13-1

Chapter 13 Phonons

Contents

Chapter 13 Phonons ...................................................................................................... 13-1

Contents .......................................................................................................................... 13-1 13.1 Crystal Vibration ................................................................................................ 13-2

13.1.1 One Atom in a Primitive cell ..................................................................... 13-2 13.1.2 Two Atoms in a Unit cell ........................................................................... 13-5

13.2 Specific Heat .................................................................................................... 13-10 13.2.1 Internal Energy......................................................................................... 13-10

13.2.2 Debye Model ............................................................................................ 13-11

Example 13.1 Atomic Size and Specific Heat .......................................................... 13-17

13.3 Lattice Thermal Conductivity .......................................................................... 13-21 13.3.1 Klemens-Callaway Model ....................................................................... 13-21 13.3.2 Umklapp Processes .................................................................................. 13-25

13.3.3 Callaway Model ....................................................................................... 13-27 13.3.4 Phonon Relaxation Times ........................................................................ 13-27

Example 13.2 Lattice Thermal Conductivity ............................................................ 13-32 Problems ....................................................................................................................... 13-34 References ..................................................................................................................... 13-36

The quantized energy of a lattice vibration is called a phonon, which is in analogy with a photon

of the electromagnetic wave.

13-2

13.1 Crystal Vibration

13.1.1 One Atom in a Primitive cell

Consider the one-dimensional elastic vibration of a crystal with one atom in a unit cell in Figure

13.1. We want to find the frequency of an elastic wave in terms of the wavevector k. There are

the precise equation if each atom were connected to its neighbors by perfect springs with spring

constant C.

Figure 13.1 One-dimensional monatomic lattice chain model. a is the distance between atoms

(lattice constant). The atoms as displaced during passage of a longitudinal wave.

We assume that the force at x is proportional to the displacement as

nnnnn xxCxxCf 11 (13.1)

Using the Newton’s second law of motion with an atom of mass m,

2

2

dt

xdmf n

n

(13.2)

Combining these two, we have

Na

a

xn-1xn xn+1 xn+2 xn+3 xn+4

13-3

nnnn xxxC

dt

xdm 2112

2

(13.3)

The general solution is found to be

tknai

n Aex (13.4)

where is the angular frequency of f, where f is the frequency. We consider the standing

wave for a crystal which has a finite number N of atoms. N is large and we simply connect the two

remote ends of the chain making a circular chain. The periodic boundary conditions requires that

1iknae (13.5)

Since knaiknaeikna sincos , we have

1cos kna (13.6)

which leads to

N

n

ak

2 , where n = 1, 2, …..N (13.7)

Note that k is changed by a2 . Considering the first Billouin zone, using periodicity we have

ak

a

(13.8)

Thus, the displacement of phonons can always be described by a wavevector within the Brillouin

zone. Taking the second derivative of Equation (13.4) gives

13-4

tknain Aedt

xd 2

2

2

(13.9)

Inserting Equations (13.4) and (13.9) into (13.3) gives

tknaiikaika

tknaitankitankitknai

eeeCA

AeAeAeCAe

2

2112

(13.10)

which leads to

21

22ikaika ee

m

C

(13.11)

Solving for , we have

2sin

4 ka

m

C

(13.12)

The solution describes a wave propagating along the chain with phase velocity sv , and group

velocity gv . The frequency is plotted against the wavevector k in Figure 13.2. Such a curve is

known as a dispersion relation. The group velocity is

kvg

(13.13)

When k is small compared with /k, is linear in k. The phase velocity is equal to the speed of

sound sv as

13-5

kvs

(13.14)

Figure 13.2 Phonon dispersion curve of a one-dimensional monatomic lattice chain for Brillouin

zone. The Debye approximation use a linear relationship between the frequency and the

wavevector.

It can be seen in Figure 13.2 that the dispersion curve in the Brillouin zone differs greatly from

dispersion curve for a continuum, which for comparison has been assumed to have the same

behavior as the discrete chain in the long-wavelength limit (small k).

13.1.2 Two Atoms in a Unit cell

Consider the one-dimensional elastic vibration of a crystal with two different mass atoms in a unit

cell, which are connected with spring constant C (Figure 13.3).

13-6

Figure 13.3 Diatomic linear chain of different atoms, connected with spring constant C.

