Phonons, Density of States, and Lattice Specific Heat

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Hello! Welcome back to our study of lattice dynamics.

In our previous lesson, we made the pivotal transition from a classical to a quantum mechanical description of lattice vibrations. By quantizing the normal modes, we introduced the concept of the phonon—a bosonic quasiparticle representing a quantum of vibrational energy. We concluded with the powerful result that the lattice Hamiltonian can be expressed as a sum over non-interacting phonon modes:
$H = \sum_{q, j} ℏ ω_{j} (q) (a_{q, j}^{†} a_{q, j} + \frac{1}{2})$
This result provides the complete energy spectrum of the crystal's vibrational excitations.

Today, we will leverage this quantum framework and the tools of statistical mechanics to compute a macroscopic, measurable property: the lattice specific heat. Our learning outcome is to calculate the phonon density of states (DOS) and derive the general expression for lattice specific heat. This will lay the essential groundwork for comparing different models of specific heat, like the Einstein and Debye models, in our next lesson.

1. Thermal Energy of the Phonon Gas

At a finite temperature $T$ , the lattice is filled with a gas of thermally excited phonons. The total internal energy $U$ is the sum of the energies of all phonon modes, weighted by their average occupation number at that temperature.

Since phonons are bosons whose number is not conserved (they can be created and destroyed by thermal fluctuations), their chemical potential is zero. Therefore, the average number of phonons in a mode $(q, j)$ with frequency $ω_{j} (q)$ is given by the Bose-Einstein distribution:
$⟨ n_{q, j} ⟩ = \frac{1}{e ^{ℏ ω_{j} (q) / k_{B} T} - 1}$
Ignoring the zero-point energy (a constant offset that does not contribute to the heat capacity), the total internal energy of the lattice vibrations is:
$U = \sum_{q, j} ⟨ n_{q, j} ⟩ ℏ ω_{j} (q) = \sum_{q, j} \frac{ℏ ω _{j} ( q )}{e ^{ℏ ω_{j} (q) / k_{B} T} - 1}$
This sum runs over all $3 p_{N} N$ normal modes, where $p_{N}$ is the number of atoms in the basis and $N$ is the number of unit cells. For a macroscopic crystal, this sum over the discrete, but very closely spaced, $q$ vectors is computationally intractable.

2. The Phonon Density of States (DOS)

To make progress, we convert the sum over modes into an integral. The key tool for this is the phonon density of states, $g (ω)$ , defined as the number of normal modes per unit frequency interval around a frequency $ω$ . With this, the sum over modes can be replaced by an integral over frequency:
$\sum_{q, j} (\dots) \to \int_{0}^{\infty} d ω g (ω) (\dots)$
where $g (ω) d ω$ is the number of modes in the frequency range $[ω, ω + d ω]$ .

Using the DOS, our expression for the internal energy becomes:
$U = \int_{0}^{\infty} d ω g (ω) \frac{ℏ ω}{e ^{ℏ ω / k_{B} T} - 1}$
The heat capacity at constant volume, $C_{V}$ , is defined as $C_{V} = (\frac{\partial U}{\partial T})_{V}$ . Differentiating the integral expression for $U$ with respect to temperature gives the general expression for the lattice specific heat:
$C_{V} = \frac{\partial}{\partial T} \int_{0}^{\infty} d ω g (ω) \frac{ℏ ω}{e ^{ℏ ω / k_{B} T} - 1}$
$C_{V} = \int_{0}^{\infty} d ω g (ω) ℏ ω \frac{- 1}{( e ^{ℏ ω / k_{B} T} - 1 ) ^{2}} (e^{ℏ ω / k_{B} T}) (- \frac{ℏ ω}{k _{B} T ^{2}})$
$C_{V} = k_{B} \int_{0}^{\infty} d ω g (ω) (\frac{ℏ ω}{k _{B} T})^{2} \frac{e ^{ℏ ω / k_{B} T}}{( e ^{ℏ ω / k_{B} T} - 1 ) ^{2}}$
This is a central result. It shows that if we know the phonon density of states $g (ω)$ , we can calculate the lattice specific heat at any temperature. The entire physics of the specific lattice structure is encoded in $g (ω)$ .

