Anisotropic exchange coupling in diluted magnetic semiconductors: Ab initio spin-density functional theory

A different scheme to calculate the exchange tensor ${\underset{͇}{J}}_{ij}$ describing in a phenomenological way the anisotropic exchange coupling of two moments in a magnetically ordered system is presented. The ab initio approach is based on spin-polarized relativistic multiple-scattering theory within the framework of spin-density functional theory. The scheme is applied to ferromagnetic CrTe as well as the diluted magnetic semiconductor system ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$. In the latter case the results show that there is a noticeable anisotropy in the exchange coupling present, although not as pronounced as those suggested in recent theoretical investigations.

Figure 1

(Color online) Isotropic exchange interaction parameters $J_{ij}$ between Cr atoms at sites $i$ and $j$ in ferromagnetic CrTe as a function of the interatomic distance $R_{ij}$. The results based on the present approach (full symbols) are compared to the results obtained using the approach of Udvardi et al. (Ref. 8) (open symbols). Circles and squares represent the coupling of a Cr atom in layer 1 (Cr1) to another Cr atom in layer 1 (Cr1) or layer 2 (Cr2), respectively.

Figure 2

(Color online) Components of the Dzyaloshinski-Moriya interaction vector ${\vec{D}}_{ij}$ (from top to bottom: $x,y,z$) between Cr atoms at sites $i$ and $j$ in ferromagnetic CrTe as a function of the interatomic distance $R_{ij}$. The results based on the present approach (full circles) are compared to the results obtained using the approach of Udvardi et al. (Ref. 8) (open squares).

Figure 3

(Color online) The same as in Fig. 2 middle but with spin-orbit coupling scaled by a factor 2.

Figure 4

(Color online) Isotropic exchange interaction $J_{ij}$ between Mn atoms at sites $i$ and $j$ in ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$, scaled by the factor ${\left(R_{ij}/a\right)}^2$, as a function of the interatomic distance $R_{ij}$. Results are given for $7\text{at}\text{.}\%$ Mn (full circles) and for $4\text{at}\text{.}\%$ Mn (open squares).

Figure 5

(Color online) Components of Dzyaloshinski-Moriya interaction vector ${\vec{D}}_{ij}$ (from top to bottom: $x,y,z$) between Mn atoms at sites $i$ and $j$ in ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$, scaled by the factor ${\left(R_{ij}/a\right)}^2$, as a function of the interatomic distance $R_{ij}$. Results are given for $7\text{at}\text{.}\%$ Mn (full circles) and for $4\text{at}\text{.}\%$ Mn (open squares).

Figure 6

(Color online) Scaled exchange interactions ${\left(R_{ij}/a\right)}^2{J}_{ij}$ in ferromagnetic ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$, where $x=0.04$, as a function of the interatomic distance $R_{ij}$ along different directions.

Figure 7

(Color online) $x$ component of Dzyaloshinski-Moriya interaction ${\vec{D}}_{ij}$ between Mn atoms at sites $i$ and $j$ in ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$, where $x=0.04$, scaled by the factor ${\left(R_{ij}/a\right)}^2$, as a function of the interatomic distance $R_{ij}$ along different directions.