The ASA Crystal Green's function program lmgf

Summary

This package implements the local spin-density-functional theory, in the Atomic Spheres Approximation using Green’s functions. The implementation is built into code lmgf, which plays approximately the same role as the LMTO-ASA band program lm. However it has functionality lm does not. It can:

• Calculate magnetic exchange interactions
• Calculate magnetic susceptibility (spin-spin, spin-orbit, orbit-orbit parts)
• Calculate properties of disordered materials, either chemically disordered or spin disorder from finite temperature, within the Coherent Potential Approximation [1], or CPA.
• Calculate the ASA static susceptibility at $q{=}0$ to help converge calculations to self-consistency.

Details about how the ASA works, and Green’s functions in particular can be found in references section of ASA overview; see in particular Turek’s book.

See Other Resources for tutorials and related programs.

Table of Contents

• Summary
• Introduction
  • Potential functions
• Structure of Green’s function program
• Energy Contours, Potential Shifts and Determination of the Fermi Level
  • Some details concerning how mode 1 works
• GF specific input
  • Energy integration
    • Energy contours and the EMESH token
  • Green’s function category
    • The GF_MODE token
    • The GF_GFOPTS token
• lmgf-specific command-line arguments
• Test cases and examples
• The Coherent Potential Approximation
  • CPA-specific input
    • SPEC category
    • GF category
  • Compatibility with other features
  • Atomic files
  • Outputs
  • Partial densities of states
  • Spectral Functions
  • CPA Test case
• Spectral function calculations with lmgf
  • How to calculate spectral functions?
  • Plotting the spectral function
  • Site-resolved spectral functions
• Footnotes
• Other Resources
• References

Introduction

The Green’s functions are constructed by approximating KKR multiple-scattering theory with an analytic potential function, described below. The approximation to KKR is essentially similar to the linear approximation employed in band methods such as LMTO and LAPW. It can be shown that this approximation is nearly equivalent to the LMTO Hamiltonian without the “combined correction” term. Implementation of the Green’s function code is accomplished through lmgf. lmgf plays approximately the same role as the LMTO-ASA band program lm: you can use lmgf to make a self-consistent density as you can do with lm. A unique potential is generated from energy moments $Q_0$, $Q_1$, and $Q_2$, in the same way as the lm code. lmgf is a Green’s function method: Green’s functions have less information than wave functions, so in one sense the things you can do with lmgf are more limited: you cannot make the bands directly, for example. However, lmgf enables you to do things you cannot with lm, as described at the beginning of this document.

Potential functions

The scattering properties of a sphere whose potential is spherically symmetric can be encapsulated in terms of a phase shift η_l of a wave scattering off the sphere. Each angular momentum l has its own phase shift, and it depends on energy. Alternatively η_l can be defined in terms of the potential function $$P_l(\varepsilon) = \frac{W\{K_l,\phi_l\}}{W\{J_l,\phi_l\}}$$

Here ${W\{K_l,\phi_l\}}$ and ${W\{J_l,\phi_l\}}$ are Wronskians of Hankel and Bessel functions $K$ and $J$ with partial waves $\phi_l(\varepsilon,r)$ at the muffin-tin sphere radius. Hankel and Bessel functions are solutions to the Schrodinger equation in the flat interstitial part of a muffin-tin potential, and Wronskians are used to match a linear combination of $K$ and $J$ to the value and the slope of $\phi_l(\varepsilon,r)$. making the wave continuous and differentiable.

By linearizing the partial wave $\phi_l(\varepsilon,r)$, the energy-dependent Hamiltonian becomes energy-independent. The Wronskians become fixed numbers, and reduce to four types thos if H and J with $\phi$ and its energy derivative, $\dot\phi$. These Wronkians are usually expressed in terms of the physically more interpretable potential parameters, the most important of which are the band center C_l and bandwith Δ_l, and also

This simplification offers a huge advantage, which is why linear methods are ubiquitous in electronic structure theory. In LMTO, it is customary to use the $\kappa=0$ ($\kappa^2$ is the energy of the Hankel and Bessel envelope functions) KKR phase shift in the following parameterization: $$1 / P_l(E) = \Delta_l / ( E - C_l ) + \gamma_l$$

$\Delta_l$, $C_l$ and $\gamma_l$ are potential parameters. they are explained in detail in the book by Turek et al; see also the description of downfolding) calculated from the partial waves inside augmentation spheres at the linearization energy.