The following document provides a formal introduction to the density of states and its role in calculating thermodynamic properties.

Introduction to the theory of lattice dynamics

The paper 'Introduction to the theory of lattice dynamics' by Dove provides a concise, formal definition of the density of states and shows how it's used to transform sums into integrals for calculating thermodynamic quantities.

Please read Section 4.2, 'Density of states' (page 135). Pay attention to how the sum for the total energy is formally rewritten as an integral over g(ω) (Equation 4.4). This section also begins the derivation for g(ω) in the low-frequency limit, which we will complete next.

3. Calculating the Density of States

The shape of $g (ω)$ is determined by the phonon dispersion curves $ω_{j} (q)$ . Regions where the dispersion curves are flat (i.e., the group velocity $\frac{d ω}{d q} \approx 0$ ) contribute a large number of states over a small frequency range, leading to sharp peaks in the DOS. These are known as van Hove singularities, analogous to those you've seen in electronic band structures.

Phonon Dispersion and Density of States — A generic example of phonon dispersion curves $\omega(\mathbf{q})$ (left) and the corresponding density of states $g(\omega)$ (right). Note how the flat bands in the dispersion, particularly at the Brillouin zone edge, result in sharp peaks in the DOS. The quadratic rise at low frequencies corresponds to the acoustic modes.

While calculating the full DOS for a real material requires a numerical approach, we can derive its form analytically in the important low-frequency (long-wavelength) limit.

In this limit ( $q \to 0$ ), only the acoustic modes are excited, and their dispersion is approximately linear: $ω = v_{s} q$ , where $v_{s}$ is the speed of sound. This is the continuum approximation, where the wavelength is much larger than the lattice spacing.

To find $g (ω)$ , we first find the density of states in $q$ -space. For a cubic crystal of side length $L$ with periodic boundary conditions, the allowed wavevectors form a grid with spacing $2 π / L$ . The number of allowed $q$ states within a spherical shell of radius $q$ and thickness $d q$ in the first Brillouin zone is:
$g (q) d q = \frac{Volume of shell in q-space}{Volume per state} = \frac{4 π q ^{2} d q}{( 2 π / L ) ^{3}} = \frac{V}{2 π ^{2}} q^{2} d q$
The density of states in frequency, $g (ω)$ , is related to $g (q)$ by the chain rule: $g (ω) d ω = g (q) d q$ , so $g (ω) = g (q) \frac{d q}{d ω}$ .
Using the linear dispersion $ω = v_{s} q$ , we have $d q / d ω = 1/ v_{s}$ . Substituting this and $q = ω / v_{s}$ into the expression for $g (q)$ , we get the DOS for a single polarization:
$g (ω) = (\frac{V}{2 π ^{2}} \frac{ω ^{2}}{v _{s}^{2}}) \frac{1}{v _{s}} = \frac{V ω ^{2}}{2 π ^{2} v _{s}^{3}}$
Finally, we account for the three acoustic polarizations (one longitudinal and two transverse). For simplicity, we can use an average speed of sound $v_{s}$ , giving a total density of states in the long-wavelength limit:
$g (ω) = \frac{3 V}{2 π ^{2} v _{s}^{3}} ω^{2}$
This $ω^{2}$ dependence is a universal feature of the phonon DOS for any 3D crystal at low frequencies.

4. The Low-Temperature Limit of Specific Heat

At low temperatures, only low-energy (low-frequency) phonons are thermally excited. This is precisely the regime where our $g (ω) \propto ω^{2}$ approximation is valid. Let's substitute this into our general expression for $C_{V}$ .

