Thermoelectrics and Thermomagnetics of novel materials and systems

Transcription

1 Thermoelectrics and Thermomagnetics of novel materials and systems August 26, 2016 Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

2 Outline 1 Motivation 2 What has been aimed in the paper? 3 Chemical and electrochemical potential 4 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients 5 Magnetization Currents 6 Mott Relation of Thermopower 7 Sondheimer Relation 8 Thomson Effect 9 Peltier effect 10 Thermal conductivity and Righi-Leduc effect 11 Limitations and Conclusion Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

3 Motivation The theory of thermoelectric(te) and thermomagnetic (TM) phenomena in metals has been built in the 1950s(Sondheimer E. H., Proc. R. Soc. London, Ser A, 193 (1948) 484.) Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

4 Motivation The theory of thermoelectric(te) and thermomagnetic (TM) phenomena in metals has been built in the 1950s(Sondheimer E. H., Proc. R. Soc. London, Ser A, 193 (1948) 484.) Essentially, it is based on the kinetic approach, where more or less complicated transport equations are formulated and solved for different systems in order to obtain the transport co-effecients characterizing the TE and TM effetcs. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

5 Motivation The theory of thermoelectric(te) and thermomagnetic (TM) phenomena in metals has been built in the 1950s(Sondheimer E. H., Proc. R. Soc. London, Ser A, 193 (1948) 484.) Essentially, it is based on the kinetic approach, where more or less complicated transport equations are formulated and solved for different systems in order to obtain the transport co-effecients characterizing the TE and TM effetcs. In the recent decades, the invention of a wide range of new materials with exotic spectra where different types of interaction can interplay (graphene and carbon nanotube being two examples) gave a boost to the studies of the most important TE and TM constants, such as Seebeck, Thomson, Nernst-Ettingshausen and Righi-Leduc co-effecients, thermal conductivity and Peltier tensors. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

6 Motivation The theory of thermoelectric(te) and thermomagnetic (TM) phenomena in metals has been built in the 1950s(Sondheimer E. H., Proc. R. Soc. London, Ser A, 193 (1948) 484.) Essentially, it is based on the kinetic approach, where more or less complicated transport equations are formulated and solved for different systems in order to obtain the transport co-effecients characterizing the TE and TM effetcs. In the recent decades, the invention of a wide range of new materials with exotic spectra where different types of interaction can interplay (graphene and carbon nanotube being two examples) gave a boost to the studies of the most important TE and TM constants, such as Seebeck, Thomson, Nernst-Ettingshausen and Righi-Leduc co-effecients, thermal conductivity and Peltier tensors. Yet the notion of a heat flow, required to find these co-effecients, becomes hardly definable in the case of systems of interacting particles, which is why one can hardly rely on kinetic approaches, in general. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

7 Motivation Such a problem does not appear if one deals with the conductivity tensor which can be always calculated using either transport equations or diagrammatic approaches. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

8 Motivation Such a problem does not appear if one deals with the conductivity tensor which can be always calculated using either transport equations or diagrammatic approaches. Some relations between the TE and TM constants and conductivity tensor are well known for non-interacting systems with simple spectra (Widemann-Franz law and Mott formula), but these relations have not been generalised to the case of interacting systems with exotic spectra so far. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

9 What has been aimed in the paper? A formulation of unified approach to the description of Thermoelectric (TE) and Thermomagnetic (TM) phenomena virtually in any electronic system Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

10 What has been aimed in the paper? A formulation of unified approach to the description of Thermoelectric (TE) and Thermomagnetic (TM) phenomena virtually in any electronic system Establish the universal links between main TE and TM co-effecients and conductivity tensor. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

11 What has been aimed in the paper? A formulation of unified approach to the description of Thermoelectric (TE) and Thermomagnetic (TM) phenomena virtually in any electronic system Establish the universal links between main TE and TM co-effecients and conductivity tensor. It is sufficient to know the temperature dependence of chemical potential to obtain Seebeck, Thomson and Peltier co-effecients. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