For large M,

nnnnn XxCXxCF 1 (13.15)

For small m,

nnnnn xXCxXCf 1 (13.16)

Using the Newton’s second law of motion, we have

𝑀𝑑2𝑋𝑛𝑑𝑡2

= 𝐶(𝑥𝑛 + 𝑥𝑛+1 − 2𝑋𝑛) (13.17)

nnnn xXXC

dt

xdm 212

2

(13.18)

The determinant should be zero and then

13-7

0242 224 ikaika eeCmMCMm (13.19)

Solving for and ,

kamMCMmCMmC

mMcos1822

2

1 222

1

(13.20)

and

kamMCMmCMmC

mMcos1822

2

1 222

2

(13.21)

When k is small, 1cos ka which leads to 01 and

mM

MmC

22

(13.22)

When k is /a at the boundary of the Brillouin zone, 1cos ka which leads to

M

C21

(13.23)

m

C22

(13.24)

In Figure 13.4, the lower branch has the same structure as the single branch we found in the

monatomic lattice: vanishes linearly in k for small k, and the curve becomes flat at the edges of

the Brillouin zone. This branch is known as the acoustical branch because its dispersion relation

is of the form = vk characteristic of sound waves at small k. The second branch starts as

13-8

mMMmC 22 at k = 0 and decreases with increasing k. This branch is known as the

optical branch because the long wavelength optical modes in ionic crystals can interact with

electromagnetic radiation, and are responsible for much of the characteristic optical behavior of

such crystals.

Figure 13.4 Phonon dispersion curve of a one-dimensional diatomic lattice chain for Brillouin

zone.

If there are p-atoms in a 3-dimensional lattice, there are 3×p branches to the dispersion relation:

three acoustical branches and (3×p-3) optical branches. For example one atom in a lattice has 3

acoustical and zero optical branches. If there are two atoms in a lattice, there will be 6 branches: 3

acoustical and 3 optical branches as shown in Figure 13.5. We have longitudinal acoustical LA

and transverse acoustical TA modes, and longitudinal optical LO and transverse optical TO modes.

The group velocity of the upper optical modes is small so that they are not very effective in

transporting energy, but they may affect heat flow by interacting with the acoustical modes which

are mainly responsible for the thermal conductivity.

13-9

Figure 13.5 Phonon dispersion relations of a three-dimensional diatomic lattice for Brillouin

zone.

Although a crystal may be said to have three acoustical branches, this does not necessary mean

that these must everywhere have different frequencies. In a cubic structure of PbTe, the two

transverse branches are degenerate, which is shown in Figure 13.6 (a). It should be noted that

“optical” is the term used to describe all branches that have a non-zero frequency at k = 0.

13-10

(a)

(b)

Figure 13.6 (a) Calculated phonon dispersion of PbTe by Zhang et al. (2009)[1], (b) The

Brillouin zone of PbTe. Points of high symmetry are denoted by , L, X etc.

13.2 Specific Heat

The heat capacity is the ratio of the amount of heat energy per unit temperature rise. The specific

heat is the heat capacity per unit mass, which is a characteristic of the material.

13.2.1 Internal Energy

In quantum statistics, the energy of a linear lattice vibration is given by

2

1, phofE

(13.25)

where ½ is the zero point energy which has no contribution to the heat capacity. is the energy

of a phonon. phof , is the thermal equilibrium occupancy (digit number) of phonons, which is

called the Bose-Einstein distribution function as

13-11

1

1,

TkphoBe

f

(13.26)

The internal energy U, which is the total vibrational energy, can be obtained by summing over all

normal modes, approximating the lattice as a continuum but with a maximum frequency.

dgEU ph

p

max

0

(13.27)

where p is pth branch (the index of phonon branches) and gph() is the phonon density of states.

13.2.2 Debye Model

In the Debye model, the lattice vibrates as if it were an elastic continuum, but the vibration

frequencies cannot exceed a certain maximum value, chosen to make the total number of modes

equal to the total number of classical degrees of freedom. The Debye model replaces all branches

of the vibrational spectrum with three branches, each with the same linear dispersion relations.

The velocity of sound is taken as constant for each branches, as it would be for a classical elastic

continuum. In the Debye model the optical modes are unceremoniously lumped into the top of the

distribution of acoustic modes, as if they were merely elastic waves of very short wavelength.

Despite its obvious crudity, the Debye approximation has the great advantage of supreme

simplicity. If any one parameter is required to measure the energy scale of the vibrations of a solid,

the Debye temperature is the most appropriate. If any one function is required to present the

distribution of the lattice frequency, the phonon density of states is the simplest.

