Angaben zur Quelle [Bearbeiten]
Autor  Philipp Kurz 
Titel  Noncollinear magnetism at surfaces and in ultrathin films 
Ort  Aachen 
Jahr  2000 
Anmerkung  Zugl.: Aachen, Techn. Hochsch., Diss., 2000 
URL  http://www.fzjuelich.de/SharedDocs/Downloads/PGI/PGI1/EN/Kurz.DrA_pdf.pdf 
Literaturverz.  ja 
Fußnoten  ja 
Fragmente  4 
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2.2 Spin Density Functional Theory
In order to describe magnetic effects the density functional theory has to be extended to the case of spin polarized electrons. This is important for systems that posses nonzero ground state magnetization, which is the case for most atoms, magnetic solids and surfaces and electronic systems exposed to an external magnetic field. The necessary extension to the HohenbergKohn theory can be formulated replacing the electron density by the electron density plus the magnetization density as fundamental variables. In this case, the variational principle becomes . (2.16) An alternative, but completely equivalent, formulation can be obtained using a four component density matrix ρ_{αβ} instead of n(r) and m(r) [117]. [117] U. von Barth and L. Hedin. A local exchangecorrelation potential for the spin polarized case: I. J. Phys. C, 5:1629, 1972.  2.3 Spin Density Functional Theory
In order to describe magnetic effects the density functional theory has to be extended to the case of spin polarized electrons. This is important for systems that posses nonzero ground state magnetization, which is the case for most atoms, magnetic solids and surfaces and electronic systems exposed to an external magnetic field. The necessary extension to the HohenbergKohn theory can be formulated replacing the electron density by the electron density plus the magnetization density as fundamental variables. In this case, the variational principle becomes . (2.15) An alternative, but completely equivalent, formulation can be obtained using a four component density matrix ρ_{αβ} instead of n(r) and m(r) [vBH72, Küb95]. [vBH72] U. von Barth and L. Hedin. A local exchangecorrelation potential for the spin polarized case: I. J. Phys. C., 5:1629, 1972. [Küb95] J. Kübler. Derivation of the singleparticle schrödinger equation: Density and spindensity functional theory and the magnetic susceptibility, noncollinear ground States, towards the curie temperature. 1995. Lecture Notes from: Workshop on Condensed Matter Physics. 


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[In order to gain a generalised] form of the KohnSham equations, it is necessary at least to introduce two component Pauli wavefunctions, that reproduce the electron and magnetization density.
[...] Applying the variational principle again yields the KohnSham equations, which now have the form of SchrödingerPauli equations. [...] The additional effective magnetic field consists of two terms. One of them is due to the variation of the exchange correlation energy with respect to the magnetization density. The second term is the external Bfield, if present. [...] In many applications, like for example ferromagnetic and antiferromagnetic solids, the magnetization is orientated along one particular direction. For these collinear cases the problem can be simplified further. The zaxis can be chosen along the direction of the magnetic field. Therefore, the Hamiltonian of equation (2.19) becomes diagonal in the two spin components of the wavefunction, i.e. the spinup and down problems become completely decoupled and can be solved independently. The energy and all other physical observables become functionals of the electron density and the magnitude of the magnetization density m(r) =  In order to gain a generalised form of the KohnSham equations, it is necessary at least to introduce two component Pauli wavefunctions, that reproduce the electron and magnetization density.
[...] Applying the variational principle again yields the KohnSham equations, which now have the form of SchrödingerPauli equations. [...] The additional effective magnetic field consists of two terms. One of them is due to the variation of the exchange correlation energy with respect to the magnetization [page 13] density. The second term is the external Bfield, if present. [...] In many applications, like for example ferromagnetic and antiferromagnetic solids, the magnetization is orientated along one particular direction. For these collinear cases the problem can be simplified further. The zaxis can be chosen along the direction of the magnetic field. Therefore, the Hamiltonian of equation 2.18 becomes diagonal in the two spin components of the wavefunction, i.e. the spinup and down problems become completely decoupled and can be solved independently. The energy and all other physical observables become functionals of the electron density and the magnitude of the magnetization density m(r) = 


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So far, no approximations have been made. The density functional formalism, outlined in the previous sections, could in principle reproduce all ground state properties of any complex manyelectron system exactly, if the exchange correlation energy Exc was known. Unfortunately, no explicit representation of this functional, that contains all manybody effects, has been found yet. Thus, approximations to Exc have to be used. The most widely used and very successful approximation is the local spin density approximation (LSDA). The underlying idea is very simple. At each point of space Exc is approximated locally by the exchange correlation energy of a homogeneous electron gas with the same electron and magnetization density. Hence, the approximate functional Exc is of the form [...] It is important to note, that εxc is not a functional, but a function of n(r) and  So far, no approximations have been made. The density functional formalism, outlined in the previous sections, could in principle reproduce all ground state properties of any complex manyelectron system exactly, if the exchange correlation energy Exc was known. Unfortunately, no explicit representation of this functional, that contains all manybody effects, has been found yet. Thus, approximations to Exc have to be used. The most widely used and very successful approximation is the local spin density approximation (LSDA). The underlying idea is very simple. At each point of space Exc is approximated locally by the exchange correlation energy of a homogeneous electron gas with the same electron and magnetization density. Hence, the approximate functional Exc is of the form [...] It is important to note, that εxc is not a functional, but a function of n(r) and 


