Spin waves and the anomalous Hall effect in ferromagnetic (Ga,Mn)As

where u(r) has the periodicity of the lattice potential and is called the periodicpart of ψ. I adopt the normalisation convention in which∫dr 3 |ψ(r)| 2 = V ,Vwhere V is the crystal volume. The matrix element of any quantum operatoris defined for the Bloch wavefunctions as〈ψ|Ô|φ〉 = 1 ∫dr 3 ψ(r) ∗ Ôφ(r) .VVThe periodic parts’ normalisation follows from the Bloch wavefunctions’normalisation,∫dr 3 |u(r)| 2 = Ω .ΩSince the Bravais lattice underlying the zincblende structure of(Ga,Mn)As (as I assumed it to be similar to GaAs within the virtualcrystalapproximation) is the fcc lattice, its reciprocal lattice Λ ⋆ is bodycentred cubic with primitive vectors b 1 = 2π a (1,1,¯1), b 2 = 2π a (¯1,1,1) andb 3 = 2π a (1,¯1,1) (the same as for diamond). The first Brillouin zone B is obtainedby bisecting these vectors with perpendicular planes (which is a verysimple example of the construction known to mathematics as the Voronoitessellation—interestingly, italsounderliesthespinodaldecompositionindilutemagnetic semiconductors and many other phenomena in various fieldsof science). It is shaped as a truncated octahedron, as illustrated in Fig. 5.1where several special points and lines are displayed. As will be explained inthe next section, they are reminiscent of the fact that Hamiltonian H (5.3)in the reciprocal space carries (through the potential term) the symmetriesof the point group of the crystal characterised in Section 3.2. Thanks tothese symmetries, the band structure calculations can be limited to the irreduciblewedge of the B, which is a fraction of only 1 48of B (marked in redin Fig. 5.1). I make use of this fact in all numerical procedures prepared forthis thesis. (The B of the diamond space group has the same geometry, butthe point group symmetries associated with the same high symmetry pointscan differ.)5.2 Band structureConsider an eigenstate ψ of the crystal Hamiltonian H (5.3) with eigenvalueE, Hψ = Eψ. The Hamiltonian H commutes with the symmetry operationst of the point group T d , henceHtψ = tHψ = tEψ = Etψ .48