PHYSICAL REVIEW B 1 MARCH 1997-IVOLUME 55, NUMBER 9
Self-consistent LCAO-CPA method for disordered alloys
Klaus Koepernik, B. Velicky´,* Roland Hayn, and Helmut EschrigMPG Research Group Electron Systems, Department of Physics, TU Dresden, D-01062, Dresden, Germany
~Received 9 August 1996!
We present a scheme for calculating the electronic structure of disordered alloys, self-consistent in thelocal-density-approximation sense. It is based on expanding the one-electron Green’s function in the basis ofmodified atomic orbitals@H. Eschrig,Optimized LCAO Method and the Electronic Structure of ExtendedSystems~Springer, Berlin, 1989!#. The two-terminal approximation introduced for the Hamiltonian and theoverlap matrix permits us to treat both the diagonal and off-diagonal disorder using an extension of theBlackman-Esterling-Berk form of the coherent-potential approximation~CPA! @Phys. Rev. B4, 2412~1971!#to a nonorthogonal basis set. Calculations using the scalar relativistic density functional for the magnetic binarytransition-metal alloys Fe-Co, Fe-Pt, Co-Pt, and for the ternary alloy Al-Fe-Mn give results comparing wellwith experimental data and calculations based on the Korringa-Kohn-Rostoker~KKR!-CPA and linear muffin-tin orbital-CPA techniques.@S0163-1829~97!03509-1#
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I. INTRODUCTION
Several powerful implementations of the coherepotential approximation1–3 ~CPA! to calculate the electronicstructure of substitutional alloys have been developed incent years. Early applications of the CPA with firsprinciples band-structure methods were mostly based onmethod of Korringa, Kohn, and Rostoker~KKR!,4 and thisKKR-CPA is still widely used, e.g., Ref. 5. In view of thmultiple-scattering formulation underlying the KKR theorthe incorporation of the CPA turns out to be quite natura
To reduce the numerical effort of the KKR method, ttight-binding-linear-muffin-tin-orbital6 ~TB-LMTO! methodwas developed. Another linear band-structure scheme islinear-combination-of-atomic-orbitals~LCAO!.7 To incorpo-rate the CPA idea into these linear band-structure methone has to start from an algebraic, matrix version of the Cinstead of using the multiple-scattering language. In tway, a TB-LMTO-CPA~Ref. 8! was developed and also firssteps towards a LCAO-CPA~Refs. 9–11! were made. Theseapproaches made it possible to apply the CPA in quite cplex structures like multilayers, surfaces or interfaces12 or inbulk materials with complex unit cells and partial disordFirst TB-LMTO-CPA calculations on partially disordered aloys were performed by Kudrnovsky´ et al.,13,14 however notfully charged and spin self-consistent. Recently, the screeKKR approach was introduced.15 It combines the pleasanfeatures of a TB formulation with the rigorousness ofmultiple-scattering approach.
In the past, special attention has been drawn to the plem of off-diagonal disorder appearing in the matrix CPA.the case of multiplicative off-diagonal disorder one can uthe approach proposed by Shiba,16 which was introducedinto the TB-LMTO-CPA.17–19 For the case of general offdiagonal disorder the procedure of Blackman-EsterliBerk20 ~BEB! is applicable. Some applications of BEBband-structure calculations were reported in Ref. 21 usintight-binding fit. But to the authors’ knowledge, the BEapproach was never used in a first-principles, charge sconsistent CPA application.
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Within the LCAO scheme one deals with a nonorthogobasis. There have been attempts to implement the CPAnonorthogonal basis schemes.9–11However, up to now therewere no charge self-consistent calculations and thediagonal disorder was only treated in virtual crystal appromation~VCA!. Despite the many similarities between LCAand TB-LMTO there are also some differences. LCAO pno restriction on the shape of the potential, as for instathe atomic-sphere approximation~ASA! of the TB-LMTO-CPA.
Our aim will be to present a fully charge self-consisteapproach to complex lattices based on a nonorthogoLCAO basis scheme without restriction to VCA. We witreat all randomness of the Hamiltonian and of the overmatrix including off-diagonal disorder at the same level. Tapproach is applicable to a wide class of alloys. It is baseda pseudospin description often used in the past~in the BEBtheory,20 in the augmented space method22!. We present ageneralization of the BEB theory to sublattices and to northogonal basis sets in the propagator formalism, embedin a charge self-consistent treatment. A variant of a propator formalism for BEB was proposed, for example, in Re23, where the analyticity of the BEB Green’s function wproved. In the context of the propagator formalism wpresent a formulation of the full multiple-scattering problewithin the LCAO approach comparable to the KKR descrtion. This leads to a quite natural introduction of the singsite approximation.
The problem of disordered alloys naturally requires qua number of approximations. In our opinion two points hato be met by every charge self-consistent CPA methodshould incorporate the major effects of disorder at leassingle-site mean-field approach and rely on a band-strucscheme which is sufficiently accurate in relation to the acracy of the CPA. When using a wave-function approach,possibility of a single-site approximation implies the usenot only a local but an effectively well localized basis. In thTB-LMTO this is achieved by the additional LMTO-TBtransformation, which is not entirely without problems. Toptimized local orbital approach,7 approved for ordered
structures, seems to us particularly appropriate to a mdirect solution. It leads naturally to the terminal-point aproximation and therefore includes the most general kindoff-diagonal disorder. Clearly the mean-field treatment ofdisorder via single-site CPA has a quite strong model chacter. However, in the case of a BEB theory the randomnof the environment is taken into account at least with respto the two terminal points of all matrices, thus giving a godescription of the densities of states as demonstratedmodel systems. Our combination of the CPA with the noorthogonal LCAO fulfills the requirement with respect to taccuracy of the whole scheme and thereby arrives directa matrix representation without a tight-binding transformtion of the Hamiltonian.
Our paper is organized as follows. First, in Sec. II, wpresent the representation of the charge density of an alloterms of conditionally averaged Green’s functions and loorbitals. This is necessary to obtain the self-consistent potial in the spirit of the local-density approximation~LDA !.Next, in Sec. III, we generalize the BEB theory to nonothogonal basis sets and to the case of several sublatticterms of the propagator formalism. In Section IV we descrthe numerical procedure and present the underlying eqtions. In Sec. V we apply the method to binary and terntransition-metal alloys. The binary examples FeCo and escially CoPt and FePt prove the applicability and accuracyour approach to alloys with a great difference in the bawidths of the constituents and to cases where relativisticfects are important. The last example FeMnAl is a ratcomplex one. It shows a rich structure of the magnetic phdiagram. Here we present an investigation of the ferromnetic phase. In that case, we obtain magnetic momentagreement with experiment for all Al concentrations, hoever, at higher Al content, the incorporation or partial ordis decisive in obtaining agreement with experiment. We dcuss the dependence of the moments on the local envment.
