Thermoelectrics and Thermomagnetics of novel materials and ...qcmjc/talk_slides/QCMJC.2016.08.25_Kingshuk.pdfOutline 1 Motivation 2 What has been "aimed" in the paper ? 3 Chemical

Thermoelectrics and Thermomagnetics of novelmaterials and systems

August 26, 2016

Thermoelectrics and Thermomagnetics of novel materials and systemsAugust 26, 2016 1 / 28

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


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 lesscomplicated transport equations are formulated and solved fordifferent systems in order to obtain the transport co-effecientscharacterizing the TE and TM effetcs.

In the recent decades, the invention of a wide range of new materialswith exotic spectra where different types of interaction can interplay(graphene and carbon nanotube being two examples) gave a boost tothe studies of the most important TE and TM constants, such asSeebeck, Thomson, Nernst-Ettingshausen and Righi-Leducco-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 interactingparticles, which is why one can hardly rely on kinetic approaches, ingeneral.


Motivation






Motivation






Motivation






Motivation

Such a problem does not appear if one deals with the conductivitytensor which can be always calculated using either transportequations or diagrammatic approaches.

Some relations between the TE and TM constants and conductivitytensor are well known for non-interacting systems with simple spectra(Widemann-Franz law and Mott formula), but these relations havenot been generalised to the case of interacting systems with exoticspectra so far.


Motivation

Such a problem does not appear if one deals with the conductivitytensor which can be always calculated using either transportequations or diagrammatic approaches.

Some relations between the TE and TM constants and conductivitytensor are well known for non-interacting systems with simple spectra(Widemann-Franz law and Mott formula), but these relations havenot been generalised to the case of interacting systems with exoticspectra so far.


What has been ”aimed” in the paper ?

A formulation of unified approach to the description ofThermoelectric (TE) and Thermomagnetic (TM) phenomena virtuallyin any electronic system

Establish the universal links between main TE and TM co-effecientsand conductivity tensor.

It is sufficient to know the temperature dependence of chemicalpotential to obtain Seebeck, Thomson and Peltier co-effecients.

The Nernst-Ettingshausen, Reghi-Leduc and thermal conductivityco-effecients can be expressed through the conductivity tensor andthermal derivatives of the chemical potential and magnetization ofthe system.

These relations allow for obtaining the TE and TM properties of novel1D and 2D systems of nomal charge carriers and Dirac fermions,elctron systems with non-trivial topological spectra, etc.






























Chemical and electrochemical potential

Chemical and electrochemical potentials have ”non-unique definitionin literature” !

One convention (electro-chemist, soft-matter physicists) assumes thatelectro-chemical potential of a system is a constant in a stationaryconditions, while a chemical potential is a local characteristic whichmay change from point to point of a system. (Bard Allen J. andFaulkner Larry R., Electrochemical Methods: Fundamentals andApplications, 2nd edition (Wiley) 2000, sect. 2.2.4(a), p. 4.)

In the solid-state physics frequently the opposite rule is postulated: achemical potential is a characteristic of the whole system, and it is aconstant in the stationary case, while the electrochemical potentialmay vary from point to point. (Ashcroft N. W. and Mermin N. D.,Solid State Physics (Brooks-Cole) 1976, p. 593.)













The paper uses the former approach following the textbooks ofMadelung (Madelung O., Introduction to Solid-State:Theory(Springer, Berlin) 1995, p. 198) and Abrikosov (Fundamentalsof the Theory of Metals (Elsevier, Amsterdam) 1989.) where theconcept of local chemical potential to solid-state systems have beenimplemented.

In this approach, the system subjected to a temperature gradient isassumed to be in thermal equilibrium locally, so that in each smallvolume of the sample one can introduce the thermodynamic potentialΩ[T (r)], r being a coordinate, the number of particles N[T (r)] andthe chemical potential

µ[T (r)] = −∂Ω

∂N(1)

The chemical potential, defined in this way, may vary in real space, ifthe temperature of the system varies.





