CPA Enhancement of the Green's function package

Preliminaries

lmgf contains an implementation of the Coherent Potential Approximation (CPA) and Disordered Local Moments (DLM) theory. It was implemented by Kirill Belashchenko and questions should be directed to him (belashchenko@unl.edu).

The CPA implementation for substitutional alloys and for spin disorder follows the formulation explained in these articles:

Particularly, see the description of the numerical implementation in Turek’s book.

CPA self-consistency is based on iterating the coherent interactor Ω, which is a spin-dependent single-site matrix defined for each CPA site at each complex energy point. The linear mixing of Ω can be interleaved with charge mixing steps. However, experience shows that much faster convergence can be achieved by iterating Ω at each z-point until its misfit reaches a sufficiently low tolerance (say, 1d-3), between charge mixing steps. In addition, it is better to skip charge mixing if sufficiently accurate charge-neutrality has not been achieved (the reason being that Ω is not Pade-adjusted). There are a few parameters controlling Ω convergence, which are summarized below along with the recommended settings that work quite well in most cases. The Ω matrices are recorded in files omegaN.ext, where N is the number of the CPA site. A human-readable version (with fewer decimal digits) is recorded in om-hrN.ext.

CPA-specific input

To turn on chemical and/or magnetic CPA, additons are required to the SPEC and GF categories in the ctrl file.

SPEC category

Chemical Disorder. Additional species must be defined for chemical CPA, and their concentrations.

SPEC ATOM CPA= and C= together turn on chemical CPA for a particular species.


They specify which species are to be alloyed with this species, and the concentrations of the other species. For example,

SPEC ATOM=Fe ... CPA=1 4 5 C=0.5 0.3 0.2


specifies that species Fe (whenever it appears in the basis (defined in SITE category) in fact refers to a disordered site composed of three kinds of elements. Numbers following CPA= refer to indices in the SPEC category: thus “CPA=1 4 5” indicate that the three elements to be identified with sites referring to this species are the 1st, 4th, and 5th species declared in the SPEC category. C= indicates the concentrations of each species; the concentrations must sum to 1. In the example given, sites with species label Fe are composite elements with with 50% of species 1, 30% of species 4 and 20% of species 5 (up to 10 species may be given).

A CPA species may refer to itself. For example, if the Fe species above is the first species to be read from the ctrl file, then CPA=1 refers to itself. All other parameters like Z, R, will be taken from this species.

Spin Disorder. No additional species are required, but the number of orientations must be specified.

SPEC ATOM NTHET= turns on spin disorder for a particular atom type.


A species with non-zero NTHET can be listed as a CPA component, and it will be included as NTHET components with different directions of the local moment.

NTHET=2 specifies that there will be two CPA-DLM components with polar angles 0 and π. NTHET=N with $N>2$ specifies a vector-DLM model, for which N polar angles for the local moment direction are selected using the Gaussian quadrature for the sphere. (Axial symmetry is always assumed and the integral over the azimuthal angle is taken analytically.)

Combined Chemical and Spin Disorder. Either spin or chemical disorder may be specified; they may also be included simultaneously. If only CPA= is chosen, that species will be treated with chemical, not spin, disorder. If only NTHET= is chosen, that species be treated with spin disorder only. Specifying both means that the CPA will include both chemical and spin disorder. For example, in the above example for CPA, if SPEC ATOM=Fe includes a tag NTHET=2 (while species 4 and 5 have NTHET=0), species Fe describes a CPA site with 4 components: 25% Fe, 25% Fe with a reversed local moment, 30% species 4 and 20% species 5.

GF category

The following token turns on the CPA and/or DLM:

GF DLM= controls what is being calculated.


At present, these values are supported:

DLM=12: normal CPA calculation; both charges and Ω's are iterated
DLM=32: no charge self-consistency; only CPA it iterated until Ω reaches
prescribed tolerance for each z-point.
DLM=112: special-purpose experimental branch (not documented)


The following are optional inputs:

GF BXY=1 turns on the self-consistent determination of the
constraining fields for vector DLM calculations.