The dispersion relation is approximated, as in Equation (13.14), to be

kvs (13.28)

13-12

For each branch, the total number of modes (degree of freedom) N is obtained by dividing a sphere

of k-space by a volume of the smallest wavevector k.

2

3

3

3

62

3

4

Vk

L

k

kN

(13.29)

where 3LV and 333

svk , we have

32

3

6 sv

VN

(13.30)

If there are N primitive cells in a specimen, the total number of acoustic phonon modes is N. Debye

treated each mode as a quantized harmonic oscillator. The Debye cutoff frequency D should be

3

1

26

V

NvsD

(13.31)

which was also shown in Figure 13.2. The phonon density of states (DOS) per each branch is

obtained by taking derivative of Equation (13.30) with respect to as

𝑔𝑝ℎ(𝜔) =𝑑𝑁(𝜔)

𝑑𝜔=

𝑉𝜔2

2𝜋2𝑣𝑠3 (13.32)

Substituting Equation (13.31) into Equation (13.32), the phonon density of states is expressed by

13-13

3

23

D

ph

Ng

(13.33)

which is a functional distribution of spectrum. Substituting Equations (13.25) and (13.33) into

(13.27), the internal energy U is expressed as

d

N

eU

DpTk

D

B

3

2

0

3

1

(13.34)

where p is the index of phonon branches (modes). The thermodynamic definition of specific heat

is

V

VdT

dU

Vdc

1

(13.35)

where d is the density and, assuming p to be 3 for bulk materials (Debye model), the specific heat

is then

de

e

Tkk

Vd

Nc

DTk

Tk

B

BV

D

B

B

3

2

0

2

2

3

13

(13.36)

The phonon density of states per branch per mode is expressed as

3

2

,

3

D

brphg

(13.37)

which was compared with experiment by Rauh (1981) [2]. The region of lower frequencies in the

experimental curve of Figure 13.7 shows the mixture of acoustical and optical branches

significantly contribute to the phonon density of states resulting in underestimate of the Debye

model, which is shown in Figure 13.6 as expected. The region of higher frequencies in the

experimental curve shows overestimate of the Debye model. Interestingly the integrated density

13-14

of states of the simple Debye model approximately matches that of the experimental density of

states.

Figure 13.7 Phonon density of states versus frequency for Bi2Te3 [3].The Debye model is

compared with experiment by Rauh (1981)[2].

Debye Temperature

The Debye temperature D is defined as

B

DD

k

(13.38)

Equivalently,

TkT B

DD

(13.39)

Specific Heat

13-15

From Equation (13.36), the specific heat Vc is finally obtained by

dx

e

exTk

Vd

Nc

T

x

x

D

BV

D

0

2

43

19

(13.40)

where

Tkx

B

(13.41)

Equation (13.40) is well known the Debye formula which is widely used in the literature.

The velocity of Sound and Atomic Size

The velocity of sound is estimated using Equations (13.31) and (13.38) as

DB

sV

Nkv

3

1

26

(13.42)

N/V can be approximately obtained assuming a cubic structure. N is the number of primitive cells

and V is the crystal volume that is the product of N and volume 3 of the primitive cell as [4]

33

1

N

N

V

N

(13.43)

where is the atomic size. Note that the Debye model assumes one atom in a primitive unit cell,

in which is the lattice constant a. However, when a crystal has more than one atom ( an ) in a

primitive cell, 𝛿 is not the lattice constant. In that case, the volume of the atomic size 3 is

expressed by

13-16

an

a33

(13.44)

Hence, the velocity of sound in Equation (13.42) can be expressed by

DB

s

kv 3

126

(13.45)

where 3 is the volume of the primitive cell. When na = 1, we have that a = as Debye model

assumed. It is seen that the vibration modes are derived based on atoms rather than lattices. We

can also approximately calculate the size of the primitive cell (atomic size) for a binary

compound using the atomic masses and the mass densities by

3133 1 yy BA , 31

AAAA dNM and

31

BABB dNM

(13.46)

yMyMM BAAB 1 (13.47)

3A

ABAB

N

Md

(13.48)

where y is the fraction of component A, NA is the Avogadro’s number, BAM , are the atomic masses

of component A or B, ABM are the mean atomic masses of component A and B, and BAd , are the

mass densities of component A or B.

13-17

Example 13.1 Atomic Size and Specific Heat

PbTe (lead telluride) is a widespread thermoelectric material with the Debye temperature D of

136 K. (a) Estimate the atomic size of the PbTe, (b) determine the velocity of sound (phase

velocity) for the PbTe, and (c) determine the specific heat at room temperature.

