SPRKKR
A spin polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) code for Calculating Solid State Properties
Hubert Ebert
User’s Guide
January 15, 2010
Universitat¨ Munchen¨
Dept. Chemie – Phys. Chemie
Butenandtstr. 5-13
D-81377 Munchen¨
c 2005 by H. Ebert, Munchen¨
H. Ebert
SPRKKR
A spin polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) code for Calculating Solid State Properties
version 5.4
Prof. Dr. Hubert Ebert
Universit¨at Munchen¨
Dept. Chemie – Phys. Chemie
Butenandtstr. 5-13
D-81377 Munchen¨
Germany
Contents
1 Introduction |
1 |
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1.1 |
The SPRKKR-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
1 |
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1.2 |
Formal Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
2 |
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1.3 |
Short Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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1.3.1 |
The graphical user interface xband . . . . . . . . . . . . . . . . . . . . |
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1.3.2 |
Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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1.3.3 |
Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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1.3.4 |
The post-processing program plot . . . . . . . . . . . . . . . . . . . . . |
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2 The SPRKKR input file |
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2.1 |
Input Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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2.2 |
General Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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2.2.1 |
File names and output control . . . . . . . . . . . . . . . . . . . . . . . |
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2.2.2 Angular momentum expansion . . . . . . . . . . . . . . . . . . . . . . . |
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2.2.3 Calculating the scattering path operator . . . . . . . . . . . . . . . . . . |
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2.2.4 |
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k-dependent KKR structure constants . . . . . . . . . . . . . . . . . . . |
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2.2.5CPA-calculations for disordered alloy systems . . . . . . . . . . . . . . 12
2.2.6Calculation mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
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2.2.7 |
Orientation of the magnetisation . . . . . . . . . . . . . . . . . . . . . . |
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2.2.8 |
Manipulating the spin-orbit coupling . . . . . . . . . . . . . . . . . . . |
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3 Electronic structure calculations |
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3.1 |
Creating self-consistent potentials . . . . . . . . . . . . . . . . . . . . . . . . . |
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3.2 |
Phase shift (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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3.3 |
Plotting of wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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3.4 |
Spin-orbit parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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3.5 |
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Dispersion relation E(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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3.6 |
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Bloch spectral function AB(E; k) . . . . . . . . . . . . . . . . . . . . . . . . . . |
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3.7 |
Density of States n(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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4 Calculating ground state properties |
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4.1 |
Magnetic Moments and Hyperfine Fields . . . . . . . . . . . . . . . . . . . . . |
40 |
4.2 |
Magnetic Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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4.3 |
Exchange coupling parameters Jij . . . . . . . . . . . . . . . . . . . . . . . . . |
43 |
5 Calculating spectroscopic properties |
48 |
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5.1 |
General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
48 |
5.2 |
X-ray absorption spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . |
48 |
5.3X-ray emission spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4X-ray magneto-optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.5Non-relativistic Appearance Potential Spectroscopy . . . . . . . . . . . . . . . 55
5.6Non-relativistic Auger Electron Spectroscopy . . . . . . . . . . . . . . . . . . . 58
5.7 Relativistic Auger Electron Spectroscopy . . . . . . . . . . . . . . . . . . . . . 59
5.8Valence Band X-ray Photoemission Spectroscopy . . . . . . . . . . . . . . . . . 62
5.9 |
Core Level X-ray Photoemission spectroscopy . . . . . . . . . . . . . . . . . . |
64 |
5.10 |
Magnetic Compton profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
66 |
5.11 |
Positron annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
68 |
5.11.1Calculation of the positron energy . . . . . . . . . . . . . . . . . . . . . 69
5.11.2Preparation of the wave-function and scattering path operator for
the positron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.11.3Calculation of the 2D-projection of the electron-positron momentum density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6 The post-processing program plot |
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7 Examples |
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8 Technical details, installation and dimensioning |
79 |
8.1 Technical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
79 |
8.2Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.2.1Mathematical libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.2.2Unpacking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.2.3 Preparing the make file . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.2.4 Compiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.3Dimensioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
License agreement for the use of the program package SPRKKR
Prof. Dr. Hubert Ebert
Universitat¨ Munchen¨
Dept. Chemie – Phys. Chemie
Butenandtstr. 5-13
D-81377 Munchen¨
Germany
Fax: +49-89-2180-77584
Email: Hubert.Ebert@cup.uni-muenchen.de
DEFINITIONS:
In the following, the term “the authors”, refers to H. Ebert at the above address. “Program” shall mean the copyrighted spin polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) code (in source and object form) comprising the computer programs known as the Munich SPRKKR-package. The additional graphical user interface xband including A. Perlov’s symmetry program findsym falls under the standard GNU license conditions.