µ[T (r)] = −∂Ω

∂N(1)






µ[T (r)] = −∂Ω

∂N(1)




The electrochemical potential is defined as

µ = µ+ eφ (2)

with φ being the electrostatic potential. This quantity remainsconstant for a whole system at stationary conditions.

Physically it means that if no electric current flows through thesystem, its electro-chemical potential is constant, while its chemicalpotential can vary.



The electrochemical potential is defined as

µ = µ+ eφ (2)

with φ being the electrostatic potential. This quantity remainsconstant for a whole system at stationary conditions.

Physically it means that if no electric current flows through thesystem, its electro-chemical potential is constant, while its chemicalpotential can vary.


Chemical Potential

Table 1: The temperature dependences of chemical potentials in the systems of different dimensionalities, for carriers havinga parabolic and linear dispersion, in the limits of Boltzmann and degenerate Fermi gases. P and D denote the expressionsobtained for parabolic and Dirac dispersion cases, respectively.

Dimensionality d = 3 d = 2 d = 1

Fermi energy µ(d)P (0)

(parabolic spectrum)µ(3)P (0) =

(3π2

)2/3 ~2

2m

[n(3)e

]2/3µ(2)P (0) = π~2

m n(2)e µ

(1)P (0) = 2π2~2

m

[n(1)e

]2

Chemical potentialfor degenerated FG(parabolic spectrum)

T ≪ µ(d)P (0)

µ(3)P (T ) = µ

(3)P (0)− π2T 2

6µ(3)P (0)

µ(2)P (T ) = µ

(2)P (0)

+T ln

[1− e−

µ(2)P

(0)

T

]

≃ µ(2)P (0)− Te−

µ(2)P

(0)

T

µ(1)P (T ) = µ

(1)P (0)− π2T 2

12µ(1)P (0)

Chemical potentialfor Boltzman FG(parabolic spectrum)

T ≫ µ(d)P (0)

µ(3)P (T ) = − 3

2T ln T

µ(3)P (0)

µ(2)P (T ) = µ

(2)P (0)

+T ln

[1− e−

µ(2)P

(0)

T

]

≃ −T ln T

µ(2)P (0)

µ(1)P (T ) = −T

2 ln π4T

4µ(1)P (0)

Fermi energy µ(d)D (0)

(Dirac spectrum)µ(3)D (0) = π~c 3

√3ne µ

(2)D (0) = ~c

√2πne µ

(1)D (0) = π~cne

Chemical potentialfor degenerated FG(Dirac spectrum)

T ≪ µ(d)D (0)

µ(3)D (T ) = µ

(3)D (0)− π2T 2

3µ(3)D (0)

µ(2)D (T ) = µ

(2)D (0)− π2T 2

12µ(2)D (0)

µ(1)D (T ) = T ln

[e

µ(1)D

(0)

T − 1

]

≃ µ(1)D (0)− Te−

µ(1)D

(0)

T

Chemical potentialfor Boltzman FG(Dirac spectrum)

T ≫ µ(d)D (0)

µ(3)D (T ) = −3T ln T

µ(3)D (0)

µ(2)D (T ) = −2T ln T

µ(2)D (0)

µ(1)D (T ) = T ln

[e

µ(1)D

(0)

T − 1

]

≃ −T ln

(T

µ(1)D (0)

)


Thermoelectric(TE) and Thermomagnetic(TM) co-effecients

Fig. 1: (Colour on-line) Schematic: geometry of the experi-ments considered in this paper.

A conductor looped via a voltmeter in the y -direction, placed in themagnetic field H oriented along the z axis, and subjected to a temperaturegradient ∇xT applied along the x axis.



In full generality, one can express the electric current density jcomponents as (

jxjy

)= σ

(Ex

Ey

)+ β

(∇xT

0

)

where σ and β are the conductivity and the thermoelectric tensors,respectively.

