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Doctoral Thesis

Influence of short range order on the electronic structure ofalloys and their surfaces

Author(s): Boriçi, Mirela

Publication Date: 1998

Permanent Link: https://doi.org/10.3929/ethz-a-002041162

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DISS. ETH Nr. 12923

Influence of Short Range Order on the

Electronic Structure of Alloys and their

Surfaces

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

for the degree of

Doctor of Natural Sciences

Presented by

Mirela Borigi

Dipl. Phys. University of Tirana

Born 11 September 1965

Citizen of Albania

accepted on the recommendation of

Prof. Dr. Danilo Pescia, examiner

PD. Dr. Rene Monnier, co-examiner

Prof. Dr. Peter Weinberger, co-examiner

1999

Contents

1 Introduction 1

1.1 The Coherent-Potential Approximation 3

1.1.1 Greens Functions and Observables 3

1.1.2 Single-Site CPA in the Multiple Scattering Formulation 5

1.2 Elements of Density Functional Theory 10

2 Tight-Binding Linear Muffin-Tin Orbital Method 15

2.1 TB-LMTO Approach for infinite systems 15

2.2 TB-LMTO Surface Green's function 20

3 The SCPA within the TB-LMTO approach 25

3.1 TB-LMTO-CPA for bulk alloys 25

3.2 TB-LMTO-CPA equations for alloy surfaces 29

3.3 Physical Quantities 31

3.4 Charge self-consistency and lattice relaxations effects 33

4 The 7 expansion method 36

4.1 GEM corrections to the CPA DOS for bulk alloys and their surfaces.... 36

4.2 Short Range Order 40

5 Results and Discussions 45

5.1 Results for bulk alloys 45

5.1.1 Ago.s7Alo.13: 46

1

5.1.2 Ago.5oPdo.5o: 48

5.1.3 Cuo.715Pdo.285: 50

5.1.4 Cuo.75Auo.25: 52

5.1.5 Pto.55Rho.45: 54

5.2 Discussion of bulk alloys results 55

5.3 Results for alloy surfaces 58

5.3.1 The homogeneous alloy surface 58

5.3.2 The inhomogeneous alloy surface 65

Acknowledgements

I would like to thank my supervisors Prof. Danilo Pescia and PD. Dr.Rene Monnier that

made possible my PhD studies. In particular, I would like to thank Rene Monnier for

his scientific support. I am grateful to him for his valuable advises and conversations on

different topics and problems at every stage of the preparation of my thesis.

I would like to thank Prof. Peter Weinberger for reading my PhD work and being co-

examiner. His hospitality during the time I stayed in his group at the Technical University

of Vienna is also gratefully acknowledged.

I am grateful to Vaclac Drchal and Josef Kudrnovsky who gave me the permission

to use their electronic band sturcture computational tools. In particular, I would like to

thank Vaclac Drchal who teached me how to use them and has helped me with many

stimulating conversations.

It is a pleasure to thank Vassili Tokar for his help in clarifying certain technical points

in the derivation of the GEM.

Financial support under the three-year PhD-student Grant No.2129-42079.94 of NFS

is gratefully acknowledged.

I would like to thank Bernd Schonfeld for many helpful discussions.

ABSTRACT

The influence of short range order and correlated scattering by pairs of atoms on the

electronic structure of a random binary alloy and its surface has been studied by using

the 7 expansion method (GEM) proposed by Tokar. The latter is implemented within a

tight-binding linear-muffin-tin orbital formulation of the (single-site) coherent potential

approximation which approximately accounts for geometric relaxation. Results are pre¬

sented for five random bulk alloy systems: Ago.87Alo.13, Cuo.715Pdo.2s5 and Cuo.75Auo.25, in

which the short range order has been measured; Ago.50Pdo.50 and Pto.55Rho.45 where it is

calculated within the GEM. For Cuo.715Pdo.2s5 and Cuo.75Auo.25 lattice relaxation effects

play an important role. In all cases we find that the density of states of the system is only

weakly affected by the short range order and that the correlated scattering of electrons

by pairs of atoms produces only minute changes to it.

Results for the (100) surface of Pto.55Rho.45 are presented for two different cases, the

seni-infinite homogeneous alloy and the one with the equilibrum concentration profile.

GEM correctionts to the layer resolved density of states due to correlated scattering of

electrons by pairs of atoms and short range order, both are-found to be of the same order

of magnitude as the corresponding bulk counterparts. They converge to the bulk result

for the deepest surface plane.

ZUSAMMENFASSUNG

Diese Doktorarbeit befasst sich mit der Untersuchung der Wirkung von Nahordnung

und korrelierter Streuung durch Atompaare auf die elektronische Struktur einer ungeord-

neten Legierung und ihrer Oberflache. Dies geschieht mit Hilfe der, von Tokar vorgeschla-

genen, sogennanten "7 Entwicklung" (GEM). Dieses Verfahren lasst sich innerhalb einer

"tight-binding linear-muffin-tin orbital" Formulierung der "(single-site) coherent poten¬

tial" Naherung, welche auch den Einfiuss der geometrischen Relaxation beriicksichtigt,

implementieren.

Es werden Ergebnisse fiir folgende ungeordneten Legierungen presentiert: AgAl, CuPd

und CuAu, fiir welche die Nahordnung experimentell bestimmt wurde, und AgPd und

PtRh, fiir die wir den Nahordnungsparameter mit Hilfe der GEM berechnet haben. Fiir

CuPd und CuAu, spielt die Gitterrelaxation eine wichtige Rolle. In alien Fallen finden

wir, dass die Zustandsdichte des Systems nur schwach durch die Nahordnung beeinflusst

wird, und dass die korrelierte Streuung der Elektronen durch Atompaare nur winzige

Veranderungen- hervorbringt.

Die Ergebnisse fiir die (100) Oberflache von PtRh sind fiir zwei verschiedene Falle

prasentiert, b.z.w. die halbunendliche homogene Legierung und die mit dem Konzen-

trationsprofil im thermodynamischen Gleichgewicht. Die GEM Korrekturen zur, nach

Atomlagen aufgelosten Zustandsdichte, die von der korrelierter Streung der Elektronen

durch Atompaare und von der Nahordnung erzeugt werden, sind beide von der selben

Grossenordnung wie im Inneren der Legierung. Sie konvergieren zu dem "bulk" Resultat

fiir die tiefste Oberflachenebene.

Chapter 1

Introduction

Nowadays, the electronic characteristics of pure metals and ordered alloys, are well stud¬

ied by efficient schemes based on density functional theory (DFT) [1, 2]. The theoretical

study of random alloys and their surfaces, due to the lack of translational invariance

(compositional inhomogeneities, violation of the translational symmetry in the direction

perpendicular to the surface), requires supplementary tools. These systems are best de¬

scribed in terms of Green's functions for which the procedure of configurational averag¬

ing, within the single-site coherent potential approximation (SCPA) [3, 4] is well defined.

Within the Green's function language the nature of the physical problems and approx¬

imations used are well understood, and equilibrium properties of random substitutional

alloys (RSA) can be computed from first principles with a level of accuracy comparable

to that for ordered compounds.

Truly random alloys, however, only exist at very high temperature and most metallic

alloy systems display a certain degree of short-range order (SRO), even in the disordered

phase. The SRO determines many material properties at high temperature and it can

have a strong impact on the surface related phenomena, (i.e., on the segregation of atoms

to the surface, crystal growth, grain bounderies, etc). It reveals ordering tendencies that

are often indicative of long-range order found at lower temperatures. It is characterized by

the so-called Warren-Cowley [5] SRO parameters, whose lattice Fourier transform, a(k),

is nothing but the coherent diffuse scattering intensity for X-rays or neutrons due to SRO

1

CHAPTER 1. INTRODUCTION 2

(in Laue units). Thus, an understanding of what gives rise to this phenomenon in terms

of a first-principles, microscopic theory is not only of great fundamental interest but also

a matter of practical importance in modern alloy-design efforts.

A few years ago, Staunton et al. [6, 7], using the Korringa-Kohn-Rostoker (KKR)-

CPA method, in combination with the theory of concentration waves [8], have derived

an explicit relation between the lattice Fourier transform of the SRO parameters, a(k),

and the electronic structure of the fully random alloy on a perfect lattice. However,

the solution to the complementary problem, i.e. how does the presence of SRO affect

the electronic structure of the alloy, cannot be given within the framework of a single-site

theory. Thus, we have to find an approximation to the configuration-averaged one-electron

Green's function which accounts for the SRO and has the necessary analytical properties

to produce a physically reasonable density of states (DOS), i.e. that is Herglotz [9].

The exponential decay of the SCPA one-electron Green's function with distance is at

the heart of the approach we have applied to solve this problem. Following Massanskii

and Tokar [10], we use the quantity ye = e~«e, where £e is the correlation length between

electrons, measured in units of the nearest- neighbor distance on the lattice, as a variable

in a series expansion of the deviations of the configuration-averaged one-electron Green's

function from the expression obtained in the SCPA. This so-called gamma expansion

method (GEM), first introduced by Tokar in the context of classical lattice statistics [11],

is implemented within the fully relativistic version of the tight-binding linear muffin-tin

orbital (TB-LMTO) code, developed by Kudrnovsky and coworkers [12,13], for calculating

the electronic structure of random bulk alloys and their surfaces within the SCPA. GEM

gives an analytic expression for the corrections to the SCPA density of states, due to SRO

and correlated scattering by more than one site.

In the following two introductory sections are given. First we derive the CPA condition

in the multiple scattering form presented by Velicky et al [4] and then we give a short

account of DFT for periodic systems and its generalization to random substitutional

alloys (RSA). A derivation of the TB-LMTO method from the KKR multiple scattering

theory for ordered systems and their surfaces is presented in the second chapter. The third

1.1.The Coherent-Potential Approximation 3

chapter is devoted to the modern theory of alloys and their surfaces within the TB-LMTO-

CPA. The next one gives the theoretical description of the GEM and its implementation

within the TB-LMTO formalism. The results and comments on our calculations are

presented in the last chapter.

1.1 The Coherent-Potential Approximation

In this section, we derive the SCPA expression for the configurationally averaged Green's

function, which can be used to calculate physical observables such as density of states and

charge densities.

1.1.1 Greens Functions and Observables

In general, any representation of the resolvent operator G(z) = (z - i/)_1, is called a

Green function, for example G(r, r', z),

G(r,r',z) = (r|(z-F)-1|r'), (1.1)

is the coordinate representation. It satisfies the inhomogeneous differential equation

[~htv'+v{T) ~ *]G(r'r''z) = ~*(r _ f,)- (L2)

This Green's function describes the way that the electron moves or propagates in the field

defined by the potential v(r) [14]. It has been shown that if all of eigenfunctions 0j(r)

and eigenvalues £, of the Hamiltonian operator H are known, the solution of eq.(1.2) on

the real energy axis is given by [15]

where the index '+' implies z = e + i8, 5 > 0. If the wave functions are normalized,

/j"* | <f>i(r) |2 d3r = 1, then it is clear that

fW,t,e)* = nx—L-. ,1.4)


The integral on the left side of the last equation is called the trace of the Green's function,

TrG(e).

Using the methods of complex variable functions theory, the Dirac S function can be

represented as [16]

Mc-a) = —Slim —-e, (1.5)

which, combined with eq.(1.4) leads to

--S / G+(r,r,e)d3r = £ 6{e - £*). (1.6)7T J .

The right side of this equation can be identified with the DOS function n(e) because its

integral from e to e + As gives the number of eigenvalues within this interval. Therefore,

the DOS n(e) for a solid described by the potential v(r) can be calculated with the

following formula

»Mr)(e) = ~9/ G+(r,r,e)d3r. (1.7)

The total charge density at a point r can be derived by using similar arguments to the

ones that led to eq.(1.7) [17], with the result

n{r) = --S f G+{r,T,e)de. (1.8)IT J

The RSA AcBi_c is described by the random potential

v(r) = 5>»(r), K(r) = p*VA(r) +rfVB(r) (1.9)n

where p®,Q = A,B are the occupation numbers defined by

A B J 1(0) if site n is occupied by species AVn Wn ) j

I 0(1) if site n is occupied by species B,

so that (p£) = c. A particular arrangement of A and B atoms, mathematically described

by the set of the occupation numbers {p%}, defines the corresponding configuration of

the alloy. The macroscopic observables of interest for the random alloy are of course

configurational averages of the microscopic ones. Thus, the most important quantity in

the theory of the random alloys is the average over all configurations of the resolvent

(G(z)) = ((z-H)-1), (1.10)


from which the DOS for the RSA can be obtained by taking its trace, that is

n(e) = -hf(G+(r,T,e))d3r, (1.11)7T J

while the total charge density becomes

n(r) = --3 / <G+(r,r,e))<fe. (1.12)IT J

Prom the definition of the conditional ensemble averages, the configurationally aver¬

aged Green's function can be written as

<G+(rn,4e)) = c(G+(rmr'n,e))(n=A) + (1 - c)(G+(rn,rn,£)){n=B), (1.13)

where (n = Q) means that in the cell n the occupation is fixed to atom Q and the average

is restricted to all configurations of the remaining N — 1 sites. rn, rn are the coordinates

of points in the cell n, measured from its center. From the conditionally averaged Green's

functions, we can derive the corresponding charge densities which, at finite temperature,

are written

1 f+0°nAiB^n) = — / 9(C7+(r„, rn, s)){n=A/B)f(£ - fie)de, (1.14)

7T J—oo

where f(e — fj,e) is the Fermi function. The density of states (DOS) associated with a

given Q = A, B atom in the alloy is found from

UAisie) = -^[Trr(G+(rn,rn,e))(n=j4/B)] (1.15)

The charge densities and DOS associated with an atom, either A or B, in the alloy are

related to the complete ensemble averages by equations similar to eq.(1.13).

