Electron-phonon interaction and spectral weight transfer in ${\text{Fe}}_{1-x}{\text{Co}}_x\text{Si}$

A comprehensive ellipsometric study was performed on ${\text{Fe}}_{1-x}{\text{Co}}_x\text{Si}$ single crystals in the spectral range from 0.01 to 6.2 eV. Direct and indirect band gaps of 73 and 10 meV, respectively, were observed in FeSi at 7 K. One of four infrared-active phonons that is energetically close to the direct absorption edge is coupled both to the electrons and to the low-energy phonon. This is evident from asymmetry in the phonon line shape and a reduction in its frequency when the absorption edge shifts across the phonon energy due to the temperature dependence of the direct band gap. As the temperature increases, the indirect gap changes sign, which manifests as a transition from a semiconductor to a semimetal. The corresponding gain of the spectral weight at low energies was recovered within an energy range of several eV. The present findings strongly support the model indicating that ${\text{Fe}}_{1-x}{\text{Co}}_x\text{Si}$ can be well described in an itinerant picture, taking into account self-energy corrections.

Figure 1

(Color online) Real parts of (a) optical conductivity ${\sigma }_1\left(\omega \right)$ and (b) dielectric function ${\epsilon }_1\left(\omega \right)$ of ${\text{Fe}}_{1-x}{\text{Co}}_x\text{Si}$ with various Co concentrations $x$ at $T=7\text{K}$. Phonons are labeled from 1 to 4.

Figure 2

(Color online) Real parts of (a) optical conductivity ${\sigma }_1\left(\omega \right)$ and (b) dielectric function ${\epsilon }_1\left(\omega \right)$ of FeSi measured at different temperatures.

Figure 3

(Color online) Real (open circles) and imaginary (open squares) parts of the measured complex dielectric response $\epsilon \left(\omega \right)$ of FeSi. The solid curves present the result of the fitting with Eq. (3) at (a) 7 and (b) 150 K ($Q_j$ values in meV).

Figure 4

(Color online) Eigenvectors for the observable IR-active modes at (a) 25.5, (b) 40.8, (c) 43.6, and (d) 56.8 meV. Length of arrows represents the scaled magnitude of the vibrations.

Figure 5

(Color online) Plot of ${\left[{\epsilon }_2\left(\omega \right){\omega }^2\right]}^2$ after subtraction of the phonon response at different temperatures for (a) FeSi and (b) ${\text{Fe}}_{0.95}{\text{Co}}_{0.05}\text{Si}$ as a function of photon energy. The intercept of the dashed line with the energy axis defines the direct energy gap. Inset: dependence of the direct gap of ${\text{Fe}}_{1-x}{\text{Co}}_x\text{Si}$ on the Co concentration $x$. The dashed line is a guide to the eyes.

Figure 6

(Color online) Temperature dependence of the direct gap $\Delta E_{\text{dir}}$ of ${\text{Fe}}_{1-x}{\text{Co}}_x\text{Si}$, for $x=0$, 0.05, and 0.1. The solid lines represent the fits using Eq. (8). The line with crosses $\left(-\times -\right)$ denotes the frequency of phonon 4 in FeSi. Variation in ${\Omega }_4$ for different $x$ is smaller than 1 meV.

Figure 7

(Color online) (a) Plot of ${\left[{\epsilon }_2\left(\omega \right){\omega }^2\right]}^{1/2}$ of FeSi after subtraction of the phonon response at different temperatures. The straight lines represent the linear relation of Eq. (10). (b) Temperature dependence of the indirect band-gap energy. Inset: energy of the transition-assisting phonon vs temperature.

Figure 8

Calculated band structure of FeSi using the density-functional theory. Both indirect (i) and direct (d) transitions are shown in the band structure.

Figure 9

(Color online) Temperature dependence of the asymmetry parameter $Q$ for phonons 1 and 4 in (a) FeSi and (b) ${\text{Fe}}_{0.95}{\text{Co}}_{0.05}\text{Si}$.

Figure 10

(Color online) Relative changes in phonon frequencies ${\Omega }_J\left(T\right)$ for the two asymmetric (A) and one Lorentzian (L) modes in (a) FeSi and (b) ${\text{Fe}}_{0.95}{\text{Co}}_{0.05}\text{Si}$ with decreasing temperature below 300 K. The solid lines represent the expected frequency change due to thermal expansion. The dashed lines are guides to the eyes (see text).

Figure 11

(Color online) Real parts of optical conductivity ${\sigma }_1\left(\omega \right)$ and dielectric function ${\epsilon }_1\left(\omega \right)$ for FeSi at 7 K and 200 K. Phonons are subtracted. Inset: normalized effective charge density at 7 K.

Figure 12

Effective charge-density difference $\Delta N_{\text{eff}}\left(\omega \right)$ between 200 and 7 K in FeSi. Absorption bands at 0.15, 0.65, and 2.0 eV are shown with arrows.