12-1

Chapter 12 Thermoelectric

Transport Properties for

Electrons

Contents

Chapter 12 Thermoelectric Transport Properties for Electrons.................................... 12-1

Contents .......................................................................................................................... 12-1 12.1 Boltzmann Transport Equation .......................................................................... 12-2

12.2 Simple Model of Metals..................................................................................... 12-4

12.2.1 Electric Current Density ............................................................................ 12-4

12.2.2 Electrical Conductivity .............................................................................. 12-5 Example 12.1 Electron Relaxation Time of Gold ...................................................... 12-7

12.2.3 Seebeck Coefficient ................................................................................... 12-8 Example 12.2 Seebeck Coefficient of Gold ............................................................. 12-10 12.2.4 Electronic Thermal Conductivity ............................................................. 12-10

Example 12.3 Electronic Thermal Conductivity of Gold ......................................... 12-12 12.3 Power-Law Model for Metals and Semiconductors ........................................ 12-12

12.3.1 Equipartition Principle ............................................................................. 12-13 12.3.2 Parabolic Single-Band Model .................................................................. 12-15

Example 12.4 Seebeck Coefficient of PbTe ............................................................. 12-17 Example 12.5 Material Parameter ............................................................................ 12-24

12.4 Electron Relaxation Time ................................................................................ 12-25 12.4.1 Acoustic Phonon Scattering ..................................................................... 12-25 12.4.2 Polar Optical Phonon Scattering .............................................................. 12-26 12.4.3 Ionized Impurity Scattering ..................................................................... 12-26 Example 12.6 Electron Mobility ............................................................................... 12-27

12.5 Multiband Effects............................................................................................. 12-28 12.6 Nonparabolicity................................................................................................ 12-30 Problems ....................................................................................................................... 12-33 References ..................................................................................................................... 12-35

12-2

12.1 Boltzmann Transport Equation

The flow of electrons in solids involves two characteristic mechanisms with opposite effects: the

driving force of the external fields and the dissipative effect of the scattering of the electrons by

phonons and defects. The interplay between the two mechanisms is described by the Boltzmann

transport equation.

The distribution function f gives the probability of finding an electron at time t, at position r,

with momentum p. The Boltzmann transport equation [1] accounts for all possible mechanisms by

which f may change. The electron flow in a metal can be affected by applied fields, temperature

gradients, and collisions (scattering). Consider the electron distribution f (t, r, p). We expand the

total derivative of f (t, r, p) as

dpp

fdr

r

fdt

t

fdf

(12.1)

With collisions, using kp , we have [2]

coll

t

f

r

fr

k

fk

t

f

dt

df (12.2)

This is the celebrated Boltzmann transport equation, which is very difficult to solve. In many

problems the collision terms may be treated by introduction of the electron relaxation time as

off

t

f

coll

(12.3)

This is called the relaxation time approximation (RTA). Here f and fo are the perturbed and

unperturbed distribution functions. The latter is the Fermi-Dirac distribution in thermal

equilibrium. The electron relaxation time is the average flight time of an electron between

successive collisions (scattering events) with electrons, phonons, or impurities.

We must solve this equation to obtain f – fo in terms of the internal electric field and temperature

gradient. It will be assumed that the conductor is isotropic and that the electric field and flows are

in the direction of the x axis so that r denotes x. And there is no magnetic field applied. The

12-3

momentum of a free electron is related to the wavevector by kmv . In an electric fieldΕ with

charge e, the force F on an electron is

Εedt

dk

dt

dvmF (12.4)

which is the Coulomb force (an external electric field Ε causes electrons to move to the opposite

direction). Using Equations (11.6) and (12.4), the force related term in Equation (12.2) is expressed

as

E

fve

k

fk

Ε (12.5)

Since f is a function of TkEE BF as shown in Equation (11.27), we introduce

TkEE BF . Then,

2

1

Tk

EE

T

E

TkT B

FF

B

(12.6)

And

x

T

T

fv

x

T

T

fv

x

fv

(12.7)

Using these equations, we have [3]

x

T

T

EE

x

Ee

E

fv

ff FFo Ε

(12.8)

We may relate the gradient of the Fermi energy xEF to the electric field Ε if the system is on

open circuit in thermal equilibrium (no temperature gradient and no electron flow, 0 off ).

Then, the gradient of the Fermi energy is notably a form of an electric field. But the temperature

12-4

gradient also causes the gradient of the Fermi energy if the system is on open circuit with a

temperature gradient which can be seen in Equation (12.8). Therefore, the total electric field Ε is

the sum of the external electric field and the gradient of the Fermi energy as (Ashcroft and Mermin

(1976) [3]

x

E

e

F

1ΕΕ (12.9)

When no external electric filed is applied, the total electric field Ε becomes

x

E

e

F

1Ε (12.10)

In general it is known that the difference (f – fo) between the perturbed and unperturbed

distributions is relatively small compared to fo. Then, the f in the right-hand side of Equation (12.8)

may be replaced by fo. It is also known that the term fo in the left-hand side will not make any

contribution. Bearing in mind that the electric field and temperature gradients lie along the x-axis,

Equation (12.8) reduces to

x

T

T

EE

x

E

E

fvf FFo (12.11)

which is a different version of the Boltzmann transport equation (BTE) with the relaxation time

approximation (RTA).