The description has been restricted to the limit of magnetic fieldsweak by a parameter ωcτ 1, where ωc 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 ωc = eHm⊥c and for DF ωc =

eHv2F

µc .




jxjy

)= σ

(Ex

Ey

)+ β

(∇xT

0

)





eHv2F

µc .




jxjy

)= σ

(Ex

Ey

)+ β

(∇xT

0

)





eHv2F

µc .



For the heat flow q components one can write a similar equation:(qxqy

)= γ

(Ex

Ey

)+ ζ

(∇xT

0

)

where γ is related to β by means of the Onsager relation:

γ(H) = −Tβ(−H) (3)

and the tensor ζ is related to σ through

ζ = −π2T

3e2σ (4)

which determines the value of thermal conductivity κ. The aboveequation is nothing but the Wiedemann-Franz law. The dimensionless

quantityπ2k2

B3e2 = 2.44× 10−8 watt− ohm/K2 is the Lorentz number.




)= γ

(Ex

Ey

)+ ζ

(∇xT

0

)


γ(H) = −Tβ(−H) (3)


ζ = −π2T

3e2σ (4)


quantityπ2k2





)= γ

(Ex

Ey

)+ ζ

(∇xT

0

)


γ(H) = −Tβ(−H) (3)


ζ = −π2T

3e2σ (4)

which determines the value of thermal conductivity κ.

The aboveequation is nothing but the Wiedemann-Franz law. The dimensionless

quantityπ2k2





)= γ

(Ex

Ey

)+ ζ

(∇xT

0

)


γ(H) = −Tβ(−H) (3)


ζ = −π2T

3e2σ (4)


quantityπ2k2





Let’s consider the case where the electric circuits are broken in boththe x and y directions, so that jx = 0Therefore the electric current density (j) equations are

jx = σxxEx + σxyEy + βxx∇xT = 0 (5)

jy = σyxEx + σyyEy + βyx∇xT = 0

The off-diagonal componensts of the thermoelectric tensor (i.e.βxy , βyx) differ from zero only due to the non-zero magnetic fieldapplied.




Let’s consider the case where the electric circuits are broken in boththe x and y directions, so that jx = 0

Therefore the electric current density (j) equations are

















The off-diagonal componensts of the thermoelectric tensor (i.e.βxy , βyx) differ from zero only due to the non-zero magnetic fieldapplied. Thermoelectrics and Thermomagnetics of novel materials and systemsAugust 26, 2016 13 / 28


M

M+ΔM

++

++

+

jy=0

X

Δ

Tx E

y

There will be magnetization currents which can be induced if themagnetization in the sample is spatially inhomogeneous.

Inhomogeneity of the magnetization is caused by the temperaturegradient.

The induced electric current contributes to the Ettingshausenco-effecient.



M

M+ΔM

++

++

+

jy=0

X

Δ

Tx E

y






M

M+ΔM

++

++

+

jy=0

X

Δ

Tx E

y






M

M+ΔM

++

++

+

jy=0

X

Δ

Tx E

y





Magnetization Currents

In this geometry βxy and βyx can be determined from fourthMaxwell’s equation or Ampere law

jmag =c

4π∇× B (6)

where B = H + 4πM, H is the spatially homogenous externalmagnetic field, and M is the magnetization.The magnetization currents are

jmagx = c

∂Mz

∂y= c

∂Mz

∂T∇yT (7)

jmagy = −c ∂Mz

∂x= −c ∂Mz

∂T∇xT

And we obtain

βxy =jx∇yT

= c∂Mz

∂T(8)

βyx =jy∇xT

= −c ∂Mz

∂T




jmag =c

4π∇× B (6)


jmagx = c

∂Mz

∂y= c

∂Mz

∂T∇yT (7)

jmagy = −c ∂Mz

∂x= −c ∂Mz

∂T∇xT

And we obtain

βxy =jx∇yT

= c∂Mz

∂T(8)

βyx =jy∇xT

= −c ∂Mz

∂T




jmag =c

4π∇× B (6)

where B = H + 4πM, H is the spatially homogenous externalmagnetic field, and M is the magnetization.