1.1.2 Single-Site CPA in the Multiple Scattering Formulation

The single-particle Hamiltonian for the random alloy can be written as

H = [Hr + Y,Vn], (1.16)n

where Hr stands for its non-random part, which is normally considered as a reference

system. To calculate its ensemble averaged Green's function we use the SCPA, which is

1.1. The Coherent-Potential Approximation 6

the best possible mean-field estimate [4] for the latter quantity. It replaces the macroscopic

alloy by a periodic array of effective scatters, which create an effective potential, optimized

in such a way that on average no extra scattering occurs if an 'effective atom' is replaced

by an atom of type A or B.

In the following, we add and subtract the required periodic potential:

Vtf^Vniz) (1.17)n

so that

H = [Hr + V(z)} + Y,(Vn - Vn(z)) = H(z) + £ vn(z) = H(z) + AV(z) (1.18)n n

where H(z) describes a periodic, possibly non-Hermitian medium. The corresponding

resolvent

G(z) = (z- H(z))~\ {G(z)) = G(z) (1.19)

is related to that of a given configuration T by the Dyson equation

G{z) = G{z) + G{z)f{z; V{z))G{z) (1.20)

where T(z) describes the total scattering due to the difference potential AV. A similar

expression can be written in terms of the reference system resolvent Gr = (z - ifr)-1,

G(z) = Gr(z) + Gr(z)T(z)Gr(z) (1.21)

where, now, T(z) describes the total scattering from the random potential v(r) in eq.(1.9).

By taking the configurational average on both sides of eq.(1.20), we immediately see

that the above mentioned optimization of the effective medium leads to

(f(z; V(z))) = 0, (1.22)

and

(G(z)) = G(z). (1.23)

This implies the following CPA condition for the average of scattering operator T(z):

G(z) = Gr(z) + Gr(z){T{z))Gr{z). (1.24)


The scattering operator f(z; V(z)) can be expressed as a multiple scattering series

where tn generates the scattered wave by the potential difference at site n:

in = vn + vnGvn + vnGvnGvn + ...= vn + vnGtn. (1.26)

The right-hand side of eq.(1.25) can be formally regrouped into a sum of contributions

over lattice sites:

f(z) = J2Qn(z) (1.27)n

with

Qn(z) = tn(z)(l + G(z) £ Qm(z)). (1.28)

The operators Qn(z) convert an incident wave into a scattered wave radiating from site n

in the presence of all other scatterers. The configurational average can be performed on

each Qn separately,

(Qn) = (in)(l + G £ (Qm)) + ((in - (tn))G £ (Qm - (Qm))), (1.29)

where

(tn) = aAn + a - C)? (i.3o)

is the average scattering operator associated with the site n.

The first term in eq.(1.29) describes the mean scattering of the average effective wave

incident on a given site, while the second term accounts for the fluctuations both in the

effective incident wave and in the scattering strength at that site.

The basic approximation in all first-principles treatments of RSA, known under the

name of the single-site approximation, is to neglect the second term. This implies, in

particular, that short-range order as well as resonant scattering by clusters of correlated

atoms are not included. The CPA condition now takes the simple form

(tn(z;V(z))) = Q; Vn (1.31)


and, because of the periodicity of the averaged quantities, it suffices to consider only a

single site. The configurational averaging restores the full translational symmetry of the

underlying lattice, making all sites equivalent for a crystal with one atom per unit cell. For

a structure with a monoatomic basis, each unit cell is charge neutral, thus the eq.(1.31) is

equivalent to the condition of the local charge neutrality for the configurationally averaged

alloy.

The CPA condition can also be written

(T(Tn,Tnte))? = c(T(rn,Tn,e))v{n=A) + (1 - c){T(xnixn,e))y{n=By (1.32)

which is derived by the combination of eq.(1.13) with eq.(1.24) in its coordinative repre¬

sentation. Although, the CPA condition eq.(1.32) is given in terms of the total scattering

operator, the single site approximation, eq.(1.31), is implicitly assumed via eq.(1.24).

Each operator in this equation can be rewritten as a multiple-scattering series of the

form of eq.(1.25) in terms of the Green's function for the reference system G+(r, r',e),

and after partial resumation we obtain

<T(r,r\e))^ = £r£(r,r',£) (1.33)ij

where W stands for any of the three potentials V, V(n=A) or V{n-B) m eq.(1.32), and

rg(r,r\*) = At*(r,r\e)*y +W j M\v,vl,e)Gt{vl^e)rkJ{r2,v ,e)dzrxd\2

(1.34)

is the scattering path operator [18], which gives the scattered wave from site j resulting

from an incident wave at site i, and carries all the information on the medium through

which the wave travels between the two sites. Af (r, r',e) is the relevant single-site scat¬

tering operator for site i in empty space, minus that for the common reference. Making

use of eq.(1.32) and eq.(1.33) the CPA condition can also be written as:

$(r,r',e) = «%„,,(r,r ,e) 4- (1 - ^^(r.r'.e). (1.35)

The practical use of this equation is, however, not straightforward. Within the KKR-

CPA, all arguments in eq.(1.34) are expanded in partial waves and after taking the matrix


element between plane waves 'on the energy shell' p2/2m = pl2/2m = e, it becomes [19]

T&(e) = AfLL,(e)<% + EE AfLLl(e)(Gr)£,L2rgiL,(e), (1.36)

The super matrix Gr = {(Gry£lL2} is called the structural Green's function of the refer¬

ence medium. The last equation can be written in a more compact form

E{[At'(e)]"% - Gf(e)(l ~ &ik)}zkj(e) = 1, (1.37)k

where each matrix element in the site indices is itself a matrix in the angular momentum

components L,L'.

Equation (1.37) is the basic equation of multiple scattering theory which, for transla-

tionally invariant systems, can be solved in Fourier space with the result

r(k,e) = [Ar1^) -Grfte)]-1. (1-38)

The physically relevant (k, e) states are those for which the Green's function of the system

has poles. The following identity, [20]

G = Gr + GriGr = (At)"1?:(At)"1 - (At)"1, (1.39)

shows that the Green's function and the scattering path operator have poles at the same

energy values on the real axis. Thus the necessary and sufficient condition for the existence

of a physically relevant (k, e) state is that:

detfAt"1^) ~ Gr(k, e)) = 0, (1.40)

which gives the poles of the scattering path operator on the real energy axis. For the

periodic system described by the coherent potential V(z) and the t matrix t<. ( relative

to that of the common reference) at every site, the latter becomes

detfc^e) - Gr(k,e)] = 0, (1.41)

that is the KKR-CPA condition.

1.2.Elements of Density Functional Theory 10

The site-diagonal matrix elements for the case where an atom of type A or B replaces

the 'CPA effective atom' at site i are given [14, 15, 21]

r!X(e) ee 4{i=A)(e) = {1 + \&{e) - fte)]^)}"1!^),

rhB{e) = 4{tmB)(e) = {1 + \&(e) - fte)^)}-1^). (1.42)

The self-consistent solution of eqns.(1.35) and (1.42), allows the construction of the con¬

ditionally averaged Green's functions of eq.(1.13), which are then used to calculate the

averaged charge densities ua/b^)- The next step is to calculate the correct potential

generated by these charge densities, which is realized in the framework of DFT.

1.2 Elements of Density Functional Theory

The Hohenberg-Kohn [1] theorem says that, at T = OK the total energy of an interacting

electronic system in an external field, is a unique functional of the electron density n(r),

and this functional has its minimum at the correct ground state density no(r). At finite

temperature, this extremum property is assumed by the grand potential fie = Ee — TSe —

fj,eNe, which is now minimal at the equilibrium density n^(r) [22]. The extremum property

allows one to obtain the ground state density by the self-consistent solution of the single

particle Kohn-Sham [2] equations

[-^V2 + ve„(r)]9a(r) = eA(r), (1.43)

where the problem of finding the optimum density for the system under consideration

has been reduced to that of finding the ground state density of a non-interacting gas of

electrons in an external effective potential,

/ x / \ 9 f rc(r') ,, , SExc\n(r)] .

A.

tyw(r) = .(r) + e'/T7i^+-^LU! (1.44)

which depends on the density,

n(r) = £/a|*a(r)|2. (1-45)a


For such a system, we know that the (internal) energy is simply the weighted sum

#e = £/«£„ (1.46)a

with the weights at temperature T given by the corresponding Fermi function

fa = f{Sa ~ He) = FT Wl. T*l , 1' (1-47)

exp[(ea - He)/kBT] + 1

In the local density approximation (LDA) the exchange-correlation energy is given in

terms of the exchange-correlation energy density, exc[n(r)], of a homogeneous electron gas

which, at the point r, has the same density, n(r), as the original system:

ExcMr)] = J exc[n(r)]n(r)rfr. (1.48)

Within the LDA, all ingredients for a self-consistent solution of the 'Kohn-Sham' eqns.(1.43)-

(1.45) are known. The starting density is usually taken as a superposition of atomic

densities centered on the sites of the lattice under investigation.

The grand potential functional of the interacting electronic gas in the external potential

v(r) can be written:

fte(T,Me,n(r)) = l{v(T)-^]n(v)d3r + jf^^dzrd3r'+G,[n(r)] + FM[n(r)], (1.49)

where Gs[n(r)] = Ks[n(r)] - TSs[n(r)] is the Helmoltz free energy of the non-interacting

electrons, with kinetic energy Ks and entropy Ss. Fxc is the exchange-correlation contri¬

bution to the free energy of the interacting system which, in the LDA is approximated

by

Fxc[n(r)] = J /xc(n(r))n(r)rf3r, (1.50)

where /xc(^(r)) is the exchange-correlation contribution to the free energy per particle of

a homogeneous electron gas of density n at temperature T. A number of approximate

calculations of fxe(n(r)) [23, 24, 25] have shown that at the temperatures of interest in

this field, this quantity can be approximated by its zero-temperature limit £,xc[ri(r)], for


which standard parameterizations to many-body perturbation theory or quantum Monte

Carlo results are already available (listed in [26]).

The kinetic energy and entropy of independent electrons are given:

Ks = £/„£«- J veff(r)n{r)d3ra

J

Ss = -£B£[/aln/a + (l-/Q)ln/a] (1.51)a

Using these last equations and after regrouping terms and replacing the sums over single-

particle states by integrals, the grand potential now is written:

ne(T, iie, n(r)) = - / N{e)f(e - »e)de + £dc[n(r)], (1.52)

where

N(e)= f n{e)de, (1.53)J—OO

is the integrated density of states at the energy e and the so-called double-counting con¬

tribution Edc[n(v)} is defined by

Edc[n(r)} = ~f ^^ld3rd3r' - j n{v)[exc{nQ{v)) - ^xc{nQv)}dzr. (1.54)

The compositional disorder renders an exact treatment of RSA within the DFT very

difficult, since the ground state density and the corresponding value of the grand potential

functional are configuration dependent. It is clear that a direct average over all configu¬

rations is unfeasible. However, Johnson et al [27] have shown that the applicability of the

DFT, within the CPA, has not been altered by the loss of the translational symmetry.

They first write, quite generally, the grand potential eq.(1.52) as

fte(T,/*e,n(r)) = - J N(e;ne)f(£- fie)d£

,[»',<[ dN(e; fjte) tl,

...

J-oo ^ J —<V—^£~^ ' ^ ®

where the second term on the right-hand side is proved [27] to be equal to EdC[n(r)].