12.2 Simple Model of Metals

12.2.1 Electric Current Density

If n electrons with charge e move in the x-direction with a velocity, the electric current density j is

12-5

xvenj (12.12)

From Equation (11.28) in thermal nonequilibrium, we have

0

dEEfEgevj x (12.13)

From Equation (12.11), we have

0

2 dEx

T

T

EE

x

E

E

fEgvej FFo

x (12.14)

Using the density of states of Equation (11.24) and also assuming a cubic isotropic structure

( mEvx 322 and 2222

zyx vvvv ), the electrical current density j is

0

2

3

32

21

3

22dE

x

T

T

EE

x

E

E

fE

emj FFo

(12.15)

12.2.2 Electrical Conductivity

The electrical conductivity is obtained from Equations (11.1) and (12.10) in the absence of a

temperature gradient.

x

E

e

jj

F

1Ε

(12.16)

Equation (12.15) leads to

0

dEE

fE o (12.17)

where

12-6

2

3

2

32

21

3

22Ee

mE

(12.18)

We use the classical asymptotic formula (Taylor series) for TkE BF >>1 (metals) in [4]

12

22

2

0

0 2n

n

F

F

nn

BnFdE

EdTkCEdE

E

fE

(12.19)

where

1

2

1

2

1

Sn

S

nS

C , so that 12

2

2

C

, 720

7 4

4

C

(12.20)

Then, Equation (12.17) can be expressed as

FEE

BFo

E

ETkEdE

E

fE

2

22

2

06

(12.21)

Neglecting the second order term (linearized assumption), the electrical conductivity is

expressed as

2

3

2

32

21

3

22FF Ee

mE

(12.22)

This tells us that the electrical conduction can take place only nearby the Fermi energy, where the

electrons in the conduction band can move from one energy state to another (partially filled). Using

Equation (11.24), Equation (12.22) can be expressed as

FEExF vEgeE

22 (12.23)

Using Equation (11.33a) and mEvx 322 , we have

12-7

nem

ne

2

(12.24)

which is known as the Drude model, and the electron mobility becomes

m

e (12.25)

Note that, in this simple model of Equations (12.24) and (12.25), all free electrons contribute to

the electrical current. The electron mobility is proportional to the relaxation time and inversely

proportional to the (effective) mass m. The relaxation time at room temperature is typically 10-14

to 10-15 sec.

Example 12.1 Electron Relaxation Time of Gold

Estimate the electron relaxation time and the electron mean free path for gold if its electrical

conductivity of 4.55 × 105 (cm)-1 is given.

Solution:

From Equation (12.22),

2

3

221

32

22

3

FEem

We assume that the effective mass in gold is equal to the electron mass and the Fermi energy EF

in gold is taken from Example 11.1, which is 8.84 × 10-19 J.

s

Ckgm 14

23192192131-

34217 1074.2

J1084.810602.1109.10922

2Js1062.631055.4

From Example 11.1, the velocity of electron is assumed to be the Fermi velocity of 1.39 × 106 m/s.

From Equation (11.36), the electron mean free path for gold is obtained as

12-8

mssmvF

9146 1021.381074.2/1039.1

Comments: This calculation is fairly crude, giving the electron mean free path of 38.2 × 10-9 m

for gold, which may be compared with the phonon mean free path in Chapter 13 (about 6.4 × 10-9

m for PbTe). Nothing can be concluded because of the different materials.

12.2.3 Seebeck Coefficient

The Seebeck coefficient (or thermopower) is obtained with j = 0 from Equation (11.1) as

xT

E

(12.26)

From Equation (12.14), using Equation (12.22), we have

0

1dE

x

T

T

EE

x

E

E

fE

ej FFo (12.27)

We approximate the second term of the right-hand side using the asymptotic formula of Equation

(12.19).

FEE

Bo

FE

ETkdE

E

fEEE

22

03

(12.28)

From Equation (12.27),

FEE

BFF

E

ETk

x

T

eTE

x

E

ej

22

3

11 (12.29)

From Equation (12.26) using Equation (12.10), the Seebeck coefficient (thermopower) is

FEE

BE

ETk

e

ln

3

22

(12.30)

12-9

This is the well-known Mott formula. In spite of the simplifications made it gives a good

interpretation and has been widely used in literature.

Using Equation (12.24), we can have a different version of the Seebeck coefficient as

FEE

BEn

EgTk

e

1

3

22

(12.31)

which is an interesting expression where the Seebeck coefficient is only meaningful near EF and

proportional to the density of states g(EF) but inversely proportional to the electron concentration

n. When we insert Equation (12.23) into Equation (12.30), we have

FEE

x

x

BE

v

vE

Eg

EgTk

e

2

2

22 11

3

(12.32)

As an approximation, we shall assume that the relaxation time can be expressed in the form of

rE0 where 0 and r are constant for a given scattering process. In many thermoelectric

materials, it seems that for scattering by acoustic-lattice vibrations, r is equal to ˗1/2. And also,

for scattering by ionized impurities, r equal to 3/2. We use the proportionality from Equations

(11.24) and so on as

2

1

EEg (12.33)

Evx 2

rE

Equation (12.32) becomes with Equation (12.33)

TkE

r

e

k

BF

B

2

3

3

2

(12.34)

12-10

Using Equation (11.33) for the Fermi energy with r = ˗1/2, the Seebeck coefficient is expressed as

3

2

2

2

33

2

nmT

e

kB

(12.35)

which is another version of the Mott formula for metals (degenerate). But it is also used for heavily

doped semiconductors (nondegenerate).

Example 12.2 Seebeck Coefficient of Gold

Estimate the Seebeck coefficient of gold at room temperature.