II. KOHN-SHAM APPROACH TO ALLOYSIN LOCAL ORBITAL REPRESENTATION
Theoretically, an alloy is described as an ensembleconfigurations of atoms, accompanied with the definitionan average of observable quantities. For each configurain the spirit of density-functional theory an effective singlparticle Hamiltonian is introduced, depending on the electdensity of that configuration. Instead of summing oversquares of Kohn-Sham orbitals, in alloy theory it is prefable to use the single-particle Green’s function in closingself-consistency cycle.
A. Local orbital representation of the single-particleGreen’s function
Given an atomic configuration, the Kohn-Sham theostarts from an effective single-particle Hamiltonian~we useatomic units\5me5ueu51):
H~rW !521
2D1V~rW ! . ~1!
re-fer-ssct
or-
at-
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with the potential being a function of atomic species apositions and a functional of the electron density. Tground-state electron density is again solely determinedthe atomic species and positions. For the sake of simpliin the notation, a possible spin dependence is understoothe following without mention. In a relativistic version,Pauli Hamiltonian or a Dirac Hamiltonian can be used likwise. To prepare for the alloy case we define the retarsingle-particle Green’s function in real spac(v15v1 id):
@v12H~rW !#G~rW,rW8;v1!5d~rW2rW8! . ~2!
From its imaginary part the electron density may be callated as
n~rW !521
pE ImG~rW,rW;v1!Q~«F2v!dv , ~3!
whereQ is the step function and«F denotes the Fermi levelAt this stage we introduce a nonorthogonal local orbi
representation7 of all real-space quantities. The basis orbitaare classified as valenceu im) and coreu ic) states with siteindex i and atomic quantum numbersm andc, respectively.The core states are assumed to be nonoverlapping andthogonal to each other, while the valence states are notget rid of the core part of the Hamiltonian, we project tvalence basis states onto the Hilbert subspace orthogonall core states:
u im&5u im)2(lc
u lc)~ lcu im! . ~4!
Core states and the corresponding density contributionsseparately treated as a first step in each cycle of sconsistency.
From now on, we consider the valence subspace inHilbert space which in practical implementations is spannby a finite number of basis states per site, but is sufficiencomplete to represent the occupied valence eigenstatesfollowing quantities like Hamiltonian, overlap matrix, anGreen’s function will be given as projected to this subspaIt may likewise be considered as spanned by a set ofthonormalized statesuk& containing the occupied eigenstatof the projected Hamiltonian:
uk&5(im
u im&aimk , ~5!
1valence5(k
uk&^ku5 (i mi 8m8
u im&(kaimk ai 8m8
k* ^ i 8m8u . ~6!
In practical implementations, completeness in this senschecked as basis set convergence. The coefficientsaim
k definethe valence states and the inverse of the overlap matrixS:
(kaimk ai 8m8
k* 5^ imu i 8m8&215Sim,i 8m821 . ~7!
Straightforwardly we deduce the Hamiltonian matrix, toverlap matrix, and from Eq.~2! the Green’s matrix~see alsoRef. 9!:
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55 5719SELF-CONSISTENT LCAO-CPA METHOD FOR . . .
H~rW !5 (im i 8m8jn j 8n8
^rWu jn&Sjn,im21 ^ imuH~rW !u i 8m8&
3Si 8m8, j 8n821 ^ j 8n8urW& ,
H5i^ imuH~rW !u i 8m8&i ,
S5i^ imu i 8m8&i ,
15~v1S2H !G1 . ~8!
G6 abbreviatesG(v6)5G(v6 id).The contravariant Green’s matrix elements in the non
thogonal basis are defined as
Gim,i 8m86 [ (
jnj 8n8
Sim, jn21 ^ jnuG~rW,rW8;v6!u j 8n8&Sj 8n8,i 8m8
21 .
~9!
Starting from the Kohn-Sham ansatz the valence part ofelectron density is given by
n~rW !5(k
^rWuk&Q~«F2«k!^kurW& , ~10!
where«F again denotes the Fermi energy.~The core part hasa similar structure and will be added, but it is simple andnot the object of consideration here.! We replace theQ func-tion by an integral over ad function, which is expressed athe difference of the retarded and advanced one-parGreen’s functions. After switching to the local basis~4! weget
n~rW !521
2p i (imi 8m8
^rWu im&E«Fdv
3@G12G2# im,i 8m8^ i 8m8urW& . ~11!
The involved integration starts below the band bottom ofvalence states. The occurrence of the advanced Green’strix is due to moving the imaginary part in Eq.~3! betweenthe ~possibly complex! local basis states.@In cases of a reaHamiltonian matrix—in the presence of inversiosymmetry—the difference in Eq.~11! reduces to the imaginary part of the retarded Green’s matrix.# To simplify thenotation, we use the abbreviation ImG[1/2i (G12G2) inthe following. We get a density representation consisting~overlapping! local site densities~for i5 i 8) and of doublyterminated terms called overlap density:
n~rW !521
p (im,i 8m8
^rWu im&E«FImGim,i 8m8dv^ i 8m8urW& .
~12!
Up to this point we considered a given configuration ofalloy. One of the main problems here is the lack of perioicity, so that there are no good quantum numbers charac
r-
e
s
le
ea-
f
-r-
izing the eigenstatesuk& of the Hamiltonian. To arrive atmeasurable quantities, we have to carry out a configuratioaverage.
We want to point out that we have no frozen-core trement, since in this approach it would not lead to a simplcation. In each cycle of charge self-consistency the cstates and the projected valence basis states will be reclated.
B. Pseudospin description of the ensemble
In this paper we discuss substitutional disordered allotaking into account the possibility of a complex unit cell. Thvectors of the underlying lattice will be denoted byRW and thebasis vectors bysW. Each site belongs to a sublattice. Thsublattices may be randomly occupied by various specieatoms or by a vacancy. Each alloy configuration is mapponto a set of pseudospins$hRW sW
Q%:
hRW sWQ
5H 1, atom of speciesQ at RW 1sW
0, otherwise,(Q
hRW sWQ
51 ,
~13!