Taking the average over all configurations the last expression becomes

Cte = - j N(s; /ie)/(e - ne)de


where N denotes the configurationally averaged integrated density of states per site. This

part of the grand potential does not yet contain the energy of the ion-ion interactions;

when these interactions are combined with £le, it becomes the total grand potential Cl of

the system. As it stands eq.(1.56) is an exact expression for the electronic part of the

grand potential for disordered systems, however, some approximation for TV is needed to

make the calculation of f2e tractable. Within KKR-CPA, such an approximation exists

and it is given by the generalized Lloyd formula [28]

N(e;ne) = N0(e) + Tr-1^{TrL[\n(4(e))]}

-7T-1£c03{TrL[ln(l + (^i(e) _ t^rf)]}, (1.57)Q

where Nq(e) is the integrated DOS for the free electrons at the energy e, and the reference

system is the empty space. The authors of ref [27] use the stationary property of N(e; fie)

with respect to variations in the single-site t matrix of the effective CPA scatterers tc [29],

to write the total energy expression (including the ion-ion interactions, ignored until now)

as follows

Cl(T,fxe,{nQ{T)}) = YtCQQQ{T,fie,nQ(r),n{T)), (1.58)Q

with the components

nQ{T, ne, nQ{r), n(r)) = - f NQ{e; fie)f(e - ne)de + Edc[nQ(r)} + EQM. (1.59)

Edc[nQ(r)] in this equation is the restriction of the double-counting term eq.(1.54) to a

cell containing an atom of type Q:

Edc[nQ(v)} =-£ I nfl)nQ}^d>rd*r' + / nQ(r)[sxc(nQ(r)) -^c(nQr)]d3r. (1.60)

L J cell I r—

r | Jcell

and the Madelung contribution E% is the sum of the intercell part of the double-counting

term and the inter-nuclear Coulomb repulsion energy:

Q_

e2^, ZQZ f n(r')nQ(r)

n -—

where Zq is the nuclear charge of species Q, n(r) and Z are the concentration weighted

averages of the densities ng(r) and nuclear charges respectively.

^UwrLj^m-f^


The last term of this equation can be expressed as a multipole expansion by means of

the Madelung constants Mffi and multipole moments q® and qf, as [30, 31]

n(r>0(r)-ru 7 U '/' =

Thus the monopole contribution to the averaged Madelung energy reads

J// |r-r' + R-|rf

rdr =2^2^Q° M0J *j (L62)jcm r r -t- xt, , ,,

.

,n

which vanishes in the SCPA because it imposes the same average local charge density at

every site, independent of its specific environment. The Madelung contribution to the

ground-state energy of a random binary alloy treated in the SCPA, thus, comes only from

multipole interactions due to the non-spherical shape of the unit cell and the angular

dependence of the electron density within the cell. For a spherical shape approximation

to the atomic charge density, it vanishes.

Chapter 2

Tight-Binding Linear Muffin-Tin

Orbital Method

2.1 TB-LMTO Approach for infinite systems

The TB-LMTO approach is best derived from a multiple scattering point of view, special¬

ized to a muffin-tin (MT) geometry. Within the MT approximation, the atomic potential

is taken as spherical inside the atom-centered muffin-tin spheres, and to be flat in the

interstitial region between the spheres, where it is weakly varying,

Q , xl viff(\Ti\) \TiHr-Ri\Kr\n.

I Vq otherwise.

For the MT potentials, the reference medium is empty space (V =' 0') and the single-site

scattering matrix is diagonal in momentum space. Its matrix elements are given in terms

of phase shifts r# [16]:

2m

tj-1(e) = -kcot rti(e) +«k, k2 = —«-(e - v0), (2.1)n

which are defined by

., v

ni{KSMT)Dt{e,SMT)-KSMTn'i{KSMT),nn,

Kcotnle) = k-71, -=—, r 77 r-. (2.2)

3i{k>Smt)Di{£, smt) - ksmtJRksmt)

15

2.1. TB-LMTO Approach for infinite systems 16

The radial logarithmic derivative function

d{ln[Ri(e,r)]} ,

1Ji{E,SmT) — SMT -j \r=SMTi \^-6)

is expressed by means of the solution of the radial Schrodinger equation inside the sphere,

Ri(e,r), at energy e, and ji and n* are the spherical Bessel and Neumann functions,

respectively. After the muffin-tin approximation is made, the KKR condition is written

in the traditional way

det[Kcotr}i{e)6Lu + BLLi(k,«)] = 0, for all z,L. (2.4)

where the usual three-dimensional KKR structure constants are given by:

BLv (k,«) = Gojll1 (k>«) - *«<5££i. (2.5)

In terms of wave functions, eq.(2.4) expresses the fact that, at the energy Ej at which a

solution to the Schrodinger equation exists, its partial wave expansion within any muffin-

tin sphere matches continuously and differentiably the solution of the Helmoltz wave

equation

[V2 + «2]*(r,ej) = 0, (2.6)

in the interstitial region.

The real space KKR equation, which after a Fourier transformation for periodic sys¬

tems leads to eq.(2.4), is

E[«cot[^(e)]Mi// + ^(«)]<L = 0, for all i,L, (2.7)

where uJ0L is the coefficient of the L-th partial wave in the MT sphere centered at site

Rj. With the energy dependent partial waves as a basis, the structure constants are

dependent on energy. Also, due to the choice of empty space as a reference medium they

are long-ranged. Both of these drawbacks are eliminated within the TB-LMTO method.

The energy dependence of the KKR matrix, eq.(2.7), through both the phase shifts

eq.(2.2) and the structure constants Bx[l,{k), is weak if the sphere packing is close [32],

2.1.TB-LMT0 Approach for infinite systems 17

that is if the wave length 2tt/k is much longer than the typical distance between nearest

MT-spheres. Within the atomic sphere approximation (ASA), which replaces the MT-

spheres by the Wigner-Seitz (WS) spheres, the energy dependence is almost completely

canceled. It has been proved that ASA gives an accurate description of the electronic

structure provided that [33]

|SR~7~d| < 0.3, d =| R - R' | (2.8)

The explicit calculation of the phase shifts at sws and for small values of k leads to:

KCoti»(e) S 2(2/ + l)^M±i±I)(n-^Sw, (2.9)

with the normalization constant TZw given as

ll'~

K(Ksws)(2i-i)\\(2i'-iyy^iU)

Replacing eq.(2.9) in eq.(2.7) and taking the k = 0 limit, one obtains the KKR ASA

equation:

• YHPL(z)khv - SIlM,l = 0, for all i,L., (2.11)J,L'

where the potential function for the site i and the canonical structure constants are given

by:

K2->0

= 2(2/+ 1)Dt(e) + (/ + !)

Dt(e)-l'

Stv = \im\nul BliLLI(K)}. (2.12)

The structure constants, eq.(2.12), are now independent of energy but they are still long

ranged. In the TB-LMTO method the reference medium is chosen in such a way that, in

the energy range of interest, no eigensolution of the Schrodinger equation exists in free

space [20]. Such a reference system can be made up of non-touching hard spheres of radii

a] centered at all the sites i, for which the phase shifts are given by [34]

co*°<i(£>=J2'~S^1)!!=2(a+i)(ff><"h,<r">»*" (2-i3»

2.1.TB-LMT0 Approach for infinite systems 18

which correspond to the single site t matrix

M)<2'+1>(2U)trl~

(2/ + l)!!(2Z-l)!!"K }

With respect to the new reference medium, the physically relevant states correspond to

the solution of the following equation [20]

YH^UcWlv ~ GZll'Kl = 0, (2.15)

where

At-^^-trr^t-^t^-t-1)-1^1 and Gr = G0(l - trGo)_1. (2.16)

Making use of equations eq.(2.12), eq.(2.14) and eq.(2.16) the screened KKR ASA equa¬

tions

EtfUWu/ " SZll'Kl = 0, for all ,\ L, (2.17)

are obtained. The potential function PQ and structure constant matrix Sa are related to

the free space counterparts by means of screening transformations

SZW = [So(l - aSo)-1]?,,, (2.18)

with

a, = 2(2TTT)^r- (2-i9)

The set of aj-s which gives the best localization for the structure constants correspond

to the so called TB representation. Their value was first found by trial and error for a

number of close packed lattices [35]. In our calculations we have used1:

aj = 0.3485 = 0„ oL = 0.05303 = f3p, otA = 0.017 = 0d, 0t = 0 for / > 2, Vz (2.20)

1using the hard sphere radius a* obtained from equating the rhs of eq.(2.19) to the value /3a = 0.3485

quoted above, for the higher partial waves as well, one would obtain 0P = 0.0564 and /?<* = 0-0164, very

close to the empirical values of Andersen and Jepsen [35]

2.1.TB-LMTO Approach for infinite systems 19

With the help of eq.(2.18), one can transform the potential function from the repre¬

sentation a to another, 6, by

PU& = PU(e)[l ~ (Si - atoPUM]'1 (2-21)

The linearization of the potential function is the last step towards the TB-LMTO

theory. It is done by its parameterization in terms of potential parameters describing the

center C\, the width A} and the distortion 7/ of the pure (i, I) bands:

KM = (e- Cf)[AJ + (7/ - ci)(e - C?)]"1. (2.22)

The choice of screening parameters a} = 7/ gives a very simple form for the potential

function,

P^M = ^p (2.23)

and transforms the screened KKR ASA equations eq.(2.17) into a eigenvalue problem in

an orthogonal basis:

B^-*M^)«U = 0, (2.24)

with

Hlju = ClSiM, + (Ai)^LL,(Ai,)* (2.25)

The Green's function corresponding to the Hamiltonian eq.(2.25) can be written as

GLM*) = (Ai)-1/2^Wz)(Aj:)-1/2 (2.26)

where

gr(z) = [P7W - S,]"1 (2.27)

is the auxiliary Green's function, equivalent to the scattering path operator in multiple

scattering theory. One can go over the most localized TB-LMTO representation /?. Using

eqs.(2.18) one can connect the auxiliary Green's functions in any two representations a

and 8 by [35],

g (z) - p^jg w pJw-(a - >W (2'28)

2.2.TB-LMTO Surface Green's function 20

In the most localized TB-LMTO representation ft the physical Green's function takes the

form

GiLjviz) = \fiM%6w + ti,L(z)g?LMzHAz) (2-29)

where the site diagonal matrices \p(z) and Vp(z) are given by

\i (z\_

1l-Pl**K) Ai + (7i-/?i)(^-Ci)

with the auxiliary Green's function g^(z)

i&jL'W = [(P/»W-S/»)-1]ii^ (2.31)

2.2 TB-LMTO Surface Green's function

The surface represents a strong perturbation for the system because the translational

symmetry in the direction orthogonal to it is violated. However, only the first few surface

layers have local physical properties which differ from those of the bulk. Assuming that

from a certain layer on, the electronic properties of all subsequent layers are identical

to those of the corresponding infinite systems, the presence of the surface can be mod¬

eled by three different regions, namely a homogeneous semi-infinite bulk, a homogeneous

semi-infinite vacuum region, coupled to each other by an intermediate region consisting

of several surface layers. The semi-infinite vacuum region is represented by empty spheres

and characterized by flat potentials. The resulting infinite system can be described math¬

ematically [36] by an infinite stack of principal layers (PL) parallel to the surface in such

a way that only the nearest-neighbor PLs are coupled by the structure constants. A PL

can include one or more atomic planes depending on the face, the lattice type, and the

spatial extent of the structure constants Sa. In the TB representation, where the struc¬

ture constants have the shortest possible extent, the fcc(OOl) surface, considered here can

be built up by PLs that consist of one atomic plane.Using the translational symmetry


parallel to the surface, we write the TB structure constants in the layer representation as

follows [37, 38]:

5f(k,,)=siom)spq+s?x(k||)<w,9+sj°(k|,)vi*' <2-32)

where

^9(kll) = £ exp(ik||R)5^(R). (2.33)it{Rpq }

k|| is a vector from the surface Brillouin zone (SBZ), and Rpg denotes the set of vectors

that connect one lattice site in the p — th layer with all lattice sites in the q — th layer. Ob¬

viously, the two- dimensional Lattice Fourier transform of the auxiliary Green's function

g13, eq.(2.31) is given by

{g"(k,i; z)}pq = {[P0(z) - S^k,,)]-1^. (2.34)

The following numbering scheme for PLs

V(vacuum) : —oo < p < 0,

S'(surface) : 1 < p < n,

B(bulk) : n + 1 < p < oo,

where n is the number of PLs in the intermediate region, is used to describe the infinite

system. This formal partitioning of the system allows one to write the inverse matrix, M,

of the auxiliary Green's function, eq.(2.34) in the following block-tridiagonal form:

