This was a post that has taken me 2 years to put together, which is kind of ridiculous given the original Spectral Energy Density (SED) paper was published in 2010. The thing is I never really could get this to be compact enough while still feeling it was informative to be useful for others to use. I've decided that I'm just going to post it as is, which is pretty complete. I will note that I did have Gemini look at it and refine some language and check things.
When you think about calculating thermal properties of crystalline materials the first thing you might encounter are the phonon dispersion and density of state (DOS) plots. But you will immediately ask your self, how do these connect to actual heat transport? The short answer is ... they don't directly. Dispersion and DOS tell you all about the allowed channels for lattice/phonon vibrations. It is telling you what the coherent, non-dissipative modes are but the actual thermal conductivity is governed by how those modes scatter via phonon-phonon, defects, and boundaries. Furthermore, standard phonon bandstructures are a ground-state properties so they assume 0 K and harmonic/quasi-harmonic1 forces. To get temperature-dependent, anharmonic phonon frequencies and lifetimes, how long a mode last, you need to use finite-temperature lattice simulations. One practical route to get this information through the simulation results is the spectral energy density (SED) method, which was developed(?)/introduced and refined by McGaughey and collaborators [1–2].
May aim with this post isto try to put together a guide on what the SED is, the main equations, what every term means, and how you get them from practical atomistic simulations. I'll stick to the formulation that is formally derived from lattice dynamics (the $\Phi$ method); an alternative $\Phi'$ was shown to misrepresent the total spectral energy and fail for lifetimes and thermal conductivity [3].
tl;dr the reciepe
Get harmonic phonon eigenvectors and $\omega_\nu(\mathbf{q})$ from the dynamical matrix; run MD and Fourier-transform mass-weighted velocities in space and time to get $\Phi(\mathbf{q},\omega)$; plot it as a heatmap for a bandstructure and fit peaks to Lorentzians to get lifetimes $\tau_\nu = 2/\Gamma_\nu$; then sum mode heat capacity $\times$ group velocity$^2$ $\times$ $\tau$ over all modes for thermal conductivity. Symbols are defined in the Notation legend at the end.
Spectral Energy Density
The SED is the distribution of kinetic energy over a wavevector $\mathbf{q}$ and the frequency (angular) $\omega$ in a crystal. To remind you the wavevector represents the reciprocal space directions in a crystal and is akin to the wave momentum. The peaks in $\Phi(\mathbf{q},\omega)$ correspond to phonon modes and their positions give the dispersion $\omega_\nu(\mathbf{q})$. The widths of the peaks correspond to the inverse lifetime (broadening) of the phonon mode at $\mathbf{q}$.
Unit Cell Phonon Eigenvectors
You need the harmonic phonon modes of the primitive (or conventional) unit cell at a set of wavevectors $\mathbf{q}$. That comes from lattice dynamics calculations where you build the dynamical matrix, diagonalize it, get eigenvalues $\omega_\nu^2(\mathbf{q})$ and eigenvectors $\mathbf{e}_{s\alpha,\nu}(\mathbf{q})$. You can do this with packages like Phonopy and GULP.
The Dynamical matrix (harmonic force constants in $\mathbf{q}$-space) is given by:
$$ \begin{equation} D_{ss'}^{\alpha\beta}(\mathbf{q}) = \frac{1}{\sqrt{m_s m_{s'}}} \sum_{\ell} \Phi_{ss'}^{\alpha\beta}(0,\ell) \, e^{i\mathbf{q}\cdot(\mathbf{R}_{s'}(\ell) - \mathbf{R}_s(0))} \label{eq:dyn_matrix} \end{equation} $$
where:
- $s,s'$: basis site indices in the primitive unit cell.
- $\alpha,\beta$: Cartesian directions.
- $m_s$: mass of atom $s$.
- $\Phi_{ss'}^{\alpha\beta}(0,\ell)$: force constant2, i.e., force on atom $(0,s)$ in direction $\alpha$ when atom $(\ell,s')$ is displaced in direction $\beta$.
- $\mathbf{R}_s(\ell)$: equilibrium position of atom $s$ in unit cell $\ell$.
This then becomes an eigenvalue problem which can be solved to get the eigenvalues $\omega_\nu^2(\mathbf{q})$ and eigenvectors $\mathbf{e}_{s\alpha,\nu}(\mathbf{q})$,
$$ \begin{equation} \sum_{s'\beta} D_{ss'}^{\alpha\beta}(\mathbf{q}) \, e_{s'\beta,\nu}(\mathbf{q}) = \omega_\nu^2(\mathbf{q}) \, e_{s\alpha,\nu}(\mathbf{q}) \label{eq:eigen} \end{equation} $$
Which gives:
- $\nu$: branch index (mode label).