~‘‘species’’ may include vacancy!. Many physical propertiesof interest are defined as configurational averages denotethe following by ^•••&. Applying it we introduce the con-centration of the speciesQ with respect to a specified sitsW in the unit cell:
^hRW sWQ
&5csWQ⇒(
QcsWQ
51 ~14!
and the condition of statistical independence of all sites:
^hRW sWQ hRW 8sW 8
Q8 &5csWQcsW 8Q8 for RW 1sWÞRW 81sW8 . ~15!
For the sake of simplicity, we use in the following a multipindex i5RW sW.
We construct now the local orbitals for an alloy with thhelp of those stochastic pseudospins:
u im)5(Q
u iQm)h iQ , local stochastic valence basis ,
~16!
u ic)5(Q
u iQc)h iQ , local stochastic core states .
~17!
The pseudospin ensures that the correct basis state is usthe right place. The orthogonalization to core states cosponds to the ‘‘Q expanded’’ equation~4!:
u im&5(Q
u iQm&h iQ , ~18!
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5720 55KOEPERNIK, VELICKY, HAYN, AND ESCHRIG
u iQm&5u iQm)2 (lcQ8
u lQ8c)h lQ8~ lQ8cu iQm! . ~19!
The core states fulfill the relation:
h iQ~ iQcu i 8Q8c8!h i 8
Q85d i i 8,QQ8,cc8h iQ , ~20!
since for iÞ i 8 and for i5 i 8, Q5Q8 they are orthonormal
andh iQh i
Q85dQQ8 at a given sitei by definition ~13!.To employ the pseudospin representation, we introd
theQ-expanded expression~12! for the density:
n~rW !521
p (iQm,i 8Q8m8
^rWu iQm&
3E«FIm@h i
QGim,i 8m8~v!h i 8Q8#dv^ i 8Q8m8urW& .
~21!
We interpret theh product of the Green’s function as a nequantity, the expanded Green’s matrix:
Gim,i 8m8QQ8 5h i
QGim,i 8m8h i 8Q8 . ~22!
The definition~22! gives the key quantity for the introduction of the BEB transformation in Sec. III.
C. Conditional average of the charge density
The expanded expression~21! for the charge density issuitable for configuration averaging, yielding the averaover the whole alloy ensemble:
n~rW !521
p (iQm,i 8Q8m8
^rWu iQm&
3E«FIm^h i
QGim,i 8m8~v!h i 8Q8&dv^ i 8Q8m8urW& .
~23!
Herein the Green’s matrix appears in a new context. Ttwofold conditional average of theh multiplied matrix is thecommonly used projected Green’s function.17 To calculatean ensemble average of the density, first the stochasticpression~21! is reduced to a sum of twofold conditionallaveraged terms, that means insertion of the two-site cotional averages of Eq.~22! in Eq. ~21!.
Now we proceed to the basic step which will permit tuse of the single-site approximation, namely we will aproximate the actual two-terminal elements of the Greematrix ~22! by their conditional component projected cofiguration averages:
h iQ^h i
QGim,i 8m8h i 8Q8&
Q8→ i 8Q→ i h i 8
Q8 . ~24!
^•••&Q8→ i 8Q→ i means averaging over all members of the e
semble of configurations with fixed occupation at the t
terminal points:h iQ5h i 8
Q851. Hence, Eq.~24! differs fromEq. ~22! by configuration averaging over all sites, excepiandi 8. These averages are self-averaging, so that all ranenvironment effects are neglected. The matrix elements~24!
e
e
e
x-
i-
-’s
-
m
still contain a stochastic dependence on two sites. To arfinally at the single-site description of an alloy, we are leda corresponding assumption about the form of the denused in the self-consistency cycle:
n~rW !'(iQ
h iQni
Q . ~25!
This means the full charge density for a configuration shbe additively decomposed into single-site components. Thwill be replaced by the self-averaging component projecdensities given by ensemble averages. Thus we neglectwo-site correlated fluctuations in the local densities. An aditional average over the right index of the overlap elemein Eq. ~24! serves to reduce Eq.~21! to a sum of single-siteexpressions:
niQ~rW !52
1
p (i 8Q8mm8
^rWu iQm&
3E«FIm^Gim,8m8
QQ8 &Q8→ i 8Q→ i dv^ i 8Q8m8urW&
3@d i i 81~12d i i 8!ci 8Q8# . ~26!
Expression~26! gives the alloy version of the local chargedensity contribution used in Ref. 7 to recalculate the cryspotential.
D. Site decomposition of the random self-consistent potential
To close the self-consistency cycle we have to calcuthe potentialV(rW) from the density~25!. In the spirit of thesingle-site approximation and in correspondence with~25!, we have to make an ansatz for the configuration depdence of the potential: we assume it to be a sum of loterms of unspecified shapes, butQ dependent and linearlydepending on the pseudospin:
V~rW !5(Q
hRW sWQVsWQ
~rW2RW 2sW ! . ~27!
This is of course a single-site approximation, there is hoever no approach available which goes beyond it, exceptextremely costly direct simulations of an ensemble of cofigurations or cluster calculations, which would reach far byond the single-site picture. Recall that every charge aspin self-consistent single-site CPA approach is boundexpress the site potential by the site density alone. Howeat variance with KKR or LMTO, here the overlap of sitdensities and potentials is not at all restricted. The acpotential construction from the density is described consing of two parts, constructing the Coulomb contribution athe exchange and correlation contribution.
Once one has calculated the electron density as a sulocal overlapping site densities one proceeds as follows.charge contribution contained innsW
Q(rW) amounts to
AsWQ
5E nsWQ
~rW !drW . ~28!
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55 5721SELF-CONSISTENT LCAO-CPA METHOD FOR . . .
Together with the nuclear chargeZsWQ we divide the total
charge density around atomsW in a neutral part and an ionipoint charge:
~29!
The neutral part gives rise to localized but overlapping Clomb potential wells via Poisson’s equation. We assumeusually in solid-state physics local charge neutrality. Tmeans that any ionicity must be totally screened withincertain cluster range. Since we have no detailed informaon that screening in an alloy, we simply use a Gaussscreening7 within a rangep21, i.e., we combine the ionicpoint chargeI sW
Q5ZsW
Q2AsW
Q with a Gaussian screening chargaccording to
I sWQS d~rW !2
p3
p3/2e2p2r2D . ~30!
