chapter iii quantum transport in nanostructures · 1 nm ↔ 10.000 k ↔ 800 mev ... source:...

Chapter III

Quantum Transportin

Nanostructures

Chapt. III - 2

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Contents:

III.1 IntroductionIII.1.1 General RemarksIII.1.2 Mesoscopic SystemsIII.1.3 Characteristic Length ScalesIII.1.4 Characteristic Energy ScalesIII.1.5 Transport Regimes

III.2 Description of Electron Transport by Scattering of WavesIII.2.1 Electron Waves and WaveguidesIII.2.2 Landauer FormalismIII.2.3 Multi-terminal Conductors

III.3 Quantum Interference EffectsIII.3.1 Double Slit ExperimentIII.3.2 Two Barriers – Resonant TunnelingIII.3.3 Aharonov-Bohm EffectIII.3.4 Weak LocalizationIII.3.5 Universal Conductance Fluctuations

III.4 From Quantum Mechanics to Ohm‘s Law

III.5 Coulomb Blockade

Chapter III: Quantum Transport in Nanostructures

Chapt. III - 3

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Literature:

1. Introduction to Mesoscopic PhysicsYoseph ImryOxford University Press, Oxford (1997)

2. Electronic Transport in Mesoscopic SystemsSupriyoto DattaCambridge University Press, Cambridge (1995)

3. Mesoscopic Electronic in Solid State NanostructuresThomas HeinzelWiley VCH, Weinheim (2003)

4. Quantum TransportYuli V. Nazarov, Yaroslav M. BlanterCambridge University Press, Cambridge (2009)

Chapter III: Quantum Transport in Nanostructures

Chapt. III - 4

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• macroscopic solid state systems

- usually consideration of thermodynamic limit 𝑁 → ∞, Ω → ∞, 𝑁/Ω = 𝑐𝑜𝑛𝑠𝑡.

• what happens if system size becomes small ?

- discrete spectrum of electronic levels

- coherent motion of electrons phase memory due to lack of inelastic scattering within system size:

system size L smaller than phase coherence length 𝐿𝜙

new interference phenomena

- validity of Boltzmann theory of electronic transport and concept of resistivity ? system size L smaller than mean free path ℓ: ballistic transport

- discreteness of electric charge and magnetic flux becomes important single electron and single flux effects

- concept of impurity ensemble breaks down sample properties show „fingerprint“ of detailed arrangement of impurities

III.1 IntroductionIII.1.1 General Remarks

Chapt. III - 5

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• mesoscopic systems (coined by Van Kampen in 1981):

- system size is between microscopic (e.g. atom, molecule) andmacroscopic system (e.g. bulk solid)

- system size L is smaller than phase coherence length 𝐿𝜙 (typically in nm - µm regime)

quantum coherent phenomena become important statistical concepts no longer applicable due to smallness of system size still coupling to environment/reservoir present

(in contrast to microscopic objects such as atoms)

study of nanostructures at low temperature

mesos (Greek): between

• properties of mesoscopic systems are usually studied at low temperatures

- phase coherence length 𝐿𝜙 decreases rapidly with increasing T

𝐿 < 𝐿𝜙 can usually be satisfied only at low T

- observation of level quantization effects require 𝑘𝐵𝑇 < Δ𝐸 ≃ 1/𝐿2

III.1 IntroductionIII.1.2 Mesoscopic Systems

Chapt. III - 6

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III.6

III.1 IntroductionIII.1.2 Mesoscopic Systems – The World of Solid State Nanostructures

Chapt. III - 7

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III.1 IntroductionIII.1.2 Mesoscopic Systems – Fabrication of Single Electron Device

Chapt. III - 8

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superconducting flux quantum circuit

Chapt. III - 9

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65 nm process2005

45 nm process2007 32 nm

2009 22 nm2011

(Source: Intel Inc.)

gate length of transistors

III.1 IntroductionIII.1.2 Mesoscopic Systems – Miniaturization of Electronic Devices

Chapt. III - 10

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Fermi wave length: 𝜆𝐹 < 1 nm (for metals)

”size” of charge carrier

electron mean free path: ℓ 10 - 100 nm

distance between (elastic) scattering events

phase coherence length: 𝐿𝜑 1 mm

loss of phase memory

sample size: 𝐿, 𝑊 0.01 - 1 mm

microscopic mesoscopic macroscopic

mesoscopic regime: L < Lj (T)

from microscopic to macroscopic systems

III.1 IntroductionIII.1.3 Characteristic Length Scales

Chapt. III - 11

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10 µm

1 mm

100 µm

1 µm

100 nm

10 nm

1 nm

0.1 nm distance between atoms

Fermi wave length in metals

mean free path in polycrystalline metal films

Fermi wave length in semiconductors

size of commercial semiconductor devices

mean free path in the Quantum Hall regime

mean free path / phase coherence length inhigh mobility semiconductors at T < 4 K

phase coherence length in clean metal films


Chapt. III - 12

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elastic impurity scattering: 𝜏𝜑 → 0 or 𝛼𝜑 → 0

electron-phonon scattering: 𝜏𝜑 ≈ 𝜏𝑒−𝑝ℎ ??

electron-electron scattering: 𝜏𝜑 ≈ 𝜏𝑒−𝑒 ??

electron-impuritiy scattering

• electron wavelength: 𝝀𝑭 =𝒉

𝟐𝒎⋆𝑬𝑭=

𝟐𝝅

𝟑𝝅𝟐𝒏𝟏/𝟑 (Fermi wavelength)

collision time

effectiveness ofcollision: 0 < am < 1

• mean free path: ℓ = 𝒗𝑭 ⋅ 𝝉𝒎 𝝉𝒎−𝟏 = 𝝉𝒄

−𝟏 ⋅ 𝜶𝒎


(with internal degree of freedom, e.g. spin)

• phase relaxation length: 𝑳𝝋 = 𝒗𝑭𝝉𝝋 𝝉𝝋−𝟏 = 𝝉𝒄

−𝟏 ⋅ 𝜶𝝋 effectiveness ofcollision in destroyingphase coherence: 0 < aj < 1

ballistic

diffusive 𝑳𝝋 = 𝑫𝝉𝝋 =𝟏

𝟑𝒗𝑭

𝟐𝝉𝒎𝝉𝝋

Chapt. III - 13

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• Altshuler, Aronov, Khmelnitsky (1982):

if ħw is characteristic energy of an inelastic process (e.g. phonon energy),then the mean-squared energy spread of electron after collision is

• question: what is the effectiveness of an inelastic scattering processregarding destruction of phase coherence ?

square of energy change

number of scattering events

• at low T: e-e scattering is dominating


Δ𝐸 2 = ℏ𝜔 2𝜏𝜑

𝜏𝑐

low-frequencyexcitationsare less effective indestroying phasecoherence !!

