3.4 Phonon density of states

Phonon modes in solids

Phonons are quantized vibrations of atoms in a crystal lattice. They carry energy and momentum, and the collective patterns of atomic motion they represent determine many of a solid's physical properties, from how it conducts heat to how it interacts with light. The phonon density of states (DOS) tells you how many of these vibrational modes exist at each frequency, and it's the central quantity connecting lattice dynamics to thermodynamics.

This topic covers the types of phonon modes, how dispersion relations give rise to the DOS, models for approximating it, and how the DOS feeds into measurable properties like specific heat and thermal conductivity.

Acoustic vs. optical phonons

In an acoustic phonon mode, all atoms in the unit cell move roughly in phase with each other. At long wavelengths, this looks just like a sound wave propagating through the crystal. Acoustic branches start at zero frequency at the zone center (the $\Gamma$ point) and increase with $k$.

In an optical phonon mode, atoms within the unit cell move out of phase. If the unit cell contains atoms of different charge (like in NaCl), this out-of-phase motion creates an oscillating dipole, which is why these modes can couple to electromagnetic radiation. Optical branches have a non-zero frequency at $\Gamma$.

• Acoustic: neighbors move in the same direction; low frequency, long wavelength behavior
• Optical: neighbors move in opposite directions; higher frequency, can be excited by light
• A crystal with $N$ atoms per primitive unit cell has 3 acoustic branches and $3(N-1)$ optical branches

Transverse vs. longitudinal modes

Phonon modes are also classified by the direction atoms move relative to the wave's propagation direction:

• Longitudinal modes: atomic displacements are parallel to the wave vector $\mathbf{k}$
• Transverse modes: atomic displacements are perpendicular to $\mathbf{k}$

In a 3D crystal, each wave vector has one longitudinal and two transverse modes. The two transverse modes are often degenerate along high-symmetry directions. Longitudinal modes typically have higher frequencies than their transverse counterparts because the restoring forces for compression/extension tend to be stiffer than for shear.

Phonon dispersion relations

Dispersion relations map out the relationship between phonon frequency $\omega$ and wave vector $\mathbf{k}$. They encode all the information about allowed vibrational modes and their energies, and the phonon DOS is derived directly from them.

Phonon dispersion curves

A dispersion curve plots $\omega$ vs. $k$ along high-symmetry directions in the Brillouin zone. Several features to note:

• The slope of a dispersion curve gives the phonon group velocity: $v_g = \frac{d\omega}{dk}$
• Acoustic branches are linear near $\Gamma$ (the slope there equals the speed of sound)
• Optical branches are relatively flat, meaning optical phonons have low group velocities
• The total number of branches equals $3N$, where $N$ is the number of atoms in the primitive unit cell

Brillouin zones

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. It's constructed by drawing perpendicular bisectors of the reciprocal lattice vectors and taking the enclosed region. Every unique phonon wave vector lives inside this zone.

The zone boundaries correspond to Bragg planes, where phonon waves satisfy the Bragg condition and the dispersion curves typically flatten out. The shape of the Brillouin zone reflects the crystal symmetry: an FCC crystal has a truncated octahedron, while a BCC crystal has a rhombic dodecahedron.

High-symmetry points

Dispersion curves are conventionally plotted along paths connecting special points in the Brillouin zone that have high symmetry. Common labels include:

• $\Gamma$: the zone center ($\mathbf{k} = 0$)
• X, L, K, W: various zone-boundary and zone-edge points (exact positions depend on the crystal structure)

Phonon frequencies at these points can be measured directly by inelastic neutron scattering, Raman spectroscopy, or infrared spectroscopy. The behavior of the dispersion curves at and between these points determines the shape of the phonon DOS.

Phonon density of states

The phonon density of states $g(\omega)$ counts the number of phonon modes per unit frequency interval per unit volume. It's the bridge between microscopic lattice dynamics and macroscopic thermodynamic quantities.

Definition of phonon DOS

Formally:

$g(\omega) = \frac{1}{V} \sum_{\mathbf{k},s} \delta(\omega - \omega_s(\mathbf{k}))$

where the sum runs over all wave vectors $\mathbf{k}$ and branches $s$, and $V$ is the crystal volume.

