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

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Figure 1.1: Examples of spintronic devices. In the GMR spin-valve (a) the currentflowing through ferromagnetic (FM) layers separated by a nonmagnetic (NM)spacer depends on the relative magnetisation of the two FM layers. The effect resultsfrom the electron scattering at the interfaces of the structure, which dependson whether the electron spin (↑ or ↓) is parallel or antiparallel to the layer magneticmoment. It can be interpreted as a parallel coupling of resistances that grow whenthe electron spin is opposing the magnetisation of the layer. S. Parkin’s racetrackmemory (b) uses a spin-coherent electric current to move magnetic domains along ananoscopic permalloy wire. The domains pass by a GMR read head positioned nearthe wire, which can retrieve bit patterns that have been encoded by a magnetisingwrite head. MRAM (c) combines a magnetic device with standard silicon-basedmicroelectronics. The magnetic tunnel junctions it employs as storage cells relyon TMR. The cells have two stable magnetic states corresponding to high or lowresistance and retain their values without any applied power. They can be read bysensing the resistance, while writing is carried out by the magnetic fields generatedfrom the current flowing in the bit and word lines. Images by Wikipedia and IBMCorporation.18
Chapter 2Motivation and outlineThe motivation for my dissertation work has been the experimental researchconducted by Professor Hideo Ohno and his group at Tohoku University.They investigated the magnetic-field and current-induced domain-wall motion[35] and achieved the magnetic state control in the field-effect transistorstructures utilising a thin-film ferromagnetic (III,Mn)V channel [36, 37].Their results are crucial for novel spintronic applications, including racetrackmemories [17], domain-wall logic circuits [38], spin-polarised currentcontrol and magnetisation characterisation.The above problems are linked to two fundamental physical effects: spinwaves and magnetic stiffness, which determine the magnetic domains’ parameters,and the anomalous Hall effect, which enables the control of magnetisationthrough electric current. These and other important characteristicsof (Ga,Mn)As, such as the Curie temperature and magnetocrystallineanisotropies, have been the focal point of my thesis work.The thesis is organised as follows.To begin with, I describe (Ga,Mn)As as a dilute magnetic semiconductor,its crystal structure, the origin of ferromagnetism in this system andthe details of the interactions shaping its band structure, in Chs. 3–5. Thedifferent band structure computational schemes that I have used in my investigationsare reviewed in Ch. 6.In Chapter 7, I analyse the hole-mediated magnetic order in dilute magneticsemiconductors [39, 40]. The presented basic statistical model givesan intuitive overall picture of ferromagnetism in these systems. To deriveit, I have used the proposed exact formula for the sum of states of classicalspins. I also provide a physically transparent perturbation-variationalmethod of treating the systems of localised magnetic moments coupled byspin carriers, based on the Löwdin calculus [41]. It is an extension of theRKKY theory [42], which—contrary to the original—is self-consistent (takesinto account the dynamics of free carriers) and can accommodate any bandstructure (of any number of bands, with the spin-orbit coupling and the19
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 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 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
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a b c022−0.2−0.41.51.5−2 −1
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It is easy to see that the basis st
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dependence on the interatomic dista
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35 spds*sps*302520151050−5−10L
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40-orbital spds ⋆ tight-binding a
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adjacentmagneticmomentsandtheirexci
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consistent and can accommodate any
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which gives for M ′ ≤ N( ) NΩ
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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
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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
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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
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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
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the coefficients of the q-dependent
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where p αβ = A µναβ q µq ν
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stiffness tensors in a similar mann
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Figure9.7: Cycloidalspinstructurein
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yφxK surf0 , l = 0 l = 1 ... l = L
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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
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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
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[12] G. Binasch, P. Grünberg, F. S
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[42] M. A. Ruderman and C. Kittel.
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[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
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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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