Mvv MVs 0

M = Msv Mss MSB

y0 MBs Mbb j

The calculation of the surface-surface block of the auxiliary Green's function of the whole

system [39], requires the following inversion

g|5(k,|; z) = [Mss - MSV{MVV)-1MVS - M5b(Mbb)-1Mb5]-1 . (2.35)


The matrix element between p and q PLs of its inverse obviously is given by

{[gSs(k||; z)Yl}pq = [Mss}PQ - [MsviMwr'Mvs]^ - [MSB(MBB)-lMBs}pq. (2.36)

The explicit expressions of the blocks of the matrix M with respect to the principal layer

indices can be written as

(Mw)M = Mv^+M01<Wi+M10<Wi>

(MssY9 = Ms^ +M01^-i + M10Wi,

(MBfl)« - MBSP,q + M01Sp,q-i + M10Sp,q+i,

(Mv,s)M = M%Ai,

(MS,flr = M°%,nSq,n+l,

(M5,vr - M10^,iô,

(MB,5)P9 = M10<W+i<W (2.37)

with diagonal blocks given by

Mv = [Pfiy{z) - Sj°(k,|)],

Ms = [/?(*) - Sj°(kn)] (l<p<n),

MB = [P/,,b(z) -S5°(k|,)]. (2.38)

The nearest neighbor PLs are coupled by structure constants ^(kn) and 5^°(k||), thus

it is found that

M01 = -Sj^k,,),

M10 = -5j°(k|,). (2.39)

Due to the tridiagonal form of the blocks of the matrix M, the infinite product of blocks

involved in eq.(2.36) are reduced to

[MsviMwy'Mvsr = [M%iM(Myv)-7'fcM01ôM >

= M10[(MvV)-1]00M01îî, (2-40)

2.2. TB-LMTO Surface Green's function 23

[M5b(MBb)-1Mb5]p<? = [M01W,«+i{(MBfl)_1}''*M\^iW,

= SfMCMflBj-^'^M'V^. (2-41)

By definition, the surface Green's function is the top PL layer projection of the Green's

function of the homogeneous semi-infinite bulk or vacuum [38]. According to this defi¬

nition, quantities [(Myy)-1]0,0 and [(M^g)]""1"1'""1"1 are nothing but the surface Green's

functions corresponding respectively to the semi-infinite vacuum and bulk. They can be

calculated from the conditions

0*V(k|,,*) = [P0y(z)-S^kll)-S}\k{l)G^(kll,z)S^(kll)}-1,

G^B{khz) = [P^zJ-fiy^J-^Ckii^fkii.^Ckii)]-1, (2.42)

wich express mathematically the idea that by adding (removing) a PL of atoms to (from)

the surface the same semi-infinite solid is recovered ref.[38]. Putting everything together,

the inverse Green's function in the surface region has the following PL representation:

{(/(k„, z))-%q = [F${z) - 5j°(k||) - rj(k„, z))Spg

where the quantities

r?(k,|,z)

rS(k,|,z)

have the meaning of the embedding potentials that couple the surface region to the vacuum

and bulk region respectively. In other words, the concept of the SGF reduces the original

problem of the infinite order in PL indices to an effective problem of finite order n.

The on-site Green's function, which is needed for the calculation of the DOS and the

charge density is obtained by integrating over the surface Brillouin zone (SBZ) the (pp)

block of the quantity ^(ky; z), namely

W^EAfth*)- (2-45)

-SjPfti^-SyCkiiJVi*, (2-43)

= Si°(kll)G^(kll,z)Sf(k{l),

= 0 for p = 2,3,...,n-l,

= Sf(kll)G^B(kll,z)Sf(kll). (2.44)


The quantity iVj| is the number of sites in a given layer. Due to the block-tridiagonal

structure of {g13(k\\; z)}-1, eq.(2.43), one finds

^(kjj;z) = [Pf{z) - Sj°(k||) - nj(k,hz) - ^(kii.z)]"1, (2.46)

where

nj(k|hz) = ^1(kll)[p|+1(z)-5j0(kll)-n^1(kll,z)]-1^°(kll),

nj(k|,,z) = ^J^iJ^W-^^iiJ-^-xCkii.^SyCkn). (2.47)

The initial values for the set of the recursive equations, eq.(2.47) are given by the embed¬

ding potentials

nj(k,|,*) = rj(k|,,z), (2.48)

nj(k,|,z) = rf(k|,,z). (2.49)

The set of eqs.(2.43 to 2.49) are most important part of this section. They will be

used as a starting point for the application of the SCPA in the case of alloy surfaces.

Chapter 3

The SCPA within the TB-LMTO

approach

The implementation of the CPA within the TB-LMTO method provides a very efficient

self-consistent Green's function approach for the calculation of electronic properties of

disordered bulk alloys, their surfaces and interfaces [12, 13, 37]. The resulting formal¬

ism, TB-LMTO-CPA, has the simplicity and physical transparency of empirical TB-CPA

schemes, while it calculates the electronic properties of completely disordered alloys with

the same level of accuracy as they are calculated by first-principles multiple-scattering

KKR-CPA [27] approach. In the following we will summarize the elements of the TB-

LMTO implementation of the SCPA for bulk alloys and their surfaces following the formu¬

lation of refs [12]. Then lattice relaxation effects due to different sizes of the component

atoms A and B will be considered. They are accounted for by a simple rescaling of the

potential parameters, which leads naturally to a trimodal (AA, AB, BB) distribution of

nearest-neighbor distances [12].

3.1 TB-LMTO-CPA for bulk alloys

The properties of individual atoms occupying the lattice sites are characterized by the

potential parameters XRL (X = C, A, and 7), which are randomly distributed on the

25

3.1.TB-LMT0-CPA for bulk alloys 26

lattice sites. The quantity of interest is the configurational average of the physical Green's

function eq.(2.26). Due to the off-diagonal randomness of the structure constant matrix

Sy, induced by the random potential parameters 7 and A, the average of eq. (2.26) cannot

be performed within the CPA. One therefore goes over to the most localized TB-LMTO

representation f3, whose screening parameters, eq.(2.20), are configuration independent.

In this representation, the structure constants Sf£L, are nonrandom and the random

quantities P^l{z)-> ^,l(z) an<^ /^f,z,(z)> which enter in the definition of G(z) are all site-

diagonal.

The configurational average of Grl,r'L' {z) is expressed as

(Grl#v(z)) = E tfA*M)***sLv + E fi?{*)tf{*))(i$«ui$jS!{*), (3-i)Q,R Q,Q'

where (g0{z^ru = {Pr9rLrl>(z)p?)i with Q,Q = ^,5 and p% the occupation num¬

ber. The first term in eq.(3.1) averages trivially, (p#) = c®, thus

E^,fw(^) = E«:^^' (3-2)Q Q

where it is assumed that R {Rp}, the group of geometrically and electronically equiva¬

lent sites, it means that after the configurational average is taken, all the sites that belong

to such a group are equivalent to each other.

To calculate the second term we use the relations:

PfiM = Y,PQRP*f(*l Y.PQR = 1. (3-3)

which lead to

Q

_A_

Pf(*)-P?(*)Pr =

APf(z)

Pr '

APj(z)• (3-4)

Now it is clear that the expression {g^(z))Q,Q ,in a matrix form, can be written

3.1.TB-LMT0-CPA for bulk alloys 27

where AP£L(z) = P^(z) - P^f(z). P*{z) is the potential function supermatrix for a

particular configuration. The sign function is defined as

sgn(Q,Q') = «

1 if Q = Q'

-1 HQ^Q'

while Q, Q' = A or B. The meaning of Q or Q is Q = B if Q = A, and Q = A if

Q = B, and similarly for Q . Dropping the site indices and after some transformations,

the eq.(3.5) can be further written as

tf(z))Q# = sga(Q,Q')[AP0(z)]-l[P%)(g^z))Pf(z)

-(P0(z)g^z))Pf(z) - P^(z)(gft(z)Pfl(z))

+(P0(z)g^z)P,(z))][AP,(z)]-K (3.6)

In the most localized TB-LMTO representation the configuration-averaged auxiliary one-

electron Green's function (g13) is related to its SCPA approximant g? through the Dyson

equation

.(g/?)=g/3 + g/?(T)^, (3.7)

where T describes the scattering of the electron by the energy dependent difference po¬

tential

W(z)=P0(z)-V0(z). (3.8)

Vp{z) is the coherent potential function, which after the configurational averaging, charac¬

terizes the scattering properties of the effective site R. The coherent potential functions

are the same in each equivalent group {Rp}. They are calculated by using the SCPA

condition eq.(1.31), namely

X &*)-*. "He, ^W.—gfcffiL. (,9)

and ^pyL(z) is the on-site element of the auxiliary Green's function eq.(2.31). Further one

uses the following identities

P0(z) = Ve(z) + (ge(z))-l-(gP(z))-i

(3.15)M*g£,M«.+A'=GM

expression,

CPAinhomogeneouscorrespondingthefindweeq.(3.14),in0=(T%9Takingformalism.

TB-LMTOthewithinfunctionGreen'sphysicallyaveragedtheforresultexactanisThis

well).as(Im)=Lindicesangular-momentumtheindiagonalcasenon-relativisticthein

(andindicessiteindiagonalareNpandMpAp,supermatricesnon-randomthewhere

(3.14)N*<T)MN*,++N*«T)g%M«

M^(^(T))pqW+MP(g\qM'+ApSp,q=(G>M

(3.13)(AP,)"1,A^=

PJ*)-1-

vfyrF-

(^=Np

obtainsone

/AP.WA„—T>P,B\-1_

wtjMf(p,B_

v,A

and

VlL(z)]}/&PlL(®.U)-nP0BL(z)[Pp0;£(z)-VlL(z)}-{nP0AL{z)[Plf{z)=Mp0,L(z)

[A/iJ)i(.)]2[(P|)L(,))-^L(,)]/[AP^(,)]2+(A^W)=AJ,L(*)

37][12,definitionstheusingand

Q'andQoversumthetakingeq.(3.1),intoitInsertingexact.is(3.11)expressionThe

(3.11)(T(z))}[AP0(z)}-1+V0(z))-+(T(z))g?(z)\Pf(z)

V/,(z)]g/>(z)(T(z))-(z)[Pf+V0(z)}-+[<P„(*)>

V0(z)}-Vfi(z)]tf(z))\Pf(z)-sgn(g,Q')[AP,(.)]-1{[P?(^)=(g^W

findto(3.7),equationDysonthefromderived

(3.10){T(z)),+-(T(z))g/>(z)Vp(z)

-V0(z)-V0(z)^(z)(T(z))

(P0(z))+V0(z)(g^z))V0(z)=<Pp{z)g?(z)T>p(z))

{^(z))V0(z)-^(z)(T(z))=tf(z)Pfi(z))

V0{z)(g^(z))-(T(z))^{z)=(P0(z)^(z))

28alloysbulkforTB-LMTO-CPA3.1.

3.2.TB-LMTO-CPA equations for alloy surfaces 29

where the SCPA auxiliary Green's function matrix element between two different sites

is given in terms of the coherent potential functions and structure constant matrix (see

eq.(2.31)) as:

g£i*i/(*) = K*V*) " S/0"W". (3-16)

The coherent potential function Vp{z) is determined from the set of CPA equations:

n,L(z) = (PlL(z)) + [PS;t(z)-VlL(z)}^L(z)

x[PP^)-VlL(z)}, (3.17)

*UZ) = wEilPM ~ Seto)-1]*} (3-18)iVP k

The quantity Sp(k) is the Bloch transform of Sffv. With the derivation of the CPA

equations, eqns.(3.15 - 3.18), the goal of this section, namely the determination of the

configurationally Green's function for a random binary alloy within TB-LMTO approach,

is achieved.

3.2 TB-LMTO-CPA equations for alloy surfaces

In a random semi-infinite binary aloy, the translational symmetry is violated not only

due to the random occupation of the sites by two different types of atoms but also by

the presence of the surface. Therefore, after the configurational average is performed,

(i.e. within the CPA), only translational symmetry parallel to the surface is restored. In

the averaged medium, all sites in a given PL are equivalent, but corresponding sites in

different PLs are of course not. The surface region of the averaged medium is coupled to

the semi-infinite bulk alloy, which is a region where all the sites are equivalent to each

other and satisfy the same bulk CPA condition eq.(3.9).

Mathematically the surface region of the random binary alloy can be described by the

auxiliary Green's function matrix, the inverse of which is given in eq.(2.43), where the

potential functions PPp(z) and the embedding potentials r^(k||,2) are random quantities.

