adiabatic pumping through a quantum dot with coulomb ...qpump/koenig.pdf · introduction: quantum...

Adiabatic Pumping through a Quantum Dotwith Coulomb Interaction

Jurgen Konig

[email protected]

Institut fur Theoretische Physik III

Ruhr-Universitat Bochum

– p.1

Collaborators & Publications

Collaborators:

Janine Splettstoesser (Bochum, Pisa)

Michele Governale (Bochum)

Rosario Fazio (Trieste, Pisa)

Fabio Taddei (Pisa)

Nina Winkler (Bochum)

Publications:

Splettstoesser, Governale, JK, Fazio, PRL 95, 246803 (2005)

Splettstoesser, Governale, JK, Fazio, PRB 74, 085305 (2006)

Splettstoesser, Governale, JK, Taddei, Fazio, cm/0612257

Winkler, Governale, JK, in preparation– p.2

Outline

Introduction

adiabatic pumping and Coulomb interaction

Nonequilibrium Green’s function approach

(Splettstoesser, Governale, JK, Fazio, PRL ’05)

relate charge to instantaneous Green’s functions

Diagrammatic transport theory

(Splettstoesser, Governale, JK, Fazio, PRB ’06)

systematic perturbation theory in tunnel coupling

Pumping through a metallic island

(Winkler, Governale, JK, in preparation)

Pumping in proximity of a superconducting lead

(Splettstoesser, Governale, JK, Taddei, Fazio, preprint ’06)

– p.3

Introduction: Mesoscopic Classical Pump

pumping through double-island system

n1 n2

U1

C1 C2

C C C

U2

U1

U2

Coulomb blockade is dominant

Q = 1 electron/cycle

H. Pothier et al., EPL 17,249 (1992)

– p.4

Introduction: Quantum Pump

Thouless, PRB 27, 6083 (1983)

µµ conductor

Mesoscopic

X

X

2

1

X

X

2

1

phase-coherent mesoscopic conductor

periodic variation of some properties X1(t), X2(t)

adiabatic pumping: Ω = pumping frequency ≪ 1lifetime

pumped charge Q = Ipump/Ω is geometric

weak, sinusoidal pumping:

δX1(t) = δX1 sin(Ωt), δX2(t) = δX2 sin(Ωt − ϕ)

Q ∝ δX1δX2 sin ϕ

– p.5

Introduction: Scattering Formalism

scattering matrix S(X1(t),X2(t))

b1

b2

= S

a1

a2

Scattering

region

1a

1b

2a

2b

emissivity: charge emitted by lead m in response to X

dnm

dX=

1

2π

∑

α∈m,β

Im

∂Sα,β

∂XS∗

α,β

Buttiker, Thomas, Pretre, Z. Phys. B 94, 133, (1994)

Brouwer’s formula

Qm =e

π

∫

AdX1dX2

∑

α∈m,β

Im

∂S∗α,β

∂X1

∂Sα,β

∂X2

Brouwer, PRB 58, R10135 (1998) – p.6

Introduction: Scattering Formalism

Brouwer’s formula

Qm =e

π

∫

AdX1dX2

∑

α∈m,β

Im

∂S∗α,β

∂X1

∂Sα,β

∂X2

Brouwer, PRB 58, R10135 (1998)

applicable when scattering matrix is known

(non-interacting systems)

what to do in presence of (strong) interaction?

– p.6

Introduction: Quantum Dot

ε

LV VR

ε + U

E = 0 ǫ ǫ 2ǫ + U

model parameters:

single level with energy ǫ + ∆ǫ(t)

charging energy U

tunnel-coupling strengths ΓL,R(t) = 2πρL,R|VL,R(t)|2

temperature T

pumping parameters: ∆ǫ(t),ΓL(t),ΓR(t)

– p.7

Outline

Introduction












– p.8

Nonequilibrium Green’s Function Approach

start: current formula for full time dependence

method: adiabatic expansion of Green’s functions

goal: current in terms of instantaneous Green functions

– p.9



Jauho, Wingreen, Meir, Phys. Rev. B, 50, 5528 (1994)

JL = −et∫

−∞dt′

∫

dωπ

Im

e−iω(t′−t)ΓL (ω, t′, t) [G< (t, t′) + f (ω) Gr (t, t′)]

with G<,r (t, t′): time-dependent dot Green’s functions



– p.9




Dyson equation:

G (t, t′) = g (t, t′) +∫

dt1dt2G (t, t1) Σ (t1, t2)g (t2, t′)

self energy: Σ (t1, t2)= Σ(t1, t2, H(τ)τ∈[t1,t2])

linearize time dependence:

H(τ) → H (t0) + (τ − t0) H (t0)

average-time approximation: τ → t1+t22

adiabatic expansion:

Σ(

t1, t2, H(τ)τ∈[t1,t2]

)

→

Σ (t1, t2, H(t0)) +(

t1+t22 − t0

) ∂Σ(t1,t2,H(t0))∂t0


– p.9






Sela and Oreg, PRL 96, 166802 (2006)