Solution:

Using Equation (12.34) assuming r = -1/2 and the Fermi energy in Example 11.1,

K

V

JC

KKJ

eE

Tk

F

oB 328.1

1084.810602.13

3001038.1

3 1919

223222

The Seebeck coefficient of -1.328 V/K for gold is small compared to the typical value of -200

V/K for semiconductor. Therefore, gold is not a good material for thermoelectrics.

12.2.4 Electronic Thermal Conductivity

The thermal conductivity k is the sum of the electronic and lattice thermal conductivities.

le kkk (12.36)

The lattice thermal conductivity will be discussed in a later chapter. The electronic thermal

conductivity is given by

12-11

xT

qk e

e

(12.37)

The heat current density (heat flux) qe is the product of electron concentration, drift velocity and

total energy transported by an electron, when the current is zero.

Fxe EEnvq (12.38)

Then, using Equation (11.28),

0

dEEfEgEEvq Fxe (12.39)

Similar to the process of the electrical conductivity , we have

0

2

1dE

x

T

T

EE

x

E

E

fEEE

eq FFo

Fe (12.40)

From Equation (12.35), using Equation (12.19), we have

FEE

FBe

ExTx

ET

e

Tkk

1

3 2

22

(12.41)

The Lorentz number Lo is defined as

22

3

e

k

T

kL Be

o

(12.42)

For metals the Lorentz number is 2.44 × 10-8 -W/K2. This is known as the Wiedemann-Franz

law, which states that the ratio of the thermal to the electrical conductivity is the same for all metals

at any particular temperature. Using Equations (12.10), (12.30) and (12.42)), we have

12-12

FEE

FBe

ExTx

ET

e

Tkk

1

3 2

22

(12.43)

Using Equation (12.26), this reduces to

TTLk oe 2 (12.44)

In metals or heavily doped semiconductors, the second term is small and usually neglected.

Example 12.3 Electronic Thermal Conductivity of Gold

Estimate the electronic thermal conductivity of gold at room temperature if the electrical

conductivity of 4.55 × 105 (cm)-1 is given.

Solution:

From Equation (12.44) and the result of Example 12.2 for KV 328.1

cmK

W

cmK

W

cmK

W

KmKVKmKWke

335.310406.2334.3

3001055.410328.13001055.4/1044.2

4

17261728

Comments: Note that the second term in Equation (12.44) is negligible. The electronic thermal

conductivity of gold of 3.335 W/cmK is large compared to the typical value of 0.01 W/cmK in

semiconductors.

12.3 Power-Law Model for Metals and Semiconductors

The Fermi energy is much greater than zero (conduction band edge) in metals while it is much less

than zero in semiconductors. The former are often called degenerate materials while the latter are

called the nondegenerate materials. In this section, we look for a generic parabolic single-band

12-13

model covering both the degenerate and nondegenerate materials. The power-law model assumes

that the electron relaxation time is a function of energy as

r

constE (12.45)

where r is called the scattering parameter, and const is independent of energy but it may be

dependent on effective mass and temperature. There are three fundamental scattering mechanisms:

r = -1/2 for the acoustic phonon scattering (most materials), r = 3/2 for ionized impurity scattering,

and r = 1/2 for polar optical phonon scattering.

12.3.1 Equipartition Principle

When we considering Equation (11.15), we may introduce the conductivity (or inertial) effective

mass

cm . The kinetic energy of an electron depends only on the temperature and is independent

of mass.

TkvmE Bc2

3

2

1 2 (12.46)

Based on the equipartition principle, we have

222

2

1

2

1

2

1zzyyxx vmvmvm (12.47)

Using p = mv = ħk and Equation (11.15), we have

z

zz

y

yy

x

xx

z

z

y

y

x

xc

c m

vm

m

vm

m

vm

m

k

m

k

m

kvm

m

kE

2222222

1

2

2222222222222

22

(12.48)

which is equal to

222

2

1

2

1

2

1zzyyxx vmvmvmE (12.49)

which, from Equation (12.47), reduces to

12-14

2

2

13 xxvmE (12.50)

We know that the average velocity is

2222

zyx vvvv (12.51)

Manipulating this,

z

zz

y

yy

x

xx

m

vm

m

vm

m

vmv

222

2 2

1

2

1

2

1

2 (12.52)

and

zyx

xxmmm

vmv111

2

12 22 (12.53)

Using Equation (12.50), we have

2

1

111

3

1

2

1v

mmmE

zyx

(12.54)

The conductivity effective mass

Im is defined as

tlzyxc mmmmmm

21

3

1111

3

11 (12.55)

and

2

2

1vmE c

(12.56)

12-15

Under an assumption of 22 3 xvv , we have

c

xm

Ev

3

22 (12.57)

The conductivity effective mass

cm has an expression in Equation (12.55) while the density-of-

states effective mass

dm has an expression in Equation (11.25). Their mathematics is rather

intractable while their quantities can be calculated as shown. Closed expressions for them is

discussed in Appendix G.

