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Theory of electronic transport in random alloys with short-range order: Korringa-Kohn-Rostoker nonlocal coherent potential approximation

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Theory of electronic transport in random alloys with short-range order:

Korringa-Kohn-Rostoker nonlocal coherent potential approximation

P. R. Tulip and J. B. Staunton

Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom S. Lowitzer, D. Ködderitzsch, and H. Ebert

Department Chemie und Biochemie, Physikalische Chemie, Universität München, Butenandtstrasse 5-13, D-81377 München, Germany 共Received 14 December 2007; revised manuscript received 6 March 2008; published 11 April 2008兲

We present anab initioformalism for the calculation of transport properties in compositionally disordered systems within the framework of the Korringa-Kohn-Rostoker nonlocal coherent potential approximation. Our formalism is based on the single-particle Kubo-Greenwood linear response and provides a natural means of incorporating the effects of short-range order upon the transport properties. We demonstrate the efficacy of the formalism by examining the effects of short-range order and clustering upon the transport properties of disordered AgPd and CuZn alloys.

DOI:10.1103/PhysRevB.77.165116 PACS number共s兲: 72.10.⫺d, 71.20.Be, 72.15.Eb

I. INTRODUCTION

The Korringa-Kohn-Rostoker coherent potential approx- imation1 共KKR-CPA兲 represents an extremely successful one-electron theory capable of describing the properties of many compositionally disordered alloy systems. In particu- lar, in combination with density functional theory2–5 共DFT兲 and the Korringa-Kohn-Rostoker method of band theory,6–8 it provides a fully first-principles description of such sys- tems. A long history of successful applications9–14attests to the utility and accuracy to which the method is capable.

Of particular relevance to our discussion here, the KKR- CPA has been used to calculate the transport properties of such alloys. Historically, the early work concerning this topic involved the use of a Boltzmann equation,15–17and although these works demonstrated remarkable agreement with ex- periment, they did suffer from two notable defects: namely, the requirement that well-defined energy bands exist within the alloy, and the neglect of vertex, or “scattering-in,” terms.

The former defect limits the application of such a theory to weak-scattering alloys only, while the latter could be ex- pected to lead to significant error in systems such as those where appreciables-p ors-d scattering manifests itself.

Velický18 developed a CPA theory for transport using the Kubo-Greenwood19,20formalism as his starting point, which was applied to a two-level tight-binding Hamiltonian. Al- though capable of yielding vertex corrections, this approach suffered from difficulties when applied to realistic systems.

For example, the necessity of assuming that the wave functions are identical on all lattice sites, irrespective of the occupying atomic species. The seminal work of Butler21 resolved these issues, as he developed a KKR-CPA theory based on the Kubo-Greenwood linear response formalism.19,20As a multiple-scattering based approach, this did not require the existence of well-defined energy bands, and further, Butler demonstrated how the vertex corrections arose quite naturally within his formalism. Although the for- mal developments of this work followed Velický’s quite closely, the use of a realistic single-electron muffin-tin Hamiltonian allowed connection with first-principles meth-

ods to be made. The method has been applied with success to a range of alloy systems22,23and it has also been successfully extended to the relativistic regime.24

Of course, all of these calculations suffer from the main drawback of the CPA; namely, that as a single-site mean- field theory, it is incapable of incorporating the effects of local-environment fluctuations in the alloy crystal potential.

The CPA therefore explicitly ignores the effects of such short-range order 共SRO兲 effects upon the physics of disor- dered alloys. As discussed by Gonis,25 these statistical fluc- tuations can be important. While in general the presence of SRO is likely to diminish the resistivity, there are examples, the so-called “Komplex”K-state alloys,26where the onset of SRO is accompanied by an increase in the alloy residual resistivity.

Recently, there have been some successful attempts at cal- culating the effects of SRO upon the electronic structure of disordered alloys and they provide the means to study trans- port properties. Mookerjee and co-workers,27–29 using a tight-binding, linear muffin tin orbitals 共TB-LMTO兲 method30 in conjunction with an augmented space formalism31,32 and real space recursion method,33 described SRO effects on alloy electronic densities of states and related quantities, while Saha et al.34 obtained spectral functions within the same framework. Recently, Tarafderet al.35devel- oped a formalism for the optical conductivity and reflectivity from the same basis and used it to study copper-zinc alloys.

