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

point group O h (of a cube) with twice as many symmetry operations. HenceT d is a subgroup of O h .The whole GaAs lattice can be generated by translating the primitivecell containing two atoms: Ga and As, which are marked by orange circlesin Fig. 3.1b. For the sake of convenience in numerical modelling, I choosethe origin at the As site. The three primitive vectors a 1 = a 2 (1,1,0), a 2 =a2 (1,0,1) and a 3 = a 2(0,1,1) (orange arrows), turn from this point towardthe centres of the faces of the cube adjacent to this site. The primitivetranslations constitute an invariant symmetry group. They generate a crystalstructure characterised by a face centred cubic Bravais lattice with thebasis given by the blue As atom at the origin plus the red Ga atom at thecentre of the tetrahedron. It can be thought of as two chemically distinctinterlocking fcc sublattices, displaced from each other by one-quarter of theunit cube main diagonal, d = a 4 (1,1,1).The combinations of the primitive translations with the rotations of theT d point group form the space group of zincblende, T 2 d (or F¯43m in theinternational notation, which can be deciphered as the cubic lattice witha four-fold rotation-inversion axis, a three-fold rotation axis, and mirrorplanes perpendicular to it). The space group is symmorphic. However, ifagain the two atoms in the primitive cell were identical, the group wouldcontain elements combining the operations of the O h point group, the primitivetranslationgroupand,additionally, anonprimitivetranslationbyvectord. Together they form the nonsymmorphic space group O 7 h (Fd¯3m), whichgenerates the diamond lattice. The group contains the inversion operation,which consists in the inversion about the origin point plus a translation byd. The lack of this particular symmetry operation in the crystal lattice mayresult in many interesting phenomena, such as the parity anomaly, currentandstrain-induced spin polarisation, spin dependent scattering or variouselectric and optical effects [77, 78]—I would venture to say, all phenomenaproduced by polar-vector perturbations. The so-called bulk inversionasymmetry will be addressed in the further course of this thesis and itseffect on magnetotransport properties in ferromagnetic (Ga,Mn)As will bedemonstrated in Ch. 10.The (Ga,Mn)As samples are usually grown epitaxially on a buffer, forminglayers of controlled thickness with abrupt interfaces. These lower thesymmetry of the samples to D 2d or C 2v , as will be explained further inSec. 5.4.2. Thicker layers are usually satisfactorily modelled as bulk crystalsof the Td 2 or O7 hsymmetry (the latter neglecting the bulk inversion asymmetry).However, in thinner samples (up to few tens of nanometres thick),the effects brought about by the presence of the interfaces may become veryimportant, like in the case of spin waves in Sec. 9.4.1 and the anomalousHall effect in Sec. 10.4.2.28