12.3.2 Parabolic Single-Band Model

Density of States

We like to use the reduced energy TkEE B and the reduced Fermi energy TkEE BFF since

the electron energy is approximately the order of TkB . The electron relaxation time in Equation

(12.45) becomes

rrr

Bconst EETk 0 (12.58)

The density of states g(E) in Equation (11.24) with the degeneracy is expressed as

2

1

2

12

3

22

2

2

ETk

mNEg B

dv

(12.59)

where Nv is the degeneracy (multiple valleys or the number of bands). The electron concentration

n in Equation (11.30) is expressed as

dE

e

ETkmNn

FEE

Bdv

0

2

1

2

3

221

2

2 (12.60)

12-16

For the sake of simplicity, define the Fermi integrals as

dEe

EF

FEE

s

s

0 1 (12.61)

Using the Fermi integrals, the electron concentration n is expressed as

21

2

3

22

2

2F

TkmNn Bdv

(12.62)

Electrical Conductivity

In a similar manner in Section 12.2, the electrical conductivity using Equations (12.57) and

(12.61) is expressed as

21

2

3

22

0

2

2

32

3

r

Bd

c

v FrTkm

m

eN

(12.63)

For this equation, we used an integral relation below:

dEE

EEfdE

E

EfE

0

0

0

0 , where 00 (12.64)

Equation (12.63) is expressed in terms of n, e, and as

21

21021

2

3

22 2

3

3

22

2 F

Fr

m

eeF

TkmN r

c

Bdv

(12.65)

Equivalently,

ne (12.66)

12-17

where is here a different form of the electron mobility as

cm

e (12.67)

where is the average relaxation time.

21

21

02

3

3

2

F

Fr

r

(12.68)

Equation (12.67) is the electron mobility for the power-law model also shown in Equation (12.25).

Seebeck Coefficient

The Seebeck coefficient is derived in a similar manner of Section 12.4 as

F

r

r

B E

Fr

Fr

e

k

21

23

2

3

2

5

(12.69)

The value of ekB is 86 V/K, which gives an idea of the order of the magnitude of the Seebeck

Coefficient in many thermoelectric materials.

Example 12.4 Seebeck Coefficient of PbTe

PbTe (lead telluride) is a widespread thermoelectric material at mid-range temperatures, which is

doped by Na (sodium) at a doping concentration of 1.3 × 1019 cm-3, having that the degeneracy

of the conduction valleys is 4 and the DOS effective mass is 0.12 me. Assuming that the acoustic

phonon scattering (r = -1/2) is a dominant mechanism of the relaxation time, determine the

Seebeck coefficient of PbTe at room temperature.

Solution:

Physical constant: 210626.6 34 Js , KJkB

231038.1 , and Ce 19106021.1

12-18

Information given: 325103.1 mn , Nv = 4, and )101.9(12.0 31kgmd

From Equation (12.69), the Seebeck coefficient is given by

F

r

r

B E

Fr

Fr

e

k

21

23

2

3

2

5

In order to calculate the Fermi integrals, we need the Fermi energy for the integrals. Since the

high doping of 1.3 × 1019 cm-3 is used, the criteria of TkEE BF >> 1 in Section 11.6 cannot

be applied and Equation (11.41) is not used. Instead, we solve Equation (12.60) for the Fermi

energy.

dE

e

ETkmNn

FEE

Bdv

0

2

1

2

3

221

2

2

We want to use Table C-1 for the Fermi integral. Therefore,

dE

e

ETkm

Nn

FEE

Bd

v 0

2

1

2

3

2

2

1761.2

22

From Table F-1, we find by interpolating the 2.761 between 2.5025 and 2.7694 for s = ½ as

2.2

FE

Then, the values of the Fermi integrals of F0 and F1with 2.2

FE can be obtained by

interpolation between the values in Table F-1. Then, we have

3051.20 F and 9571.31 F

The Seebeck coefficient is finally obtained by

K

V

C

KJ 4.1062.2

3051.22

3

2

1

9571.32

5

2

1

106021.1

1038.119

23

Comments: The absolute value of -106.4 V/K for PbTe (semiconductor) is much greater than

the value of -1.328 V/K for gold (metal). If we use Equation (11.41) for the Fermi energy (this

is not correct since the criteria of TkEE BF >> 1 cannot be applied), we have

12-19

eV

Tkm

N

nTkE Bd

v

BF

029.0210626.62

3001038.1101.912.0

42

103.1ln3001038.1

22ln

23

234

23312523

2

3

2

which leads to 136.1026.0

029.0

eV

eV

Tk

EE

B

FF

Then, the values of the Fermi integrals with 136.1

FE are obtained by interpolation between

the values in Table F-1. Then, we have

415.10 F and 992.11 F

The Seebeck coefficient is finally obtained by

K

V

C

KJ 7.144136.1

415.12

3

2

1

992.12

5

2

1

106021.1

1038.119

23

This value of -144.7 V/K differs from -106 V/K by an error of 44%, which can be seen in

Figure 12.2.

Electronic Thermal Conductivity

After a lengthy algebra, it comes up with the Lorentz number of the power-law model as

2

21

23

21

252

2

3

2

5

2

3

2

7

r

r

r

r

Bo

Fr

Fr

Fr

Fr

e

kL (12.70)

And the electron thermal conductivity of the power-law model is expressed as

12-20

oe LTk (12.71)

The Lorentz number oL is plotted as a function of the reduced Fermi energy

FE for three different

values of r in Figure 12.1. It is seen that, at the high Fermi energies in metals, they converge on

the Wiedemann-Franz law of 2.44 × 10-8 -W/K2. On the other hand, as

FE decreases below zero

in semiconductors, Lo drops to about 1.5 × 10-8 -W/K2 for r = -1/2 which is the case of acoustic-

phonon scattering. Note that Lo for the acoustic-phonon reveals the lowest electronic thermal

conductivity among the three as shown in Equation (12.71).

Figure 12.1 Lorentz number versus the reduced Fermi energy for r = -1/2, 1/2, and 3/2.