To date, however, it has not been possible to incorporate this technique fully within electronic density functional theory.

The recent work of Rowlands et al.36–38 along with Biava and co-workers et al.39,40 concern a development which is not restricted in this way. References 36 and 37 formulate and illustrate a successful method for incorporating the ef- fects of SRO within the framework of KKR-CPA theory, while implementation for realistic systems is described in Refs.37and39. The method can be readily combined with density functional theory to provide a first-principles descrip- tion of disordered alloys, as demonstrated in Refs.38and41.

This nonlocal CPA共NLCPA兲 theory is based on reciprocal space coarse-graining ideas introduced by Jarrell and

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Krishnamurthy,42originating from the dynamical cluster ap- proximation 共DCA兲.43–45 The KKR-NLCPA36,37 introduces an effective 共translationally invariant兲 disorder term ␦G, which represents an effective propagator that accounts for all nonlocal scattering correlations on the electronic propagation due to disorder configurations and modifies the structure constants accordingly. By coarse-graining reciprocal space, one naturally introduces real space periodically repeating clusters. As such, the NLCPA maps an effective lattice prob- lem to that of an impurity cluster embedded in a self- consistently determined effective medium, and thus yields a cluster generalization of the KKR-CPA that includes nonlo- cal correlations up to the range of the cluster size. Unlike other cluster approaches, such as the molecular CPA,46 it is fully translationally invariant, that is, the effective medium has the site-to-site translational invariance of the underlying lattice. It is also computationally tractable, largely on ac- count of the reciprocal space coarse-graining procedure em- ployed.

Thus far, the NLCPA has been employed to investigate the effects of SRO upon the electronic structure of a range of realistic systems by using both the muffin-tin Hamiltonian37,47and the tight-binding approach.48 Given its ability to successfully address such issues and its proven in- corporation into an electronic DFT38for disordered systems with SRO, it makes sense then to extend it to the calculation of transport properties. To this end, by invoking time depen- dent DFT49within the adiabatic approximation, in this paper, we present a formalism for the determination of the residual resistivity in the KKR-NLCPA and explicitly demonstrate the efficacy of the method through application to several realistic alloy systems. Our theoretical formalism is a careful gener- alization of that of Butler21 and where appropriate we omit the steps in the derivation which can be straightforwardly obtained from this paper. Our paper is structured as follows:

The next section gives a short overview of the conductivity tensor and then the transport coefficients available from the Kubo-Greenwood formalism. This is followed by a section containing the salient points of the KKR-NLCPA formalism including its use in implementing the density functional theory. We then develop our theory for the conductivity of disordered systems with short-range order, which includes the treatment of “vertex corrections.” The implementation strategy is outlined before calculations for the effects of SRO on the resistivity of both bcc CuZn and fcc AgPd are pre- sented.

II. CONDUCTIVITY: KUBO-GREENWOOD LINEAR RESPONSE

The Kubo-Greenwood19,20 linear response formalism states that, for a disordered system, the symmetric part of the conductivity tensor has coefficients C, which can be deter- mined from the evaluation of an expression of the form

C= Tr具O1GO2G典, 共1兲 whereG is a single-particle Green’s function, which is de- pendent on the details of the effective one-electron potential.

O1 and O2 are operators, and the angled brackets denote a

configuration average over the distribution of the potentials.

To determine the dc conductivity, we consider the follow- ing expression19–21

␮␯共EF兲= ␲ប

N⍀

mnJmn;␮Jnm;␯共EFEm共EFEn

,

共2兲 whereJmn;␮=具m兩J兩n典denotes the matrix element of the cur- rent operator in the␮th spatial direction, which is given by

J= −iបe m

r, 共3兲

with兩m典 and兩n典 denoting the eigenfunctions of a particular configuration of the disordered system. Here,N is the num- ber of atoms and⍀is the volume per atom.EFis the Fermi energy.