The normalization condition is:

$\int_0^\infty g(\omega)\, d\omega = 3N$

where $N$ is the total number of atoms. This just says the total number of modes equals three times the number of atoms (one mode per degree of freedom).

You can decompose $g(\omega)$ into partial contributions from individual branches or polarizations, which is useful for identifying which types of phonons dominate at a given frequency.

Calculation of phonon DOS

Several methods exist for computing $g(\omega)$ from dispersion relations:

1. Histogram method: Divide the Brillouin zone into a fine grid, compute $\omega(\mathbf{k})$ at each grid point, and bin the frequencies into a histogram. Simple but requires a very fine grid for smooth results.
2. Tetrahedron method: Divide the Brillouin zone into tetrahedra and interpolate the dispersion linearly within each one. This gives a much smoother DOS with fewer grid points.
3. Gaussian smearing: Replace each delta function in the DOS definition with a Gaussian of small width. Computationally straightforward and widely used.
4. First-principles (DFT) calculations: Density functional theory can compute interatomic force constants from scratch, yielding accurate dispersion relations and DOS even for complex materials with no experimental input.

Debye model approximation

The Debye model assumes all phonon branches have a linear dispersion ($\omega = v_s k$) up to a maximum cutoff called the Debye frequency $\omega_D$. Under this assumption, the DOS takes a simple parabolic form:

$g(\omega) = \frac{9N}{\omega_D^3}\,\omega^2 \quad \text{for } \omega \leq \omega_D$

and $g(\omega) = 0$ for $\omega > \omega_D$.

The cutoff is chosen so that the total mode count is correct: $\int_0^{\omega_D} g(\omega)\,d\omega = 3N$.

This model works well at low temperatures, where only long-wavelength acoustic phonons (which really do have near-linear dispersion) are thermally excited. It fails at higher frequencies where the actual dispersion curves flatten and optical branches appear. Still, the Debye model gives the correct $T^3$ low-temperature specific heat and provides a useful single-parameter ($\Theta_D$) characterization of a material's vibrational spectrum.

Van Hove singularities

Wherever the dispersion curve is flat ($\nabla_\mathbf{k}\,\omega = 0$), many modes pile up at the same frequency, producing sharp features in the DOS called Van Hove singularities. These occur at critical points in the Brillouin zone, typically at zone boundaries or saddle points.

The character of the singularity depends on dimensionality:

• 1D: step discontinuities (the DOS jumps)
• 2D: logarithmic divergences
• 3D: kinks or cusps (the DOS itself stays finite, but its derivative is discontinuous)

Van Hove singularities can have real physical consequences. A spike in the DOS at a particular frequency enhances any process that depends on the number of available modes at that energy, including electron-phonon scattering and specific heat contributions near the corresponding temperature.

Experimental techniques

Phonon dispersion relations and the DOS can be probed experimentally through the interaction of phonons with neutrons, photons, or electrons. Each technique has different strengths in terms of the energy and momentum range it can access.

Inelastic neutron scattering

Inelastic neutron scattering (INS) is the most comprehensive technique for mapping phonon dispersions. Thermal neutrons have wavelengths on the order of interatomic spacings (~1 Å) and energies comparable to phonon energies (~meV), making them ideal probes.

• A monochromatic neutron beam hits the sample; scattered neutrons gain or lose energy by creating or absorbing phonons
• By measuring the energy and angle of scattered neutrons, you extract both $\omega$ and $\mathbf{k}$ of the phonon involved
• INS can map the full Brillouin zone and is sensitive to all phonon branches
• Particularly valuable for studying acoustic phonons, effects of disorder, and anharmonic broadening

The main drawback is that INS requires large single crystals and access to a neutron source (reactor or spallation facility).

Raman spectroscopy

Raman spectroscopy uses visible laser light to probe phonons near the $\Gamma$ point (since photon momenta are negligible compared to the Brillouin zone size).

• Incident photons scatter inelastically off the crystal, shifting in frequency by the phonon frequency
• Only Raman-active modes are visible: those that modulate the electronic polarizability of the material
• Provides phonon frequency and linewidth for zone-center optical modes
• Polarized Raman measurements can identify the symmetry of individual modes

Raman is a tabletop technique requiring only small samples, making it far more accessible than INS. However, it only probes $\mathbf{k} \approx 0$ modes.

