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

to their spin; it is practically enough to derive such paradigms of physics asthe Ising, Heisenberg or Hubbard model.Similarly to spin, the exchange interaction is a purely quantum phenomenon,which has no analogy in the classical world. 2 It arises from theCoulomb interaction and is characterised by an electrostatic potential integral.It competes with the spin-orbit interaction, which couples the spin andorbital momenta of the electron (described in Sec. 5.3 in more detail). Inthis context, the two interactions are often respectively referred to as nonrelativisticand relativistic. This seems to be at odds with the fact that theexchange interaction arises from the spin statistics, which results from theLorentz invariance of the spinor field operators in the relativistic quantumfield theory. Actually, the latter being the epistemically fundamental theory,gives a tint of the relativistic character to every physical phenomenon. Onthe other hand, the relativistic character of the spin-orbit interaction resultsfrom the Lorenz invariance of the electromagnetic field.Since on the atomic scale the strength of the spin-orbit interaction ismuch lower than that of the exchange energy, ca 1Kk B vs. up to 1000Kk B ,one is tempted to think that the former has no chance to make its mark. Infact, comparing the Bohr radius, a ⋆ ≈ 0.053nm, representing the nonrelativisticscale, and the fine structure constant, α = 1/137, representing therelativistic scale, one can see that the two types of exchange become comparableat the length scales considered in solid bodies, a ⋆ /α ≈ 7nm [107].Thus, one can expect to observe the veritable cornucopia of physical effectsresulting from the interplay of these two mechanisms, many of which arediscussed in this thesis: magnetic anisotropies (Sec. 8.2) and formation ofdomain walls, effective magnetic fields of Bychkov–Rashba and Dresselhaus(Sec. 5.4) and Dzyaloshinskii–Moriya interaction, as well as their importantconsequences on physics of spin waves (Ch. 9) and the anomalous Hall effect(Ch. 10).Underlying the microscopic origin of the first Hund rule, the exchangeinteraction maximises the total moment within the half-filled Mn d shell tothe value of S = 5 2, already mentioned in Sec. 3.3. Similarly, it couples thes and d electrons on the same Mn site. On the other hand, the phenomenaof magnetic order require interaction between Mn moments, and thusthe exchange coupling acting between electrons residing at different sites.Such is the coupling between the d electrons of the Mn substituting Ga sitesand delocalised p holes which are more heavily weighted on As sites of the(Ga,Mn)As lattice. The last two interactions result from the sp–d hybridisationof atomic orbitals, and are called the s–d and p–d exchange coupling.an exceptionally beautiful topological explanation (though not a proof) [105].2 Spin has no classical analogue. The commonly shown picture of a spinning sphere isjust an understanding aid, as the electron’s size is so small that it would need to spin withequatorial speeds faster than the speed of light to have the required value of the angularmomentum, a.k.a. spin [106].34