Lattice vibrations

One-dimensional monoatomic lattice:

Equation of motion (nearest neighbours interaction only) according to Hooke’s law:

𝑑2 𝑥𝑛
M 𝑑𝑡 2 = C[(𝑥𝑛+1 – 𝑥𝑛 ) – (𝑥𝑛 – 𝑥𝑛−1 )]

M = atomic mass, C = force constant

Consider a travelling wave solution of the form, 𝑥𝑛 = A𝑒 −𝑖(𝜔𝑡 –𝐾𝑛𝑎)

where na is the equilibrium position of the n-th atom w.r.t. origin
𝑖𝐾𝑎 𝑖𝐾𝑎
𝐾𝑎 2
(-i𝜔)2 𝑥𝑛 = C𝑥𝑛 [𝑒 𝑖𝐾𝑎 + 𝑒 −𝑖𝐾𝑎 – 2] = C𝑥𝑛 (𝑒 2 – 𝑒− 2 )2 = C𝑥𝑛 (2i sin 2
)

4𝐶 𝐾𝑎
⟹ The dispersion relation is 𝜔= sin
𝑀 2

Note: we change K = K + 2m𝜋/a, m = ±1, ±2, ±3, … . . , the atomic displacements and frequency
ω do not change ⟹ these solutions are physically identical

1
Lattice vibrations

o Most of the wave described by the wavevector K are travelling waves (waves propagate
through the lattice)
o At the zone boundary, the wave becomes a standing wave (wave moves neither to the left
nor to the right)
o At K = ± 𝜋/𝑎, alternate atoms oscillate in opposite phases, so the wave does not
propagate

2
Lattice vibrations

At the boundaries of the Brillouin zone, 4𝐶

⟹ For 𝜔 < ,
𝑀
𝜋
K = ± 𝑎 ⇒ 𝑥𝑛 = A𝑒 −𝑖𝜔𝑡 (−1)𝑛
[Standing wave solution] o The lattice behaves as as a low-
pass filter (dispersive medium)
𝑣𝑔 = 0 at the boundaries of the Brillouin zone (K = ± 𝜋/𝑎)
o A number of wavelengths
⟹ no energy transfer – standing wave propagate at the same frequency

𝜔 2 𝐶 𝐾𝑎
⟹ 𝑣𝑝 = 𝐾 = 𝐾 𝑀
sin 2
𝑑𝜔 𝐶 𝐾𝑎
⟹ 𝑣𝑔 = 𝑑𝐾 = a 𝑀
cos 2

Only wavelengths longer than 2a are needed to represent the motion

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Lattice vibrations

𝜋 𝜋
⟹– ≤K≤𝑎 [K within the First Brillouin Zone]
𝑎

4𝐶
The maximum frequency is 𝜔𝑚𝑎𝑥 = 𝑀
2𝜋 2𝜋
⟹ Long wavelength limit: 𝜆 >> a, K = << ⟹ Ka << 1
𝜆 𝑎

4𝐶 𝐾𝑎 𝐶
𝜔= 𝑀
sin 2
≈ 𝑀
Ka - linear dispersion

𝐶
𝑣𝑝 = 𝑣𝑔 = 𝑀
a - sound velocity for the one-dimensional lattice

The dispersion effects are negligible at low frequencies (K → 0) and lattice behaves as a
continuum.

4
Elementary excitations in solids

5
Lattice vibrations
One-dimensional diatomic lattice:

(2n-2) (2n-1) (2n) (2n+1) (2n+2)

C C C C
M m M m M a)

2a

b)

x2n-2 x2n-1 x2n x2n+1 x2n+2

Two different types of atoms of masses M and m (M > m) are connected by

identical springs of spring constant C

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Lattice vibrations
One-dimensional diatomic lattice:

Equation of motion (nearest neighbours interaction only):

𝑑2 𝑥2𝑛
M 𝑑𝑡 2 = C[(𝑥2𝑛+1 + 𝑥2𝑛−1 –2𝑥2𝑛 ] (1)

𝑑2 𝑥2𝑛+1
m 𝑑𝑡 2 = C[(𝑥2𝑛+2 + 𝑥2𝑛 –2𝑥2𝑛+1 ] (2)

One can consider the following travelling wave solutions

𝑥2𝑛 = A𝑒 −𝑖(𝜔𝑡 –2𝑛𝐾𝑎) (3)

𝑥2𝑛+1 = B𝑒 −𝑖[𝜔𝑡 – 2𝑛+1 𝐾𝑎] (4)

