Thermoelectric properties in semimetals with inelastic electron-hole scattering Keigo Takahashi1Hiroyasu Matsuura1Hideaki Maebashi1and Masao Ogata1 2 1Department of Physics University of Tokyo 7-3-1 Hongo Bunkyo Tokyo 113-0033 Japan

2025-05-06 0 0 632.93KB 25 页 10玖币

侵权投诉

Thermoelectric properties in semimetals with inelastic electron-hole scattering

Keigo Takahashi,1, ∗Hiroyasu Matsuura,1Hideaki Maebashi,1and Masao Ogata1, 2

1Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan

2Trans-Scale Quantum Science Institute, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan

(Dated: March 31, 2023)

We present systematic theoretical results on thermoelectric eﬀects in semimetals based on the vari-

ational method of the linearized Boltzmann equation. Inelastic electron-hole scattering is known to

play an important role in the unusual transport of semimetals, including the broad T2temperature

dependence of the electrical resistivity and the strong violation of the Wiedemann-Franz law. By

treating the inelastic electron-hole scattering more precisely beyond the relaxation time approxima-

tion, we show that the Seebeck coeﬃcient when compensated depends on the screening length of the

Coulomb interaction as well as the Lorenz ratio (the ratio of thermal to electric conductivity due

to electrons divided by temperature). It is found that deviations from the compensation condition

signiﬁcantly increase the Seebeck coeﬃcient, along with crucial suppressions of the Lorenz ratio.

The result indicates that uncompensated semimetals with the electron-hole scattering have high

thermoelectric eﬃciency when the phonon contribution to thermal conductivity is suppressed.

I. INTRODUCTION

Thermoelectric eﬀect or the Seebeck eﬀect, which in-

duces the electromotive force by a temperature gradi-

ent, has attracted much attention from the perspective

of energy harvesting. The eﬃciency of the power gener-

ation due to the thermoelectric eﬀect is expressed by a

dimensionless ﬁgure of merit, ZT ≡S2σT/(κel +κph),

where S,σ,κel (κph), and Tare the Seebeck coef-

ﬁcient, electrical conductivity, thermal conductivity of

electrons (phonons), and temperature, respectively. Ma-

terials with large ZT have potential applications in power

supplies and thermoelectric cooling.

Conducting materials can be broadly classiﬁed into

three categories according to their transport properties:

metals, semiconductors, and semimetals [1]. Metals have

the highest electrical conductivity, but they also have

proportionally high thermal conductivity and usually sat-

isfy the Wiedemann-Franz (WF) law, which states that

the Lorenz ratio (L=κel/σT ) becomes the universal con-

stant L0=π2k2

B/3e2with e < 0 being the charge of an

electron. The WF law prevents metals from having large

ZT . In general, materials that exhibit high thermoelec-

tric performance belong to semiconductors with a large

Seebeck coeﬃcient. Thermoelectricity of semimetals, the

third category of conducting materials with intermediate

conductivity between that of metals and semiconductors,

has also been studied for many years [2–5], and has re-

cently attracted renewed interest [6–11].

The electronic transport due to the electron-hole scat-

tering in semimetals shows several intriguing phenomena,

even if the energy dispersion of the model is simple as

in Fig. 1. First, the electron-hole scattering gives a T2

temperature dependence of the electrical resistivity even

without Umklapp process [4,12–17]. This is because mo-

mentum conservation does not necessarily lead to veloc-

∗takahashi@hosi.phys.s.u-tokyo.ac.jp

ity conservation in the case of semimetals. Second, re-

cent experimental and theoretical studies on WP2have

revealed a downward violation of the WF law [18–22],

in which the Lorenz ratio becomes small depending on

the screening length of the Coulomb interaction. This is

due to the fact that the thermal current is more strongly

relaxed than the electrical current due to electron-hole

scattering, an eﬀect that goes beyond the relaxation time

approximation (RTA) in transport theory. Since the di-

mensionless ﬁgure of merit ZT can be rewritten as

ZT =S2

L+κph/σT ,(1)

an unusually small Lorenz ratio in semimetals can lead

to a large ﬁgure of merit.