12 What has been aimed in the paper? A formulation of unified approach to the description of Thermoelectric (TE) and Thermomagnetic (TM) phenomena virtually in any electronic system Establish the universal links between main TE and TM co-effecients and conductivity tensor. It is sufficient to know the temperature dependence of chemical potential to obtain Seebeck, Thomson and Peltier co-effecients. The Nernst-Ettingshausen, Reghi-Leduc and thermal conductivity co-effecients can be expressed through the conductivity tensor and thermal derivatives of the chemical potential and magnetization of the system. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

13 What has been aimed in the paper? A formulation of unified approach to the description of Thermoelectric (TE) and Thermomagnetic (TM) phenomena virtually in any electronic system Establish the universal links between main TE and TM co-effecients and conductivity tensor. It is sufficient to know the temperature dependence of chemical potential to obtain Seebeck, Thomson and Peltier co-effecients. The Nernst-Ettingshausen, Reghi-Leduc and thermal conductivity co-effecients can be expressed through the conductivity tensor and thermal derivatives of the chemical potential and magnetization of the system. These relations allow for obtaining the TE and TM properties of novel 1D and 2D systems of nomal charge carriers and Dirac fermions, elctron systems with non-trivial topological spectra, etc. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

14 Chemical and electrochemical potential Chemical and electrochemical potentials have non-unique definition in literature! Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

15 Chemical and electrochemical potential Chemical and electrochemical potentials have non-unique definition in literature! One convention (electro-chemist, soft-matter physicists) assumes that electro-chemical potential of a system is a constant in a stationary conditions, while a chemical potential is a local characteristic which may change from point to point of a system. (Bard Allen J. and Faulkner Larry R., Electrochemical Methods: Fundamentals and Applications, 2nd edition (Wiley) 2000, sect (a), p. 4.) Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

16 Chemical and electrochemical potential Chemical and electrochemical potentials have non-unique definition in literature! One convention (electro-chemist, soft-matter physicists) assumes that electro-chemical potential of a system is a constant in a stationary conditions, while a chemical potential is a local characteristic which may change from point to point of a system. (Bard Allen J. and Faulkner Larry R., Electrochemical Methods: Fundamentals and Applications, 2nd edition (Wiley) 2000, sect (a), p. 4.) In the solid-state physics frequently the opposite rule is postulated: a chemical potential is a characteristic of the whole system, and it is a constant in the stationary case, while the electrochemical potential may vary from point to point. (Ashcroft N. W. and Mermin N. D., Solid State Physics (Brooks-Cole) 1976, p. 593.) Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

17 Chemical and electrochemical potential The paper uses the former approach following the textbooks of Madelung (Madelung O., Introduction to Solid-State: Theory(Springer, Berlin) 1995, p. 198) and Abrikosov (Fundamentals of the Theory of Metals (Elsevier, Amsterdam) 1989.) where the concept of local chemical potential to solid-state systems have been implemented. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

18 Chemical and electrochemical potential The paper uses the former approach following the textbooks of Madelung (Madelung O., Introduction to Solid-State: Theory(Springer, Berlin) 1995, p. 198) and Abrikosov (Fundamentals of the Theory of Metals (Elsevier, Amsterdam) 1989.) where the concept of local chemical potential to solid-state systems have been implemented. In this approach, the system subjected to a temperature gradient is assumed to be in thermal equilibrium locally, so that in each small volume of the sample one can introduce the thermodynamic potential Ω[T (r)], r being a coordinate, the number of particles N[T (r)] and the chemical potential µ[t (r)] = Ω N (1) Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

19 Chemical and electrochemical potential The paper uses the former approach following the textbooks of Madelung (Madelung O., Introduction to Solid-State: Theory(Springer, Berlin) 1995, p. 198) and Abrikosov (Fundamentals of the Theory of Metals (Elsevier, Amsterdam) 1989.) where the concept of local chemical potential to solid-state systems have been implemented. In this approach, the system subjected to a temperature gradient is assumed to be in thermal equilibrium locally, so that in each small volume of the sample one can introduce the thermodynamic potential Ω[T (r)], r being a coordinate, the number of particles N[T (r)] and the chemical potential µ[t (r)] = Ω N (1) The chemical potential, defined in this way, may vary in real space, if the temperature of the system varies. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