Within the KKR approach, the electronic structure of the alloy is expressed in terms of the single-particle Green’s function, rather than in terms of eigenstates and eigenvalues of the Hamiltonian; we can introduce the Green’s function simply by using the identity50

−␲

n n典具nEEn= lim0ImGE+i, 4

while the awkward imaginary part of the Green’s function may be removed by writing

− 2i␲

n 兩n典具n兩共EEn= lim0关G共E+iG共Ei兲兴.

共5兲 Inserting this into Eq. 共2兲, yields the following for the conductivity

␮␯= 1/4 lim

0关␴˜␮␯E+,E+兲−␴˜␮␯E,E+

−␴˜␮␯E+,E兲+␴˜␮␯E,E兲兴, 共6兲 where we define the complex energies as

E+=EF+i, E=EFi,0, 共7兲 and

˜␮␯共z1,z2兲= − ប

N⍀Tr具JG共z1兲JG共z2兲典, 共8兲 wherez1 andz2are each eitherE+or E.

For nonoverlapping effective single electron potentials the Hamiltonian takes the form

H= − ប2

2mⵜ2+

i v共rRi兲, 共9兲

where the atomic positionsRiare fixed, and form a regular lattice. The potentials v共ri兲 vary from site to site 共ri=r

Ri兲and␥is a configuration label.

Within the multiple-scattering theory, the single-particle Green’s function for a given configuration␥ can be written as51

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G共E,ri,r

j兲= 2m/ប2

⌳,⌳

Z⌳,␥i 共E,ri兲␶␥;⌳,⌳ij Zj,†,␥共E,r

j

Zi,␥riJi,†,␥E,riij

, 10

where␶␥,ij,is the scattering path operator共SPO兲describing propagation between sites i and j in configuration ␥, Zi,␥E,ri兲 is the regular solution to the Schrödinger and/or Dirac equation in the cell surrounding the atom i, and Ji,␥共E,ri

兲 represents the irregular solution within the same cell共note that there should be no confusion with the current matrix elements here兲.⌳encapsulates the appropriate angu- lar momentum quantum numbers.21,24

In calculating the conductivity, the second term in the Green’s function expression 关Eq. 共10兲兴 is real and may be omitted, when calculated for a real potential at a real energy.21Thus, the conductivity may be written as

˜␮␯共z1,z2兲= − 4m2

N⍀ប3

i,j

1,2,3,4

⫻具J

12,␮

i,␥z2,z1兲␶␥;ij23z1J

34, j,␥z1,z2

⫻␶␥;⌳ji 41共z2兲典, 共11兲 with

J⌳⌳i,␥,␮共z,z

兲= −ieប m

celli

driZ⌳,␥i 共ri,z兲 ⳵

rZi,␥共ri,z

兲, 共12兲 wherecellidefines the region surrounding the sitei.

We now need to consider how to carry out the averaging over configurations implicit in Eq.共11兲. Butler21 showed in detail how to use the CPA to accomplish this. The single site nature of this effective medium theory means, however, that the potentials on the different lattices could only be treated as statistically independent. We will show how to carry out the averaging using the NLCPA whereby short-ranged correla- tions can be naturally included. To this end, in the next sec- tion, we summarize briefly the key aspects we need from the KKR-NLCPA together with its incorporation into electronic density functional theory. Full details can be found in Refs.

36–38and41

III. THE KORRINGA-KOHN-ROSTOKER NONLOCAL COHERENT POTENTIAL APPROXIMATION AND ELECTRONIC DENSITY FUNCTIONAL THEORY In general the SPO between two sitesiandj for an elec- tron, moving through an effective medium so that it mimics the average motion in a disordered system, is given by

ˆij=ij+

k

i

tˆ关G共RiRk兲+␦Gˆ共RiRk兲兴␶ˆkj. 共13兲 Here, all quantities are matrices in angular momentum space and the indices i and j run over all sites in the lattice.

