Scattering theory of nonlinear thermoelectric transport

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1 Scattering theory of nonlinear thermoelectric transport David Sánchez Institute for Cross-Disciplinary Physics and Complex Systems IFISC (UIB CSIC, Palma de Mallorca, Spain) Workshop on Thermoelectric Transport (NanoCTM) In collaboration with: Rosa López (IFISC)

2 Motivation Thermal gradients as large as 13 K/µm [R. Venkatasubramanian et al., Nature 413, 597 (2001)] Rectification effects, thermopower changes of sign Role of electron-electron interactions. Differential conductance [S. De Franceschi et al., Phys. Rev. Lett. 89, (2002)] Unknown nonlinear effects on power generators and refrigerators performance A.A.M. Staring et al., EPL 22, 57 (1993) D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transport October 21 27, / 24

3 Outline 1 Motivation 2 Nonlinear thermoelectric transport 3 Nonlinear heat transport 4 Conclusions D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transport October 21 27, / 24

4 Outline 1 Motivation 2 Nonlinear thermoelectric transport 3 Nonlinear heat transport 4 Conclusions D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transport October 21 27, / 24

5 Weakly nonlinear transport Theoretical model Expansion around equilibrium point Nonlinear coefficients depend on screening (interaction driven) response [M. Büttiker, J.Phys.: Condens. Matt. 5, 9361 (1993)] Description in terms of characteristic potentials: Particle injectivities (electric bias) Entropic injectivities (thermal bias) D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transport October 21 27, / 24

6 Electric current Voltage and temperature differences: ev α = µ α E F and θ α = T α T Scattering matrix: s αβ = s αβ [E,U( r,{v γ },{T γ })] Current: I α = 2e h β deaαβ (E)f β (E) where A αβ = Tr[δ αβ s αβ s αβ] Second-order expansion: I α = β G αβ V β + β L αβ θ β + βγ G αβγ V β V γ + βγ L αβγ θ β θ γ +2 βγ M αβγ V β θ γ. D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transport October 21 27, / 24

7 Transport coefficients Linear response: G αβ = (2e 2 /h) de A αβ E f (2e 2 /h)a αβ (E F ) L αβ = (2e/hT) de (E E F )A αβ E f (eπ 2 kb 2 T/3h) EA αβ E=EF Leading-order nonlinearities: ( G αβγ = e2 Aαβ de + A ) αγ +eδ βγ E A αβ E f h V γ V β T. Christen and M. Büttiker, EPL 35, 523 (1996) ( Aαβ L αβγ = e de E F E h T ( M αβγ = e2 EF E de h et + A αγ E E F +δ βγ E A αβ ) E f θ γ θ β T A αγ A αβ E E F δ βγ E A αβ ) E f V β θ γ T D. Sánchez and R. López, arxiv: (2012) D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transport October 21 27, / 24

8 Injectivities Charge imbalance: q = q bare +q scr Voltage bias Thermal bias Particle injectivity: Tr ν p α(e) = 1 2πi β [ s βα ] ds βα de Entropic injectivity: να(e) e = 1 [ E EF Tr 2πi T β s βα ] ds βα de D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transport October 21 27, / 24

9 Screening Screening potential (up to first order): U = U eq + α u α V α + α z α θ α where u α = ( U/ V α ) eq and z α = ( U/ θ α ) eq Assumptions: local potential and δ/δu e / E Then, q bare = e α (Dp αev α +D e αθ α ) where D p α = deν p α(e) E f, and D e α = deν e α(e) E f Internal potential response: U = U U eq Mean-field approximation: q scr = e 2 Π U where Π = α Dp α = D(E F ) is the static Lindhard function at low energies and k B T = 0 Use Poisson equation 2 U = 4π(q bare +q scr ): 2 u α +4πe 2 Πu α = 4πe 2 D p α, 2 z α +4πe 2 Πz α = 4πeD e α. D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transport October 21 27, / 24

10 Nonlinear transport coefficients From characteristic potentials: θγ A αβ = z γ δa αβ /δu ez γ E A αβ and Vγ A αβ = u γ δa αβ /δu eu γ E A αβ G αβγ = e3 de [ E A αγ u β + E A αβ (u γ δ βγ )] E f, h L αβγ = e ( de [Ξ αγ z β +Ξ αβ z γ E E F δ βγ )] E f, h T M αβγ = e2 de [ E A αβ z γ +Ξ αγ u γ Ξ αβ δ βγ ] E f, h where Ξ αβ = [(E E F )/T] E A αβ D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

11 Quantum dot (an application) Single level (E d ) coupled to two reservoirs (1 and 2) gives rise to broadening Γ = Γ 1 +Γ 2 Breit-Wigner resonance and Hartree approximation: q d = e π de Γ 1f 1 (E)+Γ 2 f 2 (E) (E E d eu) 2 +Γ 2 Charge excess δq d = e 2 D p 1 V 1 +e 2 D p 2 V 2 +ed1 eθ 1 +ed2 eθ 2 e 2 DU where: Dα p = Γ α 1 de π (E E d ) 2 +Γ 2 Ef, Dα e = Γ α de E E F 1 π T (E E d ) 2 +Γ 2 Ef Poisson equation: δq d = C(U V g ) D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

