introduction to mesoscopic physics

Introduction to mesoscopic

physics

Markus BüttikerUniversity of Geneva

NiPS Summer School 2010: Energy Harvesting at the micro and nanoscale

Avigliano, Umbro, August 1 – August 6 , (2010).

Mesoscopic PhysicsWave nature of electrons becomes important

Webb et al., 1985

Yacoby et al. 1995

2

Mach-Zehnder Interferometers

Roulleau, Portier, Glattli, Roche, Faini,

Gennser, and D. Mailly, PRL 100, 126802

(2008)

Litvin, Tranitz, Wegscheider and Strunk,

PRB 75, 033315 (2007)

Neder, Heiblum, Levinson, Mahalu, Umansky,

PRL 96, 016804 (2006)

Bieri, Schoenenberger, Oberholzer,

et al. PRB 79. 245324 (2009).

3

Probing mesoscopics on the nanoscale

M. J. Brukman and D. A. Bonnell, Physics Today, June 2008, p. 36

4

Grapehne: single and bilayer

@Jian Li unige

5

Length scales

Phase coherence length

Elastic scattering length

Inelastic scattering length

Geometrical dimension

Macroscopic conductor

Mesoscopic conductor

(size of conductor)

(distance an electron travels before suffering a phase change of

(mean free path between elastic scattering events)

(distance an electron travels before loosing an energy kT)

)

Beenakker and van Houten, 1991

6

Physics versus geometry

Mesoscopic physics = « Between mircoscopic and macroscopic »

Nano physics = on the geometrical length of a nanometer

Definition of mesoscopic physics is based on physical length scales.

In contrast, nanophysiscs, is a definition based on a geometrical length

scale.

7

Lecture contentsConductance from transmission

1. Single channel conductors

2. Multichannel conductors

3. Multiprobe conductors

Thermoelectric transport

Nonlinear transport

2. Equilibrium noise

3. Shot-noise two-probe conductors

Fluctuation relations

Noise

8

1. Basic

Conductance from Transmission

1. Single channel conductors

9

Conductance from scattering theory

Fermi energy right contact

applied voltage

Heuristic discussion

transmission probability

reflection probability

Fermi energy left contact

incident current

density

density of states

independent of material !!

10

« Landauer formula »

Drift and diffusion

at constant Einstein relation

for space dependent

11

Scattering matrix

12

scattering state

scattering matrix

current conservation S is a unitray matrix

In the absence of a magnetic field S is an orthogonal matrix

Conductance from transmission

conductance quantum resistance quantum

dissipation and irreversibility

boundary conditions

13

Persistent current(periodic boundary conditions)

Buttiker, Imry and Landauer, Phys. Lett. 96A, 365 (1983).

14

Measured in 1990 by L. Levy et al, in 1991 by Webb et al. .

Persistent currentA. C. Bleszynski-Jayich, W. E. Shanks, B. Peaudecerf, E. Ginossar, F. von Oppen,

L.Glazman, and J. G. E. Harris,, Science 326, 272 (2009).

15

Tunable wave splitterButtiker, Imry, Azbel, Phys. Rev. A30, 1982 (1984)

16

Aharonov-Bohm conductance oscillations17

Gefen, Imry, Azbel, PRL 2004

Buttiker, Imry, Azbel, Phys. Rev. A30, 1982 (1984)

Aharonov-Bohm oscillations18


2. Two-probe multi-channel conductors

19

Multi-channel conductance: leads

asymptotic perfect translation invariant potential

seprable wave function

energy of transverse motion

energy for transverse and longitudnial motion

scattering channel

channel threshold

20

Mulit-channel conductance

incident current in channel i

density in channel i

density of states in channel i

independent of channel

« Landauer formula »

21

Eigen channels

hermitian matrix; real eigenvalues

hermitian matrix; real eigenvalues

are the genetic code of

mesoscopic conductors !!

Many single channel conductors in parallel.

All the properties we discussed for single-channel two-probe conductors apply

equally to many-channel multi-probe conductors: in particular

Eigen channels 22

Conductance of a perfect wire

equilbrium electrochemical potential

number of channels with threshold

spin degeneracy Example: Single wall carbon

nanotube:

23

Quantum Point Contact

2D-electron gas

gate

gate

van Wees et al., PRL 60, 848 (1988)

Wharam et al, J. Phys. C 21, L209 (1988)

24

Quantized conductance: saddle

Saddle-point potential

Transmission probability

Buttiker, Phys. Rev. B41, 7906 (1990)25

Quantized conductance-magnetic field

magnetic field B

Buttiker, Phys. Rev. B41, 7906 (1990)

26

for symmetric cavity with

Chaotic cavity

asmmetric cavity including weak localization:

27

Baranger and Mello, 1994

Diffusive wire

Universal conductance fluctuations

28

Dorokhov-Mello-Pereyra-Kumar

Stone and Lee, Altschuler


3. Multi-probe conductors

29

Multi-probe conductors

Buttiker, PRL 57, 1761 (1986); IBM J. Res. Developm. 32, 317 (1988)

30

Four-probe resistances

Current contacts

Voltgae probes

G has eigenvalue zero!