In this paper, we systematically study the thermoelec-

tric properties of semimetals using a simple but standard

model to clarify the dependences of the electrical, ther-

mal, and thermoelectric transport coeﬃcients on (i) the

carrier numbers (compensated, electron-doped, and hole-

doped), (ii) the eﬀective masses of electrons and holes,

and (iii) the screening length of the Coulomb interaction.

In the previous studies, the Lorenz ratio in a compen-

sated semimetal was studied by exact solutions of the

Boltzmann equation [20,22]. However, this method is

not valid for the thermoelectric coeﬃcients. The ther-

moelectric coeﬃcients due to the electron-hole scattering

were studied only for the compensated case by the RTA

[22]. Therefore, the general behavior of thermoelectric

coeﬃcients for the uncompensated semimetal with the

electron-hole scattering is unclear. In addition, the RTA

is not exact for inelastic scattering [1] and the importance

of inelastic scattering in a semimetal has been discussed

[10]. Therefore, it should be testiﬁed whether the RTA

is valid or not by the analysis beyond RTA. Analysis by

the trial functions is useful to consider transports in the

presence of the inelastic scattering and employed in vari-

ous systems, such as graphene and bilayer graphene [23–

25]. Here, we apply the variational method [2] to the lin-

earized Boltzmann equation, which is more reliable than

arXiv:2210.14825v2 [cond-mat.str-el] 30 Mar 2023

FIG. 1. Two-band model consisting of electron (blue) and

hole (orange) bands.

RTA. We will show that there is a contribution to the

thermoelectric eﬀect that is not captured by the RTA

in the previous study. In the present paper, we focus on

the eﬀect of the electron-hole scattering, and the eﬀect of

phonons is out of the scope of this paper [26]. In the fol-

lowing, we study the temperature range kBT/∆.0.06

where ∆ is an energy oﬀset (see Fig.1) since the electron-

hole scattering becomes dominant in low-temperature re-

gion compared to the electron-phonon scattering.

This paper is organized as follows. In Sec. II, we

introduce the model and the Boltzmann equation. In

particular, in Sec. II A, we provide a detailed descrip-

tion of our model, illustrated in Fig. 1. In Sec. II B,

we introduce a systematic method based on the Boltz-

mann equation to calculate the transport coeﬃcients for

this model. The results and discussions are given in

Sec. III. First, we present the temperature dependence

of transport coeﬃcients. Then, we discuss the carrier-

number dependence of thermoelectric properties when

the electron-hole scattering dominates. In the compen-

sated case, the Seebeck coeﬃcient is zero if the eﬀective

masses of the electrons and holes are the same. If the

eﬀective masses are diﬀerent the Seebeck coeﬃcient be-

comes ﬁnite, but small. However, we will show that it

is sensitive to the screening length of the Coulomb in-

teraction as is the case for the Lorenz ratio. In the un-

compensated cases, we ﬁnd that slight deviations from

the compensation bring a large Seebeck coeﬃcient when

the electron-hole scattering dominates. We also estimate

ZT ≡S2σT/κel =S2/L, which gives an upper bound of

the ﬁgure of merit, in our framework, and ﬁnd that the

electron-hole scattering gives large g

ZT in the uncompen-

sated case due to the collaboration of the reduction of the

Lorenz ratio and the increase of the Seebeck coeﬃcient.

Finally, the conclusions are given in Sec. IV.

II. MODEL AND BOLTZMANN EQUATION

A. Model

We study a two-band model depicted in Fig. 1consist-

ing of electron and hole bands with three-dimensional

quadratic dispersions [20,22]

ε1,k=~2k2

2m1

, ε2,k= ∆ −~2(k−k0)2

2m2

(2)

where m1(m2) is the eﬀective mass of electrons (holes),

and ∆ is the energy oﬀset. Therefore, both carriers have

spherical Fermi surfaces.

