# Quelle:Ym/Kurz 2000

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 Autor Philipp Kurz Titel Non-collinear magnetism at surfaces and in ultrathin films Ort Aachen Jahr 2000 Anmerkung Zugl.: Aachen, Techn. Hochsch., Diss., 2000 URL http://www.fz-juelich.de/SharedDocs/Downloads/PGI/PGI-1/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 non-zero 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 Hohenberg-Kohn 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

$E[n(\mathbf{r}),\mathbf{m}(\mathbf{r})]\geq E[n_0(\mathbf{r}),\mathbf{m}_0(\mathbf{r})]$. (2.16)

An alternative, but completely equivalent, formulation can be obtained using a four component density matrix ραβ instead of n(r) and m(r) [117].

[117] U. von Barth and L. Hedin. A local exchange-correlation 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 non-zero 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 Hohenberg-Kohn 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

$E[n(\mathbf{r}),\mathbf{m}(\mathbf{r})]\geq E[n_0(\mathbf{r}),\mathbf{m}_0(\mathbf{r})]$. (2.15)

An alternative, but completely equivalent, formulation can be obtained using a four component density matrix ραβ instead of n(r) and m(r) [vBH72, Küb95].

[vBH72] U. von Barth and L. Hedin. A local exchange-correlation potential for the spin polarized case: I. J. Phys. C., 5:1629, 1972.

[Küb95] J. Kübler. Derivation of the single-particle schrödinger equation: Density and spin-density 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 Kohn-Sham 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 Kohn-Sham equations, which now have the form of Schrödinger-Pauli 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 B-field, 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 z-axis 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 spin-up 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 Kohn-Sham 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 Kohn-Sham equations, which now have the form of Schrödinger-Pauli 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 B-field, 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 z-axis 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 spin-up 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 many-electron system exactly, if the exchange correlation energy Exc was known. Unfortunately, no explicit representation of this functional, that contains all many-body 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 many-electron system exactly, if the exchange correlation energy Exc was known. Unfortunately, no explicit representation of this functional, that contains all many-body 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 so-called muffin-tins, and the remaining interstitial region (cf. Fig. 3.1). Inside the muffin-tins 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 shape-approximations. Noting that plane-waves 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 muffin-tin spheres. Hence, the APW’s form a set of continuous basis functions that cover all space. Each basis function consists of a plane-wave 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 muffin-tin 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 muffin-tin 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 plane-waves 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 so-called muffin-tins, and the remaining interstitial region (cf. Fig. 3.1). Inside the muffin-tins 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 shape-approximations. Noting that plane-waves 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 muffin-tin 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 muffin-tin 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 muffin-tin 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].

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