G共RiRk兲’s are structure constants. The effective medium is specified by single site t-matrices and effective structure

constant corrections␦Gˆ共RiRj兲. The effective mediummust be translationally invariant so that␶ˆijis given in terms of a Brillouin zone integral,

ˆij= 1

BZ

BZdk关tˆ−1G共k兲Gˆ共k兲兴−1eik·共Ri−Rj. 共14兲

In order to establish a tractable procedure for determining the effective medium the NLCPA draws its chief idea from the DCA for interacting electron systems.42,43 This is a coarse graining of␦Gˆ consistently in real and reciprocal space and a mapping to a self-consistently embedded impurity cluster problem with appropriate boundary conditions imposed.48 The full translational symmetry of the underlying lattice is preserved. The size of the cluster sets the range of correla- tions that can be included. The lattice is divided into “tiles”

centered on a superlattice vectors RC and each contains Nc

sites at positionsRC+RI,I= 1 , . . . ,Nc. The Brillouin zone is also broken intoNctiles, of volume⍀t=⍀BZ/Nc, centered on the cluster momentaKn,n= 1 , . . . ,NcandRI’s andKnsatisfy the following equation:

1 Nc

K

n

eiKn·共RI−RJ=␦IJ. 共15兲 Here,␦Gˆ共k兲is coarse grained so that it has the average value

Gˆ共Kn兲 in a tile centered on Kn and in real space, ␦Gˆ共RI

RJ兲=共1/Nc兲兺KnGˆ共Kn兲eiKn·共RI−RJ with ␦Gˆ共Kn

=兺JIGˆ共RIRJ兲e−iKn·RI−RJ. The SPO is coarse grained,

ˆ共Kn兲= Nc

BZ

tdk˜关tˆ−1G共k˜ +KnGˆ共Kn兲兴−1, 共16兲

appropriate to the reciprocal space tile of volume⍀tand in real space for multiple scattering starting and ending on clus- ter sitesIandJ, respectively,

ˆIJ= 1

BZ

K

n

t

dk˜关tˆ−1G共k˜ +Kn

−␦GˆKn兲兴−1

eiKn·共RI−RJ. 17

Note, howeik˜ ·RI−RJis taken to be⬇1 as the coarse graining is applied.43The final step is to find the SPO for an impurity cluster␶IJ describing a particular configuration␥Cof atoms, which is embedded into the NLCPA medium. By demanding that the average is equal to the SPO of the NLCPA enables the effectivetmatrix and structure constant corrections to be determined, i.e.,

C

P

CIJC=ˆIJ. 共18兲 SRO can be included by choosing the probabilities P

C ap- propriately as demonstrated in, for example, Refs.37and38 In Ref.38, in a generalization of the work of Johnsonet al.,4,5it is described how to specify a configurationally aver- aged electronic Grand potential⍀¯ in terms of KKR-NLCPA

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quantities and charge densities ␳C共rI兲and one-electron po- tentials v

C共rI兲 different for each cluster configuration. The functional minimization of⍀¯ with respect to the charge den- sities,␳C共rI兲’s, determines the total energy of the system and requires␳C共rI兲’s andv

C共rI兲’s to be found self-consistently.

Rowlands et al.38 applied this DFT to investigate how the total energy, charge densities, and densities of states are af- fected by SRO. Tulipet al.47 showed how further informa- tion can be found about the effects of SRO on the electronic structure by formulating and calculating the Bloch spectral function at the cluster momenta and averaged over tiles, while Batt and Rowlands48 explained how the spectral func- tion at any point in the Brillouin zone can be found. In the following, we build on these developments and describe the theory for a two-particle correlation function of a disordered system with SRO. The particular example is to the dc con- ductivity.

IV. ANALYTIC CONFIGURATION AVERAGING OF THE CONDUCTIVITY USING THE KORRINGA-KOHN- ROSTOKER NONLOCAL COHERENT POTENTIAL

APPROXIMATION

From the above, it is clear that the KKR-NLCPA should enable an analytical configurational average of the conduc-

tivity to be carried out. To do so, some care needs to be exercised. There are two distinct cases that we must consider in Eq.共11兲:共i兲where the two sites under consideration,iand j, lie within the same NLCPA cluster, and共ii兲when they lie in two different clusters, in which case the occupancies of the sites will be statistically independent, as the two distinct clusters will be statistically independent. This is a natural generalization of Butler’s work,21where he distinguishes be- tween the two cases ofi=j andij.

