Quantum Transport throughCoulomb-Blockade Systems

Bjorn Kubala

Institut fur Theoretische Physik IIIRuhr-Universitat Bochum

COQUSY06 – p.1

Overview• Motivation

• Single-electron box/transistor• Coupled single-electron devices

• Model and Technique• Real-time diagrammatics

• Thermoelectric transport• Thermal and electrical conductance• Quantum fluctuation effects on thermopower

• Multi-island systems• Diagrammatics for complex systems• New tunneling processes

COQUSY06 – p.2

Single-Electron Box

Gate attracts charge to island.Tunnel barrier → quantized charge

LQ

Vg

Cg

CJ

QR

box

Quantitatively:

Ech =Q2

L

2CJ

+Q2

R

2Cg

+ QRVg

=e2

2CΣ(n − nx)2 + const. ,

-1 0 1

EC

Ech

-1 0 1n

x

-2

-1

0

1

2

<n>

Ech

(-1)

Ech

(0)

Ech

(1)

Coulomb staircase

COQUSY06 – p.3

Single-Electron Transistor

Two contacts → transport

V

V

−Vt

g

t

Quantitatively:

Ech =Q2

L

2CJ

+Q2

R

2Cg

+ QRVg

=e2

2CΣ(n − nx)2 + const. ,

-1 0 1

EC

Ech

-1 0 1n

x

0.5

GV

/Gas

-2

-1

0

1

2

<n>

Ech

(-1)

Ech

(0)

Ech

(1)

Coulomb oscillationsCOQUSY06 – p.4

Simple Coupled Device

,1Vg

Vg,2

SET1

SET2+V/2 −V/2

Transistor measuresbox charge

box

Q

0.5

0.5

1.0

0.01.0nx

0.0

one step of Coulomb staircase

(Lehnert et al. PRL ’03,Schafer et al. Physica E ’03)

Box:

Vg

charge on Cc (sawtooth)input for transistor

Transistor:

V

V

−Vt

g

t

COQUSY06 – p.5

Overview• Motivation

• Single-electron box/transistor• Coupled single-electron devices

• Model and Technique• Real-time diagrammatics

• Thermoelectric transport

• Thermal and electrical conductance

• Quantum fluctuation effects on thermopower

• Multi-island systems

• Diagrammatics for complex systems

• New tunneling processes

COQUSY06 – p.6

Real-time diagrammatics for an SET(Schoeller and Schon, PRB ’94)

Hamiltonian: H = HL + HR + HI + Hch + HT = H0 + HT

charge degrees of freedom separated from fermionic degrees:

charging energy

Hch =e2

2C(N − nx)2

tunneling

HT =∑

r=R,L

∑

kln

(

T rnkl a†

krnclne−iϕ + h.c.)

e±iϕ = |N ± 1〉〈N |Time evolution of e.g. density matrix of charge governed by propagatorQ

:

Yn′1,n1

n′2,n2

= Trace| {z }

fermionic d.o.f’s

»

〈n′2|T exp

„

−i

Z t0

tdt′HT (t′)I

«

|n2〉〈n1|T exp

„

−i

Z t

t0

dt′HT (t′)I

«

|n′1〉

–

⇒ Keldysh contour

H T H T H T

H T

t t tt’

t1 2 3 4

8−

A(t)

t

^

COQUSY06 – p.7

Dyson-equation

Integrating out reservoirs/ contracting tunnel vertices ⇒each contraction ⇔ golden-rule rate:αr±(ω) =

∫

dEαr0 f±

r (E+ω)f∓(E) = ±αr0

ω−µr

e±β(ω−µr)−1with αr

0 = RK

4π2Rr.

0 −1

0 −1

L L

R−1 −2

1 0 0 1

1 1 00

01 0 1

1

R RR L LR

−1

|−1><−1|

diagram with sequential, cotunneling and 3rd order processes

Write full propagator∏

as Dyson equation:

Π

n’1

n 2

n1

n’2

n’1

n 2

n1

n’2

Π(0)

n

=

’’1

n’’2

n’1

n 2

n1

n’2

Π(0)ΣΠ+

∏

=∏(0)

+∏ ∑ ∏(0)

with free propagator (w/o tunneling)Q(0)

to calculate:

self-energy∑

COQUSY06 – p.8

Overview• Motivation

• Single-electron box/transistor• Coupled single-electron devices

• Model and Technique• Real-time diagrammatics

• Thermoelectric transport• Thermal and electrical conductance• Quantum fluctuation effects on thermopower