Since the power-law model covers the entire range from metals to semiconductors, it is worthwhile

to examine all the thermoelectric transport properties against the Fermi energy. In order to

calculate those properties, for example for PbTe, using the material inputs (r = -1/2, 4VN and

ed mm 22.0) and the constant relaxation time 0 of 6.2 × 10-14 s. Then the Seebeck coefficient

can be computed using Equation (12.69), the electrical conductivity using Equation (12.63), and

the electronic thermal conductivity ek using Equation (12.71). The results are shown in Figure

12.2. One faces the fundamental challenges in having a high dimensionless figure of merit ZT

because, as

FE decreases, increases while decreases. The net improvement in the ZT can be

examined by introducing the power factor 2 .

12-21

Figure 12.2 The Seebeck coefficient, the electrical conductivity, and the electronic thermal

conductivity as a function of the Fermi energy at room temperature for PbTe assuming a constant

relaxation time 0 of 6.2 × 10-14 s (realistic value) and r = -1/2 (see Equation (12.42)).

Dimensionless Figure of Merit and Material Parameter

The dimensionless figure of merit is given as

le kk

TZT

2

(12.72)

Using Equation (12.71),

lolo kTL

T

kTL

TZT

22

(12.73)

Inserting Equations (12.63), (12.69), and (12.71) into (12.72) yields

12-22

21

2

23

2

25

21

2

21

23

2

3

2

5

2

71

2

3

2

5

2

3

r

r

r

rF

r

r

Fr

Fr

Fr

FE

Fr

Fr

r

ZT

(12.74)

where is called the material parameter as

𝛽 =

(𝑘𝐵𝑒 )

2

𝜎0𝑇

𝑘𝑙

(12.75)

Equivalently,

lc

BBdv

km

TkTkmN

0

22

3

22

2

3

(12.76)

Note that 0 is defined without the Fermi integral 21rF , so that 0 does not depend on the reduced

Fermi energy. The dimensionless material parameter was first introduced by Chasmar and

Stratton [5].

In Figure 12.3, we show how the dimensionless figure of merit ZT varies with the reduced

Fermi energy

FE for different values of the material parameter . We have supposed that the

scattering parameter r has a value of -1/2, as for acoustic phonon scattering. We see that, as

becomes larger, the optimum value for

FE becomes more negative. Thus, if were large enough,

we could use classical statistics in our calculations. However, the best materials that are used in

today’s thermoelectric modules, is about 0.4 and we hardly expect it ever to approach the highest

valve in Figure 12.3. It can be seen that the optimum Fermi energy has a range of little more than

kBT for a wide range of values for the material parameter

12-23

Figure 12.3 The dimensionless figure of merit plotted against the reduced Fermi energy for

different values of the material parameter . The scattering parameter is r = -1/2.

It is well known that the optimal electronic performance of a thermoelectric semiconductor

depends primarily on the weighted mobility [1, 5, 7]. The weighted mobility is,

2

3

e

dv

m

mN (12.77)

This weighted mobility implies that high degeneracy Nv, high density-of-states effective mass

dm ,

and high mobility seem to improve the performance of ZT. This has been widely understood in

the literature and industry. Nevertheless, in many semiconductors acoustic-phonon scattering

dominates and that has the proportionality

2

30

1

edv mmN

(12.78)

Considering this, the material parameter is expressed as

12-24

cm

e (12.79)

For materials dominated by acoustic phonon scattering, the material parameter depends only on

the electron mobility as shown in Equation (12.79). Therefore, for anisotropic materials, the

direction of lightest

cm is preferred for thermoelectric transport (in cubic crystals,

cm is close to

dm [8]), which is actually oppose to Equation (12.77) and the following statement.

It is not sufficient to select a particular semiconductor or compound if a high ZT is required. It

is necessary to specify the carrier concentration which can, of course, be adjusted by changing the

number of donors or acceptors. In this section, we discuss the problem of achieving the optimum

concentrations of charge carriers.

It is instructional to proceed first with a calculation in which it is assumed that the

semiconductor obeys classical statistics. We also suppose that there is only one type of carrier in

a parabolic band, and we ignore the possibility of a bipolar effect. Thus, the thermoelectric

parameter can be expressed by Equations (12.63) – (12.71).

Example 12.5 Material Parameter

PbTe (lead telluride) is a thermoelectric material at mid-range temperatures with a band

degeneracy of 4, DOS effective mass of 0.12 me and acoustic-phonon scattering for electrons with

constant relaxation time of 1.55 × 10-13 s. Assuming the material has lattice thermal conductivity

of 1.2 W/mK, determine the material parameter at room temperature.

Solution:

Since the acoustic-phonon scattering for electrons is assumed, the scattering parameter should be

r = -1/2.

Information given:

4vN , ed mm 12.0, s13

0 1055.1 , and mK

Wkl 2.1

From Equation (12.76), removing the Fermi integral (cancelled) and assuming that dc mm ,

12-25

246.02

3

0

22

3

22

lc

BBdv

km

TkTkmN

Comment: Most thermoelectric materials show about 0.1 ~ 0.5 for .

12.4 Electron Relaxation Time

The relaxation time is the average flight time of electrons between successive collisions or

scattering events with the lattice or impurities. The relaxation time plays the most important role

in determining the transport properties such as electron mobility, electrical conductivity, thermal

conductivity, and Seebeck coefficient. In this section, three fundamental scattering mechanisms

are introduced to account for electron relaxation time.

12.4.1 Acoustic Phonon Scattering

The wavelength of a free electron of thermal energy is large compared with the lattice constant.

Such electrons interact only with acoustical vibration modes of comparable long wavelength. The

local deformations produced by the lattice waves are similar to those in homogeneously deformed

crystals.

