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

More documents

Recommendations

Info

6 Band structure methods 596.1 The k ·p method . . . . . . . . . . . . . . . . . . . . . . . . . 596.1.1 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.1.2 Exchange interaction . . . . . . . . . . . . . . . . . . . 626.1.3 Parametrisations . . . . . . . . . . . . . . . . . . . . . 626.2 Tight-binding approximation . . . . . . . . . . . . . . . . . . 706.2.1 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2.2 Exchange interaction . . . . . . . . . . . . . . . . . . . 746.3 Comparison of the k ·p and tight-binding approximations . . 747 Ferromagnetism in (Ga,Mn)As 797.1 Basic statistical picture . . . . . . . . . . . . . . . . . . . . . 827.1.1 Free energy of Mn spins . . . . . . . . . . . . . . . . . 837.1.2 Free energy of ion-carrier system . . . . . . . . . . . . 867.1.3 Average magnetisation and critical temperature . . . . 897.2 Self-consistent Löwdin calculus . . . . . . . . . . . . . . . . . 937.3 Interpretation of the effective Hamiltonian . . . . . . . . . . . 977.4 Small oscillations approximation . . . . . . . . . . . . . . . . 987.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018 Curie temperature and uniaxial anisotropy in (Ga,Mn)As 1038.1 Mean-field Curie temperature . . . . . . . . . . . . . . . . . . 1038.2 Uniaxial anisotropy field . . . . . . . . . . . . . . . . . . . . . 1058.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 Spin waves and exchange stiffness in (Ga,Mn)As 1099.1 Microscopic picture of spin waves . . . . . . . . . . . . . . . . 1119.2 Micromagnetic theory . . . . . . . . . . . . . . . . . . . . . . 1169.3 Bulk (Ga,Mn)As . . . . . . . . . . . . . . . . . . . . . . . . . 1189.4 Thin (Ga,Mn)As layers . . . . . . . . . . . . . . . . . . . . . 1219.4.1 Cycloidal spin structure in (Ga,Mn)As layers . . . . . 1219.4.2 Stiffness tensors in layers . . . . . . . . . . . . . . . . 1279.5 Normalised spin-wave stiffness . . . . . . . . . . . . . . . . . . 1289.6 Spin waves’ contribution to magnetisation . . . . . . . . . . . 1319.7 Comparison to experiment . . . . . . . . . . . . . . . . . . . . 1359.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14110 Anomalous Hall effect in ferromagnetic (Ga,Mn)As 14310.1 Theoretical approach . . . . . . . . . . . . . . . . . . . . . . . 14710.2 Berry curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 14810.3 Anomalous Hall conductivity . . . . . . . . . . . . . . . . . . 15010.4 Comparison to experiment . . . . . . . . . . . . . . . . . . . . 15410.4.1 Bulk (Ga,Mn)As . . . . . . . . . . . . . . . . . . . . . 15410.4.2 Thin layers of (Ga,Mn)As . . . . . . . . . . . . . . . . 15510
10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15811 Summary and outlook 16111.1 Summary of thesis work . . . . . . . . . . . . . . . . . . . . . 16111.2 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . 163A Numerical implementation 165A.1 Code structure . . . . . . . . . . . . . . . . . . . . . . . . . . 165A.2 Computational problems . . . . . . . . . . . . . . . . . . . . . 166B Acronyms and symbols 169C Odpowiedzi dla Recenzentów 173Bibliography 17711
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: ContentsPreface 41 Introduction 132
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 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
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
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
Page 102 and 103:
102
Page 104 and 105:
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
Page 122 and 123:
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
Page 152 and 153:
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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?