20 Chemical and electrochemical potential The electrochemical potential is defined as µ = µ + eφ (2) with φ being the electrostatic potential. This quantity remains constant for a whole system at stationary conditions. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

21 Chemical and electrochemical potential The electrochemical potential is defined as µ = µ + eφ (2) with φ being the electrostatic potential. This quantity remains constant for a whole system at stationary conditions. Physically it means that if no electric current flows through the system, its electro-chemical potential is constant, while its chemical potential can vary. Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

22 Chemical Potential Table 1: The temperature dependences of chemical potentials in the systems of different dimensionalities, for carriers having a parabolic and linear dispersion, in the limits of Boltzmann and degenerate Fermi gases. P and D denote the expressions obtained for parabolic and Dirac dispersion cases, respectively. Dimensionality d = 3 d = 2 d = 1 Fermi energy µ (d) P (0) µ (3) P (parabolic spectrum) (0) = ( 3π 2) [ ] 2/3 2 2/3 [ ] 2 2m n (3) (2) e µ π 2 P (0) = m n(2) e µ (1) P (0) = 2π2 2 m n (1) e µ (2) P (T ) = µ(2) [ P (0) ] µ (3) P (T ) = µ(3) P (0) π2 T 2 +T ln 1 e µ (2) P (0) T µ (1) 6µ (3) P (0) P (T ) = µ(1) P (0) π2 T 2 12µ (1) Chemical potential for degenerated FG (parabolic spectrum) T µ (d) P (0) Chemical potential for Boltzman FG (parabolic spectrum) T µ (d) P (0) Fermi energy µ (d) D (0) (Dirac spectrum) Chemical potential for degenerated FG (Dirac spectrum) T µ (d) D (0) Chemical potential for Boltzman FG (Dirac spectrum) T µ (d) D (0) µ (3) P (T ) = 3 2 T ln T µ (3) P (0) µ (2) P µ (2) P [ +T ln (0) T e µ (2) P (0) T (T ) = µ(2) P (0) ] 1 e µ (2) P (0) T T T ln µ (2) P (0) µ (1) P (T ) = T 2 ln π4 T 4µ (1) P (0) µ (3) D (0) = π c 3 3n e µ (2) D (0) = c 2πn e µ (1) D (0) = π cne [ ] (1) µ (3) D (T ) = µ(3) D (0) π2 T 2 µ (2) 3µ (3) D (0) D (T ) = µ(2) D (0) µ π2 T 2 D (T ) = T ln e µ (1) D (0) T 1 12µ (2) D (0) µ (1) (1) µ D D (0) T (0) e T [ ] (1) µ µ (3) D (T ) = 3T ln T µ (2) µ (3) D (0) D (T ) = 2T ln T D (T ) = T ln e µ (1) D (0) T 1 ( ) µ (2) D (0) T T ln µ (1) D (0) P (0) Thermoelectrics and Thermomagnetics of novel materials andaugust systems 26, / 28

23 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients Fig. 1: (Colour on-line) Schematic: geometry of the experiments considered in this paper. A conductor looped via a voltmeter in the ŷ-direction, placed in the magnetic field H oriented along the ẑ axis, and subjected to a temperature gradient x T applied along the ˆx axis. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

24 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients In full generality, one can express the electric current density j components as ( ) ( ) ( ) jx Ex = ˆσ + ˆβ x T 0 j y E y where ˆσ and ˆβ are the conductivity and the thermoelectric tensors, respectively. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

25 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients In full generality, one can express the electric current density j components as ( ) ( ) ( ) jx Ex = ˆσ + ˆβ x T 0 j y E y where ˆσ and ˆβ are the conductivity and the thermoelectric tensors, respectively. The description has been restricted to the limit of magnetic fields weak by a parameter ω c τ 1, where ω c is the cyclotron frequency, and τ is the elastic scattering time. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

