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

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3.3 Non-Collinear Magnetism 41 ⇒ e ik·Rn � i(k−q/2)r e α(k, r + Rn) ei(k+q/2)r � = e β(k, r + Rn) ik·Rn � i(k−q/2)r e α(k, r) ei(k+q/2)r � β(k, r) ⇒ α(k, r + Rn) =α(k, r), β(k, r + Rn) =β(k, r). (3.39) The fact that α and β are periodic functions is very important for the implementation of the generalized Bloch theorem into the FLAPW[38] and many other (plane-wave based) methods. 3.3.4 Non-Collinear Magnetism in FLAPW The first implementation of non-collinear magnetism in the ab-initio calculations [98, 90, 89, 99, 91, 34, 100], allowed only one direction of magnetization per atom, i.e. the direction of the magnetization density ˆm is not allowed to change within one sphere 1 , but varies only from sphere to sphere (so-called the atomic sphere approximation for the direction of magnetization). This agrees with the intuitive picture that an atom carries a magnetic moment of a certain size and only the direction of these moments differs between the atoms. Such methods describe only the inter-atomic non-collinearity. However, in general the direction of the magnetization changes continuously from site to site, though, in many cases, the deviations from the main atomic direction are only significant in a region between the atom, where the magnitude of the magnetization is rather small. The first calculation that treated the magnetization as a continuous vector quantity was published by Nordström et al. [101]. They followed the most general approach allowing the magnetization to change magnitude and direction continuously, i.e. even within an atom. Thus, their implementation, that is based on the FLAPW method, allows them to also investigate the intra-atomic non-collinearity, which is important for actinides like Pu. Our method uses a “hybrid” approach (Fig.3.4) where the magnetization is treated as a continuous vector field in the interstitial and in the vacuum regions, while inside each muffin-tin sphere we only allow for one direction of magnetization. Inside the muffin-tins, like in the collinear case, it is still possible to work with V↑ and V↓ in the non-collinear case, since we restrict the magnetization to the local quantization quantization axis. Therefore, a local spin-space coordinate-frame is introduced with the z-axis parallel to the local quantization axis. V↑ and V↓ are now spin-up and -down with respect to the local axis. Since both, the potential and the basis functions, are set up in terms of the local spin-coordinate frame, the determination of the basis functions and calculation of the integrals of these functions with the Hamiltonian inside the muffintins is completely unchanged. The changes come in, when the basis functions inside the muffin-tins are matched to the plane waves in the interstitial region, because the local spin-coordinate frame S α is rotated with respect to the global frame S g . The FLAPW method uses augmented plane waves as basis functions. Therefore, each basis function for a given k-vector, � k, can be uniquely identified by is wave vector G and the spin direction. The basis functions in the interstitial region are: 1 Within the muffin-tin spheres, however, magnetization can vary in magnitude
42 3 Magnetism of low dimensional systems Figure 3.4: Schematic illustration of the representation of the non-collinear magnetization density within the present approach. The magnetization is treated as a continuous vector field in the interstitial region and in the vacuum. Within each muffin-tin the magnetization has a fixed direction and can only vary in magnitude[38]. e i(k+G)r χ g σ (3.40) χg σ is a two component spinor. The index g has been added to notify that χg σ is the representation of this spinor in the global spin frame. This representation of the basis functions is used for both collinear and non-collinear calculations. However, the potential matrix V , and thus the Hamiltonian, is diagonal in the two spin directions in the collinear case. Therefore, the Hamiltonian can be set up and solved separately for the two spin directions. In the non-collinear case the off-diagonal part of V is not zero anymore. Hence, the full Hamiltonian for both spin directions has to be set up and solved in a single step. In the vacuum we also use the global spin frame for the representation of the basis functions. The basis set is only changed in the muffin-tins, because we use a local spin coordinate frame, which is rotated with respect to the global frame. The consequence is that, when the plane waves are matched to the functions in the muffin tin spheres, each spin direction in the interstitial region has to be matched to both, the spin-up and -down basis functions, in the sphere. Thus, the basis set has the following form. ⎧ e ⎪⎨ ϕG,σ(k, r) = ⎪⎩ i(G+k)r χ g σ Int. � A G σ (k�)u G� σ (k�,z)+B G σ (k�)˙u G � � σ (k�,z) e i(G�+k�)r� g χσ Vac. � � � A μG Lσσα(k)uσα l (r)+B μG Lσσα(k)˙uσα � l (r) YL(ˆr) χσα MTμ σ α L (3.41) where, u σα l (r) is the solution of the radial Schrödinger equation in the “local“ spin-frame σ α . The sum in the muffin-tins is over the local spin directions and L abbreviates lm. The
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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 and 40: 26 2 The FLAPW method of the operat
Page 41 and 42: 28 2 The FLAPW method a matrix expr
Page 43 and 44: 30 3 Magnetism of low dimensional s
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92 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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