2.

Quantum Theory of Solids

2A. Free Electron Model
2B. Periodic Potential and Band Structure
2C. Lattice Vibration

2C. Lattice Vibration (Bube Ch. 3)

2C.1. 1D Monoatomic Lattice
2C.2. 1D Diatomic Lattice
2C.3. Band Structure and DOS of Phonon
2C.4. Heat Capacity
− Lattice vibration or wave describes the vibrational motion of
atoms in a crystalline solid in terms of a wave passing through
the atoms of the crystal as they are displaced by their thermal
energy from their equilibrium positions.
− The thermal properties of solids and the electronic transport
are strongly related to these lattice waves. SiC at 600 K
− The behavior of lattice waves and the derivation of the suitable
wave equation can be based on the same classical mechanics
approach we have used for waves in a string.
− Many of the major characteristics of the lattice waves can be
derived from the consideration of a one dimensional crystal
lattice, which can be thought as a kind of discontinuous string.
-Thu/9/17/20
1
2C.1. 1D Monoatomic Lattice
We first consider vibrations associated with a one-dimensional crystal in which all the atoms have the same
mass (M) and the same atomic spacing of a. The spring constant, which reflects interatomic interaction, is
also a constant K. The displacement of nth atom from the equilibrium point is denoted as un(t).

Longitudinal wave

Transverse wave

Let’s consider the longitudinal vibration. The force on the nth atom is affected by the stretch
or compression of the two springs attached to it.

This is the wave equation for the discrete atoms. When a is very small, the left-hand side becomes
the second derivative of the displacement and the above equation becomes the wave equation for the
continuum.

2
 k
: dispersion relation for
acoustic wave λ=∞ λ = 2a
λ = 2a

The largest wavelength for

This wave is called the acoustic mode and an infinite string.
neighboring atoms in acoustic waves move in
the same direction. When sound waves The shortest
travel in solid, they involve this type of wavelength.
lattice oscillation near k = 0 (long
wavelength limit). Wavelengths of the sound All the atoms displaced by
wave are on the order of m. the same amount in the same
direction.

3
Since the lattice spacing is a, the shortest possible wavelength is 2a, and any wavelength shorter
than this is unphysical in this system. This means that |k| bigger than π/a practically corresponds
to k within the 1st BZ (see below). This is also true for 2D and 3D systems.

k = 12p/5a or λ = 5a/6

k = 2p/5a or λ = 5a

Kittel Chap. 4

The velocity at which traveling waves carry energy is

_
the group velocity:
1/2
dw æ K ö æ1 ö
vg = = acos ç ka÷ = sound velocity
dk çè M ÷ø è2 ø

The vgmax is obtained at the long wavelength limit, which |k|

corresponds to the sound wave. In the continuum
description,
1/2
æYö ψ(r) = eik·r u(r)
vg » ç ÷ Y = elastic modulus, ρ = density
è rø 4
Bloch Wave
~300 m/s in air

5
The transverse mode can be described in similar approach. The only difference is that atoms
displace vertically, so the restoring force is weaker than for longitudinal modes. In terms of spring
constant, KL > KT (L: longitudinal mode, T: transverse mode).

ω
1

0.9

0.8 L
0.7

0.6

0.5

0.4

0.3 T
0.2

0.1

0
Bube Fig. 3-3
│-3 -2 -1 0 1 2 │
3
k
-p/a p/a
Coupled oscillator → independent normal modes with certain oscillation frequency

6
Phonon
We have seen in the simple harmonic oscillators that the energy of an oscillator with the angular
frequency of ω is quantized as (n + 1/2)ħω (n = 0, 1, 2, ..). The same is true for the vibration in
solids: energy of each normal mode is quantized as (n + 1/2)ħω, meaning that the energy-exchange
with lattice waves occurs in integer multiples of ħω.

The quantized vibration is called phonon, similar to photon. And, the energy of phonon is
Ephonon = ħω = hν. The phonon momentum is ħk, similar to photon. However, the momentum
of the phonon is sometimes called a phonon crystal momentum because the lattice wave itself does
not have a real physical momentum.