The screening lengthp has to be chosen such that the lattisum of the added heavily overlapping Gaussian chargesities is essentially zero. This treatment is comparable todiscussion in the literature. In Refs. 24 and 25 a screeimpurity model is used to describe the effect of the Madlung potential in a disordered alloy. It is based on the obsvation that almost all of the compensating charge is locain the first coordination shell around the impurity.
The exchange and correlation potential which is less ssitive to density modulations is calculated in an ASA aproximation:
Vxc,sWQ FnsWQ1 (
RW 81sW8ÞsW,Q8csW8Q8nRW 8sW8
Q8 G ~rW !
for urWu<rASA,sWQ , ~31!
Vxc ,interstitial5const for urWu.rASA,sWQ . ~32!
This latter simplification used in our applications of Sec. Vhowever not a necessary prerequisite of the approach. Bywe complete the charge and spin self-consistency cycle,vided the average of the Green’s matrix~22! is obtained.
III. THE COHERENT-POTENTIAL APPROXIMATION
The preceding section was devoted to the constructiothe self-consistent potential from a known Green’s functin the orbital representation. Now, we come to the other pnamely to the approximate calculation of the correspondGreen’s matrix. This is an algebraic problem. Thus, unlikethe KKR approach, the multiple-scattering problem is solvin the corresponding arithmetic space, rather than in thespace.9 We have to make the fundamental two-terminal aproximation. Then, the structure of the Green’s matrix~22!and the requirement to treat both the Hamiltonian andoverlap matrix in the same manner without neglecting odiagonal disorder leads naturally to the BEB theory.
-stann
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A. Algebra of pseudospins and the Blackman-Esterling-Berkformalism
In Sec. II B we introduced the pseudospins. Now we gthem an algebraic structure, which leads in consequencthe BEB transformation. For a given sitei , the pseudospinh iQ is a column indexed byQ. For the following, it is advan-
tageous to consider it as a diagonal block of a rectangmatrix h5ih i
Qd i i 8i , to be used together with its transposhT in the definition of configuration-dependent projectox:
x[hhT, hTx5hT, xh5h . ~33!
This x is simply a square diagonal matrix with diagonelementsx i i
QQ5h iQ of Eq. ~13!. It has the properties
x25x, TrQx i i 85d i i 8 . ~34!
We expand the Hamiltonian and overlap matrices to coninformation of all possible configurations in such a way ththe formerly introduced matrices of a given configuration aobtained by projection:
H5I (QQ8
h iQ^ iQmuH~rW !u i 8Q8m8&h i 8
Q8I5hTHh ,
~35!
S5I (QQ8
h iQ^ iQmu i 8Q8m8&h i 8
Q8I5hTSh . ~36!
H and S are the expanded matrices.~The h ’s provide amapping of the product Hilbert space of all configuratioonto the valence Hilbert spaces of given configurations. T‘‘tensor products’’ x5hhT, at variance, are projectorwithin the expanded Hilbert space onto the subspaces cosponding to those configurations. In this sense the Btheory is a kind of ‘‘tensorial’’ CPA. Compare the paper Re26, where a similar language is used in the framework ofKKR-CPA.! It is worth noting thatH andS are stochasticand not translational invariant, whileH and S will be ap-proximated in the following by expressions which arelonger stochastic and are translational invariant. This strture is used in the BEB formalism20 ~the xi and yi in thatpaper are the componentsh i
A andh iB for the binary case!.
The pseudospinsh enter the matricesH andS in threeways: at the left and right ‘‘matrix terminals,’’ and via thcrystal potential and the core orthogonalization correctioWe average over theh ’s entering on sites different fromboth terminal sites of the matrices. This terminal-point aproximation preserves the fullh dependence at both terminasites and is far better than the former virtual-crysapproximation,9 since the only averaged parts are the crysfield and the orthogonalization corrections fromthird cen-ters. The expanded matrices are now split into on-siteoff-site terms:
H5H1H, Hx5xH , ~37!
S5S1S, Sx5xS, S'1 , ~38!
with the detailed structure:
5722 55KOEPERNIK, VELICKY, HAYN, AND ESCHRIG
Sim,i 8m8QQ8 'd i i 8,QQ8Fdmm82(
lcQ
~ iQmu lQc!clQ~ lQcu iQm8!G , ~39!
Sim,i 8m8QQ8 '~12d i i 8!F ~ iQmu i 8Q8m8!2(
lcQ
~ iQmu lQc!clQ~ lQcu i 8Q8m8!G , ~40!
H im,i 8m8QQ8 'd i i 8,QQ8F ~ iQmu t1Vi
Qu iQm8!1 (lÞ i ,Q
~ iQmuclQVl
Qu iQm8!2(lcQ
~ iQmu lQc!clQ« lc
Q~ lQcu iQm8!G , ~41!
H im,i 8m8QQ8 '~12d i i 8!F ~ iQmu t1Vi
Q1Vi 8Q8u i 8Q8m8!1 (
lÞ i ,i 8,Q~ iQmucl
QVlQu i 8Q8m8!2(
lcQ
~ iQmu lQc!clQ« lc
Q~ lQcu i 8Q8m8!G .~42!
poel.sonreu
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e
hf
onethe
butlex
The energy levels« lcQ are the core levelsc of atom Q. The
combination of the matrix elements of the three centertential and the core orthogonalization correction at a givsite is known as the Phillips-Kleinman pseudopotentia27
Normally, its action on smooth functions is quite small,the average over it at third centers away from the basis fution centers is not a severe approximation. Now we are ppared to formulate the multiple-scattering theory in the psdospin language.
B. The scattering problem
The random arrangement of different types of atomsthe disordered alloy and the resulting breaking of the tralational symmetry leads to an incoherent scattering ofelectrons. By defining a coherent reference medium it is psible to collect all incoherent parts of the scattering wrespect to this medium in one quantity, called the scattematrix T. Then the Green’s function of the disordered allis expressed in terms of the propagation in the coherentdium and of the scattering matrix. The coherent medium wbe defined by a self-consistency condition.
-n
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e-ll
We start byQ expanding the equation of motion of thone-particle Green’s matrix~8!:
hT~vS2H !hG51 . ~43!