Solution:

From Equation (13.46) with y = 0.5 for component Pb and the periodic tables in Appendices A-1

and A-5, the atomic size is

mcmg

gdNM PbAPbPb

10

3/1

323

311012.3

34.1110022.6

2.207

mcmg

gdNM TeATeTe

10

3/1

323

3110236.3

25.610022.6

6.127

3133 1 yy TePb = 3.179 × 10-8 cm = 3.179 Å

From Equation (13.45), the velocity of sound is

smmKJs

KJkv D

Bs /10452.110179.31366

210626.6

1038.16 3103

12

34

23

3

12

From Equation (13.47) and (13.48), the mean atomic mass and density of PbTe can be calculated

gggyMyMM TePbPbTe 4.1675.016.1275.02.2071

13-18

338233654.8

10179.310022.6

4.167

cm

g

cm

g

N

Md

A

PbTePbTe

From Equation (13.40),

dx

e

exTk

Vd

Nc

T

x

x

D

B

PbTe

V

D

0

2

43

19

The Debye integrand is obtained by interpolation from Table C-2 with 453.0TD

031.0

10

2

4

dxe

exT

x

xD

The specific heat at room temperature is

Kkg

J

K

K

mkgm

KJcV

5.147031.0136

300

10654.810179.3

1038.193

33310

23

In mole basis,

Kmol

J

Kkg

J

mol

gkgggcV

37.495.147106.1272.207 3

Comments: These values of the velocity of sound and the specific heat are comparable to the

experimental values for PbTe.

13-19

The calculated temperature-dependent specific heat for Mg2Si is plotted along with the

measurements by Gerstein (1967)[5] in Figure 13.8.

Figure 13.8 Specific heat of Mg2Si [3]. The marks are the measurements by Gerstein (1967) [5]

and the solid line is the calculation by Equation (13.40) using the Debye temperature of 417 K.

Dulong-Petit Law

Equation (13.34) with three branches in a three-dimensional crystal can be expressed as

dx

e

x

xTNkU

x

xB

0

3

3 1

33

(13.49)

where T

x D

When the temperature is much higher than the Debye temperature ( 1 Tx D ), the bracket

may be expanded as

13-20

13

....!430

1

!26

1

21

3

1

3

1

3

0

2

3

0

422

3

0

2

3

0

3

3

dxxx

dxxxx

xx

dxe

xx

xdx

e

x

x

xx

x

x

x

x

(13.50)

Then, the internal energy U becomes

TNkU B3 (13.51)

Since V

VdT

dU

Vdc

1 from Equation (13.35), the specific heat at high temperatures is

BV kVd

Nc 3

(13.52)

Here we have the famous classical Dulong-Petit law, showing that the atomic heat of all solids

tends to the same value at high temperatures. The specific heat due to the lattice vibrations is just

3 kB per atom.

Debye T3 Law

When the temperature is much lower than the Debye temperature ( 1 Tx D ), one can derive

3TcV (13.53)

This is known the Debye T3 law which is experimentally observed in Figure 13.8 at very low

temperatures.

13-21

13.3 Lattice Thermal Conductivity

13.3.1 Klemens-Callaway Model

The heat current density (heat flux) is given by

x

TkTkq lll

(13.54)

A temperature gradient causes thermal non-equilibrium. The heat current density is the product of

the energy of lattice vibrations and the group velocity as

𝑞𝑙 =∑∑𝐸(𝜔)𝑣𝑥𝑘𝑝

(13.55)

Where p is the index of phonon branches, k the wavevector, and xv the one-dimensional energy

transport velocity of the phonons (group velocity). Using Equation (13.25), the heat current density

lq is

𝑞𝑙 =∑∑ℏ𝜔𝑓𝑝ℎ(𝜔)𝑣𝑥𝑘𝑝

(13.56)

This means that the heat current density lq is the product of three terms: the energy of phonon

, the nonequilibrium (disturbed) phonon distribution number phf , and the group velocity

xv . The phonon distribution number departs from equilibrium in the presence of a temperature

gradient, and in the absence of any interactions it would vary with time at any point in the crystal.

