Thermoelectricity with cold atoms?

Transcription

1 Thermoelectricity with cold atoms? Ch. Grenier, C. Kollath & A. Georges Centre de physique Théorique - Université de Genève - Collège de France Université de Lorraine Séminaire du groupe de physique statistique Janvier 2013 () Thermoelectricity and cold atoms Nancy, Jan / 33

2 Introduction General considerations Introduction to Thermoelectricity Seebeck effect : A difference of temperature T creates a voltage V Peltier effect : An electric current I generates a heat current I Q = Π I = (T S) I Seebeck coefficient S : Entropy per carrier Stationnary effects 1 : permanent currents/differences 1 L. Onsager, Phys. Rev. 38, 2265 (1931)& Phys. Rev. 37, 405 (1931) H.B. Callen, Phys. Rev. 73, (1948) () Thermoelectricity and cold atoms Nancy, Jan / 33

3 Introduction General considerations Thermoelectricity and materials Good thermoelectrics : a recurrent interest in material physics Good Peltier cooling Energy saving purposes Idea : increase figure of merit ZT = σ el TS 2 σ th G. Snyder & E. Toberer, Nat. Mat (2008) More fundamental : access high temperature transport properties, without phonons... () Thermoelectricity and cold atoms Nancy, Jan / 33

4 Introduction General considerations Thermoelectricity and mesoscopic physics GOALS : Extract energy from fluctuating environment Optimize energy to electricity conversion (cooling purposes) Heat engines with magnons B. Sothmann & M. Büttiker, EPL (2012) Quantum limited refrigerator Timofeev et al PRL (2009) Three terminal thermoelectricity J.-H. Jiang et al Phys. Rev. B (2012) Quantum dot refrigerator Prance et al PRL (2009) () Thermoelectricity and cold atoms Nancy, Jan / 33

5 Introduction General considerations Introduction - Transport and cold atoms Spin transport (MIT, 2011) A. Sommer et al., Nature Interactions (LMU, 2012) U. Schneider et al., Nat. Phys. Disorder (Inst. d optique - LENS, 2008) J. Billy et al.-g. Roati et al., Nature Also: H. Ott et al, Phys. Rev. Lett. 92, (2004) S. Palzer et al, Phys. Rev. Lett. 103, (2009) J. Catani et al, Phys. Rev. A 85, (2012) K.K. Das et al, Phys. Rev. Lett. 103, (2009) And many others... () Thermoelectricity and cold atoms Nancy, Jan / 33

6 Introduction Experimental motivation Introduction - Experimental motivation 1 Tunable channel shape Zimmermann et al New Journal of Physics 13 (2011) Interactions Setup for 6 Li (fermions) Possibility to modify : Disorder (ballistic or diffusive) Lattice or not No DC transport (finite samples) () Thermoelectricity and cold atoms Nancy, Jan / 33

7 Introduction Experimental motivation Introduction - Experimental motivation 2 Have been observed Particle transport Brantut et al, Science 2012 Resistance drop with superfluid Stadler et al, Nature 2012 () Thermoelectricity and cold atoms Nancy, Jan / 33

8 Introduction Experimental motivation Introduction - Experimental motivation 3 ETH, 2012 (Brantut et al., Science) Realization of a two terminal transport setup Discharge of a mesoscopic capacitor (reservoirs) in a resistor (the conduction channel) Simulation of mesoscopic physics with cold atoms Question : Can this setup demonstrate offdiagonal transport? () Thermoelectricity and cold atoms Nancy, Jan / 33

9 Outline 1 General framework 2 Proposal for thermoelectricity 3 Experimental signal estimates () Thermoelectricity and cold atoms Nancy, Jan / 33

10 General framework Outline 1 General framework 2 Proposal for thermoelectricity 3 Experimental signal estimates () Thermoelectricity and cold atoms Nancy, Jan / 33

11 General framework Transport setup Transport setup Two terminal configuration Grand canonical ensemble : flow of entropy and particles between reservoirs Linear response coefficients in L () Thermoelectricity and cold atoms Nancy, Jan / 33

12 General framework Transport setup Thermodynamic coefficients 6 4 κe F /N αt F /N C µ T F /Nk B T C N T F /Nk B T T/T F () Thermoelectricity and cold atoms Nancy, Jan / 33