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3.1.1 The APW Method
Within the APW approach, space is divided into spheres centered at each atom site, the socalled muffintins, and the remaining interstitial region (cf. Fig. 3.1). Inside the muffintins the potential is approximated by a spherically symmetric shape, and in many implementations the interstitial potential is set to a constant. The restrictions to the potential are commonly called shapeapproximations. Noting that planewaves solve the Schro ̈dinger equation in a constant potential, while spherical harmonics times radial functions are the solutions in a spherical potential, suggests to expand the single particle wavefunctions in terms of the following basis functions: [...] Where k is the Bloch vector, Ω is the cell volume, G is a reciprocal lattice vector, L abbreviates the quantum numbers l and m and ul is the regular solution of the radial Schro ̈dinger equation [...] V (r) is the spherical component of the potential. The coefficients AμG(k) are determined from the requirement, that the wavefunctions have to be continuous at the boundary of the muffintin spheres. Hence, the APW’s form a set of continuous basis functions that cover all space. Each basis function consists of a planewave in the interstitial region plus a sum of functions, which are solutions of the Schro ̈dinger equation for a given set of angular momentum quantum numbers (lm) and a given parameter El inside the muffintin spheres. If the El’s were fixed, used only as a parameter during the construction of the basis, the Hamiltonian could be set up in terms of this basis. This would lead to a standard secular equation for the band energies. Unfortunately, it turns out, that the APW basis does not offer enough variational freedom if the El’s are kept fixed. An accurate description can only be achieved if they are set to the corresponding band energies. However, requiring the El’s to be equal to the band energies, the latter can no longer be determined by a simple diagonalization of the Hamiltonian matrix. Since the ul’s depend on the band energies, the solution of the secular equation becomes a nonlinear problem, which is computationally much more demanding than a secular problem. Another disadvantage of the APW method is, that it is difficult to extend beyond the spherically averaged muffintin potential approximation, because in the case of a general potential the optimal choice of El is no longer the band energy. And finally, but less serious, if, for a given choice of El’s, the radial functions ul vanish at the muffin tin radius, the boundary conditions on the spheres cannot be satisfied, i.e. the planewaves and the radial functions become decoupled. This is called the asymptote problem. It can already cause numerical difficulties if ul becomes very small at the sphere boundary. Further information about the APW method can be found in the book by Loucks [60], which also reprints several early papers including Slater’s original publication [100].  3.1.1 The APW Method
Within the APW approach, space is divided into spheres centered at each atom site, the socalled muffintins, and the remaining interstitial region (cf. Fig. 3.1). Inside the muffintins the potential is approximated to be spherically symmetric, and in many implementations the interstitial potential is set constant. The restrictions to the potential are commonly called shapeapproximations. Noting that planewaves solve the Schro ̈dinger equation in a constant potential, while spherical harmonics times a radial function are the solution in a spherical potential, suggests to expand the single particle wavefunctions in terms of the following basis functions: [...] Where k is the Bloch vector, Ω is the cell volume, G is a reciprocal lattice vector, L abbreviates the quantum numbers l and m and ul is the regular solution of the radial Schro ̈dinger equation [...] V (r) is the spherical component of the potential. The coefficients AμG(k) are determined from the requirement, that the wavefunctions have to be continuous at the boundary of the muffintin spheres. Hence, the APW’s form a set of continuous basis functions that cover all space. Where each function consists of a planewave in the interstitial region plus a sum of functions, which are solutions of the Schro ̈dinger equation to a given set of angular momentum quantum numbers lm and a given parameter El inside the muffintin spheres. If the El were kept fixed, used only as a parameter during the construction of the basis, the hamiltonian could be set up in terms of this basis. This would lead to a standard secular equation for the band energies. Unfortunately, it turns out, that the APW basis does not offer enough variational freedom if the El are kept fixed. An accurate description can only be achieved if they are set to the corresponding band energies. However, requiring the El’s to be equal to the band energies, the latter can no longer be determined by a simple diagonalization of the Hamiltonian matrix. Since the ul’s depend on the band energies, the solution of the secular equation becomes a nonlinear problem, which is computationally much more demanding than a secular problem. Another disadvantage of the APW method is, that it is difficult to extend beyond the spherically averaged muffintin potential approximation, because in the case of a general potential the optimal choice of El is no longer the band energy. And finally, but less serious, if, for a given choice of El, the radial functions ul vanish at the muffin tin radius, the boundary conditions on the spheres cannot be satisfied, i.e. the planewaves and the radial functions become decoupled. This is called the asymptote problem. It can already cause numerical difficulties if ul becomes very small at the sphere boundary. Further information about the APW method can be found in the book by Loucks [Lou67], which also reprints several early papers including Slater’s original publication [Sla37]. 