Infrared spectroscopy

Infrared (IR) spectroscopy also probes zone-center phonons, but through absorption rather than scattering.

• IR-active modes are those that produce a net oscillating dipole moment in the unit cell
• Complementary to Raman: in centrosymmetric crystals, modes that are Raman-active are IR-inactive and vice versa (mutual exclusion rule)
• Measures phonon frequencies and oscillator strengths of optical modes
• Far-infrared (terahertz) spectroscopy extends the range to lower-energy modes

Applications of phonon DOS

The phonon DOS feeds directly into calculations of thermal, electrical, and superconducting properties. Engineering the DOS through nanostructuring, alloying, or strain is an active area of materials design.

Thermal properties of solids

The phonon DOS controls how a solid stores and transports thermal energy:

• Specific heat depends on how many modes are thermally populated at a given temperature
• Thermal conductivity depends on how effectively phonons carry energy and how far they travel before scattering

Materials with many low-frequency modes tend to have higher heat capacities at low temperatures but can also have lower thermal conductivities if those modes scatter strongly.

Specific heat capacity

The specific heat at constant volume, $C_v$, quantifies how much energy a material absorbs per degree of temperature increase. Using the phonon DOS, it's calculated as:

$C_v = k_B \int_0^\infty g(\omega) \left(\frac{\hbar\omega}{k_BT}\right)^2 \frac{e^{\hbar\omega/k_BT}}{(e^{\hbar\omega/k_BT}-1)^2}\, d\omega$

In the Debye model, this simplifies to:

$C_v = 9Nk_B \left(\frac{T}{\Theta_D}\right)^3 \int_0^{\Theta_D/T} \frac{x^4 e^x}{(e^x - 1)^2}\, dx$

where $\Theta_D = \hbar\omega_D / k_B$ is the Debye temperature.

Two important limits:

• Low $T$ ($T \ll \Theta_D$): $C_v \propto T^3$ (the Debye $T^3$ law)
• High $T$ ($T \gg \Theta_D$): $C_v \to 3Nk_B$ (the classical Dulong-Petit limit, about 25 J/mol·K per atom)

Deviations from the Debye model at intermediate temperatures reveal the actual shape of the phonon DOS.

Thermal conductivity

The kinetic theory expression for lattice thermal conductivity is:

$\kappa = \frac{1}{3} C_v\, v_s\, \Lambda$

where $v_s$ is an average phonon (sound) velocity and $\Lambda$ is the phonon mean free path.

The mean free path is limited by several scattering mechanisms:

• Phonon-phonon scattering (Umklapp processes dominate at high $T$)
• Defect and impurity scattering (important at intermediate $T$)
• Boundary scattering (dominates at very low $T$ or in nanostructures)

Reducing $\Lambda$ through nanostructuring or alloying lowers $\kappa$, which is the strategy behind high-performance thermoelectric materials. The thermoelectric figure of merit $ZT$ improves when $\kappa$ decreases while electrical conductivity is maintained.

Phonon-mediated superconductivity

In conventional (BCS) superconductors, phonons mediate an attractive interaction between electrons that leads to Cooper pair formation. The superconducting transition temperature $T_c$ depends on:

• The electron-phonon coupling constant $\lambda$
• A characteristic phonon frequency scale (often related to the Debye temperature)

The McMillan formula gives an approximate relationship:

$T_c \approx \frac{\Theta_D}{1.45} \exp\left[-\frac{1.04(1+\lambda)}{\lambda - \mu^*(1+0.62\lambda)}\right]$

where $\mu^*$ is the screened Coulomb pseudopotential. Materials with a high DOS at low frequencies and strong electron-phonon coupling tend to have higher $T_c$.

Phonon-electron interactions

Phonons and electrons interact in ways that shape a material's electrical conductivity, optical absorption, and even its electronic band structure. These interactions are quantified through the electron-phonon coupling and manifest in phenomena like polaron formation and indirect optical transitions.

Electron-phonon coupling

The electron-phonon coupling strength is characterized by the dimensionless parameter $\lambda$:

$\lambda = 2 \int_0^\infty \frac{\alpha^2 F(\omega)}{\omega}\, d\omega$

Here $\alpha^2 F(\omega)$ is the Eliashberg function, which combines the phonon DOS $F(\omega)$ with the electron-phonon coupling matrix elements $\alpha^2$. This function weights each phonon frequency by how strongly it couples to electrons near the Fermi surface.

