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

Quantum Transport through Coulomb-Blockade Systems Bj¨ orn 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. -1 0 1 Tunnel barrier → quantized charge E ch (0) E C E ch (1) E ch Q Q R L V E ch (-1) g 2 1 C g box <n> C J 0 -1 Quantitatively: -2 -1 0 1 n x Q 2 + Q 2 L R E ch = + Q R V g 2 C J 2 C g e 2 Coulomb staircase ( n − n x ) 2 + const. , = 2 C Σ COQUSY06 – p.3

Single-Electron Transistor 0 -1 1 E ch (0) E C Two contacts → transport E ch E ch (1) E ch (-1) V g 2 1 <n> V −V t 0 t -1 -2 Quantitatively: 0.5 G V /G as Q 2 + Q 2 L R E ch = + Q R V g 2 C J 2 C g e 2 ( n − n x ) 2 + const. , = 2 C Σ -1 0 1 n x Coulomb oscillations COQUSY06 – p.4

Simple Coupled Device V ,1 g Box: SET1 V g SET2 +V/2 −V/2 V g ,2 charge on C c (sawtooth) Transistor measures input for transistor box charge 1.0 Transistor: box V Q 0.5 g 0.0 V −V 0.0 0.5 1.0 t t nx one step of Coulomb staircase (Lehnert et al. PRL ’03, Sch ¨ afer et al. Physica E ’03) 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 ¨ on, PRB ’94) Hamiltonian: H = H L + H R + H I + H ch + H T = H 0 + H T charge degrees of freedom separated from fermionic degrees: charging energy tunneling H ch = e 2 � krn c ln e − iϕ + h . c . � � � kl a † T rn H T = 2 C ( ˆ N − n x ) 2 r = R,L kln e ± iϕ = | N ± 1 �� N | Time evolution of e.g. density matrix of charge governed by propagator Q : Z t 0 Z t » „ « „ « – Y n ′ 1 ,n 1 2 | ˜ � n ′ dt ′ H T ( t ′ ) I dt ′ H T ( t ′ ) I | n ′ 2 ,n 2 = Trace T exp − i | n 2 �� n 1 | T exp − i 1 � n ′ t t 0 | {z } fermionic d.o.f’s H H T H T T ^ A(t) H ⇒ Keldysh contour T − t’ 8 t t t t t 1 2 3 4 COQUSY06 – p.7

Dyson-equation Integrating out reservoirs/ contracting tunnel vertices ⇒ each contraction ⇔ golden-rule rate: ω − µ r R K α r ± ( ω ) = � dEα r 0 f ± r ( E + ω ) f ∓ ( E ) = ± α r e ± β ( ω − µr ) − 1 with α r 0 = 4 π 2 R r . 0 1 0 1 0 1 0 −1 0 1 0 −1 L R L R R R L L |−1><−1| R 1 0 1 0 1 0 −1 −2 −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 ’’ n 1 1 = (0) + (0) Π Π Π Σ Π to calculate: n’ 2 n 2 n’ 2 n 2 n’ 2 n’’ n 2 2 self-energy � = � (0) + � � � (0) � with free propagator (w/o tunneling) Q (0) 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 V ,T V ,T Z Z dω ( βω/ 2) 2 dω βω/ 2 k B L L R R G V = G as sinh βω A ( ω ) ; G T = − G as sinh βω A ( ω ) e V g g V = G V g T = − e G T ; Thermoelectric transport: G as k B G as perturbative expansion to 2nd order in coupling α 0 I = G V V + G T δT V/T + g ˜ g V/T = g seq V/T + g ˜ α V/T + g cot ∆ V/T A ( ω ) = [ C < ( ω ) − C > ( ω )] / (2 πi ) g V -g T 0.1 g V/T 0.5 seq 0.04 g V/T 0.4 0.2 0 0 cot g V/T 0.3 -0.04 ∼ α g V/T 0.2 -0.1 0.1 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 ∼ 0.1 ∆ g V/T 0 0 n x n x 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 COQUSY06 – p.10

Sequential tunneling V/T + g ˜ g V/T = g seq V/T + g ˜ α ∆ V/T + g cot V/T • sequential tunneling: 8 < 1 : V β ∆ 0 / 2 g seq V/T = κ 0 with κ 0 = sinh β ∆ 0 β ∆ 0 / 2 : T : Resonances around degeneracy points ∆ n = 0 . g V -g T 0.1 g V/T 0.5 seq 0.04 g V/T 0.4 0.2 0 0 cot g V/T 0.3 -0.04 ∼ α g V/T 0.2 -0.1 0.1 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 ∼ ∆ 0.1 g V/T 0 0 n x n x 0.5 0.6 0.7 0.8 0.6 0.7 0.8 0.5 COQUSY06 – p.11

Cotunneling V/T + g ˜ g V/T = g seq V/T + g ˜ α ∆ V/T + g cot V/T • standard cotunneling: „ 1 « 2 2 π 2 1 g cot 3 ( k B T ) 2 = α 0 − V ∆ 0 ∆ − 1 „ 1 « 2 „ 1 « 8 π 4 1 1 g cot 15 ( k B T ) 3 = α 0 − + T virtual occupation ∆ 0 ∆ − 1 ∆ 0 ∆ − 1 of unfavourable dominant away from resonance | ∆ n | ≫ k B T . charged state. g V -g T 0.1 g V/T 0.5 seq 0.04 g V/T 0.4 0.2 0 0 cot g V/T 0.3 -0.04 ∼ α g V/T 0.2 -0.1 0.1 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 ∼ 0.1 ∆ g V/T 0 n x 0 n x 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 COQUSY06 – p.12

