First-principles calculations of exchange interactions, spin waves, and temperature dependence of magnetization in inverse-Heusler-based spin gapless semiconductors

Employing first-principles electronic-structure calculations in conjunction with the frozen-magnon method, we calculate exchange interactions, spin-wave dispersion, and spin-wave stiffness constants in inverse-Heusler-based spin gapless semiconductor (SGS) compounds ${\text{Mn}}_2\text{CoAl}, {\text{Ti}}_2\text{MnAl}, {\text{Cr}}_2\text{ZnSi}, {\text{Ti}}_2\text{CoSi}$, and ${\text{Ti}}_2\text{VAs}$. We find that their magnetic behavior is similar to the half-metallic ferromagnetic full-Heusler alloys, i.e., the intersublattice exchange interactions play an essential role in the formation of the magnetic ground state and in determining the Curie temperature $T_{\mathrm{c}}$. All compounds, except ${\text{Ti}}_2\text{CoSi}$, possess a ferrimagnetic ground state. Due to the finite energy gap in one spin channel, the exchange interactions decay sharply with the distance, and hence magnetism of these SGSs can be described considering only nearest- and next-nearest-neighbor exchange interactions. The calculated spin-wave dispersion curves are typical for ferrimagnets and ferromagnets. The spin-wave stiffness constants turn out to be larger than those of the elementary $3d$ ferromagnets. Calculated exchange parameters are used as input to determine the temperature dependence of the magnetization and $T_{\mathrm{c}}$ of the SGSs. We find that the $T_{\mathrm{c}}$ of all compounds is much above the room temperature. The calculated magnetization curve for ${\text{Mn}}_2\text{CoAl}$ as well as the Curie temperature are in very good agreement with available experimental data. This study is expected to pave the way for a deeper understanding of the magnetic properties of the inverse-Heusler-based SGSs and enhance the interest in these materials for application in spintronic and magnetoelectronic devices.

Figure 1

Schematic representation of the lattice structure of inverse Heusler compounds having the chemical formula $X_{2}YZ$ where $X$ and $Y$ are transition-metal atoms (with the valence of $Y$ larger than of $X$) and $Z$ is an $sp$ element. On the right we present the nearest and next-nearest neighbors of an A site. Note that the large cube contains exactly four primitive unit cells.

Figure 2

Density of states (DOS) projected on the transition-metal atoms. In the case of ${\text{Mn}}_2\text{CoAl}$, positive (negative) DOS values correspond to the majority- (minority-) spin electrons. In the case of the other two compounds, which are fully compensated ferrimagnets, spin-up (positive DOS) and spin-down (negative DOS) electrons have been chosen such that the atomic spin magnetic moments in Table 1 have the right sign. Fermi level is set to the zero-energy value.

Figure 3

Intersublattice Heisenberg exchange parameters as a function of interatomic distances for (a) ${\text{Mn}}_2\text{CoAl}$, (b) ${\text{Ti}}_2\text{MnAl}$, and (c) ${\text{Cr}}_2\text{ZnSi}$.

Figure 4

Spin-wave dispersion curves along the high-symmetry lines in Brillouin zone for (a) ${\text{Mn}}_2\text{CoAl}$, (b) ${\text{Ti}}_2\text{MnAl}$, and (c) ${\text{Cr}}_2\text{ZnSi}$. In the case of ${\text{Mn}}_2\text{CoAl}$ and ${\text{Ti}}_2\text{MnAl}$, there are three branches since there are three magnetic atoms per unit cell; the $sp$ atom is almost nonmagnetic. In the case of ${\text{Cr}}_2\text{ZnSi}$, the Zn atoms have all their $d$-valence states completely occupied and have vanishing spin magnetic moments, and thus there are two branches. In the case of ${\text{Mn}}_2\text{CoAl}$, the dispersion curve of the acoustic branch around the $\mathrm{\Gamma }$ point shows a quadratic behavior while for ${\text{Cr}}_2\text{ZnSi}$ the behavior is linear; in the case of ${\text{Ti}}_2\text{MnAl}$, it is in-between.

Figure 5

Calculated temperature dependence of the sublattice and total magnetization for (a) ${\text{Mn}}_2\text{CoAl}$, (b) ${\text{Ti}}_2\text{MnAl}$, and (c) ${\text{Cr}}_2\text{ZnSi}$. In the case of ${\text{Ti}}_2\text{MnAl}$, the total spin magnetic moment for 0 K is $-0.46{\mu }_{\mathrm{B}}$ since in the Monte Carlo calculations we ignore the $sp$ atoms as well as the interstitial region (see Table 1). With the arrow we denote the room temperature of 300 K. In the case of ${\text{Mn}}_2\text{CoAl}$, we include also the experimental results from Ref. [38].

Figure 6

Calculated temperature dependence of the susceptibility for SGSs. The maximum of the susceptibility corresponds to the critical temperature (shown in parentheses in the legends).