𝜏𝜑 is time required to acquire a phase change of ≈ 2p

Δ𝜑 ≈Δ𝐸

ℏ𝜏𝜑 ≈ 2𝜋 ⇒ 𝜏𝜑 ≈

𝜏𝑐

𝜔2

1/3

Chapt. III - 14

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• Fermi wavelength:

- if 𝝀𝑭 > 𝑳𝒙, 𝑳𝒚, 𝑳𝒛 reduction of dimension by size quantization

3D 2D 1D 0D

- for metals: n ≈ 1022 - 1023 cm-3 𝜆𝐹 ≈ 1 nm

- for semiconductors: n ≈ 1016 - 1019 cm-3 𝜆𝐹 ≈ 10 - 100 nm

• electron in a box:

- level spacing:

1 nm ↔ 10.000 K ↔ 800 meV10 nm ↔ 100 K ↔ 8 meV100 nm ↔ 1 K ↔ 0.08 meV

L

• single charge/flux effects: 𝑒2

2𝐶> 𝑘𝐵𝑇,

Φ02

2𝐿> 𝑘𝐵𝑇

III.1 IntroductionIII.1.4 Characteristic Energy Scales - Size Quantization

Δ𝐸 =ℎ2

2𝑚⋆

1

𝐿

2

𝜆𝐹 =ℎ

2𝑚⋆𝐸𝐹

=2𝜋

3𝜋2𝑛 1/3

Chapt. III - 15

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quantum wire

1-dim

bulk

3-dim

superlattice

D(E)

E

quantum well

D(E)

E

2-dim

quantum dot

0-dim

const ED )(

D(E)

E

EED )(

D(E)

E

EED /1)(

D(E)

E

)()( iEE ED

III.1 IntroductionIII.1.4 Size Quantization – DOS in 3D, 2D, 1D, and 0D

Chapt. III - 16

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L

• how long does it take for an electronto diffuse through a sample of length L

(Thouless energy)• macroscopic samples: ETh << kBT• mesoscopic samples: ETh > kBT

• low T• small L• clean samples (large D)

• mean diffusion time is related to the characteristic energy (uncertainty relation)

III.1 IntroductionIII.1.4 Characteristic Energy Scales – Thouless Energy

𝐿 = 𝐷𝑡 𝑡 =𝐿2

𝐷

𝐸𝑇ℎ =ℏ

𝑡=

ℏ𝐷

𝐿2

(vF: Fermi velocity)

• ballistic transport regime (see below):

𝑡 =𝐿

𝑣𝐹𝐸𝑇ℎ =

ℏ

𝑡=

ℏ𝑣𝐹

𝐿

𝐿 <ℏ𝐷

𝑘𝐵𝑇

Chapt. III - 17

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• meaning of Thouless energy

electrons in energy interval Δ𝐸 = 𝐸𝑇ℎ stay phase coherent in sample of length L

E+DE

E

Fv

Lt

D

Lt

2

(ballistic case)

(diffusive case)

ETh

tE

i 2 pjj

tEE

i 2 pD

jj

L

pjD 2

if 𝚫𝑬 ≤ 𝑬𝑻𝒉, acquired phase shift is less than 𝟐𝝅

III.1 IntroductionIII.1.4 Characteristic Energy Scales – Thouless Energy

w t

after length L,if Δ𝐸 = 𝐸𝑇ℎ

example: D = 10³ cm²/s, L = 1 µm ETh/kB 1 K

𝐸𝑇ℎ =ℏ

𝑡=

ℏ𝐷

𝐿2

Chapt. III - 18

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macroscopic sample mesoscopic sample

diffusive: L,W >> ℓ ballistic: L,W < ℓ

quasi-ballistic: W < ℓ

incoherent: L >> Lf coherent: L < Lf

• @ 300 K: ℓ ∼ 10 nm due to e-ph scattering• @ at low T: ℓ is limited by impurity and e-e scattering sample quality matters• 𝐿𝜙 is limited by inelastic processes: e-ph and e-e scattering:

strong T dependence: 𝐿𝜙 increases with decreasing T

𝐿𝜙 ≈ 1 µm @ 1K

III.1 IntroductionIII.1.5 Transport Regimes

Chapt. III - 19

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Contents:






III.2 Quantum Transport in Nanostructures

Chapt. III - 20

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• electrons as plane waves (true only in vacuum)

wave function

probability to find electron at position 𝐫 at time 𝑡

normalization volume

wave vector

momentum

energy

tkE

ii

Vt )(exp

1),(

rkr

),( tr

2),( tr

V

k

kp

m

kE

2

22

III.2 Description of Electron Transport by Scattering of WavesIII.2.1 Electron Waves and Waveguides

Chapt. III - 21

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• electrons as Fermions:

Pauli principle (state either occupied by single electron or empty)

density of states in k-space: (factor 2 due to spin)

fraction of filled states: f

3)2(2

p

V

)(

)(

)(

1

)2(2

density current

density energy

density

3

3

k

kv

k

J

f

e

Ekd

E

p

1)(

exp

1)(

Tk

µkEf

B

knot by qm !!

• important quantities:

• f determined by statistics:


Fermi statisticsfor electrons

Chapt. III - 22

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• ballistic conductor as waveguide: 1D free motion of charge carriers, e.g. in x-directionconfinement in y,z-direction

Source: Handouts Nazarov, TU Delft


mode index n standing waveplane wave

Ψ𝑘𝑥,𝑛 𝒓, 𝑡 = 𝜙𝑛 𝑦, 𝑧 exp[𝑖 𝑘𝑥𝑥 − 𝜔𝑡 ] 𝐸𝑛 𝑘𝑥 =ℏ2𝑘𝑥

2

2𝑚⋆ + 𝐸𝑛; 𝐸𝑛 =𝜋2ℏ2

2𝑚⋆

𝑛𝑦2

𝑎2 +𝑛𝑧

2

𝑏2

EF

2𝑚

⋆𝑎

2

𝜋2ℏ

2𝐸

𝜋𝑘𝑥/𝑎

Chapt. III - 23

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• wave guide with potential barrier

II

III

I4 unknown variables:

A,B,r,tt: transmission amplitude

r: reflection amplitude

4 equations(wave function matching

at interfaces)

d

ener

gy

U0 tr

1

I II III

x


E

𝐸 =ℏ2𝑘𝑥

2

2𝑚⋆

𝐸 − 𝑈0 =ℏ2𝜅2

2𝑚⋆

Ψ 𝑥 = 1 ⋅ exp 𝑖𝑘𝑥𝑥 + 𝑟 ⋅ exp −𝑖𝑘𝑥𝑥

Ψ 𝑥 = 𝐴 ⋅ exp 𝑖𝜅𝑥 + 𝐵 ⋅ exp −𝑖𝜅𝑥

Ψ 𝑥 = 𝑡 ⋅ exp 𝑖𝑘𝑥𝑥

Chapt. III - 24

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• wave guide with potential barrier example: rectangular barrier

thickbarrier

thinbarrier

E / U0

T

classical

resulttransmission probability/coefficient:


𝑇 𝐸 ≡ 𝑡2 =1

1 +𝑘𝑥

2 − 𝜅2

2𝑘𝑥𝜅

2

sinh2 𝜅𝑑

for 𝜅𝑑 ≫ 1:sinh2(𝜅𝑑) = exp 𝜅𝑑 − exp −𝜅𝑑 2 ≃ exp 2𝜅𝑑

Chapt. III - 28

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• modelling of nanostructures as complex waveguides:

transport channels + potential barrier

rese

rvo

ir

rese

rvo

ir

Tn

scattering region

ideal waveguides

sufficient to describe transport !!

examples:(i) adiabatic quantum transport(ii) quantum point contact

• description of transport by a set of transmission coefficients Tn


Chapt. III - 29

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• example: adiabatic quantum transport constriction as a potential barrier

a = const

a = a(x)

y

x

3 open channelsT = 1

x

E n(x

)

E

closed channelsT = 0

adiabatic waveguide:variation of dimensions occurs on length scale large compared to width waveguide walls can be assumed

parallel locally


𝐸𝑛(𝑘𝑥, 𝑥) =ℏ2𝑘𝑥

2

2𝑚⋆ +𝜋2ℏ2

2𝑚⋆

𝑛𝑦2

𝑎2(𝑥)+

𝑛𝑧2

𝑏2(𝑥)

Chapt. III - 30

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• example: quantum point contactle

ft r

ese

rvo

ir

righ

tre

serv

oir

E

µlµr

closed

open

f0 1

f 01

net current:

I = Il - Ir

p

)(2

2xlxxl kfevdkTI

p

)(2

2xrxxr kfevdkTI

x

xk

Ev

1

spin

VNh

eN

eEfEfdE

eI rlrl open

2

open

ch open

2)(2

2)()(

2

2mm

p

p

eV GQ

quantized conductance !!

= µl - µr

eV


Chapt. III - 31

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• what is the meaning of the quantityopenopen

2

22 NGNh

e

V

IG Q

open

1

open

2

1

2

1

2

1

NG

Ne

hG Qc

quantum resistance25 812.807 W = 1 Klitzing

number of available modes

• for ballistic transport and reflectionless contacts there should not be any resistance!

• where does the resistance come from ?

contact resistance from the interface between the ballistic conductor and thecontact pads

resistance is denoted as contact resistance

• GQ determined by fundamental constants, does not depend on materials properties, geometry or size of nanostructure


Chapt. III - 32

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+k states (quasi Fermi level)

ballistic conductor reservoirreservoir

µ1

µ2

k

EE

E

k

k

x

µ1

µ2-k states

average value of quasi Fermi level

voltage drop at interfaces !!


Chapt. III - 33

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• quantum point contact: experimental results

Thomas et al., PRB 58, 4846 (1998)

gate voltage narrows channel reduction of Nopen

2DEG

• first experiment byVan Wees et al. Phys. Rev. Lett. 60, 848 (1988)


T = 0.6 K

Chapt. III - 34

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conduction through a single atom !

(Elke Scheer, Univ. Konstanz)


Chapt. III - 35

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• considered examples have been too simple: T only 1 (open) or 0 (closed)

• more complicated situation: ideal sample + scattering sites

transmission probabilityof the different modes

will no longer be only 0 or 1

„dusty waveguide“

0 ≤ T ≤ 1

• T represents the average probability that an electron injected at one end will be transmitted to the other end

• treatment of the situation by a scattering matrix

III.2 Description of Electron Transport by Scattering of WavesIII.2.2 Landauer Formalism

Chapt. III - 36

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r12

r22

r32

1t12

t22

t32Nl Nr

Nl + Nr incoming amplitudes 𝒂𝒍, 𝒂𝒓

Nl + Nr outgoing amplitudes 𝒃𝒍, 𝒃𝒓

scattering regionleftreservoir

rightreservoir

ab s

scattering matrixscattering matrix

Nl x Nl matrix

Nr x Nr matrix

III.2 Description of Electron Transport by Scattering of WavesIII.2.2 Landauer Formalism: scattering matrix

𝑏𝑙

𝑏𝑟=

𝑠𝑙𝑙 𝑠𝑙𝑟

𝑠𝑟𝑙 𝑠𝑟𝑟

𝑎𝑙

𝑎𝑟= 𝑟 𝑡′

𝑡 𝑟′

𝑎𝑙

𝑎𝑟

Chapt. III - 37

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• time reversal symmetry: (sym. matrix)

• electrons do not disappear:

unitary matrix


𝑅𝑛 = 𝑟† 𝑟𝑛𝑛

𝑇𝑛 = 𝑡† 𝑡𝑛𝑛

conjugate transposeof 𝑠

Chapt. III - 38

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1 0 0 1

r´t´one channel scatterer

r

l

r

l

a

a

rt

tr

b

b

'

'

'

'

)2(

i

i

i

i

eRr

eTt

eTt

eRrR = 1 - T


• example:

𝑟, 𝑡, 𝑟′, 𝑡′ are complex numberscondition of unitarity

only three independent parameters

follows from condition of unitarity

Chapt. III - 39

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𝑟⋆ 𝑡′⋆

𝑡⋆ 𝑟′⋆⋅ 𝑟 𝑡

𝑡′ 𝑟′ =𝑟 2 + 𝑡′ 2 𝑟⋆𝑡 + 𝑡′⋆𝑟′

𝑡⋆𝑟 + 𝑟′⋆𝑡′ 𝑡 2 + 𝑟′ 2= 1

• condition of unitarity: 𝑆† 𝑆 = 1= 0

= 0


𝑟⋆𝑡 + 𝑡′⋆𝑟′ = 0

𝑅𝑒−𝑖𝜃 𝑇𝑒𝑖𝜂 − 𝑇𝑒−𝑖𝜂 𝑅𝑒𝑖 2𝜂−𝜃 =

𝑇 𝑅𝑒−𝑖 𝜃−𝜂 − 𝑇 𝑅 𝑒−𝑖 𝜃−𝜂 = 0 !!

𝑡⋆𝑟 + 𝑟′⋆𝑡′ = 0

𝑇𝑒−𝑖𝜂 ⋅ 𝑅𝑒𝑖𝜃 − 𝑅𝑒−𝑖 2𝜂−𝜃 ⋅ 𝑇𝑒𝑖𝜂 =

𝑇 𝑅𝑒𝑖 𝜃−𝜂 − 𝑇 𝑅 𝑒𝑖 𝜃−𝜂 = 0 !!

(i)

(ii)

Chapt. III - 40

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• the Hermitian matrix 𝑡† 𝑡 has a set of eigenvalues Tp (for each energy E)

this gives just the number of open channels, if Tp is either 0 or 1

Landauerfomula

Rolf Wilhelm (William) Landauergeb. 4. Februar 1927 in Stuttgart† 27. April 1999 in Briarcliff Manor, N.Y.