The magnetization currents are

jmagx = c

∂Mz

∂y= c

∂Mz

∂T∇yT (7)

jmagy = −c ∂Mz

∂x= −c ∂Mz

∂T∇xT

And we obtain

βxy =jx∇yT

= c∂Mz

∂T(8)

βyx =jy∇xT

= −c ∂Mz

∂T




jmag =c

4π∇× B (6)


jmagx = c

∂Mz

∂y= c

∂Mz

∂T∇yT (7)

jmagy = −c ∂Mz

∂x= −c ∂Mz

∂T∇xT

And we obtain

βxy =jx∇yT

= c∂Mz

∂T(8)

βyx =jy∇xT

= −c ∂Mz

∂T




jmag =c

4π∇× B (6)


jmagx = c

∂Mz

∂y= c

∂Mz

∂T∇yT (7)

jmagy = −c ∂Mz

∂x= −c ∂Mz

∂T∇xT

And we obtain

βxy =jx∇yT

= c∂Mz

∂T(8)

βyx =jy∇xT

= −c ∂Mz

∂T



And we establish when there is no transport current jtr = 0

βxy = −βyx

This equation also holds when there is transport currents.

M

M+ΔM

++

++

+

jy=0

X

Δ

Tx E

y

In full analogy with a classical Hall effect, the magnetization current inthe y-direction is compensated by the induced Nernst-Ettingshausenvoltage, which yields the electric field Emag

y = ρyy jmagy (ρyy is the

diagonal of the resistivity tensor, ρyy = ρxx)



And we establish when there is no transport current jtr = 0

βxy = −βyx

This equation also holds when there is transport currents.

M

M+ΔM

++

++

+

jy=0

XΔ

Tx E

y

In full analogy with a classical Hall effect, the magnetization current inthe y-direction is compensated by the induced Nernst-Ettingshausenvoltage, which yields the electric field Emag

y = ρyy jmagy (ρyy is the

diagonal of the resistivity tensor, ρyy = ρxx)



M

M+ΔM

++

++

+

jy=0

X

Δ

Tx E

y

The electric field along the x-direction induced by the temperaturegradient can be determined using the condition of constancy of theelectrochemical potential,

∇[eφ+ µ(T (x), n(x))] = 0

and can be expressed in terms of the full derivative of the chemicalpotential:

Ex = −∇φ =1

e(dµ

dT)∇xT

where it is assumed that (1) the electro-neutrality of the system ispreserved and (2) no volume charge is formed so that ∇xn = 0



M

M+ΔM

++

++

+

jy=0

X

Δ

Tx E

y

The electric field along the x-direction induced by the temperaturegradient can be determined using the condition of constancy of theelectrochemical potential,

∇[eφ+ µ(T (x), n(x))] = 0

and can be expressed in terms of the full derivative of the chemicalpotential:

Ex = −∇φ =1

e(dµ

dT)∇xT

where it is assumed that (1) the electro-neutrality of the system ispreserved and (2) no volume charge is formed so that ∇xn = 0



And consequently we obtain the Seebeck co-effecient

S =Ex

∇xT=

1

e

dµ

dT(9)

Now let us construct the thermoelectric tensor β. In this geometry noHall response σxy = 0 and therefore σyx = −σxy = 0, Ey 6= 0.

From the equation σxxEx + βxx(∇xT ) = 0 it is followed thatβxx = −σxx Ex

∇xT= −σxx

e ( dµdT )

Now using ∇xT = 0 and ∇yT 6= 0 one can similarly find

βyy = −σyye ( dµ

dT )




S =Ex

∇xT=

1

e

dµ

dT(9)



∇xT= −σxx

e ( dµdT )



dT )




S =Ex

∇xT=

1

e

dµ

dT(9)



∇xT= −σxx

e ( dµdT )



dT )




S =Ex

∇xT=

1

e

dµ

dT(9)



∇xT= −σxx

e ( dµdT )



dT )



We have constructed the β tensor.