Let $g (ω) = A ω^{2}$ , where $A = \frac{3 V}{2 π ^{2} v _{s}^{3}}$ .
$C_{V} = k_{B} \int_{0}^{\infty} A ω^{2} (\frac{ℏ ω}{k _{B} T})^{2} \frac{e ^{ℏ ω / k_{B} T}}{( e ^{ℏ ω / k_{B} T} - 1 ) ^{2}} d ω$
We perform a change of variables to $x = ℏ ω / k_{B} T$ . This makes the integral dimensionless.
$ω = \frac{k _{B} T}{ℏ} x and d ω = \frac{k _{B} T}{ℏ} d x$
Substituting and collecting all the temperature-dependent terms outside the integral:
$C_{V} = k_{B} A \int_{0}^{\infty} (\frac{k _{B} T}{ℏ} x)^{2} (x^{2}) \frac{e ^{x}}{( e ^{x} - 1 ) ^{2}} (\frac{k _{B} T}{ℏ}) d x$
$C_{V} = k_{B} A (\frac{k _{B} T}{ℏ})^{3} \int_{0}^{\infty} \frac{x ^{4} e ^{x}}{( e ^{x} - 1 ) ^{2}} d x$
At low temperatures, the upper limit of integration for $ω$ can be extended to $\infty$ because the Bose-Einstein factor heavily suppresses the contribution from high-frequency phonons anyway. The integral is now just a number:
$\int_{0}^{\infty} \frac{x ^{4} e ^{x}}{( e ^{x} - 1 ) ^{2}} d x = \frac{4 π ^{4}}{15}$
Plugging this constant and the expression for $A$ back in, we get:
$C_{V} = k_{B} (\frac{3 V}{2 π ^{2} v _{s}^{3}}) (\frac{k _{B} T}{ℏ})^{3} (\frac{4 π ^{4}}{15}) = (\frac{2 π ^{2} k _{B}^{4} V}{5 ( ℏ v _{s} ) ^{3}}) T^{3}$
The crucial result is that the specific heat is proportional to the cube of the temperature:
$C_{V} \propto T^{3}$
This is the celebrated Debye T³ law, a cornerstone result in solid-state physics that correctly describes the specific heat of insulating crystals at low temperatures.

Test your understanding!

How would you expect the low-temperature specific heat law to change for a 2D material (like graphene) and a 1D material (like a carbon nanotube)? Hint: Rederive the low-frequency density of states $g (ω)$ for 2D and 1D.

Show answer

The key is the dimensionality of the shell in $q$ -space.

2D: The number of states in an annulus of radius $q$ and width $d q$ is $g (q) d q \propto q d q$ . Using $ω \propto q$ , we find $g (ω) \propto ω$ . Plugging this into the integral for internal energy $U \propto \int ω \cdot ω \cdot \frac{1}{e ^{ℏ ω / k T} - 1} d ω \propto T^{3}$ . Therefore, $C_{V} = \partial U / \partial T \propto T^{2}$ .
1D: The number of states in a line segment of width $d q$ is $g (q) d q \propto d q$ . Using $ω \propto q$ , we find $g (ω) \propto ω^{0} = constant$ . Plugging this into the integral for internal energy $U \propto \int 1 \cdot ω \cdot \frac{1}{e ^{ℏ ω / k T} - 1} d ω \propto T^{2}$ . Therefore, $C_{V} = \partial U / \partial T \propto T$ .

So, at low temperatures, we expect $C_{V} \propto T^{d}$ where $d$ is the dimensionality of the system.

Conclusion

In this lesson, we established the formal connection between the microscopic quantum picture of phonons and the macroscopic thermal properties of a solid.

Key Takeaways:

The phonon density of states (DOS), $g (ω)$ , is a crucial function that bridges the microscopic dispersion relations with macroscopic thermodynamic properties.
The total internal energy and lattice specific heat can be expressed as integrals over the frequency, involving the product of the DOS and the Bose-Einstein thermal factor.
For any 3D crystal at low frequencies, the DOS of acoustic phonons follows a universal law: $g (ω) \propto ω^{2}$ .
This quadratic dependence of the DOS directly leads to the experimentally verified Debye T³ law for lattice specific heat at low temperatures.

Preview of Next Lesson:
Our derivation of the T³ law relied on a low-frequency approximation. This works well at low temperatures but must fail at high temperatures, where all modes are excited. The Debye model provides a clever interpolation between the low and high-temperature regimes by imposing a cut-off on the $ω^{2}$ spectrum. In the next lesson, we will analyze the Debye model in detail, compare it to the simpler Einstein model, and see how it successfully explains the behavior of specific heat across all temperatures.