From Eq.~22! we take the idea to multiply this equation with on the left and withhT on the right to get an equation omotion forG5hGhT:
x~vS2H !hGhT5x . ~44!
Now we use the definition ofx and the commutation rules inEqs.~37! and ~38!:
@vS2H1x~vS2H#x!hGhT5x . ~45!
When using the vacancy concept we associate at leastbasis orbital with every site, sort of a smooth Gaussian atvacancy site.
Then the expression on the left-hand side is stochasticHerglotz, so that the resolvent exists in the upper compv half plane:
n’s
G[hGhT5xhGhT ~46!
5x@vS2H1x~vS2H !x#21x
~47!
Here we introduced the Herglotza ~the stochastic potential! and the HerglotzG ~the nonstochastic nonlocal coherent Greematrix!. The next steps are aimed at collecting all stochastic quantities (x anda) into one matrix:
~48!
G5b~b2G!21G5G1G~b2G!21G[G1G T G . ~49!
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ly
ert
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55 5723SELF-CONSISTENT LCAO-CPA METHOD FOR . . .
At this stage we expressed the multiple scattering in termthe translational invariant coherent Green’s matrixG ~in theproduct Hilbert space! and the random perturbationb.
The averaging procedure now follows the well-knowscheme. The nonregular inverse potentialb, containing allrandomness, defines the scattering matrixT, whose averageis required to vanish:
^T&5^~b2G!21&[0⇒^G&5G . ~50!
In that way we defined the self-energyS. Remembering ouraim we examine some projector relations concerningtwo-site conditional average ofG. From Eq.~46! we deduce
G5xGx, x i iQQuq→ i5dqQ , ~51!
Gi i 8QQ8u q→ i
q8→ i 85Gi i 8
qq8u q→ iq8→ i 8
dQq,Q8q8 , ~52!
where the subscriptq→ i again means to fix the speciesq atsite i . We gain the insight, that theQQ8 matrix Gi i 8 ~fori ,i 8 fixed!, as expected by its definition, contains only ononzero element under the conditionq→ i ,q8→ i 8 ~no aver-ages were taken!. Furthermore the connection to the fulaveragedG is given by
G i iQQ85^Gi iQQ8&5(
qciq^Gi iQQ8&q→ idQq,Q8q
5ciQ^Gi iQQ&Q→ idQQ8 ~53!
and
G i i 8QQ85^Gi i 8
QQ8&5(qq8
ciqci 8
q8^Gi i 8QQ8& q→ i
q8→ i 8dQq,Q8q8
5ciQci 8
Q8^Gi i 8QQ8& Q→ i
Q8→ i 8. ~54!
In other words the elements ofG give the physical two-siteconditional averaged Green’s matrix connected with the dsity ~26!. This just is the gain of using the product Hilbespace.
For the caseciQ50 one gets from Eq.~48! bii
QQ50 andhenceGi iQQ50 independent ofG. That means, configuration( iQ) with ci
Q50 must be removed from the beginning: thset of speciesQ may depend on the sitei in cases of siteselectivity.
C. The single-site approximation
Except for the terminal-point approximation for the mtrix elements of the Hamiltonian and the overlap matrix,troduced in Sec. III A, we made no approximation in Sec.so far. In order to solve Eq.~50! we have to introduce anapproximation similar to the single-site approximation.1,2Werequire the self-energy to be diagonal in site indices. Tleads to a site diagonal inverse scattering potentialb and wemay decouple~50!:
T5~12t G8!21t, t i i 85d i i 8~bii2G i i !21 , ~55!
G i i 88 5~12d i i 8!G i i 8 ~56!
with the condition
of
e
n-
-I
is
^t&[0 . ~57!
After expanding the scattering series Eq.~55! and inserting itinto Eq. ~49! the averagedG becomes
^G&5G1O~^t4&! ~58!
as is easily seen in Fig. 1. The final result for the twofoldconditional averages in Eq.~26! is then
^Gi iQ&Q→ i5G i iQQ
ciQ , ^Gi i 8
QQ8& Q→ iQ8→ i 8
5G i i 8QQ8
ciQ ci 8
Q8, ~59!
except for terms of the orderO(^t4&).This may be a good place for a few comments on the BE
approximation and its quality. As shown here, in the generacase of diagonal and off-diagonal disorder, the BEB approxmation is formally exact to the same order of the multiplescattering expansion as the CPA is in the case of the diagondisorder. We have made a number of comparisons with thresults of a direct calculation of the projected DOS for onedimensional model random alloys which indicate an excelent quality of the BEB approximation in a wide range ofcomponent bandwidths and other alloy parameters. The aternative treatment of the off-diagonal disorder, the Shibmultiplicative ansatz, can be shown to be a special casethe BEB method, Ref. 28. While the self-consistent equations reduce in number for the Shiba method, an additioncalculation is needed to generate the physically and comp
tationally relevant component projected averagesG i i 8QQ8. For
a nonzero overlap matrix, the generalized Shiba matrix factoshould be the same for the Hamiltonian and the overlap mtrices, and this appears as excessively restrictive for userealistic calculations. Details of this model analysis will bepresented in a separate communication.
In the present paper, we treat consistently the caseseveral inequivalent sublattices in the elementary cell. Fothe structure of the self-energy and the validity of the singlesite approximation in this case we refer to Appendix A.
IV. THE COMPUTATIONAL SCHEME
In this brief section we summarize the basic features othe present approach, specify the orbital basis used, and gthe working equations as they were actually coded and usin the self-consistent computation cycle.
FIG. 1. The diagrams of the first four orders int.
-rithmorbipirelikraani--
,dp
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reison
eros
eosmso
fmpenot fkeeau
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5724 55KOEPERNIK, VELICKY, HAYN, AND ESCHRIG
A. General characterization
Any meaningfulab initio theory of the electronic structure of alloys should be an extension of a well tested desction of the corresponding pure component crystals. Inpresent case, this is represented by the orbital theory sumrized in Ref. 7. In this sense, we continue the earlier wRef. 9. This early paper served to demonstrate that an orCPA need not be a model toy, nor a qualitative semiemical scheme. Both conceptual and practical limitationsstricted the approach to the alloys with near components,CuNi. By contrast, our present method is sufficiently gene
~1! The present scheme is fully charge self-consistent,electron, and, in principle, not restricted by approximatioof the atomic-sphere type~additional approximations specfied below are not essential!. To achieve a wider applicability, we use the scalar relativistic model of Ref. 29.