The “scattering path operator” $P-S$, essentially the inverse of the Green’s function apart from a scaling, s given by $P$ and the structure matrix $S$. The latter are are structure constants, independent of potential, that relate the expansion Hankel function $H_L(\mathbf{r}-\mathbf{R})$ centered around a remote site $\mathbf{R})$ in Bessel functions around another site: $$H_L(\mathbf{r}-\mathbf{R}) = \sum_{L'} S_{L'L} J_L(\mathbf{r})$$

Eigenstate appear where there are poles in G or when $$\lvert P-S \rvert = 0$$

Finding poles must be done by an searching in energy (a nonlinear eigenvalue problem). But it is easily shown with linearization the eigenvalue condition becomes a linear algebraic eigenvalue problem, which can be solved by band methods. However, when thee potential is energy-dependent, as it is, e.g., in the CPA, the hamiltonian cannot be so so simplified.

Green’s function methods must resolve by energy so advantage is gained by linearization is much less (the KKR method is the LMTO Green’s function method without the linear approximation). But lmgf makes the linear approximation anyway, for consistency with the rest of the Questaal suite.

Note: The potential function $P_l$ is easily confused with the “continuous principal quantum number” which bears the same symbol and has a similar purpose. This is unfortunate; you have to infer which is meant from the context.

Structure of Green’s function program

lmgf runs in much the same way as lm, at least in its primary mode, MODE=1. The band pass routine of lm, bndasa.f, generates the eigenvalues and eigenvectors, which can in turn generate the quantities of interest. bndasa is replaced by a Green’s function routine, gfasa. gfasa can generate output moments, DOS, density-matrix, etc., in the same way as bndasa does.

In contrast to band methods (implemented in lm) where the Hamiltonian $H$ is energy independent and all the bands are generated by diagonalizing H, Green’s functions are calculated for a specific energy; information is extracted from $G$ for a particular energy.

This fact highlights the strengths and weaknesses of a Green’s function approach. Energy-integrated properties such as the moments, must be obtained by integrating over energy. Calculating $G$ explicitly at a family of energies is more cumbersome than diaonalizing a Hamiltonian. On the other hand, Green’s function methods are naturally suited to contexts where the energy-dependence is needed anyway. CPA theory yields an energy-dependent potential; Green’s functions are a natural way to implement it. Similarly, noninteracting susceptibilities can be expressed as G×G (`×’ implies either convolution or product, depending on the space you are working in).

lmgf always loops over some energy contour; what contour you use depends on the context as described below. gfasa accumulates various kinds of data for each mesh point, such as the point’s contribution from energy moments used in an ASA self-consistent cycle. Finally, an estimate for the Fermi level $E_F$ is determined from a Pade approximation. If the original guess for $E_F$ is sufficiently close, the cycle is finished as in lm. If the estimate is too far off, a new energy mesh is taken and the process is repeated.

In addition to its primary mode (MODE=1), there are other modes, notably MODE=10 and MODE=11 for computing magnetic exchange interactions within ASA-linear response.

Energy Contours, Potential Shifts and Determination of the Fermi Level

For energy-integrated properties (MODE=1), a very fine energy mesh would be required if the energy integration was carried out close to the real axis. It is much more efficient to deform the integration contour into an elliptical path in the complex plane, approaching the real axis only at the lower and upper integration limits.

To integrate quantities over occupied states, integration to the Fermi level E_F is required. E_F is not known but must be fixed by charge neutrality. Thus $E_F$ must be guessed at and iteratively refined until the charge neutrality condition is satisfied. lmgf does not vary E_F; the user specifies it at the outset. Instead lmgf looks for a global constant potential shift vconst added to the one-particle hamiltonian that allows it to satisfy charge neutrality. vconst must be found by an iterative search, as described in the subsection below. Both vconst and E_F are maintained in a file vshft.ext. Inspect this file and you may find it unecessarily complicated; it’s because you can also use it to add site-dependent shifts. vconst and E_F are stored in the first line. vshft.ext is also used by the layer Green’s function code lmpg, which requires extra information about shifts on the left and right leads; this additional information is also stored in vshft.

Metals and nonmetals are distinguished in that in the latter case, there is no DOS in the gap and therefore the Fermi level (or potential shift) cannot be specified precisely.

Metal case (set by BZ_METAL=1): once the $k$- and energy-points are summed over and the deviation from charge neutrality is determined, the code will attempt to find the potential shift that fixes charge neutrality.

This tutorial offers a working example.

