Bound-hole states in a ferromagnetic (Ga,Mn)As environment

A numerical technique is developed to solve the Luttinger-Kohn equation for impurity states directly in $\mathbf{k}$ space and is applied to calculate bound-hole wave functions in a ferromagnetic (Ga,Mn)As host. The rich properties of the band structure of an arbitrarily strained, ferromagnetic zinc-blende semiconductor yields various features which have direct impact on the detailed shape of a valence band hole bound to an active impurity. The role of strain is discussed on the basis of explicit calculations of bound-hole states.

Figure 1

(Color online) Pure GaAs band structure of the ${\Gamma }_7$ and ${\Gamma }_8$ bands calculated with two different methods. The lines which are present in the whole Brillouin zone (black) result from a tight-binding calculation (Ref. 11) while the lines which are only plotted near $\Gamma$ [red (gray)] are obtained from a $\mathbf{k}\bullet \mathbf{p}$-type calculation.

Figure 2

(Color online) The $j=3∕2$ valence bands of ferro- and nonmagnetic materials. The top figure shows the typical nonmagnetic GaAs band structure with heavy and light holes. The bottom figure is the typical ferromagnetic band structure of (Ga,Mn)As. The bands are labeled by numbers. The total spin splitting at $\Gamma$ is $6B_g$. The dotted line (red) is the energy level to which the binding energy of the bound hole is referred.

Figure 3

(Color online) Absolute values of envelope wave functions in $\mathbf{k}$ space in $jm$ representation for $\mathbf{k}\Vert \left[100\right],\left[010\right],\left[001\right]$ with (a) magnetization along [001] and (b) magnetization along [100]. The results are obtained from a calculation with 50 $\mathbf{k}$-space directions and 200 points in each direction. The abscissa ($k$ axis) is given in units of $a_0^{-1}$ with $a_0=0.565\mathrm{nm}$ the lattice constant of GaAs.

Figure 4

(Color online) Absolute values of envelope wave functions in $\mathbf{k}$ space in eigenbasis representation. As in Fig. 3, the magnetization orientation is [001] in (a) and [100] in (b). Here, the different lines represent different bands. These bands are enumerated as in Fig. 2.

Figure 5

(Color online) Fit of a hydrogen ground-state wave function model (blue) to the numerical data of ${\mid \mathbf{B}\left(\mathbf{k}\right)\mid }^2$ (red) for $\mathbf{k}$ in the magnetization direction.

Figure 6

(Color online) Shape of bound-hole wave functions. The top part is the result of a full $\mathbf{k}\bullet \mathbf{p}$ band structure model and the bottom part results from a simple effective mass treatment. On the left side, the magnetization is oriented along [001], on the right side, the magnetization is oriented along [100]. The axes are given in units of the GaAs lattice constant $a_0$.

Figure 7

(Color online) (Ga,Mn)As band structure for $\mathbf{k}$ in the [100] direction and $\mathbf{M}$ in the [001] direction. The line labeled by $m_{\mathrm{eff}}$ model (blue) refers to the parabolic top band with curvature determined by the effective mass at the $\Gamma$ point and the line labeled by $\mathbf{k}\bullet \mathbf{p}$ model (red) refers to the top band in full $\mathbf{k}\bullet \mathbf{p}$ treatment. Note that the zero point of energy is chosen as the reference energy level of the binding energy (see also Fig. 2 and text nearby).

Figure 8

(Color online) The effect of tensile (left) and compressive (right) growth strains on the impurity wave functions. The nonstrained wave function in the middle is for comparison purposes. The magnetization is along [010]. The smallest extent of the wave functions shown here is $3a_0$ and the largest extent is $6a_0$.

Figure 9

(Color online) Relative extent of the bound-hole wave functions in dependence of magnetization orientation. Each line is labeled by a particular ${ϵ}_i$. The thick black line is for ${ϵ}_i=0$. $\Delta V$ is given relative to the mean volume of the ${ϵ}_i=0$ wave function.

Figure 10

(Color online) Bound-hole wave functions near critical strain: (a) and (b) show the wave functions for ${ϵ}_i$ smaller than critical and larger than critical, respectively, with magnetization orientation along [100]. The wave function in (c) is for magnetization along [010] and strain near its critical value. (d) shows the relative difference of the extent of the bound-hole wave function in the growth direction ([001]) for magnetization in the [100] and [010] directions.

Figure 11

(Color online) $\mathbf{k}$-direction dependence of the effective mass of the first band at the $\Gamma$ point for magnetization along [100] on the two sides of the strain criticality. The axes are given in units of the free electron mass.