The number of electrons (holes) is given by n1=

V−1Pk2f0(ε1,k) (n2=V−1Pk2(1 −f0(ε2,k))) where

a factor 2 and Vindicate the spin degeneracy, and the

volume of the system, and f0(ε)=(eβ(ε−µ)+ 1)−1is the

Fermi-Dirac distribution function with β= (kBT)−1and

µis the chemical potential which keeps the net charge

e∆n=e(n1−n2) at the value of T= 0. By intro-

ducing a parameter χdeﬁned by kF,2=χkF,1, we obtain

n2=χ3n1at T= 0 and the Fermi energy (εF) is given by

εF=m2∆/(χ2m1+m2). As a typical scale of wavenum-

ber, we deﬁne kF=p2m1m2∆/~2(m1+m2), which is

the Fermi wavenumber in the case of χ= 1, which cor-

responds to the compensated case, n1=n2.

B. Boltzmann equation and variational method

The Boltzmann equation of the system is given by [2,

20,22]

−eExv(l)

k;x−(εl,k−µ)v(l)

k;x−∇xT

T−∂f0(εl,k)

∂εl,k

=∂f (l)(k)

∂t imp

+∂f (l)(k)

∂t e-h

+∂f (l)(k)

∂t e-e

(3)

where v(l)

k;x=~−1∇kxεl,k(l= 1,2) is the velocity of the

band l(l= 1,2). Exand (−∇xT/T ) are the electric

ﬁeld and the temperature gradient along the xaxis, re-

spectively. The three terms on the right-hand side of

eq. (3) represent the impurity, interband (electron-hole),

and intraband (electron-electron and hole-hole) scatter-

ing, respectively. We assume that the impurity scatter-

ing is due to the short-range impurity potential and the

inter- and intraband scattering are due to the screened

Coulomb interaction where we neglect the exchange pro-

cess [20,22]. For example, the electron-hole scattering

for the band l= 1 is given by [20,22]

∂f (1)(k)

∂t e-h

=−2X

k2,k3,k4

Se-h(k,k2;k3,k4)f(1)(k)f(2)(k2)(1 −f(1)(k3))(1 −f(2)(k4))

+2 X

k1,k2,k4

Se-h(k1,k2;k,k4)f(1)(k1)f(2)(k2)(1 −f(1)(k))(1 −f(2)(k4)),(4)

where the factor 2 is the spin degeneracy and Se-h(k1,k2;k3,k4) is given by

Se-h(k1,k2;k3,k4) = 2π

V21

4πε024πe2

|k1−k3|2+α22(2π)3

Vδ(k1+k2−k3−k4)δ(ε1,k1+ε2,k2−ε1,k3−ε2,k4).(5)

Here, ε0is the dielectric constant and αrepresents the

inverse of the Thomas-Fermi screening length, where

α2=e2(m1kF,1+m2kF,2)/π2~2ε0[20,22]. The other

scattering terms have similar forms, which are given in

Appendix A.

In the variational method [2], the distribution func-

tion is expanded as f(l)(k) = f0(εl,k) + βf0(εl,k)(1 −

f0(εl,k))Φ(l)(k) where Φ(l)is assumed to be small. Keep-

ing terms up to the ﬁrst order of Φ(l), eq. (3) can be

rewritten as [2]

X(l)=P(l)[Φ] = P(l)

imp[Φ] + P(l)

e-h[Φ] + P(l)

e-e [Φ],(6)

where −X(l)denotes the left hand side of eq. (3). Note

that P(l)

imp[Φ] and P(l)

e-e [Φ] are linear functionals of Φ(l),

while P(l)

e-h[Φ] is a linear functional of (Φ(1),Φ(2)). Explicit

forms of these scattering terms are presented in Appendix

A. Then, we assume that the trial function for the band

lis given by Φ(l)(k) = P2

i=1 η(l)

iϕ(l)

i(k),where ϕ(l)

i(k) =

v(l)

k;x(εl,k−µ)i−1and the coeﬃcients η(l)

iare determined

so as to maximize a variational functional. Using the

variational method [2,27], we obtain the expression for

η(l)

ias

η(l)

i=X

j,k

(P−1)(lk)

ij J(k)

jEx+U(k)

j−∇xT

T,(7)

where P(lk)

ij is a matrix representation of P(l)[Φ] for the

chosen basis {ϕ(l)

i(k)}and

J(l)

U(l)

i!=1

ϕ(l)

i(k)v(l)

k;xe

εl,k−µ−∂f0(εl,k)

∂εl,k.