In order to obtain phf , the Boltzmann transport equation is used as given in Equation (12.2).

coll

t

f

r

fr

k

fk

t

f

dt

df

(13.57)

13-22

Under the condition that external forces and electric field do not affect the vibrational motion of

the atoms in a crystal, k has a value of zero. Using the relaxation time approximation in Section

12.1, we have

x

T

T

fvf

ph

xcph

,0

(13.58)

where c is the combined relaxation time. Inserting this into Equation (13.56) gives

𝑞𝑙 = −∑∑ℏ𝜔𝜏𝑐𝑣𝑥2𝜕𝑓0,𝑝ℎ

𝜕𝑇

𝜕𝑇

𝜕𝑥𝑘𝑝

(13.59)

From Equation (13.54), the lattice thermal conductivity 𝑘𝑙 is expressed as

𝑘𝑙 =∑∑ℏ𝜔𝜏𝑐𝑣𝑥2𝜕𝑓0,𝑝ℎ

𝜕𝑇𝑘𝑝

(13.60)

For cubic and isotropic systems, the average velocity is 22 31 vvx . The lattice thermal

conductivity is

𝑘𝑙 =1

3∑∑ℏ𝜔𝜏𝑐𝑣

2𝜕𝑓0,𝑝ℎ

𝜕𝑇𝑘𝑝

(13.61)

This can be expressed in the form of integral using the phonon density of states as

𝑘𝑙 =1

3∑∫ ℏ𝜔𝜏𝑐𝑣

2𝜕𝑓0,𝑝ℎ

𝜕𝑇𝑔𝑝ℎ(𝜔)𝑑𝜔

𝜔𝐷

0𝑝

(13.62)

which, using Equation (13.32) per unit volume, leads to

13-23

𝑘𝑙 =1

3𝑘𝐵∑∫ 𝜏𝑐𝑣

2 (ℏ𝜔

𝑘𝐵𝑇)2 𝑒ℏ𝜔 𝑘𝐵𝑇⁄

(𝑒ℏ𝜔 𝑘𝐵𝑇⁄ − 1)2(

𝜔2

2𝜋2𝑣3)𝑑𝜔

𝜔𝐷

0𝑝

(13.63)

It is assumed that there are three acoustic branches, which are one longitudinal and two transverse

branches (p = 3). This assumption is later verified numerically by Esfarjani et al. (2011) [6]. Using

Equation (13.31), the lattice thermal conductivity finally gives

T

x

x

cBB

l

D

dxe

exTk

v

kk

0

2

43

212

(13.64)

where the phonon velocity v is very close to the velocity of sound vs and Tkx B . This is a

well-known Klemens-Callaway model of the lattice thermal conductivity. In the case that both

and v are independent of frequency , comparing Equation (13.36) for Vc , the lattice thermal

conductivity lk is expressed as

Vcl Cvk 2

3

1

(13.65)

where VC is the heat capacity (specific heat per volume), dcC VV . Using the mean free path

v ,

Vl Cvk 3

1

(13.66)

Although the above model is fairly crude, the expression for the lattice thermal conductivity is

actually a surprisingly good approximation. Very often, the above equation is used to estimate the

mean free path on the basis of experimental results for the other parameters in the equation.

13-24

Figure 13.9 Various mechanisms reducing the lattice thermal conductivity.

If the lattice vibrations are entirely the normal modes (N-processes) as shown in the specific heat,

a heat flow will transport without decaying or scattering, which implies that the lattice thermal

conductivity be infinite as harmonic oscillations. In fact, anharmonicity persists along with the N-

processes. The major mechanism of anharmonicity is called the Umklapp processes (U-processes)

which was discovered by Peierls (1929). The U-processes do not obey the conservation of

momentums while the N-processes do obey, causing the scattering of phonons. The U-processes

increase with increasing temperature.

From Equation (13.66), since the velocity of sound v is independent of temperature, the lattice

thermal conductivity lk depends on the mean free path and the heat capacity VC . At low

temperatures, the U-processes is reduced and the mean free path may reach several millimeters,

being comparable with the dimensions of specimens. The mean free path thus becomes constant

at the grain boundaries. Hence lk depends on VC that varies as T3 which is shown in Figure 13.9.

13-25

At high temperatures, VC is almost constant and T1 . Then the Umklapp processes

predominate lk so that Tkl 1 .

Phonon Relaxation Time

The scattering rate is the reciprocal of the relaxation time . The scattering may be caused by the

coexistence of different processes and a relaxation time can be defined for each process. The

combined relaxation time c can be calculated from individual relaxation time i according to

Matthiessen’s rule (1862),

i ic

11

(13.67)

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

(Ashcroft and Mermin, 1976)[7].