Einstein relation ↔ Landauer formula

singlespin DOS

diffusion constant

number of modes

transmissionprobability

• expression for the current:

Landauer formula ‘mesoscopic version’ of Einstein relation


𝐼 =2𝑒

2𝜋ℏ

𝑝

.𝑛

. 𝑑𝐸 𝑇𝑝 𝐸 ⋅ 𝑓𝑙 𝐸 − 𝑓𝑟 𝐸 = 2𝐺𝑄

𝑝

𝑛

𝑇𝑝 ⋅ 𝑉

𝜎 = 2𝑒2𝑁 𝐸𝐹 𝐷 ⇔ 𝐺 = 2𝑒2

ℎ𝑁 𝑇

Chapt. III - 41

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• consider a conductor with a single conduction channel

• reservoir biased at V sends out the following number of electrons:

spin

• the chance to pass is T0, then the passed charge is just 𝑄 𝑡 = 𝑒𝑇0𝑁 𝑡

• the average current is charge per time: 𝐼 =𝑄

𝑡= 2

𝑒2

ℎ𝑇0 𝑉

• many channels: just sum up to obtain

III.2 Description of Electron Transport by Scattering of WavesIII.2.2 Landauer Formalism: plausibility

emission frequency

𝐼 = 2𝐺𝑄

𝑝

𝑛

𝑇𝑝 𝑉

Chapt. III - 42

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t (2

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only elastic scattering (electrons pass the conductor at constant energy) no interactions between electrons

• limitations:

low temperatures and low voltages short conductors (shorter then inelastic scattering length)

III.2 Description of Electron Transport by Scattering of WavesIII.2.2 Landauer Formalism: limitations and restrictions

Chapt. III - 43

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• so far discussion of two-terminal systems, extension to multi-terminal conductors?

V1

V2

V3

I2I1

I3

reservoirideal

conductor

gate

scattering region

how to express currents in terms of voltages using the Landauer formalism ?

III.2 Description of Electron Transport by Scattering of WavesIII.2.2 Additional Topic: Multi-terminal conductors

Chapt. III - 44

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V1

V2

V3

I2

I1

I3

4

• conduction matrix Gkl

• properties of conduction matrix:

current conservation:

no current, if potential is shifted by the same amount in all leads

l

l

klk VGI

0 0 k

kl

k

k GI

0l

klG


Chapt. III - 45

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) • simplest case: two-terminal conductor

• the conduction matrix only has a single independent element:

V1 V2

I1 I2

2

1

2

1

V

V

GG

GG

I

I


Chapt. III - 46

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t (2

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) • scattering matrix for multi-terminal conductors

• number of modes: N = N1+N2+N3+ .... scattering matrix is NxN matrix

• meaning of Sbm,an: propagation amplitude

from terminal a, transport channel n, to the terminal b, transport channel m

• transmission probability:

s11,12

s12,12

s13,12

1N1

abab

ba

ab

N

n

mn

N

m

N

n

nm

N

m

sTT1

2

,

111


Chapt. III - 47

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• properties of scattering matrix:

reflection back from a: San,am

transmission from b to a: San,bm

• current conservation requires

• time reversal symmetry:

(unitary matrix)

a

bbaa

n

lmmnln ss ,

*

,

)()( ,, BsBs nmmn abba


Chapt. III - 48

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• sum rules:

• example: two-terminal conductor:

• since

b

a

ab NT

b

a

ba NT

number of modes in lead b

21

22221

11211

sum

2

1

sum21)(

NN

NTT

NTT

ET

b

b

aaba

2112121111211 TTNTTTT

transmission functionis reciprocal ! time reversal symmetry

abab

baab

N

n

mn

N

m

N

n

nm

N

m

sTT1

2

,

111


Chapt. III - 49

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• multi-terminal expression of Landauer formula relates currents to voltages viaa scattering matrix

• probability for transmission from a to b:

• plausibility check:

current conservation is satisfied (follows from unitarity) no current is flowing in equilibrium, same voltage at all terminals

(follows also from unitarity)

trace includes allpossible transport channels

ab

ababab ssT ˆˆ Tr

b

b

aba )( EfTdE

e

GI Q


Chapt. III - 50

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t (2

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) • linear transport regime:

• relation to two-terminal expression: a,b = l,r

• time reversal symmetry:

ab

ababab ssGG Q

ˆˆ Tr

ttGssGG QlrlrQlr

Trˆˆ Tr

)()( BGBG baab

this is in agreement with Onsager symmetry relations !


Chapt. III - 51

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t (2

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) • three-terminal scattering element:

• scattering matrix:

• conductance matrix:

• example: V1 = V2 = V; V3 = 0

1

2

3

02/12/1

2/12/12/1

2/12/12/1

ˆBSs

ab

12/12/1

2/14/34/1

2/14/14/3

QGG

GQV

GQV/2

GQV/2

ideal beam splitter


Chapt. III - 52

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Contents:






III.3 Quantum Interference Effects

Chapt. III - 53

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coherent charge carriers 𝐿𝜙 > 𝐿

low temperatures ( 𝐿𝜙 gets large), nanoscale samples (L gets small)

interference of multiply scattered charge carriers

corrections to the classical conductance

III.3 Quantum Interference EffectsIII.3.1 Double Slit Experiment

• macroscopic and mesoscopic samples:

weak localization (WL)

• mesoscopic samples:

Aharonov-Bohm (AB) oscillations

Universal Conductance Fluctuations (UCF)

Chapt. III - 54

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III.3 Quantum Interference EffectsIII.3.1 Revision: Characteristic Length Scales (see III.1.3)

elastic impurity scattering: 𝜏𝜑 → 0 or 𝛼𝜑 → 0

electron-phonon scattering: 𝜏𝜑 ≈ 𝜏𝑒−𝑝ℎ ??

electron-electron scattering: 𝜏𝜑 ≈ 𝜏𝑒−𝑒 ??

electron-impuritiy scattering

• electron wavelength: 𝝀𝑭 =𝒉

𝟐𝒎⋆𝑬𝑭=

𝟐𝝅

𝟑𝝅𝟐𝒏𝟏/𝟑 (Fermi wavelength)

collision time

effectiveness ofcollision: 0 < am < 1

• mean free path: ℓ = 𝒗𝑭 ⋅ 𝝉𝒎 𝝉𝒎−𝟏 = 𝝉𝒄

−𝟏 ⋅ 𝜶𝒎

(with internal degree of freedom, e.g. spin)

• phase relaxation length: 𝑳𝝋 = 𝒗𝑭𝝉𝝋 𝝉𝝋−𝟏 = 𝝉𝒄

−𝟏 ⋅ 𝜶𝝋 effectiveness ofcollision in destroyingphase coherence: 0 < aj < 1

ballistic

diffusive 𝑳𝝋 = 𝑫𝝉𝝋 =𝟏

𝟑𝒗𝑭

𝟐𝝉𝒎𝝉𝝋

Chapt. III - 55

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A B

1

2

• basic quantum mechanics: double slit experiment

• probability of propagation from point A to point B:

classical interference term:quantum mechanical

P1 P2


Chapt. III - 56

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A B

1

2

interference terms may be destructive or constructive

depends on phase shift 𝜙


problem: calculate phase shift 𝜙 as a function of geometry, electric potential, magnetic field, …