~2! We use a scattering formulation in the matrix formwhich in its general scope parallels the KKR-CPA methoIn fact there are similarities with the screened KKR aproach.
~3! In the orbital formulation leading to the energindependent Hamiltonian matrix, the present approach palels the TB-LMTO-CPA. However, the present approachnot based on the TB transformation, and includes the orboverlaps.
~4! Both the diagonal and the off-diagonal disorder aincluded on the same footing in the BEB framework. Thopens the possibility to apply it to local basis representatiin the most flexible way, including nonorthogonal bases.
~5! The method is formulated so as to incorporate sevcomponents and several sub-lattices with different comptions.
B. Orbital basis
Our computational scheme follows the lines of Ref. 7. Ware dealing with a minimal basis set of modified orbitals. Feach atom the local valence basis is represented by oneper angular momentum and spin. The maximum angularmentum is determined by the type of the element, for trantion metals we uses, p, andd states and for actinides up tf states:
with Ylm being real spherical harmonics. The radial partfdoes not depend on the magnetic quantum numberm. Asdescribed, the crystal potential~27! is chosen to be a sum ooverlapping site potentials. Additionally, we want to assuthe shape of the site potentials to be spherical. This simfication implies that all aspherical parts of the crystal pottial are claimed to be approximated by the lattice sumthose spherical site potentials. Surely this is not exact, bua wide range of applications especially for closed-pacstructures this method together with the valence basis trment as described below has been proved to give resbetween muffin-tin and full potential approaches.~Aspheri-cal site potentials could be readily incorporated.! First ineach self-consistency cycle, the species-dependent bstates are recalculated by solving an atomic Schro¨dinger orDirac equation for each atomic species with a poten
p-ea-ktalr--el.ll-s
.-
l-sal
s
ali-
rtateo-i-
eli--fordt-lts
sis
l
VRW sWQ . Details concerning the relativistic implementation of t
LCAO method are described elsewhere.29 VRW sWQ is the poten-
tial corresponding to the resulting density of the previocycle. Some atomic potential is used to start with.
While recalculating the core orbitals readily from the spotentialVRW sW
Q , the valence basis states will be modifiedmake them more suitable for the construction of Bloch wafunctions. They will be recalculated with an additional atractive potential term (r /r 0)
4 which compresses strongly thextended valence states. These compressed states areciently complete in the interstitial region and have a stronreduced overlap, compared to those calculated from the psite potential. To close the recalculation of the orbitals thoptimized valence orbitals will be orthogonalized to the costates calculated from the pure potential. By this waynumber of multicenter integrals is reduced. The paramer 0 are chosen by convergence checks of the band enerThey scale with the lattice constant and are determinedprinciple, by the lattice structure.
C. Implementation of the self-consistent cycle
For the construction of the nonstochastic and hence latsymmetric Hamilton and overlap matrix we use Bloch suof AO’s in the usual manner resulting in expressions ofform @m5( lm)#:
H sWm,sW8m8QQ8 ~k! and S sWm,sW8m8
QQ8 ~k! , ~61!
for the off-diagonal parts~40,42!. Thesek-dependent matrixelements together with the elements of the self-energybe inserted in Eq.~47! to calculate via Brillouin-zone inte-gration the on-site elements of the coherent Green’s ma
GsW,mm8
QQ85E
BZd3k@v2S1vS~k!2H~k!# QQ8
sWm,sWm8
21. ~62!
The off-diagonal elements forQÞQ8 of the Green’s matrixare kept during the self-consistency cycle, but they vanishthe end. Details concerning the crystal symmetry are giveAppendix B. The site diagonal inverse scattering potenb ~48! can be written as
bRW sWm,RW sWm8
QQ85(
qhRW sWqbsW,mm8
~q!QQ8~63!
with
bsW,mm8
~q!QQ85dQQ8,Qq~H sW
q2vS sW
q1v2S
sWq!mm8
21 . ~64!
To fulfill condition ~57! we choose the averagedt-matrixapproach. That means, we calculate from Eq.~55! the aver-aged single-site scattering matrix
^tsW,mm8
QQ8&5(
qcsWq~b
sW~q!
2GsW!QQ8mm8
21, ~65!
which should be zero. The resulting change in the senergy is dS5(11^t&G)21^t&. In an inner CPA self-consistency cycle the Green’s matrix, the self-energy andscattering matrix are recalculated until the averaged sinsite scattering matrix vanishes.
sn’thhealtolos.rge
Aitof
tilf
boraea
urtyltFecutti
ethis
t isOults.d-ola-dn-xi-u-by
chTheTothens.ithB-
eri-roves.weveeticticales-
O,
ulles
55 5725SELF-CONSISTENT LCAO-CPA METHOD FOR . . .
To perform the involvedv integration along the real axiaccurately, we employ the Herglotz property of the Greematrix and of related quantities, which allows us to usecommon trick of changing the integration contour from treal line to a semicircular path in the upper complex hplane. The integrals depend only on the end points andfunctions to be integrated are much smoother far away frthe real axis. The integration starts in energy regions bethe band bottom and ends at the Fermi energy, which habe readjusted to get the right number of occupied states
Once reaching the converged self-energy the eneintegrated Green’s matrix elements are inserted into thepression for the density~26!. In view of the spherical ap-proximation of site potentials, which is single-site CPconsistent, we compute the spherical average of site densonly. We end up at a density representation consistinglattice sum of local spherical, overlapping densities.
This representation of the density leads to the potenvia the expressions in Sec. II D. In this way, the seconsistency loop is closed.
V. ILLUSTRATIVE APPLICATIONS
The numerical tests shown below serve to test the reliaity and flexibility of our approach. It should be suitable flight and moderately heavy atoms, and for alloys of metwith widely different pure component bandwidths. Wpresent calculations and comparisons for some magneticordered binary systems and for a ternary system.