13.3.2 Umklapp Processes

We consider three-phonon processes in a lattice. For example, when the sum of two wavevectors

is within the first Brillouin zone at low temperatures, it is the normal processes as shown in Figure

13.10 (a). At high temperatures it may be the case that the sum of the two wavevectors falls outside

of the zone as shown in Figure 13.10 (b). However, there is a rule that no wavevector must be so

large as to lie outside the first Brillouin zone. In other words, the phonon wavelength cannot be

smaller than the lattice constant. In such a case, the sum of the two wavevectors is forced to lie

inside the first Brillouin zone rather than falling outside the zone, using the reciprocal lattice vector

G and losing the conservation of momentum. This is the Umklapp process or anharmonicity. The

wavevectors can be written as

13-26

Gkkk 321 (13.68)

When the process is normal, G = 0. On the other hand, when the process is Umklapp

process, 0G .

(a) (b)

(c)

Figure 13.10 (a) Normal process 321 kkk , (b) Unacceptable process 321 kkk , and (c)

Umklapp process Gkkk 321 , where G is the reciprocal lattice vector.

13-27

13.3.3 Callaway Model

The normal processes are harmonic oscillations and cannot be expressed as the scattering or the

relaxation time. On the other hand, normal process cannot be ignored because they indirectly cause

the scattering. Callaway (1959)[8] suggested an expression based on the work of Debye (1912)[9]

and Klemens (1955)[10], which is widely used. The lattice thermal conductivity is given by

21 kkkl (13.69)

T

x

x

cB

s

B D

dxe

exTk

v

kk

/

0 2

43

21

12

(13.70)

T

x

x

U

c

U

T

x

x

U

c

B

s

B

D

D

dxe

ex

dxe

ex

Tk

v

kk

/

0 2

4

2

/

0 2

4

3

22

11

1

1

2

(13.71)

where is the ratio of the Umklapp processes to the N-processes, c and U are the combined

and Umklapp relaxation times, respectively. The second Callaway term 2k is large for a pure

crystal but decrease rapidly with addition of defects so that it is dominant over a wide temperature

range for the pure crystal but becomes almost negligible for an impure crystal [11]. Most doped

crystal compounds that have intrinsic defects are impure in fact, so that 1kkl .

13.3.4 Phonon Relaxation Times

13-28

Extensive studies have been made into phonon relaxation times, including normal-mode processes

(N-processes), Umklapp processes, point defect scattering, electron-phonon scattering, and

boundary scattering. We use here the expressions suggested by Steigmeier and Abeles (1964) [12].

Vining (1991) [13] and Minnich (2009) [14] also used those scattering expressions.

Scattering by N-Processes and Umklapp Processes

The Umklapp processes are three-phonon scattering involving anharmonicity. The N-processes do

not directly cause the scattering but help the Umklapp processes contribute to the scattering.

Therefore, they can be expressed in terms of the relaxation time in 1k even when 2k is neglected

in Equation (13.70). The Umklapp scattering rate is the inverse of the relaxation time, which was

first studied by Leibfried and Schlömann (1954).[15] Later, a modified expression including the

effect of the N-processes was given by [12, 13, 16]

2

3

2

231

21

1

9

51

4

6

3

20x

T

MN

DAB

AU

(13.72)

where AN is Avogadro’s number, the Grüneisen anharmonicity parameter, ABM the atomic

mass of compounds A and B, the mean atomic size (see Equation (13.46)), and Tkx B .

The N-process scattering rate is given by

11 UN

(13.73)

where 2 is often used for a good approximation.

Scattering by Point Defects

13-29

The point defects may include vacancies, isotopes, dislocations, substitute atoms, etc. A defect

with dimensions much smaller than the phonon wavelength can be considered as a point defect

[11]. The scattering is then caused by the difference in mass and the difference in bonding between

the atoms.[11] The scattering rate for point defects is given by Klemens (1960)[17], which is

3

4

4

01

4 s

B

PDv

xTk

V

(13.74)

where 0V is the atomic volume (3 ) and the mass-fluctuation-scattering parameter for a binary

compound is given by[12]

22

1

s

ABM

Myy

(13.75)

BA MMM , )1( yMyMM BAAB , and BA (13.76)

where y is the fraction of component A, the atomic size (see Equation (13.46)) and s the

strain parameter [12].

