Quantum Transport through Coulomb-Blockade Systems Bj orn Kubala - - PowerPoint PPT Presentation

Quantum Transport through Coulomb-Blockade Systems

Bj¨

• rn Kubala

Institut f¨ ur Theoretische Physik III Ruhr-Universit ¨ at 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

L

Q V

g

Cg CJ QR box

Quantitatively: Ech = Q2

L

2CJ + Q2

R

2Cg + QRVg = e2 2CΣ (n − nx)2 + const. ,

• 1

1 EC

Ech

• 1

1

nx

• 2
• 1

1 2

<n>

Ech(-1) Ech(0) Ech(1)

Coulomb staircase

COQUSY06 – p.3

Single-Electron Transistor

Two contacts → transport

V V −V

t g t

Quantitatively: Ech = Q2

L

2CJ + Q2

R

2Cg + QRVg = e2 2CΣ (n − nx)2 + const. ,

• 1

1 EC

Ech

• 1

1

nx

0.5

GV /Gas

• 2
• 1

1 2

<n>

Ech(-1) Ech(0) Ech(1)

Coulomb oscillations

COQUSY06 – p.4

Simple Coupled Device

,1

V

g

V

g,2 SET1 SET2

+V/2 −V/2

Transistor measures box charge

box Q 0.5 0.5 1.0 0.0 1.0 nx 0.0

• ne step of Coulomb staircase

(Lehnert et al. PRL ’03, Sch ¨ afer et al. Physica E ’03)

Box:

V

g

charge on Cc (sawtooth) input for transistor Transistor:

V V −V

t 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 Sch ¨

• n, PRB ’94)

Hamiltonian: H = HL + HR + HI + Hch + HT = H0 + HT charge degrees of freedom separated from fermionic degrees: charging energy Hch = e2 2C ( ˆ N − nx)2 tunneling HT =

• r=R,L
• kln
• T rn

kl a† krnclne−iϕ + h.c.

• e±iϕ = |N ± 1N|

Time evolution of e.g. density matrix of charge governed by propagator Q:

Yn′

1,n1

n′

2,n2 = Trace

| {z }

fermionic d.o.f’s

» n′

2| ˜

T exp „ −i Z t0

t

dt′HT (t′)I « |n2n1|T exp „ −i Z t

t0

dt′HT (t′)I « |n′

1

–

⇒ Keldysh contour

H

T

H

TH T

H

T

t t t t’ t

1 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) =

• dEαr

0 f ± r (E +ω)f ∓(E) = ±αr ω−µr e±β(ω−µr)−1 with αr 0 = RK 4π2Rr .

−1 −1 L L R −1 −2 1 1 1 1 1 1 1 R R R L L R −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)

Σ Π +

• = (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 dω βω/2 sinh βω A(ω) ; GT =−Gas kB e Z dω (βω/2)2 sinh βω A(ω) gV = GV Gas ; gT = − e kB GT Gas perturbative expansion to 2nd order in coupling α0 gV/T = gseq

V/T + g ˜ α V/T + g ˜ ∆ V/T + gcot V/T

V ,T

L L

V ,T

R R

V

g

Thermoelectric transport: I = GV V + GT δT A(ω) = [C<(ω)−C>(ω)]/(2πi)

0.5 0.6 0.7 0.8

nx

0.1 0.2

• gT

0.5 0.6 0.7 0.8

nx

0.1 0.2 0.3 0.4 0.5

gV

gV/T gV/T

seq

gV/T

cot

gV/T

α ∼

gV/T

∆ ∼ 0.5 0.6 0.7 0.8

• 0.04

0.04 0.5 0.6 0.7 0.8

• 0.1

0.1

COQUSY06 – p.10

Sequential tunneling

gV/T = gseq

V/T + g ˜ α V/T + g ˜ ∆ V/T + gcot V/T

• sequential tunneling:

gseq

V/T = κ0

β∆0/2 sinh β∆0 with κ0 = 8 < : 1 : V β∆0/2 : T Resonances around degeneracy points ∆n = 0.

