Quantum Transport through Coulomb-Blockade Systems

Quantum Transport through Coulomb-Blockade Systems Björn Kubala Institut für Theoretische Physik III Ruhr-Universität Bochum COQUSY6 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 COQUSY6 p.2

Single-Electron Box Gate attracts charge to island. Tunnel barrier quantized charge E C -1 1 E ch () Q L E ch E ch (1) Q R V g E ch (-1) 2 C J box C g <n> 1 Quantitatively: E ch = Q2 L 2C J + Q2 R 2C g + Q R V g -1-2 -1 1 n x = e 2 2C Σ (n n x ) 2 + const., Coulomb staircase COQUSY6 p.3

Single-Electron Transistor -1 1 Two contacts transport E ch E C E ch () E ch (1) V g 2 E ch (-1) V t V t <n> 1-1 Quantitatively: E ch = Q2 L 2C J + Q2 R 2C g + Q R V g = e 2 2C Σ (n n x ) 2 + const., -2 G V /G as.5-1 1 n x Coulomb oscillations COQUSY6 p.4

Simple Coupled Device SET1 Vg,1 Box: +V/2 SET2 V/2 V g Vg,2 Transistor measures box charge charge on C c (sawtooth) input for transistor Q box 1..5 Transistor: V g...5 n x 1. V t V t one step of Coulomb staircase (Lehnert et al. PRL 3, Schäfer et al. Physica E 3) COQUSY6 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 COQUSY6 p.6

Real-time diagrammatics for an SET (Schoeller and Schön, PRB 94) Hamiltonian: H = H L + H R + H I + H ch + H T = H + H T charge degrees of freedom separated from fermionic degrees: charging energy H ch = e2 2C ( ˆN n x ) 2 H T = r=r,l tunneling ( ) Tkl rn a krn c lne iϕ + h.c. kln e ±iϕ = N ± 1 N Time evolution of e.g. density matrix of charge governed by propagator Q : Y n 1,n 1 n 2,n 2» = Trace n 2 T exp i {z } fermionic d.o.f s Z t t «dt H T (t ) I n 2 n 1 T exp Z t «i dt H T (t ) I n 1 t Keldysh contour H T H T H T H T ^ A(t) 8 t t t 1 2 3 4 t t t COQUSY6 p.7

Dyson-equation Integrating out reservoirs/ contracting tunnel vertices each contraction golden-rule rate: α r± (ω) = deα r f r ± (E +ω)f (E) = ±α r ω µ r e ±β(ω µ r ) 1 with αr = R K 4π 2 R r. 1 1 1 1 1 R L R R R L 1 1 1 1 L L R 1 2 1 1>< 1 diagram with sequential, cotunneling and 3rd order processes Write full propagator as Dyson equation: n 1 n 1 n 1 n 1 n 1 n 1 n 1 n 2 Π n 2 = () Π n 2 n 2 + = () + Π Σ () Π n 2 n 2 n 2 () to calculate: self-energy with free propagator (w/o tunneling) Q () COQUSY6 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 COQUSY6 p.9

Electrical and thermal conductance G V =G as Z dω βω/2 sinh βω A(ω) ; G T = G as k B e Z dω (βω/2)2 sinh βω A(ω) V L,TL V R,T R g V = G V G as ; g T = e k B G T G as perturbative expansion to 2nd order in coupling α g V /T = g seq V /T + g V α /T + g V /T + g cot V /T V g Thermoelectric transport: I = G V V + G T δt A(ω) = [C < (ω) C > (ω)]/(2πi) g V.5.4.3.2.1.1 -.1.5.6.7.8.5.6 n.7.8 x gv/t seq cot α g V/T.4 -.4.5.6.7.8.5.6 n.7.8 x -g T.2.1 COQUSY6 p.1

Sequential tunneling = g seq V/T + g α V/T + g V/T + g cot V/T sequential tunneling: g seq V /T = κ β /2 sinh β with κ = 8 < : 1 : V β /2 : T Resonances around degeneracy points n =. g V.5.4.3.2.1.1 -.1.5.6.7.8 gv/t seq cot α g V/T.4.5.6.7.8 -.4 -g T.2.1.5.6 n.7.8 x.5.6 n.7.8 x COQUSY6 p.11

Cotunneling = g seq V/T + g α V/T + g V/T + g cot V/T standard cotunneling: g cot V = α 2π 2 3 (k BT ) 2 g cot T = α 8π 4 15 (k BT ) 3 1 1 1 1 1 1 «2 «2 1 + 1 «1 dominant away from resonance n k B T. virtual occupation of unfavourable charged state. g V.5.4.3.2.1.1 -.1.5.6.7.8.5.6 n.7.8 x gv/t seq cot α g V/T.4 -.4.5.6.7.8.5.6 n.7.8 x -g T.2.1 COQUSY6 p.12

Cotunneling = g seq V/T + g α V/T + g V/T + g cot V/T standard cotunneling: g cot V = α 2π 2 3 (k BT ) 2 g cot T = α 8π 4 15 (k BT ) 3 1 1 1 1 1 1 «2 «2 1 + 1 «1 κ n = 8 < : 1 : V β n /2 : T g cot V /T = κ 1 1 2 φ 1 +κ 2 φ + κ + κ 1 φ φ 1 + 1 φ 1 φ 2 E C g V.5.4.3.2.1.1 -.1.5.6.7.8.5.6 n.7.8 x gv/t seq cot α g V/T.4 -.4.5.6.7.8.5.6 n.7.8 x -g T.2.1 COQUSY6 p.12

