum.
21
As mentioned above in Sec. 1.4.3, quantized conductance is the hallmark of ballistic
1d transport. Although the Hall bar sample is two-dimensional, a perpendicular magnetic
field discretizes the 2d Fermi sea into Landau levels (LL’s) at EN = (N + 1/2) · ~ωc, where
N (= 0, 1, 2, . . .) is the LL index, ωc = eB/m∗ is the cyclotron frequency, and m∗ is the
effective mass for electrons. Chiral transport occurs along the 1d channels where LL’s
cross the Fermi energy near the edges, in opposite directions along opposite edges. The
degeneracy of each LL, for each spin, is the same as the flux density nφ = B/φ0, therefore,
the number of filled LL’s is just the filling factor ν ≡ n/nφ. Since each LL contributes to
one 1d channel, the conductance is thus quantized to ν · e2/h for ν = integers. Due to
the chirality of edge transport, backscattering is strongly suppressed, and resistance can be
quantized to an unprecedented level such that it becomes the new standard for resistance.
Although integer QH effects are well understood in the non-interacting picture described
above, interactions are essential for fractional QH effects, as energy gaps need to open up
within the otherwise degenerate LL’s. Compared to integer QH effects, fractional QH
effects show much more interesting physics, yet they are also much less understood. In
addition to the fractional charge for quasi-particles in the fractional QH regime, as observed
in shot-noise measurements [38, 39, 40], these quasi-particles are expected to be neither
bosons nor fermions, but anyons that obey fractional or non-Abelian statistics. Interference
experiments in Fabry-Perot interferometers are proposed [62, 63, 64] to reveal their effects,
but several experiments [65, 66, 67, 68] on Fabry-Perot interferometers trying to observe
the anyonic statistics have remained inconclusive, possibly dominated by Coulomb blockade
rather than interference [69, 22]. Many other outstanding problems in the QH regime have
also remained to be answered, such as spin polarization, edge reconstruction and neutral
modes, anisotropic and reentrant states, etc.
22
Chapter 2
System for measuring auto- andcross correlation of current noise atlow temperatures
L. DiCarlo1, Yiming Zhang1, D. T. McClure1, C. M. MarcusDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138
L. N. Pfeiffer, K. W. WestAlcatel-Lucent, Murray Hill, New Jersey 07974
We describe the construction and operation of a two-channel noise detection system
for measuring power and cross spectral densities of current fluctuations near 2 MHz in
electronic devices at low temperatures. The system employs cryogenic amplification and
fast Fourier transform based spectral measurement. The gain and electron temperature are
calibrated using Johnson noise thermometry. Full shot noise of 100 pA can be resolved with
an integration time of 10 s.2
1These authors contributed equally to this work.
2This chapter is adapted with permission from Rev. Sci. Instrum. 77, 073906 (2006). c©(2006) by the American Institute of Physics.
23
2.1 Introduction
Over the last decade, measurement of electronic noise in mesoscopic conductors has suc-
cessfully probed quantum statistics, chaotic scattering and many-body effects [19, 29, 30].
Suppression of shot noise below the Poissonian limit has been observed in a wide range of
devices, including quantum point contacts [70, 71, 72], diffusive wires [73, 74], and quan-
tum dots [75], with good agreement between experiment and theory. Shot noise has been
used to measure quasiparticle charge in strongly correlated systems, including the fractional
quantum Hall regime [38, 39, 40] and normal-superconductor interfaces [76], and to inves-
tigate regimes where Coulomb interactions are strong, including coupled localized states in
mesoscopic tunnel junctions [43] and quantum dots in the sequential tunneling [77, 46] and
cotunneling [45] regimes. Two-particle interference not evident in dc transport has been
investigated using noise in an electronic beam splitter [72].
Recent theoretical work [78, 79, 80, 81] proposes the detection of electron entanglement
via violations of Bell-type inequalities using cross-correlations of current noise between
different leads. Most noise measurements have investigated either noise autocorrelation
[70, 73, 38, 82, 72, 37, 45] or cross correlation of noise in a common current [71, 83, 74, 84,
75, 43], with only a few experiments [85, 86, 87, 88] investigating cross correlation between
two distinct currents. Henny et al. [85, 86] and Oberholzer et al. [87] measured noise
cross correlation in the acoustic frequency range (low kilohertz) using room temperature
amplification and a commercial fast Fourier transform (FFT)-based spectrum analyzer.
Oliver et al. [88] measured cross correlation in the low megahertz using cryogenic amplifiers
and analog power detection with hybrid mixers and envelope detectors.
In this chapter, we describe a two-channel noise detection system for simultaneously
measuring power spectral densities and cross spectral density of current fluctuations in
24
electronic devices at low temperatures. Our approach combines elements of the two meth-
ods described above: cryogenic amplification at low megahertz frequencies and FFT-based
spectral measurement.
Several factors make low-megahertz frequencies a practical range for low-temperature
current noise measurement. This frequency range is high compared to the 1/f noise corner
in typical mesoscopic devices. Yet, it is low enough that FFT-based spectral measurement
can be performed efficiently with a personal computer (PC) equipped with a commercial
digitizer. Key features of this FFT-based spectral measurement are near real-time opera-
tion and sufficient frequency resolution to detect spectral features of interest. Specifically,
the fine frequency resolution provides information about the measurement circuit and am-
plifier noise at megahertz, and enables extraneous interference pickup to be identified and
eliminated. These two features constitute a significant advantage over both wideband ana-
log detection of total noise power, which sacrifices resolution for speed, and swept-sine
measurement, which sacrifices speed for resolution.
2.2 Overview of the system
Figure 2.1 shows a block diagram of the two-channel noise detection system, which is inte-
grated with a commercial 3He cryostat (Oxford Intruments Heliox 2VL). The system takes
two input currents and amplifies their fluctuations in several stages. First, a parallel resistor-
inductor-capacitor (RLC) circuit performs current-to-voltage conversion at frequencies close
to its resonance at fo = (2π√LC)−1 ≈ 2 MHz. Through its transconductance, a high elec-
tron mobility transistor (HEMT) operating at 4.2 K converts these voltage fluctuations into
current fluctuations in a 50 Ω coaxial line extending from 4.2 K to room temperature. A
50 Ω amplifier with 60 dB of gain completes the amplification chain. The resulting signals
25
0.3 K 4.2 K
Digitize& FFT
Cross Spectrum
PowerSpectrum 1
V1
V2
R L C 60 dB
R L C 60 dB
Device
PowerSpectrum 2
Figure 2.1: Block diagram of the two-channel noise detection system, configured to measurethe power spectral densities and cross spectral density of current fluctuations in a multi-terminal electronic device.
V1 and V2 are simultaneously sampled at 10 MS/s by a two-channel digitizer (National
Instruments PCI-5122) in a 3.4 GHz PC (Dell Optiplex GX280). The computer takes the
FFT of each signal and computes the power spectral density of each channel and the cross
spectral density.
2.3 Amplifier
2.3.1 Design objectives
A number of objectives have guided the design of the amplification lines. These include (1)
low amplifier input-referred voltage noise and current noise. (2) simultaneous measurement
of both noise at megahertz and transport near dc, (3) low thermal load, (4) small size,
allowing two amplification lines within the 52 mm bore cryostat, (5) maximum use of
commercial components, and (6) compatibility with high magnetic fields.
2.3.2 Overview of the circuit
Each amplification line consists of four circuit boards interconnected by coaxial cables,
as shown in the circuit schematic in Fig. 2.2(a). Three of the boards are located inside
26
4.2 K
Sapphire
C2
R3C3
Q1
UT-85
1.6 KVdac
300 K
FR-4UT-85
R5
R6 R7
C6
C5
C4
LOCK IN
Ih
AU1447
SS/SS
SR560R2 R4
SS/SS
CRYOAMP
DIG.SPLITTER
SINK
C1
R1L1
UT-34CRES
1 cm
0.3 K
0.3 K
R1R2R3R4R5R6R7C1C2C3C4C5C6L1
51050
15011
102x515222.22.222
2x33
(a)
(b) (c)
Figure 2.2: (a) Schematic diagram of each amplification line. Values of all passive compo-nents are listed in the accompanying table. Transistor Q1 is an Agilent ATF-34143 HEMT.(b) Layout of the CRYOAMP circuit board. Metal (black regions) is patterned by etchingof thermally evaporated Cr/Au on sapphire substrate. (c) Photograph of a CRYOAMPboard. The scale bar applies to both (b) and (c).
the 3He cryostat. The resonant circuit board [labeled RES in Fig. 2.2(a)] is mounted on
the sample holder at the end of the 30 cm long coldfinger that extends from the 3He
pot to the center of the superconducting solenoid. The heat-sink board (SINK) anchored
to the 3He pot is a meandering line that thermalizes the inner conductor of the coaxial
cable. The CRYOAMP board at the 4.2 K plate contains the only active element operating
cryogenically, an Agilent ATF-34143 HEMT. The four-way SPLITTER board operating
at room temperature separates low- and high- frequency signals and biases the HEMT.
Each line amplifies in two frequency ranges, a low-frequency range below ∼ 3 kHz and a
high-frequency range around 2 MHz.
The low-frequency equivalent circuit is shown in Fig. 2.3(a): a resistor (R1 = 5 kΩ)
to ground, shunted by a capacitor (C1 = 10 nF), converts an input current i to a voltage
on the HEMT gate. The HEMT amplifies this gate voltage by ∼ −5 V/V on its drain,
which connects to a room temperature voltage amplifier at the low frequency port of the
27
SR560
(a) (b)
150 Ω
50 Ω
5 kΩ 96 pF10 nF
66 µH
IhVdac
i i
Vh,d
AU-1447
DMM
50 ΩVh,ds
+
Figure 2.3: Equivalent circuits characterizing the amplification line in the (a) low-frequencyregime (up to ∼ 3 kHz), where it is used for differential conductance measurements, and inthe (b) high-frequency regime (few megahertz), where it is used for noise measurement.
SPLITTER board. The low-frequency voltage amplifier (Stanford Research Systems model
SR560) is operated in single-ended mode with ac coupling, 100 V/V gain and bandpass
filtering (30 Hz to 10 kHz). The bandwidth in this low-frequency regime is set by the input
time constant.
The high-frequency equivalent circuit is shown in Fig. 2.3(b). The inductor L1 = 66 µH
dominates over C1 and forms a parallel RLC tank with R1 and the capacitance C ∼ 96 pF
of the coaxial line connecting to the CRYOAMP board. Resistor R4 is shunted by C2 to
enhance the transconductance at the CRYOAMP board. The coaxial line extending from
4.2 K to room temperature is terminated on both sides by 50 Ω. At room temperature, the
signal passes through the high-frequency port of the SPLITTER board to a 50 Ω amplifier
(MITEQ AU-1447) with a gain of 60 dB and a noise temperature of 100 K in the range
0.01− 200 MHz.
2.3.3 Operating point
The HEMT must be biased in saturation to provide voltage (transconductance) gain in the
low (high) frequency range. R4, R5 + R6 and supply voltage Vdac determine the HEMT
28
operating point (R1 grounds the HEMT gate at dc). A notable difference in this design
compared to similar published ones regards the placement of R4. In previous implementa-
tions of similar circuits [89, 90, 91], R4 is a variable resistor placed outside the refrigerator
and connected to the source lead of Q1 via a second coaxial line or low-frequency wire.
Here, R4 is located on the CRYOAMP board to simplify assembly and save space, at the
expense of having full control of the bias point in Q1 (R4 fixes the saturation value of the
HEMT current Ih). Using the I-V curves in Ref. [91] for a cryogenically cooled ATF-34143,
we choose R4 = 150 Ω to give a saturation current of a few mA. This value of satura-
tion current reflects a compromise between noise performance and power dissipation. As
shown in Fig. 2.4, Q1 is biased by varying the supply voltage Vdac fed at the SPLITTER
board. At the bias point indicated by a cross, the total power dissipation in the HEMT
board is IhVh,ds + I2hR4 = 1.8 mW, and the input-referred voltage noise of the HEMT is
∼ 0.4 nV/√
Hz.
2.3.4 Passive components
Passive components were selected based on temperature stability, size and magnetic field
compatibility. All resistors (Vishay TNPW thin film) are 0805-size surface mount. Their
variation in resistance between room temperature and 300 mK is < 0.5%. Inductor L1 (two
33 µH Coilcraft 1812CS ceramic chip inductors in series) does not have a magnetic core
and is suited for operation at high magnetic fields. The dc resistance of L1 is 26(0.3) Ω at
300(4.2) K. With the exception of C1, all capacitors are 0805-size surface mount (Murata
COG GRM21). C1 (two 5 nF American Technical Ceramics 700B NPO capacitors in
parallel) is certified nonmagnetic.
29
Figure 2.4: Drain current Ih as a function of HEMT drain-source voltage Vh,ds, with theHEMT board at temperatures of 300 K (dashed) and 4.2 K (solid). These curves wereobtained by sweeping the supply voltage Vdac and measuring drain voltage Vh,d with anHP34401A digital multimeter (see Fig. 2.3(a)). From Vh,d and Vdac, Ih and Vh,ds were thenextracted. Dotted curves are contours of constant power dissipation in the HEMT board.The HEMT is biased in saturation (cross).
2.3.5 Thermalization
To achieve a low device electron temperature, circuit board substrates must handle the heat
load from the coaxial line. The CRYOAMP board must also handle the power dissipated
by the HEMT and R4. Sapphire, having good thermal conductivity at low temperatures
[92] and excellent electrical insulation, is used for the substrate in the RES, SINK and
CRYOAMP boards. Polished blanks, 0.02 in. thick and 0.25 in. wide, were cut to lengths
of 0.6 in. (RES and CRYOAMP) or 0.8 in. (SINK) using a diamond saw. Both planar
surfaces were metallized with thermally evaporated Cr/Au (30/300 nm). Circuit traces were
then defined on one surface using a Pulsar toner transfer mask and wet etching with Au
and Cr etchants (Transene types TFA and 1020). Surface mount components were directly
soldered.
The RES board is thermally anchored to the sample holder with silver epoxy (Epoxy
30
Technology 410E). The CRYOAMP (SINK) board is thermalized to the 4.2 K plate (3He
pot) by a copper braid soldered to the back plane.
Semirigid stainless steel coaxial cable (Uniform Tube UT85-SS/SS) is used between the
SINK and CRYOAMP boards, and between the CRYOAMP board and room temperature.
Between the RES and SINK boards, smaller coaxial cable (Uniform Tube UT34-C) is used
to conserve space.
With this approach to thermalization, the base temperature of the 3He refrigerator is
290 mK with a hold time of ∼ 45 h. As demonstrated further below, the electron base
temperature in the device is also 290 mK.
2.4 Digitization and FFT processing
The amplifier outputs V1 and V2 (see Fig. 2.1) are sampled simultaneously using a commer-
cial digitizer (National Instruments PCI-5122) with 14-bit resolution at a rate fs = 10 MS/s.
To avoid aliasing [93] from the broadband amplifier background, V1 and V2 are frequency
limited to below the Nyquist frequency of 5 MHz using 5-pole Chebyshev low-pass filters,
built in-house from axial inductors and capacitors with values specified by the design recipe
in Ref. [94]. The filters have a measured half power frequency of 3.8 MHz, 39 dB suppression
at 8 MHz and a passband ripple of 0.03 dB.
While the digitizer continuously stores acquired data into its memory buffer (32 MB per
channel), a software program processes the data from the buffer in blocks of M = 10 368
points per channel. M is chosen to yield a resolution bandwidth fs/M ∼ 1 kHz, and to be
factorizable into powers of two and three to maximize the efficiency of the FFT algorithm.
Each block of data is processed as follows. First, V1 and V2 are multiplied by a Hanning
window WH [m] =√
2/3[1 − cos(2πm/M)] to avoid end effects [93]. Second, using the
31
FFTW package [95], their FFTs are calculated:
V1(2)[fn] =M−1∑m=0
WH[m]V1(2)(tm)e−i2πfntm , (2.1)
where tm = m/fs, fn = (n/M)fs, and n = 0, 1, ...,M/2. Third, the power spectral densities
P1,2 = 2|V1,2|2/(Mfs) and the cross spectral density X = 2(V ∗1 · V2)/(Mfs) = XR + iXI are
computed.
As blocks are processed, running averages of P1, P2, and X are computed until the
desired integration time τint is reached. With the 3.4 GHz computer and the FFTW algo-
rithm, these computations are carried out in nearly real-time: it takes 10.8 s to acquire and
process 10 s of data.3
2.5 Measurement example: quantum point contact
In this section, we demonstrate the two-channel noise detection system with measurements
of a quantum point contact (QPC). While the investigation of bias-dependent current noise
in QPCs is the main topic of Ch. 3, we here describe the techniques used for measuring
dc transport, as well as the circuit model and the calibration (based on Johnson-noise
thermometry) that are used for extracting QPC noise.
2.5.1 Setup
A gate-defined QPC4 is connected to the system as shown in the inset of Fig. 2.5. The two
amplification lines are connected to the same reservoir of the QPC. In this case, the two input
3We have achieved fully real-time operation with a newly purchased 3.16 GHz Core 2Duo computer.
4This device is named QPC 1 in Ch. 3.
32
500nm
VdcVac
Vg1
Vg2
Figure 2.5: Inset: setup for detection of QPC current noise using cross-correlation, andelectron micrograph of a device identical in design to the one used. The QPC is defined bynegative voltages Vg1 and Vg2 applied on two facing gates. All other gates in the device aregrounded. Main: linear conductance g(Vsd = 0) as a function of Vg2 at 290 mK, measuredusing amplification line 1. Vg1 = −3.2 V.
RLC tanks effectively become a single tank with resistance R′ ≈ 2.5 kΩ, inductance L′ ≈
33 µH and capacitance C ′ ≈ 192 pF. The QPC current noise couples to both amplification
lines and thus can be extracted from either the single channel power spectral densities or
the cross spectral density. The latter has the technical advantage of rejecting any noise not
common to both amplification lines.
2.5.2 Measuring dc transport
A 25 µVrms, 430 Hz excitation Vac is applied to the other QPC reservoir and used for lock-in
measurement of g. A dc bias voltage Vdc is also applied to generate a finite Vsd. Vsd deviates
from Vdc due to the resistance in-line with the QPC, which is equal to the sum of R1/2 and
ohmic contact resistance Rs. Vsd could in principle be measured by the traditional four-
wire technique. This would require additional low-frequency wiring, as well as filtering to
prevent extraneous pick-up and room-temperature amplifier noise from coupling to the noise
33
!"
!#
!"
Figure 2.6: Power spectral densities P1 and P2, and real and imaginary parts XR and XI
of the cross spectral density, at base temperature and with the QPC pinched off (g = 0),obtained from noise data acquired for τint = 20 s. Inset: expanded view of XR nearresonance, along with a fit using Eq. (2.3) over the range 1.7 to 2.3 MHz.
measurement circuit. For technical simplicity, here Vsd is obtained by numerical integration
of the measured bias-dependent g:
Vsd =∫ Vdc
0
dV
1 + (R1/2 +Rs)g(V )(2.2)
Figure 2.5 shows linear conductance g(Vsd = 0) as a function of gate voltage Vg2, at a fridge
temperature Tfridge = 290 mK (base temperature). Here, g was extracted from lock-in
measurements using amplification line 1. As neither the low frequency gain of amplifier 1
nor Rs were known precisely beforehand, these parameters were calibrated by aligning the
observed conductance plateaus to the expected multiples of 2e2/h. This method yielded a
low-frequency gain −4.6 V/V and Rs = 430 Ω.
2.5.3 Measuring noise
Figure 2.6 shows P1, P2, XR, and XI as a function of frequency f , at base temperature and
with the QPC pinched off (g = 0). P1(2) shows a peak at the resonant frequency of the RLC
tank, on top of a background of approximately 85(78)× 10−15 V2/Hz. The background in
34
P1(2) is due to the voltage noise SV,1(2) of amplification line 1(2) (∼ 0.4 nV/√
Hz). The peak
results from thermal noise of the resonator resistance and current noise (SI,1 + SI,2) from
the amplifiers5. XR picks out this peak and rejects the amplifier voltage noise backgrounds.
The inset zooms in on XR near the resonant frequency. The solid curve is a best-fit to the
form
XR(f) =X0R
1 + (f2 − f2o )2/(f∆f3dB)2
, (2.3)
corresponding to the lineshape of white noise band-pass filtered by the RLC tank. The
fit parameters are the peak height X0R, the half-power bandwidth ∆f3dB and the peak
frequency fo. Power spectral densities P1(2) can be fit to a similar form including a fitted
background term:
P1(2)(f) = PB1(2) +
P 01(2)
1 + (f2 − f2o )2/(f∆f3dB)2
. (2.4)
2.5.4 System calibration using Johnson noise
Chapter 3 presents measurements of QPC excess noise, defined as SPI (Vsd) = SI(Vsd) −
4kBTeg(Vsd) (SI is the total QPC current noise spectral density). The extraction of SPI
from measurements of X0R requires that a circuit model for the noise detection system be
defined and that all its parameters be calibrated in situ. The circuit model we use is shown
in Fig. 2.7. Within this model,
SPI =
(X0R
G2X
− 4kBTeReff
)(1 + gRsReff
)2
, (2.5)
where GX =√G1G2 is the cross-correlation gain and Reff = 2πf2
oL′/∆f3dB is the total
effective resistance parallel to the tank6. Calibration requires assigning values for Rs, Te,
5Further below, the contribution from amplifier current noise is shown to be negligible.
6Within the model, Reff = (1/(1/g + Rs) + 1/R′)−1. The best-fit ∆f3dB to the mea-surement shown in Fig. 2.6 with the QPC pinched off (g = 0) gives R′ = 2.4 kΩ. This
35
C'L'R'
Rs
4kBTe/Rs
4kBTe/R'4kBTegSIP
g
G1
SV,1
SI,1
G2
SV,2
SI,2
Figure 2.7: Circuit model used for extraction of the QPC partition noise SPI . G1(2) is the
voltage gain of amplification line 1(2) between HEMT gate and digitizer input.
and GX . While the value Rs is obtained from the conductance measurement, GX and Te
are calibrated from thermal noise measurements. The procedure demonstrated in Fig. 2.8
stems from the relation7 X0R = 4kBTeReffG
2X , valid at Vsd = 0.
First, XR(f) is measured over τint = 30 s for various Vg2 settings at each of three elevated
fridge temperatures (Tfridge = 3.1, 4.2, and 5.3 K). X0R and Reff are extracted from fits to
XR(f) using Eq. (2.3) and plotted parametrically [open markers in Fig. 2.8(a)]. A linear fit
(constrained to pass through the origin) to each parametric plot gives the slope dX0R/dReff
at each temperature, equal to 4kBTeG2X . Assuming Te = Tfridge at these temperatures,
GX = 790 V/V is extracted from a linear fit to dX0R/dReff(Tfridge), shown in Fig. 2.8(b).
Next, the base electron temperature is calibrated from a parametric plot of X0R as a
function of Reff obtained from similar measurements at base temperature [solid circles in
Fig. 2.8(a)]. From the fitted slope dX0R/dReff [black marker in Fig. 2.8(b)] and using the
calibrated GX , a value Te = 290 mK is obtained. This suggests that electrons are well
small reduction from 2.5 KΩ reflects small inductor and capacitor losses near the resonantfrequency.
7The full expression within the circuit model is X0R = (4kBTeReff + (SI,1 +SI,2)R2
eff)G2X .
The linear dependence of X0R on Reff observed in Fig. 2.8(a) demonstrates that the quadratic
term from amplifier current noise is negligible.
36
Figure 2.8: Calibration by noise thermometry of the electron temperature Te at base fridgetemperature and the cross-correlation gain GX . (a) X0
R as function of Reff (both from fitsto XR(f) using Eq. (2.3)), at base (solid circles) and at three elevated fridge temperatures(open markers). Solid lines are linear fits constrained to the origin. (b) Slope dX0
R/dReff
(from fits in (a)) as a function of Tfridge. Solid line is a linear fit (constrained to the origin)of dX0
R/dReff at the three elevated temperatures (open markers).
thermalized to the fridge.
2.6 System performance
The resolution in the estimation of current noise spectral density from one-channel and
two-channel measurements is determined experimentally in this section. Noise data are
first sampled over a total time τtot = 1 h, with the QPC at base temperature and pinched
37
Figure 2.9: (a) X0R as a function of time t, for τint of 10 s (open circles) and 100 s (solid
circles). (b) Standard deviations σ1 and σR as a function of τint. The solid line is a fit toσR of the form Cτ
−1/2int , with best-fit value C = 0.30 × 10−15 s1/2V2/Hz. (c) σR/
√σ1σ2 as
a function of τint. The dashed line is a constant 1/√
2.
off. Dividing the data in segments of time length τint, calculating the power and cross
spectral densities for each segment, and fitting with Eqs. (2.3) and (2.4) gives a sequence of
τtot/τint peak heights for each of P1, P2 and XR. Shown in open (solid) circles in Fig. 2.9(a)
is X0R as a function of time t for τint = 10(100) s. The standard deviation σR of X0
R is
1(0.3) × 10−16 V2/Hz. The resolution δSI in current noise spectral density is given by
σR/(G2XR
2eff) [see Eq. (2.5)]. For τint = 10 s, δSI = 2.8× 10−29 A2/Hz, which corresponds
to full shot noise 2eI of I ∼ 100 pA.
The effect of integration time on the resolution is determined by repeating the analysis
38
for different values of τint. Fig. 2.9(b) shows the standard deviation σ1 (σR) of P 01 (X0
R) as a
function of τint. The standard deviation σ2 of P 02 , not shown, overlaps closely with σ1. All
three standard deviations scale as 1/√τint, consistent with the Dicke radiometer formula
[96] which applies when measurement error results only from finite integration time, i.e., it
is purely statistical. This suggests that, even for the longest segment length of τint = 10 min,
the measurement error is dominated by statistical error and not by instrumentation drift
on the scale of 1 h.
Figure 2.9(c) shows σR/√σ1σ2 as a function of τint. This ratio gives the fraction by
which, in the present measurement configuration, the statistical error in current noise spec-
tral density estimation from X0R is lower than the error in the estimation from either P 0
1 or
P 02 alone. The geometric mean in the denominator accounts for any small mismatch in the
gains G1 and G2. In theory, and in the absence of drift, this ratio is independent of τint and
equal to 1/√
2 when the uncorrelated amplifier voltage noise [SV,1(2)] dominates over the
noise common to both amplification lines. The ratio would be unity when the correlated
noise dominates over SV,1(2).
The experimental σR/√σ1σ2 is close to 1/
√2 (dashed line). This is consistent with
the spectral density data in Fig. 2.6, which shows that the backgrounds in P1 and P2 are
approximately three times larger than the cross-correlation peak height. The ratio deviates
slightly below 1/√
2 at the largest τint values. This may result from enhanced sensitivity to
error in the substraction of the P1(2) background at the longest integration times.
A similar improvement relative to estimation from either P 01 or P 0
2 alone would also
result from estimation with a weighted average (P 01 /G
21 + P 0
2 /G22)G2
X/2. The higher res-
olution attainable from two channel measurement relative to single-channel measurement
in this regime has been previously exploited in noise measurements in the kilohertz range
39
[71, 83, 84].
2.7 Discussion
We have presented a two-channel noise detection system measuring auto- and cross corre-
lation of current fluctuations near 2 MHz in electronic devices at low temperatures. The
system has been implemented in a 3He refrigerator where the base device electron temper-
ature, measured by noise thermometry, is 290 mK. Similar integration with a 3He -4He
dilution refrigerator would enable noise measurement at temperatures of tens of millikelvin.
2.8 Acknowledgements
We thank N. J. Craig, J. B. Miller, E. Onitskansky, and S. K. Slater for device fabrication.
We also thank H.-A. Engel, D. C. Glattli, P. Horowitz, W. D. Oliver, D. J. Reilly, P. Roche,
A. Yacoby, Y. Yamamoto for valuable discussion, and B. D’Urso, F. Molea and H. Steinberg
for technical assistance. We acknowledge support from NSF-NSEC, ARDA/ARO, and
Harvard University.
