1

Single Electron Tunneling – Examples

Danny Porath 2002

(Schönenberger et. al.)

“It has long been an axiom of mine that the little things are infinitely the most important”Sir Arthur Conan Doyle

Books and Internet Sites“Electronic Transport in Mesoscopic Systems”, S. Datta

“Single Charge Tunneling”, H. Grabert & M.H. Devoret

“Introduction to Mesoscopic Physics”, Y. Imry

http://www.iue.tuwien.ac.at/publications/PhD%20Theses/wasshuber/

http://vortex.tn.tudelft.nl/grkouwen/reviewpub.html

http://www.aip.org/web2/aiphome/pt/vol-54/iss-5/archive.html

http://www.ee.princeton.edu/~chouweb/newproject/research/QDT/SingElecTunnel.html

http://qt.tn.tudelft.nl/publi/papers.html

www.unibas.ch/phys-meso

....

Homework 81. Read the paper:

“Variation of the Coulomb staircase in a two-junction system by fractional electron charge”, by: M.E. Hanna and M. Tinkham, Physical Review B, 44(11), 5919 (1991).

Outline SET Examples:1. The C60 work – D.P. et. al.

a. Background

b. Sample preparation and imaging

c. SET Theory

d. I-V Spectroscopy results and fits

e. Summary (of this part)

2. Advanced material from C. Schönenbergera. Introduction

b. Law transparency – SET

c. High transparency – Fabry Perot & UCF

d. Intermediate transparency – Co-tunneling & Kondo

e. Conclusions (from this part)

Single Electron Tunneling and Level Spectroscopy of Single C60 Molecules

Danny Porath, Muin Tarabia, Yair Levi and Oded Millo

The General Experimental Scheme

I

V QD

R1,C1

R2,C2

J-1

J-2

STMtip

gold

C60Insulating

layer

2

C60 Fullerenes

~1 m ~10 Å: 1,000,000,000

Structure: (icosahedral symmetry)

60 carbon atoms. (20 hexagons, 12 pentagons)

Bacground

Before......Significant effort was devoted to studies of the interplay between Single Electron Tunneling (SET) effects and quantum size effects in isolated nano-particles.

The interplay was studied for:∆El << Ec

(Tinkham, Van-Kempen, Sivan, Molenkamp)

C60 spectroscopy was done mainly by optical means, limited by selection rules.

Background

What was new here?Extremely small QD (~8 Å)

Quantum Size effects even at RT.

Interplay between SET and discrete levels in two regimes:

Ec < ∆El

Ec > ∆El

Tunneling spectroscopy of isolated C60Electronic level splitting.

Sample Preparation Steps:Rinse C60 powder in Toluene

Evaporate dried powder on a gold substrate covered by carbon and (later) a PMMA layer

C60Evaporation

Goldsubstrate

Prepared sample

STM Images of C60 Fullerenes

70 Å 40 Å

100 Å 50 Å

STM Images of C60 Fullerenes 3-D

3

SET Effects in DBTJ

STMtip

TunnelJunction 1

TunnelJunction 2

I

QD

V

R1,C1

R2,C2

CB

CS

V

I T(V

)

I-V curves

SET effects can be observed If:1) EC = e2/2C > kBT Charging Energy > Thermal Energy2) RT > RQ = h/e2 Tunneling Resistance > Quantum Resistance

Coulomb Blockade (CB):IT = 0 up to |Vt| ≤ e/2C ; Vt depends on “fractional charge” Q0

Coulomb Staircase (CS):Sequence of steps in I-V. Each step - adding an electron to the island

Charge Quantization Leads to SET EffectsCharge quantization leads to SET effects:

Vt depends on “fractional charge” Q0.Q0 represents any offset potentials.Q = ne - Q0 ; |Q0| ≤ e/2

E

Q(e)

Q0 = e/2-1

0 1

2E

Q(e)

-1

Q0 = 0

0

-2

1

2

Ec

Vt = e/2C Vt = 0

Allowed states with n access electrons indicated on the graphs

Origin of the Fractional Charge Q0

Difference in contact potentials across the junctions:Q0 = (C1∆φ1 - C2∆φ2)/e

Note: We are interested only in Q0 mod e.

Q0 can be varied by controlling C1 through tip-sample separation. This is done by changing the STM current and bias settings.