Renormalized sequential tunneling = g seq V/T + g α V/T + g V/T + g cot V/T κ = 8 < : 1 : V β /2 : T A( ω) ω Renormalization of coupling: gv α /T = κ β /2 sinh β energy gap: g V /T =» (2φ + φ 1 + φ 1 ) + φ 1 φ 1 E C» β /2 κ (2φ φ 1 φ 1 ) sinh β sequential tunneling but spectral density A(ω) broadened and shifted. renormalized parameters for coupling: α charging energy gap: n COQUSY6 p.13

Renormalization by quantum fluctuations gv α /T = κ β /2 sinh β» (2φ + φ 1 + φ 1 ) + φ 1 φ 1 ; g E V /T = C» β /2 κ (2φ φ 1 φ 1 ) sinh β Quantum fluctuations renormalization of system parameters G(α, ) = G seq ( α, ) + cot. terms expand: G seq ( α, ) = α Gseq (α, ) G + seq (α, ) α renormalization of parameters (perturbative in α ): j α βec = 1 2α 1 + ln α π «βec = 1 2α» 1 + ln π Re Ψ Re Ψ i β «2π i β «ff 2π α and decrease logarithmically by renormalization! (for lowering temperature and increasing coupling α ) many-channel Kondo-physics COQUSY6 p.14

Renormalization effects on G V/T G(α, ) = G seq ( α, ) + cot. terms α and decrease logarithmically by renormalization: α peak structure reduced by quantum fluctuations. closer to resonance; peak broadened by quantum fluct. g V.5.4.3.2.1.1 -.1.5.6.7.8 gv/t seq cot α g V/T.4.5.6.7.8 -.4 -g T.2.1.5.6 n.7.8 x.5.6 n.7.8 x (logarithmic reduction of maximum electrical conductance (König et al. PRL 97) experimentally observed by Joyez et al. PRL 97) COQUSY6 p.15

Thermopower Thermopower: V L,TL V R,T R S = lim δt V δt = G T I= G V V g Thermoelectric transport: I = G V V + G T δt S measures average energy: S = ε et. COQUSY6 p.16

Charging energy gaps determine S E ch E C -1 1 E ch () E ch (1) n = E ch (n + 1) E ch (n) = E C [1 + 2(n n x )] A) at resonance: G V /G as.5 E ch (-1) A B C 1 peak in G V S ε = ε E F cancels B) sequential G V decays off resonance βe C /2 S ε n x S/e k B T 1 C) n x = 1 = -βe C /2-1 1 n x two levels add for G V two levels cancel for S COQUSY6 p.17

Sequential and cotunneling only /e)4 S / (k B 2 2 lower T scales with βn X S / (k B /e) 3 2 1 S seq S cot S 4.5 n.5 X SeT = ε = gseq V /2 + gv cot(k BT ) 2 / g seq V + gcot V S seq = /(2eT ) n x sawtooth sawtooth suppressed by cotunneling S cot = k B e 4π 2 5 1 β 5 1 15 2 25 β (Turek and Matveev, PRB 2) universal low-t behavior S seq+cot = S(β ) How do quantum fluctuations change this picture? COQUSY6 p.18

Renormalization effects on thermopower Low T properties governed by renormalization: 3 S seq lower T Maximum of S S / (k B /e) 2 1 S seq+cot -β max 9 8 our pert. expansion 5 1 15 β charging-energy gap <ε>/.5.48.46.44.42 seq. seq.+ cot. our perturbative expansion.1.1.1 k B T / E C reduced charging-energy gap smaller ε (measures ) 7 seq.+cot..1.1 k B T / E C SeT = ε = gseq V /2 + gv cot(k BT ) 2 / crossover from g seq V to g cot V g seq V + gcot V maximum position system closer to resonance crossover for larger max Further support for renormalization picture! COQUSY6 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 COQUSY6 p.2

Multi-island geometries: parallel setup: series setup: ( ) 1 1 ( ) 1 1 L b ( ) 1 ( 1 ) ( ) R t ( ) 1 ( ) 1 ( 1, 1 ) (, 1) ( 1, 1) L ( 2, ) ( 1, ) M (, ) 1 M Trivial changes allow application to different setups! (no changes in calculation of diagrams) COQUSY6 p.21

Algorithm for multi-island systems e.g. parallel setup: charge states: (t, b) ( 1, ) Electrostatics : E ch (t, b) = E t (t t x ) 2 + E b (b b x ) 2 + E coupl. (t t x )(b b x ) generate all diagrams: - start from any charge state - choose vertex positions - choose tunnel junctions and directions of lines - connect vertices - change charge states (t, b) (t, b) (t, b) (t, b) (t, b 1) (t, b) L b R t (t, b) (t 1, b) (t, b) L b (t, b) R t (t, b 1) (t 1, b) (t, b 1) (t, b 1) (t 1, b 1) Calculate value of diagram, contributing to Σ (t,b) (t,b 1) simple rules but plenty of diagrams (in 2nd order 2 11 per charge state) Automatically generate and calculate all diagrams! COQUSY6 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 SET 1 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 2 COQUSY6 p.23

Noise assisted tunneling generator I g limiting cases: small driving detector-cotunneling independent of I g noise assisted tunneling d detector electrostatic interaction P (n g = ) 1 2 P (n g = 1) P (n d = ) 1 P (n d = 1) by noise-assisted tunneling strong driving of generator P (E) Sg Q /E 2 generator noise Z Γ d 1 = Γ( d E ) = α de 1 e P βe ( d E) I D [pa].2.1 noise-assisted tunneling I g (instead of exponential suppression). I D [pa] 8 6 4 2 full 2nd order detector cot. P(E) theory -.5.5 1 V G sd / 2 [mv].5 1 sd V G / 2 [mv] COQUSY6 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. König, Phys. Rev. B 73, 165316 (26). B. Kubala and J. König, Phys. Rev. B 73, 195316 (26). COQUSY6 p.25