Chapt. III - 57

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• phase shifts:

• energy dependence:

dynamical phase shift e.g. by potential V along path same for time-reversed path

time of flight between points𝑥1 and 𝑥2 at energy E


local wave vector at position 𝑥

− if 𝑉 𝑥 = 𝑐𝑜𝑛𝑠𝑡., then 𝑘 = 𝑐𝑜𝑛𝑠𝑡. and hence 𝜙 = 𝑘𝐿 (geometric phase)− usually, absolute value of phase is not interesting but the relative phase shift between

different paths

𝑣 =1

ℏ

𝑑𝐸

𝑑𝑘

𝑑

𝑑𝑘𝜙 =

𝑑

𝑑𝑘∫

𝑑𝜙

𝑑𝑥𝑑𝑥 =

𝑑

𝑑𝑘∫ 𝑘 𝑑𝑥 = ∫ 𝑑𝑥

Chapt. III - 58

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• magnetic field dependence:

A B

1

2

Fcanonical momentum: p = mv + qA

phase shift accumulated along the trajectory due to magnetic field:

phase shift along closed path (electron returns to the same point):

(„normal“ flux quantum)

x2 x2

(q = - e)

(in superconductors we have 𝑞𝑠 = −2𝑒and therefore Φ0 = ℎ/2𝑒)

results in phase shift 𝜙𝑚𝑎𝑔

Stokes theorem

Chapt. III - 59

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1

𝑟tot

𝑡tot

• „classical“ expectation:

(tunneling) resistances are added

product of transmission probabilities TL ∙ TR

L R

• we consider only a single conductance channel• no magnetic field

III.3 Quantum Interference EffectsIII.3.2 Two Barriers – Resonant Tunneling

21

21

21

GG

GGG

RRR

Ohm‘s law:

R1 R2

• what is the role of quantum interference?

• how do individual scattering matrices have to be combined?

Chapt. III - 60

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1

L R

a c

b d

scattering matrix

of left barrier

scattering matrix

of right barrier

propagation

between

barriers

sacquired phase

during propagation

between barriers


𝑟tot

𝑡tot

𝝋 = 𝒌 ⋅ 𝒔

𝑟𝐿 𝑡𝐿′

𝑡𝐿 𝑟𝐿′

𝑟𝑅 𝑡𝑅′

𝑡𝑅 𝑟𝑅′

𝑒𝑖𝜑 0𝑜

0𝑜 𝑒𝑖𝜑𝑜

Chapt. III - 61

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1

L R

a aeiφ

deiφ d

outgoing modes incoming modes

outgoing modes incoming modes


𝑟tot

𝑡tot

s

𝑟𝑡𝑜𝑡

𝑎0=

𝑟𝐿 𝑡𝐿′

𝑡𝐿 𝑟𝐿′

1𝑑𝑒𝑖𝜑

𝑑𝑡𝑡𝑜𝑡0

= 𝑟𝑅 𝑡𝑅

′

𝑡𝑅 𝑟𝑅′

𝑎𝑒𝑖𝜑

0

Chapt. III - 62

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)

ji

RL ett

j i

RLRL errtt 3

RLTT

RLRL RRTT

process amplitude probability

… … …

j

i

RL

RLtot

err

ttt

21 RL

RLcl

RR

TTT

1

sum of all amplitudes: sum of all probabilities:

coherent incoherent

Ttot = |ttot|²

path can be viewedas Feynman path


cos21

2

tot

RLRL

RL

RRRR

TTt

Chapt. III - 63

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t (2

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) 1

rtot

ttot

L R

a aeiφ

deiφ d

j

i

RL

RLtot

err

ttt

21

cos21)(

2

tot

RLRL

RL

RRRR

TTtET

what is the relation to the double slit experiment?

phase accumulatedduring the round trip

𝜒 = 2𝜑 = 2𝑘𝑠


s

Chapt. III - 64

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cos21)(

2

tot

RLRL

RL

RRRR

TTtET

time for round trip

• quantum result: transmission coefficient depends on energy(not the case for classical result!)

assume 𝑇𝐿 = 𝑇𝑅 = 𝑇 ≪ 1, 𝑅𝐿 = 𝑅𝑅 = 𝑅 ≃ 1

between peaks: 𝑇 𝐸 ≈ 𝑇2

peak values: 𝑇𝑚𝑎𝑥 = 1 (@ 𝜒 = 2𝜋 ⋅ 𝑛)

resonant tunneling(or Fabry-Perot resonances)

double barrier structure behaves as an optical interferometer resonant tunneling is quantum interference effect

𝜒 = 2𝜑 = 2𝑘𝑠

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

T

/ 2p

T = 0.5

T = 0.1


Chapt. III - 65

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-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

T

/ 2p

T = 0.5

T = 0.1

cos21)(

2

tot

RLRL

RL

RRRR

TTtET

(i) 𝜒 = 0:

(ii) 𝜒 = 𝜋:

𝑇𝐿,𝑅 ≪ 1, 𝑅𝐿,𝑅 → 1:

(expanding the denominator up to linear term in TL,R)

Chapt. III - 66

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• how does the transmission 𝑇(𝐸) look like close to the transmission resonances?

D = level spacing in potential well of width s

for 𝜒 ≪ 1

• after some math:

transmission assumes Lorentzian shape

interpretation in terms of a particle that moves back and forth between the two potential wells and escapes at a certain tunneling rates Γ𝐿 and Γ𝑅

with according to uncertainty relation we obtain well-known Breit-Wigner formula


energy width of transmission resonance:

Chapt. III - 67

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Φ

two trajectories enclosingmagnetic flux

phase: 𝑖𝑘𝑥with vector potential: )(xA

qkk

2,1

2,12,1 ldAe

kL

ldAe

LLk

)( 1212

FF

pF

0

2

eldA

e

e

hF 0

flux quantum:

all quantities are periodic in

Φ/Φ0 , even if there is NO

magnetic field at the

trajectories!

