Calculation of Solid Properties Using WIEN2k Package

1
Calculation of Solid Properties Using WIEN2k
Package
2
Introduction

• Our main purpose is to calculate some solid
  properties such as lattice constants, band
  structure, optical properties, hyperfine fields,
  .
• Since all the properties depend on the total
  electronic wave function or the total
  electronic density , we need to calculate
  one of them.

3

• This can be done by two ways
• Using Sch. Equation !!!
• Using Density Functional Theory.
• Density Functional Theory is the most common
  used method in electronic structure calculation.
• The Wien2k code is based on Khon-Sham formalism
  of the density functional theory using
  Full-Potenatial Linearized Augmented Plane Wave
  (FP-LAPW) method.

4
DFT

• The main idea behind the DFT is that the energy
  is a functional of the electron density
  .
• The usual quantum mechanical approach to SE can
  be summarized as following
• While in the DFT approach the observables are
  obtained directly from the density as following

5
The Hohenberg-Kohn Theorem

• The complete many-body wavefunction ?, of an
  electronic system is a unique functional ?n(r)
  of the electronic charge density n(r). As a
  consequence, the expectation value of any
  observable is also a functional of n(r).
• The correct ground state density is obtained by
  minimizing the energy functional En(r) under
  the constraint that the integration of the
  density over all the volume gives the number of
  electrons.

6

• The energy functional can be written as
• Kinetic energy T n and electron-electron
  interaction U n are system independent.
• Hohenberg and Kohn defined a universal
  functional

7
Kohn-Sham Formalism

• The Kohn-Sham formalism of DFT provides a way to
  exactly transform the many-body problem into a
  single-body one.
• Kohn and Sham have introduced (1965) the
  following separation of the functional F n
• Using Variational Principle we get

8

• Now there are two questions
• How can we solve the KS equations?
• How can we find Excn or consequently Vxcn?
• Starting with the second question, the answer is
  we need approximations!

9
LDA

• If we have a system with non-uniform electron
  distribution where the density is slowly varying,
  one can divide the space into portions, where the
  density in each portion is almost constant.
• Each portion is a uniform electron gas.
• So if we assume that the exchange energy per
  particle in the uniform electron gas is
  then
• The corresponding exchange-correlation potential
  is

10

• Usually, is splitted into two parts
• And the correlation part can be calculated by
  Quantum Monte Carlo method.

11

• So in terms of the parameter rs we can express
  the two parts of the exchange-correlation energy
  per unit electron in a homogeneous system as
  following

12
Disadvantages of LDA

• It is a good approximation for systems with
  slowly varying density.
• It tends to overbind, which cause for example
  small lattice constants.
• It does not into account the polarization of the
  electrons.

13
LSDA

• This is a very important approximation because in
  this approximation we take into account the spin
  of the electrons and so the existence of
  polarized electrons will affect the density.
• In this case, the density of the electrons will
  be divided into two parts na, nß.
• So that

14

• We must note that the LSDA will reduce to the LDA
  in the case of full occupied orbitals, but in the
  case of half-empty orbitals the two
  approximations will be different.
• The following represents the ionization
  potentials in electron volts of some atoms.

15
GGA

• Exc is calculated just as in the LDA method, but
  enhancement is introduced by multiplying
  with a factor Fxc which takes into account the
  local variation of the density n.

16
EEX

• The exchange part is calculated exactly, and the
  correlation part still approximated using LDA or
  EEG
• Where fi are the KS orbitals which give the
  charge density distribution.

17

• These are some of the approximations that can be
  used to calculate the exchange-correlation energy
  and its functional dependence on the density of
  the system.
• Now let us consider the first question
• How can we solve the KS equations?
• There are many methods to do such solution.
• We are interested in one of these methods
  which is FP-LAPW

18
FP-LAPW

• The linearized augmented plane wave (LAPW) method
  is among the most accurate methods for performing
  electronic structure calculations for crystals.
• It is based on the density functional theory for
  the treatment of exchange and correlation and
  uses e.g. the local spin density approximation
  (LSDA).
• The LAPW method is a procedure for solving the
  Kohn-Sham equations for the ground state density,
  total energy, and (Kohn-Sham) eigenvalues of a
  many-electron system (here a crystal) by
  introducing a basis set which is especially
  adapted to the problem.

19

• Let us first start with the APW method.
• Dividing the unit cell into
• (I) non-overlapping atomic spheres (Muffin-tin
  spheres, MT) (centered at the atomic sites) and
• (II) an interstitial region (IR).
• This allows an accurate description of both, the
  rapidly changing (oscillating) wavefunctions,
  potential and electron density close to the
  nuclei as well as the smoother part of these
  quantities in between the atoms.
• In the two types of regions different basis sets
  are used

20
Figure 1 Partitioning of the unit cell into
atomic spheres (I) and an interstitial region
(II)
21

• (I) inside atomic sphere t of radius Rt a linear
  combination of radial functions times spherical
  harmonics Ylm(r) is used
• where ul(r,El) is the (at the origin) regular
  solution of the radial Schroedinger equation for
  energy El and the spherical part of the potential
  inside sphere t.
• is the energy derivative of ul with
  respect to E taken at the same energy El.
• A linear combination of these two functions
  constitute the linearization of the radial
  function the coefficients Alm and Blm are
  functions of kn .
• These coefficients can be determined by requiring
  that this basis function matches (in value and
  slope) the corresponding basis function of the
  interstitial region.