JL (t) = − eπ

∫

dω(

− ∂f∂ω

)

Re

ddt

[ΓL (t) Gr0 (ω, t)] (Gr

0 (ω, t))−1 Ga0 (ω, t)

+ vertex-correction terms

Gr,a0 (ω, t): instantaneous Green’s functions

average-time approximation exact if

U = 0

U → ∞ and linear order in Γ

T = 0 (U arbitrary)– p.9

Results for Single-Level Quantum Dot


pumped charge per period: Q =∫ T0 JL (t) dt

pumping parameters: ΓL,R (t) = Γ2 + ∆ΓL,R (t)

weak pumping: Q ∝ η =∫ T0

˙∆ΓL(t)∆ΓR(t)dt

Q = −eηΓ

πΓLΓR

∫

dω

(

−∂f

∂ω

)

∂δ(ω)

∂ΓT (ω)

phase of Green’s function Gr0(ω) = |Gr

0(ω)| exp[iδ(ω)]

transmission probability T (ω) = 2ΓLΓR/Γ · Im[Gr0(ω)]

– p.10



high-temperature regime T > TK

use equations-of-motions method for Gr0(ω)

-10 0 10

ε/Γ

0

0.01Q

.Γ2 /e

η

U infinite, (∆ΓL, ∆Γ

R )

U=0, (∆ΓL, ∆Γ

R)

interaction changes pumping characteristics

– p.10



high-temperature regime T > TK

use equations-of-motions method for Gr0(ω)

-10 0 10

ε/Γ

0

0.01

0.02

0.03

0.04

0.05Q

.Γ2 /e

η

U infinite, (∆ΓL, ∆Γ

R )

U=0, (∆ΓL, ∆Γ

R)

U=0, (∆ΓL, ∆ε)

pumping by time-dependent level renormalization

– p.10



low-temperature regime T ≪ TK

T = 0 result (exact): Q = −4eη

Γ

∂ ¯〈n〉

∂Γsin2

(

π ¯〈n〉)

use slave-boson mean-field method for ¯〈n〉

-4 -3 -2 -1 0ε/Γ

0

0.5

1

Q . Γ

2 /eη

– p.10

Outline

Introduction












– p.11

Diagrammatic Transport Theory

J.K., Schoeller, Schon, PRL ’96; J.K., Schmid, Schoeller, Schon, PRB ’96

ingredients:

– quantum dot with strong interaction

– tunnel coupling to noninteracting leads

– nonequilibrium due to finite bias voltage

– finite temperature

Hamiltonian:

H = Hleads + Hdot + HT ≡ H0 + HT

general idea:

– integrate out leads −→ reduced density matrix for dot

– expand in tunnel coupling

– treat interaction exactly

– work on Keldysh contour– p.12

Derivation of Diagrams

a) goal: calculate expectation values

〈A(t)〉 = tr[Aρ(t)] = tr[eiHtAe−iHtρ0]

ρ0

A

exp[−iHt]

exp[iHt]

time

– p.13


b) go to interaction picture H = H0 + HT

〈A(t)〉 = tr[T ei∫

dt′HT(t′)ATe−i∫

dt′HT(t′)ρ0]

ρ0

A

T

exp[ −i dt’ H (t’) ]T

exp[ i dt’ H (t’) ]

– p.13


c) expand in tunneling

〈A(t)〉 = tr[∑

n(−i)n∫

dtiTKHT(t1) . . . HT(tn)Aρ0]

H T H T H T H T H T H T H T

H TH TH TH TH TH TH TH T

TH

ρ0

A

– p.13


d) integrate out leads (noninteracting)

Wick’s theorem → contract vertices in pairs

ρ0

A

0 00

0

0

0 000

L R L R L

L R

L

reduced density matrix for quantum dot

charging energy treated exactly

– p.13

Kinetic Equation for Probabilities

= p(t )0

t

+ p(t )1

0p (t) = p(t ) ....χ

χ

Σ Σ Σ Σ

tΣ

χ

Master equation: pχ(t) =∑

χ′

t∫

t0

dt′Σχχ′(t, t′)pχ′(t′)

transition rate: Σχχ′ = Σχ χ

χχ’

’χ

= sum of irreducible diagrams

– p.14

Calculation of Current

IL(t) = e∑

s

∑

χχ′

t∫

t0

dt′sΣsLχχ′(t, t′)pχ′(t′)

pχ′ : probability for state χ′ on dot

Σχχ′ : tunnel rate from χ′ to χ

ΣsLχχ′ : rate from χ′ to χ with s electrons entering from left

– p.15

Adiabatic Expansion of Diagrams


starting point:d

dtp (t) =

∫ t

−∞dt′ W

(

t, t′)

p(

t′)

expand X(τ) → X(t) + (τ − t) ddτ

X(τ)|τ=t about final time

adiabatic expansion (Ω ≪ Γ) for

kernels: W(t, t′) → W(i)t (t − t′) + W

(a)t (t − t′)

probabilities: p(t) → p(i)t + p

(a)t

perturbation expansion in tunnel coupling Γ

instantaneous kernel: W(i)t → W

(i,1)t + W

(i,2)t + . . .

adiabatic correction: W(a)t → W

(a,1)t + . . .

instantaneous probability: p(i)t → p

(i,0)t + p

(i,1)t + . . .

adiabatic correction: p(a)t → p

(a,−1)t + p

(a,0)t + . . .