GF TEMP= supplies the spin temperature (not implemented yet)


Self-consistency in Ω is controlled by the following tags supplied in GF GFOPTS:

lotf    if present, Ω is iterated at each z-point until converged to omgtol (recommended)
nitmax= maximum number of Ω iterations (30 is usually sufficient)
omgmix= linear mixing parameter for Ω (0.4 works well in most cases)
omgtol= tolerance for Ω
padtol= same meaning as usual, but note that Ω is not mixed unless padtol is reached
(1d-3 is recommended for all CPA calculations)
dz=     special branch, in which z-points are shifted by dz along the real axis (experimental)


Recommended options:

GF GFOPTS=[...];omgmix=0.4;padtol=1d-3;omgtol=1d-3;lotf;nitmax=30


Compatibility with other features

Downfolding is supported. Note, however, that downfolding applies to the crystal Green’s function and not to individual CPA components. The downfolding options are taken from the first species appearing in the CPA list. Gamma representation is supported with a caveat. CPA does not allow random structure constants, which means that the screening parameters must be the same for all components on the same CPA site. In the present implementation, the screening parameters are taken from the first class listed for the given CPA site (for a DLM site this is angle #1).

LDA+U is not supported, and density matrices are not calculated for the components on the CPA site. However, the modes IDU=4 and IDU=5 are supported. The U and J parameters for these modes are taken from the first species appearing in the CPA list.

Broyden mixing for charged works fine if omgtol is set to a sufficiently low value. If Broyden mixing seems to act strangely, try to reduce omgtol. Charge self-consistency in CPA may sometimes be difficult for impurities with low concentrations. (Note that an isolated impurity can be described by adding it as a CPA component with zero concentration.)

Atomic files

It is important to understand the atomic file handling with CPA. For a CPA site (say, species Fe) the code creates an atomic file per each CPA component. In the above example with SPEC ATOM=Fe … NTHET=2 CPA=1 4 5 there will be four atomic files: fe#1.ext for Fe with angle 0, fe#2.ext for Fe with angle π, fe#3.ext for species type 4, and fe#4.ext for species type 5. Note that fe#3 and fe#4 will not actually correspond to Fe atoms, but to those described by species 4 and 5. Because convergence can be delicate, it is always recommended to copy appropriately prepared atomic files before attempting a CPA calculation. In the above example, converge a Fe atom and copy the atom file to fe#1.ext and fe#2.ext; then converge species type 4 and copy it to fe#3.ext, and so on. For DLM with NTHET=N, make N copies of the atomic file: say, fe#1.ext, fe#2.ext, …, fe#N.ext.

Outputs

At the beginning of the run, some debugging information is printed, listing the indexing for the CPA sites. DLMWGTS lists the polar angles (0 for non-DLM classes) and weights for all CPA classes (this is also for debugging purposes). GETZV prints the total valence charge, which in CPA is generally not integer. Output for each CPA component includes the usual information (charge, local moment, etc.). Exchange constants J0 are automatically calculated for all CPA components using the linear response formula from Liechtenstein et al. (it can not be disabled, but the computational cost in any case negligible). Off-diagonal local moments and constraining fields are always printed out, even if DLM is not used. These include the diagonal local moment as well. All these moments are output, unmixed values. In the self-consistent state the z-component should equal to the input moment.

At the end of the iteration Ω is mixed, and its misfit for each CPA site is printed out (see “Mixed Omega for site …”) The total energy is correctly calculated and printed out as ehk, as usual.

Partial densities of states

Partial DOS can be calculated as usual using contour type 2 and adding pdos to the GFOPTS tag. Note that in this case Ω needs to be converged anew at each point of the new contour. This destroys the old converged Ω file, so it is recommended to create a separate directory for a DOS calculation. The file dos.ext contains the usual information, but the data for CPA sites are averaged over components. The partial DOS for all components are separately recorded in files dosN.ext, where N is the number of the CPA site. The format of this file is the same as that of dos.ext, as if it described a system with M sites (where M is the number of CPA components). For example, for a binary CPA on site 2 with spd basis, file dos2.ext contains channels 1:6 for the first CPA component and channels 7:12 for the second CPA component. This file can be processed using pldos, as a conventional dos file.

Spectral Functions

lmgf can generate spectral functions. It is very useful way to see the broadening of states from disorder, and you can plot energy bands with it. This document explains how to make them and draw energy bands.

Test case

To familiarize yourself with a CPA case you can run the following test case

  your path to lm /lm/gf/test/test.gf fe2b