Nonmetal case (set by BZ METAL=0): lmgf will not attempt to shift the potential, or ensure charge neutrality. The user is cautioned to pay rather closer attention to deviations from charge neutrality. It can happen because of numerical integration errors, or because your assumed Fermi level does not fall within the gap. You can use METAL=1 even if the material is a nonmetal, but be advised that this scheme is not foolproof: the vanishing DOS at E_F puts a strain on the iterative search algorithm.

Some details concerning how mode 1 works

For each energy point, the BZ integration is accomplished by routine in gf/gfibz.f, which loops over all irreducible points, generating the “scattering path operator” $g$ and the corresponding $g$ for all the points in the star of $k$ to generate a properly symmetrized $g$. Within the ASA, second-generation LMTO, $g$ is converted to proper Green’s function G, by an energy scaling. The scaling is carried out in routine gf/gfg2g.f. Next the various integrated quantities sought are assembled (done by gf/gfidos.f). vconst needed satisfy charge neutrality is found by a nested iterative procedure explained in the following paragraph. The shift is stored in vshft.ext, as noted above; I/O to this file is handled by routine subs/iovshf.f.

The search for charge neutrality proceeds iteratively inside two nested loops. In the outer loop, sphere moments are integrated from $\mathrm{Im} G(\mathbf{k},E)$ for a given trial potential vconst, which yields deviation Δq from charge neutrality in the entire system. vconst is a global constant potential shift of the one-particle hamiltonian, and it must be adjusted until Δq is zero. (Alternatively the Fermi level may be adjusted keeping vconst fixed; see shftef in GFOPTS below). From various data (initially, density-of-states at E_F; later, data pairs (vconst, Δq) from prior iterations) are used to estimate a new vconst. The procedure is iterated until the change in vconst falls below a tolerance padtol.

It is possible to find a new vconst using the outer loop alone. The outer loop is simply repeated for new values of vconst until tolerance is reached. This is expensive, however, and there is additionally an inner “Pade” loop to accelerate convergence. The diagonal elements of G are fit to a Pade polynomial, and the Pade approximation to G is used in the iterative procedure described above. Once the neutrality point is found within the Pade approximation, the outer loop is repeated until the change in vconst falls below padtol. After convergence the search terminates and the updated vconst is written to file vshft.ext.

There is a second (optional) tolerance qtol, that modifies how the search terminates. If the qtol tag is missing or its value is zero, this tag has no effect. Otherwise when Δq < qtol the internal search for vconst ends. Specifically:

• A test is made before the inner loop starts. If Δq < qtol the search terminates without any Pade correction.
• Otherwise the search proeeds with inner loop (i.e. using a Pade estimate for G). If Δq < qtol for the Pade estimate for G, the inner loop terminates and the search reverts to the outer loop.

GF specific input

Energy integration

Green’s functions are always performed on some energy contour, which is discretized into a mesh of points in the complex energy plane. (A description of the various kinds of contours this code uses is documented in the comments to gf/emesh.f.) $G$ is “spikey” for energies on the real axis (it has poles where there are eigenstates).

Energy contours and the EMESH token

Note: the  EMESH  tag applies to both lmgf and lmpg.

To compute energy-integrated properties such as magnetic moments or the static susceptibility, the calculation is most efficiently done by deforming the contour in an ellipse in the complex plane.

At other times you want properties on the real axis, e.g. density-of-states or spectral functions. You specify the contour in category BZ as:

EMESH= nz mode emin emax [other args which depend on mode]

• nz number of energy points

• emin,emax the energy window (emax is usually the Fermi level when integrating over the BZ)

• mode specifies the kind of contour:

  mode=10: a Gaussian quadrature on an ellipse.

  This is the standard mode for integrating over the occupied states.

    EMESH= nz 10 emin emax ecc eps
  

  • ecc is the eccentricity of the ellipse, ranging from 0 (circle) to 1 (line)
  • eps is a ‘bunching’ parameter that, as made larger, tends to bunch points near emax.
    As a rule, eps=0 is good, or maybe eps=0.5 to emphasize points near the Fermi level.

  After the integration is completed, there will be some deviation from charge neutrality, because emax will not exactly correspond to the Fermi level. This deviation is ignored if METAL=0; otherwise lmgf or lmpg has to go through an iterative procedue to find the charge neutrality point. There is a tutorial demonstrating the process.

  mode=110: is a contour specific to nonequilibrium principal layer Green’s function calculations.

  Use this mode for iterating to self-consistency in the non-equilibrium case.