40
Chapter 3
Current noise in quantum pointcontacts
L. DiCarlo1, Yiming Zhang1, D. T. McClure1, D. J. Reilly, C. M. MarcusDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138
L. N. Pfeiffer, K. W. WestAlcatel-Lucent, Murray Hill, New Jersey 07974
M. P. Hanson, A. C. GossardDepartment of Materials, University of California, Santa Barbara, California 93106
We present measurements of current noise in QPCs as a function of source-drain bias,
gate voltage, and in-plane magnetic field. At zero bias, Johnson noise provides a measure
of the electron temperature. At finite bias, shot noise at zero field exhibits an asymme-
try related to the 0.7 structure in conductance. The asymmetry in noise evolves smoothly
into the symmetric signature of spin-resolved electron transmission at high field. Compari-
son to a phenomenological model with density-dependent level splitting yields quantitative
agreement. Additionally, a device-specific contribution to the finite-bias noise, particularly
visible on conductance plateaus (where shot noise vanishes), agrees quantitatively with a
model of bias-dependent electron heating.2
1These authors contributed equally to this work.
2This chapter is adapted from Ref. [32] and from Phys. Rev. Lett. 97, 036810 (2006)[with permission, c© (2006) by the American Physical Society].
41
3.1 Introduction
The experimental discovery nearly two decades ago [6, 28] of quantized conductance in quan-
tum point contacts (QPCs) suggested the realization of an electron waveguide. Pioneering
measurements [70, 71, 72] of noise in QPCs almost a decade later observed suppression of
shot noise below the Poissonian value due to Fermi statistics, as predicted by mesoscopic
scattering theory [97, 54]. Shot noise has since been increasingly recognized as an important
probe of quantum statistics and many-body effects [19, 29, 30], complementing dc transport.
For example, shot-noise measurements have been exploited to directly observe quasiparticle
charge in strongly correlated systems [38, 39, 40, 76], as well as to study interacting local-
ized states in mesoscopic tunnel junctions [43] and cotunneling [45] and dynamical channel
blockade [46, 33] in quantum dots.
Paralleling these developments, a large literature has emerged concerning the appear-
ance of an additional plateau-like feature in transport through a QPC at zero magnetic
field, termed 0.7 structure. Experiments [23, 98, 99, 100, 101, 102, 103] and theories [104,
105, 106, 107, 108, 109] suggest that 0.7 structure is a many-body spin effect. Its underlying
microscopic origin remains an outstanding problem in mesoscopic physics. This persistently
unresolved issue is remarkable given the simplicity of the device.
In this chapter, we review our work [31] on current noise in quantum point contacts—
including shot-noise signatures of 0.7 structure and effects of in-plane field B‖—and present
new results on a device-specific contribution to noise that is well described by a model that
includes bias-dependent heating in the vicinity of the QPC. Notably, we observe suppression
of shot noise relative to that predicted by theory for spin-degenerate transport [97, 54]
near 0.7 × 2e2/h at B‖ = 0, consistent with previous work [110, 56]. The suppression
near 0.7 × 2e2/h evolves smoothly with increasing B‖ into the signature of spin-resolved
42
Figure 3.1: (a) Linear conductance g0 as a function of Vg2 (Vg1 = −3.2 V), for B‖ rangingfrom 0 (red) to 7.5 T (purple) in steps of 0.5 T. The series resistance Rs ranging from 430 Ωat B‖ = 0 to 730 Ω at B‖ = 7.5 T has been subtracted to align the plateaus at multiplesof 2e2/h. (b,c) Nonlinear differential conductance g as a function of Vsd, at B‖ = 0 (b) and7.5 T (c), with Vg2 intervals of 7.5 and 5 mV, respectively. Shaded regions indicate the biasrange used for the noise measurements presented in Figs. 3.3(c) and 3.4.
transmission. We find quantitative agreement between noise data and a phenomenological
model for a density-dependent level splitting [109], with model parameters extracted solely
from conductance. In the final section, we investigate a device-specific contribution to
the bias-dependent noise, particularly visible on conductance plateaus (where shot noise
vanishes), which we account for with a model [71] of Wiedemann-Franz thermal conduction
in the reservoirs connecting to the QPC.
3.2 QPC characterization
Measurements are presented for two QPCs defined by split gates on GaAs/Al0.3Ga0.7As
heterostructures grown by molecular beam epitaxy. For QPC 1(2), the two-dimensional
electron gas [2DEG] 190(110) nm below the heterostructure surface has density 1.7(2) ×
1011 cm−2 and mobility 5.6(0.2)×106 cm2/Vs. Except where noted, all data are taken at the
base temperature of a 3He cryostat, with electron temperature Te of 290 mK. A magnetic
field of 125 mT, applied perpendicular to the plane of the 2DEG, was used to reduce bias-
43
dependent heating [71] (see section below). Each QPC is first characterized at both zero and
finite B‖ using near-dc transport measurements. The differential conductance g = dI/dVsd
(where I is the current and Vsd is the source-drain bias) is measured by lock-in technique
as discussed in Sec. 2.5.2. The B‖-dependent ohmic contact and reservoir resistance Rs in
series with the QPC is subtracted.
Figure 3.1 shows conductance data for QPC 1 (see micrograph in Fig. 3.2). Linear-
response conductance g0 = g(Vsd ∼ 0) as a function of gate voltage Vg2, for B‖ = 0 to 7.5 T
in steps of 0.5 T, is shown in Fig. 3.1(a). The QPC shows the characteristic quantization
of conductance in units of 2e2/h at B‖ = 0, and the appearance of spin-resolved plateaus
at multiples of 0.5 × 2e2/h at B‖ = 7.5 T. Additionally, at B‖ = 0, a shoulder-like 0.7
structure is evident, which evolves continuously into the 0.5 × 2e2/h spin-resolved plateau
at high B‖ [23].
Figures 3.1(b) and 3.1(c) show g as a function of Vsd for evenly spaced Vg2 settings
at B‖ = 0 and 7.5 T, respectively. In this representation, linear-response plateaus in
Fig. 3.1(a) appear as accumulated traces around Vsd ∼ 0 at multiples of 2e2/h for B‖ = 0,
and at multiples of 0.5×2e2/h for B‖ = 7.5 T. At finite Vsd, additional plateaus occur when
a sub-band edge lies between the source and drain chemical potentials [111]. The features
near 0.8 × 2e2/h (Vsd ∼ ±750 µV) at B‖ = 0 cannot be explained within a single-particle
picture [112]. These features are related to the 0.7 structure around Vsd ∼ 0 and resemble
the spin-resolved finite bias plateaus at ∼ 0.8× 2e2/h for B‖ = 7.5 T [99, 101].
3.3 Current noise
QPC current noise is measured using the cross-correlation technique (see Fig. 3.2) discussed
in Sec. 2.5.3. Johnson-noise thermometry allows in situ calibration of Te and the ampli-
44
Digitize& FFT
R L C 60 dB
R L C 60 dB
Vg1
Vg2500 nm
VdcCrossSpectrum
290 mK 4.2 K
Figure 3.2: Equivalent circuit near 2 MHz of the system measuring QPC noise by cross-correlation on two amplification channels. The scanning electron micrograph shows a deviceof identical design to QPC 1. The QPC is formed by negative voltages Vg1 and Vg2 appliedon two facing electrostatic gates. All other gates on the device are grounded.
fication gain in the noise detection system by the procedure previously demonstrated in
Sec. 2.5.4.
We characterize the QPC noise at finite bias by the excess noise, defined as SPI (Vsd) =
SI(Vsd)− 4kBTeg(Vsd), where SI is the total QPC current noise spectral density. Note that
SPI is the noise in excess of 4kBTeg(Vsd) rather than 4kBTeg(0) and thus differs from excess
noise as discussed in Refs. [70] and [56]. In the absence of 1/f and telegraph noise as well as
bias-dependent electron heating, SPI originates from the electron partitioning at the QPC.
Experimental values for SPI are extracted from simultaneous measurements of cross-
spectral density and of g as described in Sec. 2.5.3. With an integration time of 60 s, the
resolution in SPI is 1.4 × 10−29 A2/Hz, corresponding to full shot noise 2eI of I ∼ 40 pA.
SPI as a function of dc current I for QPC 1 with gates set to very low conductance (g0 ∼
0.04× 2e2/h) [Fig. 3.3(b)] exhibits full shot noise, SPI = 2e|I|, demonstrating an absence of
1/f and telegraph noise at the noise measurement frequency [113].
Figure 3.3(c) shows SPI (Vsd) in the Vsd range −150 µV to +150 µV [shaded regions
in Figs. 3.1(b) and 3.1(c)], at B‖ = 0 and Vg2 settings corresponding to open markers in
Fig. 3.3(a). Similar to when the QPC is fully pinched off, SPI vanishes on plateaus of linear
45
Figure 3.3: (a) Linear conductance g0 as a function of Vg2 at B‖ = 0. Solid marker and openmarkers indicate Vg2 settings for the noise measurements shown in (b) and (c), respectively.(b) SP
I as a function of dc current I with the QPC near pinch-off. The dotted line indicatesfull shot noise SP
I = 2e|I|. (c) Measured SPI as a function of Vsd, for conductances near
0 (circles), 0.5 (squares), 1 (upward triangles), 1.5 (squares), and 2 ×2e2/h (downwardtriangles). Solid lines are best-fits to Eq. (3.1) using N as the only fitting parameter. Inorder of increasing conductance, best-fit N values are 0.00, 0.20, 0.00, 0.19, and 0.03.
conductance. This demonstrates that bias-dependent electron heating is not significant in
QPC 1. In contrast, for g ∼ 0.5 and 1.5×2e2/h, SPI grows with |Vsd| and shows a transition
from quadratic to linear dependence [70, 71, 72]. The linear dependence of SPI on Vsd at
high bias further demonstrates the absence of noise due to resistance fluctuations. Solid
curves superimposed on the SPI (Vsd) data in Fig. 3.3(c) are best-fits to the form
SPI (Vsd) = 2
2e2
hN[eVsd coth
(eVsd
2kBTe
)− 2kBTe
], (3.1)
with the noise factor N as the only free fitting parameter. Note that N relates SPI to Vsd, in
contrast to the Fano factor [19, 30], which relates SPI to I. This fitting function is motivated
by mesoscopic scattering theory [97, 54, 19, 30], where transport is described by transmission
coefficients τn,σ (n is the transverse mode index and σ denotes spin) and partition noise
46
originates from the partial transmission of incident electrons. Within scattering theory, the
full expression for SPI is
SPI (Vsd) =
2e2
h
∫ ∑n,σ
τn,σ(ε)[1− τn,σ(ε)](fs − fd)2dε, (3.2)
where fs(d) is the Fermi function in the source (drain) lead. Equation (3.1) follows from
Eq. (3.2) only for the case of constant transmission across the energy window of transport,
with N = 12
∑τn,σ(1 − τn,σ). Furthermore, for spin-degenerate transmission, N vanishes
at multiples of 2e2/h and reaches the maximal value 0.25 at odd multiples of 0.5 × 2e2/h.
Energy dependence of transmission can reduce the maximal value below 0.25, as discussed
below.
While Eq. (3.1) is motivated by scattering theory, the value of N extracted from fitting
to Eq. (3.1) simply provides a way to quantify SPI (Vsd) experimentally for each Vg2. We
have chosen the bias range e|Vsd| . 5kBTe for fitting N to minimize nonlinear-transport
effects while extending beyond the quadratic-to-linear crossover in noise that occurs on the
scale e|Vsd| ∼ 2kBTe.
The dependence of N on conductance at B‖ = 0 is shown in Fig. 3.4(a), where N is
extracted from measured SPI (Vsd) at 90 values of Vg2. The horizontal axis, gavg, is the average
of the differential conductance over the bias points where noise was measured. N has the
shape of a dome, reaching a maximum near odd multiples of 0.5 × 2e2/h and vanishing
at multiples of 2e2/h. The observed N (gavg) deviates from the spin-degenerate, energy-
independent scattering theory in two ways. First, there is a reduction in the maximum
amplitude of N below 0.25. Second, there is an asymmetry in N with respect to 0.5×2e2/h,
resulting from a noise reduction near the 0.7 feature. A similar but weaker asymmetry is
observed about 1.5× 2e2/h. The reduction in the maximum amplitude can be understood
as resulting from an energy dependence of transmissions τn,σ; the asymmetry is a signature
47
!"
#
Figure 3.4: (a) Experimental N as a function of gavg at B‖ = 0 (red circles) along withmodel curves for nonzero (solid) and zero (dashed) proportionality of splitting, γn (see text).(b) Experimental N as a function of gavg in the range 0 − 1 × 2e2/h, at B‖ = 0 T (red),2 T (orange), 3 T (green), 4 T (cyan), 6 T (blue), and 7.5 T (purple). The dashed curveshows the single-particle model (γn = 0) at zero field for comparison.
of 0.7 structure, as we now discuss.
3.3.1 0.7 structure
We investigate further the relation between the asymmetry in N and the 0.7 structure by
measuring the dependence of N (gavg) on B‖. As shown in Fig. 3.4(b), N evolves smoothly
from a single asymmetric dome at B‖ = 0 to a symmetric double dome at 7.5 T. The latter
is a signature of spin-resolved electron transmission. Notably, for gavg between 0.7 and 1 (in
units of 2e2/h), N is insensitive to B‖, in contrast to the dependence of N near 0.3×2e2/h.
We compare these experimental data to the shot-noise prediction of a phenomenolog-
ical model [109] for the 0.7 anomaly. This model, originally motivated by dc transport
data, assumes a lifting of the twofold spin degeneracy of mode n by an energy splitting
∆εn,σ = σ · ρn · γn that grows linearly with 1D density ρn (with proportionality γn) within
that mode. Here, σ = ±1 and ρn =√
2m∗/h∑
σ(√µs − εn,σ +
√µd − εn,σ), where µs(d) is
the source(drain) chemical potential and m∗ is the electron effective mass. Parameters of
48
the phenomenological model are extracted solely from conductance. The lever arm convert-
ing Vg2 to energy (and hence ρn) as well as the transverse mode spacing are extracted from
transconductance (dg/dVg2) data [Fig. 3.5(a)] [112]. Using an energy-dependent transmis-
sion τn,σ(ε) = 1/(1 + e2π(εn,σ−ε)/~ωx,n) for a saddle-point potential [50, 51], the value ωx,n
(potential curvature parallel to the current) is found by fitting linear conductance below
0.5 × 2e2/h (below 1.5 × 2e2/h for the second mode), and γn is obtained by fitting above
0.5(1.5)× 2e2/h, where (within the model) the splitting is largest [see Fig. 3.5(b)]. We find
~ωx,1(2) is ∼ 500(300) µeV and γ1(2) ∼ 0.012(0.008) e2/4πε0 for the first (second) mode.
Note that the splitting 2 ·ρn ·γn is two orders of magnitude smaller than the direct Coulomb
energy of electrons spaced by 1/ρn. Using these parameters, SPI (Vsd) is calculated using
Eq. (3.2), and N is then extracted by fitting SPI (Vsd) to Eq. (3.1). The calculated values of
N (gavg) at B‖ = 0 are shown along with the experimental data in Fig. 3.4(a). For compar-
ison we include calculation results accounting for energy-dependent transmission without
splitting (γn = 0). The overall reduction of N arises from a variation in transmission across
the 150 µV bias window (comparable to ~ωx), and is a single-particle effect. On the other
hand, asymmetry of N about 0.5 and 1.5× 2e2/h requires nonzero γn.
Magnetic field is included in the model by assuming a g-factor of 0.44 and adding the
Zeeman splitting to the density-dependent splitting, maintaining the parameters obtained
above. Figure 3.5(c) shows calculated N (gavg) at B‖ corresponding to the experimen-
tal data, reproduced in Fig. 3.5(d). Including the magnetic field in quadrature or as a
thermally weighted mixture with the intrinsic density-dependent splitting gives essentially
indistinguishable results within this model. Model and experiment show comparable evolu-
tion of N with B‖: the asymmetric dome for B‖ = 0 evolves smoothly into a double dome
for 7.5 T, and for conductance & 0.7× 2e2/h, the curves for all fields overlap closely. Some
49
!"
#
Figure 3.5: (a) Transconductance dg/dVg2 as a function of Vsd and Vg2. Blue lines tracethe alignment of mode edges with source and drain chemical potentials; their slope andintersection give the conversion from Vg2 to energy and the energy spacing between modes,respectively. As two crossing points are observed between the first and second modes (themodel attributes this to spin-splitting in the first mode), we take the midpoint as thecrossing point for the blue lines. (c) Measured linear conductance (red) as a function ofVg2 at B‖ = 0, and linear conductance calculated with the model (black solid) with best-fitvalues for ωx,n and γn. Single-particle model takes γn = 0 (black dashed). (c) Model N asa function of gavg in the range 0− 1× 2e2/h, at B‖ = 0, 2, 3, 4, 6, and 7.5 T. (d) Same asFig. 3.4(b).
differences are observed between data and model, particularly for B‖ = 7.5 T. While the
experimental double dome is symmetric with respect to the minimum at 0.5 × 2e2/h, the
theory curve remains slightly asymmetric with a less-pronounced minimum. We find that
setting the g-factor to ∼ 0.6 in the model reproduces the measured symmetrical double
dome as well as the minimum value of N at 0.5×2e2/h. This observation is consistent with
reports of an enhanced g-factor in QPCs at low density [23, 101].
Recent theoretical treatments of 0.7 structure have also addressed its shot-noise signa-
ture. Modelling screening of the Coulomb interaction in the QPC, Lassl et al. [114] qualita-
tively reproduce the B‖-dependent N . Jaksch et al. [115] find a density-dependent splitting
in density-functional calculations that include exchange and correlation effects. This theory
justifies the phenomenological model and is consistent with the observed shot-noise suppres-
sion. Using a generalized single-impurity Anderson model motivated by density-functional
calculations that suggest a quasi-bound state [116], Golub et al. [117] find quantitative
50
Figure 3.6: Experimental N as a function of gavg at B‖ = 0 (red circles) for QPC 2, alongwith model curves for nonzero (solid) and zero (dashed) proportionality of splitting γn.Model calculations include bias-dependent electron heating.
agreement with the B‖-dependent N .
3.3.2 Bias-dependent electron heating
In contrast to QPC 1, noise data in QPC 2 show evidence of bias-dependent electron
heating. Figure 3.6 shows N (gavg) at B‖ = 0 over the first three conductance steps, ex-
tracted from fits using Eq. (3.1) to SPI (Vsd) data over the range |Vsd| ≤ 400 µV at 50
gate voltage settings. As in Fig. 3.4(a), a clear asymmetry in the noise factor is ob-
served, associated with enhanced noise reduction near 0.7 × 2e2/h. For this device, N
remains finite on conductance plateaus, showing super-linear dependence on plateau index.
This is consistent with bias-dependent thermal noise resulting from electron heating. Fol-
lowing Ref. [71], we incorporate into our model the bias-dependent electron temperature
T ∗e (Vsd) =√T 2e + (24/π2)(g/gm)(1 + 2g/gm)(eVsd/2kB)2, where gm is the parallel conduc-
tance of the reservoirs connecting to the QPC. This expression [71] models diffusion by
Wiedemann-Franz thermal conduction of the heat flux gV 2sd/2 on each side of the QPC and
of Joule heating in the reservoirs, assuming ohmic contacts thermalized to the lattice at Te.
51
In the absence of independent measurements of reservoir and ohmic contact resistances, we
treat 1/gm as a single free parameter.
Theoretical N curves including effects of bias-dependent heating are obtained from fits
to Eq. (3.1) of calculated SI(Vsd, T∗e (Vsd))−4kBTeg(Vsd). Parameters ωx,n = 1.35, 1.13, 0.86 meV
and γn = 0.019, 0.008, 0 e2/4πε0 for the first three modes (in increasing order) are extracted
from conductance data. To avoid complications arising from a zero-bias anomaly [101]
present in this device, γ0 is extracted from the splitting of the first sub-band edge in the
transconductance image [109], rather than from linear conductance. Other parameters are
extracted in the same way as for QPC 1. As shown in Fig. 3.6, quantitative agreement with
the N data is obtained over the three conductance steps with 1/gm = 75 Ω.
3.4 Conclusion and acknowledgements
We have presented measurements of current noise in quantum point contacts as a function of
source-drain bias, gate voltage, and in-plane magnetic field. We have observed a shot-noise
signature of the 0.7 structure at zero field, and investigated its evolution with increasing field
into the signature of spin-resolved transmission. Comparison to a phenomenological model
with density-dependent level splitting yielded quantitative agreement, and a device-specific
contribution to bias-dependent noise was shown to be consistent with electron heating.
We thank H.-A. Engel, M. Heiblum, L. Levitov, and A. Yacoby for valuable discussions,
and S. K. Slater, E. Onitskansky, N. J. Craig, and J. B. Miller for device fabrication. We
acknowledge support from NSF-NSEC, ARO/ARDA/DTO, and Harvard University.
52
Chapter 4
Tunable noise cross-correlations ina double quantum dot
D. T. McClure, L. DiCarlo, Y. Zhang, H.-A. Engel, C. M. MarcusDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138
M. P. Hanson, A. C. GossardDepartment of Materials, University of California, Santa Barbara, California 93106
We report measurements of the cross-correlation between temporal current fluctuations
in two capacitively coupled quantum dots in the Coulomb blockade regime. The sign of
the cross-spectral density is found to be tunable by gate voltage and source-drain bias.
We find good agreement with the data by including inter-dot Coulomb interaction in a
sequential-tunneling model.1
1This chapter is adapted with permission from Phys. Rev. Lett. 98, 056801 (2007). c©(2007) by the American Physical Society.
53
4.1 Introduction
Current noise cross-correlation in mesoscopic electronics, the fermionic counterpart of intensity-
intensity correlation in quantum optics, is sensitive to quantum indistinguishability as well
as many-body interactions [19, 29, 78, 79, 80, 81]. A distinctive feature of fermionic sys-
tems is that in the absence of interactions, noise cross-correlation is expected to always
be negative [54, 55]. Experimentally, negative correlations have been observed in several
solid-state Hanbury-Brown and Twiss-type noise measurements [85, 88, 86]. Since no sign
constraint exists for interacting systems, a positive noise cross-correlation in a fermi system
is a characteristic signature of interactions.
Sign reversal of noise cross-correlation has been the focus of recent theory and ex-
periment [118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 87, 130]. Theory
indicates that positive cross-correlations can arise in the presence of BCS-like interaction
[118, 119, 120], dynamical screening [121, 124], dynamical channel blockade [125, 126], and
strong inelastic scattering [127, 124, 128, 129]. Experimentally, sign reversal of noise cross-
correlation has been realized using a voltage probe to induce inelastic scattering [87], and
in a beam-splitter geometry, where the sign reversal was linked to a crossover from sub-
to super-Poissonian noise in a tunnel-barrier source [130]. This crossover was attributed
to Coulomb interaction between naturally-occurring localized states in the tunnel barrier
[113], as has been done in experiments on GaAs MESFETs [43] and stacked, self-assembled
quantum dots [44].
In this chapter, we investigate gate-controlled sign reversal of noise cross-correlation
in a simple four-terminal device. The structure consists of a parallel, capacitively coupled
double quantum dot operated in the Coulomb blockade regime. In this configuration, the
double dot acts as a pair of tunable interacting localized states, enabling a systematic study
54
280 mK 4.2 K
Digitize &Analyze
R L C
60 dB
R L60 dB
Vtl Vtc Vtr
Vbl
Vr
Vt
Vl
(a)
It
IbVb
500 nm
gb
Vbc Vbr
Stb
St
Sb
50Ω
50Ω
50Ω
50Ω
gt
(M,
N)(M,
N+1)
(M+1,
N+1)
(M+1,
N)
Figure 4.1: (a) Scanning electron micrograph of the double-dot device, and equivalentcircuit at 2 MHz of the noise detection system measuring the power spectral densities andcross spectral density of fluctuations in currents It and Ib. (b) Differential conductances gt
(yellow) and gb (magenta) as a function of Vtc and Vbc over a few Coulomb blockade peaksin each dot, at Vt = Vb = 0. Black regions correspond to well-defined charge states in thedouble-dot system. Superimposed white lines indicate the honeycomb structure resultingfrom the finite inter-dot capacitive coupling. (c) Zero-bias (thermal) noise Sb (black dots,right axis), conductance gb (magenta curve, left axis), and calculated 4kBTegb (magentacurve, right axis) as a function of gate voltage Vbc, with Vtc = −852.2 mV.
of Coulomb-induced correlation. Turning off inter-dot tunneling by electrically depleting
the connection between dots ensures that indistinguishability (i.e., fermi statistics) alone
cannot induce any cross-correlation; any cross-correlation, positive or negative, requires
inter-dot Coulomb interaction. We find good agreement between the experimental results
and a sequential-tunneling model of capacitively coupled single-level dots.
55
4.2 Device
The four-terminal double-dot device [see Fig. 4.1(a)] is defined by top gates on a GaAs/AlGaAs
heterostructure grown by molecular beam epitaxy. The two-dimensional electron gas 100 nm
below the surface has density 2 × 1011 cm−2 and mobility 2 × 105 cm2/Vs. Gate voltages
Vl = Vr = −1420 mV fully deplete the central point contact, preventing inter-dot tunneling.
Gate voltages Vtl (Vbl) and Vtr (Vbr) control the tunnel barrier between the top (bottom)
dot and its left and right leads. Plunger gate voltage Vtc (Vbc) controls the electron number
M (N) in the top (bottom) dot; for this experiment M ∼ N ∼ 100. The lithographic area
of each dot is 0.15 µm2. We estimate level spacing ∆t(b) ≈ 70 µeV in each dot, for ∼ 100 nm
depletion around the gates.
4.3 Methods
Measurements are performed in a 3He cryostat using a two-channel noise measurement
system [Fig. 4.1(a)] [131]. A voltage bias Vt (Vb) is applied to the left lead of the top
(bottom) dot, with right leads grounded. Separate resistor-inductor-capacitor resonators
(R = 5 kΩ, L = 66 µH, C = 96 pF) convert fluctuations in currents It and Ib through
the top and bottom dots around 2 MHz into voltage fluctuations on gates of high electron
mobility transistors (HEMTs) at 4.2 K, which in turn produce current fluctuations in two
50 Ω coaxial lines extending to room temperature, where further amplification is performed.
These signals are then simultaneously digitized at 10 MHz, their fast Fourier transforms
calculated, and the current noise power spectral densities St, Sb and cross-spectral density
Stb extracted following 15 s of integration, except for the data in Fig. 4.1(c), which was
averaged for 50 s per point. The total gain of each amplification line and the base electron
56
temperature Te = 280 mK are calibrated in situ using Johnson-noise thermometry at base
temperature and 1.6 K with the device configured as two point contacts [131]. Differential
conductance gt (gb) through the top (bottom) dot is measured using standard lock-in tech-
niques with an excitation of 25 (30) µVrms at 677 (1000) Hz. Ohmic contact resistances of
roughly a few kΩ, much less than the dot resistances, are not subtracted.
4.4 Double-dot characterization
Superposed top- and bottom-dot conductances gt and gb as a function of plunger voltages
Vtc and Vbc form the characteristic double-dot honeycomb pattern [132, 133], with dark
regions corresponding to well-defined electron number in each dot, denoted (M,N) (first
index for top dot), as shown in Fig. 4.1(b). Horizontal (vertical) features in gt (gb) are
Coulomb blockade (CB) conductance peaks [134], across which M (N) increases by one
as Vtc (Vbc) is raised. The distance between triple points, i.e., the length of the short
edge of the hexagon, provides a measure of the mutual charging energy U due to inter-dot
capacitive coupling. By comparing this distance to the CB peak spacing, and using the
single-dot charging energy EC = 600 µeV extracted from finite bias CB diamonds (not
shown), we estimate U ≈ 60 µeV [133]. We refer to the midpoint of the short edge of a
hexagon, midway between triple points, as a “honeycomb vertex.” Current noise Sb and
conductance gb, measured simultaneously at zero dc bias, over a CB peak in the bottom
dot (with the top dot in a CB valley) are shown in Fig. 4.1(c). Agreement between the
measured Sb and the Johnson-Nyquist thermal noise value 4kBTegb is observed.
57
Figure 4.2: Measured (a) and simulated (b) cross-spectral density Stb near a honeycombvertex, with applied bias Vt = Vb = −100 µV (e|Vt(b)| ≈ 4kBTe ≈ EC/6). Blue regions(lower-left and upper-right) indicate negative Stb, while red regions indicate positive Stb.