- $\omega_\nu(\mathbf{q})$: harmonic frequency of mode $\nu$ at $\mathbf{q}$ (rad/s).
- $\mathbf{e}_{s\alpha,\nu}(\mathbf{q})$: polarization eigenvector for atom $s$, direction $\alpha$, mode $\nu$. Usually normalized.
In practice you use a code like Phonopy or GULP with your relaxed structure and force constants to get $\omega_\nu(\mathbf{q})$ and $\mathbf{e}_{s\alpha,\nu}(\mathbf{q})$ on a grid of $\mathbf{q}$-points or along high-symmetry paths in reciprocal space. The thing is that the same primitive cell and $\mathbf{q}$-points must be consistent with the supercell you use in your MD simulations.
Computing the Power Spectrum
The power spectrum is the Fourier transform of the kinetic energy in $(\mathbf{q},\omega)$ space, in other words it is where the thermal energy is distributed in momentum space. To get this you'll typically configure your atomistic simulation to run in the canonical (e.g. NVT) or microcanonical (NVE) ensemble at the temperature of interest. For this step you need to run the simulation for sufficient enough timne as well as use a supercell so that the wavevectors you want are commensurate with the supercell (no spurious aliasing). From the simulation you get the Output positions and velocities at some sampling time interval [1–2].
Then to get the SED you need to compute the power spectrum of the kinetic energy in $(\mathbf{q},\omega)$ space and optionally project onto the eigenvectors. Let $n$ index unit cells in the supercell and $i$ the atoms within a cell (or a combined index over all atoms). The velocity of atom $i$ in cell $n$ at time $t$ is $\mathbf{v}_{ni}(t)$, then we define the mass-weighted velocity in $\mathbf{q}$-space, which is the spatial Fourier transform at each time:
$$ \begin{equation} \tilde{\mathbf{v}}_i(\mathbf{q}, t) = \sum_n \sqrt{m_i}\, \mathbf{v}_{ni}(t) \, e^{i\mathbf{q}\cdot\mathbf{R}_{ni}}, \label{eq:v_q} \end{equation} $$
here, we have
- $\mathbf{R}_{ni}$: equilibrium position of that atom or use instantaneous positions; equilibrium seems to be more typical for the SED [1–2]).
- The sum over unit cells $n$ in the supercell with $i$ running over basis atoms, so each $ni$ is a unique atom.
Again, the SED $\Phi(\mathbf{q},\omega)$ is the temporal power spectrum of the kinetic energy in $\mathbf{q}$-space [1–2, 4], and so we compute it as:
$$ \begin{equation} \Phi(\mathbf{q},\omega) = \sum_i \frac{1}{2} \bigl| \tilde{v}_i(\mathbf{q},\omega) \bigr|^2 \label{eq:sed_total} \end{equation} $$
where $\tilde{v}_i(\mathbf{q},\omega)$ is the Fourier transform of $\tilde{\mathbf{v}}_i(\mathbf{q},t)$ with respect to time (e.g. FFT over the trajectory). So $\tilde{v}_i(\mathbf{q},\omega)$ has dimensions of $m^{1/2} \cdot \text{length} / \text{time}^2$ in frequency space; $\Phi$ is energy per unit frequency per mode (e.g. $\text{J} \cdot \text{s}/\text{rad}$).
At thermal equilibrium two things hold. First, the SED $\Phi(\mathbf{q},\omega)$ is concentrated near $\omega = \omega_\nu(\mathbf{q})$: at each wavevector $\mathbf{q}$ the lattice motion is mostly a sum of the harmonic modes, so the kinetic energy in $(\mathbf{q},\omega)$-space shows peaks at those mode frequencies instead of being spread over all frequencies. Second, if you integrate $\Phi(\mathbf{q},\omega)$ over $\omega$ at fixed $\mathbf{q}$, you get the total kinetic energy in that $\mathbf{q}$-channel; when normalized correctly this integral is on the order of $\frac{1}{2} n_\text{bands} k_B T$, in other words the equipartition across the $n_\text{bands}$ branches at that $\mathbf{q}$ [4].