13 General framework Transport equations and coefficients Transport in linear response Our approach : Constriction Black box responding linearly Linear circuit picture Linear response : ( IN I S ) ( µ = L T ) ( L11 L, L = 12 L 12 L 22 ) Thermodynamics : (Quasistatic evolution) ( N S ) ( µ = M T ) ( κ α, M = α C µ T ) Equations for chemical potential and temperature difference ( ) ( ) d µ = M 1 µ L dt T T () Thermoelectricity and cold atoms Nancy, Jan / 33

14 General framework Transport equations and coefficients Transport equations and coefficients Equations for particle number and temperature difference : ( d N/κ τ 0 dt T ) ( N/κ = Λ T ) ( 1 S,Λ = S L+S 2 l l Discharge of a capacitor through a resistor, including thermal properties Global timescale τ 0 = κ L RC 11 ). Effective transport coefficients : L L 22 /L 11 (L 12 /L 11 ) 2 R/TR T Lorenz number l C µ /κt (α/κ) 2 = C N /κt Charac. of reservoirs, analogue to L S α/κ L 12 /L 11 Total Seebeck coefficient () Thermoelectricity and cold atoms Nancy, Jan / 33

15 General framework Transport equations and coefficients Transport coefficients Both reservoirs and constriction participate to transport S = 0: -Only pure thermal and mass transport -At low T : L/l 1 for a free Fermi gas Wiedemann-Franz law T and N relaxation timescales are identical L 12 = 0: -Total Seebeck S 0 Intrinsic thermoelectric effect driven by finite α i. How to reveal thermoelectric effects? ii. Smoking gun protocol? Questions: () Thermoelectricity and cold atoms Nancy, Jan / 33

16 Proposal for thermoelectricity Outline 1 General framework 2 Proposal for thermoelectricity 3 Experimental signal estimates () Thermoelectricity and cold atoms Nancy, Jan / 33

17 Proposal for thermoelectricity Proposal Solution to transport equations N(t) = { [ ] 1 [ e t/τ + e t/τ ] L + S2 e t/τ e t/τ } + N 2 l 2(λ + λ ) 0 + } {{ } Diagonal transport: exponential decrease Sκ [ e t/τ e t/τ ] + T 0,τ 1 λ + λ ± = τ 1 0 λ ± } {{ } Thermoelectric effect! ΔT/T F =0.25 ΔT 0 /T F = t/τ 0 5 Particle imbalance with and without initial temperature difference With N 0 N = 0 10% Hard to isolate thermoelectric contribution when exp. decrease is switched on () Thermoelectricity and cold atoms Nancy, Jan / 33

18 Proposal for thermoelectricity Proposal An appropriate setup Two steps i. Prepare reservoirs with equal particle number and different temperatures, with closed constriction ii. Open the constriction and monitor particle number ΔN/N 0 ΔT/T F 0.05 Particles flow in both ways during temperature equilibration : Transient analogue of the Seebeck effect t/τ 0 5 () Thermoelectricity and cold atoms Nancy, Jan / 33

19 Proposal for thermoelectricity Physical interpretation What happens? with λ ± = 1 2 Sκ [ ] N(t) = e t/τ e t/τ + T 0,τ 1 ± λ + λ = τ 1 0 λ ± ) ( ) (1 + L+S2 S ± 2 l l L+S2 2l and S = α/κ L12 /L 11. i. Particle imbalance and current proportional to T 0 and S ii. Reservoir properties participate to the effect iii. Sign of N (and I N ) given by the sign of S Entropy flows from hot to cold : 2nd principle I S = SI N σ th T }{{} Dominant () Thermoelectricity and cold atoms Nancy, Jan / 33

20 Proposal for thermoelectricity Physical interpretation Protocol-Summary Simple protocol to observe thermoelectric effects Very general up to now... Need model for : Reservoirs Constriction For practical purposes: Observability of the effect? Estimates? Which configuration(s) favor the observation of the effects? Change of sign observable? () Thermoelectricity and cold atoms Nancy, Jan / 33

21 Experimental signal estimates Outline 1 General framework 2 Proposal for thermoelectricity 3 Experimental signal estimates () Thermoelectricity and cold atoms Nancy, Jan / 33