4.5 Sign-reversal of noise cross correlation
Turning now to finite-bias noise measurements, Fig. 4.2(a) shows the measured cross-
correlation Stb as a function of plunger gate voltages Vtc and Vbc, in the vicinity of a
honeycomb vertex, with voltage bias of −100 µV applied to both dots. The plot reveals
a characteristic quadrupole pattern of cross-correlation centered on the honeycomb vertex,
comprising regions of both negative and positive cross-correlation. Similar patterns are
observed at all other honeycomb vertices. The precise symmetry of the pattern is found to
depend rather sensitively on the relative transparency of each dot’s left and right tunnel
barriers. Away from the vertices, noise cross-correlation vanishes.
58
4.6 Master equation simulation
To better understand this experimental result, we model the system as single-level dots
capacitively coupled by a mutual charging energy U , each with weak tunneling to the
leads. The energy needed to add electron M + 1 to the top dot depends on the two
plunger gate voltages as well as the electron number n ∈ N,N + 1 on the bottom dot:
Et = αtVtc + βtVbc + U · n + const., where lever arms αt and βt are obtained from the
honeycomb plot in Fig. 4.1(b) [132] and the measured EC . The energy Eb to add electron
N + 1 to the bottom dot is given by an analogous formula. Occupation probabilities for
charge states (M,N), (M +1, N), (M,N +1), and (M +1, N +1) are given by the diagonal
elements of the density matrix, ρ = (ρ00, ρ10, ρ01, ρ11)T . The time evolution of ρ is given by
a master equation dρ/dt =Mρ, where
M =
−W out00 W00←10 W00←01 0
W10←00 −W out10 0 W01←11
W01←00 0 −W out01 W10←11
0 W11←10 W11←01 −W out11
. (4.1)
Each diagonal term of M gives the total loss rate for the corresponding state: W outα =∑
βWβ←α. Off-diagonal terms give total rates for transitions between two states. For
example, W10←00 = W l10←00 + W r
10←00 is the total tunneling rate into (M + 1, N) from
(M,N), combining contributions from the top-left and top-right leads.
Rates for tunneling between a dot and either of its leads i ∈ tl, tr, bl, br depend on
both the transparency Γi of the tunnel barrier to lead i and the Fermi function fi(ε) =
1 + exp [(ε− µi)/kBTe]−1 evaluated at ε = Et(b), where µi is the chemical potential in
lead i. For example, the rates for tunneling into and out of the top dot from/to the left
lead are given by W l10←00 = Γltflt(Et) and W l
00←10 = Γlt [1− flt(Et)], respectively. As Et is
59
lowered across µlt, W l10←00 increases from 0 to Γlt over a range of a few kBTe, while W l
00←10
does the opposite.
We obtain the steady-state value of ρ, denoted ρ, by solving Mρ = 0. Following
Refs. [135, 136, 137], we define current matrices J tr and Jbr for the top- and bottom-right
leads, with elements J trmn,m′n′ = |e|δnn′(m − m′)W rmn←m′n′ and Jbrmn,m′n′ = |e|δmm′(n −
n′)W rmn←m′n′ . We next obtain the average currents 〈It(b)〉 =
∑i
[J t(b)rρ
]i
and the cor-
relator 〈It(τ)Ib(0)〉 =∑
i
[θ(τ)J treMτJbrρ+ θ(−τ)JbreMτJ trρ
]i
(θ is the Heaviside step
function). The cross-spectral density in the low-frequency limit is then given by Stb =
2∫∞−∞ [〈It(τ)Ib(0)〉 − 〈It〉〈Ib〉] dτ.
Simulation results for cross-correlation Stb as a function of plunger gate voltages are
shown in Fig. 4.2(b), with all parameters of the model extracted from experiment: U =
60 µeV, Te = 280 mK, Γtl = Γtr = 1.5×1010 s−1, and Γbl = Γbr = 7.2×109 s−1. The Γi were
estimated from the zero-bias conductance peak height using Eq. (6.3) of Ref. [138], taking
left and right barriers equal. The simulation shows the characteristic quadrupole pattern of
positive and negative cross-correlation, as observed experimentally. We note that the model
underestimates Stb by roughly a factor of two. This may be due to transport processes not
accounted for in the model. For instance, elastic cotunneling should be present since the
Γi are comparable to kBTe/~. Also, since the voltage-bias energy |eVt(b)| is greater than
the level spacing ∆t(b), transport may occur via multiple levels [139, 125, 126, 46, 33] and
inelastic cotunneling [140, 141, 45].
4.7 Intuitive explanation
Intuition for how Coulomb interaction in the form of capacitive inter-dot coupling can lead
to the observed noise cross-correlation pattern can be gained by examining energy levels in
60
U
(b)
(c) (d)
(a)
Vbc
Vtc
eVt
eVb
kT
kT
µtl
µtr
µbl
µbr
U
Figure 4.3: Energy level diagrams in the vicinity of a honeycomb vertex, with biases Vt(b) =−100 µV. (The various energies are shown roughly to scale.) The solid horizontal line inthe top (bottom) dot represents the energy Et(b) required to add electron M + 1 (N + 1)when the bottom (top) dot has N (M) electrons. The dashed horizontal line, higher thanthe solid line by U , represents Et(b) when the bottom (top) dot has N+1 (M+1) electrons.In each dot, the rate of either tunneling-in from the left or tunneling-out to the right issignificantly affected by this difference in the energy level, taking on either a slow value(red arrow) or a fast value (green arrow) depending on the electron number in the otherdot. In (a) and (d), where the occurrence of each U -sensitive process enhances the rate ofthe other, we find positive cross-correlation. In (b) and (c), where the occurrence of eachU -sensitive process suppresses the rate of the other, we find negative cross-correlation.
both dots in the space of plunger gate voltages, as shown in Fig. 4.3. With both dots tuned
near Coulomb blockade peaks, the fluctuations by one in the electron number of each dot,
caused by the sequential tunneling of electrons through that dot, cause the energy level of
the other dot to fluctuate between two values separated by U . These fluctuations can raise
and lower the level across the chemical potential in one of the leads of the dot, strongly
affecting either the tunnel-in rate (from the left, for the case illustrated in Fig. 4.3) or the
tunnel-out rate (to the right) of that dot. Specifically, the rate of the “U -sensitive” process
61
in each dot fluctuates between a slow rate (red arrow), suppressed well below Γi, and a
fast rate (green arrow), comparable to Γi. For balanced right and left Γi in each dot, the
U -sensitive process becomes the transport bottleneck when its rate is suppressed.
These U -sensitive processes correlate transport through the dots. In region (b) of
Fig. 4.3, for instance, where Stb is negative, the U -sensitive process in each dot is tunneling-
out. Here and in (c), where the U -sensitive process in each dot is tunneling-in, the U -
sensitive processes compete: occurrence of one suppresses the other, leading to negative
Stb. Conversely, in region (a) [(d)], where Stb is positive, the top [bottom] dot’s U -sensitive
process is tunneling-out, but the bottom [top] dot’s is tunneling-in. Here, the U -sensitive
processes cooperate: occurrence of one lifts the suppression of the other, leading to positive
Stb.
4.8 Some additional checks
The arguments above also apply when one or both biases are reversed. When both are
reversed, we find both experimentally and in the model that the same cross-correlation
pattern as in Fig. 4.2 appears (not shown). When only one of the biases is reversed, we find
both experimentally [as shown in Fig. 4.4(a)] and in the model that the pattern reverses
sign. In the absence of any bias, cross-correlation vanishes both experimentally [as shown
in Fig. 4.4(b)] and in the model, despite the fact that noise in the individual dots remains
finite [as seen in Fig. 4.1(c)].
62
Figure 4.4: (a) Measured Stb near a honeycomb vertex, with opposite biases Vt = −Vb =−100 µV. Note that the pattern is reversed from Fig. 4.2(a): negative cross-correlation(blue) is now found in the upper-left and lower-right regions, while positive cross-correlation(red) is now found in the lower-left and upper-right. (b) Measured Stb near a honeycombvertex, with Vt = Vb = 0. Cross-correlation vanishes at zero bias, though the noise in eachdot is finite.
4.9 Conclusion and acknowledgements
We have observed gate-controlled sign reversal of noise cross-correlation in a double quan-
tum dot in the Coulomb blockade regime with purely capacitive inter-dot coupling. Ex-
perimental observations are in good agreement with a sequential-tunneling model, and can
be understood from an intuitive picture of mutual charge-state-dependent tunneling. This
study, notable for the simplicity and controllability of the device, may be particularly useful
for understanding current noise in systems where interacting localized states occur naturally
and uncontrollably.
We thank N. J. Craig for device fabrication and M. Eto, W. Belzig, C. Bruder, E. Sukho-
rukov, and L. Levitov for valuable discussions. We acknowledge support from the NSF
through the Harvard NSEC, PHYS 01-17795, DMR-05-41988, DMR-0501796, as well as
support from NSA/DTO and Harvard University.
63
Chapter 5
Noise correlations in a Coulombblockaded quantum dot
Yiming Zhang, L. DiCarlo, D. T. McClure, C. M. MarcusDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
M. Yamamoto, S. TaruchaDepartment of Applied Physics, University of Tokyo, Bunkyoku, Tokyo 113-8656, Japan
ICORP-JST, Atsugi-shi, Kanagawa 243-0198, JapanM. P. Hanson, A. C. Gossard
Department of Materials, University of California, Santa Barbara, California 93106, USA
We report measurements of current noise auto- and cross correlation in a tunable quan-
tum dot with two or three leads. As the Coulomb blockade is lifted at finite source-drain
bias, the auto-correlation evolves from super-Poissonian to sub-Poissonian in the two-lead
case, and the cross correlation evolves from positive to negative in the three-lead case, con-
sistent with transport through multiple levels. Cross correlations in the three-lead dot are
found to be proportional to the noise in excess of the Poissonian value in the limit of weak
output tunneling.1
1This chapter is adapted with permission from Phys. Rev. Lett. 99, 036603 (2007). c©(2007) by the American Physical Society.
64
5.1 Introduction
Considered individually, Coulomb repulsion and Fermi statistics both tend to smooth elec-
tron flow, thereby reducing shot noise below the uncorrelated Poissonian limit [19, 29, 54,
55]. For similar reasons, Fermi statistics without interactions also induces a negative noise
cross correlation in multiterminal devices [19, 54, 55, 85, 88]. It is therefore surprising that
under certain conditions, the interplay between Fermi statistics and Coulomb interaction
can lead to electron bunching, i.e., super-Poissonian auto-correlation and positive cross
correlation of electronic noise.
The specific conditions under which such positive noise correlations can arise has been
the subject of numerous theoretical [142, 127, 124, 128, 129, 140, 137, 139, 125, 126, 141]
and experimental [142, 43, 44, 113, 130, 87, 45, 46, 143, 34] studies in the past few years.
Super-Poissonian noise observed in metal-semiconductor field effect transistors [43], tunnel
barriers [113] and self-assembled stacked quantum dots [44] has been attributed to interact-
ing localized states [136, 137, 43] occurring naturally in these devices. In more controlled
geometries, super-Poissonian noise has been associated with inelastic cotunneling [140] in
a nanotube quantum dot [45], and with dynamical channel blockade [139, 125, 126] in
GaAs/AlGaAs quantum dots in the weak-tunneling [46] and quantum Hall regimes [143].
Positive noise cross correlation has been observed in a capacitively coupled double dot [34]
as well as in electronic beam splitters following either an inelastic voltage probe [87, 127,
124, 128, 129] or a super-Poissonian noise source [130]. The predicted positive noise cross
correlation in a three-lead quantum dot [125, 126] has not been reported experimentally to
our knowledge.
This chapter describes measurement of current noise auto- and cross correlation in a
Coulomb-blockaded quantum dot configured to have either two or three leads. As a function
65
of gate voltage and bias, regions of super- and sub-Poissonian noise, as well as positive and
negative noise cross correlation, are identified. Results are in good agreement with a multi-
level sequential-tunneling model in which electron bunching arises from dynamical channel
blockade [139, 125, 126]. For weak-tunneling output leads, noise cross correlation in the
three-lead configuration is found to be proportional to the deviation of the auto-correlation
from the Poissonian value (either positive or negative) similar to the relation found in
electronic Hanbury Brown–Twiss (HBT)–type experiments [85, 88, 130].
5.2 Device
The quantum dot is defined by gates on the surface of a GaAs/Al0.3Ga0.7As heterostructure
[Fig. 5.1(a)]. The two-dimensional electron gas 100 nm below the surface has density 2 ×
1011 cm−2 and mobility 2 × 105 cm2/Vs. Leads formed by gate pairs Vl-Vbl, Vr-Vbr, and
Vl-Vr connect the dot to three reservoirs labeled 0, 1, and 2, respectively. Plunger gate
voltage Vbc controls the electron number in the dot, which we estimate to be ∼ 100. The
constriction formed by Vtl-Vl is closed.
5.3 Methods
A 3He cryostat is configured to allow simultaneous conductance measurement near dc and
noise measurement near 2 MHz [131]. For dc measurements, the three reservoirs are each
connected to a voltage amplifier, a current source, and a resistor to ground (r = 5 kΩ).
The resistor r converts the current Iα out of reservoir α to a voltage signal measured by
the voltage amplifier; it also converts the current from the current source to a voltage
excitation Vα applied at reservoir α. The nine raw differential conductance matrix elements
66
(a)
Digitize&
AnalyzeS12
S2
S1
0.3 K 4.2 K
60 dB
60 dB
Vtl
Vbl
VrVl
I2
I1
Vbc Vbr
V0I0
500 nm
R
R
Figure 5.1: (a) Micrograph of the device and equivalent circuit near 2 MHz of the noisedetection system (see text for equivalent circuit near dc). For the data in Figs. 5.1 and 5.2,the Vl-Vr constriction is closed and the dot is connected only to reservoirs 0 and 1. (b, c)Differential conductance g01 and current noise spectral density S1, respectively, as a functionof V0 and Vbc. (d) S1 versus |I1| data (circles) and multi-level simulation (solid curves) alongthe four cuts indicated in (b) and (c) with corresponding colors. Black solid (dashed) lineindicates S1 = 2e|I1| (S1 = 1e|I1|). (e) Data (diamonds) and multi-level simulation (solidcurves) of the modified Fano factor F along the same cuts as taken in (d). Inset: detail ofF at high |V0|.
gαβ = dIβ/dVα are measured simultaneously with lock-in excitations of 20 µVrms at 44, 20
and 36 Hz on reservoirs 0, 1 and 2, respectively. Subtracting r from the matrix g yields the
67
intrinsic conductance matrix2 g = [E + rg]−1 · g, where E is the identity matrix. Ohmic
contact resistances (∼ 103 Ω) are small compared to dot resistances (& 105 Ω), and are
neglected in the analysis. Values for the currents Iα with bias V0 applied to reservoir 0 are
obtained by numerically integrating g0α.
Fluctuations in currents I1 and I2 are extracted from voltage fluctuations around 2 MHz
across separate resistor-inductor-capacitor (RLC) resonators [Fig. 5.1(a)]. Power spectral
densities SV1,2 and cross-spectral density SV12 of these voltage fluctuations [131] are av-
eraged over 20 s, except where noted. Following the calibration of amplifier gains and
electron temperature Te using noise thermometry [131], the dot’s intrinsic current noise
power spectral densities S1,2 and cross-spectral density S12 are extracted by solving the
Langevin [19] equations that take into account the feedback [128] and thermal noise from
the finite-impedance external circuit3:
S1 = a211SV 1 + a2
21SV 2 + 2a11a21SV 12 − 4kBTe/R
S2 = a212SV 1 + a2
22SV 2 + 2a12a22SV 12 − 4kBTe/R
S12 = a11a12SV 1 + a21a22SV 2 + (a11a22 + a12a21)SV 12,
where a11(22) = 1/R − g11(22), a12(21) = −g12(21) and R is the RLC resonator parallel
resistance.
5.4 Noise in the two-lead configuration
Figure 5.1(b) shows conductance g01 as a function of Vbc and V0 in a two-lead configuration,
i.e., with the Vl-Vr constriction closed. The characteristic Coulomb blockade (CB) diamond
2See derivation in Sec. C.1 of Ap. C.
3See derivation in Sec. C.2 of Ap. C.
68
structure yields a charging energy EC = 0.8 meV and lever arm for the plunger gate
ηbc = ∆εd/(e∆Vbc) = 0.069, where εd is the dot energy. The diamond tilt ηbc/(1/2 − η0)
gives the lever arm for reservoir 0: η0 = ∆εd/(e∆V0) = 0.3. As shown in Fig. 5.1(d), current
noise S1 along selected cuts close to the zero-bias CB peak (red, orange cuts) is below the
Poissonian value 2e|I1| at all biases |I1|, while cuts that pass inside the CB diamond (green,
blue cuts) exceed 2e|I1| at low currents, then drop below 2e|I1| at high currents. At finite
Te, the current noise SP1 = 2eI1 coth(eV0/2kBTe) of an ideal Poissonian noise source at bias
V0 may exceed 2e|I1| due to the thermal (Johnson) noise contribution [140]. Accordingly, we
define a modified Fano factor F ≡ S1/SP1 . Figure 5.1(e) shows regions of super-Poissonian
noise (F > 1) when the green and blue cuts are within the CB diamond. For all cuts, F
approaches 1/2 at large bias.
Current noise can also be identified as sub- or super-Poissonian from the excess Pois-
sonian noise SEP1 ≡ S1 − SP
1 being negative or positive, respectively. Unlike F , SEP1 does
not have divergent error bars inside the CB diamond, where currents vanish. As shown in
Fig. 5.2(a), in regions where both I1 and S1 vanish, SEP1 also vanishes. Far outside the CB
diamonds, SEP1 is negative, indicating sub-Poissonian noise. However, SEP
1 becomes positive
along the diamond edges, indicating super-Poissonian noise in these regions.
We next compare our experimental results to single-level and multi-level sequential-
tunneling models of CB transport. The single-level model yields exact expressions for
average current and noise [19, 29, 141, 144]: I1 = (e/h)∫dεγ0γ1(f1 − f0)/[(γ1 + γ0)2/4 +
(ε − εd)2], S1 = (2e2/h)∫dεγ2
0γ21 · [f0(1 − f0) + f1(1 − f1)] + γ0γ1[(γ1 − γ0)2/4 + (ε −
εd)2] · [f0(1 − f1) + f1(1 − f0)]/[(γ1 + γ0)2/4 + (ε − εd)2]2, where γ0(1) is the tunneling
rate to reservoir 0(1) and f0(1) is the Fermi function in reservoir 0(1). The dot energy εd
is controlled by gate and bias voltages: εd = −eVbcηbc − eV0η0 − eV1η1 + const. For the
69
multi-level sequential-tunneling model, a master equation is used to calculate current and
noise4, following Refs. [139, 125, 126, 135]. To model transport, we assume simple filling
of orbital levels and consider transitions to and from N -electron states that differ in the
occupation of at most n levels above (indexed 1 through n) and m levels below (indexed
−1 through −m) the highest occupied level in the (N + 1)-electron ground state (level 0).
For computational reasons, we limit the calculation to n = m = 3. For simplicity, we
assume equal level spacings, symmetric tunnel barriers, and an exponential dependence of
the tunneling rates on level energy: ∆εl ≡ εld−ε0d = l×δ and γl0 = γl1 = Γ exp(κ∆εl), where
l = −3, ..., 0, ..., 3 is the level index, εld is the energy of level l, and γl0(1) is the tunneling rate
from level l to reservoir 0(1). We choose δ = 150 µeV, Γ = 15 GHz and κ = 0.001 (µeV)−1
to fit the data in Figs. 5.1(d) and 5.1(e).
Super-Poissonian noise in the multi-level model arises from dynamical channel block-
ade [139, 125, 126], illustrated in the diagrams in Fig. 5.2. Consider, for example, the energy
levels and transport processes shown in the green-framed diagram, which corresponds to the
location of the green dot on the lower-right edge in Fig. 5.2(c). Along that edge, the trans-
port involves transitions between the N -electron ground state and (N + 1)-electron ground
or excited states. When an electron occupies level 0, it will have a relatively long lifetime,
as tunneling out is suppressed by the finite electron occupation in reservoir 1 at that en-
ergy. During this time, transport is blocked since the large charging energy prevents more
than one non-negative-indexed level from being occupied at a time. This blockade happens
dynamically during transport, leading to electron bunching and thus to super-Poissonian
noise. At the location of the pink dot on the lower-left edge in Fig. 5.2(c), the transport
involves transitions between the (N + 1)-electron ground state and N -electron ground or
4The simulation source routines are provided in Ap. E.
70
-1.0
-0.5
0.0
0.5
1.0
V0
[mV
]
-1.74-1.77 Vbc [V]
-8 -4 0 4 8SEP [10-28A2/ Hz]
SUB-POISSONIAN | SUPER-POISSONIAN
EXP. (a)
1
-1.0
-0.5
0.0
0.5
1.0
V0
[mV
]
-1.745-1.76 Vbc [V]
SIM. (S.L.) (b)
-1.745-1.76 Vbc [V]
SIM. (M.L.) (c)
+3+2+10-1-2-3
Figure 5.2: (a) Excess Poissonian noise SEP1 as a function of V0 and Vbc. Red (blue)
regions indicate super(sub)-Poissonian noise. (b, c) Single-level (S.L.) and multi-level (M.L.)simulation of SEP
1 , respectively, corresponding to the data region enclosed by the whitedashed parallelogram in (a). At the four colored dots superimposed on (c), where SEP
1
is most positive, energy diagrams are illustrated in the correspondingly colored frames atthe bottom. In these diagrams, black (white) arrows indicate electron (hole) transport;the greyscale color in the reservoirs and inside the circles on each level indicates electronpopulation, the darker the higher.
excited states; a similar dynamical blockade occurs in a complementary hole transport pic-
ture. The hole transport through level 0 is slowed down by the finite hole occupation in
reservoir 0, modulating the hole transport through negative-indexed levels, thus leading to
hole bunching and super-Poissonian noise. Transport at the blue (orange) dot is similar
71
to transport at the green (pink) dot, but with the chemical potentials in reservoirs 0 and
1 swapped. Both experimentally and in the multi-level simulation, SEP1 is stronger along
electron edges than along hole edges. This is due to the energy dependence of the tunnel-
ing rates: since the positive-indexed electron levels have higher tunneling rates than the
negative-indexed hole levels, the dynamical modulation is stronger for electron transport
than for hole transport.
5.5 Noise in the three-lead configuration
We next investigate the three-lead configuration, obtained by opening lead 2 [Fig. 5.3(a)].
At zero bias, thermal noise cross correlation is found to be in good agreement with the
theoretical value5, S12 = −4kBTeg12, as seen in Fig. 5.3(b).
To minimize this thermal contribution to S12, output leads are subsequently tuned to
weaker tunneling than the input lead (g01 ∼ g02 ∼ 4g12), for reasons discussed below. Note
that as a function of Vbc and V0, S12 [Fig. 5.3(c)] looks similar to SEP1 [Fig. 5.2(a)] in the
two-lead configuration. The slightly positive S12 (∼ 0.2×10−28A2/Hz) inside the rightmost
diamond is due to a small drift in the residual background of SV 12 over the 13 h of data
acquisition for Fig. 3(c). Without drift, as in the shorter measurement of Fig. 5.3(b), S12
approaches 0 at zero bias as g12 vanishes.
Both the single-level and multi-level models can be extended to include the third
lead [144, 125, 126]. Figures 5.3(d) and 5.3(e) show the single-level and multi-level simula-
tions of S12, respectively. Similar to the two-lead case, only the multi-level model reproduces
the positive cross correlation along the diamond edges.
5At zero bias, the fluctuation-dissipation theorem requires S12 = −2kBTe(g12 + g21), butg12 = g21 at zero bias and zero magnetic field.
72
R
R
Vtl
Vbl
VrVl
I2
I1
Vbc Vbr
V0I0
500 nm
(a)
-0.3
-0.2
-0.1
0.0
[10-2
8 A2 / H
z]
-1.512 -1.509Vbc [V]
-0.04
-0.02
0.00
[e2/ h x 4k
B Te ]
S12 -4kBTe g12
(b)
-1.49-1.50 Vbc [V]
SIM. (M.L.) (e)
-1.0
-0.5
0.0
0.5
1.0
V0
[mV
]
-1.49-1.50 Vbc [V]
SIM. (S.L.) (d)
-1.0
-0.5
0.0
0.5
1.0
V0
[mV
]
-1.48-1.49-1.50-1.51 Vbc [V]
-2 -1 0 1 2S12 [10-28A2/ Hz]
EXP. (c)
Figure 5.3: (a) The device in the three-lead configuration, in which the data for this figureand for Fig. 5.4 are taken. (b) S12, integrated for 200 s, and −4kBTeg12 over a CB peak atzero bias. Left and right axes are in different units but both apply to the data. (c) S12 as afunction of V0 and Vbc. Red (blue) regions indicate positive (negative) cross correlation. (d,e) Single-level (S.L.) and multi-level (M.L.) simulation of S12, respectively, correspondingto the data region enclosed by the white dashed parallelogram in (c).
To further investigate the relationship between noise auto- and cross correlation, we com-
pare S12 to the total excess Poissonian noise, SEP ≡ S1+S2+2S12−2e(I1+I2) coth(eV0/2kBTe),
measured in the same three-lead configuration. Figure 5.4 shows SEP and S12, measured
at fixed bias V0 = +0.5 mV. The observed proportionality S12 ∼ SEP/4 is reminiscent
73
-1
0
1
[10-2
8 A2 / H
z]
-1.48-1.49-1.50-1.51Vbc [V]
S12 SEP/ 4
(a)
-4 0 4SEP [10-28A2/ Hz]
S12 vs. SEP
Slope = 1/4
(b)
Figure 5.4: (a) S12 (green) and SEP/4 (blue) as a function of Vbc at V0 = +0.5 mV [greenhorizontal line in Fig. 5.3(c)]. (b) Parametric plot of S12 (green circles) versus SEP for thesame data as in (a). The solid black line has a slope of 1/4, the value expected for a 50/50beam splitter.
of electronic HBT-type experiments [85, 88, 130], where noise cross correlation following a
beam splitter was found to be proportional to the total output current noise in excess of the
Poissonian value, with a ratio of 1/4 for a 50/50 beam splitter. In simulation, we find that
this HBT-like relationship holds in the limit g01 ∼ g02 g12 (recall that g01 ∼ g02 ∼ 4g12
in the experiment); on the other hand, when g01 ∼ g02 ∼ g12, thermal noise gives a negative
contribution that lowers S12 below SEP/4, as we have also observed experimentally (not
shown). The implications are that first, with weak-tunneling output leads, the three-lead
dot behaves as a two-lead dot followed by an ideal beam splitter, and second, the dynamical
channel blockade that leads to super-Poissonian noise in the two-lead dot also gives rise to
positive cross correlation in the three-lead dot.
5.6 Acknowledgements
We thank N. J. Craig for device fabrication and H.-A. Engel for valuable discussions. We
acknowledge support from the NSF through the Harvard NSEC, PHYS 01-17795, DMR-
05-41988, DMR-0501796. M. Yamamoto and S. Tarucha acknowledge support from the
74
DARPA QuIST program, the Grant-in-Aid for Scientific Research A (No. 40302799), the
MEXT IT Program and the Murata Science Foundation.
75
Chapter 6
Shot noise in graphene
L. DiCarlo†, J. R. Williams‡, Yiming Zhang†, D. T. McClure†, C. M. Marcus††Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA‡School of Engineering and Applied Sciences, Harvard University, Cambridge,
Massachusetts 02138, USA
We report measurements of current noise in single- and multi-layer graphene devices.
In four single-layer devices, including a p-n junction, the Fano factor remains constant to
within ±10% upon varying carrier type and density, and averages between 0.35 and 0.38.
The Fano factor in a multi-layer device is found to decrease from a maximal value of 0.33
at the charge-neutrality point to 0.25 at high carrier density. These results are compared
to theories for shot noise in ballistic and disordered graphene.1
1This chapter is adapted with permission from Phys. Rev. Lett. 100, 156801 (2008). c©(2008) by the American Physical Society.
