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

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BMPs to overlap at sufficiently low temperatures and giving rise to a ferromagneticexchange interaction between percolated polarons (Fig. 4.4). Ifthe hole localisation radius is significantly lower than the distance betweenthem, disorder effects are of crucial importance for the magnetic propertiesof the material [129].Figure 4.4: Bound magnetic polaron. Holes hop among Mn centres (red arrows)and align their d shell spins via the exchange interaction. The percolated BMPs(blue circles) can connect into ferromagnetic clusters.Some theoretical works, like the donor impurity band model, tried toapply this picture to the regime where the density of localised holes is muchsmaller than the density of Mn ions [130]. To the best of my knowledge,this model did not find unequivocal confirmation by the experimental data.4.3 Magnetic regimes in (Ga,Mn)AsThe character of long-range magnetic order in (Ga,Mn)As depends stronglyon the interplay of all the types of direct and indirect exchange interactionsdistinguished in the previous section. As it has been already mentioned,their relative prominence depends on such crucial sample parameters as thecarrier and magnetic moment concentrations. Additionally, the number ofMn moments and (or) holes participating in the ordered state may changesignificantly in presence of charge and moment-compensating defects.The following qualitative picture of different magnetic regimes in the(Ga,Mn)As dilute magnetic semiconductor emerges from the experimentaldata and their theoretical interpretations discussed above:• At very low doping levels, the average distance r between Mn impurities(or between holes bound to Mn ions) is much larger than thesize of the bound hole approximately given by the impurity effective42
Bohr radius a ⋆ . One can expect that in this regime, holes remain localisedaround isolated Mn ions, as confirmed by the measured valueof g = 2.77, which corresponds to the neutral A 0 (d 5 +hole) Mn complex.The theoretical model of exchange interactions mediated bythermallyactivatedcarriers, proposedtodescribethisverydiluteinsulatinglimit [131], did not find its confirmation in experiments, though,as the material turned out to be a paramagnet [80].• When Mn doping reaches 1%, ferromagnetism sets in [132, 133]. Thesystem is close to the Mott insulator-metal transition—the localisationlength of holes’ states bound on Mn impurities starts to extend to adegree that allows them to mediate the exchange interaction betweenneighbouring moments. Several approaches have been used to describeferromagnetisminthisregime, includingtheinteractingBMPsorholeshopping within the impurity band. Due to the hopping nature ofconduction and the mixed sp–d character of states in the impurityband, this semi-insulating regime is often regarded as an example ofdouble-exchange ferromagnetism [88].• At highest Mn concentrations, the holes become delocalised—the desiredend product, where (Ga,Mn)As becomes a ferromagnetic semiconductor,with the magnetic degrees of freedom strongly coupled tothe electronic ones. They can mediate the coupling between Mn localmoments over large distances. Thus, the dominant mechanism of magneticorder in this metallic limit can be described by the RKKY theoryor the p–d Zener model. On the other hand, in strongly compensatedsystems, where the overall magnitude of the hole-mediated exchange isweaker, antiferromagnetic superexchange can have a significant effect.Although the nature of ferromagnetism in (Ga,Mn)As and related DMSis still debated [134, 135], the above list presents what is usually done whenmodelling the physics of these materials. I will concentrate on the lastcase—the metallic regime, including some compensation effects.At first glance, it seems that substitutional Mn spins are necessary forcreating magnetic order in (Ga,Mn)As. But, as the discoveries of recentyears show [136–139], there exist other mechanisms which can produce effectivemagnetic fields in crystals. They result from the spin-orbit interactionin the presence of symmetry-breaking factors such as interfaces, electricfields, strain, 3 and crystalline directions [136]. What is more, the combinedcrystal symmetry and relativistic effects can lead to a more complicatednon-collinear magnetic ordering of spins, like the spin canting caused by theDzyaloshinskii–Moriya interaction. Some of these effects will be addressedin the further part of my thesis.3 In one of the fascinating experiments on graphene, the application of strain allowedto create an enormous effective magnetic field of 300 T [140].43
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 20 and 21: values of spin-splitting up to thos
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: Taking into account the above equat
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
Page 88 and 89: volume. For P carriers, the Fermi e
Page 90 and 91: FMFigure 7.3: Free energy F of the
Page 92 and 93:
MT1 2 3 4TFigure 7.6: Average magne
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the ions’ magnetisation changes.
Page 96 and 97:
Because of the spin-orbit coupling
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my perturbation calculus invalid. H
Page 100 and 101:
Due to the equality ∆ = NSβ/V, t
Page 102 and 103:
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
Page 114 and 115:
where excitation modes are spin wav
Page 116 and 117:
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
Page 126 and 127:
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
Page 134 and 135:
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
Page 140 and 141:
the employed Landau-Lifshitz equati
Page 142 and 143:
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
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the negative AHE conductivity is ob
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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
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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
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[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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