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

e done from first principles, where the only input parameters are atomicnumbers and positions, and the band structure is obtained using a variationalcalculation of the ground state of a many-body system. The methodsfalling in this category, such as the density functional theory, Hartree,or the Hartree-Fock approximation, require onerous iterative procedures toachieve self-consistency. That is why we often prefer the so-called semiempiricalapproaches, which assume a certain form of the potential basedon our knowledge about the crystal. Among these methods are:• the simplest one, nearly-free electron approximation, which assumesthat the lattice potential V(r) is vanishingly small and therefore its influenceon the band structure shows only in the form of the symmetriesof the Brillouin zone,• the pseudopotential method, which solves the one-electron equation inthe plane-wave basis, assuming a given form of the Fourier transformof the lattice potential,• the k ·p method,• the tight-binding (TB) approximation.The last two methods usually express V(r) in terms of parameters determinedfrom fitting experimental results, allowing the quantitative descriptionof the electron energy in a crystal (though there exist empirical pseudopotentialmethods, as well as—growing in popularity—ab initio tightbindingapproach [141]).The above procedures require that we turn the formal expression (5.1)into a quantum-mechanical operator acting on a well-defined Hilbert space,and perform numerical calculations in a concrete finite-dimensional basis.To this end, I will now analyse the general form of the electronic eigenstatesin the crystal, building upon the discussion of its symmetries from Ch. 3.5.1 Reciprocal spaceIn Chapter 3, I have described a dilute magnetic semiconductor crystal anddefined its space group according to its symmetries. Mathematically, the abstract(real crystals are not infinite) periodic potential of the crystal latticedoesnotdiffermuchfromaplanewavespreadingfrom−∞to∞. Accordingto the Heisenberg uncertainty principle such a wave can be characterised bya well-defined wavevector and momentum. Similarly, our model correspondsto precise values of a wavevector G, the set of which constitute a reciprocallattice Λ ⋆ dual to the real one. This important structure enables theapplication of analytic geometry of linear forms to coordinate systems withnon-orthonormal bases, like those of crystals of lower symmetry. Therefore,46