Lattice dynamics and electron-phonon coupling calculations using nondiagonal supercells

We study the direct calculation of total energy derivatives for lattice dynamics and electron-phonon coupling calculations using supercell matrices with nonzero off-diagonal elements. We show that it is possible to determine the response of a periodic system to a perturbation characterized by a wave vector with reduced fractional coordinates $(m_1/n_1,m_2/n_2,m_3/n_3)$ using a supercell containing a number of primitive cells equal to the least common multiple of $n_1,n_2$, and $n_3$. If only diagonal supercell matrices are used, a supercell containing $n_1{n}_2{n}_3$ primitive cells is required. We demonstrate that the use of nondiagonal supercells significantly reduces the computational cost of obtaining converged zero-point energies and phonon dispersions for diamond and graphite. We also perform electron-phonon coupling calculations using the direct method to sample the vibrational Brillouin zone with grids of unprecedented size, which enables us to investigate the convergence of the zero-point renormalization to the thermal and optical band gaps of diamond.

Figure 1

Example of the use of nondiagonal supercells to access all points on a $2\times 2$ grid sampling the BZ (red shaded area) of a square lattice (black dots). The black arrows indicate the reciprocal superlattice vectors, and the red dots indicate the points in the first BZ that are accessible to them.

Figure 2

Zero-point energy as a function of the number of normal modes (coarse grid size) for diamond (red circles) and graphite (blue squares). The solid lines are a guide to the eye.

Figure 3

Convergence with respect to coarse grid size of phonon dispersions along symmetry lines for diamond.

Figure 4

Convergence with respect to coarse grid size of phonon dispersions along symmetry lines for graphite.

Figure 5

ZPR to the thermal (red circles) and optical (red squares) band gaps of diamond as a function of the linear size of the BZ grid. The solid lines are a guide to the eye.