The first example is the bcc FeCo alloy. One main featof this compound is the transition from filled iron majoribands to lower filling by increasing the iron content, resuing in a maximum of the magnetization curve at 70%concentration. This compound is widely studied. We callated the magnetization versus concentration at a fixed la
FIG. 2. bcc-FeCo alloy: The dashed lines are the TB-LMTASA result taken from Tureket al. The full lines are our resultsexperimental values are marked by filled symbols.
se
fhemwto
y-x-
iesa
al-
il-
ls
lly
e
-
-ce
constant~Fig. 2!. For comparison with other methods wpresent also the LMTO results, published in Ref. 12. Bocurves show the characteristic maximum, while our curvecloser to the experimental one. The higher Co momenconsistent with the bcc Co moment obtained by pure LCAcalculations. The deviations to the LMTO result may resfrom a slightly higher pure Fe moment in their calculation
The above-mentioned applicability to different banwidths and relativistic cases is tested in the following twexamples. The fcc-CoPt alloy was explored by the fully retivistic KKR method.30 The magnetization of this compounexhibits a transition to zero moment with increasing Pt cocentration. Here the bandwidth of the pure Pt is appromately twice the Co bandwidth. Our scalar relativistic calclation reproduces the slightly nonlinear behavior as giventhe KKR results, in good agreement with experiment~Fig.3!.
The last binary example is the Invar alloy fcc FePt, whishows anomalies, for instance in the thermal expansion.critical region lies between 70% and 80% Pt content.investigate the nontrivial behavior of the moment andlattice constant one has to perform total-energy calculatioHere we compare the moments given by our method wthose calculated by the recently developed relativistic TLMTO-CPA ~Ref. 31! at a fixed lattice constant~Fig. 4!.~The LMTO data are taken from Ref. 32!. Both results shownearly the same local and averaged moments. The expmental data are given in Ref. 33. These three examples pthe applicability of our method in the case of binary alloy
To show an application to more complicated structuresswitch now to a ternary system. The FeAlMn alloys habeen known to show interesting and complicated magnand structural features. We want to present here theoreresults for the ferromagnetic phase. The composition invtigated has the formula Fe0.892xMn0.11Al x . When disregard-
- FIG. 3. fcc-CoPt alloy: The dashed line is calculated by the frelativistic KKR-CPA. These data and the experimental valu~filled symbols! are reported in Ebertet al.Our result is marked bya full line.
eia
l-le
ith
hisenttohe
ag-lecu-m--
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erex-
heve
entsborated
hthav-ed
t ofbyout
th
ym
ry
m
sthe
5726 55KOEPERNIK, VELICKY, HAYN, AND ESCHRIG
ing the different constituents the structure is a bcc (A2) lat-tice. Comparison is made to measurements by Bremet al.34,35 First we calculated a bcc alloy where each siterandomly occupied. The room-temperature lattice constwas taken from experiment~Bremers! and reduced by0.5% to the T50 value, estimated from the thermaexpansion coefficient. As can be seen in Fig. 5 and Tab
FIG. 4. fcc-FePt alloy: The dashed lines are calculated withfull relativistic TB-LMTO-CPA, developed by Shicket al.The fulllines are our results, experimental values are marked by filled sbols, they are taken from Wasserman.
FIG. 5. bcc-FeAlMn alloy: The moment per atom in the ternabcc alloy for theA2 structure and for theB2 structure in compari-son with experiment. For details see the text.
rssnt
I
the values of the averaged magnetization coincide well wthe experiment at low Al concentrations. Beyondx50.2 themagnetization at helium temperature is breaking down. Tphase transition is not obtained in our ternary CPA treatmof theA2 structure. To go a step further we paid attentionthe fact, that with increasing Al content experimentally tstructure seems to switch to theB2 structure. To investigatethe influence of this structural long range order on the mnetization we performed a CPA calculation with two simpcubic sublattices, displaced by (1/2,1/2,1/2). One is ocpied with pure Fe atoms, the other is occupied with the coposition Fe0.7822xMn0.22Al 2x , which corresponds to the ternary alloy with the Al contentx. This subdivision ismotivated by the observation that in the system AlFe~Ref.36! ~a DO3 derived structure atxAl,40%) and in similarsystems like SiFe~V,Mn!,37,38 the Fe atoms tend to occuptwo of the four fcc sublattices of the DO3 structure. Theselattices reduce to theB2 structure if one considers the othtwo sublattices to be equally occupied. We transfer theperimentally known occupation preferences to theB2AlFeMn by assuming one site to carry pure iron atoms. Tresults are striking. They are collected in Table II. We githe local moments of the two different sites in theB2 struc-ture and the moment per atom.
One remarkable feature is easily seen. The local momof the Fe atoms depend strongly on the nearest-neighshell. The Fe atom with eight pure neighbors has a saturmagnetization of about 2.5mB , while the atom on the Fesublattice, for which only a small percentage of the eigneighbor atoms are iron, has a reduced moment. This beior is known from the antistructure atoms in orderFe50Al50 ~Ref. 39! and from the alloys SiFe and SiFeV.13
Most important, we get a sudden transition of the momenthe pure iron sublattice to ferrimagnetic coupling, thereenormously reducing the total magnetic moment at ab30%
e
-
TABLE I. The local moments and the moment per atoat various Al concentrations for theA2 structure of theAl xFe0.892xMn0.11 alloy.
TABLE II. The local moments of both simple cubic sublatticeand the moment per atom at 10, 20, 30, and 35% Al content forB2 structure of the Fe1.0Al 2xFe0.7822xMn0.22 alloy. The numberbesides the elements refers to the sublattice.
aluminum. This reduction meets excellently the experimtally observed decrease of the magnetization~Fig. 5!.
VI. CONCLUSION
Starting from the local orbital representation of the KohSham theory we constructed a fully charge and spin sconsistent CPA scheme for calculation of the electrostructure of substitutional disordered alloys. An optimizatiresulting into a strong localization of the valence orbitalsneeded for the assignment of the atoms to the basis stand for a single atom decomposition of the potential. Evso, the basis states are nonorthogonal. For alloys with vdifferent components, it is essential to treat properly bothdiagonal and the off-diagonal disorder both in the overmatrix and in the Hamiltonian matrix. In our method, thisachieved by the terminal-point approximation, permittingto achieve the CPA solution within the BEB formalism icorporating a self-consistent recalculation of all matrix ements in each cycle. Among the advantages of the loorbital-matrix representation is a simple inclusion of the slar relativistic effects and its straightforward applicabilitystructures with complex unit cells and with a complete opartial disorder. The numerical results presented have ancuracy comparable with the KKR and LMTO CPA methocurrently in use.
