Ab initio determination of anharmonic phonon peaks

A method to compute anharmonic phonon peaks and anharmonic phonon-dispersion curves in crystals using ab initio calculated Hellmann-Feynman forces created from series of supercells is reported. The supercells are filled with atoms displaced from equilibrium positions in such a way that their configurations correspond to a given temperature. The obtained phonon-dispersion bands are able to represent the positions and shapes of the anharmonic peaks. As illustrations, the anharmonicity in cubic tungsten W, perovskite ${\mathrm{MgSiO}}_3$, and superconductor ${\text{MgB}}_2$ crystals is presented. The method can be applied to search for anharmonic potential-energy landscape of crystals. It includes electron-phonon coupling.

Figure 1

Tungsten $W$ phonon-dispersion curves along $N\text{−}\mathrm{\Gamma }\text{−}H\text{−}P\text{−}\mathrm{\Gamma }$ wave-vector path. (a) Harmonic dispersion relations. (b),(c) Anharmonic phonon maps created from averaging over 50 different sets of DP's at $T=1\text{K}$ and $T=300\text{K}$, respectively.

Figure 2

The phonon linewidth temperature behaviors of phonons in bcc cubic tungsten. The labels $N1,N2,N3$ (from lowest to higher frequency at $\mathbf{k}=N)$, $H$, and $P$ denote the high-symmetry points of bcc Brillouin zone. The phonon mode at $H$ point has large linewidth, which diminishes while decreasing the temperature.

Figure 3

${\text{MgSiO}}_3$ phonon-dispersion curves along $\mathrm{\Gamma }\text{−}X\text{−}S\text{−}Y\text{−}\mathrm{\Gamma }\text{−}Z\text{−}U\text{−}R\text{−}T\text{−}Z$ wave-vector path. (a) Harmonic phonon-dispersion relations at temperatureless regime, and pressure $P=57.3\text{GPa}$. (b) Anharmonic phonon bands created by 100 different sets of DP's at $T=2300\text{K}$ and $P=70.5\text{GPa}$. (c) Plots of all anharmonic peaks of $B_{1g}$ symmetry close to $\mathbf{k}=\mathrm{\Gamma }(0,0,0)$ [close to vertical line at (b)].

Figure 4

${\text{MgSiO}}_3$ phonon-dispersion curves along $\mathrm{\Gamma }\text{−}X\text{−}S\text{−}Y\text{−}\mathrm{\Gamma }\text{−}Z\text{−}U\text{−}R\text{−}T\text{−}Z$ wave-vector path. (a) Harmonic phonon-dispersion relations at temperatureless regime, and pressure $P=0.0\text{GPa}$. (b) Anharmonic phonon bands created by 100 different sets of DP's at $T=800\text{K}$ and $P=0.0\text{GPa}$. (c) Plots of all anharmonic peaks of $B_{1g}$ symmetry close to $\mathbf{k}=\mathrm{\Gamma }(0,0,0)$ [close to vertical line at (b)].

Figure 5

A comparison of $A_g$ anharmonic phonon modes of ${\text{MgSiO}}_3$ at $\mathrm{\Gamma }$ wave vector. (a) at $T=2300\text{K}$ and $P=70.5\text{GPa}$, (b) at $T=800\text{K}$ and $P=0.0\text{GPa}$. No smoothing is used for plots. The lines joint the points following from the histograms created by 100 different sets of DP's.

Figure 6

$\text{MgB}{}_2$ phonon-dispersion curves plotted along $\mathrm{\Gamma }\text{−}K\text{−}M\text{−}\mathrm{\Gamma }\text{−}A\text{−}L\text{−}H\text{−}A$ wave-vector path. $E_{1u},A_{2u},E_{2g}$, and $B_{1g}$ are the irreducible representation of modes at $\mathrm{\Gamma }$. (a) Harmonic phonon-dispersion relations. (b),(c) Anharmonic phonon maps created from averaging over 200 different sets of DP's at $T=1\text{K}$ and $T=300\text{K}$, respectively.

Figure 7

${\text{MgB}}_2$ anharmonic peaks plotted at (a) $M$, (b) $\mathrm{\Gamma }$, and (c) $A$ high-symmetry points in reciprocal space for $T=1\text{K}$. Labels a, b, eg, and eu correspond to notations of modes used in Table 1.