Heisenberg Hamiltonian description of multiple-sublattice itinerant-electron systems: General considerations and applications to NiMnSb and MnAs

We consider magnetic systems where the magnetic sublattices can be unambiguously separated into sublattices of inducing and induced moments. The concrete numerical calculations are performed for half-metallic ferromagnetic Heusler compound NiMnSb and hexagonal phase of MnAs. In both systems, Mn atoms possess a robust atomic moment that is much larger than the induced moments of other atoms. It is shown that the treatment of the induced moments as independent variables of the Heisenberg Hamiltonian leads to artificial features in the spin-wave spectrum. We show that the artificial features of the model do not have a dramatic influence on the estimated value of the Curie temperature. This is demonstrated within both mean-field approximation and random-phase approximation. It is shown that the calculational scheme where the induced moments are assumed to fully adjust their values and directions to the adiabatic magnetic configurations of the inducing moments is free from the artificial feature in the spin-wave spectra. In this scheme, the exchange interaction between the inducing and induced moments appears as renormalization of the exchange interactions between inducing moments. It is shown that the redistribution of the exchange interactions has strong influence on the estimated value of the Curie temperature because of the decreased number of the degrees of freedom in the thermodynamic model. Different schemes of the mapping of the systems on the Heisenberg Hamiltonian are examined. The similarities and differences in the properties of NiMnSb and MnAs are discussed.

Figure 1

(Color online) The calculated exchange parameters of NiMnSb. The parameters $J_{\mathrm{Mn}\text{−}\mathrm{Mn}}$ and $J_{\mathrm{Mn}\text{−}\mathrm{Ni}}$ are obtained with the use of the force theorem. The self-consistent SF2 scheme gives parameters that are very close to the results of the FT calculations. ${\overline{J}}_{\mathrm{Mn}\text{−}\mathrm{Mn}}$ presents the exchange parameters obtained with the use of the frozen-magnon energies calculated within the SF1 scheme. In the inset, the parameters $J_{\mathrm{Mn}\text{−}\mathrm{Mn}}$ and ${\overline{J}}_{\mathrm{Mn}\text{−}\mathrm{Mn}}$ are compared. All frozen-magnon calculations are performed for $\theta =30°$. The exchange parameters are defined for discrete values of the interatomic distances. The lines connecting the values of the exchange parameters are guides for the eye.

Figure 2

(Color online) The properties of spin waves of NiMnSb calculated with the two-sublattice Heisenberg model. Upper panel: the spin-wave dispersions ${\omega }_{1,2}$. The broken curves present the $\mathbf{q}$ dependence of the exchange parameters. Lower panel: the ratio of the deviations of the atomic moments of Mn and Ni from the magnetization axis as a function of $\mathbf{q}$ and for both spin-wave branches. In the ordinate axis, the logarithmic scale is used.

Figure 3

(Color online) The graphical solution of the RPA equations [Eqs. (17, 18)] for the Curie temperature of NiMnSb.

Figure 4

(Color online) The schematic presentation of the SF1 (top) and SF2 (bottom) schemes for the self-consistent calculation of the frozen magnons. See text for the description.

Figure 5

(Color online) The Mn moment in NiMnSb as a function of ${\theta }_{\mathrm{Mn}}$. The calculations are performed with both SF1 and SF2 schemes and for various wave vectors $\mathbf{q}$. The values of the Mn moment obtained in different calculations are close to each other, which reveals the robustness of the moment.

Figure 6

${\theta }_{\mathrm{Ni}}$ as a function of ${\theta }_{\mathrm{Mn}}$ for four different wave vectors of frozen magnons obtained within the SF1 calculation.

Figure 7

(Color online) The induced Ni moment as a function of ${\theta }_{\mathrm{Mn}}$ for four different wave vectors of frozen magnons obtained within the SF1 and SF2 calculations. The broken curves present the $\cos \theta$-type dependence.

Figure 8

(Color online) The frozen-magnon energy of NiMnSb as a function of ${\theta }_{\mathrm{Mn}}$ calculated with different schemes. The results are presented for four wave vectors. The broken curves give the results of the FT calculations. The curves with closed (open) circles are obtained within the SF1 (SF2) scheme.