MANDATORY TERMS AND CONDITIONS:
I will adhere to the following conditions upon receipt of the program:
1.All title, ownership and rights to the program or to copies of it remain with the authors, irrespective of the ownership of the media on which the program resides.
2.I will not supply a copy of the code to anyone for any reason whatsoever. This in no way limits my making copies of the code for backup purposes, or for running on more than one computer system at my institution. I will refer any request for copies of the program to the authors.
3.When leaving my present institution, I will remove all copies of SPRKKR from its computer systems unless another user has signed and submitted this license form and use has been approved by the authors.
4.I will not incorporate any part of SPRKKR into any other program system, without prior written permission of the authors.
5.I will keep intact all copyright notices.
6.I understand that the authors supply SPRKKR and xband and its documentation on an “as is” basis without any warranty, and thus with no additional responsibility or liability. I agree to report any difficulties encountered in the use of SPRKKR or xband to the authors.
7.In any publication in the scientific literature based on an application of SPRKKR I will reference the program package as follows:
The Munich SPR-KKR package, version 5.4, H.Ebert et al, http://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR; H Ebert, Fully relativistic band structure calculations for magnetic solids – Formalism and Application, in Electronic Structure and Physical Properties of Solids, editor: H. Dreysse,´ Lecture Notes in Physics, vol. 535, p. 191, Springer Berlin.
One copy of each and every such publication shall be filed in timely fashion with the author, H. Ebert.
8.It is understood that modifications of the SPRKKR or the xband code can lead to problems where the authors may not be able to help. It would be appreciated if useful modifications or major extensions are reported to the authors.
9.I agree that modifications or extensions of the program do not change the copyright; i.e. it remains with the author.
10.I understand that support for running the program can not be provided in general, except on the basis of a joint project between the authors and the research partner.
11.I accept these conditions also for future versions of SPRKKR provided by the authors and will use the updated version number when making reference to SPRKKR (see 7.).
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1 Introduction
1.1The SPRKKR-package
This manual describes the use of the spin-polarized relativistic KKR band structure program package SPRKKR. Development of this package started during a post-doc stay of H. Ebert with B. L. Gyorffy in 1986/1987 and was continued later with contributions by M. Battocletti, D. Benea, M. Kosuth, J. Minar, A. Perlov, V. Popescu and others. Several subroutines used by SPRKKR have been kindly supplied by H. Akai, S. Blugel,¨ J. J. Rehr, W. M. Temmerman, and P. Strange. P. Blaha kindly made the LATEX-files of the WIEN97 manual available, that have been used as a template for this manual.
The SPRKKR-package allows to calculate the electronic structure of arbitrary three-dimensio- nal periodic systems, including in particular systems with chemical disorder. The treatment of two dimensional periodic systems (e.g. surfaces) can be done at the moment only by using an auxiliary system having three dimensional periodicity or by making use of the cluster approximation. The electronic structure calculation can be done in a non-relativistic, scalarrelativistic as well as fully relativistic mode. In the scalar-relativistic mode paramagnetic as well as spin-polarized systems can be treated, including non-collinear spin structures and arbitrary spin spirals. In the fully relativistic mode, paramagnetic as well as spin-polarized systems with an arbitrary spin configuration can be dealt with. On the basis of the electronic structure calculation many different properties can be investigated by means of the SPRKKRpackage, with a strong emphasize on response functions and spectroscopic properties. This applies in particular for the fully relativistic mode.