26 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients In full generality, one can express the electric current density j components as ( ) ( ) ( ) jx Ex = ˆσ + ˆβ x T 0 j y E y where ˆσ and ˆβ are the conductivity and the thermoelectric tensors, respectively. The description has been restricted to the limit of magnetic fields weak by a parameter ω c τ 1, where ω c is the cyclotron frequency, and τ is the elastic scattering time. The cyclotron frequency is defined differently for normal carriers (NC) having a parabolic dispersion and for Dirac fermions (DF) having a linear dispersion. For NC ω c = eh m c and for DF ω c = ehv 2 F µc. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

27 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients For the heat flow q components one can write a similar equation: ( ) ( ) ( ) qx Ex x T = ˆγ + ˆζ 0 q y E y Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

28 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients For the heat flow q components one can write a similar equation: ( ) ( ) ( ) qx Ex x T = ˆγ + ˆζ 0 q y where ˆγ is related to ˆβ by means of the Onsager relation: E y γ(h) = T β( H) (3) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

29 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients For the heat flow q components one can write a similar equation: ( ) ( ) ( ) qx Ex x T = ˆγ + ˆζ 0 q y where ˆγ is related to ˆβ by means of the Onsager relation: E y and the tensor ˆζ is related to ˆσ through γ(h) = T β( H) (3) ˆζ = π2 T ˆσ (4) 3e2 which determines the value of thermal conductivity ˆκ. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

30 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients For the heat flow q components one can write a similar equation: ( ) ( ) ( ) qx Ex x T = ˆγ + ˆζ 0 q y where ˆγ is related to ˆβ by means of the Onsager relation: E y and the tensor ˆζ is related to ˆσ through γ(h) = T β( H) (3) ˆζ = π2 T ˆσ (4) 3e2 which determines the value of thermal conductivity ˆκ. The above equation is nothing but the Wiedemann-Franz law. The dimensionless quantity π2 k 2 B 3e 2 = watt ohm/k 2 is the Lorentz number. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

31 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients Fig. 1: (Colour on-line) Schematic: geometry of the experiments considered in this paper. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

32 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients Fig. 1: (Colour on-line) Schematic: geometry of the experiments considered in this paper. Let s consider the case where the electric circuits are broken in both the x and y directions, so that j x = 0 Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

33 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients Fig. 1: (Colour on-line) Schematic: geometry of the experiments considered in this paper. Let s consider the case where the electric circuits are broken in both the x and y directions, so that j x = 0 Therefore the electric current density (j) equations are j x = σ xx E x + σ xy E y + β xx x T = 0 (5) j y = σ yx E x + σ yy E y + β yx x T = 0 Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

34 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients Fig. 1: (Colour on-line) Schematic: geometry of the experiments considered in this paper. Let s consider the case where the electric circuits are broken in both the x and y directions, so that j x = 0 Therefore the electric current density (j) equations are j x = σ xx E x + σ xy E y + β xx x T = 0 (5) j y = σ yx E x + σ yy E y + β yx x T = 0 The off-diagonal componensts of the thermoelectric tensor (i.e. β xy, β yx ) differ from zero only due to the non-zero magnetic field Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

35 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients M+ΔM T + x + M Δ E y j y =0 X Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

36 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients M+ΔM T + x + M Δ E y j y =0 X There will be magnetization currents which can be induced if the magnetization in the sample is spatially inhomogeneous. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

37 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients M+ΔM T + x + M Δ E y j y =0 X There will be magnetization currents which can be induced if the magnetization in the sample is spatially inhomogeneous. Inhomogeneity of the magnetization is caused by the temperature gradient. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

38 Thermoelectric(TE) and Thermomagnetic(TM) co-effecients M+ΔM T + x + M Δ E y j y =0 X There will be magnetization currents which can be induced if the magnetization in the sample is spatially inhomogeneous. Inhomogeneity of the magnetization is caused by the temperature gradient. The induced electric current contributes to the Ettingshausen co-effecient. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