To summarize, the pure orbital alternative to the KKCPA and to the TB-LMTO-CPA appears as feasible ameaningful. Like the recently developed screened KKRcombines the rigor and directness of the KKR with the prtical simplicity of the LMTO; on top of that its LCAO formprovides physically illuminating ‘‘quantum chemical’’ insights.
There are several obvious directions for a future woFirst, a number of improvements, especially making the cstruction of the potential more flexible, will be neededgenerate a standard computer code, suitable for generalWithin the CPA, several extensions are inevitable, butdifficult to achieve, notably the inclusion of the spin-orbeffects, and the calculation of the total energy as a funcof the alloy parameters. A more general question whicharisen during the work is the position of BEB among othalloy formalisms, and the reasons for the unexpected quaof the BEB-CPA.
All this methodological work should not mask the fathat we have developed the present method becausecomputationally facile and suitable for the studies of coplex alloy structures. We intend to concentrate primarilythis aspect.
ACKNOWLEDGMENTS
We thank J. Kudrnovsky´ and V. Drchal for helpful dis-cussions and for giving us the TB-LMTO-CPA program fcomparative calculations. Furthermore we thank M. RichJ. Forstreuter, and A. Ernst for discussing details concernthe LCAO formalism. Finally, we wish to thank H. Bremeand J. Hesse for providing us with experimental resultsthe FeAlMn system, partially prior to publication, and fdiscussions. B.V. acknowledges financial support byMax-Planck society.
-
-f-c
ses,nryep
s
-al-
ac-
-dit-
.-
se.t
nsrty
is-n
r,g
n
e
APPENDIX A: THE SELF-ENERGY
Without approximationS is not diagonal in real spaceWe may write with diagonal invertiblel5vS2H ~droppingthe underbar!:
G5x@ l2x~Q2G21!x#21x, Q5v2S .
Corresponding to Eq.~48! one derives~with site diagonall5x l21x)
G5l@12~Q2G21!l#215G1GTG,
which gives
l5~G1GTG!@12~Q2G21!l# ,
Tl5~11TG!~Ql21!, ~^T&50! .
Now we are asking for elements with the right index~calledf ) being at a site with only one possible occupation. Th^l f&5l f holds. The averaging procedure removes terwith ^T& and yields
Q i fl f5d i f ,
S i f5d i f @v~12S f !1H f # .
Becausea thus vanishes in the case of fixed site occupatwe have to exclude these sites from the CPA equations~50!,~57!. This general peculiarity ofS is preserved in the singlesite approximation, where we set by definition all site odiagonal elements ofS to zero. It gives us hope to expecthat the CPA would be better if less stochastic sites existethe unit cell.
APPENDIX B: THE BRILLOUIN-ZONE INTEGRATION
In order to calculate the on-site elements of the coher
Green’s matrixGsW,mm8
QQ8 we perform Brillouin-zone integra-
tions overGsW,mm8
QQ8,k. To reduce this time consuming step itunavoidable to integrate only in the irreducible part of tBZ. Unfortunately, we have to symmetrize the Green’s mtrix after doing so, becauseGk does not form a unit representation of the full Brillouin-zone symmetry. In simplcases, like unit cells where each site has the full cubic symetry and with basis states up tod states, this task is very
easy. ThenGsW,mm8
QQ8 is diagonal inmm8 and the matrix ele-ments for given angular momentuml arem degenerated. Theaverage over all diagonal elements to a fixedl gives just thedesired symmetrization ofG calculated by integration ovethe irreducible part of the Brillouin zone.
For more complicated crystals we outline the treatmeLet us start with the definition of the space-group operatU5$uut%, where we used the symbol after Seitz for the oeration which takes a point atr to r 85Ur5ur1t. u meansa point-group operation, a proper or improper rotation, wht is a translation through a vector. It can be a composite oprimitive lattice translationRW and of an essential nonprimitive translationt0, which in cases of nonsymmorphic spa
in
a
s
t-
p
dti-e
-
the
ting
f
s
Eq.rm
hen
5728 55KOEPERNIK, VELICKY, HAYN, AND ESCHRIG
groups can never be made to vanish by shifting the origRef. 40. The set of symbols$uut0% with t0 being a specifiedset of nonprimitive translation vectors does not form itselfsubgroup in nonsymmorphic space groups. We therefoconsider the vectorst as general composits of primitive andnonprimitive lattice vectors. This ensures the group propeties of the symbols$uut%.
The action of these operations on the LCAO basis stategiven by
UuRW sWm&5(m8
uU~RW 1sW !m8&am8m . ~B1!
The coefficientsa are representation matrices of the poingroup elementu for an angular momentuml and provide thetransformation of the spherical harmonics included in thbasis states. This relation holds for arbitrary space grouTogether with the transformation properties of the phase fator exp(ikRW ) we deduce
GsW,mm8
QQ8,u†k5(
nn8anm* G
UsW,nn8
QQ8,k an8m8 ~B2!
holding for site diagonal elements only. This formula is valifor arbitrary space groups. Note, that a space-group operaU acts onsW, while only the corresponding point-group elementu acts onk. Now we decomposed the integral in thwhole zone into a sum over integrals in the irreducible pabut over rotated matrices,u represents all point-group operations:
b
s
c
s.
ys
:
re
r-
is
es.c-
on
rt
GsW,mm8QQ8 5E
BZd3kGsW,mm8
QQ8,k ~B3!
5(U
Eir BZ
d3kGsW,mm8.QQ8,u†k
~B4!
The sum runs over all space-group elements. Examiningsymmetry relations~B2! we get
GsW,mm8QQ8 5(
UEir BZ
d3k(nn8
anm* G UsW,nn8QQ8,k an8m8 ~B5!
5 (sW8,nn8
GsW8,nn8QQ8~ ir BZ! (
UsW5sW8anm* an8m8.
~B6!
The space-group operations can be classified by inspecthe pairs of basis vectorssW, sW8 which are transformed intoanother. There exists for eachsW8 a possibly empty subset oU ’s which transforms a given vectorsW into sW8:sW85UsWmod$RW %. For a specifiedsW, the union of the subsetbelonging to all basis vectorssW8 in the unit cell contains eachgroup operation exactly once. Thus, the second sum in~B6! runs over the set of group operations which transfosW into sW8. The superscript onG denotes thek integration overthe irreducible part. Since the last sum in Eq.~B6! is onlystructure dependent it is performed once for ever and tthe symmetrization is much faster than to calculateGk foreachk point in the whole zone.
ys.
. F
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