Phonon behaves as if it had a momentum ħk in its interactions (with electrons or photons) inside the
crystal, and is involved in the momentum conversation law.

7
2C.2. 1D Diatomic Lattice Primitive Unit Cell

Next, we consider when the unit cell of the 1D lattice contains two basis atoms. The two atoms
could be of different kind or the same species with different left and right spring constants. The
two cases give qualitatively the same results so we will assume the latter case because it is slightly
simpler. a
G K G

un−1 vn−1 un vn un+1 vn+1

un = Aei( kna-w t )
d un 2
M = - K(un - vn ) - G(un - vn-1 ) vn = Bei( kna-w t )
dt2 2 [ Mw 2 - (K + G)]A + (K + Ge-ika )B = 0
d v (K + Geika ) A + [ Mw 2 - (K + G)]B = 0
M 2n = - K(vn - un ) - G(vn - un+1 ) Normal mode
dt
(Eigenvalue
æ Mw 2 - (K + G) öæ problem of
K + Ge- ika A ö =æ 0 ö dynamical matrix)
ç ÷ ç ÷ ç ÷
çè K + Geika Mw 2 - (K + G) ÷ø è B ø è 0 ø

For non-trivial solution, the determinant should be zero.

2
[ Mw 2 - (K + G)]2 = K + Ge-ika = K 2 + G 2 + 2KGcos ka

8
 K +G 1 B K + Geika
w = 2
± K + G + 2KG cos ka,
2 2
=∓
M M A K + Geika

i) k ~ 0 (long wavelength) cos ka ~ 1−(ka)2/2

2(K + G) B Optical phonon

w= - O(ka)2 , = -1
M A

w=
KG
ka ,
B
=1
Acoustic phonon
2 M (K + G) A

ii) k = π/a (K > G)

2K B
w= , = -1 Optical phonons, especially in ionic
M A
solids, is critical in electron scattering.
This is called optical phonon because it
is activated by the electric field as the
2G B ions with different charges move in the
w= , =1
M A opposite direction under the electric field.
9
at k = p/4a

Optical T

n ≈ 1013/s
Optical L

Acoustic T

Acoustic L

│ │
-p/a p/a

10
at k = p/2a

n ≈ 1013/s

│ │
-p/a p/a

11
2C.3. Band Structure and DOS of Phonon
In real materials in 3D with p number of basis atoms, for each normal mode identified by k in
the 1st BZ, the 3p×3p dynamical matrix (like K and G) is constructed. This results in 3p normal
modes, among which three are acoustic branches. Collecting the (ω, k) gives the phonon band
structure.
Dispersion curves or phonon band structure of fcc Pb. In 3D, there are two transverse
Since there is only one basis atom, only acoustic modes for each propagating
modes appear. direction of k, they are indexed
as TA1 and TA2.
THz

TA1 and TA2 are degenerate.

The phonon band structure can be measured to high precision using neutron scattering. 12
 Phonon band of GaAs Phonon Energy
≈ 9 THz or
≈ 0.037 eV
Neutron scattering (♦) vs. Theory (⎯)

Optical

Acoustic

0
Phonon spectrum of GaAs

Courtesy by 이규현
Note that these modes are all independent.
x

13
Transverse acoustic (TA) Longitudinal acoustic (LA) Transverse optical (TO) Longitudinal optical (LO)
 2B.4
Phonon Density of States (DOS) ppt 2B-33
Like in electronic band structure, k is discrete, rather than continuous, for finite crystals, which
can be neatly handled by Born-von Karman boundary condition. The mathematical procedure is
exactly the same, which shows that k is discrete with a small spacing and the number of k points in
each band is exactly the same as the number of unit cell in the crystal. Like electronic DOS, this
results in the phonon DOS. The total number of normal modes is 3pN = 3×(basis atom)×(number of
unit cell) = 3×(total number of atoms in solid)!
Phonon DOS Dph(ω)

ωmax

~ω2

Diamond: phonon band

_ _ _ _ _ _ _ _ _ _
p p p 2mp p N N
- <k £ ®- < £ «- <m£
structure measured with 2B.4
neutron diffraction
ppt 2B-33 a a a Na a 2 14 2
 ppt 2A-8
Density of States (DOS) in Metal and Semiconductor ppt 2A-9

quantized k space

Phonon Density of States (DOS) schematically

15
2C.4. Heat Capacity of Lattice Vibrations (skip---)

