Electron-hole contribution to the apparent $s-d$ exchange interaction in III-V dilute magnetic semiconductors

Spin splitting of photoelectrons in $p$-type and electrons in $n$-type III-V Mn-based diluted magnetic semiconductors is studied theoretically. It is demonstrated that the unusual sign and magnitude of the apparent $s\text{−}d$ exchange integral reported for GaAs:Mn arises from exchange interactions between electrons and holes bound to Mn acceptors. This interaction dominates over the coupling between electrons and Mn spins, so far regarded as the main source of spin-dependent phenomena. A reduced magnitude of the apparent $s\text{−}d$ exchange integral found in $n$-type materials is explained by the presence of repulsive Coulomb potentials at ionized Mn acceptors and a bottleneck effect.

Figure 1

Theoretical values of the electron spin-splitting energies $\hslash {\omega }_s\left(B\right)/x$ (solid line) computed as a function of the magnetic field at 5 K. Dashed line represents fitting to the solid line obtained by treating the apparent $s\text{−}d$ exchange energy $N_0{\alpha }^{\left(\text{app}\right)}$ and temperature $T_{\text{eff}}$ as adjustable parameters within the model that neglects the presence of the electron-hole exchange interaction $\left(J_{\text{eh}}=0\right)$.

Figure 2

The components of the acceptor wave function (see the main body of the text for the definition of the functions $f$ and $g$).

Figure 3

The components of the acceptor wave function in the momentum representation (see the main body of the text for the definition of the functions $\tilde{f}$ and $\tilde{g}$).

Figure 4

The assumed dependence of the dielectric constant $\epsilon$ on the distance $r$ to an ionized acceptor.

Figure 5

The dependence of the ratio of the apparent and bare exchange energies $\alpha$ on $x$ for for ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{N}$ and various models of screening.

Figure 6

The dependence of the ratio of the apparent and bare exchange energies $\alpha$ on $x$ for ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$.