(8)

The explicit form of matrix P(lk)

ij is given in Appendix

B. Since P(l)

e-h[Φ] contains the distribution function of

the other band, we have the superscript (lk) in P(lk)

ij .

In the evaluation of P(lk)

ij , we analytically perform an-

gular integrals, and numerically evaluate the remaining

energy integrals. The details are given in the Supple-

mental Material (SM) [27]. Transport coeﬃcients (L11,

L12 =L21, and L22), which relate the electric (heat)

current Jx(Jq;x) to the external ﬁelds, deﬁned as

Jx

Jq;x=L11 L12

L21 L22  Ex

−∇xT/T ,(9)

are given by

L11 L12

L21 L22 = 2 X

i,l,j,k J(l)

U(l)

i!(P−1)(lk)

ij (J(k)

j, U(k)

j).

(10)

In the degenerate regime (kBTεF), these transport

coeﬃcients are approximated as

L11 '2(tJ1P−1

11 J1),(11)

L12 '2[tJ1P−1

11 U1+ (tJ2−tJ1P−1

11 P12)P−1

22 U2],(12)

L22 '2(tU1P−1

11 U1+tU2P−1

22 U2),(13)

by considering the power of kBT/εFwhere Ji=

t(J(1)

i, J(2)

i) and Ui=t(U(1)

i, U(2)

i). It should be noted

that although tU1P−1

11 U1in L22 is not the leading or-

der, we consider this term because it corresponds to the

ambipolar contribution [2,21,22].

III. RESULTS AND DISCUSSIONS

A. Temperature dependence

Figure 2shows the temperature dependences of re-

sistivity ρ=σ−1=L−1

11 , thermal conductivity κel =

(L22 −L21L12/L11)/T , and the Seebeck coeﬃcient S=

L12/T L11 for three values of χ. Lorenz ratio and Peltier

conductivity P≡Sσ are also shown in the inset of (b)

and (c), respectively. Here, we set m2= 3m1= 3me

with mebeing the electron mass and ε0so that α=kF

at χ= 1. In the following, we use ∆ = 0.2 eV as a typical

value. The strength of the impurity scattering is chosen

so that the electron-hole scattering dominates above 4 K

(see Appendix Afor the choices of parameters).

1. Electrical resistivity

The electrical resistivity ρ(Fig. 2(a)) is independent

of Tin the region of T.4 K, because the impu-

rity scattering dominates. As temperature increases, ρ

FIG. 2. Temperature dependences of (a) electrical resistivity,

(b) thermal conductivity, and (c) the absolute value of See-

beck coeﬃcient for three values of χwith m2= 3m1= 3me.

We normalize the electrical resistivity and thermal conductiv-

ity by the value at T= 0.1 K. The dotted lines in (b) repre-

sent ˜κel deﬁned in the text. In (c), the Seebeck coeﬃcient for

χ= 1.2 changes its sign at T= 10 K. The inset of (b) shows

the temperature dependence of the normalized Lorenz ratio

L/L0=κelρ/T L0(solid lines) and ˜κelρ/T L0(dotted lines)

with L0=π2k2

B/3e2. The inset of (c) indicates temperature

dependence of the Peltier conductivity P≡Sσ.

shows T2-dependence due to the electron-hole scattering

(4 K .T.30 K). On the other hand, in high tem-

perature regime (T&30 K), ρin the compensated case

(χ= 1.0) shows T2-dependence, while ρsaturates in the

uncompensated cases. This is because the contribution

to the electric current from the total momentum, which

is relaxed only through momentum dissipative scatter-

ings, is proportional to n1−n2, and this contribution

does not vanish in the uncompensated case [13] (see also

SM [27]). When the system is uncompensated and the

relative momentum is strongly relaxed by the electron-

hole scattering, the relaxation of the total momentum

by the impurity scattering governs the electric conduc-

tion [13,14,28]. Then, the saturated resistivity ρsat

obeys ρsat/ρ(T= 0) ∼[(n1+n2)/(n1−n2)]2[13,27].

