Thermoelectric transport of ultracold fermions : theory Collège de France, December 2013 Theory : Ch. Grenier C. Kollath A. Georges Experiments : J.-P. Brantut J. Meineke D. Stadler S. Krinner T. Esslinger
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 α) I i. Seebeck coefficient α : Entropy per carrier I S = αi N λ T i. Stationary effects : permanent currents/differences L. Onsager, Phys. Rev. 38, 2265 (1931)& Phys. Rev. 37, 405 (1931) H.B. Callen, Phys. Rev. 73, 1349-1358 (1948)
Thermoelectricity and materials Good thermoelectrics : a recurrent interest in material physics Good Peltier cooling Efficient wasted heat recovery : energy saving purposes Idea : increase figure of merit ZT = σ el T α 2 σ th Fundamental purposes : High temperature transport properties Understanding of electron/phonon coupling... G. Snyder & E. Toberer, Nat. Mat. 7 105 (2008)
Thermoelectricity and mesoscopic physics GOALS : Extract energy from fluctuating environment Optimize energy to electricity conversion (cooling purposes) Heat engine with quantum dots A. Jordan et al. Phys. Rev. B 87, 075312 (2013) Quantum limited refrigerator Timofeev et al PRL (2009) Nonlinear thermoelectric transport R. Whitney Phys. Rev. B, 88, 064302 (2013) Quantum dot refrigerator Prance et al PRL (2009) What about cold atoms?
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, 160601 (2004) S. Palzer et al, Phys. Rev. Lett. 103, 150601 (2009) J. Catani et al, Phys. Rev. A 85, 023623 (2012) K.K. Das et al, Phys. Rev. Lett. 103, 123007 (2009) And many others...
Introduction - Experimental motivation ETH, 2012 J.P. Brantut et al., Science Realization of a two terminal transport setup Discharge of a mesoscopic capacitor (reservoirs) in a resistor (the channel) Simulation of mesoscopic physics with cold atoms Question : Can this setup demonstrate offdiagonal transport?
Outline 1 Experimental setup and theory framework 2 Thermoelectricity with cold atoms 3 A cold atom based heat engine
Outline 1 Experimental setup and theory framework 2 Thermoelectricity with cold atoms 3 A cold atom based heat engine
Experimental procedure i. Heating of one reservoir at constant particle number (closed channel) ii. Reopen the channel iii. Monitor temperature difference and particle number imbalance
Typical results Temperatures : exp. relaxation Particle number imbalance : i. Transient evolution ii. Current from hot to cold Thermoelectricity!
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 ) And thermodynamics : ( N S ) ( µ = M T ) ( κ γ, M = C µ γ T Equations for chemical potential and temperature difference ( ) ( ) d µ = M 1 µ L dt T T )
Transport equations and coefficients Equations for particle number and temperature difference : ( d N/κ τ 0 dt T ) ( N/κ = Λ T ) ( 1 α,λ = α L+α 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 N /κt Reservoir analogue to L α α r α ch γ/κ L 12 /L 11 Total Seebeck coefficient Both thermodynamic and linear response coeffs participate to transport Competition for offdiagonal contributions
How to compute transport coefficients? Need to take care of constriction and reservoirs Reservoirs and constriction are treated separately Reservoirs Trapped Fermi gas Noninteracting fermions Constriction Geometry: Transverse harmonic trap 1-2-3 D Conduction regime : Ballistic or diffusive Response coefficients : Landauer-Büttiker formalism
Computing coefficients Thermo. coefficients Proportional to moments of DOS f/ ε + R n = dεg r (ε)( f )(ε µ)n 0 ε κ n = 0, γ 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 R. Kim et al Appl. Phys. Lett. 105, 034506 (2009) G. D. Mahan & J. O. Sofo, PNAS 93, 7436 (1996)
Transport function Φ(ε) = n x,n z ( ħk y dk y M T (k)δ ε ħ2 ky 2 ) 2M n zhν z n x hν x 200 150 Ballistic Diffusive Ballistic, approx Diffusive, approx Dimensionless Φ (ε) 100 50 Ballistic : T (k) = 1, Φ(ε) 1 2 Diffusive : T (k) l(k) L, Φ(ε) 4 15 ( ) ε ε0 ε ε0 1 + hν )(1 + x hν z τ s L 2M (ε ε 0 ) 5/2 hν x hν z 0 0 5 10 15 20 (ε-ε 0 )/h (khz) For ν z = 5 khz, ν x = 0.5 khz