β =

(−σxx

e ( dµdT ) c dMz

dT

−c dMzdT −σyy

e ( dµdT )

)

which can be written as

β = −(σI )1

e

dµ

dT+ ec

dMz

dT(10)

Henceforth we can construct the Seebeck tensor Q(H)

Q(H) = −σ(H)β(H) =1

e(dµ

dT)I − cσ−1e(

dM

dT) (11)

where

e =

(0 1−1 0

)

is the Levi-Civita tensor, I is the identity tensor.




β =

(−σxx

e ( dµdT ) c dMz

dT

−c dMzdT −σyy

e ( dµdT )

)


β = −(σI )1

e

dµ

dT+ ec

dMz

dT(10)


Q(H) = −σ(H)β(H) =1

e(dµ

dT)I − cσ−1e(

dM

dT) (11)

where

e =

(0 1−1 0

)





β =

(−σxx

e ( dµdT ) c dMz

dT

−c dMzdT −σyy

e ( dµdT )

)


β = −(σI )1

e

dµ

dT+ ec

dMz

dT(10)


Q(H) = −σ(H)β(H) =1

e(dµ

dT)I − cσ−1e(

dM

dT) (11)

where

e =

(0 1−1 0

)



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

∂µ

∂n=

~2

m

π4/3

(3n)1/3(14)

and dndT = ν ∂µ∂T with ν = mpF

π2~3 as the density of states




Q(T , 0) =1

e(dµ

dT) (12)


dµ

dT=∂µ

∂T+∂µ

∂n

dn

dT(13)


∂µ

∂n=

~2

m

π4/3

(3n)1/3(14)






Q(T , 0) =1

e(dµ

dT) (12)


dµ

dT=∂µ

∂T+∂µ

∂n

dn

dT(13)


∂µ

∂n=

~2

m

π4/3

(3n)1/3(14)





Using these expressions one finds

dµ

dT= 2

∂µ

∂T(15)

Sommerfeld expansion

µ(T ) = µ(0)− π2T 2

6

ν ′(µ)

ν(µ)(16)

In 3D ν(µ) ∼ √µ and we obtain Mott formula

Q(T , 0) = −π2T

3µ(17)

which can also be obtained from kinetic approach.




dµ

dT= 2

∂µ

∂T(15)


µ(T ) = µ(0)− π2T 2

6

ν ′(µ)

ν(µ)(16)


Q(T , 0) = −π2T

3µ(17)





dµ

dT= 2

∂µ

∂T(15)


µ(T ) = µ(0)− π2T 2

6

ν ′(µ)

ν(µ)(16)


Q(T , 0) = −π2T

3µ(17)



Sondheimer Relation

The Nernst-Ettingshausen effect is the thermal analog of the Halleffect and it consists in the appearance of an electric field Ey

perpendicular to the mutually perpendicular magnetic field H(||z) andtemperature gradient ∇x(T ). It is characterized by

ν =Ey

(−∇xT )H(18)

which can be expressed in terms of the resistivity and thethermoelectric tensors

ν = −ρxxβxy + ρxyβyyH

(19)

Substituting in the above equation the expressions for thethermoelectric tensor components

ν =σxxe2nc

(dµ

dT) + c

ρyyH

dMz

dT(20)


Sondheimer Relation



ν =Ey

(−∇xT )H(18)



(19)


ν =σxxe2nc

(dµ

dT) + c

ρyyH

dMz

dT(20)


Sondheimer Relation



ν =Ey

(−∇xT )H(18)



(19)


ν =σxxe2nc

(dµ

dT) + c

ρyyH

dMz

dT(20)