39 Magnetization Currents Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

40 Magnetization Currents In this geometry β xy and β yx can be determined from fourth Maxwell s equation or Ampere law j mag = c 4π B (6) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

41 Magnetization Currents In this geometry β xy and β yx can be determined from fourth Maxwell s equation or Ampere law j mag = c 4π B (6) where B = H + 4πM, H is the spatially homogenous external magnetic field, and M is the magnetization. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

42 Magnetization Currents In this geometry β xy and β yx can be determined from fourth Maxwell s equation or Ampere law j mag = c 4π B (6) where B = H + 4πM, H is the spatially homogenous external magnetic field, and M is the magnetization. The magnetization currents are j mag y j mag x = c M z y = c M z x = c M z T y T (7) = c M z T xt Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

43 Magnetization Currents In this geometry β xy and β yx can be determined from fourth Maxwell s equation or Ampere law j mag = c 4π B (6) where B = H + 4πM, H is the spatially homogenous external magnetic field, and M is the magnetization. The magnetization currents are j mag y j mag x = c M z y = c M z x = c M z T y T (7) = c M z T xt And we obtain j x β xy = y T = c M z T β yx = j y x T = c M z T (8) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

44 Magnetization Currents And we establish when there is no transport current j tr = 0 β xy = β yx This equation also holds when there is transport currents. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

45 Magnetization Currents And we establish when there is no transport current j tr = 0 β xy = β yx This equation also holds when there is transport currents. M+ΔM T + x + M Δ E y j y =0 X In full analogy with a classical Hall effect, the magnetization current in the y-direction is compensated by the induced Nernst-Ettingshausen voltage, which yields the electric field Ey mag = ρ yy jy mag (ρ yy is the diagonal of the resistivity tensor, ρ yy = ρ xx ) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

46 Magnetization Currents M+ΔM M + Δ + T x E y X j y =0 The electric field along the x-direction induced by the temperature gradient can be determined using the condition of constancy of the electrochemical potential, [eφ + µ(t (x), n(x))] = 0 Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

47 Magnetization Currents M+ΔM M + Δ + T x E y X j y =0 The electric field along the x-direction induced by the temperature gradient can be determined using the condition of constancy of the electrochemical potential, [eφ + µ(t (x), n(x))] = 0 and can be expressed in terms of the full derivative of the chemical potential: E x = φ = 1 e ( dµ dt ) xt where it is assumed that (1) the electro-neutrality of the system is preserved and (2) no volume charge is formed so that x n = 0 Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

48 Magnetization Currents And consequently we obtain the Seebeck co-effecient S = E x x T = 1 dµ e dt (9) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

49 Magnetization Currents And consequently we obtain the Seebeck co-effecient S = E x x T = 1 dµ e dt (9) Now let us construct the thermoelectric tensor ˆβ. In this geometry no Hall response σ xy = 0 and therefore σ yx = σ xy = 0, E y 0. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

50 Magnetization Currents And consequently we obtain the Seebeck co-effecient S = E x x T = 1 dµ e dt (9) Now let us construct the thermoelectric tensor ˆβ. In this geometry no Hall response σ xy = 0 and therefore σ yx = σ xy = 0, E y 0. From the equation σ xx E x + β xx ( x T ) = 0 it is followed that E β xx = σ x xx x T = σxx e ( dµ dt ) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

51 Magnetization Currents And consequently we obtain the Seebeck co-effecient S = E x x T = 1 dµ e dt (9) Now let us construct the thermoelectric tensor ˆβ. In this geometry no Hall response σ xy = 0 and therefore σ yx = σ xy = 0, E y 0. From the equation σ xx E x + β xx ( x T ) = 0 it is followed that E β xx = σ x xx x T = σxx e ( dµ dt ) Now using x T = 0 and y T 0 one can similarly find β yy = σyy e ( dµ dt ) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