76
6.1 Introduction
Shot noise, the temporal fluctuation of electric current out of equilibrium, originates from
the partial transmission of quantized charge [19]. Mechanisms that can lead to shot noise
in mesoscopic conductors include tunneling, quantum interference, and scattering from im-
purities and lattice defects. Shot noise yields information about transmission that is not
available from the dc current alone.
In graphene [145, 146], a zero-gap two-dimensional semi-metal in which carrier type and
density can be controlled by gate voltages [147], density-dependent shot-noise signatures
under various conditions have been investigated theoretically [148, 149]. For wide samples
of ballistic graphene (width-to-length ratio W/L & 4) the Fano factor, F , i. e., the current
noise normalized to the noise of Poissonian transmission statistics, is predicted to be 1/3 at
the charge-neutrality point and ∼ 0.12 in both electron (n) and hole (p) regimes [148]. The
value F = 1−1/√
2 ≈ 0.29 is predicted for shot noise across a ballistic p-n junction [149]. For
strong, smooth “charge-puddle” disorder, theory predicts F ≈ 0.30 both at and away from
the charge-neutrality point, for all W/L & 1 [150]. Disorder may thus have a similar effect on
noise in graphene as in diffusive metals, where F is universally 1/3 [151, 152, 153, 73, 74, 82]
regardless of shape and carrier density. Recent theory investigates numerically the evolution
from a density-dependent to a density-independent F with increasing disorder [154]. To
our knowledge, experimental data for shot noise in graphene has not yet been reported.
This chapter presents an experimental study of shot noise in graphene at low tempera-
tures and zero magnetic field. Data for five devices, including a locally gated p-n junction,
are presented. For three globally-gated, single-layer samples, we find F ∼ 0.35 − 0.37 in
both electron and hole doping regions, with essentially no dependence on electronic sheet
density, ns, in the range |ns| . 1012 cm−2. Similar values are obtained for a locally-gated
77
Digitize& FFT
60 dB
60 dB CrossSpectrum
0.3 K
2 µmI0
4.2 K (c)
Vbg
50Ω
50Ω 50Ω
50Ω
r
Figure 6.1: (a) Differential resistance R of sample A1 as a function of back-gate voltage Vbg
at electron temperature Te = 0.3 K, perpendicular field B⊥ = 0, and source-drain voltageVsd = 0. (b) Differential two-terminal conductance g(Vsd = 0) as a function of B⊥ and Vbg
in the quantum Hall regime, after subtracting a quadratic fit at each B⊥. Lines of constantfilling factors 6, 10, 14, and 18 (dashed lines) indicate a single-layer sample. (c) Equivalentcircuit near 1.5 MHz of the system measuring current noise using cross correlation of twochannels [131]. Current bias Io contains a 7.5 nArms, 20 Hz part for lock-in measurementsand a controllable dc part generating the dc component of Vsd via the shunt resistancer = 5 kΩ. False-color scanning electron micrograph of a three-lead pattern defining twodevices similar to A1 and A2. Purple indicates single-layer graphene and gold indicatesmetallic contacts.
single-layer p-n junction in both unipolar (n-n or p-p) and bipolar (p-n or n-p) regimes.
In a multi-layer sample, the observed F evolves from 0.33 at the charge-neutrality point to
0.25 at ns ∼ 6× 1012 cm−2.
6.2 Methods
Devices were fabricated by mechanical exfoliation of highly-oriented pyrolytic graphite [147].
Exfoliated sheets were deposited on a degenerately-doped Si substrate capped with 300 nm
of thermally grown SiO2. Regions identified by optical microscopy as potential single-layer
78
graphene were contacted with thermally evaporated Ti/Au leads (5/40 nm) patterned by
electron-beam lithography. Additional steps in the fabrication of the p-n junction device are
detailed in Ref. [155]. Devices were measured in two 3He cryostats, one allowing dc (lock-
in) transport measurements in fields |B⊥| ≤ 8 T perpendicular to the graphene plane, and
another allowing simultaneous measurements of dc transport and noise [131] near 1.5 MHz,
but limited to B⊥ ∼ 0.
6.3 Shot noise in single-layer devices
Differential resistance R = dVsd/dI (I is the current, and Vsd is the source-drain voltage) of a
wide, short sample [A1, (W,L) = (2.0, 0.35) µm] is shown as a function of back-gate voltage
Vbg at Vsd = 0 and B⊥ = 0 in Fig. 6.1(a). While the width of the peak is consistent with
A1 being single-layer graphene [156, 157], more direct evidence is obtained from the QH
signature shown in Fig. 6.1(b). The grayscale image shows differential conductance g = 1/R
as a function of Vbg and B⊥, following subtraction of the best-fit quadratic polynomial
to g(Vbg) at each B⊥ setting to maximize contrast. Dashed lines correspond to filling
factors nsh/eB⊥ = 6, 10, 14, and 18, with ns = α(Vbg + 1.1 V) and lever arm α =
6.7×1010 cm−2/V. Their alignment with local minima in δg(Vbg) identifies A1 as single-layer
graphene [158, 159]. The Drude mean free path ` = h/2e2 ·σ/kF [160], where kF =√π|ns|,
is found to be ∼ 40 nm away from the charge-neutrality point using the B⊥ = 0 conductivity
σ = (RW/L)−1 [Fig. 6.2(a) inset].
Current noise spectral density SI is measured using a cross-correlation technique de-
scribed in Ref. [131] [see Fig. 6.1(c)]. Following calibration of amplifier gains and electron
temperature Te using Johnson noise thermometry (JNT) for each cooldown, the excess
noise SeI ≡ SI − 4kBTeg(Vsd) is extracted. Se
I(Vsd) for sample A1 is shown in Fig. 6.2(a).
79
Figure 6.2: (a) Inset: Conductivity σ = (RW/L)−1 calculated using R(Vbg) data inFig. 6.1(a) and W/L = 5.7. Solid black circles correspond to σ(Vsd = 0) at the Vbg settingsof noise measurements shown in (b). Main: Excess noise Se
I as function of Vsd near thecharge-neutrality point, Vbg = −0.75 V. The solid red curve is the single-parameter bestfit to Eq. (6.1), giving Fano factor F = 0.349 (using Te = 303 mK as calibrated by JNT).(b) Best-fit F at 25 Vbg settings across the charge-neutrality point for electron and holedensities reaching |ns| ∼ 1.4× 1012 cm−2. (c) R (left axis) and σ (right axis) of sample A2as a function of Vbg (W/L = 1.4), with Vsd = 0, at 0.3 K (solid markers) and at 1.1 K (openmarkers). (d), (e) Crossover width Tw (normalized to JNT-calibrated Te) and F , obtainedfrom best-fits using Eq. (6.1) to Se
I(Vsd) data over |Vsd| ≤ 350(650) µV for Te = 0.3(1.1) K.
80
Linearity of SeI at high bias indicates negligible extrinsic (1/f or telegraph) resistance fluc-
tuations within the measurement bandwidth. For these data, a single-parameter fit to the
scattering-theory form (for energy-independent transmission) [97, 55],
SeI = 2eIF
[coth
(eVsd
2kBTe
)− 2kBTe
eVsd
], (6.1)
gives a best-fit Fano factor F = 0.349. Simultaneously measured conductance g ≈ 22.2 e2/h
was independent of bias within ±0.5% (not shown) in the |Vsd| ≤ 350 µV range used for the
fit. Note that the observed quadratic-to-linear crossover agrees well with that in the curve
fit, indicating weak inelastic scattering in A1 [73, 74], and negligible series resistance (e. g.,
from contacts), which would broaden the crossover by reducing the effective Vsd across the
sample.
Figure 6.2(b) shows similarly measured values for F as a function of Vbg. F is observed
to remain nearly constant for |ns| . 1012 cm−2. Over this density range, the average F is
0.35 with standard deviation 0.01. The estimated error in the best-fit F at each Vbg setting
is ±0.002, comparable to the marker size and smaller than the variation in F near Vbg = 0,
which we believe results from mesoscopic fluctuations of F . Nearly identical noise results
(not shown) were found for a similar sample (B), with dimensions (2.0, 0.3) µm and a QH
signature consistent with a single layer.
Transport and noise data for a more square single-layer sample [A2, patterned on the
same graphene sheet as A1, with dimensions (1.8, 1.3) µm] at Te = 0.3 K (solid circles) and
Te = 1.1 K (open circles) are shown in Figs. 6.2(c-e). At both temperatures, the conductivity
shows σmin ≈ 1.5 e2/h and gives ` ∼ 25 nm away from the charge-neutrality point. That
these two values differ from those in sample A1 is particularly notable as samples A1 and
A2 were patterned on the same piece of graphene. Results of fitting Eq. (6.1) to SeI(Vsd)
for sample A2 are shown in Figs. 6.2(d) and 6.2(e). To allow for possible broadening of
81
the quadratic-to-linear crossover by series resistance and/or inelastic scattering, we treat
electron temperature as a second fit parameter (along with F) and compare the best-fit
value, Tw, with the Te obtained from Johnson noise. Figure 6.2(d) shows Tw tracking the
calibrated Te at both temperatures. Small deviations of Tw/Te from unity near the charge-
neutrality point at Te = 0.3 K can be attributed to conductance variations up to ±20%
in the fit range |Vsd| ≤ 350 µV at these values of Vbg. As in sample A1, F is found to be
independent of carrier type and density over |ns| . 1012 cm−2, averaging 0.37(0.36) with
standard deviation 0.02(0.02) at Te = 0.3(1.1) K. Evidently, despite its different aspect
ratio, A2 exhibits a noise signature similar to that of A1.
6.4 Shot noise in a p-n junction
Transport and noise measurements for a single-layer graphene p-n junction [155], sample
C, are shown in Fig. 6.3. The color image in Fig. 6.3(a) shows differential resistance R as
a function of Vbg and local top-gate voltage Vtg. The two gates allow independent control
of charge densities in adjacent regions of the device [see Fig. 6.3(c) inset]. In the bipolar
regime, the best-fit F shows little density dependence and averages 0.38, equal to the average
value deep in the unipolar regime, and similar to results for the back-gate-only single-layer
samples (A1, A2 and B). Close to charge neutrality in either region (though particularly
in the region under the top gate), SeI(Vsd) deviates from the form of Eq. (6.1) (data not
shown). This is presumably due to resistance fluctuation near charge neutrality, probably
due mostly to mobile traps in the Al2O3 insulator beneath the top gate.
82
12
Figure 6.3: (a) Differential resistance R of sample C, a single-layer p-n junction, as afunction of back-gate voltage Vbg and top-gate voltage Vtg. The skewed-cross pattern definesquadrants of n and p carriers in regions 1 and 2. Red lines indicate charge-neutrality linesin region 1 (dotted) and region 2 (dashed). (b) Se
I(Vsd) measured in n-p regime with(Vbg, Vtg) = (5,−4) V (solid dots) and best fit to Eq. (6.1) (red curve), with F = 0.36. (c)Main: Best-fit F along the cuts shown in (a), at which ns1 ∼ ns2 (purple) and ns1 ∼ −4 ns2(black). Inset: Schematic of the device. The top gate covers region 2 and one of thecontacts.
6.5 Shot noise in a multi-layer device
Measurements at 0.3 K and at 1.1 K for sample D, of dimensions (1.8, 1.0) µm, are shown
in Fig. 6.4. A ∼ 3 nm step height between SiO2 and carbon surfaces measured by atomic
force microscopy prior to electron-beam lithography [161] suggests this device is likely multi-
layer. Further indications include the broad R(Vbg) peak [162] and the large minimum
conductivity, σmin ∼ 8 e2/h at B⊥ = 0 [Fig. 6.4(a)], as well as the absence of QH signature
for |B⊥| ≤ 8 T at 250 mK (not shown). Two-parameter fits of SeI(Vsd) data to Eq. (6.1)
show three notable differences from results in the single-layer samples [Figs. 6.4(b) and
83
Figure 6.4: (a) Differential resistance R (left axis) and conductivity σ (right axis) of sampleD as a function of Vbg, with Vsd = 0, at 0.3 K (solid markers) and at 1.1 K (open markers).(b),(c) Best-fit Tw (normalized to JNT-calibrated Te) and F to Se
I(Vsd) data over |Vsd| ≤0.5(1) mV for Te = 0.3(1.1) K. Inset: Sublinear dependence of Se
I on Vsd is evident in datataken over a larger bias range. Solid red curve is the two-parameter best fit of Eq. (6.1)over |Vsd| ≤ 0.5 mV.
6.4(c)]: First, F shows a measurable dependence on back-gate voltage, decreasing from
0.33 at the charge-neutrality point to 0.25 at ns ∼ 6 × 1012 cm−2 for Te = 0.3 K; Second,
F decreases with increasing temperature; Finally, Tw/Te is 1.3-1.6 instead of very close to
1. We interpret the last two differences, as well as the sublinear dependence of SeI on Vsd
(see Fig. 6.4 inset) as indicating sizable inelastic scattering [151, 152] in sample D. (An
alternative explanation in terms of series resistance would require it to be density, bias, and
84
temperature dependent, which is inconsistent with the independence of g on Vsd and Te).
6.6 Summary and acknowledgements
Summarizing the experimental results, we find that in four single-layer samples, F is in-
sensitive to carrier type and density, temperature, aspect ratio, and the presence of a p-n
junction. In one multi-layer sample, F does depend on density and temperature, and SeI(Vsd)
shows a broadened quadratic-to-linear crossover and is sublinear in Vsd at high bias. We
may now compare these results to expectations based on theoretical and numerical results
for ballistic and disordered graphene.
Theory for ballistic single-layer graphene with W/L & 4 gives a universal F = 1/3 at
the charge-neutrality point, where transmission is evanescent, and F ∼ 0.12 for |ns| & π/L2,
where propagating modes dominate transmission [148]. While the measured F at the charge-
neutrality point in samples A1 and B (W/L = 5.7 and 6.7, respectively) is consistent with
this prediction, the absence of density dependence is not: π/L2 ∼ 3×109 cm−2 is well within
the range of carrier densities covered in the measurements. Theory for ballistic graphene
p-n junctions [149] predicts F ≈ 0.29, lower than the value ∼ 0.38 observed in sample C
in both p-n and n-p regimes. We speculate that these discrepancies likely arise from the
presence of disorder. Numerical results for strong, smooth disorder [150] predict a constant
F at and away from the charge-neutrality point for W/L & 1, consistent with experiment.
However, the predicted value F ≈ 0.30 is ∼ 20% lower than observed in all single-layer
devices. Recent numerical simulations [154] of small samples (L = W ∼ 10 nm) investigate
the vanishing of carrier dependence in F with increasing disorder strength. In the regime
where disorder makes F density-independent, the value F ∼ 0.35− 0.40 is found to depend
weakly on disorder strength and sample size.
85
Since theory for an arbitrary number of layers is not available for comparison to noise
results in the multi-layer sample D, we compare only to existing theory for ballistic bi-layer
graphene [163]. It predicts F = 1/3 over a much narrower density range than for the single
layer, and abrupt features in F at finite density due to transmission resonances. A noise
theory beyond the bi-layer ballistic regime may thus be necessary to explain the observed
smooth decrease of F with increasing density in sample D.
We thank C. H. Lewenkopf, L. S. Levitov, and D. A. Abanin for useful discussions.
Research supported in part by the IBM Ph.D. Fellowship program (L.D.C.), INDEX, an
NRI Center, and Harvard NSEC.
86
Chapter 7
Distinct Signatures For CoulombBlockade and Aharonov-BohmInterference in ElectronicFabry-Perot Interferometers
Yiming Zhang, D. T. McClure, E. M. Levenson-Falk, C. M. MarcusDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
L. N. Pfeiffer, K. W. WestBell Laboratories, Alcatel-Lucent Technologies, Murray Hill, New Jersey 07974, USA
Two distinct types of magnetoresistance oscillations are observed in two electronic
Fabry-Perot interferometers of different sizes in the integer quantum Hall regime. Mea-
suring these oscillations as a function of magnetic field and gate voltages, we describe three
signatures that distinguish the two types. The oscillations observed in a 2.0 µm2 device
are understood to arise from a Coulomb blockade mechanism, and those observed in an
18 µm2 device from an Aharonov-Bohm mechanism. This work clarifies, provides ways to
distinguish, and demonstrates control over these distinct mechanisms of oscillations seen in
electronic Fabry-Perot interferometers. 1
1This chapter is adapted from Ref. [22], which has been accepted for publication inPhys. Rev. B as a Rapid Communication.
87
7.1 Introduction
Mesoscopic electronics can exhibit wave-like interference effects [4, 10, 11, 164], particle-like
charging effects [3], or a complex mix of both [20]. Experiments over the past two decades
have investigated the competition between wave and particle properties [21], as well as
regimes where they coexist [14, 15, 165, 20]. The electronic Fabry-Perot interferometer
(FPI)— a planar two-contact quantum dot operating in the quantum Hall regime—is a
system where both interference and Coulomb interactions can play important roles. This
device has attracted particular interest recently due to predicted signatures of fractional [62]
and non-Abelian [64, 63, 166] statistics. The interpretation of experiments, however, is
subtle, and must account for the interplay of charging and interference effects in these
coherent confined structures.
The pioneering experimental investigation of resistance oscillations in an electronic FPI
[167] interpreted the oscillations in terms of an Aharonov-Bohm (AB) interference of edge
states, attributing the magnetic field dependence of the field-oscillation period to a chang-
ing effective dot area. More recent experiments [168, 169, 170, 171, 69] have observed
frequencies of integer multiples of the fundamental AB frequency; in particular, a pro-
portionality of field frequency to the number of fully-occupied Landau levels (LL’s) has
been well established [172, 170, 171, 69] in devices up to a few µm2 in size. Both experi-
mental [173, 168, 170, 171, 69] and theoretical [172, 174, 175] investigations indicate that
Coulomb interaction plays a critical role in these previously observed oscillations—as a
function of both magnetic field and electrostatic gate voltage—suggesting an interpretation
in terms of field- or gate-controlled Coulomb blockade (CB). The questions of whether it
is even possible to observe resistance oscillations that arise from pure AB interference in
FPI’s, and if so, in what regime, and how to distinguish the two mechanisms, have yet to
88
be answered to our knowledge.
In this chapter, we report two different types of resistance oscillations as a function of
perpendicular magnetic field, B, and gate voltage in FPI’s of two different sizes. The type
observed in the smaller (2.0 µm2) device, similar to previous results [167, 173, 168, 169,
170, 171, 69], is consistent with the interacting CB interpretation, while that observed in
the larger (18 µm2) device is consistent with noninteracting AB interference. Specifically,
three signatures that distinguish the two types of oscillations are presented: The magnetic
field period is inversely proportional to the number of fully occupied LL’s for CB, but field-
independent for AB; The gate-voltage period is field-independent for CB, but inversely
proportional to B for AB; Resistance stripes in the two-dimensional plane of B and gate
voltage have a positive (negative) slope in the CB (AB) regime.
7.2 Device and measurement
The devices were fabricated on a high-mobility two-dimensional electron gas (2DEG) resid-
ing in a 30 nm wide GaAs/AlGaAs quantum well 200 nm below the chip surface, with Si
δ-doping layers 100 nm below and above the quantum well. The mobility is ∼ 2, 000 m2/Vs
measured in the dark, and the density is 2.6×1015 m−2. Surface gates that define the FPI’s
are patterned using electron-beam lithography on wet-etched Hall bars [see Fig. 7.1(a)].
These gates come in from top left and bottom right, converging near the middle of the Hall
bar. Figures 7.1(b) and (c) show gate layouts for the 2.0 µm2 and 18 µm2 interferome-
ters. All gate voltages except VC are set around ∼ −3 V (depletion occurs at ∼ −1.6 V).
Voltages, VC, on the center gates are set near 0 V to allow fine tuning of density and area.
Measurements are made using a current bias I = 400 pA, with B oriented into the
2DEG plane as shown in Fig. 7.1(a). The diagonal resistance, RD ≡ dVD/dI is related to
89
(a) B
VTVTL VTR VR
VBVL VC1 µm
2.0 µm2 device(b)
1 µm
VTVTL VTRVL
VBRVB VRVBLVC
18 µm2 device(c)
RD
I
Figure 7.1: Measurement setup and devices. (a) Diagram of the wet-etched Hall bar, surfacegates, and measurement configuration. Diagonal resistance, RD, is measured directly acrossthe Hall bar, with current bias, I. Subsequent zoom-ins of the surface gates are also shown;the red box encloses the detailed gate layouts for the device shown in (c). (b,c) Gate layoutsfor the 2.0 µm2 and 18 µm2 devices, respectively. The areas quoted refer to those underVC.
the dimensionless conductance of the device g = (h/e2)/RD [176]. Here, VD is the voltage
difference between edge states entering from the top right and bottom left of the device.
7.3 Resistance oscillations in the 2.0 µm2 device
Figure 7.2(a) shows RD as a function of B measured in the 2.0 µm2 device, displaying
several quantized integer plateaus. Figures 7.2(b) and (c) show the zoom-ins below the
g = 1 and 2 plateaus, respectively, displaying oscillations in RD as a function of B, with
periods ∆B = 2.1 mT and 1.1 mT. This ∆B of 2.1 mT corresponds to one flux quantum,
90
1
1/2
1/3
1/41/51/61/8
RD [
h / e
2 ]
87654321B [ T ]
f 0 = 4 3 2 1
(a)
0.50
0.47
RD [ h / e
2 ]
3.703.66B [ T ]
ΔB = 1.1 mT
(c)
0.98
0.94R
D [ h / e2 ]
7.307.26 B [ T ]
ΔB = 2.1 mT
(b)
Figure 7.2: Oscillations in RD as a function of magnetic field, B, for the 2.0 µm2 device.(a) RD as a function of B, showing well-quantized integer plateaus. Different colored back-grounds indicate different numbers of fully-occupied LL’s, f0, through the device. (b, c)Zoom-ins of the data in (a), at f0 = 1 and 2, respectively, showing oscillations in RD, andtheir B periods, ∆B.
φ0 ≡ h/e, through an area A = 2.0 µm2, which matches the device design; hence 1.1 mT
corresponds to φ0/2 through about the same area. This is indeed the field-period scaling
observed previously [167, 170, 171, 69], where for f0 number of fully occupied LL’s in the
constrictions, ∆B is expected to be given by (φ0/A)/f0. Thus, in Fig. 7.3(a) we show ∆B
at each 1/f0, and a linear fit constrained through the origin, demonstrating the expected
relationship.
We emphasize that this field-period scaling is inconsistent with simple AB oscillations,
which would give a constant ∆B corresponding to one flux quantum through the area of the
device. This can, however, be understood within an intuitive picture presented in a recent
theoretical analysis [174] that considers a dominant Coulomb interaction within the device.
In this picture, on the riser of RD where f0 < g < f0 + 1, the (f0 + 1)th and higher LL’s
91
2.0 μm2 device
2.0
1.5
1.0
0.5
0.0
Δ B [
mT
]
ΔB = 2.1 mT • [ ]
(a)
3
2
1
0
ΔVT, Δ
VC [
mV
]
ΔVT = 3.1 mV ΔVC = 0.18 mV
(e) 0.98
0.94
RD [ h / e
2 ]
210ΔVC = 0.17 mV
(f) B = 7.2 T
0.47
0.45
RD [ h / e
2 ]
210ΔVC = 0.18 mV
(g) B = 3.6 T
0.24
0.22
RD [ h / e
2 ]
210VC [ mV ]
ΔVC = 0.18 mV
(h) B = 1.8 T
0.96
0.92R
D [ h / e2 ]
7.137.12ΔB = 2.1 mT
(b)
0.42
0.40
RD [ h / e
2 ]
3.443.43ΔB = 1.1 mT
(c)
0.24
0.23
RD [ h / e
2 ]
1.821.81B [ T ]
ΔB = 0.54 mT
(d)
112
14
16
180
f 0
1
f 0
1
112
14
16
180
f 0
1
Figure 7.3: Magnetic field and gate voltage periods at various f0, for the 2.0 µm2 device.(a) ∆B as a function of 1/f0, and a best-fit line constrained through the origin. (b-d) RD
oscillations as a function of B, at f0 = 1, 2, and 4, respectively. (e) ∆VT (diamonds) and∆VC (circles) as a function of 1/f0, and their averages indicated by horizontal lines. (f-h)RD oscillations as a function of VC, at f0 = 1, 2, and 4, respectively.
will form a quasi-isolated island inside the device that will give rise to Coulomb blockade
92
effects for sufficiently large charging energy,
EC =1
2C(ef0 ·BA/φ0 + eN − CgVgate)2, (7.1)
where N is the number of electrons on the island, C is the total capacitance, and Cg is the
capacitance between the gate and the dot. The magnetic field couples electrostatically to
the island through the underlying LL’s: when B increases by φ0/A, the number of electrons
in each of the f0 underlying LL’s will increase by one. These LL’s will act as gates to
the isolated island: Coulomb repulsion favors a constant total electron number inside the
device, so N will decrease by f0 for every φ0/A change in B, giving rise to f0 resistance
oscillations.
Further evidence for the CB mechanism in the 2.0 µm2 device is found in the resistance
oscillations as a function of gate voltages. Figures 7.3(f-h) show RD as a function of center
gate voltage VC, for f0 = 1, 2 and 4, respectively. Figure 3(e) summarizes gate voltage
periods ∆VT and ∆VC at various f0, and shows they are independent of f0. This behavior
is consistent with the CB mechanism, because, as can be inferred from Eq. (7.1), gate-
voltage periods are determined by the capacitance Cg, which should be independent of
f0.
7.4 Resistance oscillations in the 18 µm2 device
Having identified CB as the dominant mechanism2 for resistance oscillations in the 2.0 µm2
device, we fabricated and measured an 18 µm2 device, an order of magnitude larger in size,
2Although the existence of interference in small devices cannot be ruled out, we emphasizethat Coulomb charging alone is sufficient to explain all data observed in small devices.A recent preprint (Ref. [177]) interprets magneto-oscillations in a small dot in terms ofan interfering AB path, quantized to enclose an integer N . However, as will be seen inFig. 7.4(e), gate voltage periods can change continuously by an order of magnitude in theAB regime, suggesting that N is not quantized in the AB regime.
93
18 μm2 device0.3
0.2
0.1
0.0
ΔB [
mT
]
2.01.51.00.50.01 / B [ T-1 ]
ΔB = 0.244 mT
(a)
14
12
10
8
6
4
2
0
ΔVT, Δ
VC [
mV
]
2.01.51.00.50.01 / B [ T-1 ]
2.34
2.32
RD [ h / e
2 ]
-3.004 -3.000ΔVT = 0.74 mV
(f) B = 6.2 T
0.562
0.559
RD [ h / e
2 ]
-3.00 -2.98ΔVT = 3.0 mV
(g) B = 2.5 T
0.184
0.180
RD [ h / e
2 ]
-3.03 -3.00 -2.97VT [ V ]
ΔVT = 9.4 mV
(h) B = 0.72 T
2.27
2.24R
D [ h / e2 ]
6.1986.194ΔB = 0.24 mT
(b)
0.555
0.552
RD [ h / e
2 ]
2.5142.510
ΔB = 0.24 mT
(c)
0.185
0.181
RD [ h / e
2 ]
0.7280.724B [ T ]
ΔB = 0.25 mT
(d)
ΔVT = 6.7 mV • [ ]
ΔVC = 1.3 mV • [ ]
(e) 1TB1TB
Figure 7.4: Magnetic field and gate voltage periods at various B, for the 18 µm2 device.(a) ∆B as a function of 1/B, and their average indicated by a horizontal line. (b-d) RD
oscillations as a function of B, over three magnetic field ranges. (e) ∆VT (diamonds) and∆VC (circles) as a function of 1/B, and best-fit lines constrained through the origin. (f-h)RD oscillations as a function of VT, at B = 6.2 T, 2.5 T, and 0.72 T, respectively.
hence an order of magnitude smaller in charging energy. The center gate covering the whole
device, not present in previous experiments [167, 173, 168, 169, 170, 171, 69], also serves
94
to reduce the charging energy. In this device, RD as a function of B at three different
fields is plotted in Figs. 7.4(b-d), showing nearly constant ∆B. The summary of data in
Fig. 7.4(a) shows that ∆B, measured at 10 different fields ranging from 0.5 to 6.2 T, is
indeed independent of B; its average value of 0.244 mT corresponds to one φ0 through an
area of 17 µm2, close to the designed area. This is in contrast to the behavior observed in
the 2.0 µm2 device, and is consistent with simple AB interference. Gate voltage periods are
also studied, as has been done in the 2.0 µm2 device. Figures 7.4(f-h) show RD as a function
of VT at three different fields, and Fig. 7.4(e) shows both ∆VT and ∆VC as a function of
1/B. In contrast to the behavior observed in the 2.0 µm2 device, ∆VT and ∆VC are no
longer independent of B, but proportional to 1/B. This behavior is consistent with AB
interference, because the total flux is given by φ = B · A and the flux period is always φ0;
assuming that the area changes linearly with gate voltage, gate-voltage periods would scale
as 1/B for AB.