Figure 9

(Color online) The frozen-magnon energies of NiMnSb as a function of $\mathbf{q}$ for the (001) direction of the Brillouin zone. The empty (filled) circles present the results of the SF1 (SF2) calculation. The calculations are performed for ${\theta }_{\mathrm{Mn}}=30°$. To be compared with the spin-wave energies presented in Fig. 2, the frozen-magnon energies should be divided by ${\sin }^2\theta$ [Eq. (7)] that for $\theta =30°$ gives the multiplication factor of 4.

Figure 10

(Color online) The Curie temperature of NiMnSb as a function of the maximal distance between Mn atoms whose interaction is taken into account. The Mn-Mn exchange parameters are calculated within the frozen-magnon approach with ${\theta }_{\mathrm{Mn}}=30°$. Three calculational schemes are used: FT, SF1, and SF2.

Figure 11

(Color online) The dependence of the leading Mn-Mn exchange parameters on the angle ${\theta }_{\mathrm{Mn}}$ used in the frozen-magnon calculations. The parameters obtained within both SF1 and SF2 schemes are shown.

Figure 12

(Color online) The calculated Curie temperature of NiMnSb as a function of the angle ${\theta }_{\mathrm{Mn}}$ used in the frozen-magnon calculations. The left panel presents the results of the mean-field calculations, whereas the results in the right panel are obtained within the random-phase approximation. In addition to the data obtained with the SF1, SF2 and FT schemes, we present the results of the SF1-type calculations that take into account the nonspherical part of the electron potential (these results are labeled as FP-SF1). The broken horizontal line at $730\mathrm{K}$ shows the experimental value of the Curie temperature.

Figure 13

${\theta }_{\mathrm{As}}$ as a function of ${\theta }_{\mathrm{Mn}}$ for three different wave vectors of frozen magnons obtained within the SF1 calculation.

Figure 14

(Color online) The Mn moment (upper panel) and induced As moment (three lower panels) as a function of ${\theta }_{\mathrm{Mn}}$ for three different wave vectors of frozen magnons. The calculations are performed within the SF1 and SF2 schemes. For Mn moment, all calculations give very close values that cannot be distinguished in the scale of the figure. The reversed value of the induced As moment is presented. The filled circles show the results of the SF1 calculation, whereas the empty circles are the results of the SF2 calculations.

Figure 15

(Color online) The frozen-magnon energy of MnAs as a function of ${\theta }_{\mathrm{Mn}}$ calculated with different schemes. The results are presented for three wave vectors. The broken curves give the results of the FT calculations. The curves with closed (open) circles are obtained within the SF1 (SF2) scheme.

Figure 16

(Color online) The calculated Mn-Mn exchange parameters of MnAs. The parameters are obtained within non-self-consistent FT scheme and two self-consistent SF1 and SF2 schemes. The frozen-magnon approach has been used. Presented are the results for ${\theta }_{\mathrm{Mn}}=30°$. The inset shows the difference between exchange parameters obtained within SF1 and SF2 schemes.

Figure 17

(Color online) The Curie temperature of MnAs as a function of the maximal distance between Mn atoms whose interaction is taken into account. The Mn-Mn exchange parameters are calculated within the frozen-magnon approach with ${\theta }_{\mathrm{Mn}}=30°$. Three different calculational schemes are used: FT, SF1, and SF2.

Figure 18

(Color online) The calculated Curie temperature of MnAs as a function of the angle ${\theta }_{\mathrm{Mn}}$ used in the frozen-magnon calculations. The left panel presents the results of the mean-field calculations, whereas the results in the right panel are obtained within the random-phase approximations. In addition to the data obtained with the SF1, SF2 and FT schemes, we present the results of the SF1-type calculations that take into account the nonspherical part of the electron potential (these results are labeled as FP-SF1). The broken horizontal line at $400\mathrm{K}$ shows the experimental value of the Curie temperature obtained by the extrapolation of the magnetization as a function of temperature to the zero-magnetization point.