• quantum interference effects in multiply connected conductors, e.g. rings• phase shift due to magnetic field

(in superconductors we have 𝑞𝑠 = −2𝑒 andtherefore Φ0 = ℎ/2𝑒)

III.3 Quantum Interference EffectsIII.3.3 Aharonov-Bohm Effect

(q = -e)

Chapt. III - 68

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Φ1

r

t

b1

a1

c1

d1

b2

a2

c2

d2

1

1

1

1

1

212121

212121

21210

c

a

d

b

r

• description of Aharonov-Bohm ring by two beam splitters


Chapt. III - 69

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Φ1

r

t

b1

a1

c1

d1

b2

a2

c2

d2

212121

212121

21210

02121

212121

212121

left beam splitter right beam splitter

upper arm

lower arm

0/2

2/

FFpf

j kL

• description of Aharonov-Bohm ring by two beam splitters


Chapt. III - 70

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Φ1 t

classical

clockwise counter clockwise

extra turn

0/2

2/

FFpf

j

AB

kL

A B

interference contributions

twiceshorterperiod


universal conductance fluctuations weak localization

Chapt. III - 71

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Φ1 t

2AB

2

AB

)cos1)(2/1(2cos2sin

)²cos1)(2cos1(

fjj

fjT

2AB

AB

]cos3[ 5.0

)²cos1(

42

f

f

pj T

Aharonov-Bohm effect: flux dependent transmission

f

T

2j p/4

0/2

2/

FFpf

j

AB

kLsumming up (without closed loops):

+ + + ….


Chapt. III - 72

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Aharonov-Bohm (AB) oscillations:

• period: h/e• amplitude: 2e2/h• one channel in Landauer model

Fourier analysis shows that there are also weak oscillations with half period

higher order interferences:Altshuler-Aronov-Spivak (AAS)oscillations

• period: h/2e• exactly same traces• constructive interference for B = 0• coherent backscattering

R. Webb et al, PRL 54, 2696 (1985)


Chapt. III - 73

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• AB oscillations vanish in an ensemble of small ring (phases 2𝜋Φ/Φ0 are random)• AAS oscillations survive ensemble averaging

test of ensemble averaging:• Ag loops• area 940 x 940 nm2

• width of wires 80 nm

Umbach et al, PRL 56, 386 (1986)


Chapt. III - 74

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) Cu ring on Si, width 80 nm

F. Pierre et. a., PRL 89, 206804 (2002)

• conductance of a Cu ring in units ofe2/h, as a function of magnetic fieldat T = 100 mK.

• narrow AB oscillations ΔB ≈2.5mT are superimposed on larger andbroader universal conductancefluctuations.


Chapt. III - 75

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8.6 km

Benzene ring: ring accelerator:

Large Electron Positron Collider at CERN (Geneva)

0.5 nm

1013

AB effect: one flux quantum (h/e) through ring area:

T 5000 2

p r

ehT107

23

2

p r

eh


Chapt. III - 76

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) magnetoresistance of a Mg film (d = 8.4 nm) as a function of the magnetic field H. [Physics Reports, 107, 1 (1984), G. Bergmann]

• classically: resistance would be completely field independent because 𝜔𝑐𝜏 ≪ 1

• magnetoresistance would increase with the magnetic field, relative increase of order 𝜔𝑐𝜏 2

• classical theory could not explain the observed behavior

III.3 Quantum Interference EffectsIII.3.4 Weak Localization

Chapt. III - 77

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Weak localization:

interference of time reversed

electron paths


Chapt. III - 78

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Chapt. III - 79

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) 2

*

1

2

2

2

1

2

21 Re2 AAAAAAPAB

classicalinterference term

quantum mechanical

P1 P2

A1

A2

special trajectories:

consider now a closed loop with 1 = 2

then the amplitude A2 is just

a time reversal of A1. Hence

2

1

2*

11

2

21 4 AAAAA

• the backscattering probability is enhanced by factor 2 !!!

• this is a predecessor of localization.

jcos||2 21 AA

A B

0cos j

does averaging overmany paths destroyinterference effects

in diffusive conductor ?


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• magnetic field dependence of WL:

A1

A2

F

ff dsAe

AA

212

loss of constructive interference for:

0

42

12 F

pff

FBdsA

eAA

F = area of the enclosed loopB∙F = flux enclosed in the loop

characteristic field:

pff 212 AA

𝐹 ≃ 𝐿𝜙222

*f

eL

B


calculate phase difference of

time reversed paths:

Chapt. III - 81

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• coherent backscattering: called the weak localization (the relative number of contributing closed loops is small)

• effect is important, since it is sensitive to weak magnetic fields:

• small fields: contributions of large rings oscillate rapidly, phase difference in small rings almost unchanged

• the larger the field, the fewer loops/rings contribute to constructive backscattering

• resistance drops to classical value for large fields, if phase shift in smallest rings is about 2p

• WL has to be distinguished from strong localization (due to strong disorder)


Chapt. III - 82

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weak localizationin SiGe2-dimensionalquantum well withhole gas

• requirement: sample larger than elastic scattering length: 𝐿 > ℓ

conductivity reduced by ≈ 2e2/h for B = 0

• large B: Shubnikov de-Haas oscillations

V. Senz, PhD thesis, ETH Zürich (2002)


Chapt. III - 83

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• dependence of magnitude of WL on the coherence time 𝜏𝜙 ∼ 𝐿𝜙2 /𝐷 is known:

weak localization experiments can be used to determine tf

Senz et al., PRB 61, 5082 (2000)


Chapt. III - 84

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-8 -6 -4 -2 0 2 4 6 8

-0.5

0.0

0.5

1.0

1.5

200mK

B(T)

DG

(e2/h

)

800mK

fff

'

)(

'

2

2

'

pp

i

pp

p

p

p

i

pippp eAAAeAT

A

B

influence of magnetic field on conductance of simply connected conductor

F

random phase shifts position of scatters

becomes important

III.3 Quantum Interference EffectsIII.3.5 Universal Conductance Fluctuations

Chapt. III - 85

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• irregular conductance variations as a function of magnetic field (B), carrier

density (n), and voltage (V)

• conductance variations are symmetric with respect to B (2 probe setup)

• different in each individual sample (”magnetic fingerprint”)

• caused by irregular quantum interference

• fluctuations characterize impurity configuration

• no sample size dependence

• (border & impurity scattering)

• amplitude of conductance variations is of the order e2/h

• not noise

• theory based on ergodicity theorem

Experimental observations:


Chapt. III - 86

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phase shift of individual electron trajectories depends on

• magnetic field B• voltage V• Fermi energy EF (carrier density) • impurity (scatterer) configuration

consider an ensemble of macroscopically identical but microscopically different samples (different configurations of scattering centers)

variance of ensemble conductance:

2

2

42

mn

mn

mn

mn TTh

eGG

2

mnmn tT → complicated calculation


Chapt. III - 87

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1 2 3 4 5 6 7-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

G -

<G

> (

e2/h

)

B (T)

50 nm

Au

Au 𝑳𝝓

ℓ ≫ 𝝀𝑭

(a) (b)

(c)


T = 20 mK

Walther-Meißner-Institut

red and bluecurve takenat different days withoutwarming upthe sample

Chapt. III - 88

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H. Hegger, Ph.D. Thesis, Universität zu Köln (1997)

UCF in gold nanowire

L = 600 nm

W = 60 nm


Chapt. III - 90

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data fromHeinzel (2003)


Chapt. III - 91

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Contents:






Chapt. III - 92

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• two different points of view:

quantum transport(electron waves, scattering matrix)

classical transport(electric currents, charged particles, friction due to scattering, Ohm‘s law)

What is the bridge between these limiting cases ??