Cotunneling V/T + g ˜ g V/T = g seq V/T + g ˜ α ∆ V/T + g cot V/T • standard cotunneling: „ 1 « 2 2 π 2 1 g cot 3 ( k B T ) 2 = α 0 − V ∆ 0 ∆ − 1 8 „ 1 « 2 „ 1 « 8 π 4 < 1 : V 1 1 g cot 15 ( k B T ) 3 = α 0 − + κ n = T ∆ 0 ∆ − 1 ∆ 0 ∆ − 1 β ∆ n / 2 : T : V/T = κ − 1 ∆ − 1 ∂ 2 φ − 1 + κ 0 ∆ 0 ∂ 2 φ 0 + κ 0 + κ − 1 · φ 0 − φ − 1 + ∆ − 1 ∂φ − 1 − ∆ 0 ∂φ 0 g cot 2 E C g V -g T 0.1 g V/T 0.5 seq 0.04 g V/T 0.4 0.2 0 0 cot g V/T 0.3 -0.04 ∼ α g V/T 0.2 -0.1 0.1 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 ∼ 0.1 ∆ g V/T 0 n x 0 n x 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 COQUSY06 – p.12

Renormalized sequential tunneling 8 < 1 : V V/T + g ˜ g V/T = g seq κ 0 = V/T + g ˜ α V/T + g cot ∆ β ∆ 0 / 2 : T : V/T ω ω A( ) • Renormalization of coupling: » – β ∆ 0 / 2 ∂ (2 φ 0 + φ − 1 + φ 1 ) + φ − 1 − φ 1 g ˜ α V/T = κ 0 sinh β ∆ 0 E C sequential tunneling but spectral density A ( ω ) energy gap: broadened and shifted. » – ∂ β ∆ 0 / 2 ˜ ∆ ⇓ g V/T = κ 0 (2 φ 0 − φ − 1 − φ 1 ) ∂ ∆ 0 sinh β ∆ 0 renormalized parameters for coupling: ˜ α charging energy gap: ˜ ∆ n COQUSY06 – p.13

Renormalization by quantum fluctuations » – » – β ∆ 0 / 2 ∂ (2 φ 0 + φ − 1 + φ 1 ) + φ − 1 − φ 1 ∂ β ∆ 0 / 2 ˜ g ˜ α ∆ V/T = κ 0 ; g V/T = κ 0 (2 φ 0 − φ − 1 − φ 1 ) sinh β ∆ 0 E C ∂ ∆ 0 sinh β ∆ 0 Quantum fluctuations ⇒ renormalization of system parameters α, ˜ G ( α 0 , ∆ 0 ) = G seq (˜ ∆) + cot. terms α∂G seq ( α 0 , ∆ 0 ) ” ∂G seq ( α 0 , ∆ 0 ) “ α, ˜ ˜ G seq (˜ expand: ∆) = ˜ + ∆ − ∆ 0 ∂α 0 ∂ ∆ 0 renormalization of parameters (perturbative in α 0 ): „ βE C  « » „ «–ff α ˜ iβ ∆ 0 = 1 − 2 α 0 − 1 + ln − ∂ ∆ 0 ∆ 0 Re Ψ α 0 π 2 π „ βE C » « „ «– ˜ ∆ iβ ∆ 0 = 1 − 2 α 0 1 + ln − Re Ψ ∆ 0 π 2 π α and ˜ many-channel ∆ decrease logarithmically by renormalization! ˜ ⇔ Kondo-physics (for lowering temperature and increasing coupling α 0 ) COQUSY06 – p.14

Renormalization effects on G V/T α, ˜ G ( α 0 , ∆ 0 ) = G seq (˜ ∆) + 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. g V -g T 0.1 g V/T 0.5 seq 0.04 g V/T 0.4 0.2 0 0 cot g V/T 0.3 -0.04 ∼ α g V/T 0.2 -0.1 0.1 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 ∼ ∆ 0.1 g V/T 0 n x 0 n x 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 (logarithmic reduction of maximum electrical conductance (K ¨ onig et al. PRL ’97) experimentally observed by Joyez et al. PRL ’97) COQUSY06 – p.15

Thermopower Thermopower: V ,T V ,T L L R R � V = G T � S = − lim � δT G V δT → 0 � I =0 V g Thermoelectric transport: S measures average energy: I = G V V + G T δT S = −� ε � eT . COQUSY06 – p.16

Charging energy gaps determine S 0 -1 1 ∆ n = E ch ( n + 1) − E ch ( n ) E ch (0) E C = E C [1 + 2( n − n x )] ∆ 0 E ch E ch (1) A) at resonance: • peak in G V E ch (-1) • S ∝ � ε � = 0 A 0.5 ε ≷ E F cancels ∆ B G V /G as 0 C B) sequential ∆ −1 • G V decays off resonance β E C /2 • S ∝ � ε � ∝ ∆ 0 ∝ n x ∆ 0 S/e 0 k B T ∆ C) n x = 0 ⇔ ∆ − 1 = − ∆ 0 −1 - β E C /2 • two levels add for G V -1 0 1 n x • two levels cancel for S COQUSY06 – p.17