To calculate the electronic structure of the system we use the inhomogeneous SCPA

3.2.TB-LMTO-CPA equations for alloy surfaces 30

expression for the averaged physically Green's function eq.(3.15), where the coherent

potential function Vp(z) is determined from the set of the coupled CPA equations similar

to eq.(3.17), with the on-site SCPA Green's function given by

*t>M = ]J-IX(klh*). (3-19)II fen

^(k„, z) = MM - s?(kn) - (nj(k,i, z)) - <n£(klh z)))~l. (3.20)

The SCPA expressions for quantities (n^(k||, z)) and (n£(k||, z)), needed in the eq.(3.20),

are given by replacing the potential functions P/?,p+i and P/3,p-i in eqs.(2.47) by their

coherent potential functions counterparts, and their initial values by

<nj(k||, z)) = (rj(k||, z)) = sfik^m, z)sl°(kll),

(nf(k,|,z)) = {r?(kl|,,)) = 5j°(k||)^v(kll,,)5j1(k||), (3.21)

where Q^a(k\\, z) and Q^v{k\\, z) are the CPA expressions for the SGFs of the bulk alloy

and of the vacuum. They are given by the self-consistent equations

S*°(k||,*) = [VU*)-Sf{k\\)-G^{khz)\-\

QW{khz) = [V,,v(z)-Sf(kll)-g^v(kll,z)}-\ (3.22)

where Vp,a(z) is the coherent potential function which satisfy the CPA bulk condition,

and Vpy(z) is the corresponding vacuum quantity.

The presence of the surface makes that all the quantities entering in the eq.(3.19) and

eq.(3.20) to be layer dependent, which is a reflection of the fact that the CPA averaging

restores only the translational symmetry parallel to the surface. It is clear from eqs.(3.22)

that the surface region is coupled with the averaged semi-infinite bulk alloy, it means that

all the sites in the bulk are equivalent and satisfy the same bulk CPA condition.

The eqs.(3.19 to 3.22), are the main result of this section. They are used to calculate

the DOS and charge density that are needed in the LDA charge self-consistency.

3.3.Physical Quantities 31

3.3 Physical Quantities

The last step toward the complete TB-LMTO-CPA theory for random alloys and their

surfaces is the calculation, within this approach, of the charge densities and densities of

states for each alloy constituent. These are the simplest quantities obtained from the

on-site elements of the averaged Green's function (see sec.1.1.1).

The total DOS is related to the Green's function of the system by eq.(l.ll). Combining

this equation with eq.(3.1) and using the fact that A^'j- is a real quantity, the total DOS

of the random alloy within the TB-LMTO method becomes

nP(s) = -- EES°^W',?(£ + «)], (3-23)^

Q L

where

Op

The DOS eq.(3.23) helps one to determine the Fermi energy £f of the random alloy, which

is defined from:

r^=EEW- (3-25)J-°°

P Q

where Z® is the number of valence electrons corresponding to the alloy component Q

located on a lattice site that belongs to the group Rp of equivalent sites, (i.e., on the p-th

PL). The projected local DOS on an atom of type Q on the site p then is given:

«?(*) = - E $*^mi(e + «'*)]• (3-26)

In order to perform the charge self-consistent alloy calculations in the ASA within

the DFT, one needs the electron densities njp(r), which are related to the conditionally

averaged Green's functions by eq.(1.14). Within the TB-LMTO method the electron

density, spherically averaged (r =| r |) in its own sphere and corresponding to an atom Q

on the site p is [12, 37]

= ^ E ff^(£'r)^ W520O1 &, (3-27)

3.3.Physical Quantities 32

where the wave functions (Ppti(£,r) = <Pi(e, | r |)F£,(f) obey the Schrodinger equation

with the effective one-electron potential given by [37]

VpQ(\ r I) = -J*f + *?•*["?(! r I)] + Vp%[n?(\ r |)] + V?«, (3.28)

where the first term is the Coulomb attraction to the nucleus with charge ZqjP, the second

term is the Hartree potential due to the spheridized charge density n^(\ r |), and the third

term is the exchange-correlation contribution. The last term in eq.(3.28) is the Madelung

term given by eq.(1.62). It was menntioned previously (see sec. 1.2) that for spherical shape

approximations to the atomic charge density, the Madelung contribution from multi-pole

interactions vanishes and the sum in eq.(1.62) is reduced to the 11 = s term and that

within the SCPA it vanishes. A treatment which goes beyond the approximation of local

charge neutrality implicit in the SCPA for bulk random alloys and mimics the geometrical

relaxation due to the different sizes of the component atoms [12] will be presented in the

next sectionls Is.

For the alloy surfaces, Skriver and Rosengaard [31] have observed that in the neigh¬

borhood of the surface the charge density within the spheres deviates strongly from the

spherical symmetry. Therefore, a large V = (1,0) = pz dipole component, perpendicular

to the surface, has to be included in order to obtain the correct behavior at infinity. The

Madelung contribution to the one-electron potential, for alloy surfaces, thus becomes [37]

VMad = £(M;^ + Jif»gj), (3.29)Q

where M*sq and M** are the surface Madelung constants, qsq and qzq are averaged multipole

moments given by

# = £«? Winh jf M'n(f)»?w*-zp%}.

3.4.Charge self-consistency and lattice relaxations effects 33

3.4 Charge self-consistency and lattice relaxations ef¬

fects

When different atoms are brought together to form an alloy or compound, the redistri¬

bution of charge densities as compared to their atomic densities may lead to a transfer of

electronic charge from one atom to another, Each configuration of the completely disor¬

dered alloy is distinguished from other configurations by the arrangement of atoms about

each atomic site, namely the local environment. Therefore the alloy constituent atoms of

one given type can be crystallographically inequivalent. This inequivalence causes, for ex¬

ample, different net charges on chemically equivalent but crystallographically inequivalent

atoms.

The different sizes of atoms, on the other hand, cause some structural deformations

(geometrical relaxation) as eg observed when a large impurity atom is embedded into a

matrix of smaller atoms. The lattice expansion around the impurity has been confirmed

by extended x-ray-absorption fine structure (EXAFS) measurements, (e.g., for Cu-rich

alloys [40]). The chemically equivalent but crystallographically inequivalent atoms of the

completely disorderd alloys relax differently, leading to a distribution of bond lengths

different from a unimodal distribution.

The SCPA, which replaces the real atomic environment of a site by a homogeneous

average medium of identical effective scatters, can not account for those physical properties

(i.e. charge transfer and the distribution of the bond length) that are related to the local

environment. Furthermore, it can not answer the question as to how these properties

affect the electronic structure of the random alloy. Kudrnovskyand Drchal [12] have

shown however, how to approximately account for these effects within the TB-LMTO-

CPA scheme, and we reproduce their treatment below.

It is based on the obsrvation that within the ASA, the atomic sphere radii can be

chosen in such a way that the spheres are approximately charge neutral. The constraints

are that the spheres fill in all the space and that the validity of the ASA, eq.(2.8), is

preserved. A first guess for the sphere radii, for alloys obeying Vegard's law reasonably

3.4.Charge self-consistency and lattice relaxations effects 34

well, is given by the radii for the pure metals. The generalization to the case, when

Vegard's law is not satisfied, which is the case if the binding in the alloy is different from

that in the pure crystals, is also possible. If VQ and Vq are the actual and pure crystals

WS-sphere volumes, the preservation of the alloy volume VMoy requires

cVA + (1 - c)VB = VMoy, (3.30)

and, assuming linear pressure-volume relations with bulk moduli B® for the elements,

(VA - VA)/VA : (VB - VB)/VB = BB : BA. (3.31)

Solving the last two equations one finds:

va _

BBVall°y + (l-c)VB(BA-BB) AV ~

cVABB + (1 - c)V0BBAK°' {3M)

B_

BAVM°y - cVA{BA - BB) BV ~

cV0AB0B + (l-c)VBBAV° (3'33)

If Vegard's law is satisfied, Valloy = cV0A + (1 - c)V0B, the solution is, VQ = V?; oth¬

erwise, the sphere radii for pure metals, s® = (3V®/4-ir)1>/3, and the new ASA radii,

s<9 = (3V2/47T)1/3, are used to determine new potential parameters [33] through:

^Llnf^dlnsQ [s<7? = 7o^ + T^oln^]

A? = A& + [4rnnA?AnnsQ (3-34)so

After the extrapolation of the potential parameters to the new radii has been made,

account is taken of the fact that the WS radius of the alloy, walloy = {Walloy/4ir)1/z, is

different from sQ's, by multiplying the potential parameters A^ and j% by (sQ/wall°y)2l+1

[33]. Starting with this choice we perform CPA calculations and determine the local DOS

on each component and the total DOS as well. We then determine the alloy Fermi level

EF and calculate the local charges qQ by integrating the local DOS up to EF. The cor¬

responding deviations from the sphere neutrality are 6qQ = Z® — qQ = —4Tr(sQ)2nQ6sQ,

3.4. Charge self-consistency and lattice relaxations effects 35

where nQ is the electron density evaluated at WS radius sQ. Then, new potential pa¬

rameters are calculated for the new radii sQ + SsQ and all the steps are repeated until

8qQ <C 0.01 electron per site.

To incorporate the lattice relaxations the structure constant for the deformed (re¬

laxed) lattice between the sites R and R' occupied by atoms Q and Q' respectively is

approximated by:...alloy

coQQ'_ q0QQ' r

w]l+l'+l (o qc\

2rL,R>L'~

^RL,R'L'[rsQsQ>y/2i' ^"30>'

where SRQ^R,V is the structure constant of a perfect, unrelaxed lattice. To derive eq.(3.35)

we have used the common behavior of the structure constant matrix within the ASA [34],

Srl,R'L' CXI R _ R,' IJ+Z'+l

Yl+l',m-m'{R - R')> (3.36)

Yi+ti,m-mi (R - R') being the spherical harmonic comming from the partial wave expansion

of the free space propagator, and expressed the relative change of distance d9Q' between

the sites R and R' occupied by atoms Q and Q', in the form dQQ' = d0[(sQ + sQ')/2walloy].

Here d0 is the corresponding average distance between the the points R and R! in an un¬

relaxed lattice with the averaged WS radius Walloy. For values of sQ typical for transition

metals, the quantity (sQ + sQ')/2walloy is close to (sQsQ')1/2/walloy, from which follows

eq.(3.35). It is also assumed that eq.(3.35) holds locally, i.e. it is not influenced by the

occupation of sites other than R and R'. It is clear that factors ^sQsQ')ll2/waUov\l+l'+l

coming from (A?)i(A#)*, and [wall°v/(sQsQ'y/2]l+r+\ coming from SRl%L, in the ex¬

pansion of the second term on the rhs of eq.(2.25), cancel each other. As a result, the

unrelaxed structure constant SRLR,L, and the potential parameters given by eq.(3.34) can

be used to treat approximately the effects of the charge transfer and the lattice relax¬

ations. This is certainly an approximation, but as we shall see later on, it gives results

that are in good agreement with experiment.

Chapter 4

The 7 expansion method

4.1 GEM corrections to the CPA DOS for bulk alloys

and their surfaces

The GEM rests on the observation that, because V^(z) is complex, gp{z) will decay with

the distance | i — j | between two sites as 7J,1--''- This observation remains valid also for

the surface region Green's function, g%q(z) because the coherent potential, V^^v[z\ that

enters in its definition eqs.(3.20) is a complex quantity as well.