2. Thermal conductivity

The thermal conductivity κel (Fig. 2(b)) is propor-

tional to Tat low temperatures. As shown in the in-

set, WF law holds in this temperature region. In the

intermediate temperature region, κel decreases slightly

slower than to T−1. This temperature dependence is ap-

proximately consistent with WF law (κel ∝T σ), but the

normalized Lorenz ratio is less than 1 and temperature

dependent.

For the compensated case (χ= 1), this result is consis-

tent with previous studies [20–22]. In particular, T3de-

pendence is due to the ambipolar eﬀect [21,22]. Actually,

˜κel ≡2(tU2P−1

22 U2)/T (dotted lines in Fig. 2(b)), which

does not include the ambipolar contribution, does not

show the increase but instead has the T−1-dependence

in a wider range of temperatures. For T > 60 K, ˜κel

upwardly deviates from T−1due to the subleading tem-

perature dependence [27].

In contrast, in the uncompensated case, κel does not

follow T3-dependence and is smaller than κel for χ= 1.

As a result, L/L0(inset) is small even in the high tem-

perature region. This means that the ambipolar contri-

bution is not large when uncompensated. The ambipo-

lar contribution is associated with the transport of the

compensated electrons and holes moving in the same di-

rection under the temperature gradient giving no electric

current as discussed for semiconductors [2]. This ambipo-

lar contribution is also present in semimetals [21,22] and

is weak in the uncompensated case. We can show [27]

that, in κel = (L22 −L21L12/L11)/T , the enhancement

of L12 in the uncompensated case cancels the ambipo-

lar contribution in L22 leading to the suppression of the

Lorenz ratio.

3. Seebeck coeﬃcient

The Seebeck coeﬃcient Sis negative and almost in-

dependent of χin the low temperature region. This is

because the transport property is mainly determined by

the electrons that have a smaller eﬀective mass than the

holes. In this low temperature region, the impurity scat-

tering is dominant and thus the Mott formula is valid.

At higher temperatures, Sfor χ= 1.2 changes its sign to

positive at T∼10 K. This can be understood as follows.

In the high temperature region, the relative momentum

between electrons and holes is strongly relaxed by the

electron-hole scattering, and thus the electric current is

mainly carried by the total momentum proportional to

e(n1−n2) [13]. As a result, the sign of Sis determined

by the holes for the case of χ= 1.2 (n2> n1). In the

temperature region above 40 K, Sbecomes again almost

linear in T. Apparently, the coeﬃcient of the linear T

term for χ6= 1 is about ten times larger than that at low

temperatures.

First, let us study the Seebeck coeﬃcient for the com-

pensated case (χ= 1) more closely. We plot in Fig. 3

FIG. 3. Temperature dependences of |S/T |for three screening

lengths, or equivalently three dielectric constants for (a) χ= 1

and (b) χ= 0.8. The dielectric constants are chosen so that

α= 0.5kF,1.0kF,and 2.0kFat χ= 1. Colored dotted lines in

(a) show results when considering the impurity and intraband

scattering.