  The nonequilibrium Green’s function requires additional information for the energy window between the left and right leads. (The nonequilibrium Green’s function is implemented for the layer geometry in lmpg.) Thus the integration proceeds in two parts: first an integration on an elliptical path is taken to the left Fermi level (as in mode=10). Then an integration over is performed on the nonequilibrium contour, i.e. the energy window from the left to the right Fermi level. This integration is performed on a uniform mesh close to the real axis, as in mode=0. For the nonequilibrium contour, three additional pieces of information must be supplied:

  • nzne number of (uniformly spaced energy points on the nonequilibrium contour
  • vne difference in fermi energies of right and left leads, E_F(R)minus;E_F</i>(L)
  • delne Im-z on the nonequilibrium contour

  Parameters are specified as

  EMESH= nz 110 emin ef(L) ecc eps nzne vne delne [delend]
  

  The last argument delend plays the role of delne specifically for computing the self-energy of the left and right leads. There is an incompatibility in the requirements for $\mathrm{Im}\,z$ in the central and end regions. See Footnote 1.

  mode=0: a uniformly space mesh of points between emin and emax, with a constant imaginary component.

  Use this mode when you want quantities resolved by energy on the real axis, e.g. spectral functions or transmission in the lmpg case.

  mode=1: is the same contour as mode=0.

  The only difference is that the sign of Im z is flipped.

  mode=2: is the same contour as mode=0.

  The only difference is that weights for each energy are set to unity instead of the spacing between energy points. Whether to choose mode 0 or mode 2 depends on if the energy-resolved property of interest, .e.g. density-of-states, is to be given as is, or to be weighted for an integration over energy.

   EMESH= nz 0|1|2 emin emax Im-z [... + possible args for layer geometry.]
  

  Im-z is the (constant) imaginary component.

  For the Principal layer code lmpg, use one of:

  EMESH= nz 0 emin emax delta xx xx xx xx delend                 [equilibrium]
  EMESH= nz 110 emin ef(L) ecc eps nzne vne delne delend         [non-equilibrium]
  

  • emin,emax the energy window (emax is usually the Fermi level when integrating over the BZ)
  • delta is Im z for the central region
  • xx has no meaning but are present for compatibility with the contour used in nonequilibrium calculations.
  • nzne, vne, delne are for nonequilibrium calculations (see mode 110).
  • delend is ${\mathrm{Im}}\, z$ used to make the self-energy of the leads; see Footnote 1.

  mode=310: Alternative to Pade approximant in finding the Fermi level.

  Acts like mode 10 for the elliptical contour to emax, then switches to a uniform mesh to fine the Fermi level.

  This mode integrates with a Gaussian quadrature on an ellipse to a trial emax, as in mode 10. However, the search for the Fermi level is not done by Pade approximant, as in mode 10. Instead, a second integration proceeds along a uniform mesh from emax to some (Fermi) energy which satisfies charge neutrality. This procedure is not iterative.

  EMESH= nz 310 emin emax e1 e2 delz
  

  +e1 and e2 are just as in mode 10 +delz is the spacing between energy points for the second integration on the uniform mesh.

Green’s function category

lmgf requires a GF-specific category.

GF  MODE=1 GFOPTS=options

The GF_MODE token

MODE=n controls what lmgf calculates. Options are MODE=1, MODE=10, MODE=11, MODE=26, described below.

MODE=1 goes through the self-consistency cycle, calling gfasa. It performs a function analogous to bndasa in the band program, generating output density, moments, and other quantities such as density-of-states.

Taken with the special integration contour mode=2 (see EMESH above), the density-of-states $D(E)$ and its integral are computed and tabulated over the window specified.

MODE=10 invokes a special branch that computes magnetic exchange interactions using a linear response technique. (The source code has its entry point in gf/exasa.f.)

gf/test/getJq0z

gf/test/test.gf co 5

--sites:pair:site-list

MODE=11 is an exchange branch that is run after MODE=10. It prints out the $J_{ij}$ and does several other analyses.

Switch --sites:pair:site-list applies to mode 11 as well as mode 10; see command-line arguments.

The GF_GFOPTS token

The GF category has a token GFOPTS=tag;tag;…, which causes lmgf to perform a variety of special purpose functions.

Options are entered as a sequence of tags delimited by a semicolons:  tag1;tag2;… .