0.5 0.6 0.7 0.8

nx

0.1 0.2

• gT

0.5 0.6 0.7 0.8

nx

0.1 0.2 0.3 0.4 0.5

gV

gV/T gV/T

seq

gV/T

cot

gV/T

α ∼

gV/T

∆ ∼ 0.5 0.6 0.7 0.8

• 0.04

0.04 0.5 0.6 0.7 0.8

• 0.1

0.1

COQUSY06 – p.11

Cotunneling

gV/T = gseq

V/T + g ˜ α V/T + g ˜ ∆ V/T + gcot V/T

• standard cotunneling:

gcot

V

= α0 2π2 3 (kBT)2 „ 1 ∆0 − 1 ∆−1 «2 gcot

T

= α0 8π4 15 (kBT)3 „ 1 ∆0 − 1 ∆−1 «2 „ 1 ∆0 + 1 ∆−1 « dominant away from resonance |∆n| ≫ kBT. virtual occupation

• f unfavourable

charged state.

0.5 0.6 0.7 0.8

nx

0.1 0.2

• gT

0.5 0.6 0.7 0.8

nx

0.1 0.2 0.3 0.4 0.5

gV

gV/T gV/T

seq

gV/T

cot

gV/T

α ∼

gV/T

∆ ∼ 0.5 0.6 0.7 0.8

• 0.04

0.04 0.5 0.6 0.7 0.8

• 0.1

0.1

COQUSY06 – p.12

Cotunneling

gV/T = gseq

V/T + g ˜ α V/T + g ˜ ∆ V/T + gcot V/T

• standard cotunneling:

gcot

V

= α0 2π2 3 (kBT)2 „ 1 ∆0 − 1 ∆−1 «2 gcot

T

= α0 8π4 15 (kBT)3 „ 1 ∆0 − 1 ∆−1 «2 „ 1 ∆0 + 1 ∆−1 « gcot

V/T = κ−1∆−1∂2φ−1+κ0∆0∂2φ0+ κ0 + κ−1

2 ·φ0 − φ−1 + ∆−1∂φ−1 − ∆0∂φ0 EC κn = 8 < : 1 : V β∆n/2 : T

0.5 0.6 0.7 0.8

nx

0.1 0.2

• gT

0.5 0.6 0.7 0.8

nx

0.1 0.2 0.3 0.4 0.5

gV

gV/T gV/T

seq

gV/T

cot

gV/T

α ∼

gV/T

∆ ∼ 0.5 0.6 0.7 0.8

• 0.04

0.04 0.5 0.6 0.7 0.8

• 0.1

0.1

COQUSY06 – p.12

Renormalized sequential tunneling

gV/T = gseq

V/T + g ˜ α V/T + g ˜ ∆ V/T + gcot V/T

• Renormalization of

coupling: g ˜

α V/T = κ0

β∆0/2 sinh β∆0 » ∂ (2φ0 + φ−1 + φ1) + φ−1 − φ1 EC – energy gap: g

˜ ∆ V/T =

∂ ∂∆0 » κ0 β∆0/2 sinh β∆0 – (2φ0 − φ−1 − φ1) κ0 = 8 < : 1 : V β∆0/2 : T

A( ) ω ω

sequential tunneling but spectral density A(ω) broadened and shifted. ⇓ renormalized parameters for coupling: ˜ α charging energy gap: ˜ ∆n

COQUSY06 – p.13

Renormalization by quantum fluctuations

g ˜

α V/T = κ0

β∆0/2 sinh β∆0 » ∂ (2φ0 + φ−1 + φ1) + φ−1 − φ1 EC – ; g

˜ ∆ V/T =

∂ ∂∆0 » κ0 β∆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 Re Ψ „ iβ∆0 2π «–ff ˜ ∆ ∆0 = 1 − 2α0 » 1 + ln „ βEC π « − Re Ψ „ iβ∆0 2π «– ˜ α and ˜ ∆ decrease logarithmically by renormalization! (for lowering temperature and increasing coupling α0) ⇔ many-channel Kondo-physics

COQUSY06 – p.14

Renormalization effects on GV/T

G(α0, ∆0) = Gseq(˜ α, ˜ ∆) + cot. terms ˜ α and ˜ ∆ decrease logarithmically by renormalization:

• ˜

α ց − → peak structure reduced by quantum fluctuations.