Sondheimer Relation

For 3D metal the magnetization currents are negligible and(kF l)

−1 1 with kF being the Fermi wave vector, l being the meanfree path. Therefore neglecting the second term one easily reproduces

ν =σxxe2nc

(dµ

dT) = − σxx

e2nc

π2T

3µ= −π

2T

3µ

τ

mc(21)

Sondheimer relation

ν = −π2T

3µ

τ

mc(22)

Quasiparticle Nernst signal N = νH is linear with increasing magneticfield as found in various experiemnts. For metal ν ∼ T

TFis very small

10−3 − 10−2 µVkBT

, can be large in degenrated semiconductors.


Sondheimer Relation



ν =σxxe2nc

(dµ

dT) = − σxx

e2nc

π2T

3µ= −π

2T

3µ

τ

mc(21)

Sondheimer relation

ν = −π2T

3µ

τ

mc(22)


TFis very small

10−3 − 10−2 µVkBT



Sondheimer Relation



ν =σxxe2nc

(dµ

dT) = − σxx

e2nc

π2T

3µ= −π

2T

3µ

τ

mc(21)

Sondheimer relation

ν = −π2T

3µ

τ

mc(22)


TFis very small

10−3 − 10−2 µVkBT



Sondheimer Relation

In fluctuating superconductors

ν =σxxe2nc

(dµ

dT) + c

ρyyH

dMz

dT(23)

the second term ”saves” the third law of thermodynamics (ν → 0 asT → 0) in the vicinity of second critical field Hc2(0)

The present approach of obtaining ν aslo gives ”remarkableagreement” for graphene Nernst-ettingshausen oscillation as claimedin Lukyanchuk I. A., Varlamov A. A. and Kavokin A. V., Phys. Rev.Lett., 107 (2011) 016601

0 1 2 3 4 5 6 7 8 9 10−100

−50

0

50

100

Inverse magnetic field H0/H

NE

oscilla

tio

ns

ν/ν

0

NC: m*=0.04m, µ=0.02eV, H0=6.7T, Γ=10K

3D: t=2meV

2D: t=0

0 2 4 6 8 10

x 1012

−40

−20

0

20

40

Carriers concentration n (cm−2

)

NE

oscilla

tio

nν/ν

0 NC: m*=0.04m, H=3T, Γ=30K

0 2 4 6 8 10

x 1012

−40

−20

0

20

40


)

NE

oscilla

tio

nν/ν

0 DF: v=10 cm/s, H=3T, Γ=30K

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%1Þ for DF has the same profile as for NC but

shifted on half period. Vertical lines shows the quantization

condition (1).


Sondheimer Relation

In fluctuating superconductors

ν =σxxe2nc

(dµ

dT) + c

ρyyH

dMz

dT(23)

the second term ”saves” the third law of thermodynamics (ν → 0 asT → 0) in the vicinity of second critical field Hc2(0)The present approach of obtaining ν aslo gives ”remarkableagreement” for graphene Nernst-ettingshausen oscillation as claimedin Lukyanchuk I. A., Varlamov A. A. and Kavokin A. V., Phys. Rev.Lett., 107 (2011) 016601

0 1 2 3 4 5 6 7 8 9 10−100

−50

0

50

100

Inverse magnetic field H0/H

NE

oscilla

tio

ns

ν/ν

0

NC: m*=0.04m, µ=0.02eV, H0=6.7T, Γ=10K

3D: t=2meV

2D: t=0

0 2 4 6 8 10

x 1012

−40

−20

0

20

40


)

NE

oscilla

tio

nν/ν

0 NC: m*=0.04m, H=3T, Γ=30K

0 2 4 6 8 10

x 1012

−40

−20

0

20

40


)

NE

oscilla

tio

nν/ν

0 DF: v=10 cm/s, H=3T, Γ=30K

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%1Þ for DF has the same profile as for NC but

shifted on half period. Vertical lines shows the quantization

condition (1).


