Theory

This section provides a brief introduction to the Kubo-Greenwood formalism, explaining the theoretical background so that users can then better understand the following exercises.

Kubo-Greenwood Formula

By using linear response theory in the Kubo formalism and expressing the band-like current operator in terms of effective, single-particle eigenstates \(\ket{{\bf k}n}\), one obtains the following Kubo-Greenwood (KG) formula for the real, diagonal part of the frequency-dependent conductivity:

$$ \Re(\sigma_{\alpha\alpha}(\omega)) = \frac{2\pi q^2\hbar^2}{m^2V\omega}\sum_{\textbf{k}mn} (\braket{\textbf{k}n|\nabla_\alpha|\textbf{k}m}\braket{\textbf{k}m|\nabla_\alpha|\textbf{k}n}) (f_{\textbf{k}n}-f_{\textbf{k}m}) \delta(\epsilon_{\textbf{k}m}-\epsilon_{\textbf{k}n}-\hbar\omega) $$ Here, \(\alpha\) denotes a Cartesian axis, \(f_{{\bf k}n}\) and \(\epsilon_{{\bf k}n}\) are the occupation number and the energy eigenvalue of the eigenstate \(\ket{{\bf k}n}\), \(\braket{\textbf{k}m|\nabla_\alpha|\textbf{k}n}\) are the gradient matrix elements in this basis, and \(V\) is the cell volume. In the FHI-aims implementation, the latter elements are obtained via
\begin{equation} \braket{\textbf{k}m|\nabla_\alpha|\textbf{k}n} = \sum_{ij} [C_{m}^{i}(\textbf{k})]^*C_{n}^j(\textbf{k}) \sum_{\textbf{N}} e^{i\textbf{k}\cdot\bf{T( \textbf{N})}} \braket{\phi_{i\bf{0}}(r)|\nabla_\alpha|\phi_{j\textbf{N}}(r)} \ , \end{equation}

whereby the real-space gradient matrix elements \(\braket{\phi_{i\bf{0}}(\vec{r})|\vec{\nabla}|\phi_{j\textbf{N}}(\vec{r})}\) are evaluated using the numeric atom-centered orbitals \(\phi_{j\textbf{N}}(r)\) and \(C^j_n({\bf k})\) are the expansion coefficients of the energy eigenstates. From the KG formula above, we can also define the electron / hole conductivity (\(\sigma_{e/h}\)) by restricting the sum to only cover the conduction or valence band, respectively.

The carrier mobility \(\mu\) is defined via:

$$ \mu = \frac{\sigma}{qn} $$ where \(n\) is the charge carrier density. In part 3 of this tutorial, we will discuss in more detail why it is often advantageous to calculate mobilities instead of conductivities.

Fourier Interpolation

In crystalline materials the band structure is typically dispersive, i.e., band energies and occupation numbers vary significantly as function of the wave vector \(\bf k\). As a consequence, dense \({\bf k}\)-grids are usually needed to obtained converged results from the KG formula.

To alleviate the associated numerical cost, we exploit the fact that the electronic density and, in turn, the associated real-space Hamiltonian, can typically be converged within a relatively sparse \(\bf{k'}\)-grid during the self-consistent field (SCF) cycle. Due to the locality of the real-space matrix elements, the converged real-space Hamiltonian \({\bf H}_{i{\bf 0},j\textbf{N}}\) and overlap matrix \({\bf S}_{i{\bf 0},j\bf{N}}\) can then be Fourier-interpolated to arbitrary dense \({\bf k}\)-grids via:

\[\begin{eqnarray} {\bf H}_{ij}(\textbf{k}) &=& \sum_{\textbf{N}} e^{i\textbf{k}\cdot\textbf{T(N)}} {\bf H}_{i{0},j\textbf{N}} \\ {\bf S}_{ij}(\textbf{k}) &=& \sum_{\textbf{N}} e^{i\textbf{k}\cdot\textbf{T(N)}} {\bf S}_{i{0},j\textbf{N}} \;. \end{eqnarray}\]

These interpolated quantities are used to obtain new associated energy eigenstates by diagonalization. The KG formula can then be evaluated on the Fourier interpolated dense \(\bf k\)-grid. Note that this approach is not unique to the KG formula, but is commonly used in FHI-aims, e.g., in band-structure calculations.

General KG Workflow

The key advantage of the KG formalism is its ability to capture all orders of anharmonicity of nuclei vibrations in the calculation of conductivities and mobilities. To this end, it is necessary to appropriately cover the portion of the potential-energy surface (PES) that is explored at different thermodynamic conditions, e.g., via ab initio molecular dynamics. A typical workflow for calculating the vibration-limited carrier conductivity at a desired temperature \(T\) with the KG formula consists of the following steps:

1. Generate representative geometric configurations (samples) that cover the phase space. To account for all orders of anharmonic effects in the nuclear dynamics, ab initio molecular dynamics (aiMD) must be used. For very harmonic materials, for which anharmonic effects can be neglected, these samples can be generated by approximative models, e.g., via harmonic sampling according to the phonon modes.
2. Perform a SCF calculation to obtain the electronic structure of each sample.
3. Perform Fourier interpolation to compute the frequency dependent conductivity \(\sigma(\omega)\) of each sample on a dense \(\bf k\)-grid.

4. Calculate the conductivity / mobility by ensemble averaging over all \(N\) samples:

   \[\braket{\sigma(\omega)}_T = \frac{1}{N} \sum_I \sigma^I(\omega)\]

   where \(I\) is the sample index.

5. Extrapolate to the DC limit to calculate the DC mobility \(\mu(\omega = 0)\).

Note that several computational parameters need to be converged to ensure that the obtained result is converged. These include

• Supercell size

  In the KG formalism, the nuclear motion and lattice vibrations are sampled in real space supercell. To accurately capture long wavelength contributions, a sufficiently large supercell is needed in the aiMD and consequent aiKG calculations.

• Number of samples

  The thermodynamic phase space is sampled using discrete geometries, the samples. Obviously, enough samples have to be used to converge the thermodynamic average.

• K-grid density and Broadening width

  In evaluation of the KG formula, the delta function is represented by a broadening function with a broadening width \(\eta\), which has to be evaluated in the limit of vanishing width. Accordingly, dense k-grids have to be chosen to appropriately resolve such a narrow broadening function.