• Multi-island systems

• Diagrammatics for complex systems

• New tunneling processes

COQUSY06 – p.9

Electrical and thermal conductance

GV =Gas

Z

dωβω/2

sinh βωA(ω) ; GT =−Gas

kB

e

Z

dω(βω/2)2

sinh βωA(ω)

gV =GV

Gas; gT = −

e

kB

GT

Gas

perturbative expansion to 2nd order in coupling α0

gV/T = gseqV/T

+ gαV/T

+ g∆V/T

+ gcotV/T

V ,TL L V ,TR R

Vg

Thermoelectric transport:

I = GV V + GT δT

A(ω) = [C<(ω)−C>(ω)]/(2πi)

0.5 0.6 0.7 0.8nx

0

0.1

0.2

-gT

0.5 0.6 0.7 0.8nx

0

0.1

0.2

0.3

0.4

0.5

gV g

V/T

gV/T

seq

gV/T

cot

gV/T

α∼

gV/T∆∼

0.5 0.6 0.7 0.8

-0.04

0

0.04

0.5 0.6 0.7 0.8-0.1

0

0.1

COQUSY06 – p.10

Sequential tunneling

gV/T = gseqV/T + gα

V/T + g∆V/T + gcot

V/T

• sequential tunneling:

gseqV/T

= κ0β∆0/2

sinh β∆0with κ0 =

8<

:

1 : V

β∆0/2 : T

Resonances around degeneracy points ∆n = 0.

0.5 0.6 0.7 0.8nx

0

0.1

0.2

-gT

0.5 0.6 0.7 0.8nx

0

0.1

0.2

0.3

0.4

0.5

gV g

V/T

gV/T

seq

gV/T

cot

gV/T

α∼

gV/T∆∼

0.5 0.6 0.7 0.8

-0.04

0

0.04

0.5 0.6 0.7 0.8-0.1

0

0.1

COQUSY06 – p.11

Cotunneling

gV/T = gseqV/T + gα

V/T + g∆V/T + gcot

V/T

• standard cotunneling:

gcotV = α0

2π2

3(kBT )2

„1

∆0−

1

∆−1

«2

gcotT = α0

8π4

15(kBT )3

„1

∆0−

1

∆−1

«2 „1

∆0+

1

∆−1

«

dominant away from resonance |∆n| � kBT .

gcotV/T = κ−1∆−1∂2φ−1+κ0∆0∂2φ0+

κ0 + κ−1

2·φ0 − φ−1 + ∆−1∂φ−1 − ∆0∂φ0

EC

virtual occupationof unfavourablecharged state.

κn =

8<

:

1 : V

β∆n/2 : T

0.5 0.6 0.7 0.8nx

0

0.1

0.2

-gT

0.5 0.6 0.7 0.8nx

0

0.1

0.2

0.3

0.4

0.5

gV g

V/T

gV/T

seq

gV/T

cot

gV/T

α∼

gV/T∆∼

0.5 0.6 0.7 0.8

-0.04

0

0.04

0.5 0.6 0.7 0.8-0.1

0

0.1

COQUSY06 – p.12

Cotunneling

gV/T = gseqV/T + gα

V/T + g∆V/T + gcot

V/T

• standard cotunneling:

gcotV = α0

2π2

3(kBT )2

„1

∆0−

1

∆−1

«2

gcotT = α0

8π4

15(kBT )3

„1

∆0−

1

∆−1

«2 „1

∆0+

1

∆−1

«

dominant away from resonance |∆n| � kBT .

gcotV/T = κ−1∆−1∂2φ−1+κ0∆0∂2φ0+

κ0 + κ−1

2·φ0 − φ−1 + ∆−1∂φ−1 − ∆0∂φ0

EC

virtual occupationof unfavourablecharged state.

κn =

8<

:

1 : V

β∆n/2 : T

0.5 0.6 0.7 0.8nx

0

0.1

0.2

-gT

0.5 0.6 0.7 0.8nx

0

0.1

0.2

0.3

0.4

0.5

gV g

V/T

gV/T

seq

gV/T

cot

gV/T

α∼

gV/T∆∼

0.5 0.6 0.7 0.8

-0.04

0

0.04

0.5 0.6 0.7 0.8-0.1

0

0.1

COQUSY06 – p.12

Renormalized sequential tunneling

gV/T = gseqV/T + gα

V/T + g∆V/T + gcot

V/T

• Renormalization of

coupling:

gαV/T = κ0

β∆0/2

sinh β∆0

»

∂ (2φ0 + φ−1 + φ1) +φ−1 − φ1

EC

–

energy gap:

g∆V/T =

∂

∂∆0

»

κ0β∆0/2

sinh β∆0

–

(2φ0 − φ−1 − φ1)

κ0 =

8<

:

1 : V

β∆0/2 : T

A( )ω ω

sequential tunneling butspectral density A(ω)

broadened and shifted.