The configuration-averaged DOS is, as usual given by

n(e) = -- limImTrL(G(e + iS))m, (4.1)

while in the SCPA, it takes the form

n{e) = -- limImTrLG0o(e + iS), (4.2)7T 5—>0

where Goo is any site diagonal element of the physical one-electron Green's function su-

permatrix in the SCPA, eq.(3.15). For an inhomogeneous system, it depends on the site

p under consideration:

np(s) = — limImTri(G(£ + %S))„. (4.3)

36

4.1.GEM corrections to the CPA DOS for bulk alloys and their surfaces 37

Its SCPA counterpart can be written as

np(e) = — lim ImTrLGpp(e +15), (4.4)

where Gpp is the site diagonal element of the physical one-electron Green's function in

the SCPA. The corrections to the DOS due to SRO and/or correlated scattering from

clusters of atoms corresponding to the homogeneous and inhomogeneous cases then are

given respectively by:

8n(e) = n(e) - n(e)

= --/mTrL[(Mg0(T)g^M)oo + (Mg*<T)N)oo7T

+(N(T)g^M)00 + (N(T)N)00],

Snp(e) = rip(e) - np(e)

= -i/m'ftL{(Mg/,<T>g^M)w + (Mg^(T>N)w

+(N{T)^M)pp + (N(T)N)ff} • (4.5)

The energy argument on the rhs of eqs.(4.5) have been dropped for convenience. Our

treatment now follows that given by Masanskii and Tokar for a single-band tight-binding

model with site-diagonal disorder [10]. We start by dividing the coherent auxiliary Green's

function into a site-diagonal and a non-diagonal part:

g^gj + gin (4-6)

introduce the single-site scattering supermatrix

t = (1 - WgJj^W, (4.7)

where W has been defined in eq.(3.8), and expand (T) in powers of t:

(T) = (t) + (tgfdt) + (t&tg&t) + ... , (4.8)

where the first term vanishes in the SCPA. Inserting the above multiple scattering series

in eq.(3.7) we get the expansion of the configuration-averaged auxiliary Green's function


(g^), in terms of the small parameter je. By construction, the trace of (g^)^ is analytical

in the complex energy plane except for cuts on the real axis at all levels of truncation,

and its real (imaginary) part is symmetric (antisymmetric) when reflected through the

real axis. The last property which (g0) has to satisfy in order to be Herglotz, namely

that the trace of the imaginary part of (g/3)d be negative or zero as the energy approaches

the real axis from above, cannot be rigorously guaranteed a priori and must be explicitly

verified for each case. It is satisfied for all systems we have investigated.

To lowest order in g^, where i and j are site indices, the site-diagonal and off-diagonal

matrix elements of the configuration-averaged scattering operator are given by

(Tu) = £W$**>.

(Ty> = (t%V). (4.9)

Prom eqns.(4.9) follows that the leading term in the gamma expansion of the off-

diagonal matrix element of (T) is linear in %, whereas its diagonal matrix element is of

order 7^. After some trivial manipulations, we obtain:

7\ = c(l - c)aiAt(sfAt),

To = Zic(l-c)ai(A*flf)2(«A + *B) (4-10)

where the subscript 0 denotes the diagonal matrix element and 1 the one between nearest

neighbors. The number of nearest neighbors is Zi, a\ is the Warren-Cowley parameter

for the first coordination shell, and c is the alloy concentration.

In inhomogeneous systems the nearest neighbor pair of sites can be of different kinds,

i.e., they can belong to the same group of equivalent sites or to the different groups,

it means that for the fee (001) face considered here, three different kinds of nearest

neighbor pairs can be distinguished, (p,p) the sites belong to the same plane, (p,p+ 1)

and (p,p — 1) if one site belongs to the plane of the origin and the other site to one of

its nearest neighbour planes respectively. Therefore, the matrix elements between nearest

neighbors of the SCPA auxiliary Green's function will be noted as <7p,p, t^p+n and <7p,p_i,

while its diagonal matrix element is written as g^°. They can be calculated via inverse


Fourier transform of the g^g(k\i,z). The same notations will be used for the corresponding

scattering matrix elements, i.e, Tp° for the diagonal matrix element and Tpp,Tp\p+1, and

Tpp_x for the one between two nearest neighbors. Some trivial manipulations, similar to

the ones that lead to eq.(4.10), are needed to obtain the configuration-averaged scattering

matrix elements to lowest order in g^q

Tl,v = Cpil-cJalpAtpgÂtp,

Tp,p+i = Cp(l ~ cp+i)ap,p+iAtp9pp+i&tp+i,

Tp,p-, = cp(l - Cp-Jal^&tpfyUAtp^ , (4.11)

Tp° = Acp[{\-Cp)alpAtptfp*Atpgp\l +

+ (1 - Cp+JalpÂtpgÂtp+xg^p

+ (1 - Cp-JalÂtpgÂtp^g^p

x (t} + if), (4.12)

where af9<5p_g)(o;±i) is the SRO parameter between nearest neighbors and Atp = tp — tp.

In the absence of SRO, ai vanishes, and the expansion eq.(4.8) has to be carried out

to higher order. The first non-vanishing contributions to Ty and Ta, due to the correlated

scattering by pair of atoms in the fully random alloy, are then given by:

<Ty> = {eg^gftgft),

(Ta) = YJ$tP$MMJ)- (4-13)

To lowest order in je, the expressions for 7\ and T0 are in this case:

Tx = c2(l - c)2At(g?At)3

T0 = ZlC2(l-c)2(l-2c)At(g?At)4. (4.14)

For the inhomogeneous case, to lowest order in %, one finds

Tp\p = CpCp(l-Cp)(l-Cp)AtpgÂtpgÂtpgÂtp,

Tp\p+i = cpCp+i(l - cp+i)(l - Cp)Atpg^p+1Atp+1g^ltPAtpg^p+1Atp+l,

Tp\p-i = CpCp_l{l-cp^{l-Cp)Atp^_lAtp^\pAtpg^_xAtp.l, (4.15)

4.2.Short Range Order 40

T» = 4cp(l - cp)(l - 2cp) [cp(l - cp)AtpfpiAtpfpiAtpg^Atpg^ +

+ Cp_1(l-Cp_1)Aip5p5;p1_1Atp+1^l\)pAip^1A<p+1pp}l11)p] x A*p, (4.16)

In terms of To and T\ the leading corrections to the DOS for homogeneous systems read:

6n(e) = -hmTrL{M[gST0f0+Z1(flT1gS + g0^T1f1)}M

+M[g0oTo + Z1(gT1)]N

+N[T0g§ + Z1(T1g)]M

+NTQN}. (4.17)

Replacing the explicit expressions for T0 and Ti, given either by eq.(4.10) or by eq.(4.14),

in the eq.(4.17) the correction to the CPA density of states due to SRO or correlated

scattering by pairs of atoms, respectively, is obtained.

In the case of inhomogeneous sytems, the leading corrections to the layer resolved

CPA density of states, in terms of Tp and T1„#p,g(g=o,±i), become

Snp(e) = -UmTxL{M* [f/Tffi0 + ^+iTUp + tPlv + tUTliM

+ *9p P(T}+iJpi+i + T}^i + T^gH.,)] M*

+ M? [f/T; + 4(^p1+1Tp1+1,p + ~g%Tlv + tUTUP)\ ^

+ Np [TZf/ + i(Tp\hp~g^+1 + TijH + Tl^g^)) M*

+ NpT°Np}. (4.18)

4.2 Short Range Order

As shown formally by Massanskii and Tokar [10] the application of GEM to the grand

potential of the disordered alloy makes possible the derivation of an explicit expression

for the SRO parameters. The idea is to expand the internal energy of the electronic grand

potential in terms of the parameter 7e, and to go beyond the one point approximation for

the configuration entropy of the alloy.


The grand potential of the RSA can be written

0 = ne - TS (4.19)

where Qe is the internal energy part of the electron subsystem averaged over the configu¬

rations, S is the configuration entropy of the alloy, and T is the temperature at which the

system has been equilibrated. The equilibrium value of the short-range order parameter

a can be found from the condition

fj-l (4.20)

Now let us treat the terms on the right-hand side of eq.(4.19) separately. The electronic

part of the grand potential eq.(1.59) consists of three conributions, the band energy, the

double counting term and the ion-ion interaction or Madelung term. The double counting

term, which corrects the intra-atomic electrostatic and exchange-correlation energy, con¬

tributes only to that part of the grand potential which is independent on the correlations

between two and more sites. The Madelung term in principle, contrary to the double

counting term, should give a contribution to that part of the grand potential which is de¬

pendent on SRO parameters. However, we have shown that within the TB-LMTO-ASA

method, it is possible to vary the atomic radii in such a way that the charge neutrality at

each site is imposed while preserving the total volume of the system (see sec.2.4). This

condition satisfies the single-site approximation condition, namely that the potential at

any site is independent of its environment. Thus the Madelung contribution to the ground

state energy of a random binary alloy treated in the single-site CPA vanishes in the present

version of the LMTO-ASA. As a result, from all the parts of the grand potential, only

the band term remains to be treated which, at (T = 0) becomes

a l(T = 0,eF) = - TF N(e)de, (4.21)J—00

where the integrated density of states N(e) in the TB-LMTO approximation is given

[12. 41]:

N(e) = -^Tr^lnpP^)] +hV(z)). (4.22)


The first term compensates the extra singularities in lng^(z) which originate from the

poles of P/?(z). In the tight-binding representation /5, as a rule, these singularities lie well

outside the occupied part of the spectrum and need not be considered [12, 41]. Therefore,

the integrated density of states of the random alloy can be written

N{e) = -$s{Tr\ngfi(z)). (4.23)7T

To handle the last equation we use the identity [42]

Tr lng^(z) = Trg^) - Tr ln(l - Wgf(z)) - Trln(l - fnd(z)t). (4.24)

When this identity is averaged over the configurations within the CPA, the last term on

its right-hand side vanishes[42]. Taking this fact into account and expanding the final

logarithm in a series, we obtain

00 -I

(Trlng^)) = Tv\n^(z))CPA + £ - £ {th tim)(fnd)ilh (fnd)imh (4.25)„^ o lib -

m—Z i\....im

This relation leads to an expansion of the integrated DOS eq.(4.23) and the electronic

part of the grand potential eq.(4.21) in powers of je.

For the configuration entropy we can write down an expansion in a series with respect

to the correlations [43]:

-NS = M££PnQlnPnQ + ^££^f'ln^f7^^')Q i

Z-QQ" ij

'<?" ijk

where

*&?"* = (Ph--Pt) (4-26)

is the m-site probability. This expansion can be written in the form

S = 50+£5m, S0 = -fcB^£ifm/f, (4.27)m—2 in

Sm = -j£^ £ £ Pt£mMPt£mlPt£m), (4-28)'

Q\—Qm h—im


where S0 is the mixing entropy of the ideal mixture, and P®*J®mis the m-site probability

in the corresponding generalized superposition approximation of Kirkwood [43], which for

a cluster of three sites is given

PQ,Q' PQ,Q" PQ',Q"pQ,Q'Q"

_

rhi rh,k rj,k (A 9QxriJ,k ~

pQpQ'pQ"•

V*'**)

If we introduce the set of short-range order parameters by means of the relation

ottm = i - it:£mIPt£m (4-30)

then we finally obtain

kB

N-m\Sm = -jr^ E E 3?.S?m(i - <£::£") • Hi - <£::£m) (4.31)

Qi—Qm h—im

The population numbers satisfy the conditions

£*>? = 1, EP? = *Q (4-32)i i

where N® is the total number of atoms of species i. These conditions lead to relations for

the probabilities:

EpQi—Qi-iQiQi+i—Qm _pQi—Qi-iQi+i—Qm

*

»l...tj_l»/»/+l...»Tn *i...*j_iij+i...imQi

EpQl-Ql-lQlQl+l-Qm_

ArQpQl—Qj-lQ(+l—Qm /^ oo\

Ml...tj_i*itj+i...»ro— "/V rtl...»|_i»J+i...tm ^.OOJ

The probability P^J®m possesses the same properties [43]. This means that the param¬

eters of short-range order are not completely independent, so

E^:fe9m-«?^gm = ° (4-34)Qi

These conditions allow all the parameters a^.'.'/fm of the short-range order corresponding

to a fixed set of sites ix,..., im to be expressed in terms of any one of them. Expanding the

cluster entropy eq.(4.31) in terms of the SRO parameters to the second order we obtain

{zi\•

m.) Ql_Q < -im


Choosing the parameters oc^\\\fm as an independent variable and using eq.(4.34), the sum

over Qi variables becomes

rA

Qi...Qm 1=0 vm */•*• c

(^)m«:t)2(l +4 (4.36)

Now the m-sites cluster entropy eq.(4.35) writes:

kn ,cA- 2N.J^mZ«::Zf- (4-37)

We stop the expansion of the grand potential up to the clusters of pairs and write only

a dependent part of it with the result

«(*) = l*Y,Ur WWl&*^tJ*)<*£* - k-f£)£«)2, (4-38)7T • Z J —oo Z C „•„•

l\l2 ll%2

where Af1 =t%}—f^ and tlQ is defined in eq.(3.9). This expansion of the grand potential

is to the lowest order in the small parameter 7e if only the nearest neighbors pairs are

taken into account. Using the equilibrium condition eq.(4.20), we obtain the following

expression for the SRO parameter a\

ai = ~^fc{1~c)Vu (439)

where V\ is the nearest neighbor pair interaction in the generalized perturbation method

[42, 41]:

Vx = --S fF TrL[A^fA^f] ds, (4.40)7T 7-oo

where <?f is the Green's function matrix element between two nearest-neighbors sites.