52 Magnetization Currents We have constructed the ˆβ tensor. ( σ xx ˆβ = e ( dµ c dmz dt dt ) c dmz dt σyy e ( dµ dt ) ) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

53 Magnetization Currents We have constructed the ˆβ tensor. ( σ xx ˆβ = e ( dµ which can be written as c dmz dt ˆβ = (ˆσÎ ) 1 e dt ) c dmz dt σyy e ( dµ dt ) dµ dt + êc dm z dt ) (10) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

54 Magnetization Currents We have constructed the ˆβ tensor. ( σ xx ˆβ = e ( dµ which can be written as c dmz dt ˆβ = (ˆσÎ ) 1 e dt ) c dmz dt σyy e ( dµ dt ) dµ dt + êc dm z dt ) (10) Henceforth we can construct the Seebeck tensor Q(H) where Q(H) = ˆσ(H) ˆβ(H) = 1 e ( dµ dt )Î c ˆσ 1 ê( dm dt ) (11) ê = ( 0 ) is the Levi-Civita tensor, Î is the identity tensor. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

55 Mott Relation of Thermopower At zero magnetic field H = 0 Q(T, 0) = 1 e ( dµ dt ) (12) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

56 Mott Relation of Thermopower At zero magnetic field H = 0 Q(T, 0) = 1 e ( dµ dt ) (12) The chemical potential derivative can be explicitly written as dµ dt = µ T + µ n dn dt (13) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

57 Mott Relation of Thermopower At zero magnetic field H = 0 Q(T, 0) = 1 e ( dµ dt ) (12) The chemical potential derivative can be explicitly written as dµ dt = µ T + µ n dn dt (13) For isotropic 3D metal with parabolic spectrum in zero magnetic field and dn dt = ν µ T with ν = mp F π 2 3 µ n = 2 π 4/3 m (3n) 1/3 (14) as the density of states Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

58 Mott Relation of Thermopower Using these expressions one finds dµ dt = 2 µ T (15) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

59 Mott Relation of Thermopower Using these expressions one finds dµ dt = 2 µ T (15) Sommerfeld expansion µ(t ) = µ(0) π2 T 2 6 ν (µ) ν(µ) (16) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

60 Mott Relation of Thermopower Using these expressions one finds dµ dt = 2 µ T (15) Sommerfeld expansion µ(t ) = µ(0) π2 T 2 6 ν (µ) ν(µ) (16) In 3D ν(µ) µ and we obtain Mott formula Q(T, 0) = π2 T 3µ (17) which can also be obtained from kinetic approach. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

61 Sondheimer Relation The Nernst-Ettingshausen effect is the thermal analog of the Hall effect and it consists in the appearance of an electric field E y perpendicular to the mutually perpendicular magnetic field H( z) and temperature gradient x (T ). It is characterized by ν = E y ( x T )H (18) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

62 Sondheimer Relation The Nernst-Ettingshausen effect is the thermal analog of the Hall effect and it consists in the appearance of an electric field E y perpendicular to the mutually perpendicular magnetic field H( z) and temperature gradient x (T ). It is characterized by ν = E y ( x T )H which can be expressed in terms of the resistivity and the thermoelectric tensors ν = ρ xxβ xy + ρ xy β yy H (18) (19) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

63 Sondheimer Relation The Nernst-Ettingshausen effect is the thermal analog of the Hall effect and it consists in the appearance of an electric field E y perpendicular to the mutually perpendicular magnetic field H( z) and temperature gradient x (T ). It is characterized by ν = E y ( x T )H which can be expressed in terms of the resistivity and the thermoelectric tensors ν = ρ xxβ xy + ρ xy β yy H (18) (19) Substituting in the above equation the expressions for the thermoelectric tensor components ν = σ xx e 2 nc ( dµ dt ) + c ρ yy dm z H dt (20) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

64 Sondheimer Relation For 3D metal the magnetization currents are negligible and (k F l) 1 1 with k F being the Fermi wave vector, l being the mean free path. Therefore neglecting the second term one easily reproduces ν = σ xx e 2 nc ( dµ dt ) = σ xx π 2 T e 2 nc 3µ = T τ π2 3µ mc (21) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