⇓

renormalized parameters

for coupling: α

charging energy gap: ∆n

COQUSY06 – p.13

Renormalization by quantum fluctuations

gαV/T = κ0

β∆0/2

sinh β∆0

»

∂ (2φ0 + φ−1 + φ1) +φ−1 − φ1

EC

–

; g∆V/T =

∂

∂∆0

»

κ0β∆0/2

sinh β∆0

–

(2φ0 − φ−1 − φ1)

Quantum fluctuations ⇒ renormalization of system parameters

G(α0, ∆0) = Gseq(α, ∆) + cot. terms

expand: Gseq(α, ∆) = α∂Gseq(α0, ∆0)

∂α0+

“

∆ − ∆0

”∂Gseq(α0, ∆0)

∂∆0

renormalization of parameters (perturbative in α0):α

α0= 1 − 2α0

−1 + ln

„βEC

π

«

− ∂∆0

»

∆0 ReΨ

„

iβ∆0

2π

«–ff

∆

∆0= 1 − 2α0

»

1 + ln

„βEC

π

«

− ReΨ

„

iβ∆0

2π

«–

α and ∆ decrease logarithmically by renormalization!(for lowering temperature and increasing coupling α0)

⇔many-channelKondo-physics

COQUSY06 – p.14

Renormalization effects on GV/T

G(α0, ∆0) = Gseq(α, ∆) + cot. terms

α and ∆ decrease logarithmically by renormalization:• α ↘ −→ peak structure reduced by quantum fluctuations.

• ∆ ↘ −→ closer to resonance; peak broadened by quantum fluct.

0.5 0.6 0.7 0.8nx

0

0.1

0.2

-gT

0.5 0.6 0.7 0.8nx

0

0.1

0.2

0.3

0.4

0.5

gV g

V/T

gV/T

seq

gV/T

cot

gV/T

α∼

gV/T∆∼

0.5 0.6 0.7 0.8

-0.04

0

0.04

0.5 0.6 0.7 0.8-0.1

0

0.1

(logarithmic reduction of maximum electrical conductance (Konig et al. PRL ’97)experimentally observed by Joyez et al. PRL ’97)

COQUSY06 – p.15

Thermopower

V ,TL L V ,TR R

Vg

Thermoelectric transport:

I = GV V + GT δT

Thermopower:

S = − limδT→0

V

δT

∣

∣

∣

∣

I=0

=GT

GV

S measures average energy:

S = −〈ε〉

eT.

COQUSY06 – p.16

Charging energy gaps determine S

-1 0 1

EC

Ech

0.5

GV

/Gas

-1 0 1n

x

-βEC/2

0

βEC/2

S/e

kBT

Ech

(-1)

Ech

(0)

Ech

(1)

AB

C0

∆−1

∆

∆ 0

−1∆

0∆

∆n = Ech(n + 1) − Ech(n)

= EC [1 + 2(n − nx)]

A) at resonance:

• peak in GV

• S ∝ 〈ε〉 = 0

ε ≷ EF cancels

B) sequential

• GV decays off resonance

• S ∝ 〈ε〉 ∝ ∆0 ∝ nx

C) nx = 0 ⇔ ∆−1 = −∆0

• two levels add for GV

• two levels cancel for S

COQUSY06 – p.17

Sequential and cotunneling only

−0.5 0 0.5n

n

X

X

−4

0

2

4

S / (

k /e

)

−2

B

lower T

scales with β

−SeT = 〈ε〉 =gseq

V ∆0/2 + gcotV (kBT )2/∆0

gseqV + gcot

V

• Sseq = −∆0/(2eT ) ∝ nx → sawtooth

• sawtooth suppressed by cotunneling

Scot = −kB

e

4π2

5

1

β∆0

0 5 10 15 20 25−β∆0

0

1

2

3

S / (

k B/e

) S

Sseq

Scot

(Turek and Matveev, PRB ’02)

‘universal low-T behavior’

Sseq+cot = S(β∆0)

How do quantum fluctuations change this picture?COQUSY06 – p.18

Renormalization effects on thermopowerLow T properties governed by renormalization:

0 5 10 15−β∆0

0

1

2

3

S / (

k B/e

)

lower T

Sseq+cot

Sseq • Maximum of S

0.001 0.01k

BT / E

C

7

8

9

-β∆m

ax our pert. expansion

seq.+cot.