The most interesting, in my opinion is the mode-resolved SED (projection onto eigenvectors) which allows you to assign energy to a specific branch $\nu$ and hence mode specific thermal conductivity [1–2]. To do this you project the mass-weighted velocities onto the harmonic eigenvector $\mathbf{e}_{s\alpha,\nu}(\mathbf{q})$ before taking the time Fourier transform. For atom index $i$ mapping to basis $s$ and Cartesian $\alpha$, the mode amplitude is
$$ \begin{equation} Q_{\mathbf{q}\nu}(t) \propto \sum_{ni} \sqrt{m_i}\, v_{ni,\alpha}(t) \, e_{s\alpha,\nu}(\mathbf{q}) \, e^{-i\mathbf{q}\cdot\mathbf{R}_{ni}} \label{eq:mode_proj} \end{equation} $$
also doing so with proper normalization and phase convention. Then the mode-resolved SED $\Phi_\nu(\mathbf{q},\omega)$ is the power spectrum of $Q_{\mathbf{q}\nu}(t)$. In practice you can compute $\Phi(\mathbf{q},\omega)$ first and then decompose by $\nu$ using the eigenvectors, or compute $\Phi_\nu$ directly; both approaches are used in the literature [1–2, 4]. You then get the the following pieces:
- $\mathbf{v}_{ni}(t)$: from your MD trajectory with velocities.
- $\mathbf{q}$: set of wavevectors commensurate with the supercell (e.g. $\mathbf{q} = \sum_i (n_i/N_i)\mathbf{b}_i$).
- $\mathbf{e}_{s\alpha,\nu}(\mathbf{q})$: eigenvectors from earlier.
- $\omega_\nu(\mathbf{q})$: frequencies from earlier.
SED Bandstructure and Mode Lifetimes
A natural way to visualise the SED is like a phonon bandstructure, i.e., plot $\Phi(\mathbf{q},\omega)$ or $\Phi_\nu(\mathbf{q},\omega)$ along a path in the Brillouin zone (e.g. $\Gamma \to X \to K \to \Gamma$) on one axis and frequency $\omega$ (e.g. in THz) on the other. What you get looks like the phonon bandstructure but with bright ridges following the branches. The heatmap intensity comes from the spectral energy density and the spread is related to the mode lifetimes, so at what we see is at finite temperature ( and with anharmonicity) the branches can shift and broaden compared to the standard ground-state quasi-harmonic phonon dispersion. An example is shown in the Figure 1.
In our SED bandstructure each peak in $\Phi_\nu(\mathbf{q},\omega)$ at a given $\mathbf{q}$ can be modeled as a damped harmonic oscillator. As result, the frequency domain the power spectrum has a Lorentzian-like form [4–5]:
$$ \begin{equation} S_\nu(\omega) \propto \frac{\Gamma_\nu \, \omega_\nu^2}{(\omega^2 - \omega_\nu^2)^2 + (\Gamma_\nu \omega)^2} \label{eq:lorentzian} \end{equation} $$
where:
- $\omega_\nu$: central frequency peak position of the mode.
- $\Gamma_\nu$: full width at half maximum (FWHM).
With this then we can compute the phonon lifetime using the following equation:
$$ \begin{equation} \tau_\nu = \frac{2}{\Gamma_\nu} \label{eq:lifetime} \end{equation} $$
which corresponds to the decay of the oscillator envelope going like $e^{-t/\tau_\nu}$.
To get the lifetime, we would then fit the SED peak for mode $\nu$ at each $\mathbf{q}$ to a Lorentzian in order to extract $\omega_\nu$ and $\Gamma_\nu$ and then compute $\ref{eq:lifetime}$. These fits are typically done per branch and per $\mathbf{q}$; so it becomes challenging with overlapping branches or when there are multiple modes at the same frequency.
 |
| Figure 1. SED Bandstructure from [1] |
Mode-Specific Thermal Conductivity
Within the relaxation-time approximation of the phonon Boltzmann transport equation, the lattice thermal conductivity tensor can be written as a sum over modes $\mathbf{q},\nu$ [6]:
$$ \begin{equation} \kappa^{\alpha\beta} = \frac{1}{V} \sum_{\mathbf{q}\nu} C_{\mathbf{q}\nu} \, v_{\mathbf{q}\nu}^\alpha \, v_{\mathbf{q}\nu}^\beta \, \tau_{\mathbf{q}\nu} \label{eq:kappa_tensor} \end{equation} $$
lets break down these terms in $\ref{eq:kappa_tensor}$:
- $V$: volume of the crystal (supercell or primitive cell).
- $C_{\mathbf{q}\nu}$: mode heat capacity (J/K per mode), assuming haromonic approx: $C_{\mathbf{q}\nu} = k_B \, x^2 e^x / (e^x - 1)^2$ with $x = \hbar\omega_{\mathbf{q}\nu}/(k_B T)$ (Einstein heat capacity per mode).