The following are specific to the the layer code

The following are specific to the CPA:

See Table of Contents

lmgf-specific command-line arguments

-ef=# overrides upper limit of energy integration (Fermi level) and assigns to #

The following are specific to the exchange calculation modes 10 and 11:

--sites[:pair]:_site-list_  Make the exchange parameters J_ij only for sites in _site-list_, which is a standard [Questaal integer list](/docs/numerics/integerlists).

Example: running lmgf using MODE=10 with this command line argument

--sites:pair:1,3,5,7

generates J connecting sites 1, 3, 5 and 7 to all neighbors.

This switch also works in mode 11, but the meeaning is different: lmgf will print out the exchange interacations only from between in

Running lmgf using MODE=11 with the same --sites argument will print out the exchanges just between pairs of these sites.

Running lmgf using MODE=11 without any --sites argument will print out the exchanges between these sites and all neighbors.

--wrsj[:fn=name][:scl=#][:tol=#]  (mode 11 only)
  Writes the Heisenberg exchange parameters in a standard format, suitable for use
  in spin dynamics simulations.
  fn=name writes to file 'name'  (default name is rsj)
  scl=#   scales the parameters by #
  tol=#   writes only parameters with energy > tol

--rcut=#
  Truncates the range of the R.S exchange parameters ...
  useful to assist in the determination of the effect distant neighbors.

--2xmsh
  When integrating over the BZ to estimate Tc from Tablikov formula, this option doubles the k-mesh.
  Can be helpful in testing k-convergence of the singular q->0 limit entering into the formula.

--amoms=mom1,mom2,...
--amom=mom1,mom2,...
  This switch overrides ASA moments (which are automatically generated).
  The first switch reads a vector of nbas moments, one for each site.
  The first switch reads a vector of nclass moments, one for each class.

Sphere magnetic moments are tabulated in the printout at the end of mode 10, and the start of mode 11. If you are importing exchange parameters (file jr.ext , e.g. from the full-potential code, you will want to supply the moments calculated from that program.)

See Table of Contents

Test cases and examples

This script:

gf/test/test.gf --all

carries out a number of tests, which also demonstrate various branches of the code. To see the materials and corresponding tests try

gf/test/test.gf --list

The Coherent Potential Approximation

The CPA implementation for substitutional alloys and for spin disorder follows the formulation explained in these References [1,2,3]. 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.

CPA Test case

To familiarize yourself with a CPA case you go through the CPA tutorial.

You can also run the following test case

  ~/lm/gf/test/test.gf fe2b

Spectral function calculations with lmgf

Spectral function have been implemented in lmgf v7.10 by Bhalchandra Pujari and Kirill Belashchenko (belashchenko@unl.edu). The details of the theory in the CPA case can be found in Ref [1].

How to calculate spectral functions?

The spectral function can be calculated both with and without CPA. The calculation is performed in 3 steps:

• Charge self consistency. (The spectral function can be calculated for any potential, but it is usual to work with the self-consistent one).
• CPA self-consistency in the coherent interactor $\Omega$ (CPA only). Since $\Omega$ is energy dependent, it has to be calculated for the energy points where the spectral function is needed. For drawing spectral functions this is usually a uniform mesh of points close to the real axis.
• Calculation of the spectral function on some contour, usually a uniform mesh close to the real axis.

Charge self consistency is performed in the usual manner, for example with the following options:

BZ       EMESH=31 10 -.9 0 .5 .0
GF       MODE=1 DLM=12 GFOPTS=p3;omgmix=1.0;padtol=1d-3;omgtol=1d-5;lotf;nitmax=50

Note the EMESH mode (contour type) is elliptical (type 10). If CPA is used, the coherent interactors $\Omega$ for all CPA sites are also iterated to self-consistency during this calculation, but this is done for the complex energy points on the elliptical contour. The following additional step is needed in this case to obtain self-consistent $\Omega$ at those points where the spectral function will be calculated.

Omega self-consistency is turned on by setting DLM=32. In this mode only Ω for each CPA site is converged, while the atomic charges are left unchanged. It is important to converge Ω to high precision. Typically omgtol=1d-6 is a good criterion. The contour type should be set to 2. Example input for this step is

BZ       EMESH=150 2 -.25 .25  .0005  0
GF       MODE=1 DLM=32  GFOPTS=p3;omgmix=1.0;padtol=1d-3;omgtol=1d-6;lotf;nitmax=50

The highlighted parameters are of particular importance. lotf is required to iterate $\Omega$ for convergence (and it is recommended to keep it enabled in all CPA calculations, including charge self-consistency). It is also necessary to monitor the output file (set --pr41) and make sure that the required precision has been achieved for all energy points. If convergence appears to be problematic, try to start with a larger imaginary part for the complex energy or reduce the mixing parameter omgmix.

