First-principles studies of the Gilbert damping and exchange interactions for half-metallic Heuslers alloys

Heusler alloys have been intensively studied due to the wide variety of properties that they exhibit. One of these properties is of particular interest for technological applications, i.e., the fact that some Heusler alloys are half-metallic. In the following, a systematic study of the magnetic properties of three different Heusler families ${\mathrm{Co}}_2\mathrm{Mn}\mathrm{Z},{\text{Co}}_2\text{Fe}\text{Z}$, and ${\mathrm{Mn}}_2\mathrm{V}\mathrm{Z}$ with $\text{Z}=\left(\text{Al,}\text{Si,}\text{Ga,}\text{Ge}\right)$ is performed. A key aspect is the determination of the Gilbert damping from first-principles calculations, with special focus on the role played by different approximations, the effect that substitutional disorder and temperature effects. Heisenberg exchange interactions and critical temperature for the alloys are also calculated as well as magnon dispersion relations for representative systems, the ferromagnetic ${\mathrm{Co}}_2\mathrm{Fe}\mathrm{Si}$ and the ferrimagnetic ${\mathrm{Mn}}_2\mathrm{V}\mathrm{Al}$. Correlation effects beyond standard density-functional theory are treated using both the local spin density approximation including the Hubbard $U$ and the local spin density approximation plus dynamical mean field theory approximation, which allows one to determine if dynamical self-energy corrections can remedy some of the inconsistencies which were previously reported for these alloys.

Figure 1

Total density of states for different exchange correlation potentials with the dashed line indicating the Fermi energy, subfigures (a) and (b) when LSDA is used for ${\text{Co}}_2\text{Mn}\text{Si}$ and ${\text{Co}}_2\text{Fe}\text{Si}$, respectively. Subfigures (c) and (d) show the DOS when the systems $({\mathrm{Co}}_2\mathrm{MnSi}$ and ${\mathrm{Co}}_2\mathrm{FeSi}$, respectively) are treated with $\mathrm{LSDA}+U$. It can be seen that the half-metallicity of the materials can be affected by the shape of the potential and the choice of exchange correlation potential chosen.

Figure 2

(a) Adiabatic magnon spectra for ${\text{Co}}_2\text{Fe}\text{Si}$ for different exchange correlation potentials. In the case of FP-LSDA and $\mathrm{LSDA}+\mathrm{DMFT}\left[\mathrm{\Sigma }\left(0\right)\right]$ the larger deviations are observed in the case of high energies, with the DMFT curve having a lower maximum than the LSDA results. In (b) a comparison of the adiabatic magnon spectra (solid lines) with the dynamical structure factor $S\left(\mathbf{q},\omega \right)$ at $T=300\text{K}$, when the shape of the potential is considered to be given by the atomic sphere approximation and the exchange correlation potential to be given by LSDA, some softening can be observed due to temperature effects specially observed at higher $q$ points.

Figure 3

Adiabatic magnon dispersion relation for ${\text{Mn}}_2\text{V}\text{Al}$ when different exchange correlation potentials are considered. In general only a shift in energy is observed when considering LSDA or GGA with the overall shape being conserved.

Figure 4

Temperature dependence of the Gilbert damping for ${\text{Co}}_2\text{Mn}\text{Si}$ for different exchange correlation potentials. For LSDA, GGA, and $\mathrm{LSDA}+U\left[\mathrm{AMF}\right]$ exchange correlation potentials the damping increases with temperature, whilst for $\mathrm{LSDA}+U\left[\mathrm{FLL}\right]$ the damping decreases as a function of temperature.

Figure 5

Gilbert damping for different Heusler alloys at $T=300\text{K}$ as a function of density of states at the Fermi energy for LSDA exchange correlation potential. In general the damping increases as the density of states at the Fermi energy increases (the dotted line is to guide the eyes).

Figure 6

Gilbert damping for the random alloy ${\text{Co}}_2{\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ as a function of the Fe concentration at $T=300\text{K}$ when (a) only atomic displacements are considered and (b) when both atomic displacements and spin fluctuations are considered.

Figure 7

Dependence of the Gilbert damping for the alloys ${\mathrm{Co}}_2{\mathrm{MeAl}}_x{\mathrm{Si}}_{1-x}$ and ${\mathrm{Co}}_2{\mathrm{MeGa}}_x{\mathrm{Ge}}_{1-x}$ with Me denoting Mn or Fe under the LSDA exchange correlation potential.

Figure 8

DOS in ${\mathrm{Co}}_2\mathrm{FeSi}$ (top panel) and ${\mathrm{Co}}_2\mathrm{MnSi}$ (bottom panel) obtained in different computational setups.

Figure 9

Top panel: DOS in ${\mathrm{Co}}_2\mathrm{MnSi}$ projected onto Mn and Co $3d$ states of different symmetry. Middle and bottom panels: orbital-resolved spin-up and spin-down imaginary parts of the self-energy. The results are shown for the “$\mathrm{\Sigma }\left(0\right)$” DC.

Figure 10

Calculated exchange parameters in ${\mathrm{Co}}_2\mathrm{MnSi}$ within LSDA and LSDA+DMFT for different choice of DC.

Figure 11

Exchange interactions for ${\mathrm{Co}}_2\mathrm{FeSi}$ within LSDA and $\mathrm{LSDA}+U$ schemes and a full potential approach for different DC choices.