- $v_{\mathbf{q}\nu}^\alpha = \partial\omega_{\mathbf{q}\nu}/\partial q_\alpha$: group velocity (m/s) of branch $\nu$ at $\mathbf{q}$. From the harmonic dispersion $\omega_\nu(\mathbf{q})$ (Step 1) or from the peak positions of the SED at neighboring $\mathbf{q}$-points.
- $\tau_{\mathbf{q}\nu}$: relaxation time (s) from the Lorentzian fit (Step 3), $\tau_{\mathbf{q}\nu} = 2/\Gamma_{\mathbf{q}\nu}$.
For an isotropic material you often report a scalar $\kappa$ and use $\frac{1}{3}\operatorname{tr}(\kappa)$ or one diagonal component:
$$ \begin{equation} \kappa = \frac{1}{3V} \sum_{\mathbf{q}\nu} C_{\mathbf{q}\nu} \, v_{\mathbf{q}\nu}^2 \, \tau_{\mathbf{q}\nu} \label{eq:kappa_scalar} \end{equation} $$
with $v_{\mathbf{q}\nu}^2 = |\mathbf{v}_{\mathbf{q}\nu}|^2$. So: you take harmonic $C_{\mathbf{q}\nu}$ and $\mathbf{v}_{\mathbf{q}\nu}$ (or SED-derived frequencies), and anharmonic $\tau_{\mathbf{q}\nu}$ from the SED. That combination is the usual SED-based route to $\kappa$ [1–2].
Pseudocode Summary
I do have python code that achieves all of this below in conjuction with LAMMPS and Phonopy but they are old and messy at the moment. In addition the package dynasor appears to be well maintained and has good tutorial for how to use it; pySED is a Python package that implements the SED method to compute kinetic-energy-weighted phonon dispersion and extract phonon lifetimes from MD trajectories [7]. What I'll share is is the basic recap of the algorithm:
- Primitive cell + force constants $\to$ dynamical matrix $D(\mathbf{q})$ for $\mathbf{q}$ on desired grid/path.
- Diagonalize $D(\mathbf{q})$ $\to$ $\omega_\nu(\mathbf{q})$, $\mathbf{e}_{s,\nu}(\mathbf{q})$ (eqs. \ref{eq:dyn_matrix}, \ref{eq:eigen}).
- Build supercell (integer repetition); run NVT or NVE MD at target $T$; dump positions and velocities every $\Delta t$.
- For each commensurate $\mathbf{q}$:
- a. Compute $\tilde{\mathbf{v}}_i(\mathbf{q},t) = \sum_n \sqrt{m_i}\, \mathbf{v}_{ni}(t)\, e^{i\mathbf{q}\cdot\mathbf{R}_{ni}}$ (eq. \ref{eq:v_q}).
- b. FFT in time $\to$ $\tilde{v}_i(\mathbf{q},\omega)$.
- c. $\Phi(\mathbf{q},\omega) = \sum_i \frac{1}{2} |\tilde{v}_i(\mathbf{q},\omega)|^2$ (eq. \ref{eq:sed_total}).
- (Optional: project onto $\mathbf{e}_\nu(\mathbf{q})$ $\to$ $\Phi_\nu(\mathbf{q},\omega)$.)
- Plot $\Phi(\mathbf{q},\omega)$ along BZ path $\to$ heatmap bandstructure.
- For each $(\mathbf{q},\nu)$: fit SED peak to Lorentzian $\to$ $\omega_\nu$, $\Gamma_\nu$; $\tau_\nu = 2/\Gamma_\nu$ (eq. \ref{eq:lifetime}).
- From harmonic calc: $C_{\mathbf{q}\nu}$, $\mathbf{v}_{\mathbf{q}\nu}$. From SED: $\tau_{\mathbf{q}\nu}$. (eq. \ref{eq:kappa_scalar})
- $\kappa = \frac{1}{3V} \sum_{\mathbf{q}\nu} C_{\mathbf{q}\nu}\, v_{\mathbf{q}\nu}^2\, \tau_{\mathbf{q}\nu}$ (eq. \ref{eq:kappa_scalar}).
Practical Caveats
There are some practical caveats to be aware of with calculating the SED:
- Commensurability: $\mathbf{q}$ must belong to the reciprocal lattice of the supercell; otherwise the SED is polluted by aliasing [4].
- Sampling: Long trajectories and fine $\Delta t$ improve frequency resolution and the maximum $\omega$; run length should be much longer than $\tau_\nu$ for reliable lifetime extraction.
- Temperature: SED reflects the ensemble you simulate (NVT/NVE at $T$) so anharmonic shifts and broadenings are included implicitly in the MD.
