Ferromagnetic (Ga,Mn)As Layers and ... - OPUS Würzburg

18 2. The (Ga,Mn)As Material System
2.2 Band Structure Calculations in (Ga,Mn)As
The most widely used theoretical approach to describe ferromagnetism in zinc blende
magnetic semiconductors in general and (Ga,Mn)As in particular is the p-d mean
field Zener model description of carrier mediated ferromagnetism. This model was
originally proposed by T. Dietl in 2000 [Diet 00] and subsequently developed more
thoroughly by [Diet 01] and [Abol 01]. A detailed treatise on this subject can also be
found in [Schm 06].
We are mainly interested in the valence band structure around the Γ-point (k = 0),
which is why we will use a k · p technique, which is essentially a theory, that is exact
to the order of k 2 near Γ [Lutt 55]. All included effects need to be expressed in the
form of Hamiltonians, which we will discuss individually in the following.
Kohn-Luttinger Hamiltonian
The first contribution is the band structure of pure GaAs, without the inclusion of
Mn atoms into the lattice. Its valence band structure for the first six valence bands
is given by the 6×6 Kohn-Luttinger Hamiltonian H KL , whose full form is provided in
Appendix A.1. It contains the three phenomenological Kohn-Luttinger parameters,
γ 1 , γ 2 , and γ 3 . Their values are accurately known for common semiconductors and in
GaAs are (γ 1 , γ 2 , γ 3 ) = (6.85, 2.1, 2.9)[Vurg 01]. H KL takes into account the spin-orbit
interaction H so for the orbital angular momentum L and the spin S. This interaction
can be expressed as
H so = λ L · S, (2.5)
where λ is the spin-orbit coupling. The eigenfunctions of H so are eigenstates of the
total angular momentum J = L + S. For the quantum numbers l = 1 and s = 1,
2
j can take the values 3 and 1 . At the zone center, the spin-orbit interaction splits
2 2
these two states by a factor ∆ so = 3λ , which is determined experimentally. The widely
2
accepted value for the split-off energy gap in GaAs is ∆ so = 0.341 eV[Vurg 01]. This
splitting leaves the fourfold degenerate heavy-hole (hh) plus light-hole (lh) bands of
Γ 8 symmetry with j = 3 and a doubly degenerate split-off (so) band of Γ 2 7 symmetry
with j = 1. The pure Kohn-Luttinger band structure dispersion along k 2 x for the first
six valence bands of GaAs is shown in Fig. 2.3 (a).
pd Exchange Interaction
With the incorporation of Mn atoms into the lattice, we need to consider the exchange
interaction between the valence band holes and the localized moments of the Mn 2+
acceptors. We assume that the effect of the 5d-electrons in the Mn core on the holes
can be approximated by an effective field which is proportional to the magnetization
M and couples to the spin angular momentum S of the valence band holes:
H pd = 6B g M · S (2.6)
The coupling constant B g characterizes the magnitude of the band splitting due to
the pd interaction. It has a positive value for antiferromagnetic coupling, which is the
case in (Ga,Mn)As, and a negative value for ferromagnetic coupling. In this work we