Mean thermal energy of normal modes

i) Semiclassical approach
At finite T, normal modes vibrate with certain amplitude and carry thermal energies. For a certain
normal mode with the angular frequency of ω, the mean energy at T can be calculated using
Boltzmann factor. The probability Pn for the oscillator in the nth quantum state is proportional to
the Boltzmann factor: Pn ∝ exp(−En/kT) where En = nħω. (Here the zero-point energy is
neglected.) From normalization condition,

¥ ¥
x
å nx = å nx
n n
=
(1- x)2
n=0 n=1

16
 ii) Quantum approach

Another way is to consider a normal mode as the bosonic system and each energy quanta as one
phonon particle. That is to say, a certain normal mode is a state that can be occupied by phonons.
According to the Bose-Einstein statistics, the number of bosons occupying a state with the energy
of E is
1
f (E) = ( E- m )/kT
e -1

The chemical potential is zero for phonons or photons because the total number of particles are
not conserved and so cannot be a constraint for f(E). (Note that for electrons, μ or EF was
determined by the total number of electrons.) Therefore, the occupation number for the normal
mode with the energy of ħω is

The mean energy of the normal mode is then

which is the same as the previous result.

For acoustic modes near the zone center, the average energy is kT which is equal to the classical
equipartition theorem. 17
 100
300 K
90 600 K
900 K

80

70
Occupation number

60

50

40

30

20

10

0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Phonon energy (eV)

18
Total energy of lattice vibration (phonons)

The total energy of phonons is given as follows:

In Debye approximation, phonon DOS is simplified by assuming a linear dispersion (ω = υ|k|). This
results in Dph(ω) ∝ ω2. (υ is the effective sound velocity.) ωmax for the Debye model is determined
by the fact that the total number of modes up to ωmax should be 3×(total number of atoms in solid). It
is called the Debye frequency. The Debye temperature (TD) is given by TD = ħωmax/kB. For detailed
expression of the model, please refer to Kittel or Kasap.
Dph(ω)

Density of states for phonons in copper. The solid

curve is deduced from experiments on
neutron scattering. The broken curve is the three-
dimensional Debye approximation, scaled such that
the areas under the two curves are the same.
This requires that ωmax ~ 4.5×1013 rad s-1, or a
Debye characteristic temperature TD = 344 K.

19
 20
Phonon energy
×

001
K 003
K 006 09
K 009
08
Occupation number
07
Occupation number
06
05
04
03
02
01
0
40.0 530.0 30.0 520.0 20.0 510.0 10.0 500.0 0

×
)Ve( ygrene nonohP
Phonon DOS Dph(ω)

Phonon DOS
ω
The υ in the Debye model is the effective sound velocity. It is given by the average of
three acoustic modes as follows:

1 1æ 1 2ö
= ç 3 + 3÷
u 3 è u L uT ø
3

where υL and υT are sound velocities of longitudinal and transverse modes that are
experimentally measured.

More correct Debye approximation

From Kittel

(NA: total number of atoms)

and υ is related to the hardness (Young’s modulus). Therefore,

TD reflects the hardness.
21
 By putting the analytic expression of Dph(ω) from the Debye model into the total energy formula
and differentiating with respect to temperature gives the following formula for the Debye molar heat
capacity.
3 C » 3R = 24.9 (at high T > TD)
L
æT ö TD /T x 4e x dx
CL (T ) » 9R ç ÷
è TD ø
ò
0 (e x -1)2 æT ö
3

CL µ ç ÷ (at low T)
è TD ø

CL = 3R

Debye constant-volume molar heat capacity

curve. The dependence of the molar heat
capacity Cm on temperature with respect to
CL/3R the Debye temperature: CL vs. T/TD. For Si,
TD = 625 K so that at room temperature (300
K), T/TD = 0.48 and CL is only 0.81(3R).

22
 (skip)

Cm: molar heat capacity

cs: specific heat capacity

23