Interplay Between SET Effects and Discrete Energy Levels

∆EL << EC: (Level Spacing << Charging Energy)

e

Ec

1 2

∆EL << EC

(Amman et al.)

No levels

Additional sub-steps on top of a CS step

Interplay Between SET Effects and Discrete Energy Levels

∆EL ~ EC: (Level Spacing ~ Charging Energy)

CS steps are no more equidistantly spaced.

zero bias gap is not suppressed for Q0 = e/2

when the CB is suppressed.

Pronounced asymmetry of I-V curves due to

different levels on each side.

I-V Measurements

STMtip 1

2

I

QV

R1,C1

R2,C2

(a)

(c)-0.6 -0.3 0.0 0.3

-2

-1

0

1

2Single moleculeT = 300 K

Tunn

elin

g C

urre

nt [n

A]

Tip-Bias [V]

(b)

-0.6 0.0 0.6-1.6

0.0

1.6

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

0

2

4

6

8

10 Single moleculeT = 4.2 K

Tunn

elin

g C

urre

nt [n

A]

Tip-Bias [V]

Tunn

elin

g C

urre

nt [n

A]

(d)-1.0 -0.5 0.0 0.5 1.0

-4

-2

0

2

4 Two molecules

Tip Bias [V]

T = 300 K

T = 4.2 K

0.0 0.4 0.80.3

0.6

I-V measurements are done while disconnecting the feed-back loop.Note:

1. Non-vanishing gap.2. Pronounced asymmetry and different steps.3. Negative differential resistance (NDR).

4

I-V Measurements

-1.0 -0.5 0.0 0.5 1.0

-0.8

-0.4

0.0

0.4

I (nA

)

Tip Voltage (V)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

(c)

(b)

(a)

dI/d

V (a

.u.)

0.0

0.2

0.4

0.6

0.8

1.0

T = 4.2 K

T = 4.2 K

dI/d

V (a

.u.)

0.0 0.3 0.6 0.9 1.20.0

0.3

0.6

0.9

dI/d

V

Tip Voltage (V)

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0T = 300 K

dI/d

V (a

.u.)

Tip Voltage (V)

HOMO+1 (?)

0.0 0.3 0.6 0.9 1.2

0.0

0.3

0.6

0.9

dI/d

V

Tip Voltage (V)

12

HOMO

LUMO

LUMO+1

EF

12

Tunneling through LUMO+1

12

Tunneling through LUMOn=2 on the dot

Local Fields (Sarid et. al.)

Different electric fields (carbon field, STM tip) effect the molecular orbitals differently (Stark effect).

Molecule Orientation (Guntherodt et. al.)

Tunneling channel (depending on the molecule orientation) effects the level splitting.

Jahn - Teller effect (Bendale et. al.)

A splitting or a shift of the energy levels due to structural distortion.The C60

+ hu reduces by ~ 0.6 eV

Theoretical Fits (Orthodox theory-Averin & Likharev)

Γi±(n) - Tunneling rate to/from the dot from/to the electrode.

f - Fermi function.

Di(E), Ei - DOS/Fermi energy at electrode i.

Dd(E), Ed - DOS/Fermi energy at the dot.

Ti(E) - Tunneling matrix element.

Schematic of the circuit:

∫ −−−−−=± )]dEEf(E)[1E(E)DE)f(EE(ED(E)T2π(n)Γ dddiii2

ii h

I

QD

V

R1,C1 R2,C2

The tunneling rate on the dot:

5

Theoretical Fits (Orthodox theory-Averin & Likharev)

P(n) – The probability to find n electrons on the dot

We choose the density of states Dd(E):

(n)]Γ(n)P(n)[Γe(n)]Γ(n)P(n)[ΓeI(V) 1122+−−+ ∑∑ −=−=

The tunneling current is:

Energy Level Calculations

Experimental Results – Asymmetrical Curves

-1.6 -0.8 0.0 0.8 1.6-12

-9

-6

-3

0

3

6

9 (a)

Fits

Exp.

Tunn

elin

g C

urre

nt [n

A]

Bias [V]-1.6 -0.8 0.0 0.8 1.6

-9

-6

-3

0

3

6

9(b)

Fits

Exp.

dI/d

V[a

.u.]