65 Sondheimer Relation For 3D metal the magnetization currents are negligible and (k F l) 1 1 with k F being the Fermi wave vector, l being the mean free path. Therefore neglecting the second term one easily reproduces ν = σ xx e 2 nc ( dµ dt ) = σ xx π 2 T e 2 nc 3µ = T τ π2 3µ mc (21) Sondheimer relation ν = π2 T 3µ τ mc (22) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

66 Sondheimer Relation For 3D metal the magnetization currents are negligible and (k F l) 1 1 with k F being the Fermi wave vector, l being the mean free path. Therefore neglecting the second term one easily reproduces ν = σ xx e 2 nc ( dµ dt ) = σ xx π 2 T e 2 nc 3µ = T τ π2 3µ mc (21) Sondheimer relation ν = π2 T 3µ τ mc (22) Quasiparticle Nernst signal N = νh is linear with increasing magnetic field as found in various experiemnts. For metal ν T T F is very small µv k B T, can be large in degenrated semiconductors. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

67 Sondheimer Relation In fluctuating superconductors ν = σ xx e 2 nc ( dµ dt ) + c ρ yy dm z H dt the second term saves the third law of thermodynamics (ν 0 as T 0) in the vicinity of second critical field H c2 (0) (23) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

68 Sondheimer Relation In fluctuating superconductors ν = σ xx e 2 nc ( dµ dt ) + c ρ yy dm z H dt the second term saves the third law of thermodynamics (ν 0 as T 0) in the vicinity of second critical field H c2 (0) The present approach of obtaining ν aslo gives remarkable agreement for graphene Nernst-ettingshausen oscillation as claimed in Lukyanchuk I. A., Varlamov A. A. and Kavokin A. V., Phys. Rev. Lett., 107 (2011) NC: m*=0.04m, µ=0.02ev, H 0 =6.7T, Γ=10K (23) NE oscillations ν/ν NE oscillation ν/ν D: t=2mev 2D: t= Inverse magnetic field H 0 /H 0 NC: m*=0.04m, H=3T, Γ=30K Carriers concentration n (cm 2 ) x NE oscillation ν/ν DF: v=10 cm/s, H=3T, Γ=30K Carriers concentration n (cm 2 ) x FIG. 1 (color online). Normalized Nernst-Ettingshausen (NE) oscillation as a function of the inverse magnetic field and carriers concentration for normal carriers (NC) and Dirac fermions (DF). Dependence ðh 1 Þ for DF has the same profile as for NC but shifted on half period. Vertical lines shows the quantization condition (1). Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

69 Thomson Effect The Thomson co-effecient describes alternatively heating or cooling of a current carrying conductor is expressed as ˆT = T dq dt = T e ( d 2 µ dt 2 )Î ct ˆσ 1 ê( d 2 M dt 2 ) (24) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

70 Thomson Effect The Thomson co-effecient describes alternatively heating or cooling of a current carrying conductor is expressed as ˆT = T dq dt = T e ( d 2 µ dt 2 )Î ct ˆσ 1 ê( d 2 M dt 2 ) (24) In the absence of magnetic field ˆT = T dq dt = T e ( d 2 µ )Î (25) dt 2 Using the expressions for the chemical potential sumcoefficient behaves quite differently for Dirac and normal while for 2D Dirac carriers it differs not only by its tem- (0) the Thomson coefficient for the norsign and grows in its absolute value linearly with temof the Thomson coefficient is non-monotonous for normal Fig. 2: (Colour on-line) Schematic: Thomson coefficient vs. temperature in the 2D case. The Thomson coefficient is positive and non-monotonous for carriers with a parabolic dispersion, while it is negative and saturates in the case of a Dirac spectrum. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

71 Peltier effect The Peltier tensor which describes the heat generation by electric current, is given by the Kelvin relation and can also be expressed in terms of conductivity and thermoelectric tensors, ˆΠ(T, H) = T ˆQ(T, H) = T ˆσ 1 (H) ˆβ(H) (26) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