• charging-energy gap

0.001 0.01 0.1k

BT / E

C

0.42

0.44

0.46

0.48

0.5

<ε>

/∆0

seq.

seq.+cot.

our perturbative expansion

reduced charging-energy gap

⇒ smaller 〈ε〉 (measures ∆)

−SeT = 〈ε〉 =gseq

V ∆0/2 + gcotV (kBT )2/∆0

gseqV + gcot

V

crossover from gseqV to gcot

V ⇒ maximum position

system closer to resonance ⇒ crossover for larger ∆max

⇒ Further support for renormalization picture!

COQUSY06 – p.19

Overview• Motivation

• Single-electron box/transistor• Coupled single-electron devices

• Model and Technique• Real-time diagrammatics

• Thermoelectric transport• Thermal and electrical conductance• Quantum fluctuation effects on thermopower

• Multi-island systems• Diagrammatics for complex systems• New tunneling processes

COQUSY06 – p.20

Multi-island geometries:

parallel setup: series setup:

( )11

( )10

( )01

( )11 ( )0

1 ( )00

( )01

LbRt

M

ML

( , )1 1 ( , )0 1

( , )0 1( , )1102( , ) ( , )01

Trivial changes allow application to different setups !(no changes in calculation of diagrams)

COQUSY06 – p.21

Algorithm for multi-island systemse.g. parallel setup:

• charge states: (t,b) ( , )01• Electrostatics :

Ech(t, b) = Et(t − tx)2 + Eb(b − bx)2 + Ecoupl.(t − tx)(b − bx)

• generate all diagrams:

- start from any charge state- choose vertex positions- choose tunnel junctions and

directions of lines- connect vertices- change charge states

Lb

Lb

Rt

Rt

(t, b−1)

(t−1, b)

(t−1, b)

(t, b−1)(t, b−1)

(t, b−1)(t−1, b−1)

(t, b)

(t, b)

(t, b)

(t, b)

(t, b)

(t, b)

(t, b)

(t, b)

• Calculate value of diagram, contributing to Σ(t,b)→(t,b−1)

simple rules but plenty of diagrams (in 2nd order 211 per charge state)

⇒ Automatically generate and calculate all diagrams !COQUSY06 – p.22

New processes in coupled SETs

-before:

• study building blocks separately

• link blocks together:e.g., average charge of one SET

→ input for other SET

full treatment:

• complete 2nd order theory

• quantum fluctuations

• backaction of SET on box

• new class of processes:

double-island cotunnelingwith energy exchange

⇒ noise ↔ P (E) theory(SET1 noisy environment for SET2)

cotunneling

SET 1

SET 2

COQUSY06 – p.23

Noise assisted tunneling

electrostaticinteraction

I

detector

generator

noise−assistedtunneling

d∆

g

P (ng = 0) ≈ 12≈ P (ng = 1)

P (nd = 0) ≈ 1

P (nd = 1) 6= 0by noise-assisted tunneling

limiting cases:

– small drivingdetector-cotunneling independent of Ig

– strong driving of generator

P (E) ∝ SQg /E2 ⇐ generator noise

Γd01 = Γ(∆d) = α0

Z

dEE

1 − e−βEP (−∆d−E)

noise-assisted tunneling ∝ |Ig|(instead of exponential suppression).

0 0.5 1V

G

sd/ 2 [mV]

0

2

4

6

8

I D [

pA]

full 2nd orderdetector cot.P(E) theory

-0.5 0 0.5 1V

G

sd/ 2 [mV]

0

0.1

0.2

I D [

pA]

COQUSY06 – p.24

Conclusions• Higher-order tunneling effects beside cotunneling• Thermoelectric properties of an SET

• Quantum fluctuations renormalize system parameters• Electrical and thermal conductance renormalized similarly• Thermopower measures average energy ⇒

logarithmic (Kondo-like) reduction of charging-energy gap

• General scheme to analyze multi-island systems• All 2nd order diagrams computed automatically• Detailed study of mutual influence of coupled SETs possible,

backaction and quantum fluctuations• New tunneling processes exchange energy between islands

B. Kubala, G. Johansson, and J. Konig,Phys. Rev. B 73, 165316 (2006).

B. Kubala and J. Konig,Phys. Rev. B 73, 195316 (2006).

COQUSY06 – p.25