Strong coupling ($\lambda > 1$) leads to significant effects: renormalization of the electronic effective mass, polaron formation, and enhanced superconductivity. The coupling strength can be computed from first principles using DFT-based methods.

Polarons

A polaron is a quasiparticle formed when an electron (or hole) distorts the surrounding lattice and becomes "dressed" by a cloud of virtual phonons. The electron drags this lattice distortion along as it moves.

Consequences of polaron formation:

• Increased effective mass (the electron is heavier because it carries the lattice distortion)
• Reduced mobility compared to bare electrons
• Modified optical absorption (polaron absorption bands appear)

Polaron effects are most pronounced in ionic and polar materials (like transition metal oxides and organic semiconductors) where the electron-lattice coupling is strong. In weakly coupled systems, the polaron correction is small and can be treated perturbatively.

Phonon-assisted electronic transitions

In indirect bandgap semiconductors like silicon and germanium, the conduction band minimum and valence band maximum occur at different $\mathbf{k}$-points. A photon alone can't bridge this momentum gap because photon momentum is negligible on the Brillouin zone scale.

A phonon can supply the missing momentum. The process involves simultaneous absorption (or emission) of a photon and a phonon:

• Photon provides the energy
• Phonon provides the momentum

This makes optical absorption and emission in indirect-gap materials much weaker than in direct-gap materials (like GaAs), which is why silicon is a poor light emitter despite being an excellent absorber at sufficient photon energies. The transition probability depends on the electron-phonon coupling strength and the phonon DOS at the relevant momentum transfer.

Anharmonic effects

Everything discussed so far assumes the harmonic approximation: atoms sit in perfectly parabolic potential wells, phonons don't interact with each other, and phonon lifetimes are infinite. Real crystals have anharmonic terms in the interatomic potential (cubic, quartic, etc.), and these become increasingly important at higher temperatures.

Phonon-phonon scattering

Anharmonicity allows phonons to scatter off each other, redistributing energy and momentum among modes. Two categories:

• Normal (N) processes: Total crystal momentum is conserved. These redistribute phonons among modes but don't directly create thermal resistance.
• Umklapp (U) processes: Total crystal momentum changes by a reciprocal lattice vector. These are the dominant source of intrinsic thermal resistance at high temperatures.

Scattering rates can be computed using third-order perturbation theory with anharmonic force constants. The temperature dependence of phonon lifetimes from these processes is directly observable as linewidth broadening in Raman and INS experiments.

Thermal expansion

In a perfectly harmonic potential, the average atomic position doesn't change with temperature, so there would be no thermal expansion. Thermal expansion arises because real potentials are asymmetric: the repulsive wall at short distances is steeper than the attractive tail at long distances.

The Grüneisen parameter $\gamma$ quantifies how phonon frequencies shift with volume:

$\gamma_i = -\frac{V}{\omega_i}\frac{\partial \omega_i}{\partial V}$

The overall thermal expansion coefficient $\alpha$ is related to a weighted average of mode-specific Grüneisen parameters through:

$\alpha = \frac{\gamma C_v}{B V}$

where $B$ is the bulk modulus. Most materials have positive $\gamma$ and expand on heating, but some (like ZrW$_2$O$_8$) exhibit negative thermal expansion due to specific low-frequency modes with negative Grüneisen parameters.

Phonon lifetime and linewidth

A phonon in a perfectly harmonic crystal would oscillate forever. Anharmonicity gives phonons a finite lifetime $\tau$, related to the spectral linewidth $\Gamma$ (FWHM) by:

$\Gamma = \frac{1}{\tau}$

Shorter lifetimes mean broader spectral lines. The lifetime is set by the rate of phonon-phonon scattering (and, at lower temperatures, by defect scattering).

Why this matters:

• The phonon mean free path is $\Lambda = v_g \tau$, which directly enters the thermal conductivity
• Linewidth measurements from Raman or INS experiments provide a direct probe of anharmonicity
• In electron-phonon coupling calculations, finite phonon lifetimes affect the spectral function and can influence superconducting properties

At high temperatures, phonon lifetimes decrease (linewidths increase) roughly as $1/T$ due to the growing population of phonons available for scattering.