Exchange interactions and Curie temperatures in Cr-based alloys in the zinc blende structure: Volume- and composition-dependence from first-principles calculations

We present calculations of the exchange interactions and Curie temperatures in Cr-based pnictides and chalcogenides of the form $\text{Cr}X$, with $X=\text{As}$, Sb, S, Se, and Te, and the mixed alloys ${\text{CrAs}}_{50}{X}_{50}$, with $X=\text{Sb}$, S, Se, and Te. The calculations are performed for zinc blende structure for 12 values of the lattice parameter between 5.44 and $6.62\text{Å}$, appropriate for some typical II-VI and III-V semiconducting substrates. Electronic structure is calculated via the linear muffin-tin-orbitals (LMTOs) method in the atomic sphere approximation (ASA) using empty spheres to optimize ASA-related errors. Whenever necessary, the results have been verified using the full-potential version of the method, FP-LMTO. The disorder effect in the As-sublattice for ${\text{CrAs}}_{50}{X}_{50}$ $\left(X=\text{Sb},\text{S},\text{Se},\text{Te}\right)$ alloys is taken into account via the coherent-potential approximation. Exchange interactions are calculated using the linear-response method for the ferromagnetic (FM) reference states of the alloys as well as the disordered local moments (DLM) states. These results are then used to estimate the Curie temperature from the low- and high-temperature sides of the ferromagnetic-paramagnetic transition. Estimates of the Curie temperature are provided based on the mean field and the more accurate random-phase approximations. Dominant antiferromagnetic exchange interactions for some low values of the lattice parameter for the FM reference states in CrS, CrSe, and CrTe prompted us to look for antiferromagnetic (AFM) configurations for these systems with energies lower than the corresponding FM and DLM values. Results for a limited number of such AFM calculations are discussed, identifying the AFM[111] state as a likely candidate for the ground state for these cases.

Figure 1

Spin-resolved densities of states in ZB ${\text{CrAs}}_{50}{\text{Sb}}_{50}$ for lattice parameters (a) $5.55\text{Å}$, (b) $5.65\text{Å}$, (c) $5.76\text{Å}$, and (d) $5.87\text{Å}$, respectively.

Figure 2

Spin-resolved densities of states in ZB ${\text{CrAs}}_{50}{\text{Se}}_{50}$ for lattice parameters (a) $5.55\text{Å}$, (b) $5.65\text{Å}$, (c) $5.76\text{Å}$, and (d) $5.87\text{Å}$, respectively.

Figure 3

A comparison of the total densities of states in ZB CrAs with lattice parameter $5.65\text{Å}$ for the DLM and FM states.

Figure 4

Total densities of states in CrS in the DLM and FM states for lattice parameters (a) $5.44\text{Å}$, (b) $5.55\text{Å}$, (c) $5.65\text{Å}$, and (d) $5.76\text{Å}$, respectively.

Figure 5

Magnetic moment in ZB CrAS and CrSb as a function of lattice parameter in the FM and DLM states. In all cases studied (Figs. 5, 6, 7, 8), FM calculations produce induced moments on non-Cr spheres representing the $X$ atoms $\left(X=\text{As},\text{Sb},\text{S},\text{Se},\text{Te}\right)$ and one set of empty spheres. The DLM calculations produce no such induced moments, i.e., the moments reside on the Cr atoms only. See text for discussion.

Figure 6

A comparison of the moment/f.u. to the local Cr moment in the FM state as well as the DLM state for ZB CrAs (upper panel) and CrSb (lower panel).

Figure 7

Magnetic moments in ZB CrS, CrSe, and CrTe as a function of lattice parameter in the FM and DLM ground states.

Figure 8

A comparison of the moment/f.u. to the local Cr moment in the FM state as well as the DLM state for ZB CrS (top panel), CrSe (middle panel), and CrTe (bottom panel).

Figure 9

A comparison of the moment/f.u. to the local Cr moment in the FM state as well as the DLM state for ZB (a) ${\text{CrAs}}_{50}{\text{Sb}}_{50}$, (b) ${\text{CrAs}}_{50}{\text{S}}_{50}$, (c) ${\text{CrAs}}_{50}{\text{Se}}_{50}$, and (d)${\text{CrAs}}_{50}{\text{Te}}_{50}$. Circle: moment/f.u.-FM state. Diamond: local Cr-moment–FM state. Triangle: local Cr-moment–DLM state.

Figure 10

(Color online) Distance dependence of the exchange interaction $J_{ij}$ between the Cr atoms in CrAs for various lattice parameters $a$, calculated for the FM and DLM reference states. The distance between the Cr atoms is given in units of the lattice parameter $a$ (the same applies to Figs. 11, 12, 13). The main plot in Fig. 10 shows the distance dependence up to $2.25a$, while inset shows the values between $2.25a$ and $5a$. Although the individual values of $J_{ij}$ are small beyond about $2.25a$, their cumulative effects on the total exchange constant and the Curie temperature cannot be neglected (see text for details). Comparison of the insets for the FM and DLM cases shows that the interactions are more damped for the DLM case, being at least 1 order of magnitude smaller for distances beyond $\sim 2.25–2.5a$ or 15–20 neighbor shells. Similar comments apply to the interactions presented in Figs. 11, 12, 13.

Figure 11

(Color online) Distance dependence of the exchange interactions between the Cr atoms in CrSe for the FM reference state. See caption of Fig. 10.

Figure 12

(Color online) Distance dependence of the exchange interactions between the Cr atoms in CrTe for the FM reference state. See caption of Fig. 10.

Figure 13

(Color online) Distance dependence of the exchange interactions between the Cr atoms in CrSe for the DLM reference state. See caption of Fig. 10.

Figure 14

(Color online) Lattice Fourier transform of the Cr-Cr exchange interactions in CrAs, calculated for the FM and DLM reference states.

Figure 15

(Color online) Lattice Fourier transform of the Cr-Cr exchange interactions in CrS, calculated for the FM and DLM reference states.

Figure 16

(Color online) Lattice Fourier transform of the Cr-Cr exchange interactions in CrTe, calculated for the FM and DLM reference states.

Figure 17

Curie temperatures in ZB CrAs and CrSb compounds for FM and DLM reference states. For comparison, the results for the mixed alloy ${\text{CrAs}}_{50}{\text{Sb}}_{50}$ are also shown.

Figure 18

Curie temperatures in ZB CrS, CrSe, and CrTe calculated for DLM reference states. For CrS and CrSe, the ground state is antiferromagnetic (see text for details) for some low values of the lattice parameter. The values shown for those lattice parameters should be discarded.

Figure 19

Variation of Curie temperature as a function of lattice parameters in ZB ${\text{CrAs}}_{50}{X}_{50}$ alloys, with $X=\text{S}$, Se, and Te. For comparison, the results for CrAs, CrSb, and ${\text{CrAs}}_{50}{\text{Sb}}_{50}$ are also shown. All results shown are for DLM reference states and, as such, should be considered as upper limits for $T_c$.