Thermoelectrics: A theoretical approach to the search for better materials

Transcription

1 Thermoelectrics: A theoretical approach to the search for better materials Jorge O. Sofo Department of Physics, Department of Materials Science and Engineering, and Materials Research Institute Penn State

2 Abram F. Ioffe The basics

3 The devices

4 The performance T 1 T 2 Q φ = = W φ MAX 2 1 ( 2 1) κ I R /2 2 SI( T2 T1) + I R SIT T T T ZT T2 / T 1 = ( T2 T1) 1+ ZT + 1 Z + T1 ( T T) 2 1 Z = σ S κ 2

5 Conductivity 101 Drude + Sommefeld σ = e 2 n m τ -q q

6 Conductivity 101 y x J = e f ( ) 0 ε v = 0 J = e f v 0

7 f f f H H + = t p r r p f t f {, } H f = t coll d dt ρ 1 i ( t) = H + H ( t), ρ ( t)

8 f f f H H + = t p r r p f t f {, } H f = t coll d dt ρ 1 i ( t) = H + H ( t), ρ ( t)

9 f f f f dr d + + = t dt r dt t coll

10 f f f f dr d + + = t dt r dt t coll f t = 0 f r = 0

11 d dt f f = t coll

12 d dt f f = t coll d 1 dp ee = = dt dt

13 d dt f f = t coll d 1 dp ee = = dt dt ( ε ) ( ε ) ( ε ) f f ε f = = ε ε v

14 d dt f f = t coll d 1 dp ee = = dt dt f t ( ε ) ( ε ) ( ε ) f f ε f ( ε ) 0 coll = = = ε ε f f τ v

15 d dt f f = t coll d 1 dp ee = = dt dt f t ( ε ) ( ε ) ( ε ) f f ε f ( ε ) 0 coll = = = ε ε f f τ v ( ) f 0 f = f 0 ε + e τ v E ε

16 d dt f f = t coll d 1 dp ee = = dt dt f t ( ε ) ( ε ) ( ε ) f f ε f ( ε ) 0 coll = = = ε ε f f τ v ( ) f 0 f = f 0 ε + e τ v E ε

17 ( ) f 0 f = f 0 ε + e τ v E ε

18 J = e f v ( ) f 0 f = f 0 ε + e τ v E ε

19 0 0 ( ) f f f e v E ε τ ε = + J e f v = 2 0 f J e vv E τ ε =

20 0 0 ( ) f f f e v E ε τ ε = + J e f v = 2 0 f J e vv E τ ε = E J = σ Ω

21 J Ω = σ J = e f v ( ) f 0 f = f 0 ε + e τ v E ε E f 2 0 J= e τ vv E ε σ = e f 2 0 τ vv ε

22 S σ = e κ e f J = σ E σs T J = σst E κ T Q τ v ε B = σ f ε τ v 0 2 ( ε µ ) T B ( ε µ ) 2 f = τ v el B ε T B Z = κ σ 0 el + S 2 κ κ κ σ ph 0 2 el = el ST

23 e 2 f f0 σ = e τ v = e dε Σ( ε) ε ε ( ε ) µ eb f ( ε µ ) ( ) f τ ε ε T B S = v = d Σ σ ε T B σ ε B 2 2 ( ε ) µ f ( ε µ ) () 2 f κel = B τ v = B dε Σ ε ε T B ε T B Σ = 2 ( ε) τ vδε ( ε) Transport distribution σ[ Σ ] S[ Σ] κel [ Σ]

24 2 f0 σ = e ( ε ) ε Σ dε S κ el ( ) B f0 ε µ = Σ( ε ) dε σ ε T B Z 2 = σ S Z[ ] 0 κ + κ = Σ Σ best el ph ( ε µ ) 2 2 f0 = B Σ( ε ) ε T B ( ε ) = Cδ ( ε ε 0) dε Σ max Z[ Σ ] = Z[ Σ ] best Σ T ε B The best thermoelectric, G. D. Mahan and J. O. Sofo Proc. Nat. Acad. Sci. USA, 93, 7436 (1996) best

25 The Best Thermoelectric ) ( ) ( ε ε δ τ ε = Σ v v ) ( ) ( ) ( 2 ε τ ε ε v N = ) ( ) ( 0 ε ε δ ε = Σ C best T ε B = ds N ε ε 1 ) ( v ε ε = ) (

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30 J. O. Sofo, G. D. Mahan, Thermoelectric figure of merit of superlattices, Appl. Phys. Lett. 65, 2690 (1994).

31 J. O. Sofo, G. D. Mahan, Thermoelectric figure of merit of superlattices, Appl. Phys. Lett. 65, 2690 (1994).

32 Using Boltzman with ab-initio f 0 σ 2 = e ε τ τ v v v 1 ε 1 = p = pˆ m m C. Ambrosch-Draxl and J. O. Sofo Linear optical properties of solids within the full-potential linearized augmented planewave method Comp. Phys. Commun. 175, 1-14 (2006)

33 q q τ First Born Approximation Defect scattering Crystal defects Impurities Neutral Ionized Alloy Carrier-carrier scattering Lattice scattering Intravalley Acoustic Deformation potential Piezoelectric Optic Non-polar Polar Intervalley Acoustic Optic

34 B. R. Nag Electron Transport in Compound Semiconductors

35 B. R. Nag Electron Transport in Compound Semiconductors

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38 T. J. Scheidemantel, C. Ambrosch-Draxl, T. Thonhauser, J. V. Badding, and J. O. Sofo. Transport Coefficients from First-principles Calculations. Phys. Rev. B 68, (2003) Bi 2 Te 3

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40 Georg Madsen s

41 Careful Doping: rigid band Gap problem Temperature dependence of the electronic structure. Alloys. Single site approximations do not wor. Many -points Correlated materials? Connection with magnetism and topology?

