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

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values of spin-splitting up to those encountered in (III,Mn)V ferromagnets).Employing this method, I obtain the effective Hamiltonian for the latticespins in a dilute magnetic semiconductor—the starting point for my furtherinvestigations.Chapter 8 contains the application of the obtained Hamiltonian to thecalculations of the Curie temperature and uniaxial anisotropy, using thedifferent band structure computational schemes described in Ch. 6. In particular,I compare the known results of the 6-band k·p method [40] with thenew approaches utilising the 8-band k ·p and two tight-binding (20-orbitalsps ⋆ and 40-orbital spds ⋆ ) models. I obtained good agreement between allof them.To model spin waves in Ch. 9, I diagonalise the effective Hamiltonian usingtheBogoliubovtransformadaptedtosystemswithbrokeninversionsymmetry,thus extending the symmetric case solved by König et al. [43]. ThenI turn to the macroscopic theory of these systems and describe their micromagneticexchange and anisotropy constants. I find a relativistic correctionto the exchange stiffness constant resulting from the spin-orbit coupling, theanisotropic exchange stiffness constant T, and suggest new physical effectsit may induce. In thin layers of (Ga,Mn)As I show that the spins in theground-state form a cycloid which accounts for the surface anisotropy ofthe diagonal in-plane directions. Next, I discuss the spin-wave contributionto magnetisation and propose its self-consistent theory which amendsthe flaws of the previous approach [44]. Finally, I introduce a normalisedspin-wave stiffness parameter D nor , which may be a convenient tool for theexperimentalists to estimate the spin-wave stiffness value D given the Curietemperature and the magnetisation of the sample, and use it to reconstructthe experimental data.Moving on to the second major theme of my dissertation, Chapter 10deals with the anomalous Hall effect. I begin by reviewing the theory of theAHE. Next, I compare the results for the Berry curvature and the AHE conductivityobtained using different band structure computational schemes,discussing the differences between the 8-band k · p and two tight-bindingparametrisations and the previously employed 6-band k · p approach [45].The anomalous Hall effect is shown to exhibit qualitative dependence onthe details of the band structure beyond the six hole bands. In particular,the inversion asymmetry of the zincblende lattice is shown to lead to thenegative anomalous conductivity sign. This contrast between the previouslyrevealed robustness of the 6-band model when applied to the Curie temperature,magnetocrystalline anisotropy field and spin waves will prompt me todistinguish between static and dynamic properties of DMS, the simplest k·papproaches being sufficient for the first but not for the other. To conclude,I compare the obtained theoretical results with the experimental data forbulk and thin samples, and discuss possible origins of the observed discrepancies,including the additional mechanisms occurring in thin layers. The20
AHE simulations in the latter were the most laborious part of my thesiswork, which unfortunately was not finalised due to the lack of time.The work presented in this dissertation is summarised in Ch. 11, whichalso provides suggestions for future research. To model (Ga,Mn)As numerically,I developed a modularised, object-oriented library employing thedifferent computational schemes (Ch. 6). The details of its implementationare listed in Appendix A.Significant part of the results has been already published in Refs. [46],[37] and [47]. The first paper contains the comparison of the various bandstructure calculation methods with respect to the mean-field Curie temperatureand uniaxial anisotropy (Ch. 8) and the results from Ch. 10 for theAHE in bulk samples. The second one, published in Physical Review Letterstogether with the group of Professor Hideo Ohno at Tohoku University,contains the analysis of the experimental measurements for the AHE in thinlayers supported by my theoretical results (Sec. 10.4.2). Finally, Ref. [47]includes the exposition of the proposed self-consistent Löwdin calculus, thederivation of the effective Hamiltonian for lattice spins and its applicationto the spin waves in (Ga,Mn)As, gathering the results for the exchange stiffnessand anisotropy constants (including the comparison with experiment),cycloidal ground-state spin structure, and the spin waves’ contribution tomagnetisation, as described in Chs. 7 and 9.21
Page 1 and 2: Polish Academy of SciencesInstitute
Page 3 and 4: AcknowledgementsI would like to tha
Page 5 and 6: Curriculum VitaeEducation and Train
Page 7 and 8: Jaszowiec, Poland, 2005, RKKY and Z
Page 9 and 10: ContentsPreface 41 Introduction 132
Page 11: 10.5 Summary . . . . . . . . . . .