Longitudinal acoustic phonons may deform the electric band structure leading to electron

scattering due to the deformation potential. The deformation potential is determined by a shift in

the energy bands with dilations of the crystal produced by thermal vibration. The theory of

acoustic-phonon scattering was originally provided by Bardeen and Shockley (1950)[9]. The

relaxation time for the acoustic-phonon scattering is

2

1

02

1

232

24

2

2

EE

Tkm

dv

Bda

sa

(12.80)

where E is the reduced energy ( TkEE B ), d the mass density and a the acoustic

deformation potential.

12-26

12.4.2 Polar Optical Phonon Scattering

Polar optical phonon scattering is of considerable importance at low electron concentrations,

although its effect is expected to diminish at high electron concentrations because free-electron

screening will reduce the electron-phonon interaction. Polar materials are partly ionic compounds.

When two atoms in a unit cell are not alike, the longitudinal optical phonons produce a crystal

polarization that scatters free electrons. The interaction between electrons and optical phonons

cannot generally be represented in terms of a relaxation time. However, Ehrenreich (1961)[10]

developed an expression for the relaxation time assuming that at high temperatures ( DT ) the

energy change after collision is small compared to the electron energy. This allows the use of the

relaxation time. With a formula of Callen (1949)[11] for the effective ionic charge, the relaxation

time by the optical polar phonons is expressed as

2

1

11212

2

2

8

E

Tkme oBd

po

(12.81)

where o and are the static and high frequency permittivities. Equation (12.81) has been widely

used, giving good estimates for even lower than D . The 8 is added to the magnitude of the

Ehrenreich’s formula from the work of Nag (1980)[12] and Lundstrom (2000)[13].

12.4.3 Ionized Impurity Scattering

Ionized impurity scattering becomes most important at low temperatures, where phonon effects

are small. An ionized impurity produces a long-range (larger than the phonon wavelength)

Coulomb field, which forms a screening and scatters electrons. Conwell and Weisskopf

(1950)[14], Brooks (1951)[15], Blatt (1957)[16], and Amith (1965)[17] studied the ionized

impurity scattering suggesting the Brooks-Herring formula to take account of the screening effect

as

𝜏𝐼 =(2𝑚𝑑

∗ )1 2⁄ 𝜀02(𝑘𝐵𝑇)

3 2⁄

𝜋𝑁𝐼(𝑍𝑒2)2 [𝑙𝑛(1 + 𝑏) −𝑏

1 + 𝑏] (12.82)

12-27

where

22

26

ne

Tkmb Bdo

, IN is the concentration of ionized impurities, n is the electron

concentration and Z is the vacancy charge. The impurity concentration and electron concentration

are assumed to be equal.

Total Electron Relaxation Time

The electron scattering rate is the reciprocal of the relaxation time. The total relaxation time can

be calculated from individual relaxation times according to Matthiessen’s rule as

i i

11 (12.83)

Matthiessen’s rule assumes that the scattering mechanisms are independent of each other.

Example 12.6 Electron Mobility

PbTe (lead telluride) is a widespread thermoelectric material at mid-range temperatures, which is

doped by Na (sodium) at doping concentration of 1.3 × 1019 cm-3, having the following features:

the velocity of sound is 1.45 × 105 cm/s, the mass density is 8.65 g/cm3, the deformation

potential is 11.4 eV, the degeneracy of the conduction valleys is 4 and the DOS effective mass is

0.12 me. Assuming that the acoustic phonon scattering is a dominant mechanism for the

relaxation time, show that the electron mobility for the PbTe at room temperature is 1270

cm2/Vs.

Solution:

Physical constant: Js3410626.6 , KJkB

231038.1 , and Ce 19106021.1

Information given: 325103.1 mn , smvs

31045.1 , 331065.8 mkgdPbTe , eVa 4.11 ,

Nv = 4, and )101.9(12.0 31kgmd

The relaxation time with r = -1/2 for the acoustic phonon scattering is assumed. From Equation

(12.80),

s

Tkm

dv

Bda

s 13

232

24

0 1055.12

2

12-28

From Equations (12.67) and (12.68), the mobility is expressed as

21

21

02

3

3

2

F

Fr

m

e r

c

In order to calculate the Fermi integral, the reduced Fermi energy should be provided. From

Example 12.4, the Fermi energy is

2.2

FE

Then, the values of the Fermi integrals of F0 and F1/2 with 2.2

FE can be obtained by

interpolation between the values in Table C-1. Then, we have

3051.20 F and 7694.22/1 F

Now, assuming that dc mm , the electron mobility is calculated as

Vs

ms

kg

C

F

F

m

e

c

213

31

19

21

00 127.0

7694.2

3051.2

3

21055.1

)101.9(12.0

106021.1

2

3

2

1

3

2

Comments: The reduced Fermi energy of 2.2 for this type of semiconductor (PbTe) shows a

positive value because of the high doping concentration of 1.3 × 1019 cm-3. The mobility of 1270

cm2/Vs is comparable to the experimental value of 1100 cm2/Vs at room temperature.

