Double-exchange mechanisms for Mn-doped III-V ferromagnetic semiconductors

A microscopic model of indirect exchange interaction between transition metal impurities in dilute magnetic semiconductors (DMS) is proposed. The hybridization of the impurity $d$-electrons with the heavy hole band states is mainly responsible for the exchange of electrons between the impurities, whereas the Hund rule for the electron occupation of the impurity $d$-shells makes it spin selective. The model is applied to such systems as $n$-type (Ga,Mn)N and $p$-type (Ga,Mn)As, $p$-type (Ga,Mn)P. In $n$-type DMS with ${\mathrm{Mn}}^{2+∕3+}$ impurities the exchange mechanism is rather close to the kinematic exchange proposed by Zener for mixed-valence Mn ions. In $p$-type DMS ferromagnetism is governed by the kinematic mechanism involving the kinetic energy gain of the heavy hole carriers caused by their hybridization with $3d$ electrons of ${\mathrm{Mn}}^{2+}$ impurities. Using the molecular field approximation, the Curie temperatures $T_C$ are calculated for several systems as functions of the impurity and hole concentrations. Comparison with the available experimental data shows a good agreement.

Figure 1

Left panel: graphical solution of Eq. (18) for the bonding CFR and Eq. (14) for the antibonding DBH levels. Right panel: energy levels in (Ga,Mn)As. The CFR $d$-levels $d^5∕d^4$ (denoted by $R_{1\left(2\right)}$) of each impurity, lie below the $hh$ band. The DBH levels (energies ${\epsilon }_{1\sigma }$, ${\epsilon }_{2\sigma }$) are split from the $hh$ band and form localized (acceptor) levels in the energy gap. The CFR levels $d^6∕d^5$ $R_{-1(-2)}$ lie high in the conduction band. The impurity band is shown by the shaded area together with the position of the Fermi level as a function of the width of the impurity band (horizontal axis).

Figure 2

Calculated dependence of $T_C$ on the manganese concentration $x$ in (Ga,Mn)As based on the experimental data for the hole density (Ref. 48) $p_h\left(x\right)$. Solid squares (circles) stand for experimental $T_C\left(x\right)$ of annealed (as-grown) samples. Broken lines take into account the error bars of the hole density.

Figure 3

The dependence of the Curie temperature on the hole density $p$ in (Ga,Mn)As. Solid and dashed-dotted lines are the theoretical curves for the two values of the hybridization parameter ($V=1.22$ and $1.24\mathrm{eV}$, respectively). Experimental points (filled squares) are taken from Ref. 48.

Figure 4

Calculated $T_C$ (triangles connected by a solid line) vs the annealing temperature $T_a$ in (Ga,Mn)As. The data for the hole density and the experimental values of $T_C\left(T_a\right)$ (closed squares) are taken from Ref. 51. The parameters used in the calculations are: $V=1.22\mathrm{eV}$, $\overline{z}=4.0$, $E_{\mathrm{CFR}}=-2.6\mathrm{eV}$.

Figure 5

Left panel: graphical solution of Eq. (18) for the antibonding CFR level in (Ga,Mn)N. Right panel: energy levels in Ga(Mn)N. The CFR $d$-levels $d^5∕d^4$ (denoted by $R_{1\left(2\right)}$) of each impurity, lie within the energy gap. The CFR levels $d^6∕d^5$ $R_{-1(-2)}$ lie high in the conduction band.

Figure 6

Dependence of the kinematic exchange, measured in units of ${\Delta }_0$ on the Mn content $\left(x\right)$, and on the position of the Fermi energy relative to the level $E_{\mathrm{CFR}}^0$ of an isolated Mn impurity. We took ${\kappa }^{-1}=d_0$.