42 Ryogo Kubo

43 Linear Response Theory (Kubo) Valid only close to equilibrium 2 i ne σαβ ω δαβ αβ ω ω m 0 ( ) ( + q, = +Π q, + 0 ) β 1 iωτ n Π αβ ( qi, ωn ) = dτe Tj τ α ( q, τ) jβ ( q, τ) V 0 However Does not need well defined energy bands It is easy to incorporate most low energy excitations of the solid Amenable to diagrammatic expansions and controlled approximations Equivalent to the Boltzmann equation when both are valid.

44 Summary Loo for narrow transmission channels with high velocity Tool to explore new compounds, pressure, negative pressure. Prediction of a new compound by G. Madsen. Easy to expand adding new Scattering Mechanisms Limited to applications on non-correlated semiconductors. Magneto-Thermoelectric effects are beginning to be explored.

45 A final comment: EXPERIMENT<-> SIMULATION<->THEORY

46 Simulations describe complexity. Our theoretical wor is to mae it simple

47 Than you! Z 2 = σ S Z[ ] 0 κ + κ = Σ el ph Σ = v 2 ( ε) τ δε ( ε) Transport distribution

48 J = σ E σs T J = σst E κ T Q eq ( ) ( ) B t B L X t = µ µ µν ν ν eq ( ) χ ( ) ( ) ( ) = + Φ B t B X t dt t t X t µ µ µν ν µν ν ν ν ( s) lim Φ = µν 0 + s t eq ( ) χ ( ) ( ) ( ) = + Φ B t B X t ds s X t s µ µ µν ν µν ν ν 0 ν + + µν = χµν + Φµν 0 ( ) L ds s M. Toda, R. Kubo, N. Saitō, N. Hashitsume, Statistical Physics II, Nonequilibrium Statistical Mechanics. (Springer, Berlin, 2nd. Edition., 1992).

49 ( ) = δ δ ( ) X t t t 1 ν µκ eq ( ) = χ δ ( ) +Φ ( ) θ( ) B t B t t t t t t µ µ µκ 1 µκ 1 1 M. Toda, R. Kubo, N. Saitō, N. Hashitsume, Statistical Physics II, Nonequilibrium Statistical Mechanics. (Springer, Berlin, 2nd. Edition., 1992).

50 ( ) = δ θ( ) X t X t t ν νκ 1 B µ ( t) = + Φ > + eq Bµ + χµκ + ds Φ µκ ( s) for t < t1 0 + eq B µ ds µκ ( s) for t t1 t t 1 + ( t) ds ( s) Ψ = Φ µκ µκ t M. Toda, R. Kubo, N. Saitō, N. Hashitsume, Statistical Physics II, Nonequilibrium Statistical Mechanics. (Springer, Berlin, 2nd. Edition., 1992).

51 Onsager definition of heat flux to define the thermal conductivity Coarse graining in time Appearance of non locality through coarse graining projectors

52 Deformation Potential Calculations Bardeen, J., and W. Shocley. Deformation Potentials and Mobilities in Non-Polar Crystals. Phys. Rev. 80, (1950). D q ( ) = Van de Walle, Chris G. Band Lineups and Deformation Potentials in the Model-solid Theory. Phys. Rev. B 39, (1989). dε q dlnv Wagner, J.-M., and F. Bechstedt. Electronic and Phonon Deformation Potentials of GaN and AlN: Ab Initio Calculations Versus Experiment. Phys. Status Solidi (b) 234, (2002) Lazzeri, Michele, Claudio Attaccalite, Ludger Wirtz, and Francesco Mauri. Impact of the Electronelectron Correlation on Phonon Dispersion: Failure of LDA and GGA DFT Functionals in Graphene and Graphite. Physical Review B 78, no. 8 (August 26, 2008):

53 Relaxation time from e-p interaction f ( p) 2π dq t ( 2π ) { ( ) 1 ( ) 2 = M 3 q f p f p+ q e p ( N q 1) δ ( ε p ε p+ q ω q) N qδ ( ε p ε p+ q ω q) f p+ q f p N N ( ) 1 ( ) ( 1) ( ) ( ) } q δ ε p ε p+ q ω q qδ ε p ε p+ q ω q 1 2π = τ t dq 3 ( ) ( 2π ) 1 0 ( ) M q 2 q 2 ( ) ( ) ( N ( )) q + f + q δ ε ε ω q q + N q + f0 + q δ ( ε ε + ω + + q q) q M q = D 2 ρω q

54 The materials n-type p-type J.-P. Fleurial, DESIGN AND DISCOVERY OF HIGHLY EFFICIENT THERMOELECTRIC MATERIALS Download Design and Discovery, Jet Propulsion Laboratory/California Institute of Technology, 1993.

55 Limitations of the Boltzman Equation Method Also nown as the Kinetic Method because of the relation with classical inetic theory f f f f dr d + + = t dt r dt t According to Kubo, Toda, and Hashitsume (1) cannot be applied when the mean free path is too short (e.g., amorphous semiconductors) or the frequency of the applied fields is too high. However, it is very powerful and can be applied to non linear problems. (1) R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Non-equilibrium Statistical Mechanics (Springer-Verlag, Berlin, 1991) p. 197 coll f H f H f f f + = { H, f } t p r r p t t coll