Increasing thermoelectric efficiency towards the Carnot limit

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2 Increasing thermoelectric efficiency towards the Carnot limit Carlos Mejía-Monasterio Département de Physique Théorique, Université de Genève mejia/ ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 1/33

3 Outline 1. Introduction. 2. Microscopic mechanism for η η Carnot. 3. Classical instance: ergodic dilute polyatomic gas. 4. Numerical test: Gas of caffeine molecules in a Lorentz lattice. 5. Final remarks. Giulio Casati, Como. Toma z Prosen, Ljubljana. arxiv: ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 2/33

4 1 Thermoelectric effect Thermoelectricity concerns the conversion of temperature differences into electric potential or vice-versa. Thomas J. Seebeck (1821) ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 3/33

5 1 Thermoelectric effect It can be used to perform useful electrical work or to pump heat from cold to hot place, thus performing refrigeration s Abram Ioffe doped semiconductors exhibit relatively large thermoelectric effect. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 4/33

6 1 Thermoelectric effect It can be used to perform useful electrical work or to pump heat from cold to hot place, thus performing refrigeration s Abram Ioffe doped semiconductors exhibit relatively large thermoelectric effect. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 4/33

7 1 Thermoelectric effect It can be used to perform useful electrical work or to pump heat from cold to hot place, thus performing refrigeration s Abram Ioffe doped semiconductors exhibit relatively large thermoelectric effect. As a result of these efforts the thermoelectric material Bi 2 Te 3 was developed for commercial purposes. However, thermoelectric refrigerators have still poor efficiency as compared to compressor based refrigerators. [equipments in medical applications, space probes etc.]. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 4/33

8 1 Enviromental concerns ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 5/33

9 1 Waste heat recovery ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 6/33

10 1 Thermoelectric efficiency The suitability of a thermoelectric material for energy conversion or electronic refrigeration is evaluated by the thermoelectric figure of merit Z, Z = σs2 κ, σ is the coefficient of electric conductivity. S is the Seebeck coefficient. κ is the thermal conductivity. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 7/33

11 1 Thermoelectric efficiency The suitability of a thermoelectric material for energy conversion or electronic refrigeration is evaluated by the thermoelectric figure of merit Z, Z = σs2 κ, T H Q. T C { L ij}.. Q W Ẇ ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 7/33

12 1 Thermoelectric efficiency The suitability of a thermoelectric material for energy conversion or electronic refrigeration is evaluated by the thermoelectric figure of merit Z, Z = σs2 κ, The thermodynamic efficiency η of converting the heat current J Q (between two baths at temperatures T H and T C ) into the electric power P is η = Ẇ Q = η carnot ZT ZT Good thermoelectrics large values of the ZT. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 7/33

13 1 Thermoelectric efficiency The suitability of a thermoelectric material for energy conversion or electronic refrigeration is evaluated by the thermoelectric figure of merit Z, Z = σs2 κ, At room temperature 1950 s s ZT 1 ( T 100K, η 5) Quantum well devices (2000) ZT = 2.0 Superlattice thin-films (2001) ZT = 2.4 Compressor based refrigerators ZT 3 ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 7/33

14 2 Irreversible thermodynamics In the linear response regime the heat current J Q and electric current J e through an homogeneous sample subjected to a temperature gradient x T and a electrochemical potential gradient x µ are J Q = κ x T TσS x µ, J e = σs x T σ x µ. The electrochemical potential is the sum of a chemical and an electric part µ = µ + µ e, µ e = eφ, e is the particle s charge. E = x φ is the external electric field. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 8/33

15 2 Irreversible thermodynamics In the linear response regime the heat current J Q and electric current J e through an homogeneous sample subjected to a temperature gradient x T and a electrochemical potential gradient x µ are J Q = κ x T TσS x µ, J e = σs x T σ x µ. The usual phenomenological relations follow Ohm s law, ( x T = 0), J e = σe, Fourier s law, (J e = 0) J Q = κ x T, κ = κ TσS 2 Seebeck coefficient (J e = 0) x µ = S x T ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 8/33

16 2 Irreversible thermodynamics We study thermoelectricity from an energy transport point of view. Consider the energy and particle density currents J u and J ρ respectively. To linear order ( J u = L uu 1 ) ( ) x T + Luρ x µ T, ( J ρ = L ρu 1 ) ( ) x T + Lρρ x µ T, Onsager reciprocity relations: L is symmetric, L uρ = L ρu. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 9/33