Two contributions to thermopower : Simple estimates for Seebeck(s) Channel : α ch = L 12 L 11 Reservoirs : α r = γ κ At low T Mott-Cutler formula : α ch π2 k 2 B T 3 Φ (µ) Φ(µ) and α r π2 k 2 B T 3 g r (µ) g r (µ) Reservoirs 3D harmonic traps : g r ε 2 α r 2π2 k 2 B T 3µ I. Diffusive: Φ ε 5/2 -> α ch 5π2 k 2 B T 6µ II. Ballistic: Φ ε 2 -> α ch 2π2 k 2 B T 3µ M. Cutler and N. F. Mott, Phys. Rev. 181, 1336 (1969) Two cases for the channel: : Channel dominates : Need corrections, small effect expected
Resulting Seebeck 0.6 Experimental α ch -α r 0.4 Regime Ballistic,ν z = 6 khz Ballistic,ν z = 8 khz Diffusive,ν z = 8 khz 0.2 0 0 0.5 1 T/T F
Outline 1 Experimental setup and theory framework 2 Thermoelectricity with cold atoms 3 A cold atom based heat engine
Comparison data-theory in the ballistic case Particle imbalance vs. time for : A 3.5kHz and B 9.3kHz Characteristics vs confinement (ab-initio predictions): τ 0 : Good agreement with theory vs ν z Response R max = T F N(max) N tot T 0
Nanostructuration with harmonic confinement T /T F = 0.1 T /T F = 0.3 Hicks and Dresselhaus, Phys. Rev. B 47 12727 (1993) Hicks and Dresselhaus, Phys. Rev. B 47 16631 (1993)
En route for diffusive conduction Unified description : T (ε) = l(ε) L + l(ε) On the experimental side : Speckle potential l(ε) : mean free path, l(ε) = τ s v(ε) Low disorder l L : T 1 Ballistic transport Strong disorder l L : T l L Diffusive transport Energy-independent τ s Universal regime at strong speckle Seebeck coefficient is a ratio of conduction coefficients
Enhancing thermoelectric response Thermoelectric effect grows with disorder At strong disorder The effect saturates : Constant τ s Seebeck coefficient Conductivity Rescaled evolution of particle imbalance : Universal regime
Systematic comparison between ballistic and diffusive conduction 0.6 R max 0.4 0.2 Diffusive, ν z = 3.5 khz Ballistic 0 0 0.5 1 τ 0 (s) Comparison of ballistic and diffusive channel Disorder more efficient than geometry Resistance thermopower
Outline 1 Experimental setup and theory framework 2 Thermoelectricity with cold atoms 3 A cold atom based heat engine
The setup as a heat engine Hot source Cold source Particles go up the potential hill: Work! Reservoirs Hot and Cold sources Channel : converts heat into (chemical) work QUESTION : Efficiency of the process? Evolution in the µ N plane Access to thermodynamic evolution Extraction of work
Efficiency No DC regime compare work, not power Expression for the efficiency : compare output chemical work to heat η Work Heat = evolution µ d N 0 = dt µ I N evolution T d S 0 dt T I S Solution to transport equations η in terms of transport coefficients : l, L, α αα r η = l + L + α 2 αα r i. Comparison to data? ii. Relation to channel properties? iii. Output power?
Results : Efficiency and ZT η grows with confinement and speckle Slow dynamics most efficient For the channel only : Thermoelectric figure of merit ZT = α2 ch L ZT 2.4 : > than any material
Power and efficiency 0.4 For the channel only : Relative efficiency 0.3 0.2 ZT = 1 ZT = 2 ZT = 2.4 ZT = 4 0.1 0 0 0.25 0.5 0.75 1 P/P max Optimize power optimize efficiency
Results : Output Power Cycle averaged power W τ 0 Optimize power Optimizing efficiency Strong confinement/speckle : Slow dynamics Low power
Conclusions-Outlook Thermoelectricity with cold atoms! Transport : combination of reservoir and channel properties Control on the effect via geometry ( nanostructuration) or transport regime (ballistic -> diffusive) High figure of merit High-T transport without phonons What s next? Superfluid : Thermomechanical (fountain) effects Interactions : improvement of thermopower Lattice? J.P- Brantut, CG, J. Meineke, S. Krinner, D. Stadler, C. Kollath, T. Esslinger & A. Georges Science 342, 713-715 (2013) CG, C. Kollath & A. Georges arxiv:1209.3942
Outlook - Cooling by transport : Peltier effect Peltier effect injection of electron and holes around µ : Rectification of the Fermi distribution Evaporation Here : Design transport properties Injection in a chosen energy window Goal : improve evaporative cooling
Thermodynamic coefficients 6 Experimental 4 2 Regime κe F /N γt F /N C µ T F /Nk B T C N T F /Nk B T 0 0 0.5 1 T/T F
Linear response 16 0.8 Maximal effect (in %) 12 8 4 0.6 τ 0 (s) 0 0 0.2 0.4 0.6 0.8 ΔT 0 /T F 0.4
Lorentz et al 6 (A) S S, trap -S c 4 -S c, trap S r 2 0 (B) 3 L L, trap l 2 0 0.25 0.5 T/T F 0.75 1