7.5 One more signature
As shown above, the magnetic field and gate voltage periods have qualitatively different B
dependence in the 2.0 µm2 and 18 µm2 devices, the former consistent with CB, and the
latter consistent with AB interference. Based on these physical pictures, one can make
another prediction in which these two mechanisms will lead to opposite behaviors. In the
CB case, increasing B increases the electron number in the underlying LL’s, thus reducing
the electron number in the isolated island via Coulomb repulsion. This is equivalent to
applying more negative gate voltage to the device. On the other hand, for the AB case,
increasing B increases the total flux through the interferometer, and applying more positive
gate voltage increases the area, thus also the total flux; therefore, higher B is equivalent to
95
1.0
0.8
0.6
0.4
0.2
0.0
VC [
mV
]
1.7661.7651.7641.763B [ T ]
-5
0
5
δRD [ 10
-3 h / e2 ]
2.0 μm2 device
-8
-6
-4
-2
0
2
4
VC [
mV
]
0.5460.5450.544B [ T ]
2
0
-2
δRD [ 10
-3 h / e2 ]
18 μm2 device
(a)
(b)
Figure 7.5: (a) δRD, i.e. RD with a smooth background subtracted, as a function of Band VC, for the 2.0 µm2 device. (b) Same as in (a), but for the 18 µm2 device.
more positive gate voltage. As a result, if RD is plotted in a plane of gate voltage and B,
we expect stripes with a positive slope in the CB case and a negative slope in the AB case.
Figures 7.5(a,b) show RD as a function of VC and B for the 2.0 µm2 and 18 µm2
devices, respectively. As anticipated, the stripes from the 2.0 µm2 device have a positive
96
slope, consistent with the CB mechanism, while stripes from the 18 µm2 device have a
negative slope, consistent with AB interference. This difference can serve to determine the
origin of resistance oscillations without the need to change magnetic field significantly.
7.6 Discussion
The three distinct signatures that we observe between CB and AB interference in this work
can also shed light on some of the previous experiments and their interpretations. A few
recent experiments studying fractional charge and statistics in FPI’s [65, 66, 67, 68] interpret
resistance oscillations as arising from AB interference while taking each gate-voltage period
as indicating a change of a quantized charge. However, as shown in Fig. 7.4(e), the gate
voltage periods observed in the big device change by more than an order of magnitude over
the field range that we study, and are inversely proportional to 1/B, suggesting that charge
is not quantized in the AB regime. Also in Ref. [67], the authors have observed that the
magnetic field period stay constant between filling factor 1 and 1/3, but the gate voltage
period at filling factor 1/3 is only 1/3 the size at filling factor 1. Although these observations
can be interpreted as a result of fractional statistics, as the authors have done, there are at
least two other possible interpretations: integer AB interference and CB with a charge of
e/3. We consider clear identification of the mechanisms leading to oscillations—for instance
using the method of Fig. 7.5—to be crucial for interpreting future experiments, particularly
as the quantum states under investigation become more subtle.
97
7.7 Acknowledgements
We acknowledge J. B. Miller for device fabrication and discussion, R. Heeres for his work
on the cryostat, and I. P. Radu, M. A. Kastner, B. Rosenow, N. Ofek, I. Neder and B. I.
Halperin for helpful discussions. This research is supported in part by Microsoft Corporation
Project Q, IBM, NSF (DMR-0501796), and the Center for Nanoscale Systems at Harvard
University.
98
Chapter 8
Edge-State Velocity and Coherencein a Quantum HallFabry-Perot Interferometer
D. T. McClure†, Yiming Zhang†, B. Rosenow†‡, E. M. Levenson-Falk†, C. M. Marcus††Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA‡Max-Planck-Institute for Solid State Research, Heisenbergstr. 1, D-70569 Stuttgart,
GermanyL. N. Pfeiffer, K. W. West
Bell Laboratories, Alcatel-Lucent Technologies, Murray Hill, New Jersey 07974, USA
We investigate nonlinear transport in electronic Fabry-Perot interferometers in the in-
teger quantum Hall regime. For interferometers sufficiently large that Coulomb blockade
effects are absent, a checkerboard-like pattern of conductance oscillations as a function of
dc bias and perpendicular magnetic field is observed. Edge-state velocities extracted from
the checkerboard data are compared to model calculations and found to be consistent with
a crossover from skipping orbits at low fields to ~E × ~B drift at high fields. Suppression of
visibility as a function of bias and magnetic field is accounted for by including energy- and
field-dependent dephasing of edge electrons. 1
1This chapter is adapted from Ref. [178].
99
8.1 Introduction
The electronic Fabry-Perot interferometer (FPI), implemented as a quantum dot in the
quantum Hall (QH) regime, has attracted theoretical [62, 63, 64, 174, 166] and experimen-
tal [66, 179, 67, 69, 68, 22] interest recently, especially in light of the possibility of observing
fractional [62] or non-Abelian [64, 63, 166, 26] statistics in this geometry. Earlier experi-
ments reveal that Coulomb [167, 173, 180] and Kondo [181, 182] physics can play important
roles, as well. With such a rich spectrum of physics in these devices, a thorough under-
standing of the mechanisms governing transport even in the integer QH regime remains
elusive.
While most work on electronic FPI’s to date has focused on transport at zero dc bias,
finite-bias measurements have proved to be a useful tool in understanding the physical
mechanisms important in other interferometer geometries. In metallic [9] and semiconduct-
ing [132] rings interrupted by tunnel barriers, oscillations in transmission as a function of
magnetic field and dc bias, forming a checkerboard pattern, have been observed. These fea-
tures, attributed to the electrostatic Aharonov-Bohm (AB) effect [183, 184, 185], were used
to measure the time of flight and dephasing in these devices. Similar checkerboard-like lobe
structures have also been observed in Mach-Zehnder interferometers [186, 164, 187]. In that
case, the pattern of oscillations is not readily explained within a single-particle picture and
remains the subject of continued theoretical study [188, 189, 190, 191]. In electronic FPI’s,
conductance oscillations as a function of dc bias have been investigated theoretically [62]
and provide a means of extracting the edge-state velocity from the period in dc bias. Edge-
state velocity measurement without the use of high-bandwidth measurements [192, 193]
will likely be useful in determining appropriate device parameters to probe exotic statistics
beyond the integer regime. This approach was recently used [179] to measure the edge-state
100
velocity at ν = 1/3, though in a small (∼ 1 µm2) device where Coulomb interactions, absent
in the theory, may be expected to play a dominant role [69, 22].
In this chapter, we present measurements of finite-bias conductance oscillations in an
18 µm2 electronic FPI whose zero-bias behavior is consistent with AB interference without
significant Coulomb effects [22]. We find a checkerboard-like pattern of conductance oscil-
lations as a function of dc bias and magnetic field, in agreement with the predictions of
Chamon et al. [62]. Measuring the period in dc bias allows the velocity of the tunneling
edge state to be extracted over a range of magnetic fields, yielding a low-field saturation
consistent with a crossover from ~E × ~B drift to skipping orbits. High-bias fading in the
checkerboard pattern is quantitatively consistent with a dephasing rate proportional to en-
ergy and magnetic field. Zero-bias oscillations in a 2 µm2 device of similar design, where
Coulomb effects are significant [22], do not evolve periodically with dc bias; instead, plots of
conductance versus bias and magnetic field reveal diamond-like regions of blockaded trans-
port in the weak-forward-tunneling regime that become more smeared out with stronger
forward tunneling.
8.2 Device and measurement
Devices are fabricated on GaAs/AlGaAs quantum-well structures with a two-dimensional
electron gas (2DEG) of density n = 2.7× 1015 m−2 and mobility µ = 2, 000 m2/Vs located
200 nm below the surface. Hall bars are wet-etched as shown in Fig. 8.1(a), and metal
surface gates are patterned by electron-beam lithography as in Fig. 8.1(b). Interferometers
are defined by negative voltages (∼ −3V) applied to all gates except VC, and samples are
cooled in a dilution refrigerator to ∼ 20 mK. A current bias I, consisting of a dc component
of up to 30 nA and a 135-Hz component of 400 pA, gives rise to the diagonal voltage VD
101
(a) B VD
I
(b)
1 µm
VTVTL VTRVL
VBRVB VVBLVC R
(c)
a
Figure 8.1: Measurement setup and the electronic Fabry-Perot device (a) With a currentbias I applied at one end of the Hall bar, voltage VD is measured directly across its width.Surface gates are shown in increasing detail, with a red box indicating the region shownin (b). (b) Gate layout of the 18 µm2 device, which is operated as an interferometer bydepleting all gates except VC. (c) Schematic diagram of possible transmission paths throughthe device in the quantum Hall regime.
across the device, measured directly across the width of the Hall bar [Fig. 8.1(a)]. Lock-
in measurements of diagonal conductance, GD ≡ dI/dVD, are used to study changes in
interferometer transmission as a function of both VD and perpendicular magnetic field B.
As shown in Fig. 8.1(c), the current-carrying chiral edge states can be partially reflected
at each constriction, leading to interference between the different possible trajectories as a
function of the phase accumulated by encircling the interferometer.
102
8.3 Checkerboard pattern and interpretation
A typical measurement of GD as a function of B and VD in the 18 µm2 device is shown in
Fig. 8.2(a), where a smooth background has been subtracted. A checkerboard-like pattern
of oscillations periodic in both B and VD is observed, with reduced amplitude at high bias.
Similar patterns are seen at fields B = 0.22−1.26 T; over this range the Landau level index,
N , of the tunneling edge ranges from 4 to 1, but the field period of oscillations is always
∆B ≈ 0.25 mT, independent of both field and bias.
Magnetoconductance oscillations in this device reflect AB interference of partially trans-
mitted edge states [22], with a phase shift ∆ϕ = 2πΦ/Φ0, where Φ = BA is the flux enclosed
(in area A) by the interfering edge, and Φ0 ≡ h/e is the magnetic flux quantum. The ob-
served field period corresponds to A ≈ 17 µm2, consistent with the dot area after subtracting
a depletion length of roughly the 2DEG depth. The sinusoidal lineshape of the oscillations
seen here suggests that coherent transport is dominated by two trajectories that differ in
length by one traversal of the dot perimeter.
When a dc bias is added to VD, an additional phase shift appears between interfering
trajectories, associated with the energy-dependent wave vector of the contributing edge-
state electrons; we will refer to this as the Fabry-Perot phase. The wave vector changes
with energy as δk = δε/~v, where v denotes the edge-state velocity. Following the analysis
of non-interacting electrons in Ref. [62], in which bias is assumed to affect mainly the
chemical potential, we assign an additional relative phase 2aε/~v to an electron traversing
the perimeter at energy ε above the zero bias Fermi level, where a ∼ 2√A = 8.2 µm
denotes the path length between constrictions [Fig. 8.1(c)]. For a symmetrically applied
dc bias (relative to the gate voltages), and neglecting contributions from multiply-reflected
103
-60
-40
-20
0
20
40
60
VD [ m
V ]
1.51.00.50.0
d B [ mT ]
Calc
ula
ted d
GD [ a
. u. ]
(b)
-60
-40
-20
0
20
40
60
VD [ m
V ]
1.51.00.50.0
d B [ mT ]
-0.05
0.00
0.05
dG
D [ e
2 / h ]
(a)
Figure 8.2: Nonlinear magnetoconductance in an Fabry-Perot interferometer (a) GD as afunction of B and VD in the 18 µm2 device near B = 0.47 T, with a smooth backgroundsubtracted. (b) δGD calculated from Eq. (8.1), multiplied by the damping factor fromEq. (8.2), with ∆B = 0.25 mT, ∆VD = 56 µV, and α = 0.2.
trajectories, the expected differential conductance has the form
δGD(Φ, VD) = δG0 cos(2πΦ/Φ0) cos(eVDa/v~), (8.1)
where the amplitude δG0 does not depend on field or dc bias. Note that in this model,
the contributions of AB and Fabry-Perot phase separate into a product of two cosines,
yielding a checkerboard pattern, as observed in the experimental data, Fig. 8.2(a). Ref. [62]
104
predicts that when the bias is only applied to one contact, with the other contact held at
ground (again, relative to the gates), the two phase contributions from bias and field instead
appear as arguments of a single cosine, yielding a diagonal stripe pattern. Experimentally,
the bias is always applied only at one end of the Hall bar, with the other end grounded;
however, interaction effects within the dot are likely to effectively symmetrize the applied
bias [194]. Alternatively, a model in which the bias mainly affects the electrostatic (rather
than chemical) potential [195] also yields Eq. (8.1) without the need for a symmetric bias. In
either interpretation, the bias period corresponds to the edge velocity via ∆VD = (h/e)(v/a).
We account for the reduced amplitude of oscillations at high bias by multiplying the right
side of Eq. (8.1) by a damping factor, e−2πα |VD|/∆VD , where (2πα)−1 gives the number of
periods over which the amplitude falls to 1/e of its zero-bias value. Lacking theory for edge-
state dephasing in FPI’s, this form is motivated by the observation in related experiments
of a dephasing rate proportional to energy [132, 196]. We thus identify a voltage-dependent
dephasing rate, τ−1ϕ (VD) = α|eVD|/2~, which reduces amplitude by e−2to/τϕ , where 2to =
2a/v is the time of flight around the interferometer. To extract interference and dephasing
parameters, the form
δG(VD) = δG0e−2πα |δx| cos(2π δx), (8.2)
where δx = (VD−Voff)/∆VD and Voff is a bias offset, is fit to cuts of the data in Fig. 8.2(a),
which yields a period ∆VD = 56 µV and dephasing parameter α = 0.2. These values,
along with ∆B = 0.25 mT are then used to produce the plot shown in Fig. 8.2(b). Fig-
ures 8.3(a,b) show vertical cuts from data along with fits of Eq. (8.2) at B = 0.22 T and
1.26 T, respectively, representing a trend toward smaller ∆VD and larger α at higher fields,
the details of which we now study.
105
Figure 8.3: Magnetic field dependence of extracted velocity and damping factor (a) GD
as a function of VD (black dots) at a field of B = 0.22 T, with a fit of Eq. (8.2) (redcurve) yielding ∆VD = 76 µV and α = 0.063. (b) Same as (a) but at B = 1.26 T andyielding ∆VD = 47 µV and α = 0.34. (c) Black dots indicate edge velocities (left axis)determined from measured ∆VD (right axis) as a function of 1/B. The red curves indicatetheoretical calculations: at low 1/B, the diagonal dashed line indicates the drift velocitycorresponding to E = 8× 104 V/m; at high 1/B, the top and bottom solid curves indicatethe predicted skipping-orbit velocities corresponding to the lowest and highest constrictiondensities, respectively. (d) Best-fit damping parameter α as a function of B, with a linearfit of slope 0.26 T−1 constrained through the origin. Inset: γ = 2πα/e∆VD as a function ofB, with a linear fit of slope 31 (meV · T)−1 constrained through the origin.
8.4 Edge-state velocity and energy-dependent dephasing
The black circles in Fig. 8.3(c) indicate the best-fit ∆VD (right axis) and corresponding edge
velocity (left axis) as a function of 1/B. The velocities appear roughly proportional to 1/B
106
before saturating at v ∼ 1.5 × 105 m/s for 1/B & 2 T−1. Red curves indicate calculations
based on single-particle models of edge velocities in two regimes. In the high-field limit,
where the cyclotron radius is much smaller than the length scale on which the confining
potential changes by the cyclotron gap, ~E × ~B drift gives a velocity vd = E/B, where E
is the local slope of the confining potential. The data in this regime are consistent with
a value E ∼ 8 × 104 V/m, which is reasonable given device parameters. At low fields,
where the cyclotron radius exceeds the length scale set by E, electron velocities can be
estimated from a skipping-orbit model. For hard-wall confinement, the skipping velocity
would be proportional to the cyclotron frequency and radius: vs ∼ ωcrc. Here, we have
performed a detailed semi-classical calculation assuming a more realistic confining potential
that vanishes in the bulk and grows linearly near the edge. In this regime, the predicted
velocity depends on not only B and E but also on the Landau level index, N , resulting in
a discrete jump in velocity for every change in N . Since the density in the constrictions
(which along with B determines N) varies over the course of the experiment, two theoretical
curves are plotted in this regime: the top one corresponds to the lowest observed constriction
density of 2.8× 1014 m−2, and the bottom one corresponds to the highest, 9.5× 1014 m−2,
both estimated from GD and B.
Figure 8.3(d) shows the best-fit damping parameter α as a function of B, revealing
rough proportionality: a straight line constrained to cross the origin describes the data well
with a best-fit slope of 0.26 T−1. In analogy to dephasing in 2D diffusive systems [197],
we suggest that coupling to compressible regions in the bulk may lead to dephasing with
the VD-dependence τ−1ϕ ∝ RVD, where R is the resistance per square in the bulk. Over
the field range of our data, the bulk longitudinal resistivity RXX (not shown) is on average
roughly proportional to B; taking RXX as an estimate of R would then lead to a predicted
107
dephasing rate proportional to both energy and magnetic field, consistent with the data.
Despite this agreement, we emphasize that Ref. [197] was not developed for edge states or
FPI’s, and a theory of dephasing in this regime remains lacking.
Alternatively, defining the damping factor as simply e−γ|eVD|, one also finds rough pro-
portionality between γ and B, as shown in the inset of Fig. 8.3(d). Here the best-fit slope
for a straight line constrained through the origin is 31 (meV ·T)−1. The damping parameter
γ is related to α and to the dephasing length, `ϕ = vτϕ, by γ = αto/~ = 2a/|eVD|`ϕ; there-
fore, since to varies with field, at most one of α and γ can be proportional to B. Physically,
the latter case would correspond to `−1ϕ being the quantity that is linear in B instead of
τ−1ϕ . Experimental scatter prevents us from distinguishing these two possibilities.
8.5 Nonlinear magnetoconductance in a 2 µm2 device
Measurements on a 2 µm2 device of similar design, whose zero-bias oscillations have previ-
ously been demonstrated as consistent with Coulomb-dominated behavior [22], do not yield
regular oscillations as a function of bias. Figure 8.4 shows GD as a function of B and VD
in a regime of weak forward-tunneling, where diamond-like features appear. Interpreting
these features as the result of Coulomb blockade yields a charging energy of roughly 25 µeV,
reasonable given the device size, 2DEG depth, and the large capacitance afforded by the top
gate. In regimes of stronger forward tunneling, the diamond edges become more smeared
out, but in contrast to the behavior in the 18 µm2 device, periodic oscillations as a function
of dc bias are not seen.
108
-50
0
50
VD [ m
V ]
3.7723.7703.7683.7663.764
B [ T ]
2.15
2.10
2.05
GD [ e
2 / h ]
Figure 8.4: GD as a function of B and VD in the 2 µm2 device.
8.6 Conclusion
In conclusion, quantum Hall FPI’s large enough that Coulomb charging is negligible are
found to display both AB and Fabry-Perot conductance oscillations. The combination of
these two effects yields a checkerboard-like pattern of oscillations from which the edge-
state velocity and dephasing rate can be extracted, and both are found to be consistent
with theoretical calculations. Although this pattern resembles that seen in Mach-Zehnder
interferometers, the dependence of its characteristics on magnetic field is evidently quite
different from what has been observed in those devices [186, 187], providing experimental
evidence that the underlying mechanisms for oscillations with bias in the two types of
devices may be quite different.
109
8.7 Acknowledgements
We are grateful to R. Heeres for technical assistance and to B. I. Halperin, M. A. Kastner,
C. de C. Chamon, A. Stern, R. Gerhardts, J. B. Miller and I. P. Radu for enlightening
discussions. This research has been funded in part by Microsoft Corporation Project Q,
IBM, NSF (DMR-0501796), Harvard University, and the Heisenberg program of DFG.
110
Chapter 9
Unpublished results
In this last chapter, I will be describing several results that are not published nor submitted
for publication. These results are not fully understood and further experimental studies
are needed to bring them to the quality level considered publishable in the Marcus group.
Although they will generally not appear in a Ph.D. dissertation, I believe they are well
worth the presentation here because, on one hand, it can still be of value to publicize them
in case some one seeing them would be able to help uncover the mystery underlying these
data; on the other hand, I think it can be instructive for new graduate students to see some
unpublished data and to learn the specific reasons why some results remain unpublished.
Since these results are still for one reason or another incomplete, and not submitted
for publication, their presentation is much more casual, and the references to relevant work
are not as rigorously checked as those published results. To some degree, they are more
or less like peeks into an experimentalist’s lab notebook. I would also like to thank my
collaborators, Doug McClure, Leo DiCarlo, Eli Levenson-Falk, as well as my advisor Charlie
Marcus for their work on these projects.
In the following sections, I will be describing four such results: the first section stud-
ies current noise through a quantum dot, modulated by the charge noise of another dot
nearby, showing huge super-Poissonian noise; the second section studies temperature and
111
bias dependent transport through a quantum point contact in the quantum Hall regime
between filling factors 2 and 3, observing both quantized fractional plateaus and quasi-
particle tunneling; the third section reports the experimental observation of an unexpected
3/2 quantized conductance plateau in a constriction; and the final section studies high-bias
magnetoresistance in the bulk, showing rich structures that might be related to reentrant
or anisotropic integer quantum Hall effects.
9.1 Current noise modulated by charge noise
In non-interacting electronic systems, current noise is expected to be always below the
Poissonian value due to Fermi statistics [19, 29, 54, 55]. Interactions, however, can raise
the current noise above the Poissonian limit, and the specific conditions under which such
super-Poissonian noise can arise has been the subject of numerous studies recently [43,
113, 44, 136, 137, 140, 45, 139, 125, 126, 46, 143, 33]. Super-Poissonian noise observed
in metal-semiconductor field effect transistors [43], tunnel barriers [113] and self-assembled
stacked quantum dots [44] has been attributed to interacting localized states [136, 137, 43]
occurring naturally in these devices. In more controlled geometries, super-Poissonian noise
has been associated with inelastic cotunneling [140] in a nanotube quantum dot [45], and
with dynamical channel blockade [139, 125, 126] in GaAs/AlGaAs quantum dots [46, 143,
33].
In essence, these types of super-Poissonian noise arise because the current through
the device is switching between two or more levels, leading to random telegraph noise
(RTN). The most natural system to realize such RTN in a controllable fashion is perhaps
a charge sensor measuring charge fluctuations of a nearby quantum dot. Charge sensors
can be realized in the form of quantum dots (also called single electron transistors) [12,
112
13, 198], or quantum point contacts [199, 200, 201]. In this section, we have realized such
controlled system using a double quantum dot, with one dot acting as a charge sensor, and
the other dot acting as a controlled charge noise generator. We have observed huge super-
Poissonian current noise through the charge sensing dot, when the other dot is near the
charge degeneracy point, and the current noise depends non-monotonically on the tunneling
rates of the charge noise generator.
The double quantum dot device is the same one as that used in Ch. 4, with the bottom
dot acting as a charge sensor, and the top dot acting as a charge noise generator and always
operating at zero dc bias [see Fig. 9.1(a)]. The central point contact is always depleted
by gate voltages Vl = Vr = −1420 mV, preventing inter-dot tunneling. Gate voltages Vtl
(Vbl) and Vtr (Vbr) control the tunnel barrier between the top (bottom) dot and its left and
right leads. Plunger gate voltage Vtc (Vbc) controls the electron number in the top (bottom)
dot. For convenience, the variable, Vts ≡ [(Vtr + 614 mV) + (Vtl + 577 mV)]/√
2, is used to
control the left and right barriers of the top dot simultaneously. Differential conductance
gt (gb) through the top (bottom) dot is measured using standard lock-in techniques with
an excitation of 25 (30) µVrms at 677 (1000) Hz, and the current noise through the bottom
dot, Sb, is measured near 2 MHz [131]. The dc current Ib through the bottom dot at finite
Vb is obtained by numerically integrating gb.
Shown in the inset of Fig. 9.1(b) are conductances gt and gb, as a function of Vts,
which controls the barrier transparencies of the top dot, as well as its energy. When Vts is
tuned more negative, the top dot tunnel barriers are getting higher, and electrons are being
pushed out one by one, leading to a series of Coulomb blockade (CB) peaks in gt that are
getting smaller in height. In the meantime, as the bottom dot is configured near a charge
degeneracy point, its conductance also evolves through a full CB peak due to the coupling
113
[ mV ]
280 mK 4.2 K
Digitize &
Analyze
R R L L C C
60 dB
R R L L 60 dB
Vtl Vtc Vtr
Vbl
Vr
Vt
Vl
(a)
(b)
It
Ib Vb
500 nm
gb
Vbc Vbr
Stb
St
Sb
50Ω
50Ω
50Ω
50Ω
gt
1.5
1.0
0.5
0.0
g [e
2 / h]
-40 -30 -20 -10 0 10Vts [ mV ]
gt gb
-850
-800
-750
Vtc
[ m
V ]
-100 -80 -60 -40 -20 0Vts
0.60.40.20.0gb [e
2 / h]
1.51.00.50.0 gt [e2 / h]
gt
gb
Figure 9.1: (a) Micrograph of the device and equivalent circuit near 2 MHz of the noisedetection system. (b) Conductance through the top and bottom dot, gt and gb, as a functionof Vts and Vtc (see main text for the definition of Vts). Inset: gt and gb as a function of Vts.
between Vts and the bottom dot. In addition, whenever the electron number in the top dot
changes by one, gb exhibits a jump, even when the CB peak in gt is getting unmeasurably
small. This demonstrates the charge sensing of the top dot by the bottom dot.
Figure 9.1(b) shows gt and gb as a function of Vts and Vtc, tracking three consecutive
CB peaks in gt, while the top dot barrier is getting higher with more negative Vts. The
plunger gate Vtc is used to compensate the energy shift caused by changing Vts so that the
dot electron numbers remain unchanged. Although the peaks in gt is vanishing into the
background noise passing Vts ∼ −15 mV, the conductance jumps in gb clearly tracks the
114
-848
-847
-846
-845
-844
Vtc
[ m
V ]
0.030.00 gt [ e2 / h ] 0.100.02 gb [ e2 / h ]
-848
-847
-846
-845
-844
Vtc
[ m
V ]
-1257 -1256 -1255 -1254Vbc [ mV ]
1.00.0 Sb [ nA x 2e ]
-1257 -1256 -1255 -1254Vbc [ mV ]
0.5-0.5 SEPb [ nA x 2e ]
(a)
(c)
(b)
(d)
Figure 9.2: Conductance and current noise near a honey-comb vertex: gt (a), gb (b), Sb
(c), and SEPb (d), as a function of Vtc and Vbc.
charge transitions of the top dot. In later studies of the dependence of Sb on Vts, Vtc is
always used to compensate the energy shift by Vts in this way, making sure that the same
CB peak is studied.
Setting the top dot so that its conductance is barely measurable, at Vts = −15 mV, and
applying 100 µV dc bias to the bottom dot and zero bias to the top dot, both conductance
and bottom-dot current noise are measured as a function of Vtc and Vbc, as shown in
Fig. 9.2. We find that Sb is greatly enhanced at two points in the plane of Vtc and Vbc [see
Fig. 9.2(c)], corresponding to the two sides of the CB peak in gb, where gb is most sensitive
to nearby charge fluctuations, intersecting with the CB peak in gt. Plotting in Fig. 9.2(d)
the excess Poissonian noise, SEPb ≡ Sb − 2e|Ib|, through the bottom dot, we find that the
115
two Sb maxima indeed exceeds the Poissonian limit1. The right maximum of Sb is as large
as 1.2 nA× 2e, while the dc current at that point is only 0.2 nA, corresponding to a Fano
factor of 6.