Hereon, we use lower case letters to denote general sites in the lattice,i,j, . . ., upper case C denotes tiles containing the clusters, and upper case letters, with the exception ofC, denote sites within clusters. So for a site at positionRi, we useRi=RC+RIandI,J, . . . label sites within tileC,I

,J

, . . . sites within tileC

, etc.

We accordingly write

˜␮␯共z1,z2兲=␴˜␮␯0 共z1,z2兲+␴˜␮␯1 共z1,z2兲, 共19兲

˜␮␯0 共z1,z2兲= − 4m2

␲ប3J

C

1,2,3,4

具J

12;␮

I 共z2,z1兲␶IJ23共z1兲J

34;␯

J 共z1,z2兲␶JI41共z2兲典, 共20兲 and

˜␮␯1 共z1,z2兲= − 4m2

␲ប3

CC

JC

1,⌳2,⌳3,⌳4

具J

12;␮

I 共z2,z1兲␶I,C2+J3 共z1兲J

34;␯

J⬘ 共z1,z2兲␶C4+J1,I共z2兲典, 共21兲

where ␴˜0 includes sites J within the same NLCPA cluster 共denotedC兲as our reference siteI and˜1 includes all sites lying outside this cluster. Note that in writing these equa- tions, we have utilized the translational invariance of the averaged system to remove the second sum appearing in Eq.

共11兲.

We now introduce response functions, such that we can write

˜␮␯0 共z1,z2

= − 4m2

␲ប3

C

P

C

1,⌳2

J

12;␮

I,␥C 共z2,z1兲K

2,1; I;C,␥C 共z1,z2兲,

共22兲 K

2,1;␯

I;C,␥C 共z1,z2

=J

C

3,4

具␶IJ23共z1兲J

34

J,␥C,␯共z1,z2兲␶JI41共z2兲典C,␥C. 共23兲 Here,K

2,1;

I;C,␥C 共z1,z2兲involves an average over all configura-

tions, except the configuration is fixed in clusterCto be␥C. The single site quantityJ

12;␮

I,␥C, 共z2,z1兲is set up depending on what kind of element occupies site I and the one-electron potentialv

C共rI兲that is dependent on the configuration␥C. Similarly, for the intercluster contributions to the conduc- tivity, we can introduce the following:

˜␮␯1z1,z2兲= − 4m2

␲ប3

CC

C

P

C

C

P

C

1,2

J

12;␮

I,␥C 共z2,z1兲L

2,⌳1;␯

I,C,␥C;C,␥C共z1,z2兲, 共24兲 and

L

2,1;

I,C,␥C;C,␥C共z1,z2兲=

JC

3,⌳4

具␶I,C2+J3共z1兲J

34; J,␥C共z1,z2

⫻␶C4+J1,I共z2兲典C,␥C;C,␥C, 共25兲 where the notation is similar to before but the average now

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fixes clusterCto be occupied by configuration␥Cand cluster C

to be loaded with configuration␥C⬘.

In order to evaluate the ensemble averages contained in Eqs.共22兲–共25兲, it is helpful to express the SPO␶for a par- ticular configuration as the SPO in the NLCPA medium plus corrections. Again, this is the direct generalization of But- ler’s approach.21 We can write

ij=ˆij+

k,l

ˆikTklˆlj, 共26兲 where the effective medium path SPO is denoted␶ˆ as before andT is the total scattering matrix relevant to the specific configuration. The double summation is taken over all lattice sites.␶ijsatisfies the following equation:

k 关tk−1ikG共RiRk兲兴kj=ij, 共27兲

where all quantities are matrices in angular momentum space. We consider fluctuations about the NLCPA medium to obtain

k 关共tk−1−1ik+Gˆ共RiRk+−1

−␦Gˆ共RiRk兲−G共RiRk兲兴␶kj=ij, 共28兲 which can be rearranged to yield

ij=ˆij

k,l

ˆik⌬mkllj. 共29兲 with

⌬mkl=共tk

−1−1兲␦kl−␦Gˆ共RkRl兲 共30兲 and have used the fact that the effective medium SPO may be written as the inverse of the matrix with elements关tˆ−1ij

−␦Gˆ共RiRj兲−G共RiRj兲兴.