31


Sub-determinants of conductance matrix

D is a sub-determinant of rank three of the conductance matrix.

All sub-determinants are (up to a sign) equal.

Proof: Expand total determinant into sub-determinants:

The only solution without current at any terminal requires that all applied voltages

are equal.

32

Multi-probe conductors: scattering matrix

magnetic field symmetry

33


Reciprocity

From and

34

Reciprocity: Benoit et al.Benoit, Washburn, Umbach, Laibowitz, Webb, PRL 57, 1765 (1986)

35

Reciprocity: van Houten et al.

skipping orbit

electron focusing

van Houten et al. , Phys. Rev. B39, 8556 (1989)

36

Historical remarks

J. Frenkel, Phys. Rev. 36, 1604 (1930)

W. Ehrenberg and H. Hoenel, Z. f. Physik 68, 289 (1931)

A. Sommerfeld and H. Bethe, Handbuch der Physik (1945)

R. Landauer, IBM J. Res. Developm. 1, 223 (1957)

Plane-parallel barriers

Single-channel transport

R. Landauer, Phil. Mag. 21, 863 (1970)

Multi-channel conductors

H. L. Engquist and P. W. Anderson, Phys. Rev. B24, 1151 (1981)

Anderson, Economou and Soukoulis, Azbel, Fisher and Lee,

Buttiker, Imry and Landauer, Buttiker…

37

Success and limitations

Magntic field symmetry : Reciprocity relations

Success:

Widely applied to ballistic, chaotic and metallic diffusive

relatively open conductors

Theory of the Quantum Hall effect (edge state transport):

probably the most stringent test of the approach

Range of application probably the same as DFT (!!)

Limitations:

Kondo effects, conductance anomalies, ..

extensions to incorporate inelastic scattering, dephasing,

time-dependent potentials, etc. exist

however

Negative four probe resistances, « uphill voltages »

39

Thermoelectric Transport

40

Energy current

H. L. Engquist and P. W. Anderson, Phys. Rev. B24, 1151 (1981)

Energy flux in a quantum channel: reservoirs at T1 and T2:

Small temperature difference

Thermal quantum (independent of electron or channel properties!!)

40

Lorentz factor (Sommerfeld theory)

Heat current

Heat current in perfect quantum channel (linear response)

Heat current (elastic backscattering , linear response)

Thermoelectric transport

41

Thermoelectric transport Fluxes in response to potentials

Current and temperature differences as driving forces

R resistance

S thermopower

Peltier

thermal conductance

Multi-terminal expressions:

P. N. Butcher , J. Phys.: Condensed Matter 2, 4869 (1990).

42

Thermopower

S. F. Godijn, S. Möller, H. Buhmann, L. W. Molenkamp,

S. A. van Langen PRL 82, 2927–2930 (1999)

Cutler-Mott-formula

zero temperature limit

Probability distribution of the

thermopower of a chaotic cavity

one channel leads

S. A. van Langen, P. G. Silvestrov,

C. W. J. Beenakker, Supperlattice and

Microstructures, 23, 691 (1999).

43

Nonlinear transport

44

Rectification

• Scattering matrix:

• Weakly nonlinear transport:

where

M. Büttiker, J. Phys.: Condens. Matter 5, 9361 (1993);

T. Christen and M. Büttiker, Europhys. Lett. 35, 523 (1996)

18 elements

45

Characteristic potentials

• Voltage

• partial DOS:

• Magnetic-field symmetry:

• Poisson equation: injectivity is source of

injectivity

emissivity

M. Buttiker, J. Phys. Condensed Matter 5, 9361 - 9378 (1993).

46

Magnetic field asymmetry of rectification

Naive expectation: since T is even in the two-probe case, nonlinear I-V is also even

Correct only in linear regime: reciprocity of s-matrix hinges on symmetry of U

AWAY FROM EQUILIBRIUM:

Interaction effect

At equilibrium microreversibilty is sufficient to dictate symmetry of transport

coefficients: Away from equilibrium boundary conditions become important

EQULIBRIUM

Elastic transport:

47

Second order conductance of a chaotic dot

Numerical RMT

D. Sanchez and M. Buttiker, PRL 93, 10602 (2004)

Unitray limit

48

M. Polianski and M. Buttiker, PRL 96, 1056804 (2006)

Rectification: experiments I

Carbon nanotubes

J. Wei et al., PRL 95,

256601 (2005)