22

• original APW formulation the plane-waves are
  augmented to the exact solutions of the
  Schrödinger equation within the MT at the
  calculated eigenvalues.
• Problems
• The basis set have explicit energy dependence.
• Linearity problem.
• Highly computational effort.
• Solution
• The linearized APW method the energy dependence
  of the basis set is removed by using a fixed set
  of suitable MT radial functions.

23

• (II) In the interstitial region a plane wave
  expansion is used
• where knkKn Kn are the reciprocal lattice
  vectors and k is the wave vector inside the first
  Brillouin zone. Each plane wave is augmented by
  an atomic-like function in every atomic sphere.
  

24

• The solutions to the Kohn-Sham equations are
  expanded in this combined basis set of LAPW's
  according to the linear variation method
• The coefficients cn are determined by the
  Rayleigh-Ritz variational principle.
• The convergence of this basis set is controlled
  by a cutoff parameter RmtKmax 6 - 9, where Rmt
  is the smallest atomic sphere radius in the unit
  cell and Kmax is the magnitude of the largest K
  vector.

25

• In its general form the LAPW method expands the
  potential in the following form
• And the charge densities analogously.
• Thus no shape approximations are made, a
  procedure frequently called the full-potential
  LAPW (FLAPW) method.

26
The structure of the WIEN2k-code
27

• The SCF cycle of the WIEN code consist of five
  independent programs
• 1. LAPW0 generates the potential from a given
  charge density
• 2. LAPW1 computes the eigenvalues and
  eigenvectors
• 3. LAPW2 computes the valence charge density
  from the eigenvectors
• 4. CORE computes the core states and densities
• 5. MIXER mixes the densities generated by LAPW2
  and CORE with the density of the previous
  iteration to generate a new charge density.
• From these programs LAPW1 and LAPW2 are the most
  time consuming, while the time needed to run CORE
  and MIXER are basically negligible.

28
(No Transcript)
29
General Remarks on WIEN2k

• WIEN2k consists of many independent F90 programs,
  which
• are linked together via C-shell scripts.
• Each case runs in his own directory
  ./case
• The master input is called
  case.struct
• Initialize a calculation
  init_lapw
• Run scf-cycle
  run_lapw (runsp_lapw)
• You can run WIEN2k using any www-browser and
  the w2web
• interface, but also at the command line of
  an xterm.
• Input/output/scf files have endings as the
  corresponding
• programs
• case.output1lapw1 case.in2lapw2
  case.scf0lapw0
• Inputs are generated using STRUCTGEN(w2web) and
• init_lapw

30
w2web The web-interface of WIEN2k

• Based on www
• WIEN2k can be managed
  remotely via w2web
• Important steps
• start w2web on all your hosts
• login to the desired
  host (ssh)
• w2web (at first startup
  you will be
  asked for
  username/password,
  
  port-number,
  
  (master-)hostname.
  
  creates /.w2web directory)
• use your browser and connect to the (master)
  hostportnumber
• mozilla http//fp98.zserv10000
• create a new session on the desired host (or
  select an old one)

31

• Structure generator
• Spacegroup selection
• import cif file
• step by step initialization
• symmetry detection
• automatic input generation
• SCF calculations
• Magnetism (spin-polarization)
• Spin-orbit coupling
• Forces (automatic geometry
  optimization)
• Guided Tasks
• Energy band structure
• DOS
• Electron density
• X-ray spectra
• Optics

32
K mesh generation

• x kgen (generates k-mesh and reduces to
  irreducible wedge using symmetry)
• always add inversion except in magnetic
  spin-orbit calculations
• time inversion holds and E(k) E(-k)
• always shift the mesh for scf-cycle
• gaps often at G ! (might not be in your mesh)
• small unit cells and metals require large k-mesh
  (1000-100000)
• large unit cells and insulators need only 1-10
  k-points
• mesh is good if nothing changes and scf
  terminates after few (3) iterations
• use an even finer meshes for DOS, spectra,
  optics,

33
Properties with WIEN2k

• Energy bands
• classification of irreducible representations
• character-plot (emphasize a certain
  band-character)
• Density of states
• including partial DOS with l and m-character (eg.
  px, py, pz)
• Electron density, potential
• total-, valence-, difference-, spin-densities, ?
  of selected states.
• 1-D, 2Dand 3D-plots (Xcrysden).
• X-ray structure factors
• Hyperfine parameters
• hyperfine fields (contact dipolar orbital
  contribution)
• Isomer shift
• Electric field gradients

34

• Total energy and forces
• optimization of internal coordinates, (MD,
  BROYDEN)
• cell parameter only via Etot (no stress tensor)
• elastic constants for cubic cells
• Phonons via supercells
• Spectroscopy
• core level shifts
• X-ray emission, absorption, electron-energy-loss
  (with core holes)
• optical properties (dielectric function in RPA
  approximation, JDOS including momentum matrix
  elements and Kramers-Kronig)
• Fermi surface (2D, 3D)

35
WIEN2k- hardware/software

• WIEN2k runs on any Unix/Linux platform from PCs,
  workstations, clusters to supercomputers
• Pentium-IV or higher with fast memory bus (1-2
  Gb memory, 100Mbit net).
• Fortran90 (dynamical allocation, modules)
• web-based GUI w2web (perl)
• f90 compiler, BLAS-library (ifort9mkl), perl5,
  ghostscript(jpg), gnuplot(png), Tcl/Tk
  (Xcrysden), pdf-reader,www-browser, octave,
  opendx

36
(No Transcript)
37
(No Transcript)
38
Examples Using WIEN2k Code

• Structural Properties

39

• Magnetic Properties

40

• Hyperfine fields

41
(No Transcript)
42
(No Transcript)
43

• Thank You