We can thus write down

l Tklˆlj=

l mkllj. 31

If we now substitute for ␶lj using Eq. 共26兲, label the sites according to clusters,C and sites within those clusters共up- percase letters兲, we obtain

TC+K,C+L=xKLC,C⬘+

CC

MN

xKMˆC+M,C+NTC+N,C+L, 共32兲 where we have introduced the matrix x associated with a single cluster of sites given by

xIJ=

K

共1 +⌬m␶ˆIK

−1⌬mKJ. 共33兲

Our results here are a direct cluster generalization of Butler.21 Note also that these results are consistent with the cluster CPA conductivity formalism of Hwang et al.52 共al- though that is phrased in terms oft-matrices, rather than the

xmatrix that we use in this work兲. Further, the special case of a single-site cluster recovers the more familiar CPA re- sults.

The NLCPA amounts to writing 具TC+K,C+LNLCPA=具xKL典␦C,C⬘+

CC

MN

具xKM典␶ˆC+M,C+N

⫻具TC+N,C+LNLCPA, 共34兲 and if we choose that具xIJ典= 0, which is another way of ex- pressing the NLCPA ansatz关Eq.共18兲兴, then具T典NLCPA= 0, and we obtain具␶典NLCPA=␶ˆ 关Eq.共18兲兴. Of course, in writing this, we have made the approximation that

xKMˆC+M,C+NTC+N,C+L

⬇ 具xKM典␶ˆC+M,C+NTC+N,C+L典, 共35兲 which is analogous to the usual CPA-type averaging approxi- mation.

Using Eq.共26兲and closely following a cluster generaliza- tion of Butler’s derivation, which refers to fluctuations about the single site CPA medium, we find 共suppressing angular momentum labeling兲

KI;C,␥C=

M,N

DIMC

MN;

C,␥CDNI†␥C, 共36兲

with

IJ;␯C,␥C=

KLˆIK˜JKL;␯C ˆLJ+

CC

K,L⬙␶ˆI,C+KCC,␥;KCL;␯ˆC+L,J, 共37兲 and

LI,C,␥C;CC=

M,N

DIMC˜L

C,␥MN;␯C;C,␥CDNI†␥C, 共38兲 where

MN;␯C,␥C;C,␥C=

K,L

ˆM,C+K˜J

KL;␯

CˆC+L

+

C⫽共C,C

K,L

共␶ˆM,C+KCC,␥,KC;C,L,␥;CˆC+L,N兲.

共39兲 In Eqs.共36兲and共38兲, we use the NLCPA projectorD共D兲 which isD=共1 +⌬m␶ˆ−1 关共1 +␶ˆ⌬m兲兴 found in Eqs.共30兲and 共33兲. The NLCPA ansatz can be rewritten in terms of them, i.e.,兺CP

CDC= 1. We have also defined the current quanti- ties

˜J

KL;␯C =

N DKN†␥CJN,␥CDNLC 40

in Eqs. 共37兲 and 共39兲. Finally, we have introduced vertex functions ⌫CC,␥,KC,L;␯ and ⌫CC,␥,KC;C,L,␥;␯C which are the NLCPA analogs of the vertex functions derived by Butler.21We now show how to calculate these quantities.