Electron-Phonon Scattering

The scattering of phonons by electrons will be active when the band degeneracy temperature is

comparable with the temperature of the lattice [18]. Ziman (1956) derived an expression for

electron-phonon scattering, which is

13-30

216

216

4

3

,

2

1

,

,

2

,

2

1

1ln

4 x

E

xEE

x

E

xEE

rc

sida

iEP

ircFrc

ircFrc

e

ex

dE

vm

(13.77)

where a is the acoustic deformation potential, TkvmE Bsidirc 22

,,

and Tkx B .

Boundary Scattering

The phonon-boundary scattering rate is assumed independent of temperature and frequency. The

boundary scattering rate with an assumption of purely diffuse scattering is given by[4]

L

vsB 1

(13.78)

where L is the effective length of the sample. The total scattering time can be approximated by

adding the scattering rates in accordance with Matthiessen’s rule in Equation (13.67),

BEPPDUNc

111111

(13.79)

In order to visualize the individual effect of the phonon scattering mechanisms, the

weighted phonon relaxation times of 24 1 xx eex is plotted in Figure 13.11 for

PbTe. It is seen that the Umklapp processes dominates the phonon relaxation time at

low frequencies and the point defects become dominant at high frequencies. The

electron-phonon scattering has a minimal effect on the relaxation time.

13-31

Figure 13.11 Weighted phonon relaxation time versus frequency for PbTe [3].

Since we have developed the expressions for the different relaxation times, we can calculate the

lattice thermal conductivity versus temperature using of Equation (13.64) along with the

electronic thermal conductivity as shown in Figure 13.12 for PbTe. It is seen that the lattice and

electronic contributions to the combined thermal conductivity are almost equal. The total

(combined) thermal conductivity shows a good agreement with the experimental data by Pei at

al. (2012)[19].

13-32

Figure 13.12 The electronic, lattice, and combined thermal conductivities versus temperature for

PbTe. The electron doping concentration of 1.8 × 1019 cm-3 was used [3].

Example 13.2 Lattice Thermal Conductivity

PbTe (lead telluride) is a widespread thermoelectric material with Debye temperature D 136 K.

Assuming that the Umklapp processes are the dominant scattering mechanism for PbTe, (a)

determine the Debye cutoff frequency f, not angular frequency ( fD 2 ), (b) determine the

lattice thermal conductivity using the Grüneisen parameter = 1.5 at room temperature, and (c)

estimate the phonon mean free path assuming the Dulong-Petit law for the heat capacity.

Solution:

From Equation (13.46) with y = 0.5 and the periodic tables in Appendices A-1 and A-5, the atomic

size is estimated as shown Example 13.1,

mcmg

gdNM PbAPbPb

10

3/1

323

311012.3

34.1110022.6

2.207

mcmg

gdNM TeATeTe

10

3/1

323

3110236.3

25.610022.6

6.127

3133 1 yy TePb = 3.179 × 10-8 cm = 3.179 Å

From Equation (13.45), the velocity of sound is

smmKJs

KJkv D

Bs /10452.110179.31366

210626.6

1038.16 3103

12

34

23

3

12

13-33

From Equation (13.38), since fD 2 , the Debye cutoff frequency is

1121084.22

sk

f DBD

=2.84 THz

From Equation (13.76), the mean atomic mass is

gyMyMM TePbPbTe 4.167)1(

From Equation (13.72), using = 2, the Umklapp relaxation time that is here the combined one as

1

2

3

2

231

2

1

9

51

4

6

3

20

x

T

MNx

DPbTePbTe

AU

From Equation (13.64), the lattice thermal conductivity is

T

x

x

UB

s

Bl

D

dxe

exx

Tk

v

kk

0

2

43

212

The equation can be rewritten as

T

x

x

DPbTePbTe

AB

s

Bl

D

dxe

exx

T

MN

Tk

v

kk

0

2

42

1

3

2

231

23

211

9

51

4

6

3

20

2

13-34

The Debye integral with 453.0TD can be obtained by interpolation from Table F-2.

450.0

10

2

2

T

x

xD

dxe

ex

The lattice thermal conductivity is finally obtained as

mK

Wkl 0.4

Since KTKD 300136 , the heat capacity of the Dulong-Petit law from Equation (13.52)

can be used as

Km

Jkk

V

NC B

BV 3

6

310289.1

33

From Equation (13.66), the phonon mean free path is

nmmCv

k

Vs

l 4.6104.63 9

Comments: The value of 4.0 W/mK for the lattice thermal conductivity seems high compared to

the experimental value of about 1.3 W/mK in Figure 13.12. This value may be improved if we use

the Matthiessen’s rule with all the phonon scattering mechanisms.