Calculation of the spectral function should be done with EMESH set to the same mesh as used for $\Omega$ self-consistency, e.g.

BZ       EMESH=150 2 -.25 .25  .0005  0
GF       MODE=1 DLM=12 GFOPTS=p3;omgmix=1.0;padtol=1d-3;specfun

Important: If there are sites treated in CPA, the contour specified by EMESH should be kept exactly as in the previous step when CPA self-consistency was performed.

In order to start the calculation, invoke lmgf with the --band flag referring to the symmetry-line file (same format as used for band structure calculation with lm):

lmgf «sys» --band:fn=syml

where «sys» is the extension of the ctrl file. Once completed, the program will generate a spf.«sys» file containing the complete spectral function along the lines given in the syml.«sys» file. Other options included with --band are currently not used.

Note: Spectral function calculations can be run with MPI parallelization.

Plotting the spectral function

Upon successful completion of calculation spf.«sys» file will be generated. The file has the following format:

ZP        KP         SF_UP         SF_DN
.
.
-0.250   0.137      5.708839      4.742153
-0.250   0.160     10.658114      4.844647
.
.

where ZP are the points on the energy contour, KP are the points of the K-mesh and SF_UP, SF_DN are the spectral functions for Up and Down channel respectively. The very first line of the file indicate the location of the high symmetry points of the Brillouin zone. User can utilize this information to visualize the spectral function using any desired graphics package. A small bash utility, SpectralFunction.sh, is given for the sake of convenience. This bash script uses Gnuplot to view and save the spectral function. Users with standard Linux/unix distro should be able to use it without special prerequisites. Typical output of the script is shown in the figure. Please see SpectralFunction.sh -h for usage. (The file spf.«sys» should be renamed specfun.«sys» to be read by this script. See below why.)

Site-resolved spectral functions

Site-resolved spectral functions can be obtained by including the option specfr in GFOPTS. (It supersedes the option specfun.) A series of output files will then be generated, which contain the spectral function on each basis site . The file names are spfN.«sys», where N is the index of the site. The format of this file is the same as for spf.«sys», and it can be also plotted using the SpectralFunction.sh script (first rename to specfun.«sys»).

Footnotes

¹ When computing Green’s functions near the real axis via the Landauer-Buttiker formalism for transmission through the active region, or the nonequilibrium part of the contour in nonequilibrium calculatations, or in special modes that search for the Fermi energy by integrating points on the real axis, there is a problem in how to choose ${\mathrm{Im}}\,z$. A small ${\mathrm{Im}}\,z$ is needed for a reliable calculation of the transmission coefficient, but choosing a small ${\mathrm{Im}}\,z$ to determine the surface Green’s function may not succeed because G can become long range and the iterative cycle used to generate it may not be stable. To accommodate these conflicting requirements, a surface-specific ${\mathrm{Im}}\,z$ should be used. It is entered as an element delend in EMESH.

The mode=0 mesh is specified as

When computing transmission coefficients via the Landauer-Buttiker formalism, one chooses a contour as in mode=0. However, there is a problem in how to choose Imz{\mathrm{Im}}\, zImz. A small Imz{\mathrm{Im}}\, zImz is needed for a reliable calculation of the transmission coefficient, but using a small Imz{\mathrm{Im}}\, zImz to determine the surface Green’s function may not succeed because the GF can become long range and the iterative cycle used to generate it may not be stable. To accommodate these conflicting requirements, a surface-specific Imz{\mathrm{Im}}\, zImz should be used, called delend. The mode=0 mesh is specified as

Other Resources

An overview of the Atomic Spheres Approximation can be found here.

If you haven’t already done so, you are advised to go through the tutorial for the band code lm) before doing this one. lm and lmgf share many feature in common, but lm is somewhat easier to use.

See this page for an Introductory tutorial for lmgf, and this page for a tutorial explaining how lmgf implements the CPA in practice.

There is a related program lmpg, that has a similar function but is designed for layer geometries, enabling Landauer-Buttiker transport.

References

1. I. Turek et al., Electronic strucure of disordered alloys, surfaces and interfaces (Kluwer, Boston, 1996).
2. J. Kudrnovsky and V. Drchal, Phys. Rev. B 41, 7515 (1990).
3. J. Kudrnovsky, V. Drchal, and J. Masek, Phys. Rev. B 35, 2487 (1987).