The data suggest that the enhanced noise in Sb to be RTN induced by the top-dot charge
noise: when the top dot comes into the charge degeneracy point, its charge fluctuates
between two possible values, leading to the bottom-dot current switching between two
values. The power spectrum for RTN associated with two-level switching is given by the
Lorentzian form [36]: SRTN(f) = (∆I)2 ·4Γ01Γ10/(Γ01 +Γ10)/[(Γ01 +Γ10)2 +(2πf)2], where
Γ01 (Γ10) is the transition rate from state 0 (1) to state 1 (0), and ∆I is the difference in
current between state 0 and state 1. Since the top dot is operated at zero-bias, we can write
Γ01 = ftΓt and Γ10 = (1 − ft)Γt, where ft is the Fermi function of the top-dot reservoirs
evaluated at the dot energy level, and Γt = Γtl + Γtr is the total tunneling rate to both
reservoirs of the top dot. The maximum noise occurs when ft = 1/2, corresponding to a
top-dot CB peak, and the maximally enhanced Sb is simply given by:
Smaxb (f) = (∆Ib)2 · Γt/[Γ2
t + (2πf)2]. (9.1)
Extraction of ∆Ib can be performed by measuring Ib as a function of Vtc, at the Vbc
setting that corresponds to the right maximum in Sb, as shown in the inset of Fig. 9.3.
Without any fitting parameters, the theoretical prediction for Smaxb (f = 2 MHz), using
Eq. (9.1), as a function of Γt is shown in Fig. 9.3(b). The predicted Smaxb depends non-
monotonically on Γt, with a maximum at Γt = 2π · 2 MHz. The maximum Sb as a function
of the top-dot barrier transparencies is studied by changing Vts while compensating the
1Super-Poissonian Sb is also observed away from the top-dot charge degeneracy point.This is the related to dynamical channel blockade from multi-level transport of the bottomdot, as has been studied in Ch. 5. Here, we focus on the super-Poissonian noise induced bythe top-dot charge noise, much larger than the intrinsic noise of the bottom dot.
116
8
6
4
2
0S
bmax
[ nA
x 2
e ]
-100 -80 -60 -40 -20 0Vts [ mV ]
(a) Data8
6
4
2
0104 106 108 1010
Γt [ Hz ]
(b) Theory
ΔIb = 232 pA
0.4
0.2
0.0
I b [
nA ]
-848 -844Vtc [ mV ]
Figure 9.3: (a) Measured maximum current noise, Smaxb , as a function of Vts. (b) Theory
for Smaxb as a function of top dot tunneling rate Γt. Inset: Ib as a function of Vtc, and
measurement of ∆Ib.
energy shift with Vtc, as has been done in Fig. 9.1(b). Shown in Fig. 9.3(a) are Smaxb at
13 settings of Vts from 0 to −110 mV. Indeed, Smaxb depends non-monotonically on Vts,
and approaching the single-dot values at high and low limits of barrier transparencies. Yet
unexpectedly, Smaxb exhibits two peaks for intermediate values of Vts, one near −40 mV and
the other near −70 mV. So far, this has remained a mystery to us.
In conclusion, we have observed huge super-Poissonian noise through the bottom quan-
tum dot when charge sensing the top dot. The enhanced noise depends non-monotonically
on the barrier transparencies of the top dot. It is also the first time that we have studied
frequency-dependent current noise, when the time scale of the dynamics is comparable or
slower than our noise measurement frequency at 2 MHz. In contrast to the simple expecta-
tion for RTN, however, Smaxb exhibits two peaks as a function of Vts, which remains to be
understood.
117
9.2 Quasi-particle tunneling between filling factor 2 and 3 in
a constriction
This section is mainly a follow-up experiment of the quasi-particle tunneling experiment
by Radu et al. [202]. Radu et al. have found strong nonlinear I − V characteristics in
a quantum point contact (QPC) at filling factor 5/2, associating them with tunneling of
the 5/2 quasi-particles. Both the width and height of the tunneling conductance peaks are
found to scale with temperature, allowing an extraction of quasi-particle charge e∗ and the
interaction parameter g within the weak tunneling framework [203].
This section reports a similar measurement of conductance through a QPC in the quan-
tum Hall regime between filling factors 2 and 3. The new contributions of this section
include the observation of exact quantized plateaus at filling factors of 7/3, 5/2, and 8/3, as
well as two groups of tunneling peaks between these three plateaus. Both groups of peaks
are studied as a function of bias, temperature, and the high-bias background resistance,
and they are found to scale with temperature. Following the methods used in Ref. [202], e∗
and g are extracted as a function of the high-bias background resistance.
The QPC device is fabricated on a high-mobility two-dimensional electron gas (2DEG)
residing in a 30 nm wide GaAs/AlGaAs quantum well 200 nm below the chip surface.
The 2DEG mobility is ∼ 2, 000 m2/Vs measured in the dark, and the density is 2.6 ×
1015 m−2. Surface gates that define the QPC have a separation of 1.2 µm, and are patterned
using electron-beam lithography on wet-etched Hall bars. Bulk Hall resistance RXY, bulk
longitudinal resistance RXX, and diagonal resistance across the device RD are measured
with current bias I applied through the entire Hall bar and perpendicular magnetic field B
oriented into the 2DEG plane [see Fig. 9.4]. In the quantum Hall regime, RXY (= 1/νB·h/e2)
118
RDRXY
RXX
VG
VG 150 µm
I
B
Figure 9.4: Optical micrograph of the device and measurement setup. The QPC understudy is highlighted in orange, and has a width of 1.2 µm. Resistances RXY, RXX, andRD are measured across the indicated pairs of ohmic contacts, with current bias I appliedthrough the entire Hall bar. (Image courtesy of Ref. [176]).
and RD (= 1/νD ·h/e2) give independent measurements of the filling factor in the bulk, νB,
and that in the constriction, νD [176].
Shown in Fig. 9.5 are RD as a function of dc I, at B values from 3.0 to 4.4 T and
evenly spaced by 10 mT, for 7 different temperatures from T = 13 mK to 66 mK. This
magnetic field range covers the range in νD from 2 to 3. In this representation, resistance
plateaus as a function of B would appear as accumulation of traces. As can be seen from
Fig. 9.5, in addition to two strong integer plateaus at νD = 2 and 3, fractional plateaus at
7/3 and 5/2 appear at all temperatures, and the plateau at 8/3 is also visible at the lowest
temperatures. Between these three fractional plateaus, two groups of resistance peaks show
up as a function of I, with the peaks getting higher and narrower at lower temperatures.
We denote the group of peaks between 3/7 and 5/2 plateaus “Group 1”, and those between
5/2 and 8/3 “Group 2”.
Following Ref. [202], we interpret these peaks as tunneling between counter-propagating
119
R
1/2
1/3
2/5
3/7
3/8
D [
h / e
2 ]
-10 -5 0 5 10I [nA]
13 mK
-10 -5 0 5 10I [nA]
26 mK
-10 -5 0 5 10I [nA]
33 mK
-10 -5 0 5 10I [nA]
40 mK
-10 -5 0 5 10I [nA]
50 mK
-10 -5 0 5 10I [nA]
58 mK
-10 -5 0 5 10I [nA]
0.50
0.48
0.46
0.44
0.42
0.40
0.38
0.36
0.34
66 mK
Gro
up 1
Gro
up 2
Figure 9.5: Diagonal resistance RD as a function of I, at magnetic fields evenly spaced by10 mT, for 7 different temperatures from 13 mK to 66 mK.
fractional edge states, and use the weak tunneling formula [203] to analyze them. Due to
some small but random device drift over the time of these measurements, instead of B, the
high-bias limit RD value, R0D, is chosen as the metric for picking the tunneling peaks of
the same constriction filling for different temperatures. This choice is justified, because not
only it eliminates the small drift, but also R0D is found to be proportional to B in the range
of interest, thus would reflect the true constriction filling without the effect of tunneling
near zero bias.
The red curves shown in Fig. 9.6 are resistance peaks of all temperatures at R0D =
0.414 h/e2, plotted as a function of the diagonal voltage VD, which is extracted by numerical
integration of RD(I). The data set at each R0D are then fit to the weak tunneling formula:
RD = R0D +A · T 2g−2 · F
(e∗VD
kBT, g
), (9.2)
120
0.424
0.422
0.420
0.418
0.416
0.414
RD [
h / e
2 ]
-60 0 60VD [ µV ]
13 mK
-60 0 60VD [ µV ]
26 mK
-60 0 60VD [ µV ]
33 mK
-60 0 60VD [ µV ]
40 mK
-60 0 60VD [ µV ]
50 mK
-60 0 60VD [ µV ]
58 mK
-60 0 60VD [ µV ]
66 mK
Figure 9.6: An example of the best fit (black), using the weak tunneling formula [Eqs. (9.2)and (9.3)], to bias- and temperature-dependent RD data (red) at R0
D = 0.414 h/e2, givinge∗ = 0.23 e and g = 0.44.
F(x, g) = B(g + i
x
2π, g − i x
2π
)·π cosh
(x2
)− 2 sinh
(x2
)· Im
[Ψ(g + i
x
2π
)], (9.3)
where A is the tunneling amplitude, e∗ and g are the quasi-particle charge and the interac-
tion parameter, respectively. B(x, y) is the Euler beta function, and Ψ(x) is the digamma
function.
Treating A, e∗ and g as free parameters, and simultaneously fitting to all seven data
curves shown in Fig. 9.6, the weak tunneling formula yields a satisfactory fit, and gives
best-fit e∗ = 0.23 e and g = 0.44. Again following Ref. [202], we can prefix e∗ and g over
a grid of possible values, and only allow A as the free parameter to fit to the data. Shown
in Fig. 9.7 is the mean squared fit error as a function of prefixed e∗ and g, when fitting
to the same data with R0D = 0.414 h/e2. As discussed in Ref. [202], this plot yields more
information on the uncertainty of these measured e∗ and g values, and would facilitate
comparison to various theories.
The constriction filling that corresponds to R0D = 0.414 h/e2 is near midway between
7/3 and 5/2 states, and we would like to analyze the peaks with different constriction fillings
121
0.8
0.6
0.4
0.2
g [d
imle
ss]
0.60.50.40.30.20.1e* [e]
321 Fit error [ a. u. ]
Figure 9.7: Mean squared fit error as a function of prefixed e∗ and g, for fitting to the datashown in Fig. 9.6.
0.35
0.30
0.25
0.20
e* [e
]
Group2 Group1
0.60
0.55
0.50
0.45g [d
imle
ss]
0.420.410.400.39RD
0 [ h / e2 ]
Group2 Group1
Figure 9.8: Best-fit e∗ and g as a function of R0D for both groups of tunneling peaks.
within “Group 1” in the same way, and also compare to the peaks within “Group 2”. Shown
in Fig. 9.8 are the best-fit e∗ and g, as a function of R0D, from both groups of peaks.
In a simplistic picture, the peaks within “Group 1” corresponds to either backscattering
of the 5/2 edge across the 5/2 liquid, or forward tunneling of the 5/2 edge over the 7/3
122
liquid; the peaks within “Group 2” corresponds to either backscattering of the 8/3 edge
across the 8/3 liquid, or forward tunneling of the 8/3 edge over the 5/2 liquid. Further
assuming that tunneling is enhanced at zero-bias and suppressed at high bias, since peaks
in RD correspond to increased backscattering, we can eliminate the possibility of forward
tunneling, which would lead to dips in RD. Therefore, we expect the peaks in “Group
1” to exhibit the characteristics of the 5/2 state, and those in “Group 2” to exhibit the
characteristics of the 8/3 state. As shown in Fig. 9.8, in “Group 1”, the extracted e∗ is
between 0.2 and 0.25 e, and g ranges from 0.44 to 0.54, in reasonable agreement with the
predictions e∗ = 0.25 e and g = 0.5 of the anti-Pfaffian state [204, 205, 202]. In “Group 2”,
however, the extracted e∗ and g show a large variation at different R0D, making it hard to
draw any conclusions.
In conclusion, we have observed well quantized fractional plateaus at filling factors of
7/3, 5/2, and 8/3, as well as two groups of tunneling peaks between them. Best-fit e∗
and g as a function of constriction filling are extracted using the weak tunneling formula,
for both groups of peaks. The extracted values from peaks in “Group 1”, assumed to be
related to tunneling of the 5/2 edge, is consistent with predictions of the anti-Pfaffian state;
however, the extracted values from peaks in “Group 2”, presumably related to tunneling of
the 8/3 edge, show a large variation. Another drawback of this analysis is that the peaks
can overshoot the value of the next plateau at the lowest temperatures [see the 13 mK data
in Fig. 9.5], putting the theoretical assumption of weak tunneling into question.
123
9.3 The 3/2 quantized plateau in quantum point contacts
Since the original discovery of the 1/3 fractional quantum Hall effect over two decades
ago [27], a wealth of fractional quantum Hall states have been discovered, and vast majority
of them are odd-denominator fractions [206, 24]. The odd denominator is a result of Fermi
statistics, which requires the wave function to be antisymmetric under particle exchange;
consequently, any fraction with an even denominator would require an explanation quite
different from those with odd denominators. The 5/2 [207] and 7/2 states in the first excited
Landau level (LL) are the only known even-denominator states in a single-layer system; yet
their origin, possibly being a paired state [25], is still under investigation. In bilayer systems,
1/2 [208, 209] and 3/2 [210] states in the lowest LL have been observed and understood to
arise from the so-called (3, 3, 1) state [211, 212].
In this section, we report observation of the 3/2 plateau, precisely quantized to within
0.02 %, through quantum point contacts (QPCs). Evidence show that they possess a
different origin from that of the single-layer 5/2 and 7/2 states, or of the bilayer 1/2 and
3/2 states: they are never seen in the bulk and only through the QPCs; they appear when
the constriction filling is at 5/3, suggesting their relationship with the 5/3 state. Studying
their temperature and bias dependence, we found that they survive up to 80 mK or over
8 nA.
The QPC devices are formed with top gates in two devices, labeled as device 1 and
device 2, shown in the lower right insets of Figs. 9.9(a) and (b), respectively. These two
devices are fabricated on different two-dimensional electron gas (2DEG) wafers of the same
design2, with a mobility of ∼ 2, 000 m2/Vs and a density of 2.6 × 1015 m−2. As explained
2Device 1 is the 2.0 µm2 dot also used in Chs. 7 and 8. Device 2 is fabricated on thesame wafer as the 18 µm2 dot studied in Chs. 7 and 8.
124
1
1/2
1/3
1/41/51/61/8
3/4
3/52/3
R [
h / e
2 ]
108642 B [ T ]
(a) Device 1: RXY RD
0.70
0.68
0.66
0.64
RD [
h / e
2 ]6.66.4
B [ T ]
2/3
1
108642 B [ T ]
(b) Device 2: RXY RD
0.68
0.66
0.64
0.62
RD [
h / e
2 ]
7.06.86.6B [ T ]
2/3x 1.26
VTVTL VTR VR
VBVL VC1 µm 2 µm
VC VRVBRVBLVB
VL VT VTRVTL
Figure 9.9: (a) Bulk Hall resistance RXY (red) and diagonal resistance RD (black) as afunction of magnetic field B measured from device 1, whose gate layout is shown in thelower right inset. The QPC is formed with VT = VL = −1.06 V. The grey curve is the RD
data scaled horizontally by a factor of 1.26, to match the filling factor in the bulk. Theupper left inset shows that RD exhibits a 3/2 quantized plateau. (b) Similar data takenfrom device 2. The QPC is formed with VT = VB = −2.15 V and VC = +0.6 V, where theconstriction density matches that of the bulk.
in the previous section and in Chs. 7 and 8, bulk Hall resistance RXY (= 1/νB · h/e2) and
diagonal resistance RD (= 1/νD · h/e2) give independent measurements of the filling factor
in the bulk, νB, and that in the constriction, νD, with the perpendicular magnetic field B
applied into the 2DEG plane.
Figure 9.9(a) shows RXY and RD as a function of B measured from device 1. In
addition to well quantized integer plateaus, RXY shows fractional plateaus at 5/3, 4/3, etc.;
yet, RD shows a plateau that is quantized at 2/3 h/e2, corresponding to a conductance
of 3/2 channels through the QPC. This 3/2 plateau is precisely quantized to within the
measurement noise of 0.02 %, as shown in the upper left inset of Fig. 9.9(a). The plateaus
in RXY and in RD are not aligned in B due to different densities in the bulk and in the
QPC; therefore, we have scaled the RD data horizontally by a factor of 1.26, to align it with
the RXY trace. Surprisingly, the 3/2 plateau in RD appears at the same filling factor as the
125
1/2
3/4
3/5
2/3
RD [
h / e
2 ]
-8 -4 0 4 8I [ nA ]
(b)
1/2
3/5
2/3
RD [
h / e
2 ]
5.04.94.84.74.64.5B [ T ]
13 mK 20 mK 40 mK 60 mK 80 mK
(a)
Figure 9.10: (a) Temperature dependence of the 3/2 plateau in a QPC formed with VB =VTL = −1.6 V in device 1. (b) RD as a function of I, at magnetic fields evenly spaced by20 mT from 5.0 to 6.6 T, measured in a QPC formed with VB = VTL = −2.7 V in device 1.
5/3 plateau in RXY, suggesting that they are somehow related.
Figure 9.9(b) shows RXY and RD as a function of B measured from device 2. This
QPC is formed with negative voltages on two facing gates VT = VB = −2.15 V, and positive
voltage on a center gate VC = +0.6 V, matching the density in the QPC to the bulk. Again,
RD as a function B shows a 3/2 plateau, also precisely quantized to within the measurement
noise. Similar to the behavior of device 1, the position of the 3/2 plateau in RD is the same
as the 5/3 plateau in RXY.
Temperature and bias dependence of the 3/2 plateau are also studied in a QPC formed
with gates VB and VTL in device 1. Figure 9.10(a) shows RD as a function of B at five
different temperatures up to 80 mK, showing that the 3/2 plateau can survive up to 80 mK.
Figure 9.10(b) shows RD as a function of dc I, at B settings from 5.0 to 6.6 T evenly spaced
by 20 mT. The 3/2 plateau, showing up as accumulation of traces at RD = 2/3 h/e2, persists
to at least 8 nA, or equivalently 140 µV.
The origin of the 3/2 plateau in RD, and indeed how it is related to the 5/3 plateau in
126
RXY are quite puzzling, especially because the 3/2 plateau has an even denominator. Some
possibilities, however, can be ruled out. Edge reconstruction of the possibly complicated
5/3 edge does not seem to be able to explain the even denominator of the 3/2 plateau. The
precision of quantization also eliminates the possibility of it being some other nearby state
with an odd denominator. Because the 3/2 plateau is absent in the bulk, and appears at
the same filling factor as the 5/3 state in the bulk, it is unlikely to share the same origin
as the 5/2 and 7/2 states, nor as the 1/2 and 3/2 states in bilayer systems. Since the 3/2
plateau has been reproduced in three different QPCs on two wafers, we do not believe it
to be some form of experimental artifacts, and further experimental and theoretical studies
would be needed to reveal the physics underlying this intriguing finding.
127
9.4 Non-linear transport in N ≥ 2 Landau levels
It has been known for a over decade that the N ≥ 2 Landau levels (LL’s) exhibit large
anisotropy and reentrant integer states [213, 214]. Their origin, although believed to be
charge density waves with stripe or bubble phases [215, 216, 217], remains an open question.
This section provides nonlinear transport data in this regime, showing rich structures that
are present only at high bias. These data provide new information on these anisotropic and
reentrant states, and should help resolve their precise nature.
The two Hall bars studied in this section are fabricated on the GaAs/AlGaAs two-
dimensional electron gas (2DEG) wafers also used in the previous two sections and in Chs. 7
and 8. These two Hall bars are oriented perpendicular to each other with respect to the
GaAs/AlGaAs crystal direction. The bulk Hall resistance, RXY, and the bulk longitudinal
resistance, RXX, are the quantities of interest here.
We first reproduce the results of Refs. [213, 214] in Fig. 9.11. The bulk transport, RXX
and RXY as a function of B, covering filling factors νB = 2 to 8, are studied as a function
of crystal direction and temperature. The data in Figs. 9.11(a) and (b) [(c) and (d)] are
measured from a Hall bar oriented with crystal direction 1 (2), which are perpendicular
to each other. The blue traces in Fig. 9.11 are obtained at the fridge base temperature of
13 mK, while the red traces are obtained when the fridge were just cooled down and the
electrons are not yet thermalized to the fridge. Comparing the RXX data near 5/2 to those
in Ref. [176], the electron temperature for the red traces should be around 40 − 50 mK.
Similar to previous results [213, 214], reentrant features in RXY are clearly visible for N ≥ 2
(νB ≥ 4)when the electrons are warm, but they merge with the main plateaus and RXY show
only simple steps when the electrons are cold. In the meantime, RXX show large anisotropic
behaviors near 9/2, 11/2, etc.: when the electrons are cold, RXX almost vanishes for crystal
128
0.04
0.03
0.02
0.01
RX
X [
h / e
2 ]
Warm Cold
(a)
0.001/2
1/3
1/4
1/51/61/8
RX
Y [ h
/ e2 ]
5432 B [ T ]
Warm Cold
(b)
0.08
0.06
0.04
0.02
0.00
Warm Cold
(c)
1/2
1/3
1/4
1/51/61/8
5432 B [ T ]
Warm Cold
(d)
Crystal Direction 1 Crystal Direction 2
Figure 9.11: RXX (a,c) and RXY (b,d) as a function of B, at the base temperature (blue)and an elevated temperature (red). The data in (a,b) are measured from a Hall bar that isoriented perpendicular to the Hall bar used for obtaining the data in (c,d).
direction 1, but shows sharp peaks for crystal direction 2; when the electrons are warm, the
RXX data exhibit side peaks around their minima near half fillings for crystal direction 1,
but they show peaks at half fillings for crystal direction 2.
We move on now to nonlinear transport measurements. Figure 9.12 show RXX and RXY
as a function of dc I and B, at filling factors νB = 4 to 9, for crystal direction 1 and cold
electrons. Although at zero bias, RXX almost vanishes, and RXY only show simple steps,
several intriguing features appear in both RXX and RXY at high bias. Around 11/2, for
example, four leaf-shaped regions of non-zero RXX appear in each bias direction. A similar
behavior is seen near 9/23, but there are only two such regions, which are getting closer
at lower B, around 13/2, 15/2, and 17/2. The boundaries of these non-zero RXX regions
correspond to sharp transitions in RXY. Furthermore, there are small and quite regularly
3There is possibly another leaf (in each bias direction) that the measurement did notcover at higher B.
129
1/4
1/5
1/6
1/7
1/8
RX
Y [ h / e2 ]
2.42.22.01.81.61.4 B [ T ]-100
-50
0
50
100
I [ n
A ]
0.26
0.24
0.22
0.20
0.18
0.16
0.14
0.12
RX
Y [ h / e2 ]
-100
-50
0
50
100
I [ n
A ]
0.025
0.020
0.015
0.010
0.005
0.000
RX
X [ h / e2 ]
8 7 6 5 4
(a)
(b)
Figure 9.12: RXX (a) and RXY (b) as a function of I and B at the base temperature, withcrystal direction 1.
spaced bright spots in RXX along these boundaries, and corresponding features are also
present in RXY.
So far, the nonlinear transport data are obtained only for crystal direction 1 and cold
electrons, and it would be very interesting to study the temperature- and crystal-direction-
dependence of such data. It would be equally interesting to extend the bias range since the
observed features seem to extend far beyond the bias range of study. At this point, I am
not aware of any interpretations for these features, but my guess is that they are somehow
related to the anisotropic and reentrant states for N ≥ 2 LL’s. Considering the amount of
information nonlinear transport gives at these high LL’s, similar measurements for N = 0
or N = 1 LL’s, possibly with fractional states, may give surprising results as well.
130
Appendix A
Fridge Wiring: Thermal Anchoringand Filtering
Well designed and implemented electrical wiring is an essential part of low-temperature
transport experiments, yet wiring is often quite tricky as well, because:
• when transmitting electrical signals to and from the sample, the wires can also bring
down significant amount of heat to the lowest temperature stage of the fridge and the
sample, raising base fridge and electron temperatures;
• long and exposed dc wires are like antennas, picking up radiation and unwanted noise,
which would degrade the signal, and also heat up the electrons.
The usual approach [218] for dc wiring used in the Marcus lab has been to use long re-
sistive wires, eg. twisted-pair constantan (55% Cu, 45% Ni) wires from Oxford Instruments,
thermally anchored at each stage of the fridge with copper posts and GE varnish. At the
lowest-temperature stage, 3He pot or mixing chamber, a bank of resistors are inserted to
the lines. The resistors and stray capacitance from the wires to ground form simple RC
filters and effectively filter out most of the radiation from room temperature.
This simple and robust approach works quite well for bare dc lines down to 50 mK
or so, yet for the noise measurement circuit in the 3He fridge, and for the dc lines in the
131
Figure A.1: Photograph of a bank of simple RC filters
Microsoft dilution fridge that we aim at an electron temperature of 20 mK, additional
thermal anchoring and filtering are needed. Specifically, for the noise measurement circuit,
each gate line needs to be heavily filtered to prevent the thermal noise from dc wires to
couple to the 2DEG at 2 MHz, for which we used simple RC filters. We have also used
sapphire heat sinks and circuits boards for both the noise measurement circuits and the dc
lines in the Microsoft dilution refrigerator. In addition, we have used Mini-Cuicuits VLFX
filters in the Microsoft fridge with great success. Recently, we have replaced bare dc wires
with thermocoax lines in the Microsoft fridge. I will describe each of these techniques in
detail below.
A.1 Simple RC filters
For the noise measurement operating near 2 MHz, one source of extraneous noise is from
the thermal noise of gate lines, coupled capacitively to the 2DEG. Here, we simply used RC
filters, with resistance R = 5 kΩ, and capacitance C = 22 nF, giving a 3 dB cut-off frequency
of 1.4 kHz. Due to the large number of lines that need to be filtered, we have designed a
132
compact bank of RC filters, with four layers of 0805-sized surface mount components on
three boards stacked together, and soldered to two 37-pin D-sub connectors, as shown in
Fig. A.1.
A.2 Sapphire heat sinks and circuit boards
The inner conductor of the coaxial lines used for the noise circuit is insulated from the outer
conductor in teflon, and has to be thermalized before it reaches the sample. As described
in Ch. 2, we insert thin meandering lines evaporated on sapphire boards to heat sink them
without adding much resistance to the lines. In addition, all circuit boards inside the 3He
fridge are made on sapphire boards, for maximal thermalization of coaxial lines at each
stage. Given the success with sapphire heat sinks used for the noise circuit, we also make
use of it for heat sinking all dc lines in the Microsoft fridge.
Sapphire has been chosen because it has among the best thermal conductivity for insu-
lators at low temperatures [92], second only to quartz below 10 K, yet it is a lot less brittle
than quartz, making it much easier to work with than quartz. Unlike conventional circuit
boards, often made commercially, we fabricate sapphire circuit boards in house starting
with blank sapphire pieces. The latest recipe of fabrication is detailed below:
1. Blank sapphire pieces are first cut into the desired dimensions with Automatic Dicing
Saw (CNS facility at Harvard) at the lowest cutting speed of 0.1 mm/s;
2. Then 30/300 nm of Cr/Au or Ti/Au are evaporated on the top surface, and annealed
in the Rapid Thermal Annealer for better adhesion and releasing any strain in the
film. We have simply used the same annealing recipe as that for making ohmics. The
ideal temperature and annealing time have not been investigated, and is probably not
133
so important anyway;
3. There are several ways to pattern the traces. The simplest one, which works well for
large features, is to just draw traces with a Sharpie. For better control and smaller
features, one can print the designed pattern on a special toner transfer paper (Pulsar
toner transfer mask), and then iron the pattern onto the surface. For very fine features
and best results, one can use photo lithography to define the patterns;
4. After the traces are covered with masks (from Sharpie ink to photo resist), the rest
of the metal films are etched in Au and Cr (or Ti) etchants. After etching, the masks
can be easily washed off in acetone;
5. Cr/Au or Ti/Au (30/300 nm) are then evaporated on the back surface;
6. Surface-mount components are soldered to the circuit traces using In solder at the
lowest soldering temperature (350 F). Soldering these tiny components with In solder
on gold traces are tricky, because: In can easily creep over and create shorts between
traces; In does not stick to gold surfaces very well, and may develop breaks over
time; higher temperatures can readily peel off the gold traces on sapphire. Therefore,
extreme care must be taken and a lot of practice are needed. Sometimes when a break
does develop, which occurs quite often, it can be fixed with a tiny touch of silver paint
or Transene silver bond. An easier and more robust recipe is surely desired, but has
not been developed at this point;
7. Finally, the back surface are soldered to copper braids, or silver epoxied to the desired
surface, for best thermalization of the sapphire boards.