At the moment the SPRKKR-package is restricted to the atomic sphere approximation (ASA) geometry for the charge density and potential. This restriction will be removed in the near future allowing in addition full potential type calculations.
The SPRKKR-package is based on the KKR-Green’s function formalism that makes use of multiple scattering theory. This implies that the information on the electronic structure of a system is not expressed in terms of Bloch wave functions and eigenvalues but the corresponding Green’s function, leading to an extreme flexibility of the method. An introduction to band structure theory, including the KKR method, can be found for example in a review article by S. Faulkner [1]. A more detailed presentation of the KKR method is given in the monographies of P. Weinberger [2], A. Gonis and W. Butler [3, 4], that also deal with the
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CHAPTER 1. INTRODUCTION |
spin-polarized relativistic case. The KKR-formalism for this case will be sketched in short in the next section. More information and many applications can be found in two review articles [5, ?].
The next chapter describes the use of the SPRKKR-package for a number of standard applications. This is followed by a description of the use of the plot that allows to reformat and modify calculated data for display and analysis. Technical information and a guideline for installation of the SPRKKR-package can be found in chapter 8. Finally some references connected with applications of the SPRKKR-package are collected at the end. For a more complete list of references see [5] and [6].
1.2Formal Background
A short description of the spin-polarized relativistic Korringa-Kohn-Rostoker Green’s function method that supplies the formal background for the SPRKKR-package is given in the following. Throughout the conventions introduced in the books of M. E. Rose will be used [7, 8].
The spin-polarized relativistic Korringa-Kohn-Rostoker Green’s function method is based on the following Dirac-Hamiltonian for a spin-polarized system
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c~ r~ + mc2 + Ve (~r) + ~ B~e (~r)i i(~r) = i i(~r) |
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with |
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Be (~r) = Bext(~r) + |
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(1.1)
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The spin dependent potential is constructed within a relativistic version of spin density
functional theory. With the potentials functions and ~ available, the first step
Ve (~r) Be (~r)
is to solve the corresponding single site Dirac equation for every atom type. In principle,
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the effective magnetic field Be (~r) may point in any direction (non-collinear magnetism). |
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However, calculations are simplified for Be (~r) = Be (r)~ez. Therefore the single site Dirac
equation is solved in a local frame of reference for which this condition is fulfilled. For this purpose the ansatz
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is used with the partial waves chosen to have the same form as the linearly independent solutions for a spherical symmetric potential:
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if (r; E) (~r^) ! |
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g (r; E) (~r) |
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This leads to the following set of coupled radial differential equations:
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V ] P + B |
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In practice the coupling is restricted to 0 = 0 and l l0 = 0 with two functions P and P 1 coupled for j j < j and no coupling present otherwise.
Next, the single site t-matrix t 0 is calculated and transformed from the local to the global frame of reference (if necessary). With all t-matrices available, the multiple scattering problem is solved either by performing a BZ-integration
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(E) = BZ |
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d3k[t 1(E) G(~k; E)] 10 eik(R~n R~n0 ) |
(1.7) |
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or using the real space cluster approach. This means that is obtained by inverting the real space KKR-matrix:
= [m G] 1 :
the scattering path operator nn00 (E)
(1.8)
If there is substitutional disorder present in the system, this is accounted for by making use of the Coherent Potential Approximation (CPA) alloy theory. This implies that the CPA equations
xA nn;A + xB nn;B |
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nn;CPA 1 + t 1 tCPA1 nn;CPA |
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(1.10) |
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have to be solved iteratively.
After the multiple scattering problem has been solved, the resulting scattering path operator has to be transformed back to the local frame of reference in case of a non-collinear spin configuration. Finally, the Green’s function is given by:
G(~r; ~r 0; E) = |
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(E)Zn0 (~r 0; E) |
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X[ Zn(~r; E)Jn (~r 0; E) (r0 r) |
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+Jn(~r; E)Zn (~r 0; E) (r r0) ] nn0 : |
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3