Chapt. III - 93

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• consider two conductors with transmission probabilities T1 and T2 connected in series

T1R1 T2R2

• what is the transmission probability T12 ?

• problem: if we assume T12 = T1 T2 , then we do not take into account multiple reflections

to obtain the correct result we have to add the probabilities of multiplyreflected paths


• If T12 = T1 T2 , then for a chain of scatterers we would expect the transmissionprobability to drop exponentially with the length of the chain:

no Ohm‘s law

)/exp()( 0LLLT ex1 ex2 = ex1+x2

Chapt. III - 94

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2

2

2

121 RRTT

21TT

2121 RRTT

..... .....

transmissionprobabilities

+

+

+

21

2112

1 RR

TTT

(incoherent processes)

with T1 = 1 - R1 and T2 = 1 – R2

2

2

1

1

12

12 111

T

T

T

T

T

T

additive property

two scatterers in series


Chapt. III - 95

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N scatterers in series:

T

TN

NT

NT

1

)(

)(1

TTN

TNT

)1()(

• number of scatterers in conductor of length L can be written as N = n L,where n is the linear density

with0

0)(LL

LLT

)1(0

T

TL

n

)1(

1

Tn

linear densityof scatterers

scattering probability

0)1(

1L

T

n (for T close to 1)

• L0 is of the order of the mean free path ℓ


Chapt. III - 96

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quantum conductance for N channels:

• wide conductor with M ≈ kFW/p modes:h

kT

WeTM

h

eG F2

222

p

2

2

2

1

mn

p

h

k

m

kmnv FF

F

22

2

12

p

FnvTWe

G 2

p

0LL

WG or

W

L

W

L

W

LL

GR

00

11

resistanceobeying Ohm‘s law

length independent interface resistance

• 2D density of tranverse modes:

p

0

2

0

LvneLL

WG F

≈ diffusion constant

≈ (Einstein relation)

• using yields:


0

0)(LL

LLT

Chapt. III - 97

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conclusions:

• Ohm‘s law is obtained from the expression for the quantum conductance

by summing up probabilities of multiply reflected paths

note that by summing up probabilities coherence effects are neglected(of course these are not contained in Ohm‘s law, incoherent transport)

• sample size L >> phase coherence length Lf: large phase shifts(also affected by disorder)

formally identical samples: - very different phase shifts, - but same ohmic resistance, since interference

effects average out for L >> Lf

• L < Lf: interference effects play important role deviation from Ohm‘s law different resistance for formally identical samples due to different

impurity configurations


Chapt. III - 98

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Where is the resistance ??

TMh

eG 2

2

• expression for quantum conductance:

scatterers give rise to resistance by reducing T

• example: waveguide with M modes and a single scatterer

scatterer resistance determined by properties of scatterer via its transmissivity

T

T

Me

h

Me

h

G

1

²2²2

1

„scatterer“ resistance„interface“ resistance

• remaining questions:

can we associate a resistance with the scatterer ? what about the potential drop ? Does it occur across the scatterer ? what about Joule heating ? Dissipation at the scatterer ?


Chapt. III - 101

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Contents:






Chapt. III - 102

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Charge quantization and charging energy:

• electric charge is quantized for an isolated island

Q = n e

E

C

eEEn

C

en

C

QE CC

2 ²

2

²²

2

22

[nm]

eV 10

0

2

LL

eEC

typically in meV regime for 100 nm-sized samples

[nm]

eV 13LN

E

atom

F typically in µeV regime for 100 nm-sized samples

EC

level splitting

• charging energy:

• how large is EC for island of size L (bring charge e from ∞ to island)

• level splitting in nm-sized island:


Chapt. III - 103

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Single Electron Box:

islandsource

gate: induces charge CGVG

VG

CG

tunnelingbarrier

C

+Q2-Q2+Q1-Q1

• electrostatic energy:

work done by the voltage source

• boundary conditions:

charge quantization on island

voltage drops over two capacitors

• with induced charge Q = CgVg:


Chapt. III - 104

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• electrostatic energy:

EC

constant term (independent of N)is omitted

Q = CgVg


thermalfluctuations

ground state

𝑘𝐵𝑇

Chapt. III - 107

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Islands and Barriers:

islandmetal metal

tunneling barriers (characterized by tunneling resistance R)

• weak coupling of island to metallic leads (reservoirs)

too weak: no electron transfer

too strong: strong leakage, no conservation of charge number

too little just right too much


Chapt. III - 108

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Requirements for the Observation of the Coulomb Blockade:

• thermal fluctuations must be small enough:

K 1 @ fF 12

²

Tk

eCTkE

B

BC

fF 1 at µV 802

C

eVeVEC

EC

EC

DE

DE

DE

k 25²

Wt

QC Re

hR

RCE

quantumresistance

DEbroadening of

levels

• quantum fluctuations must be small enough:

• requirement for voltage:


Chapt. III - 109

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Single Electron Transistor (SET):

gCCCC 21

ggg CVCVCVQ 2211

• energy change by adding a single electron:


V1 V2

equivalent circuit:

source drain

island

gate

n

Chapt. III - 110

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Electron Tranfer Processes:

islandn

metal1

metal2

at a given n: four differentelectron transfer processes are possible

1. from the left: n n+1: DEFL(n) = E(n+1) – E(n) – eV1

2. to the left: n n-1: DETL(n) = E(n-1) – E(n) + eV1

3. from the right: n n+1: DEFR(n) = E(n+1) – E(n) – eV2

4. to the right: n n-1: DETR(n) = E(n-1) – E(n) + eV2

e(V1 –V2)

eV1

eV2


Chapt. III - 111

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• T > 0: all transfer processes are allowed (by thermal activation)

Coulomb blockade

DEFL,TL,FR,TR (n) > 0

single electron tunneling

DEFL (n) < 0 DETR (n) < 0

DEFL (n+1) > 0

no second additional or missingelectron on island !!

Electron Tranfer Processes:

n+1

n+2

n

n-1

• T = 0: only transfer processes with DE < 0 are allowed


DETR (n-1) > 0

n+1

n

n-1

n-2

Chapt. III - 112

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electron tranfer processes at varying gate voltage:

• in which range of the gate voltage is the electron transport blocked ?

