Phonons

Total heat capacity of a crystal

$$C_𝑉 =𝐶_𝑉 ^\text{phonons} + 𝐶_𝑉^\text{electrons} +C_V^\text{magnetic}$$

$C_𝑉 ^\text{phonons}$ applies for all solids,

$𝐶_𝑉^\text{electrons}$ applies only for metals,

$C_V^\text{magnetic}$ applies only for magnets.

Heat capacity

$C_𝑉 =\left(\frac{𝜕\varepsilon} {𝜕𝑇} \right)_ 𝑉$ for a non-magnetic insulator, where $C_V$ is only due to phonons.

Dulong and Petit law

Classically, for a solid with $𝑁_\text{atoms}$ atoms in 3D, the equipartition theorem leads to $C_𝑉 =3𝑁_\text{atoms}𝑘_𝐵$.

This law holds for some solids at room temperature, but fails at lower temperatures where QM is necessary.

Single QHO energy eigenvalues and equilibrium occupancy

Energy eigenvalues are given by $\varepsilon_n =\left(𝑛+\frac{1} 2\right )ℏ𝜔$.

Equilibrium occupancy is given by the Bose-Einstein distribution: $\langle n\rangle=\frac{1}{e^{\hslash\omega/k_{B}T}-1}$ (the mean number of excitations present in thermal equilibrium at temperature $T$).

These can be used to find the average energy via $\bar \varepsilon =\left(\langle 𝑛\rangle +\frac{1} 2\right )ℏ𝜔$ and then the $C_V$ via $C_𝑉 =\left(\frac{𝜕\bar \varepsilon} {𝜕𝑇} \right)_ 𝑉$.

Equipartition theorem

Total energy of a lattice

The total energy of a lattice is calculated by summing over all modes $\vec k$, but as the spacing between the states $Δ𝑘 = 2𝜋/𝐿$ is very small compared to the width of the BZ, can write the sum as an integral:

$$\varepsilon=\int^\infty_0g(\omega)\hslash\omega\left(n(\omega)+\frac{1}2\right)\,\mathrm{d}\omega$$

Density of states

$g(\omega)$ is the density of states, where $g(\omega)\mathrm{d}\omega$ is the number of states between $\omega$ and $\omega+\mathrm{d}\omega$.

Density of states per unit wavenumber (2D derivation)

In 2D, the interval between wavenumbers is $\Delta k_x=\Delta k_y=2\pi/L$, so each allowed state occupies an area of $(2\pi/L)²$.

The annulus of area $2\pi k\mathrm{d}k$ contains a number of $g(k)\mathrm{d}k=2\pi k\mathrm{d}k\frac{L^2}{(2\pi)^2}$.

The density of states per unit wavevector in 2D can then be written as $g(k)=\frac{L^2}{2\pi}k$.

Density of states per unit wavenumber (3D derivation)

In 3D, the interval between wavenumbers is $\Delta k_{x}=\Delta k_{y}=\Delta k_z=2\pi/L$ , so each allowed state occupies an area of $(2\pi/L)³$.

The number of states in a shell of volume $4\pi k^2\mathrm{d}k$ is $g(k)\mathrm{d}k=4\pi k^2\mathrm{d}k\frac{V}{(2\pi)^3}=\frac{V}{2\pi^2}k^2\mathrm{d}k$

giving $g(k)$ the density of states per unit wavevector.

Assumptions of Debye theory

• The crystal is harmonic (so the phonon modes are independent)

• Elastic waves are non-dispersive ($𝜔=𝑣_𝑠𝑘$)

• The crystal is isotropic (so that frequency depends only on $|\vec k|$, and not the direction of $\vec k$)

• There is a high-frequency cut off $𝜔_𝐷$ chosen in such a way that the total number of modes is correct, i.e., that per branch $∫^{ 𝜔_𝐷 }_0 𝑔(𝜔)\,𝑑𝜔=𝑁_\text{atoms}$, the total number of modes should be the number of atoms in the crystal.

• Assumes all the modes lie in the acoustic branches, and so the optical branches are ignored.

How is the Debye frequency calculated?

Using $∫^{ 𝜔_𝐷 }_0 𝑔(𝜔)\,𝑑𝜔=𝑁_\text{atoms}$ per branch, and using $\omega=v_sk$ to find $dk/d\omega$.