Bias [V]

C1 ~ C2 ~ 10-19, EHOMO - ELUMO ~ 0.7 eV, ∆EL ~ 0.05 eV

Source of AsymmetryC1<C2 ==> V1>V2 ==> Onset of tunneling at junction 1

(for the level configuration at hand)

12

12

Positive bias

12

Negative bias

Experimental Results – Symmetrical Curves

C1 ~ C2 ~ 10-19, EHOMO - ELUMO ~ 0.7 eV, ∆EL ~ 0.05 eV

-1.6 -0.8 0.0 0.8 1.6-5

-4

-3

-2

-1

0

1

2

3

4(a)

Fits

Exp.

Tunn

elin

g C

urre

nt [n

A]

Bias [V]-1.6 -0.8 0.0 0.8 1.6

-4

-3

-2

-1

0

1

2

3

(b)

Fits

Exp.

dI/d

V [a

.u.]

Bias [V]

Symmetrical Barriers (C1~C2)12

Positive bias

2 1

Negative bias

Tunneling through the set of levels closer to EF - in

our case the 3 LUMO levels.

6

Checking The Model - Experimentally

-4

-2

0

2

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-8

-4

0

4

8

B

A

C1 < C2

Vs = 0.75 VIs = 1.6 nA

Tunn

elin

g C

urre

nt (n

A)

dI/d

V (a

.u.)

dI/d

V (a

.u.)

Vs = 0.75 VIs = 6 nA

C1 > C2

Tunn

elin

g C

urre

nt (n

A)

Tip Voltage (V)

The Triple Barrier Tunnel Junction (TBTJ)

STMtip

PMMA

J-1J-2

goldC

C60

J-3

I

QDV

R1,C1

R2,C2

R3,C3

QD

Tunn

elin

g C

urre

nt [n

A]

-1.0 -0.5 0.0 0.5 1.0

-4

-2

0

2

4 Two molecules

Tip Bias [V]

T = 300 K

T = 4.2 K0.0 0.4 0.8

0.3

0.6

Note:Non Vanishing gap.Negative differential resistance (NDR).Non-ohmic curves everywhere on the sample.

Summary - C60 STM SpectroscopyA C60 molecule is an ideal “quantum dot” for studies of single electron tunneling and quantum size effects.

The interplay between SET effects and molecular levels is clearly observed at 4.2 K as well as at RT for the double barrier tunnel junction configuration.

The degeneracy of the HOMO, LUMO and LUMO+1 levels was fully resolved, and the degree of splitting is consistent with the J-T effect.

Quantum Dot Physics

nenbergeröC. Sch

Quantum dotsplanar dot

planar double-dot vertical dot

(„conventional“) Quantum dots

7

quantum dot ?

2 µm

Different energies (what is 0d ?)

δE=h/τround-trip

in addition: eV and kT

example: 0d with respect to quantum-size effects,provided: kT, eV, and Γ < δE

1. low transparencylow transparency single-electron tunnelingdetermined by single-electron chargingeffects, e.g. Coulomb blockade (0d limit)

2. intermediate transparencyintermediate transparency charging effectsbut co-tunneling, e.g. Kondo effect

3. high transparencyhigh transparency quantum interference ofnon-interacting electrons, e.g. Fabry-Perot resonancesand UCF (disordered limit)

Contacts matter...!1. low transparencylow transparency single-electron tunneling

determined by single-electron chargingeffects, e.g. Coulomb blockade (0d limit)

Contacts matter...!

Single-electron tunneling

„greyscale plot“ of dI/dV (Vgate,Vbias)

here: black = low differential conductance

Coulomb blockade

Vg

Vsd

(mV)

tutorial on dI/dV plots

Change Vsd Change Vg

single-electron chargingenergy UC

∆∆E E addadd addition energyaddition energy, i.e. sum of:

level-spacing δE

8

filling of statesaccording to

S = 1/2 0 1/2 ...

even number of electrons:∆Ε add = UC + δE

odd number of electrons:∆Ε add = UC

even even evenodd odd odd

single-electron-tunneling

if tunneling probability pof each junction is „small“:

current determinedby accessible levels in dot

i.e. by level spacing andCoulomb charging energy

„uncorrelated“ sequential tunnelingdominates. Current I ∝ p

1. low transparencylow transparency single-electron tunnelingdetermined by single-electron chargingeffects, e.g. Coulomb blockade (0d limit)

2. intermediate transparencyintermediate transparency charging effectsbut co-tunneling, e.g. Kondo effect

3. high transparencyhigh transparency quantum interference ofnon-interacting electrons, e.g. Fabry-Perot resonances and UCF (disordered limit)

Contacts matter...!