17 2 Irreversible thermodynamics We study thermoelectricity from an energy transport point of view. Consider the energy and particle density currents J u and J ρ respectively. To linear order ( J u = L uu 1 ) ( ) x T + Luρ x µ T, ( J ρ = L ρu 1 ) ( ) x T + Lρρ x µ T, Onsager reciprocity relations: L is symmetric, L uρ = L ρu. Identifying J e = ej ρ and using the energy conservation J u = J Q + µj ρ ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 9/33

18 2 Irreversible thermodynamics σ = e2 T L ρρ κ = 1 T 2 det L L ρρ S = 1 et ( ) Luρ L ρρ µ ZT = (L uρ µl ρρ ) 2 det L Carnot s limit ZT = is reached when the energy density current and the electric current become proportional, since then det L = 0. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 10/33

19 2 Increasing thermoelectric efficiency Ex: both energy and charge are carried only by non-interacting particles, like in a dilute gas. The microscopic instantaneous currents per particle at position x and time t, are j u (x, t) = E(t)v x (x(t), t)δ(x x(t)), j e (x, t) = ev x δ(x x(t)), E is the energy of the particle, x its position and v x its velocity along the field. The thermodynamic averages of the two currents become proportional precisely when the variables E and v x are un-correlated J u = j u = E v x = E e j e = E J ρ. ZT = : the average particle s energy E does not depend on the thermodynamic forces. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 11/33

20 3 Ergodic Billiards Noninteracting free particles moving inside a box H(p, q) = p2 2m + V (q), V (q) = { 0 q Ω q / Ω Ergodic dynamics take almost all sets of the phase space all over the phase space ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 12/33

21 3 Thermochemical baths Bath System P n (v n ) = m T v n exp ( ) mv2 n 2T P t (v t ) = ( m 2πT exp mv2 t 2T ) Thermochemical reservoirs ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 13/33

22 3 Heat and particle transport n L, T L n R, T R ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 14/33

23 3 Heat and particle transport n L, T L n R, T R p t γ L p r γ L ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 14/33

24 3 Heat and particle transport n L, T L p t γ L p r γ L n R, T R p r γ R pt γr ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 14/33

25 3 Heat and particle transport n L, T L n R, T R p t γ L p t γ R J ρ = p t (γ L γ R ) ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 14/33

26 3 Heat and particle transport Heat and particle flux J ρ = p t (γ L γ R ) J u = p t (ε L ε R ) γ particle injection rate ε energy injection rate ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 14/33

27 3 Heat and particle transport Heat and particle flux J ρ = p t (γ L γ R ) J u = p t (ε L ε R ) For noninteracting particles: p t is a property of the geometry of the billiard only. γ particle injection rate ε energy injection rate ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 14/33

28 3 Heat and particle transport Heat and particle flux J ρ = p t (γ L γ R ) J u = p t (ε L ε R ) For noninteracting particles: p t is a property of the geometry of the billiard only. γ particle injection rate ε energy injection rate does not depend on the density or the temperature of the particles. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 14/33

29 3 Heat and particle transport Heat and particle flux J ρ = p t (γ L γ R ) J u = p t (ε L ε R ) For noninteracting particles: p t is a property of the geometry of the billiard only. γ particle injection rate ε energy injection rate does not depend on the density or the temperature of the particles. has no spatial dependence: p t;lr = p t;rl. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 14/33

30 3 Heat and particle transport Heat and particle flux J ρ = p t (γ L γ R ) J u = p t (ε L ε R ) For noninteracting particles: p t is a property of the geometry of the billiard only. γ particle injection rate ε energy injection rate does not depend on the density or the temperature of the particles. has no spatial dependence: p t;lr = p t;rl. If particles interact then none of these properties is true p t p t (n(x), T(x)) J.-P. Eckmann, C. Mejia-Monasterio, E. Zabey; JSP, 123, (2006) J.-P. Eckmann, C. Mejia-Monasterio PRL, 97, (2006) ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 14/33

31 3 Injection rates Number of particles that are injected into the system with a velocity v in a time interval dt ρσv cos θdt. BATH x SYSTEM vcos dt θ θ y v σ z ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 15/33

32 3 Injection rates W(v)dvdt = 2π 0 dφ π/2 0 dθv 2 sinθ (ρσv cos θ) ( m 2πkT ) 3/2 e mv2 2kT dvdt, BATH x SYSTEM v θ σ z vcos dt θ y ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 15/33