Substituting (3) and (4) in (1) and (2)

(M𝜔2 – 2C) A + 2BC cos Ka = 0 (5)

(m𝜔2 – 2C) B + 2AC cos Ka = 0 (6)
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Lattice vibrations
To have non-vanishing solutions for A and B

(M𝜔2 – 2C) 2C cos Ka

=0
2C cos Ka (m𝜔2 – 2C)

(M𝜔2 – 2C) (m𝜔2 – 2C) – 4C2 cos2 Ka = 0

Mm𝜔4 – 2C𝜔2(M + m) + 4C2 (1 – cos2 Ka) = 0

2 1 1 1 1 2 4 𝑠𝑖𝑛2 𝐾𝑎
⟹ 𝜔 =C + ±C + − (Dispersion relation)
𝑀 𝑚 𝑀 𝑚 𝑀𝑚

For Acoustic branch at K = 0, 𝜔 0 = 0, A = B

⟹ the two atoms in the cell have the same amplitude and the phase dispersion is linear for
small K

1 1
For optical branch at K = 0, 𝜔 0 = 2C + , MA + mB = 0
𝑀 𝑚
⟹ the centre of mass of the atoms remains fixed. The two atoms move out of phase.

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Lattice vibrations
𝜔

M→∞
Optical branch

Forbidden gap
𝐾
– 𝜋/2𝑎 + 𝜋/2𝑎
Acoustical branch

o As heavier mass increases, optical

branch flattens and acoustical
branch turns downwards
o For M → ∞, atoms behave as if they
are completely isolated and vibrate
independent to one another

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Lattice vibrations
Transverse optical and acoustical modes (K → 0):

o The acoustic branch has this name because it gives rise to long wavelength vibrations –
speed of sound.
o The optical branch is a higher energy vibrations (since the frequency is higher, you need a
certain amount of energy to excite this mode)
o The term optical comes from how these were discovered – note that if atom 1 is +ve and
atom 2 is –ve, the charges are moving in opposite directions.

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Lattice vibrations

PHONONS PHOTONS
• Quanta of lattice vibrations • Quanta of electromagnetic
• Energies of phonons are radiation
quantized • Energies of photons are
quantized as well

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Phonons

o The regular lattice of atoms are tied together with bonds, so they can’t vibrate
independently. The vibrations take the form of collective modes which propagate
through the material.

o There should be energy associated with the vibrations of these atoms, which is
quantized, the quantum of the vibration energy is a “phonon”. A phonon is an
excited state in the quantum mechanical quantization of the modes of vibrations
for elastic structures of interacting particles.

o The vibrational energies of molecules are quantized and treated as quantum

1
harmonic oscillators with 𝜖 = (𝑛 + )ℏ𝜔, when the mode is excited to quantum
2
number n.

o Such propagating lattice vibrations can be considered to be sound waves, and

their propagation speed is the speed of sound in the material

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Phonon momentum

13
Phonon momentum

14
Specific Heat in solids

Dulong-Petit law fails for certain light elements such as B, Be and C, for which
CV is 3.34, 3.85, and 1.46 kcal/kmol⋅K, respectively at 20°C

15
Specific Heat in Solids

Specific heats of all solids drop sharply at low temperatures and approach 0
as T ➔ 0 K
16
Specific Heat in Solids (Einstein’s Quantum theory)

ℏω
𝜖 ҧ = ℏω 𝑓 𝜖 =
𝑒 ℏω/𝑘𝑇 − 1

3𝑁𝑜 ℏω
𝐸 = 3𝑁𝑜 𝜖 ҧ =
𝑒 ℏω/𝑘𝑇 − 1

𝑒 ℏω/𝑘𝑇
2
𝜕𝐸 ℏω
𝐶𝑉 = = 3𝑅 2
𝜕𝑇 𝑉
𝑘𝑇 𝑒 ℏω/𝑘𝑇 − 1

o At high temperatures, ℏω << kT, CV ≈ 3R, the Dulong-Petit value

ℏω 2 −ℏ𝑘𝑇ω
o At low temperatures, ℏω >> kT, CV ≈ 3R 𝑘𝑇 𝑒
o While the agreement is reasonably good at T ➔ 0 K, experimentally CV
∝ T3 at low temperatures
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Specific Heat in Solids (Einstein’s Quantum theory)

Molar heat capacity of diamond from Einstein’s original 1907 paper.