12.5 Multiband Effects

So far it has been assumed that only one type of charge carrier (electron or hole) is present in the

conductor. We now consider a conductor in which there are both electrons and holes. In an intrinsic

semiconductor there are equal numbers of negative electrons and positive holes. Similarly, if an

impurity semiconductor is taken up to a high enough temperature, a certain number of electron-

hole pairs are excited across the forbidden gap. From the electrodynamic relations (Equation

(11.1)), we have

Tj

Ε (12.84)

or

12-29

x

T

-Εj (12.85)

Considering two bands, the problem will be discussed for both electron and hole

carriers (represented by the subscripts 1 and 2, respectively)

x

T111 -Εj and

x

T222 -Εj

(12.86)

The total current must be

21 jj j (12.87)

Then,

x

T

221121 -Εj (12.88)

Therefore, comparing this with Equation (12.85), the total electrical conductivity is expressed as

21 (12.89)

and also the total Seebeck coefficient is expressed as

21

2211

(12.90)

From the electrodynamic relations (Equation (11.2)), we have

x

TkTjq ee

(12.91)

For two bands, we have

12-30

x

TkTjq ee

1111 and

x

TkTjq ee

2222 (12.92)

The total electronic thermal conductivity is expressed as

x

TkTj

x

TkTjqqq eeeee 22211121 (12.93)

Inserting Equation (12.86) into (12.93) and using Equation (12.34) finally gives

Tkkk eee

2

12

21

2121

(12.94)

The remarkable feature of Equation (12.94) is that the total electronic thermal conductivity is not

merely the sum of the thermal conductivities of the separate carriers. There is an additional term

associated with the bipolar flow that is the third term in the right-hand side of the equation.

12.6 Nonparabolicity

Nonparabolic Density of States

For the simple parabolic model the energy dispersion is given as

𝐸 =ℏ2

2(𝑘𝑥2

𝑚𝑥+

𝑘𝑦2

𝑚𝑦+𝑘𝑧2

𝑚𝑧) (12.95)

where mx, my, and mz are the principal effective masses in the x-, y-, and z-directions and here k is

the magnitude of the wavevector.

𝑘2 = 𝑘𝑥2 + 𝑘𝑦

2 + 𝑘𝑧2 (12.96)

For high applied fields, electrons may be far above the conduction band edge, and the higher order

terms in the Taylor series expansion cannot be ignored. For the nonparabolic model the energy

dispersion is given by

12-31

𝐸 (1 +𝐸

𝐸𝐺) =

ℏ2

2(𝑘𝑥2

𝑚𝑥+

𝑘𝑦2

𝑚𝑦+

𝑘𝑧2

𝑚𝑧)

(12.97)

which is known as Kane model. Let 𝛾(𝐸) = 𝐸(1 + 𝐸 𝐸𝐺⁄ ) and introduce a new wavevector 𝑘′ and

an effective mass 𝑚′ as

𝛾(𝐸) = 𝐸 (1 +𝐸

𝐸𝐺) =

ℏ2𝑘′2

2𝑚′=

ℏ2

2𝑚′(𝑘′𝑥

2 + 𝑘′𝑦2 + 𝑘′𝑧

2) (12.98)

Equating Equations (12.97) and (12.98), we have a relationship between the original wavevector

and the new wavevector as

𝑘𝑥 = 𝑘𝑥′√

𝑚𝑥

𝑚′

(12.99)

𝑘𝑦 = 𝑘𝑦′√

𝑚𝑦

𝑚′

𝑘𝑧 = 𝑘𝑧′√

𝑚𝑧

𝑚′

In k-space of Figure 12.4, we have

𝑑𝑘 = 𝑑𝑘𝑥𝑑𝑘𝑦𝑑𝑘𝑧 = √𝑚𝑥𝑚𝑦𝑚𝑧

𝑚′3𝑑𝑘𝑥

′ 𝑑𝑘𝑦′ 𝑑𝑘𝑧

′ = √𝑚𝑥𝑚𝑦𝑚𝑧

𝑚′34𝜋𝑘′2𝑑𝑘′

(12.100)

12-32

(a) (b)

Figure 12.4 A constant energy surface in k-space: (a) three-dimensional view, (b) lattice points

for a spherical band in two-dimensional view.

The volume of the smallest wavevector in a crystal of volume L3 is (2/L)3 since L is the largest

wavelength. The number of states between k and k + dk in three-dimensional k-space is then

obtained (see Figure 12.4)

𝑁(𝑘)𝑑𝑘 =2 ∙ 4𝜋𝑘′2

(2𝜋 𝐿⁄ )3√𝑚𝑥𝑚𝑦𝑚𝑧

𝑚′3𝑑𝑘′

(12.101)

where the factor of 2 accounts for the electron spin (Pauli Exclusion Principle). Now the density

of states g(k) is obtained by dividing the number of states N by the volume of the crystal L3.

𝑔(𝑘)𝑑𝑘 =𝑘′2

𝜋2√𝑚𝑥𝑚𝑦𝑚𝑧

𝑚′3𝑑𝑘′

(12.102)

From Equation (12.98), we have

𝑑𝑘′

𝑑𝛾=(2𝑚′)

12

2ℏ𝛾−

12

(12.103)

𝑑𝑘′

𝑑𝐸=𝑑𝑘′

𝑑𝛾

𝑑𝛾

𝑑𝐸=(2𝑚′)

12

2ℏ𝛾−

12𝛾′ =

(2𝑚′)12

2ℏ(𝐸 +

𝐸2

𝐸𝐺)

−12

(1 +2𝐸

𝐸𝐺)

(12.104)

Using 𝑚𝑑∗ = (𝑚𝑥𝑚𝑦𝑚𝑧)

1 3⁄ and including the degeneracy of valleys 𝑁𝑣 , 𝑚′ is cancelled out. The

nonparabolic density of states is finally obtained as

12-33

𝑔(𝐸) =𝑁𝑣

2𝜋2(2𝑚𝑑

∗

ℏ2)

32

(𝐸 +𝐸2

𝐸𝐺)

12

(1 +2𝐸

𝐸𝐺)

(12.105)

Electron Group Velocity

From Equation (11.10), the group velocity of electrons is given by

𝜐(𝐸) =1

ℏ

𝜕𝐸

𝜕𝑘′=1

ℏ

𝜕𝛾 𝜕𝑘′⁄

𝜕𝛾 𝜕𝐸⁄

(12.106)

Using Equations (12.48), (12.51) and (12.55), the group velocity of electrons is obtained from

𝑣𝑥2 =

2 (𝐸 +𝐸2

𝐸𝐺)

3𝑚𝑐∗ (1 +

2𝐸𝐸𝐺

)2

(12.107)

where

𝑚𝑐∗ = [

1

3(1

𝑚𝑥+

1

𝑚𝑦+

1

𝑚𝑧)]

−1

(12.108)

The nonparabolic two-band model of thermoelectric transport properties and scattering rates for

electrons and phonons are further discussed in Chapters 15 and 16.