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V. VERTEX FUNCTIONS AND THE CONDUCTIVITY Our two vertex functions are slightly different.⌫CC,␥;KC,L;␯, which appears in Eq.共37兲for the intracluster component of the conductivity, concerns the connection between the con- figurational occupation in one clusterCwith that of another C

, whereas the vertex function⌫CC,␥,KC;C,L,␥;Cfor the interclus- ter contribution relates the contents of clusterC

with that in two others, the reference oneCand anotherC

. To facilitate the derivation of a closed set of equations, we introduce an approximation, analogous to that in Ref.21, and assume that the dependence of the latter on the contents of clusterCmay be neglected. Thus,

CC,␥,KC;C,L,␥;␯C=⌫CC,K,␥C,L;␯, 共41兲 leading to

MN;

C,␥C;C,␥C=

MN;

C,␥C in Eq. 共39兲. Using Eq. 共26兲, the NLCPA condition关Eq.共18兲兴 共or its equivalent renditions兲 and tracking the steps in Ref.21, we obtain

C,IJ;␯C,␥C=

K,L,M,N

具xIK,␥CˆK,C+M˜J

MN;␯

CˆC+N,LxLJ,␥CC,␥C

+

C⫽共C,C

K,L,M,N

具xIK,␥CˆK,C+MCC,M,␥N;␯

⫻␶ˆC+N,LxLJ,␥CC,␥C, 共42兲

which, if we compare to Eq. 共39兲, allows us to write the vertex function in terms of the response function

C,IJ;C,␥C=

K,L

具xIK,␥C˜L

KL;

C,␥CxLJ,␥CC,␥C. 共43兲 This yields a closed set of equations for the conductivity.

We may now write Eqs.共22兲and共24兲as

˜␮␯0 共z1,z2兲= − 4m2

␲ប3⍀Nc

C

P

C

1,⌳2

I,J

˜J

JI,⌳2,⌳1;␮

Cz2,z1

IJ,⌳1,⌳2;␯

C,␥Cz1,z2兲, 共44兲

and

˜␮␯1z1,z2兲= − 4m2

␲ប3⍀NcI

C

CC

JC

C,␥C

P

CP

C

1,2

I,J

˜J

JI,⌳2,⌳1;␮

C 共z2,z1兲L˜

IJ,⌳1,⌳2;␯

C,␥C 共z1,z2兲. 共45兲 The response functions that determine the conductivity are given by

IJ,1,2;␯

C,␥C =

K,L

3,4

ˆ

13 IK ˜J

KL,34;␯

CˆLJ42+

CC

K,M,N,L

3,⌳4,⌳5,⌳6

ˆI,C1+K3 具x

34

C,KM

MN,⌳4,⌳5;␯

C,␥C x

56

C,NL⬘典C,␥CˆC6+L2,J, 共46兲

IJ,⌳1,⌳2;␯

C,␥C =

K,L

3,⌳4

ˆI,C1+K3 ˜JKL,

3,4;

CˆC4+L2,J

+

C⫽共C,C

K,L,M,N

3,4,5,6

ˆI,C1+K3 具x

34

C,KM˜L

M,N,⌳4,⌳5;␯

C,␥C x

56

C,N,L⬙典C,␥CˆC6+L2,J. 共47兲

VI. SOLUTION OF TRANSPORT EQUATION

The conductivity is determined by solution of the trans- port equation关Eq.共47兲兴. We first see that in the key Eqs.共44兲 and共45兲, we require the response function˜L

IJ;␯

C,␥C averaged over configurations␥C⬘that can be assigned to a clusterC

in a NLCPA tile located at position RC⬘ and also that we require this to be summed over all clusters C

. We rewrite Eq. 共47兲, omitting the quantum numbers, ⌳1, ⌳2, etc., for brevity, and denote the averages 兺CP

C˜L

IJ;␯

C,␥C and 兺CP

C˜J

K,L;␯

C as LIJ;␯C andJK,L;␯, respectively, as well as summing over clustersC

,

C

LIJ;␯C =

C

K,L⬘␶ˆI,C+KJK,L;ˆC+L,J

K,L⬘␶ˆI,KJK,L;␯ˆL,J +

C

CC

K,L,M,N

ˆI,C+KwK,L,M,N

⫻␶ˆC+L,JLCC⬙⬘,M,N;, 共48兲

wherewK,L,M,N=具xK,LxM,N典andLIJ;␯C is defined as zero.

We now write the SPOs in terms of their lattice Fourier transforms, i.e.,

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