– p.16

Results: Current in Low-Order Tunneling


current in zeroth-order in Γ

I(0)L (t) = −e

ΓL

Γ

d

dt〈n〉(i,0)

ε

nonzero only if level position is time dependent

current in first order in Γ

I(1)L (t) = −e

d

dt

(

〈n〉(i,broad,L))

+ΓL

Γ

d

dt〈n〉(i,ren)

only level renormalization contributes to pumping

– p.17

Results: Pumping with Level & One Barrier


lowest-order contribution

Q(0)ΓL,ǫ = e

ΓR

Γ2η1

d

dǫ¯〈n〉

(i,0)

η1 =∫ T0

∂∆ΓL

∂t∆ǫdt: area enclosed in parameter space

-10 0 10

ε/Γ0

0.025

0.1

Q Γ L

,ε[e

η 1/Γ2 ]

U=0U=4ΓU=8ΓU=50Γ

– p.18

Results: Pumping with Two Barriers


lowest-order contribution due to level renormalization

ǫ → ǫ + σ (ǫ,Γ, U) with σ ∝ Γ ln UmaxkBT,ǫ

Q(1)ΓL,ΓR

= eη2

Γ2

d

dǫ

(

¯〈n〉(i,0)

)

σ(

ǫ, Γ, U)

η2 =∫ T0

∂∆ΓL

∂t∆ΓRdt: area enclosed in parameter space

-40 -20 0ε/Γ

-0.05

-0.03

0

0.03

0.05

QΓ L

,ΓR[e

η 2/Γ2 ]

U=0.1ΓU=4ΓU=20ΓU=30Γ pure interaction effect

access to level renormalization

different sign for the two peaks

– p.19

Outline

Introduction












– p.20

Results: Current in Low-Order Tunneling

Winkler, Governale, JK, in preparation

current in zeroth-order in α0

I(0)L (t) = −e

αL0

α0

d

dt〈n〉(i,0)

α 0L

∆

α 0R

nonzero only if charging gap ∆ is time dependent

current in first order in α0

I(1)L (t) = −e

d

dt

(

〈n〉(i,broad,L))

+αL

0

α0

d

dt〈n〉(i,ren)

only gap renormalization contributes to pumping

– p.21

Results: Pumping with Two Barriers

Winkler, Governale, JK, in preparation

lowest-order contribution due to gap renormalization

∆ → ∆ + σ (∆,Γ, EC) with σ ∝ −∆ln EC

maxkBT,∆

Q(1)

αL

0,αR

0

= eη2

α20

d

d∆

(

¯〈n〉(i,0)

)

σ(

∆, Γ, EC

)

η2 =∫ T0

∂∆αL

0

∂t∆αR

0 dt: area enclosed in parameter space

pure interaction effect

access to gap renormalization

– p.22

Outline

Introduction












– p.23

Pumping in Proximity of Superconductor


Γ ΓSN

dotN N/S

transport takes place via Andreev reflection

compare N-dot-N to N-dot-S system

study U = 0 and U = ∞ limit

– p.24


current in terms of instantaneous Green’s functions:

JL (t) = −e

2π

∫

dω

(

−∂f

∂ω

)

Re

Tr

[

τ3d

dt

[

ΓL (t) Gr0 (ω, t)

] (

Gr0 (ω, t)

)−1Ga

0 (ω, t)

]

average-time approximation neglects

JcorrL =

e

2π

∫

dω

∫

dω′

πRe

Tr[

Ga0(ω

′, t)τ3ΓL (ω, t) Gr0(ω

′, t)

Σcorr,< (ω′, t) + f (ω′)(

Σcorr,r (ω′, t) − Σcorr,a (ω′, t))

ω′ − ω − i0+

– p.25

Pumping through Noninteracting Dot

Wang, Wei, Wang, Guo, APL 79, 3977 (2001)

Blaauboer, PRB 65, 235318 (2002)


ratio QS/QN for T = 0

0 0.5 1 1.5 2 2.5 3

ΓS/Γ

N

0

1

2

3

4

5Q

S/Q

N

for all choices of pumping parameters X,Y ∈ ΓN,ΓS, ǫ– p.26

Pumping through Noninteracting Dot


ratio QS/QN for T ∼ ΓN

0 1 2 3 4

ΓS/Γ

N

0

5

10

15

20

QS/Q

N

ΓN

,ΓS

ΓS,ε

ΓN

, ε

depends on choice of pumping parameters

– p.27

Outline

Introduction












– p.28

adiabatic pumping through a quantum dot with coulomb ...qpump/koenig.pdf · introduction: quantum...

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