W. Liang et al., Nature 411, p 665 (2001)

5 mV

-5 mV

gate voltage -2 ... +2 V

bias

vol

tage

T~1 limit (interaction-free wire)

T~1 limit (wire)

W. Liang et al., Nature 411, p 665 (2001)

)2cos(21)( 2

RCC

C

RRTET

δ+Θ++=

{ })2/()2/(2

eVTeVTeVI

−+=∂∂

)( −+ −=Θ kkL round-trip phasedepends on E

Fabry-Perot (interference)

simplemodel

5 mV

-5 mV

gate voltage -2 ... +2 V

their data

9

1. low transparencylow transparency single-electron tunnelingdetermined by single-electron chargingeffects, e.g. Coulomb blockade (0d limit)

2. intermediate transparencyintermediate transparency charging effectsbut co-tunneling, e.g. Kondo effect

3. high transparencyhigh transparency quantum interference ofnon-interacting electrons, e.g. Fabry-Perot resonances and UCF (disordered limit)

Contacts matter...!„co-tunneling“

if tunneling probability pof each junction is „small“:

„uncorrelated“ sequential tunnelingdominates. Current I ∝ p

if tunneling probability p of each junction is „large“:

coherent 2nd (and higher) orderprocesses add substantially

∝ p2

we call this „co-tunneling“

even even evenodd odd odd

black regions =very low conductance G

G is suppressed due toCoulomb blockade (CB)

I jump back to low transparent tunneling contacts for reference

Remainder Coulomb blockade

Vg

Vsd

(mV

)

Elastic co-tunneling Inelastic co-tunneling

δE ~ 0.55 meV UC ~ 0.45 meV

When the number of electrons on the quantum dot is odd, spin-flip processes (which screen the spin on the dot) lead to the formation of a narrow resonance in the density-of-states at the Fermi energy of the leads.

Related work: J.Nygard et al, Nature 408, 342 (2000)

Vg

Vsd

(mV

)

This is called the Kondo effect

Kondo effectresistance R(T) of a piece of metal

„ideal“ metal, e.g. Au wire

superconductor, e.g. Pb

magnetic impurities in „ideal“ metale.g. (ppm)Fe:Au Kondo system

)/log( KTT−

TK

TK

conductance G(T) through a singlemagnetic impurity

(e.g. an spin ½ quantum dot)

2e2/hunitary limit

10

S=1/2 Kondo in Q-dots

from L. Kouwenhoven & L. Glazman, Physics World, June 2001

at 50 mK

Mark Buitelaar and Thomas Nussbaumer

why interesting?

it‘s manymany--body physicsbody physics (all orders are relevant)

more precisely: it‘s manymany--electron physicselectron physics

in Condensed Matter Physics:Superconductivity

Luttinger liquid (non-Fermi liquids)Kondo physics

because it is complicated

Superfluidity

Kondo effect Superconductivity

∆∆

Al

Kondo effect and superconductivity are many-electron effects• can Kondo and superconductivity coexist or do they exclude each other ?

Kondo physics + superconductivity

spin 1/2 Kondo + S-leads

UEF

Kondo effect is the screeningof the spin-degree of the dot spinby exchange with electrons fromFermi-reservoirs (the leads)

normal case superconducting case

1. a gap opens in the leads

2. Cooper pairs have S=0

Hence: Kondo effect suppressed,but ....Cooper pair

A cross-over expected at kb TK ~ ∆

S = 0Energy scale : ~ ∆

Cooper pairS = 0

Energy scale : ~ kb TK

Kondo effect Superconductivity

∆∆

11

Conclusions

nanotubes can serve as a model systemmodel systemto study excitingexciting physics

Kondo physics (co-tunneling)Interplay between Kondo physics & superconductivity

www.unibas.ch/physwww.unibas.ch/phys--mesomeso group Web-page

www.nccrwww.nccr--nano.orgnano.org NCCR on Nanoscience

nanotubes may be one part of the toolboxof „nanotechnology“