CV (cal/K-mol

Einstein’s theory

T/Θ𝐸

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Lattice Specific Heat in Solids (Debye’s Model)

2
𝜕𝐸 𝑇𝐸 𝑒 𝑇𝐸 /𝑇
𝐶𝑉 = = 3𝑅
𝜕𝑇 𝑉
𝑇 𝑒 𝑇𝐸 /𝑇 − 1 2

o Low-temperature heat capacity of Ag is underestimated by the Einstein model.

o Instead of considering each atom as an independent harmonic oscillator, Peter
Debye considered the sound waves in a material - the collective motion of atoms -
as independent harmonic oscillators.
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 Lattice Specific Heat in Solids (Debye’s Model)
o In the Debye approximation, the velocity of sound is taken as constant for each
polarization type, as it would be for a classical elastic continuum. The frequency of
the sound wave is related to k through the dispersion relation
𝜔 = vs∣k∣,
where vs is the sound velocity of a material.

o Instead of having 3N oscillators with the same frequency 𝜔𝑜 as in the Einstein

model, we now have 3N oscillators (the vibrational modes) with frequencies
depending on k through the dispersion relation ω(k)= 𝜔 = vs∣k∣.

o The heat capacity C is a macroscopic property: it should not depend on the

material's shape and only be proportional to its volume. Therefore, one can
consider a material with a simple shape to make the calculation of C easier.

o Periodic boundary conditions: Instead of having a 1D sample of length L with

fixed ends, we imagine that the two ends are connected together making the
sample into a ring. The periodic boundary condition means that, any wave in this
sample eikr is required to have the same value for a position r as it has for r + L. This
2𝜋𝑛
then restricts the possible values of k to be (n is an integer)
𝐿
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 Density of vibrational modes
Density of states in 1D case

Number of modes in the frequency range ω to ω + dω:

𝐿
D(𝜔)𝑑𝜔 = 2𝜋 dk

⇒ Density of states in the frequency range 𝜔 to 𝜔 + d𝜔

𝐿 1
D(𝜔) = 2𝜋 𝑑𝜔/𝑑𝑘

21
Density of vibrational modes

When we consider larger and larger box

sizes, L →∞, the volume per allowed mode
becomes smaller and smaller.

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Density of vibrational modes

Number of modes in the spherical shell between the radii k to k + dk:

3
𝐿 𝑉
4𝜋𝑘 2 𝑑𝑘 = 3 4𝜋𝑘 2 𝑑𝑘
2𝜋 2𝜋

⇒ Density of states in the frequency range 𝜔 to 𝜔 + d𝜔

𝑉𝑘 2 1
D(𝜔) = 2𝜋2 𝑑𝜔/𝑑𝑘

𝜕𝐸
⇒ Lattice specific heat of solids: 𝐶𝑉 = 𝜕𝑇 𝑉

⇒ The total energy of the phonons at a temperature T in a crystal may be written as

the sum of the energies over all phonon modes, here indexed by the wavevector k and
polarization index p
E =σ𝑘,𝑝 𝑛𝑘,𝑝 ℏ𝜔𝑝 (𝑘),

23
Lattice Specific Heat in Solids (Debye’s Model)

o Debye model assumes that the acoustic modes give the dominant contribution
to the heat capacity
o Within the Debye approximation the velocity of sound is taken a constant
independent of polarization (as in a classical elastic continuum)

𝑉𝑘 2 1 𝑉𝜔 2
D(𝜔) = 2𝜋2 𝑑𝜔/𝑑𝑘
=
2𝜋2 𝑣 3
o If there are N primitive cells in the specimen, the total number of phonon modes
is 3N. A cutoff frequency 𝜔𝐷 is determined by

Debye frequency

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Lattice Specific Heat in Solids (Debye’s Model)

kD
kD K ≤ kD

The factor 3 comes from the number of

possible polarizations in 3D (two transversal,
one longitudinal).

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Lattice Specific Heat in Solids (Debye’s Model)

𝑒 𝑥 ≈ 1, (𝑒 𝑥 −1) ≈ 𝑥
26
Lattice Specific Heat in Solids (Debye’s Model)

27
Lattice Specific Heat in Solids (Debye’s Model)

At low temperatures the phonon

heat capacity is proportional to T3.

o The Debye model assumes that atoms in materials move in a collective fashion,
described by quantized normal modes with a dispersion relation, 𝜔 = vs∣Ԧ𝐤∣.
o The total energy and heat capacity are obtained by integrating the contribution of
the individual modes over k-space.
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Density of modes

The density of states for Cu

D(𝜔)

Debye’s model

D(𝜔) Einstein’s model

29