Problems

13.1. Derive Equation (13.12).

13-35

13.2. Derive Equations (13.22) and (13.23) with a brief explanation.

13.3. Derive Equation (13.40).

13.4. Estimate the atomic size for a faced centered cubic structure of Mg2Si using both

methods: a) lattice constant = 6.338 Å, and (b) atomic masses and mass densities for the

binary compound (y = 0.33).

13.5. Plot the theoretical phonon density of states versus frequency as shown Figure 13.7 for

Bi2Te3 with the Debye temperature D of 165 K and the fraction of components and the

mass densities.

13.6. Mg2Si (magnesium silicide) is a widespread thermoelectric material with the Debye

temperature D of 417 K. (a) Estimate the atomic size, (b) determine the velocity of sound

(phase velocity), and (c) determine the specific heat for Mg2Si.

13.7. Provide the theoretical curve for the specific heat of Mg2Si with the Debye temperature D

of 417 K as shown in Figure 13.8.

13.8. Derive the equation for the classical Dulong-Petit law.

13.9. Derive Equation (13.64) and (13.66).

13.10. Mg2Si (magnesium silicide) is a widespread thermoelectric material with the Debye

temperature D of 417 K. Assuming that the Umklapp processes are a dominant scattering

mechanism, (a) determine the Debye cutoff frequency f, not angular frequency ( fD 2 ),

(b) determine the lattice thermal conductivity using Grüneisen parameter = 1.67 at room

13-36

temperature, and (c) estimate the phonon mean free path assuming the Dulong-Petit law

for the heat capacity.

References

1. Zhang, Y., et al., Thermodynamic properties of PbTe, PbSe, and PbS: First-principles

study. Physical Review B, 2009. 80(2).

2. Rauh, H., et al., Generalised phonon density of states of the layer compounds Bi2Se3

Bi2Te3 Sb2Te3 and Bi2(Te05Se05)3. J. Phys. C: Solid State Phys., 1981. 14: p. 2705-2712.

3. Lee, H., A Theoretical Model of Thermoelectric Transport Properties for Electrons and

Phonons. Journal of Electronic Materials, 2016. 45(2): p. 1115-1141.

4. Ziman, J.M., Electrons and phonons. 1960, London: Oxford University Press.

5. Gerstein, B.C., Thermal Study of Groups II—IV Semiconductors. Lattice Heat Capacities

and Free Energies of Formation. Heat Capacity of Mg2Si from 15°—300°K. The Journal

of Chemical Physics, 1967. 47(6): p. 2109.

6. Esfarjani, K., G. Chen, and H.T. Stokes, Heat transport in silicon from first-principles

calculations. Physical Review B, 2011. 84(8).

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

Winston.

8. Callaway, J., Model for Lattice Thermal Conductivity at Low Temperatures. Physical

Review, 1959. 113(4): p. 1046-1051.

9. Debye, P., The theory of specific heats. Ann. Phys. Lpz. (4), 1912. 39: p. 798.

10. Klemens, P.G., The scattering of low-frequency lattice wavesby static imperfections. Proc.

Phys. Soc., 1955. 68(12): p. 1113-1128.

11. Berman, R., Thermal conduction in solids. 1976, Oxford: Clarendon Press.

12. Steigmeier, E. and B. Abeles, Scattering of Phonons by Electrons in Germanium-Silicon

Alloys. Physical Review, 1964. 136(4A): p. A1149-A1155.

13. Vining, C.B., A model for the high-temperature transport properties of heavily doped n-

type silicon-germanium alloys. Journal of Applied Physics, 1991. 69(1): p. 331.

14. Minnich, A., et al., Modeling study of thermoelectric SiGe nanocomposites. Physical

Review B, 2009. 80(15).

15. Leibfried, G. and E. Schlomann, Warmeleitung in elektrisch isolierenden Kristallen.

Nachrichten der Akademie der Wissenschaften in Gottingen, lia, Mathematisch-

Physikalische Klasse, 1954: p. 71-93.

16. Klemens, P.G., Thermal conductivity and lattice vibrational modes. Solid StatePhysics

Chapter 7. 1958, New York: Academic Press Inc.

17. Klemens, P., Thermal Resistance due to Point Defects at High Temperatures. Physical

Review, 1960. 119(2): p. 507-509.

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18. Ziman, J.M., The effect of free electrons in lattice conduction. The Philosophical Magazine,

1956. 1(2): p. 191-198.

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

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