Problems

12.1 Derive Equation (12.2) of coll

t

f

r

fr

k

fk

t

f

dt

df .

12.2 Derive Equation (12.10) of

x

T

T

EE

x

E

E

fvf FFo .

12.3 Estimate the relaxation time for copper that has an fcc lattice with lattice constant of 3.61

Å if its electrical conductivity of 5.88 × 105 (cm)-1 is given.

12-34

12.4 Derive Equation (12.23), FEExF vEgeE

22 .

12.5 Derive Equation (12.24),

nem

ne

2

.

12.6 Derive in detail Equation (12.30),

FEE

BE

ETk

e

ln

3

22

.

12.7 Derive Equation (12.31),

FEE

BE

l

n

EgTk

e

1

3

22

.

12.8 Derive Equation (12.35), 3

2

2

2

33

2

nmT

e

kB

.

12.9 Estimate the Seebeck coefficient of copper that has an fcc lattice with lattice constant of 3.61

Å at room temperature.

12.10 Derive in detail Equation (12.42), TTLk oe 2 .

12.11 Provide thermoelectric properties (, , and k) versus electron concentration curves from

1017 to 1021 cm-3 for Sn (tin) with the effective mass of 1.3 me at room temperature using the

constant relaxation time of 10-14 sec (Figure P12.11). Hint: you may use Equations (11.33),

(12.23), (12.34)with r = 0, and (12.44) with neglecting the second term for a metal. For the

plot, you may use vector notation, where n+[+i gives ni of vector notation rather than just

subscript in Mathcad to determine the logarithmic interval scale for the plots such as i =

1,2… 30, ni = 1017+0.14×i. Use V/K for , (cm)-1 for andW/cmK for k.

12-35

Figure P12.11. TE properties vs. electron concentration.

12.12 Provide the thermoelectric transport property curves against the Fermi energy for PbTe as

shown in Figure 12.2 The Seebeck coefficient, the electrical conductivity, and the electronic

thermal conductivity as a function of the Fermi energy at room temperature for PbTe

assuming a constant relaxation time 0 of 6.2 × 10-14 s (realistic value) and r = -1/2 (see

Equation (12.42))..

12.13 Derive Equation (12.74) with (12.75).

12.14 Derive the nonparabolic density of states, Equations (12.105), and the group velocity,

Equation (107).

12.15 Plot Figure 12.3 with a brief explanation.

References

1. Goldsmid, H.J., Thermoelectric Refrigeration. 1964, New York: Plenum Press. 240.

2. Kittel, C., Introduction-to-Solid-State-Physics 8th Edition. 2005: Wiley.

3. Ashcroft, N.W. and N.D. Mermin, Solid state physics. 1976, New York: Holt, Rinehart and

Winston.

4. Wilson, A.H., The theory of metals, 2nd ed. 1953, Cambridge: Cambridge University Press.

|a| &s k

n

s

ak

12-36

5. Chasmar, R.P. and R. Stratton, The thermoelectric figure of merit and its relation to

thermoelectric generators. J. Electron. Control., 1959. 7: p. 52-72.

6. Goldsmid, H.J., Introduction to thermoelectricity. 2010, Heidelberg: Springer.

7. Rowe, D.M., CRC handbook of thermoelectrics. 1995, Boca Raton London New York:

CRC Press.

8. Pei, Y., et al., Low effective mass leading to high thermoelectric performance. Energy &

Environmental Science, 2012. 5(7): p. 7963.

9. Bardeen, J. and W. Shockley, Deformation Potentials and Mobilities in Non-Polar

Crystals. Physical Review, 1950. 80(1): p. 72-80.

10. Ehrenreich, H., Band Structure and Transport Properties of Some 3–5 Compounds. Journal

of Applied Physics, 1961. 32(10): p. 2155.

11. Callen, H., Electric Breakdown in Ionic Crystals. Physical Review, 1949. 76(9): p. 1394-

1402.

12. Nag, B.R., Electron Transport in Compound Semiconductors. Springer Series in Solid

Sciences. 1980, New York: Springer.

13. Lundstrom, M., Fundamentals of carrier transport, 2nd ed. 2000, Cambridge: Cambridge

University Press.

14. Conwell, E. and V. Weisskopf, Theory of Impurity Scattering in Semiconductors. Physical

Review, 1950. 77(3): p. 388-390.

15. Brooks, H., Scattering by ionized impurities in semiconductors. Phys. Rev., 1951. 83: p.

879.

16. Blatt, F.J., Theory of mobility of electrons in solids, in Solid State Physics, F. Seitz and D.

Turnbull, Editors. 1957, Academic Press: New York.

17. Amith, A., I. Kudman, and E. Steigmeier, Electron and Phonon Scattering in GaAs at High

Temperatures. Physical Review, 1965. 138(4A): p. A1270-A1276.