Chapter 5

Results and Discussions

5.1 Results for bulk alloys

In this section the results of our calculations for five different alloys are presented. They

are obtained with the help of the fully relativistic version of TB-LMTO-CPA code, devel¬

oped by Kudrnovsky and coworkers [12, 13] with our implementation of the lowest order

GEM corrections to the SCPA. The exchange-correlation part of the energy functional

is treated in the LDA using Perdew and Zunger's parametrization [44] of Ceperley and

Alder's [45] Monte Carlo simulations.

We have chosen Ago.87Alo.13, Ago.50Pdo.50 and Pto.55Rho.45 as systems where the atomic

radii of the two components differ by less than 1 percent, so that lattice relaxation ef¬

fects on the electronic structure are expected to be small. In the other two systems,

Cuo.715Pdo.285 and Cuo.75Auo.25, the large difference in the atomic radii of the components

is expected to lead to a sizeable local geometric relaxation, which, in turn, strongly affects

the electronic structure [87]. Therefore for these latter two systems the electronic structure

is calculated for both, the relaxed and the unrelaxed geometry, whereas for Ago.s7Alo.13,

Ago.50Pdo.50 and Pto.55Rho.45 only calculations for the unrelaxed case are performed. The

problem of charge self-consistency is treated by imposing local charge neutrality for each

component of the alloy through an appropriate choice of atomic sphere radii, (see sec.3.4),

45

5.1.Results for bulk alloys 46

[12].

From a total energy minimization within the TB-LMTO-CPA, we obtain the theoreti¬

cal equilibrium lattice constants and bulk moduli, which together with their experimental

counterparts (where available) are presented in TABLE.5.1. Measured and computed

system a{°A)th ^(,-^Jexp B(GPA)th B(GPA)exp

Ag(87)Al(l3) 4.075 4.069t46l 115 105t46]

Ag(50)Pd(50) 4.024 3.978I47! 163

Pt(55)Rh(45) 3.933 3.870t49l 292

CU(72)Pd(28) 3.684

(3.684)

3.708^ 189

(193)

CU(75)AU(25) 3.835

(3.747)

3.754I48] 270

(176)

148[50]

Table.5.1 Calculated and measured values of lattice constants and bulk moduli

for the systems under consideration. Values in parenthesis refer to the unrelaxed

configuration.

lattice constants lie within less than 2 percent of each other for all five systems. Of the

two measured bulk moduli, the one for Ag(87)Al(i3) lies within 10 percent of the theoret¬

ical value. For Cu(75)Au(25) a large discrepancy is found, which we have no explanation

for. We attribute the strong increase of the calculated bulk modulus upon relaxation to

the concomitant narrowing of the DOS (see sect.5.1.4) in the Au bonding region, which

implies that a larger Coulomb energy has to be paid to compress the system.

The DOS calculations are performed at the theoretical equilibrium lattice constants.

5.1.1 Ago.s7Alo.13.*

AgAl presents a common-band behavior in the energy range ep to ep — O.GRy. The DOS

of the random alloy at the calculated equilibrium volume is shown in Fig.5.la.


30

I JE

I20a

i10 J ^

'"\

(b)

f0.01 / i i

itom aI \ A'fl ,y\A

*«-0.01

JgQ

\T '•»/•

-0.03

-0.3

E(Ry)

-0.6 -0.3

E(By)

Figure 5.1: Density of states (DOS) of Ago.s7Alo.1z obtained in the single-

site coherent potential approximation as the concentration-weighted average

of the local DOS on the Ag (dash-dotted line) and Al (dotted line) atoms

(a). Correction to the DOS due to correlated scattering from pairs ofatoms;

nearest neighbors: continuous line; next-nearest neighbors: dotted line (b).

Correction to the DOS due to short range order (SRO parameters from ref.

[46]); nearest neighbors: continuous line; next-nearest neighbors: dotted line

(c). The Fermi level is given by the thin vertical line.

The short-range order in this alloy has recently been investigated by Yu et al. [46],

using diffuse X-ray scattering on a single crystal quenched from an aging temperature

of 673 K, i.e. about 90K above the phase boundary separating the solid solution from


a two phase mixture [51]. We have used their measured SRO parameters to calculate

the relative correction to the DOS due to SRO. The result is reproduced in Fig.5.1c, and

is an order of magnitude larger than the corrections due to the correlated scattering of

electrons by pairs, as can be seen in Fig.5.1b. It induces a shift of the Fermi level by -1.15

mRy.

5.1.2 Ago.50Pdo.50:

Due to the significant separation between the Pd and Ag d levels in the free atoms, this

system is expected to present a so-called 'split band behavior', where the states in the

alloy preserve to a large extent the characteristics of the pure phases. This expectation is

illustrated in Fig.5.2a where the SCPA DOS of the random alloy and the local DOS on

the Ag and Pd atoms at the calculated equilibrium volume are shown.

The order-disorder phenomenon in this alloy has been investigated theoretically [52,

53, 54], but there is a lack of experimental data, which are limited to rather high tem¬

peratures [56], where the fee solid solution phase exists. To our knowledge, no diffuse

x-ray scattering experiments on the disordered alloy exist. Its components have almost

the same form factor which makes such experiments difficult. For that reason it is not yet

clear if this alloy has a tendency towards short-range order or whether it phase separate

at low temperatures. Using the generalized perturbation method [42, 41] and GEM, to

lowest order in %, in our calculations we find a small positive value for the short-range

order parameter, a\ — 0.054, at T = 800K which indicates a clustering tendency and

is consistent with the expectation [57] that the binary alloys of all late transition metals

should rather phase separate than order. However, Lu et al [52] have found that ordering

could be also possible. They have calculated the mixing enthalpy for the c = | AgPd

random alloy and for the ordered Lli structure with result, AHT(mdmn = —S8.2meV/atom

and AH(Lli) = —GOAmeV/atom. The negative sign of the AH for the random alloy

implies a tendency toward ordering, as does the fact that AH(Lli) < AHran<iom •Saha

et al [54] came to the same conclusion by calculating the variation of the band structure


45

L

,1.

1

(a)*

\

f1

§30 A !

I / V/\*31 ^K H

" \

815 i

VI

I \

1 \

1 \

J \

J V*-**".—„.

0.01 (b) ADC

AI*

0 4 \LA -"V

W

-0.01

V

-0.6 -0.3

E(Ry)

-0.6 -0.3

E(Ry)

* I=; 0.05 ,

£ A AE A Ao 1

/1 n1 i \ ii

1- /a o

~g~V / W1 (

/(0 1

8°-0.05

.

-0.6 -0.3

E(Ry)

Figure 5.2: Density of states (DOS) of Ago.50Pdo.50 obtained in the single-


of the local DOS on the Ag (dash-dotted line) and Pd (dotted line) atoms

(a). Correction to the DOS due to correlated scattering from pairs ofatoms;

nearest neighbors:(b). Correction to the DOS due to short range order (SRO

parameters calculated); nearest neighbors: (c). The Fermi level is given by

the thin vertical line.

energy with respect to the SRO parameter. Given the small value we obtain for ai at

T = 800K, a sign reversal due to volume dependent effects as the temperature is lowered

is not unlikely [55], so that our finding is not in contradiction with the above.

Corrections to the DOS due to the short-range order and correlated scattering by pairs


of sites are shown respectively in Fig.5.2c and Fig.5.2b. The corrections to the DOS due

to the correlated scattering (Fig.5.2b) give only minute changes to the total DOS. The

same can be said for the corrections due to the presence of SRO. The shift of the Fermi

energy coming from these corrections is only 0.08 mRy.

5.1.3 Cuo.7l5Pdo.285*

While Ag-Pd shows split-band behavior, Cu-Pd is a common-band system, where the Cu

and Pd atoms features are intermixed. This common-band behavior is expected from the

fact that the DOS of the elemental constituents overlap. The calculated total DOS of the

random alloy and the local DOS on the Cu (dash-dotted line) and Pd (dotted line) atoms

shown in Fig.5.3a, confirm this expectation.

The effects of geometrical lattice relaxations are illustrated in Fig.5.3b, which shows

the difference between the total DOS of the alloy Cuo.715Pdo.2s5 with and without taking

into account lattice relaxation within the formalism of ref.[12], (see sec.3.4). Our result

is in good agreement with that of Lu et al [58], which have found that because of the

relaxation, the DOS, near the region where the deepest Pd state is located, diminishes.

The SRO in Cui_xPds has been extensively investigated, both experimentally [59],

and theoretically [6, 58]. While in their work, Staunton et al [6] investigate the origins

of ASRO, Lu et al [58] demonstrate how the latter affects the DOS. Although they use

the special quasirandom structure method (SQS), which makes possible the description

of the electronic structure of the alloy in the presence of ASRO, their result for the

unrelaxed total DOS is in good agreement with KKR-CPA result of Ginatempo et al [60].

This fact is also confirmed by our calculations which show that the effect of correlated

scattering, Fig.5.3c, is insignificantly small, while SRO produces a correction to the DOS

in the percent range, Fig.5.3d, also in accord with the findings of Takano et al.[61] for

the composition x = 0.5. The shift of the Fermi energy due to the SRO amounts to -1.25

mRy.

To perform these calculations we have used our self-consistent potential parameters

5 1.Results for bulk alloys 51

60

1

<l

',1(«) 'A.

! 1 »

UIO 1 SA*

f 30

B

* noo

/' A

10I! l\

^J V

Q-0 003

-0* -0J

E(Ry)

(c)

VYV1

-0.6 -03

E(Ry)

-0.6 -0.3

E(Ry)

Figure 5.3: Density of states (DOS) of Cuo.n5Pdo.285 obtained in the single-


of the local DOS on the Cu (dash-dotted line) and Pd (dotted line) atoms,

including the effects of lattice relaxation (a). Difference between the DOS

in the relaxed and unrelaxed geometries (b). Correction to the DOS due

to correlated scattering from pairs of atoms; nearest neighbors: continuous

line; next-nearest neighbors: dotted line (c). Correction to the DOS due

to short range order (SRO parameters from ref. [59]); nearest neighbors :

continuous line; next-nearest neighbors : dotted line (d). The Fermi level is

given by the thin vertical line.


for the relaxed configuration at the theoretical equilibrium lattice constant and the values

«! = —0.157 and a2 = 0.171 quoted by Saha et al. [59] for a sample aged at 1023 K.

5.1.4 Cuo.75Auo.25:

The split-band behavior characterizes the DOS of disordered Cuo.75Auo.25 alloy as illus¬

trated in Fig.5.4a. This is consistent with the fact that the Au projected local DOS is

isolated mostly at the high binding energy range, whereas the Cu projected local DOS is

peaked in the energy range between -0.2 to -0.5 Ry.

Contrary to the CuPd alloy, the calculations with and without taking into account

lattice relaxations lead to the equilibrium lattice constants that differ by 2.3%. The DOS

of Cuo.75Auo.25 is strongly affected by the geometrical lattice relaxations. Our calculations

are in good agreement with those of Lu et al [58] and show that unrelaxed calculations

overestimate the bandwidth by around 40 mRy. This is illustrated in Fig.5.4b where

the DOS obtained in the single-site coherent potential approximation for the relaxed

(continuous line) and unrelaxed (dashed-dotted line) geometries is presented. It is clear

that (see Fig.5.4b) geometrical relaxation narrows the Cuo.75Auo.25 DOS in the Au bonding

region.

Using our self-consistent potential parameters for the relaxed configuration and the

values ai = -0.134 and a2 = 0.158 measured by Buttler and Cohen [62] at T = 703K,

we find the corrections of Fig.5.4c, due to correlated scattering by pairs and of Fig.5.44d,

due to SRO, respectively. Again the effect of correlated scattering is insignificantly small,

while SRO produces a correction in the percent range. In this case, the shift of the Fermi

energy due to the SRO is -1.4 mRy.

5.1.Results for bulk alloys

0.2

eE 0.1

S

I£„ o

-0.1

E(By)

"

III

In V5,1

11 *

-0 6 -0.3

E(Ry)-0.6 -0.3

E(Ry)

Figure 5.4: Density of states (DOS) of Cuo.75AU0.25 obtained in the single-


of the local DOS on the Cu (dash-dotted line) and Au (dotted line) atoms,

including the effects oflattice relaxation (a). DOS in the relaxed (continuous

line) and unrelaxed (dash-dotted line) geometries (b). Correction to the

DOS due to correlated scattering from pairs of atoms; nearest neighbors:

continuous line; next-nearest neighbors: dotted line (c). Correction to the

DOS due to short range order (SRO parameters from ref. [62]); nearest

neighbors : continuous line; next-nearest neighbors : dotted line (d). The

Fermi level is given by the thin vertical line.