Some examples of sapphire circuit boards and heat sinks are shown in Fig. A.2.
134
(a) (d)
(b)
(c)
Figure A.2: Photographs of sapphire circuit boards and heat sinks: (a) the HEMT circuitboard, (b) the SINK board, (c) the RES circuit board, and (d) the sapphire heat sink fordc lines used in the Microsoft fridge, with the pattern defined by photo lithography.
A.3 Mini-circuit VLFX filters
When pushing to the lowest electron temperature of a dilution refrigerator, say below 30 mK,
the main source of heating is high-frequency (GHz or higher) radiation carried by dc lines
from higher-temperature stages. Simple RLC filters usually stop working at these fre-
quencies due to their parasitic capacitance and inductance. Metal powder filters are often
used [219, 220] for this purpose, but they are usually home-made and sometimes tend to
break over time after a few thermal cycles. Another type of filters, called Frossati filters, are
effective, robust, and commercially available from Leiden Cryogenics, yet they cost a few
hundred dollar per filter, making the cost prohibitively high. Another approach is to use a
special type of coax from Thermocoax Co., which will be described in the next section.
Here, we have used a type of low-cost ($40 a piece) commercially available LC filter
that works very well for blocking radiation: Mini-Circuits VLFX filters. These VLFX filters
feature 21 sections of filtering, guaranteed 40 dB of isolation up to 20 GHz, and temperature
stable structure. Shown in Fig. A.3 are 18 Mini-circuits VLFX-80 filters installed with the
cold finger. These filters, together with the sapphire heat sinks described in the previous
section, have allowed the electrons to be cooled to below 20 mK, as inferred from the bulk
transport data shown in Fig. A.4, in the Microsoft fridge at a fridge temperature of 13 mK.
135
(a) (b)
Figure A.3: 18 Mini-circuits VLFX-80 filters assembled with the cold finger: (a) side view,(b) top view.
1/2
1/3
2/5
3/7
3/8
RX
Y [
h / e
2 ]
RXY
0.10
0.08
0.06
0.04
0.02
0.00
RX
X [
h / e
2 ]
5.04.84.64.44.24.03.8Bperp [ T ]
RXX
Figure A.4: Quantum Hall bulk transport, showing well developed 7/3, 5/2, 8/3 states,and reentrant integer features, consistent with an electron temperature below 20 mK whencompared to the data in Ref. [176].
A.4 Thermocoax cables
Recently, we have decided to try a special type of coaxial cable [221] from Thermocoax
Co., to replace the dc lines in the Microsoft fridge. The cable used has an outer diameter
of 0.5 mm, NiCr (80/20) alloy inner conductor, stainless steel outer conductor, and highly
compacted magnesium oxide powder as the dielectric material. Its mechanism for blocking
high-frequency radiation is the same as that for metal powder filters: skin-effect damping in
136
the very large surface area of the powder. The additional benefits include that the filtering
is built into the cable itself, thus no additional filters are needed, and the coaxial geometry
prevents pickups that bare dc lines are prone to.
Soldering connectors to thermocoax cables need special attention because the stainless
steel outer conductor is hard to solder to without stainless steel flux, and the powder
dielectric tend to absorb moisture and flux, creating dc voltage or MΩ leak between the
inner and outer conductors. After doing a lot of practice and learning from errors, Doug
and I have developed a specific set of steps for soldering connectors to thermocoax cables.
The following steps are for soldering SMP connectors, but they are similar for other types
of connectors as well:
1. Use a 30-gauge wire-stripper to cut the outer connector roughly 5 mm from the end;
rotate the cutter in a circle to be sure the outer is completely scored. Use a pair of
pliers to grab the section of cable between the cut and the end. Holding the cable
with two fingers just on the other side of the cut, use the pliers to bend the end back
and forth until you feel the outer completely breaks;
2. Squeeze the end section with the pliers to flatten it; repeat at a 90-degree angle to
flatten it the other way. Doing this, you should notice some white powder coming
out. Squeeze the end just enough to make it roughly round again, and try to pull it
straight off with the pliers. If this fails, repeat the previous step and then try again;
3. Heat the exposed end for a minute or so with a heat gun above 100 C to evaporate
any water that has been absorbed. Immediately seal the exposed dielectric using some
epoxy, eg. the Huber-Suhner blue epoxy. Test for shorts using a DMM, and allow the
epoxy to completely set before proceeding;
137
4. Use a sharp blade to gently clean the inner conductor. Hold the center pin of the
SMP connector with a little clamp or vice, heat and let a small amount of solder to
flow into the small hole at the back end of the center pin. Make sure the outer surface
of the center pin is clean. With the soldering iron still heating the center pin, push
the exposed inner conductor into the small hole; take off the iron, and let the solder
to cool and get hardened.
5. Hold the cable vertically with the center pin pointing upwards. Use a Q-tip to apply
a small amount of stainless steel flux to the outer connector ∼ 2 mm below the cut.
Make sure that the flux is not reaching the exposed powder. Apply some flux to the
tip of the solder wire as well, then tin the outer conductor. Clean the outer with
Q-tips and IPA.
6. Push the cable and center pin into the main part of the SMP connector, and solder
the outer conductor to the SMP connector outer. This should complete the process,
and check again with a DMM to see if there is any voltage or MΩ leak between the
inner and outer.
Shown in Fig. A.5 are 28 Thermocoax cables, terminated with SMP connectors and
assembled with Doug’s new silver colder finger on the Microsoft fridge. At the time of
writing, initial tests suggest that the electrons are getting close to the fridge temperature,
yet the fridge base temperature is higher than before, at ∼ 25 mK. The specific reasons for
the higher base temperature is still under investigation.
138
Figure A.5: Thermalcoax cables terminated with SMP connectors and assembled on theMicrosoft fridge
139
Appendix B
Igor implementation of virtualDACs
As Charlie once boasted in one group meeting, the use of “Igor Pro” has been a secret
weapon of the Marcus lab. Indeed, this little known yet powerful program has been the
primary platform for almost all computer related tasks for experiments, from controlling
instruments, data acquisition and processing, to making illustrations for final publication.
Even some theoretical simulations done in this thesis have made use of Igor Pro.
While most people in the group are using the ”Alex Igor Suite”, developed by Alex
Johnson [218], for controlling experiments and acquiring data, the ”Noise Team”, and later
the “5/2 Team” (excluding Jeff) have been using our own set of Igor routines. Many features
are shared between Alex’s procedures and ours, including universal interfaces of acquiring
various types of data and controlling independent parameters, universal 1d and 2d sweeps,
and automatically saving data waves and related parameters, etc. One feature that has
been greatly enhanced in our procedures is what we call “virtual DACs”, which I would like
to share here in my thesis. It is a framework that allows easy definition and simultaneous
control of a linear combination of multiple independent parameters, and is designed to be
140
AO6 DacUnitsPoint DacNames Defined-inRow
Used-inRow
0 Vbias 0.000 mV 0 -1
1 Vc -0.000 mV -1 -1
2 Vcfine 0.000 mV -1 -1
3 Vrt 0.000 mV 3 -1
4 Vrb -0.000 mV 3 -1
5 Vlt 0.000 mV 1 -1
6 Vmc 0.000 mV -1 -1
7 Vstill 0.000 mV -1 -1
8 Vlb -0.000 mV 1 -1
9 Vt 0.000 mV 2 -1
10 Vb -0.000 mV 2 -1
11 Vl1 0.000 mV 5 -1
12 Vl2 0.000 mV 6 -1
13 BVout 0.000 mV 0 -1
14 Vr1 -0.000 mV 5 -1
15 Vr2 -0.000 mV 6 -1
16 Bperp 0.000 mT -1 -3
17 Vls -0.000 mV 4 1
18 Vla 0.000 mV -1 1
19 Vs 0.000 mV -1 2
20 Va 0.000 mV -1 2
21 Vrs 0.000 mV 4 3
22 Vra 0.000 mV -1 3
23 Vss 0.000 mV -1 4
24 Vsa -0.000 mV -1 4
25 V1s 0.000 mV -1 5
26 V1a 0.000 mV -1 5
27 V2s -0.000 mV -1 6
28 V2a 0.000 mV -1 6
29 DeltaB 0.000 mT -1 0
30 Idc 0.000 nA -1 0
Figure B.1: Channel definition table
fully compatible with any previously written codes.1
B.1 Igor implementation of virtual DACs
For historical reasons, the digital-to-analog converter (DAC) outputs are stored in a wave
named “AO6” [see Fig. B.1], and are controlled by the function setdac(chan,value). The
idea is to overwrite this function, and define virtual DAC channels to allow controlling a
combination of real or virtual channels. Another type, called special channel, can also be
defined by providing the names of “set” and “get” functions in the text waves DACSetFunc-
1One only needs to change setdac() to sd() and rampdac() to rd() in any old procedures.
141
tion and DACGetFunction, for the ultimate flexibility. Channel 16, Bperp, which controls
the perpendicular magnetic field, is one example of such special channel.
Each pair of new virtual DAC channels, VChan1 and VChan2, is defined by a linear
combination of two other channels, RChan1 and RChan2, as follows:
V Chan1 = (RChan1−ROfs1)× V 1R1 + (RChan2−ROfs2)× V 1R2
V Chan2 = (RChan1−ROfs1)× V 2R1 + (RChan2−ROfs2)× V 2R2
RChan1 = V Chan1×R1V 1 + V Chan2×R1V 2 +ROfs1
RChan2 = V Chan1×R2V 1 + V Chan2×R2V 2 +ROfs2
where, V1R1, V1R2, V2R1, V2R2, ROfs1, and ROfs2 are user-given coefficients, and
are used to calculate values of R1V1, R1V2, R2V1, and R2V2. All coefficients for defining
one pair of virtual channels are stored in one row of the 2d wave, VDACparams, as shown
in Fig. B.2.
After initializing the procedures by running SetupVDACs(), and setting up default pa-
rameters by SetDefaultParams(), a pair of new virtual DACs can be defined by the function
DefineNewVDAC(VChan1,VChan2,RChan1,RChan2,VChan1Name,VChan2Name). By de-
fault, VChan1 is defined as the average of RChan1 and RChan2, and VChan2 is the differ-
ence between RChan1 and RChan2, but one can easily assign any conversion coefficients by
using SetNewCoef(row,V1R1,V1R2,V2R1,V2R2,ROfs1,ROfs2) for the pair of virtual DACs
defined in the row “row” of the wave VDACparams. To remove the definition of a pair,
simply use RemoveRow(row).
These definitions can be cascaded so that simultaneous control of many channels can be
142
Row x y VirtualCh1 VirtualCh2 RealCh1 RealCh2 V1R1 V1R2 V2R1 V2R2 Real ofs1 Real ofs2 R1V1 R1V2 R2V1 R2V20 30 29 0 13 0.01 0 0 -0.002316 0 0 100 -0 -0 -431.648
1 17 18 5 8 0.5 0.5 1 -1 0 0 1 0.5 1 -0.5
2 19 20 9 10 0.5 0.5 1 -1 0 0 1 0.5 1 -0.5
3 21 22 3 4 0.5 0.5 1 -1 0 0 1 0.5 1 -0.5
4 23 24 17 21 0.5 0.5 1 -1 0 0 1 0.5 1 -0.5
5 25 26 11 14 0.5 0.5 1 -1 0 0 1 0.5 1 -0.5
6 27 28 12 15 0.5 0.5 1 -1 0 0 1 0.5 1 -0.5
7 -1 -1 -1 -1 0.5 0.5 1 -1 0 0 1 0.5 1 -0.5
Figure B.2: Virtual DACs parameters table
easily achieved. For example, as shown in Figs. B.1 and B.2, the virtual channels Vls/Vla
(Vrs/Vra) are defined from Vlt/Vlb (Vrt/Vrb), and then Vls/Vrs are used to define Vss/Vsa.
All interdependent channels are updated automatically when the value of any channel is
changed. For example, when Vls is changed, the channels Vlt/Vlb as well as Vss/Vsa are
updated.
Here, I provide the source codes for these procedures:
/////////////////////////////////////////////////////////////////////////////////////
//Functions and operations for working with user defined virtual DAC channels
/////////////////////////////////////////////////////////////////////////////////////
constant VDAC_VCHAN1 = 0 //Virtual DAC channel 1
constant VDAC_VCHAN2 = 1 //Virtual DAC channel 2
constant VDAC_RCHAN1 = 2 //Real DAC channel 1
constant VDAC_RCHAN2 = 3 //Real DAC channel 2
constant VDAC_V1R1 = 4 //Real 1 to virtual 1 conversion coefficient
constant VDAC_V1R2 = 5 //Real 2 to virtual 1 conversion coefficient
constant VDAC_V2R1 = 6 //Real 1 to virtual 2 conversion coefficient
constant VDAC_V2R2 = 7 //Real 2 to virtual 2 conversion coefficient
constant VDAC_ROFS1 = 8 //Real DAC channel 1 offset
constant VDAC_ROFS2 = 9 //Real DAC channel 2 offset
constant VDAC_R1V1 = 10 //Virtual 1 to real 1 conversion coefficient
constant VDAC_R1V2 = 11 //Virtual 2 to real 1 conversion coefficient
constant VDAC_R2V1 = 12 //Virtual 1 to real 2 conversion coefficient
constant VDAC_R2V2 = 13 //Virtual 2 to real 2 conversion coefficient
//Conversion rules:
//VChan1 = (RChan1 - ROfs1) * V1R1 + (RChan2 - ROfs2) * V1R2
//VChan2 = (RChan1 - ROfs1) * V2R1 + (RChan2 - ROfs2) * V2R2
//RChan1 = VChan1 * R1V1 + VChan2 * R1V2 + ROFS1
//RChan2 = VChan1 * R2V1 + VChan2 * R2V2 + ROFS2
//R1V1, R1V2, R2V1 and R2V2 are calculated from V1R1, V1R2, V2R1 and V2R2
//Setup waves for defining and using virtual DACs
//Each row of VDACparams defines a pair of virtual DACs from a pair of real or virtual DACs
//UsedinRow specifies which row the channel is used to define virtual DACs; if the channel
// is not used for defining virtual channels, UsedinRow = -1.
//DefinedinRow specifies which row the virtual channel is defined; if the channel is a real
// DAC, DefinedinRow = -1; if the channel is a real special channel, DefinedinRow = -3; if
// the channel is not defined, DefinedinRow = -2
function SetupVDACs ( )
variable/G NumRowVDAC = 32
make/d/o/n=(NumRowVDAC,14) VDACparams //Each row defines a conversion pair
SetDimLabel 1, VDAC_VCHAN1, $"VirtualCh1", VDACparams
143
SetDimLabel 1, VDAC_VCHAN2, $"VirtualCh2", VDACparams
SetDimLabel 1, VDAC_RCHAN1, $"RealCh1", VDACparams
SetDimLabel 1, VDAC_RCHAN2, $"RealCh2", VDACparams
SetDimLabel 1, VDAC_V1R1, $"V1R1", VDACparams
SetDimLabel 1, VDAC_V1R2, $"V1R2", VDACparams
SetDimLabel 1, VDAC_V2R1, $"V2R1", VDACparams
SetDimLabel 1, VDAC_V2R2, $"V2R2", VDACparams
SetDimLabel 1, VDAC_ROFS1, $"Real ofs1", VDACparams
SetDimLabel 1, VDAC_ROFS2, $"Real ofs2", VDACparams
SetDimLabel 1, VDAC_R1V1, $"R1V1", VDACparams
SetDimLabel 1, VDAC_R1V2, $"R1V2", VDACparams
SetDimLabel 1, VDAC_R2V1, $"R2V1", VDACparams
SetDimLabel 1, VDAC_R2V2, $"R2V2", VDACparams
NVAR NumChan //Number of DAC Channels, defined in ingotDAC.ipf
make/o/n=(NumChan) UsedinRow, DefinedinRow
make/o/n=(NumChan)/t DACSetFunction,DACGetFunction //Only used for real special channels
end
//Set Default Parameters
function SetDefaultParams()
wave VDACparams, UsedinRow, DefinedinRow
VDACparams[][VDAC_VCHAN1] = -1 //Undefined if Chan < 0
VDACparams[][VDAC_VCHAN2] = -1
VDACparams[][VDAC_RCHAN1] = -1
VDACparams[][VDAC_RCHAN2] = -1
VDACparams[][VDAC_V1R1] = 1/2
VDACparams[][VDAC_V1R2] = 1/2
VDACparams[][VDAC_V2R1] = 1
VDACparams[][VDAC_V2R2] = -1
VDACparams[][VDAC_ROFS1] = 0
VDACparams[][VDAC_ROFS2] = 0
CalcBackConversionCoef ( )
UsedinRow = -1 //Not used
DefinedinRow = -2 //Not defined
NVAR NumRealChan
DefinedinRow[0,NumRealChan-1] = -1 //Real channels
end
//Return DAC number for a given DAC name
function Name2Num(channame)
string channame
wave/T DacNames
variable i=0
do
if(stringmatch(DacNames[i],channame))
return i
endif
i+=1
while (i < numpnts(DacNames))
print "ERROR: Unknown channel",channame
Abort
end
//Return 1 if row is used; otherwise, return 0
function RowUsed( row )
variable row
NVAR NumRowVDAC
if( row<0 || row > NumRowVDAC-1 )
printf "Warning! Row %g out of range.\r", row
return 0
endif
144
wave VDACparams
return ( VDACparams[row][VDAC_VCHAN1] > 0 && VDACparams[row][VDAC_VCHAN2] > 0 )
end
//Return 1 if chan is defined; otherwise, return 0
function ChannelDefined ( chan )
variable chan
wave DefinedinRow
NVAR NumChan
if( chan<0 || chan > NumChan-1 )
printf "Warning! Channel %g out of range.\r", chan
return 0
endif
return ( DefinedinRow[chan] != -2 )
end
//Return 1 if chan is used to define virtual DACs; otherwise, return 0
function ChannelUsed ( chan )
variable chan
wave UsedinRow
NVAR NumChan
if( chan<0 || chan > NumChan-1 )
printf "Warning! Channel %g out of range.\r", chan
return 0
endif
return ( UsedinRow[chan] > -0.5 )
end
//Check if chan is a real or special channel
function ChannelReal ( chan )
variable chan
return ( ChannelRealDAC ( chan ) || ChannelRealSpecial ( chan ))
end
//Check if chan is a real DAC channel
function ChannelRealDAC ( chan )
variable chan
wave DefinedinRow
NVAR NumChan
if( chan<0 || chan > NumChan-1 )
printf "Warning! Channel %g out of range.\r", chan
return 0
endif
return ( DefinedinRow[chan] == -1)
end
//Check if chan is a special channel
function ChannelRealSpecial ( chan )
variable chan
wave DefinedinRow
NVAR NumChan
if( chan<0 || chan > NumChan-1 )
printf "Warning! Channel %g out of range.\r", chan
return 0
endif
return ( DefinedinRow[chan] == -3)
end
//Define a new pair of virtual DACs
//If successful, return the row number that defines them
//If unsuccessful, a negative error code is returned
function DefineNewVDAC(Vchan1,Vchan2,Rchan1,Rchan2,Vchan1Name,Vchan2Name)
variable Vchan1,Vchan2,Rchan1,Rchan2
string Vchan1Name,Vchan2Name
wave VDACparams
145
variable numrows = dimsize(VDACparams,0)
//Make sure Rchan1, Rchan2 are already defined
if( ! ChannelDefined( Rchan1 ) || ! ChannelDefined( Rchan2 ) )
printf "Error! Rchan1 %g or Rchan2 %g are not defined.\r", Rchan1, Rchan2
return -1
endif
//Make sure Rchan1, Rchan2 are not used to define other virtual DACs
if( ChannelUsed( Rchan1 ) || ChannelUsed( Rchan2 ) )
printf "Error! Rchan1 %g or Rchan2 %g are used to define other virtual DACs.\r", Rchan1, Rchan2
return -1
endif
//Make sure Vchan1, Vchan2 are not defined
if( ChannelDefined( Vchan1 ) || ChannelDefined( Vchan2 ) )
printf "Error! Vchan1 %g or Vchan2 %g are already defined.\r", Vchan1, Vchan2
return -2
endif
wave/T DacNames
wave UsedinRow, DefinedinRow
variable row
for(row = 0; row < numrows; row += 1)
if( ! RowUsed( row ) )
VDACparams[row][VDAC_VCHAN1] = Vchan1
VDACparams[row][VDAC_VCHAN2] = Vchan2
VDACparams[row][VDAC_RCHAN1] = Rchan1
VDACparams[row][VDAC_RCHAN2] = Rchan2
UpdateVirtualDACs ( row )
UsedinRow[Rchan1] = row
UsedinRow[Rchan2] = row
DefinedinRow[Vchan1] = row
DefinedinRow[Vchan2] = row
DacNames[Vchan1] = Vchan1Name
DacNames[Vchan2] = Vchan2Name
printf "Success! Vchan1 %g and Vchan2 %g have been defined from Rchan1 %g and
Rchan2 %g in row %g.\r", Vchan1, Vchan2, Rchan1, Rchan2, row
return row
endif
endfor
printf "Error! Cannot find an unused row. Increase the number of rows in SetupVDACs( ).\r"
return -3
end
//Remove a row that defines a pair of Virtual DACs
function RemoveRow( row )
variable row
wave/T DacNames
wave VDACparams, UsedinRow, DefinedinRow, AO6
if( ! RowUsed( row ) ) //Check if the row is used
printf "Warning! Row %g is not used, no need to remove.\r", row
return -1
endif
//Check if the virtual channels defined in this row are used elsewhere
if( ChannelUsed( VDACparams[row][VDAC_VCHAN1] ) || ChannelUsed( VDACparams[row][VDAC_VCHAN1] ) )
printf "Failure! The virtual channels defined in this row %g are used to define other
virtual channels.\r", row
return -2
endif
UsedinRow[ VDACparams[row][VDAC_RCHAN1] ] = -1 //Rchan1 is no longer used
UsedinRow[ VDACparams[row][VDAC_RCHAN2] ] = -1 //Rchan2 is no longer used
146
DefinedinRow[ VDACparams[row][VDAC_VCHAN1] ] = -2 //Vchan1 is no longer defined
DefinedinRow[ VDACparams[row][VDAC_VCHAN2] ] = -2 //Vchan2 is no longer defined
DacNames[ VDACparams[row][VDAC_VCHAN1] ] = "" //Remove Vchan1 label
DacNames[ VDACparams[row][VDAC_VCHAN2] ] = "" //Remove Vchan2 label
AO6[ VDACparams[row][VDAC_VCHAN1] ] = nan
AO6[ VDACparams[row][VDAC_VCHAN2] ] = nan
VDACparams[row][VDAC_VCHAN1] = -1 //row is no longer used
VDACparams[row][VDAC_VCHAN2] = -1
VDACparams[row][VDAC_RCHAN1] = -1
VDACparams[row][VDAC_RCHAN2] = -1
end
//Set new coefficients to the pair of virtual DACs defined in row
Function SetNewCoef( row, v1r1, v1r2, v2r1, v2r2, ROfs1, ROfs2 )
variable row, v1r1, v1r2, v2r1, v2r2, ROfs1, ROfs2
if( ! RowUsed( row ) )
printf "Warning! The row %g does not define any virtual DACs. Nothing is updated.\r", row
return -1;
endif
wave VDACparams
VDACparams[row][VDAC_V1R1] = v1r1
VDACparams[row][VDAC_V1R2] = v1r2
VDACparams[row][VDAC_V2R1] = v2r1
VDACparams[row][VDAC_V2R2] = v2r2
VDACparams[row][VDAC_ROFS1] = ROfs1
VDACparams[row][VDAC_ROFS2] = ROfs2
CalcBackConversionCoef ( )
UpdateAllVirtualDACs ( )
end
//Calculate R1V1, R1V2, R2V1 and R2V2 from V1R1, V1R2, V2R1 and V2R2
function CalcBackConversionCoef ( )
wave VDACparams
variable numrows = dimsize(VDACparams,0)
variable row, v1r1, v1r2, v2r1, v2r2, det
for(row=0; row<numrows; row+=1)
v1r1 = VDACparams[row][VDAC_V1R1]
v1r2 = VDACparams[row][VDAC_V1R2]
v2r1 = VDACparams[row][VDAC_V2R1]
v2r2 = VDACparams[row][VDAC_V2R2]
det = v1r1 * v2r2 - v1r2 * v2r1
VDACparams[row][VDAC_R1V1] = v2r2 / det
VDACparams[row][VDAC_R1V2] = - v1r2 / det
VDACparams[row][VDAC_R2V1] = - v2r1 / det
VDACparams[row][VDAC_R2V2] = v1r1 / det
endfor
end
//Update Virtual DAC values
function UpdateVirtualDACs ( row )
variable row
wave VDACparams
wave AO6, UsedinRow
if( ! RowUsed( row ) )
printf "Warning! The row %g does not define any virtual DACs. Nothing is updated.\r", row
return -1;
endif
variable vchan1,vchan2,rchan1,rchan2
vchan1 = VDACparams[row][VDAC_VCHAN1]
147
vchan2 = VDACparams[row][VDAC_VCHAN2]
rchan1 = VDACparams[row][VDAC_RCHAN1]
rchan2 = VDACparams[row][VDAC_RCHAN2]
AO6[vchan1] = ( AO6[rchan1] - VDACparams[row][VDAC_ROFS1] ) * VDACparams[row][VDAC_V1R1]
+ ( AO6[rchan2] - VDACparams[row][VDAC_ROFS2] ) * VDACparams[row][VDAC_V1R2]
AO6[vchan2] = ( AO6[rchan1] - VDACparams[row][VDAC_ROFS1] ) * VDACparams[row][VDAC_V2R1]
+ ( AO6[rchan2] - VDACparams[row][VDAC_ROFS2] ) * VDACparams[row][VDAC_V2R2]
//If VChan1 is used to define other virtual channels, need to update them as well
if( ChannelUsed ( vchan1 ) )
UpdateVirtualDACs ( UsedinRow[vchan1] )
endif
//If VChan2 is used to define other virtual channels, need to update them as well
if( ChannelUsed ( vchan2 ) )
UpdateVirtualDACs ( UsedinRow[vchan2] )
endif
end
function UpdateAllVirtualDACs ( )
wave VDACparams
variable numrows = dimsize(VDACparams,0)
variable row
for(row=0; row<numrows; row+=1)
if( RowUsed( row ) )
UpdateVirtualDACs ( row )
endif
endfor
end
function UpdateRealSpecialChan( chan )
variable chan
wave AO6
wave/T DACGetFunction
string cmdstr
variable/g tempvar
if ( ChannelRealSpecial ( chan ) ) //If the channel is a real special channel
cmdstr = "tempvar = " + DACGetFunction[chan] + "()"
Execute cmdstr
AO6[chan] = tempvar
return 1
else
return 0
endif
end
//V1 is the real part, V2 is the imaginary part
function/C RealtoVirtual ( row, r1, r2 )
Variable row, r1, r2
wave VDACparams, AO6
if( ! RowUsed ( row ) ) //If row is not used
printf "Error! Row %g is not used.\r", row
return nan
endif
variable v1, v2
v1 = ( r1 - VDACparams[row][VDAC_ROFS1] ) * VDACparams[row][VDAC_V1R1]
+ ( r2 - VDACparams[row][VDAC_ROFS2] ) * VDACparams[row][VDAC_V1R2]
v2 = ( r1 - VDACparams[row][VDAC_ROFS1] ) * VDACparams[row][VDAC_V2R1]
+ ( r2 - VDACparams[row][VDAC_ROFS2] ) * VDACparams[row][VDAC_V2R2]
return cmplx(v1,v2)
end
//R1 is the real part, R2 is the imaginary part
function/C VirtualToReal ( row, v1, v2 )
148
Variable row, v1, v2
wave VDACparams, AO6
if( ! RowUsed ( row ) ) //If row is not used
printf "Error! Row %g is not used.\r", row
return nan
endif
variable r1, r2
r1 = v1 * VDACparams[row][VDAC_R1V1] + v2 * VDACparams[row][VDAC_R1V2] + VDACparams[row][VDAC_ROFS1]
r2 = v1 * VDACparams[row][VDAC_R2V1] + v2 * VDACparams[row][VDAC_R2V2] + VDACparams[row][VDAC_ROFS2]
return cmplx(r1,r2)
end
//Read virtual DAC values, using GetDAC
function ReadVDac(chan)
variable chan
wave DefinedinRow, VDACparams, AO6
variable row = DefinedinRow[chan]
if ( ChannelRealSpecial ( chan ) )
return AO6[chan]
elseif ( ChannelRealDAC ( chan ) )
return GetDAC(chan)
else
variable rchan1 = VDACparams[row][VDAC_RCHAN1]
variable rchan2 = VDACparams[row][VDAC_RCHAN2]
variable vchan1 = VDACparams[row][VDAC_VCHAN1]
variable vchan2 = VDACparams[row][VDAC_VCHAN2]
if( chan == vchan1 ) //VChan1 is to be ramped
return real( RealtoVirtual ( row, ReadVDac(rchan1), ReadVDac(rchan2) ) )
else //Vchan2 is to be ramped
return imag( RealtoVirtual ( row, ReadVDac(rchan1), ReadVDac(rchan2) ) )
endif
endif
end
//Extended version of rampdac(), for ramping virtual DACs as well
function rdEX(chan, destV[, stepsize, tau])
variable chan, destV
variable stepsize, tau
if ( ChannelRealSpecial ( chan ) ) //If it’s a real special channel, call sdEX() directly
sdEX( chan, destV )
return 1
endif
if ( ParamIsDefault ( stepsize ) )
stepsize = 5 //in DacUnits
endif
if ( ParamIsDefault ( tau ) )
tau = 0.1 //in secs
endif
wave AO6
//Change default stepsize and tau for arbitrary channel here
if ( chan == 30 ) //For Idc, go much slower!