Summary
The SED method bridges harmonic lattice dynamics and MD to give you mode resolved thermal conductivity. You use harmonic eigenvectors and dispersions to define modes and group velocities, and then MD trajectories to get the power spectrum in terms of $(\mathbf{q},\omega)$. From that you read off anharmonic frequencies and lifetimes via Lorentzian fits, and plug them into the relaxation-time formula for mode-resolved thermal conductivity. This can now all of this can be done with standard codes (e.g. Phonopy, LAMMPS, dynasor, pySED [7], or in-house scripts) with the main equations are \ref{eq:dyn_matrix} -- \ref{eq:kappa_scalar} driving things.
Footnotes
-
In the harmonic approximation the potential is quadratic in atomic displacements and force constants are fixed at the equilibrium structure, which is typically 0 K. Quasi-harmonic keeps the potential quadratic at each volume but lets force constants and hence phonon frequencies depend on volume (e.g. via thermal expansion), giving temperature-dependent dispersions without full anharmonicity (no cubic or higher-order terms in the potential). ↩
-
The force constants can be obtained from a supercell using frozen-phonon or density-functional perturbation theory (DFPT), or from a fitted force-constant model. ↩
References
Notation legend
| Indices | Meaning |
|---|---|
| $s,s'$ | Basis (atom) indices in primitive unit cell |
| $\ell$, $n$ | Unit-cell index (primitive / supercell) |
| $i$ | Atom index within cell (or combined $ni$) |
| $\alpha,\beta$ | Cartesian directions |
| $\nu$ | Phonon branch (mode) index |
| Space & structure | Meaning |
|---|---|
| $\mathbf{q}$ | Wavevector in reciprocal space |
| $\mathbf{R}_s(\ell)$, $\mathbf{R}_{ni}$ | Equilibrium position of atom $s$ in cell $\ell$, or atom $i$ in cell $n$ |
| $m_s$, $m_i$ | Mass of atom $s$ or $i$ |
| Harmonic lattice dynamics | Meaning |
|---|---|
| $D_{ss'}^{\alpha\beta}(\mathbf{q})$ | Dynamical matrix (force constants in $\mathbf{q}$-space) |
| $\Phi_{ss'}^{\alpha\beta}(0,\ell)$ | Force constant: force on $(0,s)$ along $\alpha$ when $(\ell,s')$ displaced along $\beta$ |
| $\omega_\nu(\mathbf{q})$ | Harmonic angular frequency of mode $\nu$ at $\mathbf{q}$ (rad/s) |
| $\mathbf{e}_{s\alpha,\nu}(\mathbf{q})$ | Polarization eigenvector (atom $s$, direction $\alpha$, mode $\nu$) |
| MD & SED | Meaning |
|---|---|
| $\mathbf{v}_{ni}(t)$ | Velocity of atom $ni$ at time $t$ |
| $\tilde{\mathbf{v}}_i(\mathbf{q},t)$, $\tilde{v}_i(\mathbf{q},\omega)$ | Mass-weighted velocity in $\mathbf{q}$-space; its time FT |
| $\Phi(\mathbf{q},\omega)$ | Spectral energy density (total kinetic energy in $(\mathbf{q},\omega)$) |
| $\Phi_\nu(\mathbf{q},\omega)$ | Mode-resolved SED for branch $\nu$ |
| $Q_{\mathbf{q}\nu}(t)$ | Mode amplitude (projection onto $\mathbf{e}_{\nu}(\mathbf{q})$) |
| Lifetimes | Meaning |
|---|---|
| $\Gamma_\nu$, $\Gamma_{\mathbf{q}\nu}$ | FWHM of SED peak (angular frequency, rad/s) |
| $\tau_\nu$, $\tau_{\mathbf{q}\nu}$ | Phonon lifetime $\tau = 2/\Gamma$ (s) |
| $S_\nu(\omega)$ | Lorentzian power spectrum of mode $\nu$ |
| Thermal conductivity | Meaning |
|---|---|
| $V$ | Crystal volume |
| $C_{\mathbf{q}\nu}$ | Mode heat capacity (J/K per mode $(\mathbf{q},\nu)$) |
| $v_{\mathbf{q}\nu}^\alpha$, $\mathbf{v}_{\mathbf{q}\nu}$ | Group velocity $\partial\omega_{\mathbf{q}\nu}/\partial q_\alpha$ |
| $\kappa^{\alpha\beta}$, $\kappa$ | Thermal conductivity tensor; scalar $\kappa = \frac{1}{3}\operatorname{tr}(\kappa)$ |
| $n_\text{bands}$ | Number of phonon branches |