Role of conduction electrons in mediating exchange interactions in Mn-based Heusler alloys

Because of the large spatial separation of the Mn atoms in Heusler alloys $(d_{\mathrm{Mn}\text{−}\mathrm{Mn}}>4\mathrm{\AA })$, the $\mathrm{Mn}3d$ states belonging to different atoms do not overlap considerably. Therefore, an indirect exchange interaction between Mn atoms should play a crucial role in the ferromagnetism of the systems. To study the nature of the ferromagnetism of various Mn-based semi- and full-Heusler alloys, we perform a systematic first-principles calculation of the exchange interactions in these materials. The calculation of the exchange parameters is based on the frozen-magnon approach. The Curie temperature is estimated within the mean-field approximation. The calculations show that the magnetism of the Mn-based Heusler alloys depends strongly on the number of conduction $sp$ electrons, their spin polarization, and the position of the unoccupied $\mathrm{Mn}3d$ states with respect to the Fermi level. Various magnetic phases are obtained depending on the combination of these characteristics. The magnetic phase diagram is determined at zero temperature. The results of the calculations are in good agreement with available experimental data. Anderson’s $s\text{−}d$ model is used to perform a qualitative analysis of the obtained results. The conditions leading to a diverse magnetic behavior are identified. If the spin polarization of the conduction electrons at the Fermi energy is large and the unoccupied $\mathrm{Mn}3d$ states lie well above the Fermi level, a Ruderman-Kittel-Kasuya-Yoshida-type ferromagnetic interaction is dominating. On the other hand, the contribution of the antiferromagnetic superexchange becomes important if unoccupied $\mathrm{Mn}3d$ states lie close to the Fermi energy. The resulting magnetic behavior depends on the competition of these two exchange mechanisms. The calculation results are in good correlation with the conclusions made on the basis of the Anderson $s\text{−}d$ model which provides useful framework for the analysis of the results of first-principles calculations and helps to formulate the conditions for high Curie temperature.

Figure 1

$C{1}_b$ and $L{2}_1$ structures adapted by the half- and full-Heusler alloys. The lattice consists from four interpenetrating fcc lattices. In the case of the half-Heusler alloys $\left(XYZ\right)$, one of the four sublattices is vacant. In VCA, the $Z$ site is occupied by a pseudoatom with a fractional number of valence electrons.

Figure 2

(Color online) Left panel: Spin projected atom-resolved density of states of $\mathrm{Pd}\mathrm{Mn}Z$ and $\mathrm{Cu}\mathrm{Mn}Z$ $(Z=\mathrm{In},\mathrm{Sn},\mathrm{Sb},\mathrm{Te})$ for stoichiometric compositions. The shadow areas show the DOS of the $Z$ constituent. The broken vertical lines denote the Fermi level. Right panel: The same for full-Heusler compounds ${\mathrm{Pd}}_2\mathrm{Mn}Z$ and ${\mathrm{Cu}}_2\mathrm{Mn}Z$.

Figure 3

(Color online) (a) Calculated atom-resolved and total spin moments (in ${\mu }_B$) in $\mathrm{Pd}\mathrm{Mn}Z$ and $\mathrm{Cu}\mathrm{Mn}Z$ as a function of the $sp$-electron number of the $Z$ constituent. (b) The same for full-Heusler alloys ${\mathrm{Pd}}_2\mathrm{Mn}Z$ and ${\mathrm{Cu}}_2\mathrm{Mn}Z$.

Figure 4

(Color online) (a): Calculated Mn spin moment in PdMnSn and CuMnSn as a function of $\theta$ for spin spiral with $\mathbf{q}=\left(00\frac{1}{2}\right),\left(001\right)$ in units of $2\pi ∕a$. Lower panel: The corresponding total energies $\Delta E(\theta ,\mathbf{q})=E(\theta ,\mathbf{q})-E(0,0)$. For comparison, the results of the force theorem calculations (broken lines) are presented. $J_{\mathbf{q}}\left(\theta \right)$ stand for $J_{\mathbf{q}}\times M_{\theta }^2$; where $M$ is the magnetic moment. (b) The same for ${\mathrm{Pd}}_2\mathrm{Mn}\mathrm{Sn}$ and ${\mathrm{Cu}}_2\mathrm{Mn}\mathrm{Sn}$ full-Heusler compounds.

Figure 5

(Color online) (a): First six nearest neighbor Mn-Mn exchange parameters in $\mathrm{Pd}\mathrm{Mn}Z$ and $\mathrm{Cu}\mathrm{Mn}Z$ as a function of the sp-electron number of the $Z$ constituent. Also given are the number of atoms within corresponding coordination spheres. (b): The same for full-Heusler alloys ${\mathrm{Pd}}_2\mathrm{Mn}Z$ and ${\mathrm{Cu}}_2\mathrm{Mn}Z$.

Figure 6

(Color online) (a): Mn-Mn exchange interactions in ${\mathrm{Pd}}_k\mathrm{Mn}\mathrm{In}$ $(k=1,2)$ as a function of distance up to the $10a$. Lower panel: RKKY-type oscillations in exchange parameters for corresponding compounds. (b) The same for ${\mathrm{Cu}}_k\mathrm{Mn}\mathrm{In}$ $(k=1,2)$.

Figure 7

(Color online) (a): The $sp$-electron spin polarization (in ${\mu }_B$) of the $X=\mathrm{Pd}$, Cu and $Z$ constituents. The total polarization is given as the sum of $X$ and $Z$ spin polarizations. Lower panel: Mean-field estimation of the Curie temperature in $\mathrm{Pd}\mathrm{Mn}Z$ and $\mathrm{Cu}\mathrm{Mn}Z$. For comparison, available experimental $T_C$ values taken from Refs. 1, 2 are presented. FM, NC, and AFM stand for ferromagnetic, noncollinear, and antiferromagnetic ordering, respectively. (b) The same for the full-Heusler alloys ${\mathrm{Pd}}_2\mathrm{Mn}Z$ and ${\mathrm{Cu}}_2\mathrm{Mn}Z$.

Figure 8

(Color online) (a): The parameters of the Mn-Mn exchange interactions in ${\mathrm{Ni}}_2\mathrm{Mn}Z$ and ${\mathrm{Pd}}_2\mathrm{Mn}Z$ $(Z=\mathrm{In},\mathrm{Sn},\mathrm{Sb},\mathrm{Te})$ for two different lattice constants. (b): Spin projected total DOS (left hand side) and $sp$-electron DOS of $X$ and $Z$ atoms (right hand side) for ${\mathrm{Ni}}_2\mathrm{Mn}Z$ and ${\mathrm{Pd}}_2\mathrm{Mn}Z$ $(Z=\mathrm{In},\mathrm{Sn},\mathrm{Sb},\mathrm{Te})$ for the lattice constants of $6.38\mathrm{\AA }$.

Figure 9

(Color online) The total DOS of ${\mathrm{Ni}}_2\mathrm{Mn}Z$ and ${\mathrm{Pd}}_2\mathrm{Mn}Z$ $(Z=\mathrm{In},\mathrm{Sn},\mathrm{Sb},\mathrm{Te})$ above the Fermi level.