1. from the left: DEFL(0) = E(1) – E(0) – eV1 = 2EC (1/2 + Qg /e) – eV/2

2. to the left: DETL(0) = E(-1) – E(0) + eV1 = 2EC (-1/2 + Qg /e) + eV/2

3. from the right: DEFR(0) = E(1) – E(0) – eV2 = 2EC (1/2 + Qg /e) + eV/2

4. to the right: DETR(0) = E(-1) – E(0) + eV2 = 2EC (-1/2 + Qg /e) – eV/2

blockade regimes:„Coulomb diamonds“

• @ T = 0: I = 0 for

C

g

E

Ven

e

Q

421

Qg/e

eV /2EC

FR TR

TLFL

½ -½

1

-1

• assumptions: C1 = C2 V1 = -V2 = V/2

• we use


0 +1 -1

Chapt. III - 113

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Single Electron Transistor – Coulomb Diamonds:

Quelle: ETH Zürich

Vg (mV)

V (m

V)

G (µS)

blue regions of vanishing conductance correspond tothe Coulomb blockade regime (no

current flow)


Chapt. III - 114

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Current-Voltage Characteristics (IVCs):

• facts: (i) charging state is determined by n(ii) no quantum coherence between different states

n

n

TLFLL pnneI )()(

n

n

FRTRR pnneI )()( RL III : tunneling rates

• if we know pn for stationary state, we get currents

)()1()()1()()()()( 11 tpntpntpnntpdt

dnFnTnTFn

tunneling from and toisland withn electrons

tunneling toisland with

n-1 electrons

tunneling fromisland with

n+1 electrons

FRFLF TRTLT

• probability pn(t) to find system in state n at time t: given by Master equation

III.5 Coulomb Blockade(additional topic)

Chapt. III - 115

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Tunneling Rates for Single Tunnel Junction:

tunneling without CB

VGI T

tunneling rate:

eVe

G

e

I T

²

tunneling with CB

eV

energy intervalfor available final states

tunneling rate:

CT

C EeVe

GEeV

² :

0 : CEeV

eVEC

energy intervalfor available final states

eV- EC

blockade regime


Chapt. III - 116

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IVC for tunneling with Coulomb Blockade:

R = 0R sufficiently large

no quantumfluctuations

large quantumfluctuations


Chapt. III - 117

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Tunneling Rates and IVC:

• electrostatic energy changes as electron tunnels determine tunneling rate at electron energy change of DE: Fermi‘s Golden Rule

EEEfHi iftfi Dp

2

||2

)(1)( )()( || 2

)(2

21 EEfEEDEfEDfHidEE fit DDp

D

empty finalstates

occupied initialstates

2

DE

1source island

• total transition rate from conductor 1 (source) to 2 (island): tunneling rate proportional to density of states D occupation probability given by Fermi functions f(E) integration over all energies


Chapt. III - 118

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t (2

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4 -

20

15

)

• simplifying assumptions: Ht is energy independent D is energy independent

TkE

EEfEfEEfEf

B/exp1

)()()(1)(

D

DD

²||²2

2

DfHie

R tp

with

)()( 12122121 DD EEeI

DD

n

nEnEnpeI )()()( 121221211: source2: island

1²

1)(21

DD

D TkE Be

E

ReE

Fermi functions ≈ step functions

(equivalent expression for current from island to drain)


• net current

• current from current source (1) to island (2) in steady state


-4 -2 0 2 40

1

2

3

4

5

e2 R

DE / kBT

Chapt. III - 119

R. G

ross

and

A. M

arx

, © W

alth

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itu

t (2

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4 -

20

15

)

-40 -20 0 20 400

10

20

30

40

50

e

2R

DE / kBT

-4 -2 0 2 40

1

2

3

4

5

e

2R

DE / kBT

high T

low T


Chapt. III - 120

R. G

ross

and

A. M

arx

, © W

alth

er-

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ißn

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t (2

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4 -

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15

)

Quelle: ETH Zürich

Coulombdiamonds



Chapt. III - 121

R. G

ross

and

A. M

arx

, © W

alth

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ißn

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itu

t (2

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4 -

20

15

)

Quelle: lt.px.tsukuba.ac.jp

V

Imovie shows variation of IVC with varyinggate voltage

Tunneling Rates and IVC – Coulomb Staircase:

1st step in IVC

2nd step in IVC

ground

Vg

V/2

V/2

groundV/2V/2

Vg


Chapt. III - 122

R. G

ross

and

A. M

arx

, © W

alth

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ißn

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t (2

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)

Variation of the Gate Voltage – Coulomb Oscillations:

eV/2

- eV/2

eV/2

- eV/2

• gate voltage shifts up and down the energy levels of the island

• at small voltages: conductance can be varied considerably by gate voltage

Coulomb Oscillations


Chapt. III - 123

R. G

ross

and

A. M

arx

, © W

alth

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ißn

er-

Inst

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t (2

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4 -

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15

)

Coulomb Oscillations – Variation of the Gate Voltage

eV/2EC =

note:• large dI / dVG

use as ultra-sensitive electrometer


Chapt. III - 124

R. G

ross

and

A. M

arx

, © W

alth

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t (2

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)

J. Schuler, Ph.D. (WMI 2005)

experimental data onAl/AlOx /Al/AlOx /Al - SET

V (source-drain)varied for different curves

Coulomb Oscillations – Variation of the Gate Voltage


Chapt. III - 125

R. G

ross

and

A. M

arx

, © W

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Coulomb Oscillations – Effect of Single Fluctuating Background Charges

J. Schuler, Ph.D. (WMI 2005)

Al/AlOx/Al/AlOx/Al - SET

shift of I(Vg)curve due tofluctuatingbackgroundcharge


Chapt. III - 126

R. G

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and

A. M

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, © W

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)

SET fabrication:

resist mask structure resist mask structure

resist mask first layer second layer tunnel junction

ghost

structures

small

junctions

large

junction

J. Schuler, Ph.D Thesis (2005)

fabrication of sub-µm

Josephson Junctions by

shadow evaporation

technique

resist mask


Chapt. III - 127

R. G

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and

A. M

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, © W

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t (2

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)

SET fabrication – Optical Lithography

(a) (b) (c)

two-layer

e-beam resist

Si substrate

SET fabrication – Electron Beam Lithography


Chapt. III - 128

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, © W

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shadow evaporation

substrate

resist layer 1

resist layer 2

first evaporation of Aloxidation of 1st Al layer

2nd evaporation of Al

after liftoff

Chapt. III - 129

R. G

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and

A. M

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, © W

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t (2

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)

gate

source drain

tunnel

junctions

gate

islandAl Al

1µm

tunnel junctions

III.5 Coulomb BlockadeSET Fabrication

Chapt. III - 130

R. G

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and

A. M

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, © W

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)

Applications:

• sensitive electrometers: Q/Q ≈ 10-5 e

• electron pumps

transporting electrons one by one: counting of electrons

current standard: I = e f application of oscillating gate voltage

• charge Qubits

basic element for quantum information systems

Quantum oscillations in two coupled charge qubitsYu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin and J. S. TsaiNature 421, 823-826(20 February 2003)

PTB


http://www.nature.com/nature/journal/v421/n6925/full/nature01365.html

Chapt. III - 131

R. G

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A. M

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)


frequency

voltage current𝑽 = 𝑹𝑲 ⋅ 𝑰

quantum metrological

triangle

• 1990:

KI-90 = 1/e not yet availabe

chapter iii quantum transport in nanostructures · 1 nm ↔ 10.000 k ↔ 800 mev ... source:...

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