Spectral permittivity tensor and density functional theory calculations on the Heusler compound ${\mathrm{Co}}_2{\mathrm{FeAl}}_{0.5}{\mathrm{Si}}_{0.5}$

The Heusler compound ${\mathrm{Co}}_2{\mathrm{FeAl}}_{0.5}{\mathrm{Si}}_{0.5}$ is an interesting material for spintronic applications due to its predicted half-metal ferromagnetic character with a large band gap in the minority states. In this work, we present the optical and magneto-optical properties of a ${\mathrm{Co}}_2{\mathrm{FeAl}}_{0.5}{\mathrm{Si}}_{0.5}$ bulk sample at room temperature. The spectral permittivity is found to be dominated by an absorption peak at 1.4 eV, originating mostly from absorptions by minority electrons. The best agreement between the experimentally obtained spectral permittivity tensor and the ab initio calculated one is for the local density approximation potential, closely followed by the generalized gradient approximation potential. However, the half-metallic ferromagnet character with $100\%$ spin polarization is preserved only in calculations employing the Coulomb interaction correction (+U). This suggests that ${\mathrm{Co}}_2{\mathrm{FeAl}}_{0.5}{\mathrm{Si}}_{0.5}$ is probably not a half metal as a single minority $d$-band dips below the Fermi level.

Figure 1

The SQUID magnetization hysteresis curve of CFAS at room temperature.

Figure 2

Ellipsometric spectra of CFAS in the form of the pseudodielectric function ${\varepsilon }_{\mathrm{pseudo}}^{\left(0\right)}$. Solid (a) red and (b) green lines represent experimental $\mathrm{Re}\left({\varepsilon }_{\mathrm{pseudo}}^{\left(0\right)}\right)$ and $\mathrm{Im}\left({\varepsilon }_{\mathrm{pseudo}}^{\left(0\right)}\right)$, respectively. Solid and dashed black lines represent B-spline and oscillator model fits, respectively. In inset (c), the magnified part of the spectra around 2 eV is shown, where the deviation of the oscillator model was largest. Insets (d) and (e) show the corresponding ellipsometry spectra expressed by ellipsometric parameters $\mathrm{\Psi }$ and $\mathrm{\Delta }$ with the respective fits. Only spectra for the angle of incidence ${60}^{\circ }$ are presented.

Figure 3

(a) Real $\left[\mathrm{Re}\left({\varepsilon }^{\left(0\right)}\right)\right]$ and (b) imaginary $\left[\mathrm{Im}\left({\varepsilon }^{\left(0\right)}\right)\right]$ part of the diagonal permittivity of CFAS. The solid orange line corresponds to the B-spline model and the black dash-dotted line corresponds to the oscillator model. Dashed lines correspond to the individual contributions of the oscillator model. The sum of the Gaussian and Tauc-Lorentz (T-L) oscillator describing the contribution of the minority spin electrons' interband transitions around the energy gap is also shown.

Figure 4

Black line: Measured reflectivity of ${\mathrm{Co}}_2{\mathrm{FeAl}}_{0.5}{\mathrm{Si}}_{0.5}$ by FTIR (Fourier-transform infrared spectroscopy). Green line: Reflectivity calculated from the oscillator model extrapolated to the IR region.

Figure 5

(a) Spectral Kerr rotation and (b) Kerr ellipticity obtained by polar MOKE measurement at 1.2 T and 3 ${}^{\circ }$ incident angle. The spectra are compared to ab initio calculated Kerr angles.

Figure 6

(a) Real and (b) imaginary parts of the magneto-optical permittivity $K$ determined from the fit to the experimental MOKE spectra (green) and from ab initio calculations (orange and purple).

Figure 7

Ab initio calculated (a) real and (b) imaginary diagonal permittivity ${\varepsilon }^{\left(0\right)}$ for different exchange-correlation potentials compared to the experimentally obtained permittivity (orange line and dashed black line) and permittivity with omitted Drude term (green line).

Figure 8

Calculated band structure of ${\mathrm{Co}}_2{\mathrm{FeAl}}_{0.5}{\mathrm{Si}}_{0.5}$ for (a) majority and (b) minority spin electrons by GGA. The color code denotes projection to electron $d$-states.

Figure 9

Calculated band structure of ${\mathrm{Co}}_2{\mathrm{FeAl}}_{0.5}{\mathrm{Si}}_{0.5}$ for (a) majority and (b) minority spin electrons by $\mathrm{GGA}+$U. The color code denotes projection to electron $d$-states.

Figure 10

Angular momentum-selective DOS obtained by (a) GGA and (b) $\mathrm{GGA}+$U calculations, respectively. The $p$-states are also shown, multiplied by a factor of 5 for clarity. The majority spin DOS is represented by positive numbers and the minority spin DOS is represented by negative numbers.

Figure 11

Schematics of different gaps with solid vertical arrows indicating optical band gaps. (a) The direct gap in the electronic band structure equals the optical band gap. (b) Indirect band gap in the band structure. The optical gap is larger in energy, and in this schematic appears at point $k$. (c) No electronic band gap at the Fermi level and the optical band gap is from the valence band to the conduction band at the Fermi level. (d) No electronic band gap at the Fermi level and the optical band gap is from the conduction band at the Fermi level to the higher conduction band.

Figure 12

Spectra of ab initio calculated unbroadened permittivity and its square root for minority spin electrons obtained by (a) GGA and (b) $\mathrm{GGA}+$U potentials. The electronic band gap, optical band gap, and absorption surge are highlighted by vertical dashed lines.

Figure 13

Dependence of magnetic moments, band-gap energies, and main peak positions for various values of $U_{\mathrm{eff}}$ for Co. The ratio of the U values of Co and Fe is kept constant.