33 3 Injection rates W(v)dvdt = 2π 0 = πρσ dφ π/2 0 ( m 2πkT dθv 2 sinθ (ρσv cos θ) ) 3/2 v 3 e mv2 2kT dvdt. ( m 2πkT ) 3/2 e mv2 2kT dvdt, BATH x SYSTEM vcos dt θ θ y v σ z ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 15/33

34 3 Injection rates W(v)dvdt = 2π 0 = πρσ dφ π/2 0 ( m 2πkT dθv 2 sinθ (ρσv cos θ) ) 3/2 v 3 e mv2 2kT dvdt. ( m 2πkT ) 3/2 e mv2 2kT dvdt, W(E)dEdt = ρσ E (2πmkT) 1/2 kt e E kt dedt. BATH v x SYSTEM θ σ z vcos dt θ y ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 15/33

35 3 Injection rates In terms of W(E) the injection rates can be written as follows: and the energy injection rate is γ = 0 W(E)dE, ε = γ E eff, with E eff the mean energy w.r.t. W(E) E eff = 0 EW(E)dE 0 W(E)dE. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 16/33

36 3 Injection rates Let d = D + D int be the number of degrees of freedom. Then γ = λ (2πm) 1/2 ρ(kt)1/2, ε = d λ (2πm) 1/2 ρ(kt)3/2. Chemical potential of a d-d.o.f. ideal gas µ = ktln c dρ (kt) d/2, c d = constant. or ρ = (kt)d/2 c d e (µ/kt). ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 17/33

37 Particle flux 3 Fluxes J ρ = p t (γ L γ R ) ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 18/33

38 Particle flux 3 Fluxes J ρ = p t (γ L γ R ) = λp t (2πm) 1/2 ( ) ρ L T 1/2 L ρ RT 1/2 R ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 18/33

39 Particle flux 3 Fluxes J ρ = p t (γ L γ R ) = λp t (2πm) 1/2 = λp t L 1/2 (2πm) ( ) ρ L T 1/2 L ρ RT 1/2 R (ρt 1/2) ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 18/33

40 Particle flux 3 Fluxes J ρ = p t (γ L γ R ) = λp t (2πm) 1/2 = λp t L 1/2 (2πm) = λp tl c d (2πm) 1/2 ( ) ρ L T 1/2 L ρ RT 1/2 R (ρt 1/2) (T (d+1)/2 e (µ/t)) ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 18/33

41 Particle flux 3 Fluxes J ρ = p t (γ L γ R ) = λp t (2πm) 1/2 = λp t L 1/2 (2πm) = λp tl c d (2πm) 1/2 = λp tl c d (2πm) 1/2 ( ) ρ L T 1/2 L ρ RT 1/2 R (ρt 1/2) (T (d+1)/2 e (µ/t)) ( d T (d 1)/2 (T) + T (d+1)/2 ( µ T ) ) e (µ/t) ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 18/33

42 Particle flux 3 Fluxes J ρ = p t (γ L γ R ) = λp t (2πm) 1/2 = λp t L 1/2 (2πm) = λp tl c d (2πm) 1/2 = λp tl c d (2πm) 1/2 = λp tl (2πm) 1/2 ( ) ρ L T 1/2 L ρ RT 1/2 R (ρt 1/2) (T (d+1)/2 e (µ/t)) ( d + 1 ( µ 2 T (d 1)/2 (T) + T (d+1)/2 T ( d + 1 ( µ ) ) 2 ρt 1/2 (T) + T 1/2 T ) ) e (µ/t) ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 18/33

43 Particle flux 3 Fluxes J ρ = p t (γ L γ R ) = λp t (2πm) 1/2 ( ) ρ L T 1/2 L ρ RT 1/2 R (ρt 1/2) = λp t L 1/2 (2πm) = λp tl (T c d (2πm) (d+1)/2 e (µ/t)) 1/2 = λp ( tl d + 1 ( µ ) ) c d (2πm) 1/2 2 T (d 1)/2 (T) + T (d+1)/2 e (µ/t) T = λp ( tl d + 1 ( µ ) ) (2πm) 1/2 2 ρt 1/2 (T) + T 1/2 T = λp ( ( ) tl d ( (2πm) 1/2 2 ρt 3/2 + ρt 1/2 µ ) ) T T ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 18/33