Page 14 and 15: to look far to find another example
Page 16 and 17: 1990 was a watershed event in the f
Page 18 and 19: Figure 1.1: Examples of spintronic
Page 23 and 24: Chapter 3(Ga,Mn)As as a dilutemagne
Page 25 and 26: in the 1970s and became commonly us
Page 27 and 28: GaAsMna 2a 3a 1dFigure 3.1: (Ga,Mn)
Page 29 and 30: 3.3 Mn impuritiesThe (Ga,Mn)As crys
Page 31 and 32: values too large to be explained by
Page 33 and 34: Chapter 4Origin of magnetism in(Ga,
Page 35 and 36: They lead to dramatic spin-dependen
Page 37 and 38: alignment of the Mn moments. The el
Page 39 and 40: Although J(r) describes exchange in
Page 41 and 42: Taking into account the above equat
Page 43 and 44: Bohr radius a ⋆ . One can expect
Page 45 and 46: Chapter 5Band structure of(Ga,Mn)As
Page 47 and 48: I know how to decompose the potenti
Page 49 and 50: k zk xΓΛLΣ K∆QWUZSXk yFigure 5
Page 51 and 52: or antibonding. In semiconductors l
Page 53 and 54: of the Brillouin zone are illustrat
Page 55 and 56: 5.4.2 Structure inversion asymmetry
Page 57 and 58: of electrons on the ion’s electro
Page 59 and 60: Chapter 6Band structure methodsIn t
Page 61 and 62: the higher-lying ones can be taken
Page 63 and 64: Parameter Kohn-Luttinger Kane (set
Page 65 and 66: and Q ǫ and R ǫ are the following
Page 67 and 68: ThematrixM kso describesthek-depend
Page 69 and 70: a b c022−0.2−0.41.51.5−2 −1
Page 71 and 72:
It is easy to see that the basis st
Page 73 and 74:
dependence on the interatomic dista
Page 75 and 76:
35 spds*sps*302520151050−5−10L
Page 77:
40-orbital spds ⋆ tight-binding a
Page 80 and 81:
adjacentmagneticmomentsandtheirexci
Page 82 and 83:
consistent and can accommodate any
Page 84 and 85:
which gives for M ′ ≤ N( ) NΩ
Page 86 and 87:
One can associate each lattice spin
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volume. For P carriers, the Fermi e
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FMFigure 7.3: Free energy F of the
Page 92 and 93:
MT1 2 3 4TFigure 7.6: Average magne
Page 94 and 95:
the ions’ magnetisation changes.
Page 96 and 97:
Because of the spin-orbit coupling
Page 98 and 99:
my perturbation calculus invalid. H
Page 100 and 101:
Due to the equality ∆ = NSβ/V, t
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102
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of the lattice ions. It ignores the
Page 106 and 107:
52with angular momentum L = 0 do no
Page 108 and 109:
in the vicinity of the Γ point, in
Page 110 and 111:
shinskii-Moriyaexchange. Someofthem
Page 112 and 113:
emaining terms).I want to obtain th
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where excitation modes are spin wav
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SPIN−WAVE DISPERSION ω q(meV)654
Page 118 and 119:
the coefficients of the q-dependent
Page 120 and 121:
where p αβ = A µναβ q µq ν
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stiffness tensors in a similar mann
Page 124 and 125:
Figure9.7: Cycloidalspinstructurein
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yφxK surf0 , l = 0 l = 1 ... l = L
Page 128 and 129:
two pictures is especially apparent
Page 130 and 131:
due to the multiplicity of the vale
Page 132 and 133:
hh COMPOSITIONso COMPOSITION0.80.60
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mean-field Brillouin function (8.1)
Page 136 and 137:
ters for numerical simulations, I s
Page 138 and 139:
EXCHANGE STIFFNESS (pJ/m)0.40.30.20
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the employed Landau-Lifshitz equati
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netisation. The basic theoretical m
Page 144 and 145:
M z [242] and has a weak anisotropy
Page 146 and 147:
the periodic parts of the modified
Page 148 and 149:
model used must also have enough ro
Page 150 and 151:
The multiband tight-binding methods
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AH CONDUCTIVITY σ xy(S/cm)10080604
Page 154 and 155:
AH CONDUCTIVITY σ xy(S/cm)14012010
Page 156 and 157:
the negative AHE conductivity is ob
Page 158 and 159:
Figure 10.10: Hall conductivity vs.
Page 160 and 161:
160
Page 162 and 163:
crystals, which provide full contro
Page 164 and 165:
form of statistical DMFT (statDMFT)
Page 166 and 167:
FigureA.1: Relationsbetweensoftware
Page 168 and 169:
168
Page 170 and 171:
a str strained lattice constanta
Page 172 and 173:
U Dzyaloshinskii-Moriya vectorV cry
Page 174 and 175:
gdzieL ′ = F ′ +2G M = H 1 +H 2
Page 176 and 177:
której wartość pola średniego
Page 178 and 179:
[12] G. Binasch, P. Grünberg, F. S
Page 180 and 181:
[42] M. A. Ruderman and C. Kittel.
Page 182 and 183:
[69] H. Munekata, A. Zaslavsky, P.
Page 184 and 185:
[92] J. Zemen, J. Kučera, K. Olejn
Page 186 and 187:
[118] C. Zener. Interaction between
Page 188 and 189:
[143] J. Jancu, R. Scholz, F. Beltr
Page 190 and 191:
[171] T. E. Ostromek. Evaluation of
Page 192 and 193:
[199] C. Gourdon, A. Dourlat, V. Je
Page 194 and 195:
[221] H. B. Callen. Green function
Page 196 and 197:
[249] G. Sundaram and Q. Niu. Wave-
Page 198 and 199:
[275] K. Y. Wang, K. W. Edmonds, R.
Page 200 and 201:
[300] D. J. Garcia, K. Hallberg, an
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Spin waves and the anomalous Hall effect in ferromagnetic (Ga,Mn)As

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