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

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Chapter 2 The FLAPW method In DFT there are several methods to solve Kohn-Sham equations and determine the material electronic structure. Basis-set based methods can be classified into three major approaches to calculate the Schrödinger-like equation[72]: Plane wave, Localized atomic(like) orbitals, and atomic sphere methods. Plane waves play an important role in all widely used methods, and still is the basis of choice for many new developments. Atomic sphere methods are efficient approaches. The efficiency comes from the representation of the atomic-like features, which vary rapidly close to each nucleus and smoothly, between the atoms. Close to the nucleus, the basis sets are presented as smooth functions which are ”augmented” near each nucleus by solving the Schrödinger-like equation in the sphere at each energy, and then match it to the outer wave function, usually plane wave. Then the method is called ”Augmented Plane Wave” (APW) approximation. This approach was introduced by Slater in 1937[73]. The eigen-states of an independent-particle Schrödinger equation is expanded in terms of basis functions each is represented according to two characteristic regions illustrated in figure 2.1. 2.1 The generalized eigenvalue problem The region around each atom in figure 2.1 is presented as sphere, or ”muffin-tin”, where the potential around each atom is similar to the potential of the atom. On the other hand, the potential is smooth in the interstitial region between the atoms. One approach is to expand the unknown one-electron wave function ψ(r) in a set of predefined basis functions ϕ(r). A quite general expression for such an expansion, for a system with translational symmetry with reciprocal lattice vector G and Bloch vector k , would be ψν,k(r) = � cνjkϕjk(r). (2.1) j where ν is a band index, the sum on j is finite in practice and the set of functions {ϕjk(r)} are in general neither orthonormal nor complete unless plane waves are used as basis 15
16 2 The FLAPW method 1 0 0 1 Figure 2.1: The division of space in the APW method. The muffin-tin spheres (MT) are surrounded by the interstitial region (I). functions. The expansion coefficients cijk are determined from the secular equation, via the Rayleigh-Ritz principle [74]: � [Hjj ′(k) − ɛνkSjj ′(k)]cνjk =0, (2.2) j where � Hjj ′(k) = ϕjk(r)[ Ω −∇2 ↑(↓) + Veff 2 ]ϕj ′ k(r)d 3 r (2.3) are the elements of the Hamilton matrix and � Sjj ′(k) = ϕjk(r)ϕj Ω ′ k(r)d 3 r (2.4) are the so-called overlap matrix elements. The integrals are evaluated over the volume of the unit cell (Ω). The type of functions {ϕjk} chosen, may or may not be energy-dependent, determines the detailed solution of the secular equation. The eigenvalues always follow from the condition det |Hjj ′(k) − ɛνkSjj ′(k)| =0. (2.5) If plane waves are chosen as basis functions. They are orthogonal, diagonal in momentum and any power of momentum and the implementation of planewave based methods is rather straightforward because of their simplicity. The APW is powerful approach but requires a solution on non-linear equations due to the matching between augmented functions the plane waves. This problem is explained in the next section, and it will be shown how it is useful to linearize the equations around
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Page 6 and 7: Abstract In this thesis, we investi
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116 rather subtle. To shed more lig
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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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