stepsize = 40
tau = 0.05
endif
variable stepdelta
if ( destV < AO6[chan] )
stepdelta = - stepsize
149
else
stepdelta = stepsize
endif
variable numsteps = floor( abs( AO6[chan] - destV ) / stepsize )
variable v = AO6[chan] + stepdelta
variable i
for ( i = 0; i < numsteps; i += 1 )
sdEx ( chan, v )
wait( tau )
v += stepdelta
endfor
sdEx ( chan, destV )
end
//Extended version of setdac(), for ramping virtual DACs as well
function sdEX(chan, destV)
variable chan, destV
if( ! ChannelDefined ( chan ) ) //Make sure the channel is defined
printf "Channel %g is not defined.\r", chan
return -1
endif
wave AO6, VDACparams, DefinedinRow
variable row = DefinedinRow [chan]
if( ChannelReal ( chan ) ) //If the channel is real
setchan(chan, destV)
UpdateAllVirtualDACs()
else //Otherwise, it is a virtual DAC, defined in DRow
UpdateAllVirtualDACs() //Update first, in case user aborted a ramp
variable rchan1 = VDACparams[row][VDAC_RCHAN1]
variable rchan2 = VDACparams[row][VDAC_RCHAN2]
variable vchan1 = VDACparams[row][VDAC_VCHAN1]
variable vchan2 = VDACparams[row][VDAC_VCHAN2]
variable/c Rdest
if( chan == vchan1 ) //VChan1 is to be ramped
Rdest = VirtualToReal ( row, destV, AO6[vchan2] )
else //Vchan2 is to be ramped
Rdest = VirtualToReal ( row, AO6[vchan1], destV )
endif
sdEX( rchan1, real(Rdest) )
sdEX( rchan2, imag(Rdest) )
UpdateVirtualDACs( row )
endif
end
//Setting real DAC or special channels
function setchan(chan, dest)
variable chan, dest
NVAR NumRealChan
wave AO6
wave/T DACSetFunction
string cmdstr
if ( ChannelRealDAC ( chan ) )
setdac(chan, dest)
elseif ( ChannelRealSpecial ( chan ) )
cmdstr = DACSetFunction[chan] + "(" + num2str(dest) + ")"
Execute cmdstr
AO6[chan] = dest
else //If not real channel, return 0
return 0
endif
150
end
//Shortcuts
function sd(chan,value)
variable chan,value
sdEX(chan,value)
end
function rd(chan,value[,stepsize,tau])
variable chan,value
variable stepsize,tau
if ( ! ParamIsDefault ( stepsize ) )
if ( ! ParamIsDefault ( tau ) )
rdEX(chan,value,stepsize=stepsize,tau=tau)
else
rdEX(chan,value,stepsize=stepsize)
endif
else
if ( ! ParamIsDefault ( tau ) )
rdEX(chan,value,tau=tau)
else
rdEX(chan,value)
endif
endif
end
151
Appendix C
Effects of external impedance onconductance and noise
C.1 Effects of external impedance on conductance
The effects of finite-impedance external circuits need to be subtracted to obtain the intrinsic
properties of the device. The simplest case is a two-terminal devices in series with an
external resistor, r [see Fig. C.1(a)], and we would like to measure the intrinsic device
resistance, R. As the voltage applied, V is different from the real voltage V dropped across
the device, we will need to subtract the series resistance r from the measured resistance
r +R to obtain the intrinsic device resistance.
A more formal approach is needed to subtract series resistance connected to a multi-lead
device, as shown in Fig. C.1(b). We consider that at lead i, the series resistance is ri, the
out-flowing current is Ii, the real voltage at the lead is Vi, and the applied voltage is Vi. As
in Ch. 5, we denote the raw conductance matrix as g, and the device intrinsic conductance
matrix as g, such that:I1
I2
I3
I4
= g ·
V1
V2
V3
V4
and
I1
I2
I3
I4
= g ·
V1
V2
V3
V4
.
152
2 1
43Device
r
R
I
V(b)(a)
V~
r1 I1
V1
V1~
r2I2
V2
V2~
r3I3
V3
V3~
r4 I4
V4
V4~
Figure C.1: Circuit schematics for calculating intrinsic conductance in (a) a two-lead deviceand (b) a multi-lead device.
Since Vi − Vi = riIi, we arrive at:I1
I2
I3
I4
= g·
V1 − r1I1
V2 − r2I2
V3 − r3I3
V4 − r4I4
=⇒
I1
I2
I3
I4
=
E + g ·
r1
r2
r3
r4
−1
·g·
V1
V2
V3
V4
,
where E is an identity matrix. Comparing to the expression for g, we get:
g =
E + g ·
r1
r2
r3
r4
−1
· g. (C.1)
For all ri = r, we simply have g = [E + rg]−1 · g, which is the expression used in Ch. 5.
C.2 Effects of external impedance on current noise
External impedance on current noise has two effects [19, 128]: one is the feedback effect
similar to the conductance case, and the other is the added thermal noise. Here, we again
consider a two-lead device first, as shown in Fig. C.2(a), with both leads connected to
external impedance. Following the approach used in Ref. [19], we model current noise, δIi,
with a current source injected at lead i. They induce voltage fluctuation ∆Vi at lead i, and
current fluctuation ∆Ii through the external resistance ri. What we measure is the voltage
noise SV i ∝ 〈∆V 2i 〉t, and the current noise that we wish to extract is SIi+4kBTe/ri ∝ 〈δI2
i 〉t
and SI12 ∝ 〈δI1δI2〉t, where 〈..〉t denotes time average.
153
r1r2ΔI2
δI2
ΔI1
ΔV1ΔV2
δI1
12
43Device
r1r2
R
ΔI2
δI2
ΔI1
ΔV1ΔV2 δI1
(a) (b)
Figure C.2: Circuit schematics for calculating intrinsic current noise in (a) a two-lead deviceand (b) a multi-lead device.
Since ∆Vi = ri∆Ii, we have:(∆I1
∆I2
)=(δI1
δI2
)+(g11 g21
g12 g22
)·(
∆V1
∆V2
)=(
∆V1/r1
∆V2/r2
)
=⇒(δI1
δI2
)=(
1/r1 − g11 −g21
−g12 1/r2 − g22
)·(
∆V1
∆V2
). (C.2)
As in Ch. 5, defining a11(22) = 1/r1(2) − g11(22), a12(21) = −g12(21), we then get
SI1 = a211SV 1 + a2
21SV 2 + 2a11a21SV 12 − 4kBTe/r1 (C.3a)
SI2 = a212SV 1 + a2
22SV 2 + 2a12a22SV 12 − 4kBTe/r2 (C.3b)
SI12 = a11a12SV 1 + a21a22SV 2 + (a11a22 + a12a21)SV 12. (C.3c)
To confirm the correctness of Eq. (C.3), consider the device being a simple resistor,
R, in which case we expect SI1,2 = 4kBTe/R and SI12 = −4kBTe/R. Solve for SV 1 in
terms of SI1,2 and SI12 by inverting Eq. (C.3). After working through some algebra, we get
SV 1 = 4kBTer1(R+ r2)/(r1 + r2 +R)—the expected thermal noise measured at lead 1.
Understanding how to extract intrinsic current noise in a two-lead device, we now move
on the the multi-lead device. Here, however, we only consider the case where two of the leads
are attached to external impedance, and the others are grounded, as shown in Fig. C.2(b).
This is the case we are mostly interested in, since we only have a two-channel noise measure-
ment system so far, but the more general case can also be worked out in a similar fashion.
154
Similar to the two-lead case, we can write:∆I1
∆I2
∆I3
∆I4
=
δI1
δI2
δI3
δI4
+
g11 g21 g31 g41
g12 g22 g32 g42
g13 g23 g33 g43
g14 g24 g34 g44
·
∆V1
∆V2
∆V3
∆V4
.
Also, ∆V1,2 = r1,2∆I1,2 and ∆V3,4 = 0. Note here, however, that we will arrive at the
same expression as in Eq. (C.2). Consequently, we will obtain the same expression as in
Eq. (C.3) for extracting the intrinsic current noise of the device, as has been used in Ch. 5.
Since Eq. (C.3) makes use of some elements in the intrinsic conductance matrix that are
not commonly measured, it also motivates the measurement of the full conductance matrix,
for which a multi-channel digital lock-in has been developed and used. I will describe the
details of operations of the multi-channel digital lock-in in the next Appendix.
155
Appendix D
Conductance matrix measurementand multi-channel digital lock-in
D.1 Simultaneous conductance matrix and current noise mea-
surement
As we have discussed in the previous Appendix, in order to extract the intrinsic current noise
auto- and cross correlations of a multi-lead device, one needs to know certain elements of
the conductance matrix, eg. g12, which are not measured with the setup described in Ch. 2.
For this purpose, we have developed the circuit that allows simultaneous measurement
of the two-channel current noise, and the full conductance matrix of a multi-lead device.
The circuit, shown in Fig. D.1, combines the effective circuit shown in Fig. C.1(b) for
conductance matrix measurement near dc, and the effective circuit shown in Fig. C.2(b) for
current noise measurement at low MHz.
At low MHz, the effective circuit is similar to what has been described in Ch. 2, except
that the parallel resistance of the resonant circuit is now 5 kΩ in parallel with two other
50 kΩ resistors, making it ∼ 4.2 kΩ. Near dc, in addition to a resistor r = 5 kΩ to ground,
there are two tapped and low pass filtered lines at each lead, one connected to a current
source, and one connected to a single-ended voltage preamplifier. The resistor r converts
156
2 1
43Device
50kΩ
50kΩ 5kΩ
5kΩ
5nF
5nF
50kΩ
50kΩ 5kΩ
5kΩ
5nF
5nF
10nF
5kΩ
50kΩ
50kΩ5kΩ
5kΩ
5nF
5nF
10nF
5kΩ
96pF5kΩ10nF
66μH
50kΩ
50kΩ5kΩ
5kΩ
5nF
5nF
96pF5kΩ 10nF
66μH
to HEMTto HEMT
Ibias1
Vprobe1
Vprobe4
Vprobe2
Vprobe3
Ibias4
Ibias2
Ibias3
Figure D.1: Circuit schematics for simultaneously measuring conductance matrix and two-channel current noise.
the current Ii out of lead i to a voltage signal measured by the voltage amplifier; it also
converts the current from the current source to a voltage excitation Vi applied at lead i.
The excitations applied at different leads are at different frequencies, and the each current
measured should contain signals at all these frequencies, which can then be used to measure
the full conductance matrix.
D.2 Multi-channel digital lock-in
Although the conductance matrix can be measured with conventional stand-alone lock-ins,
one would need up to 16 of these single-channel lock-ins to measure the full 4×4 conductance
matrix, making it practically impossible. Therefore, we1 have decided to develop in-house
a multi-channel digital lock-in using a National Instruments AD/DA card.
The latest implementation employs a National Instruments PXI-6259 card, which pro-
vides 32 16-bit analog inputs at 1 MS/s, and four 16-bit analog outputs at 2.8 MS/s. One
1This is a collective work by Doug, Reinier and myself.
157
Figure D.2: Multi-channel digital lock-in control panel
such card is sufficient to implement a 4× 42 multi-channel digital lock-in. The four analog
outputs are used to generate four sinusoidal excitations at different frequencies, and they
are also fed into channels 1 through 4 of the analog inputs as reference signals. The response
signals are measured by four voltage preamplifiers, and are then fed into channel 5 through
8 of the analog inputs.
Making use of the NI-DAQmx driver, routines of Igor Pro can directly control and
acquire data from the PXI-6259 card, with most settings controllable from the panel shown
in Fig. D.2. The panel consists of three configuration tabs: the “Lock-in” tab, the “Input
Setup” tab, and the “Output Setup” tab. In the “Lock-in” tab, one can select which
conductance matrix elements are to be measured, the lock-in time constant (meas. tau),
controls to start/stop the lock-in, and perform automatic phase adjustment, etc. In the
“Input Setup” tab, we can set the sampling frequency and the input sensitivity for each
analog inputs. In the “Output Setup” tab, we can set the frequency and amplitude of each
analog outputs, and also controls to start/stop these reference signals.
2Actually, a conductance matrix of size up to 4× 28 can be measured with this card.
158
The digital processing implemented here makes use of fast Fourier transform (FFT), in
a similar way as that for calculating current noise auto- and cross correlations. We first
acquire all reference and response signals for a length of time given by the chosen lock-in time
constant. Then, the FFTs of each acquired waves are calculate. The auto-correlation of each
reference signal, as well as pairwise cross correlation between each reference and response
waves are calculated. As shown in the code given below, the amplitude of the reference
wave are calculated by integrating its power spectrum within a narrow bandwidth around
the given reference frequency. The real (imaginary) component of the cross correlation
integrated within the same band and then divided by the amplitude of the reference wave
gives the in-phase (out-of-phase) component of the response wave. When the parameter
pshift is given when the routine is called, it will return the amplitude of the response wave
projected along the angel set by pshift; otherwise, it will return the polar angel set by the
in-phase (real) and out-of-phase (imaginary) component of the response wave relative to
the reference. Both the amplitude of the reference waves, as well as the in-phase component
of the response waves relative to each references are stored in a 2d wave named “currvals”.
//Calculate the amplitude of the spectral component in sigwave at the frequency ’freq’,
// given a phase offset of ’pshift’ between the sigwave and refwave
//If pshift is not given, figure out the phase shift automatically
function DLockinFFT(refwave,sigwave,freq[,pshift])
wave refwave,sigwave
variable freq
variable pshift
variable getphase = 0
if(ParamIsDefault(pshift)) //If pshift is not given, figure out the phase shift
getphase = 1
endif
NVAR BW = root:lockin:measbw //average over the range freq +/- BW, where BW is in points
variable df = deltax(refwave)
variable xv,yv
variable numpts = 2*BW + 1
make/n=(numpts)/o PR,Xr,Xi
Setscale/P x,0,df,PR //Power of the refwave
Setscale/P x,0,df,Xr
Setscale/P x,0,df,Xi
variable lockinpt = x2pnt(refwave,freq) //The center point
159
PR = magsqr(refwave[lockinpt+p-BW])
Xr = real(refwave[lockinpt+p-BW])*real(sigwave[lockinpt+p-BW])
+ imag(refwave[lockinpt+p-BW])*imag(sigwave[lockinpt+p-BW])
Xi = real(refwave[lockinpt+p-BW])*imag(sigwave[lockinpt+p-BW])
- imag(refwave[lockinpt+p-BW])*real(sigwave[lockinpt+p-BW])
variable/g refV = sqrt(area(PR)) //reference voltage
variable rv = area(Xr)/refV //in-phase component as the reference wave
variable iv = area(Xi)/refV //out-of-phase componentas the reference wave
if (getphase == 0) //do phase shift
xv = rv*cos(pshift) + iv*sin(pshift)
yv = - rv*sin(pshift) + iv*cos(pshift)
return xv //return the x component
else
return imag(r2polar(cmplx(rv,iv))) //return the phase shift
endif
end
The digital lock-in can run in two different modes, local mode or remote (listen) mode.
In the local mode, each time the function DoLockin() is called, it acquires and processes the
data, and updates the wave “currvals”. In the remote mode, it establishes a serial connection
with another computer, and listens to any commands sent over the serial connection. The
commands can either set parameters of the lock-in, or ask for the most recent “currvals”
wave. In this way, the processing computer, together with the NI PXI-6259 card, serves as
a standalone multi-channel digital lock-in.
160
Appendix E
The master equation calculation ofcurrent and noise in a multi-lead,multi-level quantum dot
In this Appendix, I provide the Igor routines for the master equation calculation of cur-
rent and noise in a multi-lead, multi-level quantum dot, mainly following Ref. [135]. These
routines allow setting of arbitrary number of leads and arbitrary number of levels within
the dot, and are used for the calculations in Ch. 5. Also calculated are frequency-dependent
charge and current noise. Since these routines are quite general within the sequential tun-
neling limit, I hope they can also be useful for anyone wishing to learn transport, both
conductance and noise, through a Coulomb blockaded quantum dot.
//////////////////////////////////////////////////////////////////////////////////////////////////
// The Master equation calculation of current and noise in a multi-lead, multi-level quantum dot
// by Yiming Zhang and Leo DiCarlo, Noise Team, Marcus Lab, Harvard University
//////////////////////////////////////////////////////////////////////////////////////////////////
function initmodel()
//Parameters describing system complexity
NVAR nleads,ne,nh,nlevels
nleads=3; //Number of leads
ne = 3; //Number of electron excited levels
nh = 3; //Number of hole excited levels
nlevels = ne + nh + 1
//Constants
variable/g ec = 1.6e-19; variable/g kb = 1.38e-23
variable/g lt = 0; variable/g lb = 1; variable/g rb = 2; variable/g rt = 3
make/d/o/t leadlabels = "lt","lb","rb","rt"
//Physical parameters:
variable/g Te //in K, electron temperature
161
variable/g Vg //in V, gate voltage
variable/g Vg0 //in V, gate voltage reference
variable/g alphaVg //lever arm for gate voltage. alphaVg = Cgate / Ctot
variable/g f0 //in Hz, measurement frequency
make/d/o/n=(nlevels,nleads) GammaMat //in Hz, the bare tunneling rates from each level to each lead
make/d/o/n=(nlevels) Elevels //in eV, energy of single particle levels
Elevels = 150e-6 * (ne - p) //The ne^th level is the ground level
make/d/o/n=(nlevels) Edotlevels //in Joule, energy of single particle levels, affected by Vg
make/d/o/n=(nleads) TeL //in K, electron temperature in each lead
make/d/o/n=(nleads) Vbias //in uV, Bias voltages on the leads
make/d/o/n=(nleads) MuLead //in Joule, Chemical potentials in the leads
variable/g Beta //Gating ratio of the bias voltages
make/d/o/n=(nleads) BetaL //Gating ratio of the bias voltages for different leads
variable/g NumStates0,NumStates1,NumStates
NumStates0 = nComb(nh,nlevels) //Number of states with nh electrons
NumStates1 = nComb(nh+1,nlevels) //Number of states with nh+1 electrons
NumStates = NumStates0 + NumStates1
make/d/o/n=(NumStates0,nlevels) StateConfig0 = nan //nh electron state configurations
make/d/o/n=(NumStates1,nlevels) StateConfig1 = nan //nh+1 electron state configurations
genConfig0(0,0,nh,nlevels)
genConfig1(0,0,nh+1,nlevels)
//Auxiliary waves, need to be generated once, and used in some functions
make/d/o/n=(nlevels) diffvec
make/d/o/n=(nleads) MvsLead
make/d/o/n=(1,NumStates) eT = 1
make/d/o/n=(NumStates1,NumStates0) StateDiffmat
StateDiffmat = StateDiff(p,q)
//M-matrix
make/d/o/n=(NumStates,NumStates) Mmat //Note: Mmat[i][j] is the rate j->i
make/d/o/n=(NumStates1,NumStates0,nleads) Mmat0t1 //Mmat subblock, rates from eh to eh+1 electrons
make/d/o/n=(NumStates0,NumStates1,nleads) Mmat1t0 //Mmat subblock, rates from eh+1 to eh electrons
//Current matrices, U, D matrices, and steady state vectors
make/d/o/n=(NumStates,NumStates,nleads) Jcube
make/d/o/n=(NumStates,NumStates) Jmat,Jmat1,Jmat2,U,D
make/d/c/o/n=(NumStates,NumStates) Dp,Dm
make/d/o/n=(NumStates,1) rho_o,eigenM
//Number matrix
make/d/o/n=(NumStates,NumStates) Nd
Nd = p==q ? (p>=NumStates0 ? nh+1 : nh) : 0
SetParams()
end
// Set parameters here
function SetParams()
NVAR Te, Vg, Vg0, alphaVg, f0, Beta, ne
wave GammaMat,Elevels,Vbias,BetaL,TeL
NVAR lt,lb,rb,rt
Te = 0.34 //in K, electron temperature
TeL = Te //in K, electron temperature in each lead
Vg0 = -1.7525 //in V, gate voltage reference
Vg = Vg0 //in V, gate voltage
alphaVg = 0.0692308
f0 = 0 //in Hz, measurement frequency, not used now
GammaMat[][lt] = 10e9 //in Hz, the bare tunneling rates from each level to each lead
GammaMat[][lb] = 10e9
// GammaMat[][rb] = 2.5e9
Elevels = 150e-6 * (ne - p) //The ne^th level is the ground level
Vbias = 0 //in uV, Bias voltages on the leads
162
Beta = 0.3 //Gating ratio of the bias voltages
BetaL = Beta //Gating ratio of the bias voltages for different leads
end
//Calculate the M-matrix and steady state vector
function PrepMandRho()
NVAR Te,Vg,Vg0,alphaVg,Beta,f0,nleads,ne,nh,nlevels,NumStates0,NumStates1,NumStates
wave GammaMat,Elevels,Edotlevels,Vbias,MuLead,BetaL,TeL
wave Mmat,Mmat0t1,Mmat1t0,StateDiffmat,eT
wave Jcube,U,D,rho_o,eigenM
wave/c Dp,Dm
wave M_R_eigenVectors,W_eigenValues
variable loc_rho_o, tempvar
variable ec = 1.6e-19
//Measurement frequency
variable Omega
Omega = 2 * pi * f0 //f0 is the main parameter, Omega will be updated according to f0
//Chemical potentials in the leads
MuLead = - 1e-6 * ec * Vbias //in Joule, Chemical potentials in the leads
//Energy levels in the dot
variable Edot0 = - 1e-6 * MatrixDot(Vbias,BetaL) - alphaVg * (Vg - Vg0)
Edotlevels = ec * (Edot0 + Elevels) //in eV, dot levels
TeL = Te
// //Implement bias dependent electron heating here
// TeL = sqrt(Te^2 + 1.7e-8*Vbias[p]^2)
//Prepare Rate matrices
// Is it forbidden ?Yes: [ relavant level ][lead]
Mmat1t0 = StateDiffmat[q][p]==-1 ? 0 :
GammaMat[StateDiffmat[q][p]][r] * (1 - FermiF(Edotlevels[StateDiffmat[q][p]],MuLead[r],TeL[r]))
Mmat0t1 = StateDiffmat[p][q]==-1 ? 0 :
GammaMat[StateDiffmat[p][q]][r] * FermiF(Edotlevels[StateDiffmat[p][q]],MuLead[r],TeL[r])
//Prepare M-matrix, first get off-diagonal elements.
// (( top left block ) || ( bottom right block ) ? 0 :
// ( top right block ? eh+1->eh transition :
// eh->eh+1 transition )
Mmat = ((p<NumStates0 && q<NumStates0) || (p>=NumStates0 && q>=NumStates0)) ? 0 :
(p<NumStates0 && q>=NumStates0 ? sumoverlayer(Mmat1t0,p,q-NumStates0) :
sumoverlayer(Mmat0t1,p-NumStates0,q))
MatrixOp/O sumMvec = eT x Mmat //For a given colomn, sum all the rows of the Mmat
Mmat = p==q ? - (sumMvec[0][q] - Mmat[q][q]) : Mmat[p][q] //Get diagonal elements
// Steady state probability vector
loc_rho_o = null(Mmat) //null also calculates the eigen-values and eigen-vectors
U = M_R_eigenVectors //Matrix U is the modal matrix - its columns are the eigenvectors of Mmat
eigenM = real(W_eigenValues[p]) //eigenM contain the eigen values
rho_o = U[p][loc_rho_o]
tempvar = sum(rho_o)
rho_o /= tempvar
//Prepare matrix D
MatrixOP/O D = Diagonal(eigenM)
variable m=0
do
if(m==loc_rho_o)
D[m][m] = 0
Dp[m][m] = 0
Dm[m][m] = 0
else
Dp[m][m] = 1/cmplx(eigenM[m],Omega)
Dm[m][m] = 1/cmplx(eigenM[m],-Omega)
163
D[m][m] = 1/eigenM[m]
endif
m+=1
while(m<NumStates)
//Current matrix
// (( top left block ) || ( bottom right block )) ? 0 :
// ( top right block ? eh+1->eh transition : eh->eh+1 transition )
Jcube = ((p<NumStates0 && q<NumStates0) || (p>=NumStates0 && q>=NumStates0)) ? 0 :
(p<NumStates0 && q>=NumStates0 ? Mmat1t0[p][q-NumStates0][r] : -Mmat0t1[p-NumStates0][q][r])
FastOp Jcube = (-ec) * Jcube
end
//Get DC current, in A
function getI(LeadIndex)
variable LeadIndex
wave Jcube,Jmat,rho_o
Jmat = Jcube[p][q][LeadIndex]
MatrixOP/O tempwave = Jmat x rho_o;
return sum(tempwave)
end
//Get frequency-dependent charge noise spectral density, in e^2/Hz
function getSN()
wave U,Dp,Dm,rho_o,Nd
MatrixOP/O tempwave = Nd x U x Dp x Inv(U) x Nd x rho_o
+ Nd x U x Dm x Inv(U) x Nd x rho_o
return real(-2*sum(tempwave))
end
//Get frequency-dependent current noise auto- and cross-correlation spectral density, in A*2e
function getSI(lead1,lead2)
variable lead1,lead2
wave Jcube,Jmat1,Jmat2,U,D,Dp,Dm,rho_o,Nd
variable ec = 1.6e-19; // in Coulomb
if(lead1==lead2) //auto-correlation
Jmat1 = Jcube[p][q][lead1]
MatrixOP/O tempwave = Jmat1 x U x Dp x Inv(U) x Jmat1 x rho_o
+ Jmat1 x U x Dm x Inv(U) x Jmat1 x rho_o
- ec * (Nd x Jmat1 - Jmat1 x Nd) x rho_o
return real(-2*sum(tempwave)/2/ec)
else //cross-correlation
Jmat1 = Jcube[p][q][lead1]
Jmat2 = Jcube[p][q][lead2]
MatrixOP/O tempwave = Jmat1 x U x Dp x Inv(U) x Jmat2 x rho_o
+ Jmat2 x U x Dm x Inv(U) x Jmat1 x rho_o
return real(-2*sum(tempwave)/2/ec)
endif
end
164
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