the temperature dependence of |S/T |for three screening

lengths. We can see that the coeﬃcient of the linear-T

term gradually increases as a function of Tand reaches

some value, which depends on α. The black dashed line in

Fig. 3(a) shows ˜

S/T ≡tJ1P−1

11 U1/T 2(tJ1P−1

11 J1), which

is equivalent to the RTA as shown in the SM [27]. Appar-

ently, ˜

S/T does not depend on α, which is consistent with

the previous study [22] showing that the Seebeck coeﬃ-

cient in the RTA does not depend on the relaxation time

by the electron-hole scattering (see also SM [27]). This

indicates that the RTA does not explain the αdepen-

dence of Sin Fig. 3for the compensated case. To see the

eﬀect of the electron-hole scattering, we show the results

(colored dotted lines in Fig. 3) in which the electron-hole

scattering is neglected and only the impurity and intra-

band scattering are considered. In this case, |S/T |in the

high temperature region does not depend on α, which

means that the interband scattering plays an important

role in αdependence of S. Since the ﬁrst term of the

right-hand side of eq. (12) corresponds to the RTA [27],

the αdependence comes from the other terms in eq. (12),

i.e., from the terms including P12 and P22. In particular,

P12 is nonzero for the electron-hole scattering unlike the

intraband scattering.

For the uncompensated cases (χ6= 1), the situation

is diﬀerent. Figure 3(b) shows the temperature depen-

dence of |S/T |for three screening lengths in the case

of χ= 0.8. In this case, the enhancement of |S/T |at

high temperatures is larger than that for χ= 1. In

the uncompensated cases, tJ1P−1

11 U1in eq. (12) gives

FIG. 4. χdependences of (a) Seebeck coeﬃcient and (b)

ZT =S2σT/κel =S2/L at 40 K. ε0is the same as in Fig. 2.

The inset of (b) shows the Lorenz ratio. Black lines in (a) are

proportional to (m1+χm2)(χ2m1+m2)/(χ3−1).

the major contribution when the electron-hole scattering

dominates [27]. As a result, Scan be approximated as

S=tJ1P−1

11 U1/T (tJ1P−1

11 J1), which is equivalent to the

RTA. In fact, ˜

S/T , which are shown in the black dashed

lines in Fig. 3(b), reproduce S/T at high temperatures.

B. Carrier-number dependence

To understand the eﬀect of doping and the diﬀerence

in eﬀective masses, we plot in Fig. 4the χdependence

of the Seebeck coeﬃcient, the Lorenz ratio (inset), and

ZT =S2σT/κel =S2/L for three values of m2/m1at

T= 40 K. We can see that the absolute value of S

increases and the Lorenz ratio drastically decreases in

the uncompensated case. As a result, g

ZT (Fig. 4(b))

drastically increases in the uncompensated case. For the

case of m2/m1= 1, g

ZT is almost symmetric with respect

to χ, while g

ZT is larger for the case with χ < 1, i.e.,

n1> n2, than for the case with χ > 1 when m2/m1>1.

As we can see from Fig. 2(c), Sat T= 40 K is almost

the same for χ= 0.8 and χ= 1.2. Thus, the diﬀerence

in g

ZT comes from the diﬀerence in the Lorenz ratio.

In the limit of vanishing the impurity scattering,

the Seebeck coeﬃcient behaves as S'˜

S∼(m1+

χm2)(χ2m1+m2)/(χ3−1) ∝(n2−n1)−1[27]. We see

this dependence in the black lines of Fig. 4(a) for χfar

away from 1. This means that a slight deviation from

compensation can lead to a large Seebeck coeﬃcient. As

discussed in Ref. [14] in connection to the Hall coeﬃcient,

we can interpret the behavior of Sas follows: the elec-

trons and holes are locked by the electron-hole scattering

and they can be treated as a single carrier with charge

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Thermoelectricpropertiesinsemimetalswithinelasticelectron-holescatteringKeigoTakahashi,1,HiroyasuMatsuura,1HideakiMaebashi,1andMasaoOgata1,21DepartmentofPhysics,UniversityofTokyo,7-3-1Hongo,Bunkyo,Tokyo113-0033,Japan2Trans-ScaleQuantumScienceInstitute,UniversityofTokyo,7-3-1Hongo,Bunkyo,Tokyo113-00...

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Thermoelectric properties in semimetals with inelastic electron-hole scattering Keigo Takahashi1Hiroyasu Matsuura1Hideaki Maebashi1and Masao Ogata1 2 1Department of Physics University of Tokyo 7-3-1 Hongo Bunkyo Tokyo 113-0033 Japan

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