Ab initio investigations of magnetic properties of ultrathin transition ...

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2.4 The Kohn-Sham-Dirac Equation 27 up and spin down contributions. The radial functions gl(r) and ϕl(r) (the index κ has been replaced by l) can now be calculated from the following set of differential equations. ∂ ∂r gl(r) = 2Mcϕl(r) (2.41) ∂ ∂r ϕl(r) � 1 l(l +1) = 2Mc r2 � 1 + (V (r) − E) gl(r) − c 2 r ϕl(r) (2.42) The energy derivative of these yields a set of equations for ˙gl(r) and ˙ϕl(r), which are the relativistic analog of ˙ul(r). For numerical reasons the functions gl(r) and ϕl(r) are replaced by pl(r) =rgl(r) and ql(r) =crϕl(r). In the implementation of FLAPW, the radial wavefunctions are normalized according to �� gl ϕl �� gl ϕl �� = � RMT (g 0 2 l (r)+ϕ 2 l (r))r 2 dr = 1 (2.43) The energy derivatives of the radial functions have to be made orthogonal to the radial functions. �� gl �� ˙gl = 0 (2.44) ϕl ˙ϕl Thus, the scalar relativistic FLAPW basis set is ϕG (r) = �G⊥ ⎧ ⎪⎨ � ⎪⎩ 1 √ Ω e i(G �+k �)r � e iG⊥z AG �G⊥ uG � (z)+BG �G⊥ ˙uG � (z) � α lm � Int. e i(G �+k �)r � V ac. A αGk � � gl(r) lm Ylm(ˆr)+B ϕl(r) αGk � � ˙gl(r) lm Ylm(ˆr) MT ˙ϕl(r) (2.45) Note that the Pauli-spinors have been omitted, since the spin up and down problems are solved independently within the scalar relativistic approximation. Rewriting (2.40) � � � � gl(r) gl(r) HSP = E (2.46) ϕl(r) ϕl(r) with HSP = ⎛ ⎜ ⎝ 1 2M l(l +1) ∂ + V (r) −2c − c r2 r ∂r c ∂ ∂r −2mc 2 ⎞ ⎟ ⎠ + V (r) (2.47)
28 2 The FLAPW method a matrix expression for the scalar relativistic Hamiltonian including only the spherically averaged part of the potential can be obtained. For completeness, the radial charge density is defined by �� gl �� RMT gl ρl(r) = = fl fl 0 (g 2 l (r)+f 2 l (r))r 2 dr. (2.48)
Page 1 and 2: Mitglied der Helmholtz-Gemeinschaft
Page 4 and 5: Forschungszentrum Jülich GmbH Inst
Page 6 and 7: Abstract In this thesis, we investi
Page 10: ”A human being is part of the who
Page 13 and 14: II Contents 4.2.1 Relaxations and m
Page 15 and 16: 2 INTRODUCTION (ao =3.80 ˚A) is in
Page 17 and 18: 4 INTRODUCTION understanding of our
Page 19 and 20: 6 INTRODUCTION
Page 21 and 22: 8 1 Density functional theory (DFT)
Page 23 and 24: 10 1 Density functional theory (DFT
Page 29 and 30: 16 2 The FLAPW method 1 0 0 1 Figur
Page 31 and 32: 18 2 The FLAPW method nonlinear pro
Page 33 and 34: 20 2 The FLAPW method metal systems
Page 35 and 36: 22 2 The FLAPW method Then one can
Page 37 and 38: 24 2 The FLAPW method Evac is the v
Page 39: 26 2 The FLAPW method of the operat
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78 5 Fe monolayers on hexagonal non
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100 5 Fe monolayers on hexagonal no
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102 5 Fe monolayers on hexagonal no
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104 6 Co MCA from monolayers to ato
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116 rather subtle. To shed more lig
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118
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120 where, the relation � i H =
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122 Figure 6.8: The real (left) and
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124 E(q) = −M 2 ( 2J2 +2J6 +cos(
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126 For a 2D hexagonal lattice, the
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128 Bibliography [12] P. Ferriani,
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130 Bibliography [39] A. Dallmeyer,
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132 Bibliography [68] J. Perdew and
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134 Bibliography [101] L. Nordströ
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136 Bibliography [130] O. Le Bacq,
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138 Bibliography [160] P. M. Marcus
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140 Bibliography [190] R. Germar, W
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Acknowledgement I don’t know how
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Schriften des Forschungszentrums J
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