72 Peltier effect The Peltier tensor which describes the heat generation by electric current, is given by the Kelvin relation and can also be expressed in terms of conductivity and thermoelectric tensors, ˆΠ(T, H) = T ˆQ(T, H) = T ˆσ 1 (H) ˆβ(H) (26) In the absence of magnetic field it has only diagonal components ˆΠ(T, 0) = T e ( dµ dt ) (27) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

73 Peltier effect The Peltier tensor which describes the heat generation by electric current, is given by the Kelvin relation and can also be expressed in terms of conductivity and thermoelectric tensors, ˆΠ(T, H) = T ˆQ(T, H) = T ˆσ 1 (H) ˆβ(H) (26) In the absence of magnetic field it has only diagonal components ˆΠ(T, 0) = T e ( dµ dt ) (27) Interestingly the whole system is out of thermal equilibrium in the presence of electric current, the Peltier tensor can still be linked to the thermal derivative of the chemical potential! Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

74 Thermal conductivity and Righi-Leduc effect The thermal conductivity tensor ˆκ describes the ability of a material subject to a temperature gradient to conduct heat. It can be expressed through the elements of TE tensor and electric conductivity, (κ = q x T when j = 0) ˆκ = π2 T 3e2 ˆσ(Î 3e2 π 2 [ ˆβˆσ 1 ] 2 ) (28) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

75 Thermal conductivity and Righi-Leduc effect The thermal conductivity tensor ˆκ describes the ability of a material subject to a temperature gradient to conduct heat. It can be expressed through the elements of TE tensor and electric conductivity, (κ = q x T when j = 0) ˆκ = π2 T 3e2 ˆσ(Î 3e2 π 2 [ ˆβˆσ 1 ] 2 ) (28) The Righi-Leduc effect describes the heat flow resulting from a perpendicular temperature gradient in the absence of electric current. It can be obtained from x T = LHq y when j = 0 L = σ xx enc κ yy (29) Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

76 Limitations and Conclusion There are certain limitations of this simple approach where largely the electro-neutrality condition x n = 0 is used, which may fail in certain semiconductor systems where volume charge effects are important. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

77 Limitations and Conclusion There are certain limitations of this simple approach where largely the electro-neutrality condition x n = 0 is used, which may fail in certain semiconductor systems where volume charge effects are important. For the same reason, this approach fails to account for electric currents induced by a phonon drag. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

78 Limitations and Conclusion There are certain limitations of this simple approach where largely the electro-neutrality condition x n = 0 is used, which may fail in certain semiconductor systems where volume charge effects are important. For the same reason, this approach fails to account for electric currents induced by a phonon drag. In conclusion, the crucial function which governs all the TE and TM constants as discussed is the temperature derivative of the chemical potential dµ dt. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

79 Limitations and Conclusion There are certain limitations of this simple approach where largely the electro-neutrality condition x n = 0 is used, which may fail in certain semiconductor systems where volume charge effects are important. For the same reason, this approach fails to account for electric currents induced by a phonon drag. In conclusion, the crucial function which governs all the TE and TM constants as discussed is the temperature derivative of the chemical potential dµ dt. This simple observation opens the way to the prediction of the TE and TM coeffecients in new structures and materials. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28

80 Limitations and Conclusion There are certain limitations of this simple approach where largely the electro-neutrality condition x n = 0 is used, which may fail in certain semiconductor systems where volume charge effects are important. For the same reason, this approach fails to account for electric currents induced by a phonon drag. In conclusion, the crucial function which governs all the TE and TM constants as discussed is the temperature derivative of the chemical potential dµ dt. This simple observation opens the way to the prediction of the TE and TM coeffecients in new structures and materials. In particular, this simple approach shows that the TE and TM properties may be strongly different in systems with Dirac fermions and normal carriers. Thermoelectrics and Thermomagnetics of novel materials and August systems 26, / 28