12 Screening potential Internal potential: U = e2 D p 1 C +e 2 D }{{} u 1 V 1 + e2 D p 2 C +e 2 D }{{} u 2 V 2 + ede 1 C +e 2 D }{{} z 1 θ 1 + ede 2 C +e 2 D }{{} z 2 C θ 2 + } C +e {{ 2 D } u g Charge neutral limit (C = 0) and symmetric bias (V 1 = E F +V/2, V 2 = E F V/2, T 1 = T +θ/2, and T 2 = T θ/2) Self-consistent potential E d =1 E d =0 E d =-1 u (a) Applied voltage ev E d =1 z(e d =1) E d =0, z(e d =0) E d =-1 z(e d =-1) (b) Temperature difference θ D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / V g

13 Nonlinear current For θ = 0, I = G 11 V +G 111 V 2 +O(V 3 ) where G 111 = e3 h EA 11 E=EF (1 2u 1 ). For V = 0, I = L 11 θ +L 111 θ 2 +O(θ 3 ) where L 111 = eπ2 k 2 B T 3h Current (in units of 2eΓ L Γ R /hγ) (a) ( E A 11 E=EF 2z 1 T 2 E A 11 E=EF ) T η=0 η=0.5 η= T Applied voltage ev Temperature difference θ 2 L E 0 =1 E 0 =0 E 0 =-1 D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24 (b) 2 L

14 Thermopower Linear response: S 0 = V/θ I=0 = L 11 /G 11 { (π 2 k 2 B T/6) E lna 11 E=EF T low T ( k B /e)(e d E F )/T T 1 high T Nonlinear regime: S = S 0 +S 1 θ+o(θ 2 ), where the sensitivity S 1 = 1 G11 3 [G 111 L L 111G11 2 +G 11L 11 (M 121 M 111 )], S 0 (units of k B /e) ~T ~1/T (a) T=0.05 T=0.07 T= T Temperature difference θ D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24 (b) Thermopower S (units of k B /e)

15 Experiments InAs nanowire QD [S. Fahlvik Svensson, H. Linke et al., unpublished (2012)] Heating current produces thermal gradient Coulomb-blockade regime D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

16 Experiments D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

17 Experiments D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

18 Outline 1 Motivation 2 Nonlinear thermoelectric transport 3 Nonlinear heat transport 4 Conclusions D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

19 Weakly nonlinear heat transport Heat current J α = 1 h β de(e µα )A αβ (E)f β (E) J α = β R αβ V β + β K αβ θ β + βγ R αβγ V β V γ + βγ K αβγ θ β θ γ +2 βγ H αβγ V β θ γ Linear response: R αβ = 2e h de(e EF )A αβ E f = TL αβ and K αβ = 2 ht de(e EF ) 2 A αβ E f Leading-order nonlinearities: R αβγ = e2 2h de ( Aαβ (E E F ) + A αγ ev γ ev β [ f(e) ] [ δ αγa αβ +δ αβa αβ E ( ) δ βγ (E E F ) A ) ] αβ E +A αβ D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

20 Weakly nonlinear heat transport K αβγ = g θ 2 [ de (E E ( Aαβ F) 2 (E) T θ γ ( (E EF ) A αβ + δ βγ T E + A αβ T + A ) αγ(e) θ β ) ) ] ( f E H αβγ = eg θ 2 { ( Aαγ de(e E F ) + E E ) F A αβ A αβ δ αγ θ β T ev γ T ] } ( f ) E [ (E EF ) A αβ + δ βγ T E + A αβ T R. López and D. Sánchez, unpublished (2012) D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

21 Peltier effect Generalization to the nonlinear case: Π = (dj Q /di) θ=0 Two-terminal case: J 1 = a v I +b v I 2, where a v = R 11 R 12 2G 11 b v = R 111+R 122 2R 121 G 2 11 a v α G 11 and Π 0 (units of k B /e) ~T 2 ~const (a) T=0.01 T=0.02 T=0.04 (b) Nonlinear Peltier Π (units of k B /e) T I/I 0 D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

22 Wiedeman-Franz law Generalization to the nonlinear case: Λ = djα dθ V=0 diα dvα θ=0 ( ) Λ L Two-terminal case: 0 L 0 2 K111 θ K 11 G 111V G 11 1 (Λ L 0 )/L η=0.2 η=0.3 η=0.4 η= (Λ L 0 )/L (a) E 0 (b) Temperature D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

23 Outline 1 Motivation 2 Nonlinear thermoelectric transport 3 Nonlinear heat transport 4 Conclusions D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

24 Conclusions Main idea I = G 1 V +G 2 V 2 +L 1 θ +L 2 θ 2 +MVθ Self-consistent calculation of nonlinear response coefficients Particle and entropic injectivities Screening charge: U depends on both V and θ Thermal rectification effects and nonlinear thermovoltage Nonlinear heat transport: Peltier effect and Wiedemann-Franz law deviations D. Sánchez (IFISC) Scattering theory of nonlinear thermoelectric transportoctober 21 27, / 24

Three-terminal quantum-dot thermoelectrics

Three-terminal quantum-dot thermoelectrics Björn Sothmann Université de Genève Collaborators: R. Sánchez, A. N. Jordan, M. Büttiker 5.11.2013 Outline Introduction Quantum dots and Coulomb blockade Quantum