Total number of modes is $3N_\text{atoms}$ for 3D.

Dispersion relation for Einstein’s model

where $C_V\propto e^{-\Theta_E/T}$ at low $T$, which is not accurate to reality.

Dispersion relation for Debye’s model

where $C_V\propto (T/\Theta_D)^3$ at low $T$, which is much better than Einstein’s model.

When can the harmonic approximation not be used for phonons?

The harmonic approximation works well for small deviations from the equilibrium positions.

However, for large displacements (high 𝑇) it no longer works well, and anharmonic terms in $U( r)$ must be included.

• Terms beyond second order are anharmonic.

Consequences of including anharmonic terms

1. Thermal expansion: The average interatomic distance 〈𝑟〉 > 𝑎 (it costs less energy to expand than to contract).

2. Phonons interact with each other, and they are no longer pure normal modes: In an anharmonic crystal, a phonon causes momentary local contraction/expansion of the lattice. This causes a second phonon to ‘see’ a different spring constant and the modes are therefore not independent; phonons scatter off each other and this determines the thermal conductivity of the solid.

   • i.e., there are collisions between phonons.

Phonon scattering events

Normal scattering event

In a Normal scattering event, the resulting wavevector $\vec k_3$ lies within the 1BZ.
In an N event, two phonons travelling to the right combine to produce a phonon travelling to the right.

Umklapp scattering event

In an Umklapp event, the resulting wavevector $\vec k_3$ lies outside the 1BZ, but can be translated into the 1BZ by a reciprocal lattice vector $\vec G$.
In a U event, two phonons travelling to the right combine to produce a phonon travelling to the left.
In the event, the total crystal momentum of the phonons changes by $\hslash$ times a non-zero reciprocal lattice vector.

How is heat flow affected by different scattering events?

Heat flow carried by phonons is unaffected by N events, but is impeded by Umklapp scattering. Hence U events contribute to thermal resistivity (i.e., 1/thermal conductivity).

Conserved quantities in phonon-phonon collisions

Crystal momentum: $\hslash\vec k_1+\hslash\vec k_2=\hslash\vec k_3$

Energy: $\hslash\omega_1+\hslash\omega_2=\hslash\omega_3$

Thermal conductivity

Thermal conductivity 𝜅 relates the steady-state flow of heat across a solid to the temperature gradient, i.e.,$j_𝑢 = −𝜅\frac{𝑑𝑇}{ 𝑑𝑥}$

• where thermal current density has units [Wm$^{−2}$], thermal conductivity has units [Wm$^{−1}$K$^{−1}$], and temperature gradient has units [Km$^{−1}$].

From kinetic theory of gases, $\kappa=\frac{1} 3 \bar 𝑣\,𝑙\, \tilde𝐶_𝑉$ where $\bar 𝑣$ is the average particle velocity, 𝑙 is the mean free path, and $\tilde 𝐶_𝑉$ is the specific heat per unit volume.

How is mean free path related to phonons?

The mean free path 𝑙 is inversely proportional to the number of phonons.

High temperature limit: value of $\tilde C_V$

High T ($𝑇 ≫ Θ_𝐷$): all phonon modes up to $ℏ𝜔_𝐷$ are excited; there are many phonons with large enough $| 𝑘|$ to produce U-events.

• The number of phonons is proportional to the thermal energy ($𝑘_𝐵𝑇$), hence $𝑙 ∝ 𝑇^{−1}$.

• The heat capacity $\tilde C_V$ is approximately constant (classical value).

Intermediate temperature limit: value of $\tilde C_V$

Intermediate T ($𝑇\backsim Θ_𝐷$): U-processes start to freeze out, as average energy of phonons decreases.

• The number of phonons $\backsim\exp(−Θ_𝐷/𝑇 )$, hence $𝑙 ∝ \exp(Θ_𝐷/𝑇)$.

• The heat capacity $\tilde C_V$ is approximately constant (classical value).

Low temperature limit: value of $\tilde C_V$

Low T ($𝑇 ≪ Θ_𝐷$): all U-processes stop.

• 𝑙 is constant, and is limited by the size and shape of the crystal.

• The temperature dependence comes entirely from $\tilde 𝐶_𝑉\backsim(𝑇/Θ_𝐷)^3$.

Thermal conductivity with temperature graph shape