Thomson Effect

The Thomson co-effecient describes alternatively heating or cooling ofa current carrying conductor is expressed as

T = TdQ

dT=

T

e(d2µ

dT 2)I − cT σ−1e(

d2M

dT 2) (24)

In the absence of magnetic field

T = TdQ

dT=

T

e(d2µ

dT 2)I (25)

Using the expressions for the chemical potential sum-

coefficient 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 nor-

sign and grows in its absolute value linearly with tem-

of 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 posi-tive and non-monotonous for carriers with a parabolic disper-sion, while it is negative and saturates in the case of a Diracspectrum.


Thomson Effect

The Thomson co-effecient describes alternatively heating or cooling ofa current carrying conductor is expressed as

T = TdQ

dT=

T

e(d2µ

dT 2)I − cT σ−1e(

d2M

dT 2) (24)

In the absence of magnetic field

T = TdQ

dT=

T

e(d2µ

dT 2)I (25)

Using the expressions for the chemical potential sum-

coefficient 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 nor-

sign and grows in its absolute value linearly with tem-

of 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 posi-tive and non-monotonous for carriers with a parabolic disper-sion, while it is negative and saturates in the case of a Diracspectrum.


Peltier effect

The Peltier tensor which describes the heat generation by electriccurrent, is given by the Kelvin relation and can also be expressed interms of conductivity and thermoelectric tensors,

Π(T ,H) = TQ(T ,H) = −T σ−1(H)β(H) (26)

In the absence of magnetic field it has only diagonal components

Π(T , 0) =T

e(dµ

dT) (27)

Interestingly the whole system is out of thermal equilibrium in thepresence of electric current, the Peltier tensor can still be linked tothe thermal derivative of the chemical potential!


Peltier effect


Π(T ,H) = TQ(T ,H) = −T σ−1(H)β(H) (26)


Π(T , 0) =T

e(dµ

dT) (27)



Peltier effect


Π(T ,H) = TQ(T ,H) = −T σ−1(H)β(H) (26)


Π(T , 0) =T

e(dµ

dT) (27)



Thermal conductivity and Righi-Leduc effect

The thermal conductivity tensor κ describes the ability of a materialsubject to a temperature gradient to conduct heat. It can beexpressed through the elements of TE tensor and electric conductivity,(κ = q

∇xTwhen j = 0)

κ = −π2T

3e2σ(I − 3e2

π2[βσ−1]2) (28)

The Righi-Leduc effect describes the heat flow resulting from aperpendicular temperature gradient in the absence of electric current.It can be obtained from ∇xT = LHqy when j = 0

L =σxxenc

κyy (29)


Thermal conductivity and Righi-Leduc effect

The thermal conductivity tensor κ describes the ability of a materialsubject to a temperature gradient to conduct heat. It can beexpressed through the elements of TE tensor and electric conductivity,(κ = q

∇xTwhen j = 0)

κ = −π2T

3e2σ(I − 3e2

π2[βσ−1]2) (28)

The Righi-Leduc effect describes the heat flow resulting from aperpendicular temperature gradient in the absence of electric current.It can be obtained from ∇xT = LHqy when j = 0

L =σxxenc

κyy (29)


Limitations and Conclusion

There are certain limitations of this simple approach where largely theelectro-neutrality condition ∇xn = 0 is used, which may fail in certainsemiconductor systems where volume charge effects are important.

For the same reason, this approach fails to account for electriccurrents induced by a phonon drag.

In conclusion, the crucial function which governs all the TE and TMconstants as discussed is the temperature derivative of the chemicalpotential dµ

dT .

This simple observation opens the way to the prediction of the TEand TM coeffecients in new structures and materials.

In particular, this simple approach shows that the TE and TMproperties may be strongly different in systems with Dirac fermionsand normal carriers.






dT .








dT .








dT .








dT .




Documents

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