44 3 Fluxes J ρ = λp tl (2πm) 1/2 J u = d ( ( d ρt 3/2 T λp t L (2πm) 1/2 ) ( ( d ρt 5/2 T + ρt 1/2 ) ( µ T + ρt 3/2 ) ) ( µ T ) ) ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 19/33

45 J ρ = 3 Fluxes λp tl (2πm) 1/2 J u = d ( ( d ρt 3/2 T λp t L (2πm) 1/2 ) ( ( d ρt 5/2 T + ρt 1/2 ) ( µ T + ρt 3/2 ) ) ( µ T ) ) L ρρ = λp tl 1/2 ρt 1/2 (2πm) λp t L L ρu = L uρ = d L uu = (d + 1)(d + 3) 4 3/2 ρt 1/2 (2πm) λp t L 5/2 ρt 1/2 (2πm) In the absence of ballistic contribution p t 1/L. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 19/33

46 3 Transport coefficients Taking (µ/t) = e φ/t : σ = e2 L ρρ T κ = det L T 2 L ρρ = = e2 λp t Lρ (2πmT) 1/2, λp tlρ (2πm) 1/2 ( ) d + 1 T 1/2, 2 Wiedemann-Franz law S = L uρ etl ρρ = d + 1 (2e). κ σ = d + 1 2e 2 T. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 20/33

47 3 Dilute polyatomic ideal gas For an ergodic gas of non-interacting particles with D int internal degrees of freedom enclosed in a D dimensional container ZT = d + 1 2, d = D + D int e.g. ZT = 2 for dilute mono-atomic gas in 3 dimensions. ZT is independent of the sample size L. It is also temperature independent as expected for billiard-type systems. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 21/33

48 4 Polyatomic Lorentz gas T L, µ L T R, µ R ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 22/33

49 4 Polyatomic Lorentz gas ω 1 ω 2 ω 3... ω Dint Compound molecule: Each particle" of mass m can be imagined as a stack of D int small identical disks of mass m/d int and radius r R, rotating freely and independently at a constant angular velocity ω i, i = 1,...D int. The center of mass of the particle moves with velocity v = (v x, v y ). ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 23/33

50 ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 24/33 4 Energy mixing v n v t ω 1 ω 2.. ω D = Dη 1+Dη 2η 1+Dη 2η 1+Dη 2η 1+Dη Dη 1 2 D(1+Dη) 2 D(1+Dη) 2 D(1+Dη) Dη 2 D(1+Dη) 1 2 D(1+Dη) 2 D(1+Dη) Dη 2 D(1+Dη) 2 D(1+Dη) 1 2 D(1+Dη) v n v t ω 1 ω 2.. ω D Here η = Θ/mR 2

51 4 Polyatomic Lorentz gas 10 8 L uu L ab L uρ d L ρρ ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 25/33

52 4 Polyatomic Lorentz gas ZT 10 5 ZT L d ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 26/33

53 4 Polyatomic Lorentz gas P(ε) P(ξ i ) ξ i ε ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 27/33

54 4 Heat Engine T H V T C ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 28/33

55 4 Heat Engine ξ TH ξ TC ξ TH ξ TC E ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 29/33

56 4 Power W E Q H is the heat injected by the hot bath into the system W = J e E ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 30/33

57 4 Interactions ZT ~ d figure of merit η ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 31/33

58 5 Final remarks We have found a simple general theoretical mechanism for high-efficiency thermoelectricity. Surprinsingly, the thermoelectric efficiency of a 3D-monoatomic ideal gas is comparable with the efficiencies of nano-structured complex materials. Even though the case of an ionized polyatomic gas may seem a little artificial in this context, there may be other important instances where each charge would be carried by many effectively classical degrees of freedom. We have also performed the first numerical computation of ZT from deterministic microscopic equations of motion. Our method can easily be implemented for more realistic models where also quantum effects can be taken into account. ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 32/33

How future may look like... http://www.zts.

59 How future may look like... ODYN-III, Lille, March 15, 2008 Thermoelectric efficiency - p. 33/33

Introduction to a few basic concepts in thermoelectricity

Introduction to a few basic concepts in thermoelectricity Giuliano Benenti Center for Nonlinear and Complex Systems Univ. Insubria, Como, Italy 1 Irreversible thermodynamic Irreversible thermodynamics