22 Experimental signal estimates Models for reservoirs and channel How to compute transport coefficients? Need to take care of channel and reservoirs Reservoirs and constriction are treated separately Reservoirs Trapped Fermi gas Noninteracting fermions Constriction Geometry: trap, no trap, (1-2-3) D Conduction regime : Ballistic or diffusive Response coefficients : Landauer-Büttiker formalism Transport () Thermoelectricity and cold atoms Nancy, Jan / 33

23 Experimental signal estimates Models for reservoirs and channel Computing coefficients Thermo. coefficients Proportional to moments of DOS f/ ε + R n = dεg(ε)( f )(ε µ)n 0 ε κ n = 0, α n = 1 C µ T n = 2 Transport coefficients Proportional to moments of Φ f/ ε + T n = dεφ(ε)( f )(ε µ)n 0 ε L 11 n = 0, L 12 n = 1 L 22 n = 2 Φ : transport function DOS velocity transmission Differential conductance 2 2 R. Kim et al Appl. Phys. Lett. 105, (2009) G. D. Mahan & J. O. Sofo, PNAS 93, 7436 (1996) () Thermoelectricity and cold atoms Nancy, Jan / 33

24 Experimental signal estimates Models for reservoirs and channel Transport function Φ(ε) = n x,n z dk ħk M T (k)δ(ɛ ħ2 k 2 2M ħω zn z ħω x n x ) Φ(ε) (a.u.) Diffusive,3D Ballistic,3D Approx, Ballistic Approx,Diffusive Ballistic : T (k) = 1, Φ(ε) 1 2 Diffusive : T (k) λ(k) L, Φ(ε) 2 5 ( )( ) 1 + ε ħω 1 + ε x ħω z ε 5/2 ħω x ħω z (ε-ε 0)/h (khz) For ν z = 9.8kHz, ν x = 5.1kHz () Thermoelectricity and cold atoms Nancy, Jan / 33

25 Experimental signal estimates Results Results : Transport coefficients 1 Lorenz number L and l Simple case : 3D channel, noninteracting fermions Comparison of different regimes and densities of states Diffusive Ballistic W-F law 3 W-F law 3 L L, trap l 2 L L, trap l T/T F T/T F () Thermoelectricity and cold atoms Nancy, Jan / 33

26 Experimental signal estimates Results Results : Transport coefficients 2 Reservoirs, constriction contributions and total Seebeck Simple case : 3D channel, noninteracting fermions Comparison of different regimes and densities of states Diffusive Ballistic 6 4 S S, trap -S c -S c, trap S r 6 4 S S, trap -S c -S c, trap S r T/T F T/T F () Thermoelectricity and cold atoms Nancy, Jan / 33

27 Experimental signal estimates Results Thermoelectric visibility When are thermoelectrics effects visible? GOAL : Find appropriate parameter range Recall: [ N(t) SκT N = F e 0 N 0 (λ + λ ) t/τ e +] t/τ T0 T F Maximum at t max t max 2.5 t/τ 0 5 Visibility defined as η = N(t max ) N 0 / T 0 T F () Thermoelectricity and cold atoms Nancy, Jan / 33

28 Experimental signal estimates Results Results : Thermoelectric visibility Simple case : 3D channel, noninteracting fermions Comparison of different regimes and densities of states η 0 No trap, diffusive trap,diffusive No trap, ballistic trap,ballistic T/T F Compensation of reservoir and channel effect for ballistic+trap Reservoir dominant when no trapping Channel dominant for diffusive + trap energy dependence of Φ : channel contribution () Thermoelectricity and cold atoms Nancy, Jan / 33

29 Conclusion Conclusions-Perspectives Thermoelectricity with cold atoms! Transport : combination of reservoir and channel properties Intrinsic thermoelectricity Protocols to reveal offdiagonal transport with cold atoms Sizeable (10 to 20%) effects High-T transport without phonons What s next? Interactions : improvement of thermopower Lattice in the constriction Pumping reservoirs : AC transport? Peltier cooling of reservoirs CG, C. Kollath, A. Georges arxiv: () Thermoelectricity and cold atoms Nancy, Jan / 33