Cavities

Rings

R. Leturcq et al., PRL 96, 126801 (2006)

D. M. Zumbuhl et al, PRL 96, 206802 (2006)

49

Theory agrees with experiment for N > 4

Rectification: experiments II

D. Hartmann, L. Worschech, A. Forchel, PRB 78, 113306 (2008).

50

Current Noise in Mesoscopic

Conductors

1. Basics

51

Fundamental sources of noise

Thermal fluctuations of occupation numbers in the contacts

Nyquist-Johnson noise

Quantum partition noise: kT = 0 occupation numbers:

incident beam

transmitted beam

reflected beam

averages:Each particle can only be either transmitted or reflected:

Buttiker, PRB 46, 12485 (1992)

Blanter and Buttiker, Phys. Rep. 336, 1 (2000)

52

Occupation number and current amplitudes

Incident current at kT = 0

Incident current at kT > 0

Occupation number

< > = statistical average Creation and annihilation operators

«Incident current » « Current amplitude »

Buttiker, PRB 46, 12485 (1992)

53

Noise spectral density

zero-frequency spectrum (white noise limit)

equilibrium fluctuation-dissipation theorem

Spectral density S (noise power)

quantum statistical average of four creation and annihilation op.

non-equilibrium shot-noise

Buttiker, PRL 65, 2901 (1990); PRB 46, 12485 (1992)

54


Conductors

2. Equilibrium Noise

55

Thermal current fluctuations

Use

with for all auto-correlation

cross-correlation

QHE-plateau N:

56


Conductors

3. Shot Noise: Two-probe conductors

57

Shot-noise: two-terminal

Consider kT = 0, V>0, and a two-terminal conductor:

Quantum partition noise

If all Shottky (Poisson)

Fano factor Khlus (1987)

Lesovik (1989)

Buttiker (1990)

60

Shot-noise: Qunatum point contact

Ideally only one channel contributes

• Kumar, L. Saminadayar, D. C. Glattli,

Y. Jin, B. Etienne, PRL 76, 2778 (1996)

M. I. Reznikov, M. Heiblum, H. Shtrikman,

D. Mahalu, PRL 75, 3340 (1996)

61

Crossover from thermal to shot noise

tunnel junction

H. Birk et al., PRL 75, 1610 (1995)

62


Conductors

4. Shot Noise: Correlations

63

Shot-noise correlations

Consider multi-terminal conductor at kT = 0,

M source contacts with distribution

All other contacts grounded at

voltage

voltage

M =1, partition noise

M =2, exchange effects,

two paricle Aharonov-Bohm effect,

orbital entanglement, violation of Bell inequality

Correlation measured bewteen two grounded contacts:

Samuelsson, Sukhorukov, Buttiker, PRL 92, 026805 (2004)

Buttiker, Samuelsson, Sukhorukov, Physica E20, 33 (2003)

64

Oberholzer et al. Physica E6, 314 (2000)

Bias configuration:

65Beam splitter with noisy input state

Here

Oberholzer et al, Physica E6, 314 (2000)

See also: Henny, et al., Science 284, 296 (1999); Oliver et al. Science 284, 299 (1999)

Experiment of Oberholzer et al. 66

Review on Shot Noise

« Shot Noise in Mesoscopic Conductors »

Ya. M. Blanter and M. Buttiker,

Phys. Rep. 336, 1 (2000)

67

Fluctuation relations 68

Nonlinear transport and noise

Fluctuation dissipation theorem

Fluctuation relation of Forster and Buttiker (microreversible only at eq.)

Fluctation relation of Saito and Utsumi

H. Forster and M. Buttiker, PISA, arXiv: 0903.1431

[General case: H. Forster and M. Buttiker, PRL 101, 136805 (2008) ]

69

Nonlinearity and noise H. Forster and M. Buttiker, arXiv: 0903.1431

Negative excess noise

70

Kobayashi’s experimentNakamura, Yamauchi, Hashisaka, Chida, Kobayashi, Ono, Leturcq, Ensslin,

Saito, Utsumi, and Gossard, Phys. Rev. Lett. 104, 080602 (2010)

71

Nongaussian noise on macroscopic scalesNagaev, Ayvazyan, Sergeeva, and Buttiker, arXiv: 1004.5310

macroscopic!!

potential dependence of conductance cyclotron-frequency times scattering time

[Saito and Utsumi, 2008]

72

Summary

Transport theory for coherent electron transport

Conductance

Thermal transport

Non linear transport

Noise

Correlations

Fluctuation relations

introduction to mesoscopic physics

Documents