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

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and, lowering the carriers’ energy. The delocalisation length and densityof the carriers should be sufficient to enable each of them to interact witha few localised spins, so that they can mediate the magnetic order betweenthem.Thecelebratedp–dZenermodel[39,40]describesthevalencebandstructureof the zincblende and wurzite semiconductors with the 6 × 6 Kohn–Luttinger k·p matrix [124]. It includes the p–d exchange coupling betweenthe valence band holes’ spins and the local Mn moments (S = 5 2and g = 2)within two approximations: the already mentioned virtual crystal approximation,which replaces the random distribution of Mn with their averagedensity per each lattice site, and the mean field approximation, which neglectsthe fluctuations of their spins.The Mn magnetisation M dependence on the magnetic field H and temperatureT is given (in the absence of carriers) by the Brillouin function[ ]gµ B HM = gµ B x eff N 0 SB S .k B (T +T AF )The effective Mn concentration x eff reflects the effect of compensation due tounintentional defects (Sec. 3.3), while the empirical parameter T AF accountsfor the short range antiferromagnetic superexchange between the Mn ionsmediated by the p–d exchange coupling with the occupied electron bands(Sec. 4.2.1). It can be safely neglected in (Ga,Mn)As with the dominantlong-range ferromagnetic interactions.The mean-field approach is based on the minimisation of the Ginzburg-Landau free-energy functional F(M). It can be split into the hole dependentcontribution F c (M), calculated from the Kohn–Luttinger k · p matrix andp–d exchange energy, and the local ions contribution F S (M). The latter isexpressed asF S (M) =∫ M0dM ′ H(M ′ )The minimisation of the total free-energy F(M) = F c (M)+F S (M) leadsto the solution of the mean-field equation[ ]gµB (−∂F c [M]/∂M +HM = gµ B x eff N 0 SB Sk B (T +T AF )Near the Curie temperature, where the magnetisation M is small, it canbe expected that F c (M)−F c (0) ∼ M 2 . This can be related to the carriermagnetic susceptibility χ s = A F (gµ B ) 2 ρ s , where ρ s is the spin density ofstates and A F accounts for the carrier-carrier interactions, asF c (M) = F c (0)− A Fρ s M 2 β 28(gµ B ) 2 .40.
Taking into account the above equations, the series expansion of theBrillouin function with respect to M yields the mean-field expression forthe Curie temperature in the following form:T C = x effS(S +1)β 2 A F ρ s (T C )12N 0 k B−T AF . (4.2)The above formula reflects the nature of the carrier-mediated ferromagnetismin (Ga,Mn)As—the Curie temperature is proportional both to theconcentrationandspinofthemagneticions, aswellastothedensityofstatesof the mediating holes and to the square (due to the indirect exchange process)of the local coupling constant. It should be noted, however, that theobtained mean-field Curie temperature increases without limit, while in realitythis should be stopped by such effects as localisation induced by strongexchange interactions or spin-frustration due to high carrier concentrations.Furthermore, a large density of states at the Fermi energy may also resultin a variation of the sign of the RKKY interaction, which may lead outsideof the mean-field limit.Although the mean-field model overestimates the T C values as comparedto experiment [28], it highlights the requirements toward room temperatureferromagnetic semiconductors (see e.g. [65] and references therein).It also explains the anomalous magnetic circular dichroism observed in(Ga,Mn)As [125], as well as the strain dependence of the magnetic easyaxis [126].On approaching the metal-insulator transition, the extended carrierstates mediating ferromagnetism become localised [39]. According to theuniversal scaling theory of this transition [127], the localisation radius decreasesgradually from infinity at the transition point toward the Bohr radiusdeep in the insulator phase. Within this radius the carrier wavefunctionretains an extended character, and hence can mediate the long-range interactionsbetween localised spins according to the mechanism of the p–d Zenermodel.4.2.6 Bound magnetic polaronsIn the insulating regime, an alternative, bound magnetic polaron (BMP)model is sometimes invoked, in which the carriers are quasi-localised statesin an impurity band [128]. According to its theory, a localised hole in(Ga,Mn)As is coupled by antiferromagnetic exchange mechanism with anumber of magnetic impurities within its localisation radius, which leads tothe creation of a BMP, as shown in Fig. 4.4. Such quasiparticles, in turn,interact ferromagnetically with each other, in contrast to the antiferromagneticinteractions which led to their formation. The effective radius of thepolaron increases with the ratio of the exchange and thermal energy, causing41
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: Although J(r) describes exchange in
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
Page 88 and 89: volume. For P carriers, the Fermi e
Page 90 and 91:
FMFigure 7.3: Free energy F of the
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MT1 2 3 4TFigure 7.6: Average magne
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the ions’ magnetisation changes.
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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
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102
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of the lattice ions. It ignores the
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52with angular momentum L = 0 do no
Page 108 and 109:
in the vicinity of the Γ point, in
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shinskii-Moriyaexchange. Someofthem
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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
Page 126 and 127:
yφxK surf0 , l = 0 l = 1 ... l = L
Page 128 and 129:
two pictures is especially apparent
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due to the multiplicity of the vale
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hh COMPOSITIONso COMPOSITION0.80.60
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mean-field Brillouin function (8.1)
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ters for numerical simulations, I s
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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
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M z [242] and has a weak anisotropy
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the periodic parts of the modified
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model used must also have enough ro
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The multiband tight-binding methods
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AH CONDUCTIVITY σ xy(S/cm)10080604
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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.
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160
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crystals, which provide full contro
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form of statistical DMFT (statDMFT)
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FigureA.1: Relationsbetweensoftware
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168
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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
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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.
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[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
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[171] T. E. Ostromek. Evaluation of
Page 192 and 193:
[199] C. Gourdon, A. Dourlat, V. Je
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[221] H. B. Callen. Green function
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[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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