Magnetism of mixed quaternary Heusler alloys: ${\left(\text{Ni},T\right)}_2\text{MnSn}$ $\left(T=\text{Cu},\text{Pd}\right)$ as a case study

The electronic properties, exchange interactions, finite-temperature magnetism, and transport properties of random quaternary Heusler ${\text{Ni}}_2\text{MnSn}$ alloys doped with Cu and Pd atoms are studied theoretically by means of ab initio calculations over the entire range of dopant concentrations. While the magnetic moments are only weakly dependent on the alloy composition, the Curie temperatures exhibit strongly nonlinear behavior with respect to Cu doping in contrast with an almost linear concentration dependence in the case of Pd doping. The present parameter-free theory agrees qualitatively and also reasonably well quantitatively with the available experimental results. An analysis of exchange interactions is provided for a deeper understanding of the problem. The dopant atoms perturb electronic structure close to the Fermi energy only weakly and the residual resistivity thus obeys a simple Nordheim rule. The dominating contribution to the temperature-dependent resistivity is due to thermodynamical fluctuations originating from the spin disorder, which, according to our calculations, can be described successfully via the disordered local moments model. Results based on this model agree fairly well with the measured values of spin-disorder-induced resistivity.

Figure 1

Total- and component-resolved densities of states, per formula unit and spin, for nonmagnetic Heusler alloys: (a) ordered ${\text{Cu}}_2\text{MnSn}$, (b) ordered ${\text{Ni}}_2\text{MnSn}$, and (c) ordered ${\text{Pd}}_2\text{MnSn}$.

Figure 2

Total- and component-resolved densities of states, per formula unit and spin, for ferromagnetic Heusler alloys. Top panel: (a) ordered ${\text{Ni}}_2\text{MnSn}$, (b) disordered ${\left({\text{Ni}}_{50},{\text{Cu}}_{50}\right)}_2\text{MnSn}$, and (c) ordered ${\text{Cu}}_2\text{MnSn}$. Bottom panel: (a) ordered ${\text{Ni}}_2\text{MnSn}$, (b) disordered ${\left({\text{Ni}}_{50},{\text{Pd}}_{50}\right)}_2\text{MnSn}$, and (c) ordered ${\text{Pd}}_2\text{MnSn}$. Majority (minority) densities of states are shown in upper (lower) parts of each figure.

Figure 3

Averaged magnetic moment (full circle) and local magnetic moments of Mn atoms (diamonds) as a function of composition for ${\left(\text{Ni},\text{Cu}\right)}_2\text{MnSn}$ alloys. Related results for ${\left(\text{Ni},\text{Pd}\right)}_2\text{MnSn}$ are shown as inset. Experimental data (Refs. 9, 10) are denoted by empty circles.

Figure 4

Concentration dependence of exchange integrals in ${\left({\text{Ni}}_{1-x},{\text{Cu}}_x\right)}_2\text{MnSn}$ Heusler alloys as estimated from the disordered local magnetic reference state: (a) the first six leading (Mn,Mn)-exchange integrals and (b) the same but with exchange integrals multiplied by their shell degeneracies. We have also added the seventh shell term with a large degeneracy.

Figure 5

The same as in Fig. 4 but for ${\left({\text{Ni}}_{1-x},{\text{Pd}}_x\right)}_2\text{MnSn}$ Heusler alloys.

Figure 6

The mean-field Curie temperature of ${\left({\text{Ni}}_{1-x},{\text{Cu}}_x\right)}_2\text{MnSn}$ for $x=0$, 0.2, and 0.8 illustrating the fulfillment of the saturation of results with respect to the number of shell. Note that curves for $x=0.2$ and 0.8 were shifted to fit the frame.

Figure 7

The concentration dependence of Curie temperatures for ${\left({\text{Ni}}_{1-x},{\text{Cu}}_x\right)}_2\text{MnSn}$. The calculated values (RPA) are compared with experimental data (Ref. 9). The MFA values of Curie temperatures are shown in the inset. Filled circles denote the calculated MFA and RPA Curie temperatures for ${\text{Cu}}_2\text{MnSn}$.

Figure 8

Comparison of calculated exchange integrals (present paper, theory) with those obtained from the fit to neutron spin-wave-scattering experiments (Table I of Ref. 44, fit to 8 NN): (a) ${\text{Ni}}_2\text{MnSn}$ and (b) ${\text{Pd}}_2\text{MnSn}$.

Figure 9

The same as in Fig. 7 but for ${\left({\text{Ni}}_{1-x},{\text{Pd}}_x\right)}_2\text{MnSn}$. The calculated values (RPA) are compared with experimental data (Ref. 10). The MFA values of Curie temperatures are also shown.

Figure 10

The concentration dependence of residual resistivities in ${\left({\text{Ni}}_{1-x},T_x\right)}_2\text{MnSn}$ alloys $\left(T=\text{Cu},\text{Pd}\right)$ evaluated for the ferromagnetic ground state at the zero temperature. Note the factor 10 multiplying resistivities for $T=\text{Pd}$ alloy.

Figure 11

The concentration dependence of the resistivity of ${\left({\text{Ni}}_{1-x},T_x\right)}_2\text{MnSn}$ alloys $\left(T=\text{Cu},\text{Pd}\right)$ due to the spin-disorder evaluated for temperatures above the Curie temperature. Magnetic state is represented by the DLM state.

Figure 12

(Color online) Reduced resistivity $\rho \left(T\right)/{\rho }_c$ vs reduced temperature $T/T_c$ (the suffix $c$ relates to the Curie temperature) for two models of the temperature dependence of the magnetization: (i) as the temperature dependence of the Sn hyperfine field in ${\text{Ni}}_2\text{MnSn}$ (squares, Ref. 49) and (ii) as the temperature dependence of the Fe hyperfine field in bcc Fe (circles, Ref. 50). The experimental values including the phonon part (triangles) are from. Ref. 24. The dashed line is a simple quadratic curve, showing the magnon part of the reduced resistivity. Being in the reduced form, it is actually free of the constant $B$: $\rho /{\rho }_c={\left(T/T_c\right)}^2$. The straight line is used as a guide for eyes. The inset shows the variation in the reduced magnetization $M/M_0$ as a function of $T/T_c$ determined from Ref. 49 (squares) and Ref. 50 (circles). $M_0$ is the saturation magnetization.

Figure 13

(Color online) The temperature dependence of the resistivity determined as described in the text for ${\text{Ni}}_2\text{MnSn}$ (triangles) and ${\text{Pd}}_2\text{MnSn}$ (circles). The empty and full symbols denote the calculated and experimental (Ref. 24) results, respectively.