Ab initio investigation of the anomalous phonon softening in FeSi

The anomalous softening of the acoustic phonon peak in FeSi has recently received considerable experimental attention. In our work, we investigate the effect of thermal disorder on the lattice dynamics and the filling of the narrow band gap of FeSi using density functional theory. We show, by comparing the phonon density of states from temperature-independent and temperature-dependent force constants, that thermal structural disorder together with thermal expansion explains the anomalously strong renormalization of the acoustic phonons. Furthermore, we find an intricate interplay between thermal disorder and volume in gap closure.

Figure 1

99% probability ellipsoid for Fe (red) and Si (black) atoms in FeSi at 750 K. The unit cell was shifted by $1.3\times (1,1,1)$ Å with respect to the crystallographic unit cell. Figure created with ortep-iii. [39]

Figure 2

Outline of the calculation procedure.

Figure 3

FeSi phonon band structure and density of states. Black: PBE equilibrium volume and force constants from sequential displacements. Green: $V_{\text{QHA}}\left(750\mathrm{K}\right)$ nonthermalized force constants from sequential displacements. Red: $V_{\text{QHA}}\left(750\mathrm{K}\right)$ thermalized force constants corresponding to 750 K. The density of states have been normalized to the number of atoms in the unit cell.

Figure 4

MSD along the principal axes of the equal probability ellipsoids. Full circles and crosses correspond to $\langle u_{11}^2\rangle$. Open circles and plus signs correspond to $\langle u_{33}^2\rangle$. Circles show experimental values taken from Ref. [44]. Green corresponds to iron and blue corresponds to Si. Crosses and plus signs are calculated from thermalized force constants obtained at four volumes using the experimental temperatures (upper $x$ axis) corresponding to these volumes.

Figure 5

Experimental volume-temperature curve [44] (dots) and calculated with QHA (solid line). The inset shows the MSD of Si calculated using the QHA volume at the given temperature.

Figure 6

Weighted phonon DOS of FeSi. Black: $V_{\text{exp}}(0$ K) and nonthermalized force constants. Green: $V_{\text{exp}}(750$ K) and nonthermalized force constants. Blue/red: $V_{\text{exp}}(750$ K) and using thermalized force constants evaluated at 750 and 1250 K. The calculated projected phonon DOS is weighted with the coherent scattering lengths, 4.15 and 9.45 fm, and masses, 28.09 and 55.85 amu, for Si and Fe, respectively [46]. A Gaussian smearing of $\sigma =0.4$ was used. The intensity was scaled to the 10 K low-temperature experimental acoustic peak. The same scaling was then applied to all the calculated phonon DOS. The experimental data are taken from Ref. [23].

Figure 7

Electron density of states for FeSi at (a) $0.94\times V_0$ and (b) $V_0$. For each temperature, the DOS of two sets of thermal disordered structures were calculated and averaged. The label corresponds to the temperature used to generate the thermalized displacements.