• ˜

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

0.5 0.6 0.7 0.8

nx

0.1 0.2

• gT

0.5 0.6 0.7 0.8

nx

0.1 0.2 0.3 0.4 0.5

gV

gV/T gV/T

seq

gV/T

cot

gV/T

α ∼

gV/T

∆ ∼ 0.5 0.6 0.7 0.8

• 0.04

0.04 0.5 0.6 0.7 0.8

• 0.1

0.1

(logarithmic reduction of maximum electrical conductance (K ¨

• nig et al. PRL ’97)

experimentally observed by Joyez et al. PRL ’97)

COQUSY06 – p.15

Thermopower

V ,T

L L

V ,T

R R

V

g

Thermoelectric transport: I = GV V + GT δT Thermopower: S = − lim

δT→0

V δT

• I=0

= GT GV

S measures average energy:

S = −ε eT .

COQUSY06 – p.16

Charging energy gaps determine S

• 1

1 EC

Ech

0.5

GV /Gas

• 1

1

nx

• βEC/2

βEC/2

S/e

kBT Ech(-1) Ech(0) Ech(1)

A B C

∆−1 ∆ ∆ 0

−1

∆ ∆

∆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 0.5

n

n

X

X

−4

2 4

S / (k /e)

−2

B

lower T scales with β

−SeT = ε = gseq

V

∆0/2 + gcot

V (kBT)2/∆0

gseq

V

+ gcot

V

• Sseq = −∆0/(2eT) ∝ nx → sawtooth
• sawtooth suppressed by cotunneling

Scot = − kB e 4π2 5 1 β∆0

5 10 15 20 25

−β∆0

1 2 3

S / (kB /e)

S S

seq

S

cot (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 thermopower

Low T properties governed by renormalization:

5 10 15

−β∆0

1 2 3

S / (kB /e)

lower T S

seq+cot

S

seq

• Maximum of S

0.001 0.01

kBT / EC

7 8 9

• β∆

max

• ur pert. expansion

seq.+cot.

• charging-energy gap

0.001 0.01 0.1

kBT / EC

0.42 0.44 0.46 0.48 0.5

<ε>/∆0

seq. seq.+cot.

• u

r p e r t u r b a t i v e e x p a n s i

• n

reduced charging-energy gap ⇒ smaller ε (measures ˜ ∆) −SeT = ε = gseq

V

∆0/2 + gcot

V (kBT)2/∆0

gseq

V

+ gcot

V

crossover from gseq

V

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:

( )

1 1

( )

1

( )

1

( )

1 1

( )

1

( ) ( )

1

L

b

R

t

M M L

( , ) 1 1 ( , ) 0 1 ( , ) 0 1 ( , ) 1 1 2 ( , ) ( , ) 1

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

COQUSY06 – p.21

Algorithm for multi-island systems

e.g. parallel setup:

• charge states: (t, b)

( , ) 1

• 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

L

b

L

b

R

t

R

t

(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 cotunneling with energy exchange ⇒ noise ↔ P(E) theory

(SET1 noisy environment for SET2)

cotunneling SET 1 SET 2

COQUSY06 – p.23

Noise assisted tunneling

electrostatic

interaction

I detector generator

noise− assisted tunneling

d

∆

g

P(ng = 0) ≈ 1

2 ≈ P(ng = 1)

P(nd = 0) ≈ 1 P(nd = 1) = 0 by noise-assisted tunneling limiting cases: – small driving detector-cotunneling independent of Ig – strong driving of generator P(E) ∝ SQ

g /E2 ⇐ generator noise

Γd

01 = Γ(∆d) = α0

Z dE E 1 − e−βE P(−∆d−E) noise-assisted tunneling ∝ |Ig|

(instead of exponential suppression).

0.5 1 VG

sd/ 2 [mV]

2 4 6 8 ID [pA] full 2nd order detector cot. P(E) theory

• 0.5

0.5 1 VG

sd/ 2 [mV]

0.1 0.2 ID [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. K ¨
• nig,
• Phys. Rev. B 73, 165316 (2006).
• B. Kubala and J. K ¨
• nig,
• Phys. Rev. B 73, 195316 (2006).

COQUSY06 – p.25