5.1.5 Pto.55Rho.45:

At high temperatures, platinum and rhodium form fee solid solutions across the entire

concentration range but it is not yet clear whether these systems order or phase separate.

On the basis of tight-binding d-band theories, PtRh would be predicted to separate in

two phases [63, 64], at low temperatures, which agrees with extrapolations by Raub [65],

based on the observations of a miscibility gap in the phase diagrams of Pdlr, Ptlr and

PdRh. However, later experimental work by Raub and Falkenburg [49] and ab initio

total-energy calculations combined with a cluster expansion approach of Klein et al [66],

have shown that PtRh binary alloy systems, at low temperatures, will order rather than

phase separate. Their conclusion is consistent with our result on the SRO parameter

at T = 800K oj1=-0.164, which is negative and thus indicates an ordering tendency.

Unfortunately, to our knowledge, until now no SRO measurements on PtRh alloys have

been reported.

As shown in Fig.5.5a, this system displays a typical common band behavior. The effect

of the correlated scattering is presented in Fig.5.5b and is found to be again insignificantly

small. SRO produces a correction in the percent range Fig.5.5c and shifts the Fermi level

by -1.5 mRy.

5.2.Discussion of bulk alloys results

25(»)

i\ i •> J\\

i \ i W \~ \£ i * X >\a

IUO '/\/ '4a 1 ',1 7 V ' "mm ,\1 15

• / ' >\I IS /

,»

* / /

g

s

II

L-0.6 -0J

E(By)

-0.6 -0J

E(Ry)

-0.6 -0.3

E(Ry)

Figure 5.5: Density ofstates (DOS) ofPto.55Rio.45 obtained in the single-site

coherent potential approximation as the concentration-weighted average of

the local DOS on the Pt (dash-dotted line) and Rh (dotted line) atoms (a).

Correction to the DOS due to correlated scattering from pairs of nearest

neighbor atoms (b). Correction to the DOS due to short range order (cal¬

culated SRO parameters for nearest neighbors) (c). The Fermi level is given

by the thin vertical line.

5.2 Discussion of bulk alloys results

We have described above how to incorporate the dominant effects of short range order

into the electronic structure of a random binary alloy. From our five illustrative examples

5.2.Discussion of bulk alloys results 56

we deduce that SRO only weakly affects the DOS, confirming the reliability of the SCPA,

provided lattice relaxation effects are taken into account for systems with components of

very different atomic radii. Terms involving clusters of larger size than the pairs considered

here are of higher order in the off-diagonal matrix elements of the Green's function and

involve higher powers of the SRO parameters, and will therefore contribute even less to

the DOS.

Ideally, one would like to determine both the degree of SRO and its influence on the

electronic structure in one self-consistent cycle. We have shown that within the GEM,

an explicit expression can be obtained to lowest order in 7e for the SRO parameter ai,

which involves only the band contribution of the free energy of the homogeneously random

alloy (see sec.4.2). If we apply eq.(4.39), for Ago.s7Alo.13, we obtain ai=-0.054 compared

to the experimental value ai =-0.079 [46], for Ag0.5oPd0.5o «i =-0.054, for Cuo.715Pdo.2s5

c*i=-0.161 compared to the experimental value ai=-0.157 [59], for Cuo.75Auo.25 «i =-0.454

compared to the experimental value ai=-0.134 [62] and for Pto.55Rho.45 ai=-0.164. The

SRO parameters for Cu based alloys are calculated for the relaxed geometry. The compar¬

ison between calculated and measured values (where available) of SRO parameters shows

that while for Ago.s7Alo.13 and Cuo.715Pdo.2s5, the sign and the order of magnitude of c*i

are well reproduced, for Cuo.75Auo.25 only the sign is well reproduced. The calculated

value of the SRO parameter qx in this case even lies above the maximum possible value

of — I expected for the ordered LI2 phase. The temperature, t = 703-K", at which a^

for Cuo.75Auo.25 was calculated, is very close to the transition temperature, To = 663if,

where the so called long period structures (CU3AU II structures) [47, 75] are expected to

develop. Due to their presence, other clusters than pairs of sites could be important for

this system [76]. Indeed our calculations show that, except for the nearest neighbor pair

interaction, the triple and pair interactions are of the same order of magnitude.

To draw some conclusions on these facts we refer back to eq.(4.39) which was derived

by considering only clusters of pairs in the expansion of the electronic part of the grand

potential and the entropy, eq.(4.2). The calculated SRO parameter ai that results from

this expansion is, therefore, proportional to the nearest-neighbor pair interaction Vi cal-

5.2.Discussion of bulk alloys results 57

culated within the GPM, which is based on the expansion of the band-energy contribution

only [42] in terms of the finite local concentration fluctuations. In the present approach,

which imposes local charge neutrality on average for each component of the alloy through

an appropriate choice of atomic sphere radii, the band-energy term is, in fact, the only

one which contributes to the total electronic energy of the system. Therefore, we conclude

that for Cuo.715Pdo.285 and Ago.50Alo.50 alloys the expansion of the electronic part of the

grand potential to the clusters of pairs, eq.(4.25), is valid.

The discrepancy between the calculated and the measured value of SRO parameter

for the Cuo.75Auo.25 shows, however, that this is not always the case. Instead of taking

clusters of three and more sites in the expansion eq.(4.2), for this alloy, which presents

the long period structures, it would be more appropriate to use the the concentration

wave approximation [6, 7], which allows the study of short range order effects by using

the theory of the linear response to infinitesimal fluctuations [77]. The GEM on the other

hand is similar to the GPM, which treats the nonlinear concentration fluctuations and

the short range order rigorously [77]. Therefore, systems like Cuo.75Auo.25 are not suited

for a GEM treatment to lowest order in 7.

5.3. Results for alloy surfaces 58

5.3 Results for alloy surfaces

In this section the results of our calculations for the (100) surface of Pto.55Rho.45 are

presented. They were obtained with the help of the fully relativistic version of TB-

LMTO-CPA code for surfaces, developed by Kudrnovsky and coworkers [12, 37] with our

implementation of the lowest order GEM corrections to the SCPA layer resolved DOS.

The GEM corrections have been calculated by making use of eq.(4.18) and by replacing in

it the corresponding expressions for diagonal and off-diagonal scattering matrix elements.

The corrections to the CPA DOS are of course on-site quantities, however, they can be

separated into contributions from pairs of nearest-neighbor sites that are located to the

same plane, on-plane corrections, or from those located to the nearest-neighbors planes,

off-plane corrections. This separation is allowed by the structure of the diagonal and

off-diagonal scattering matrix, eq.(4.11) - eq.(4.16), and the one of eq.(4.18).

It is now generally accepted that under true equilibrium conditions Pt always enriches

the surface layer of the PtRh alloy surface. The Pto.55Rho.45 alloy surface has been very

well studied experimentally by Florencio et al [78]. They have reported a strong segre¬

gation of Pt atoms to the surface layer and an oscillatory concentration profile. In the

following, the results of the electronic structure calculations performed for the homoge¬

neous semi-infinite alloy, in which all planes parallel to the surface have the same (bulk)

composition and for the one with the equilibrium concentration profile are reported.

5.3.1 The homogeneous alloy surface

In this section we consider the case of the homogeneous alloy surface in which the con¬

centrations of atoms on the surface's planes are the same as those of the semi-infinite

bulk. The system is modelled by a semi-infinite bulk, an intermediate region made up of

four layers of atomic spheres and one layer of vacuum spheres (the actual "surface" where

the electron density is determined selfconsistently), and a semi-infinite vacuum consisting

of empty spheres with a fiat potential. The layer resolved DOS, calculated within the

CPA, is presented in Fig.5.6. It should be noticed the overall narrowing of the DOS in

5.3.Results for alloy surfaces 59

0>) / \/\/l /

£

IN 11

t1

DOS

5 ^

28

(•)/ V\ i \

I A/ M/iIN /« 1j /

slal /

DOS /5 J I

E(Ry)

Figure 5.6: Layer resolved CPA DOS for each of four surface atomic planes,

or PL, for the homogeneous concentration profile (a) - (d). DOS for the bulk

alloy, (e). The Fermi level is given by the thin vertical line.

the first surface layer compared to the bulk due to reduction of nearest neighbors number

from 12 to 8 at the surface. The lowest order on-plane GEM corrections to the SCPA

DOS, coming from the correlated scattering by pairs, for each of the four atomic planes

in the intermediate region, are presented, Fig.5.7. Except for the first plane of atoms, the


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to.1tti

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-0.6 -0.3 0

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V-0.03

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|0.« i•

i

1i

* ji

*

{

M i*%'I sy Mw \r 1/ '

i

S

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V-0.02

Figure 5.7: On-plane GEM corrections to the SCPA DOS due to correlated

scattering by pairs of sites for each of four surface atomic planes, or PL, for

the homogeneous concentration profile (a) - (d). GEM corrections to the

DOS for the bulk alloy (full line), and for the deepest surface plane (dashed

line), (e). The Fermi level is given by the thin vertical line.

corrections are found to be very small. They become more similar with the bulk result for

the deepest surface plane, Fig.5.7e. The corresponding result for the bulk alloy, presented


by the full line, is the same as the one presented in Fig.5.5b.

The off-plane GEM corrections due to correlated scattering by pairs are presented

in Fig.5.8. Again, they are found to be very small, however, it is to be noted that

contributions coming by pairs of type (p,p + 1) and (p,p — 1), become very similar for

planes that are very near to the homogeneous semi-infinite bulk alloy, showing that the

influence of the surface becomes weaker on those planes, as expected.

The GEM corrections to the layer resolved SCPA DOS due to the presence of the

short range order are found to be of the same order of magnitude as the corresponding

corrections for the bulk alloy. The SRO parameters for on-plane and off-plane nearest

neighbors are calculated according to eq.(4.39). The on-plane and off-plane nearest neigh¬

bor pair interaction V\ is calculated within the generalized perturbation method according

to eq.(4.40) [42, 41]. The calculated on-plane SRO parameter, at T = 800K, for the deep¬

est surface plane is found to be a|4 = -0.195, 3% higher compared to the corresponding

bulk value. The differences of the same order are found also between other on-plane and

off-plane SRO parameters and the corresponding bulk value. The negative sign indicates

again an ordering tendency, while its value explains the slightly difference in the ampli¬

tudes of peaks for both curves in Fig.5.9e. SRO produces, on each plane Fig.5.9(a - d), a

correction in the percent range.


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i

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1 f^-vs

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v

,-0.02

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10.5 -0.3

E(Ry)

Figure 5.8: Off-plane GEM corrections to the SCPA DOS due to correlated

scattering by pairs of sites for the homogeneous concentration profile. Con¬

tributions coming from pairs of type (p,p+ 1), (a): p = 1 ( firstplane of

atoms); (c):p = 2; (e):p = 3. The ones coming from pairs of type (p,p— 1),

(b):p = 2; (d):p — 3; (f):p = 4. The Fermi level is given by the thin vertical

line.


E

I A A/ \.

lw

<b) IIfME

1

1

II 0

s

-0.2

-0.6 -03

EfRy)

0

0.2 M (\£

| ,

A ^ \3 V \ a/i \ \r« V 11 vXo

-0.2 1-0.6 -04

E(Ry)

-0.6 -0.3

E(Ry)

Figure 5.9: On-plane GEM corrections to the SCPA DOS due to short range

order (calculated on-plane SRO parameters for nearest neighbors) for each

of four surface atomic planes, or PL, for the homogeneous concentration

profile (a) - (d). GEM corrections to the DOS for the bulk alloy (full line),

and for the deepest surface plane (dashed line), (e). The Fermi level is given

by the thin vertical line.


EfRy)

£"" ft

statas/atom>

A,

<• \3

w

v A \ \ /ftw

s M V

-0.2 IE(Ry)

Figure 5.10: Off-plane GEM corrections to the SCPA DOS due to corre¬

lated scattering by pairs of sites for the homogeneous concentration profile

(calculated off-plane SRO parameters for nearest neighbors). Contributions

coming from pairs of type (p,p + 1), (a); (c); (e). The ones coming from

pairs of type (p,p — 1), (b); (d); (f). The Fermi level is given by the thin

vertical line.


5.3.2 The inhomogeneous alloy surface

The inhomogeneous alloy surface is modelled similarly to the homogeneous one. In the

intermediate region, however, each of the four layers of atomic spheres has a different

concentration, which differs from the bulk concentration as well. The chosen concentration

profile is very close to the equilibrium concentration profile reported by Florencio et al

[78], which is an oscillatory profile with a Pt-enriched first layer. The density of states

for each layer in the intermediate region is presented in Fig.5.11. The lowest order on-

plane GEM corrections to the SCPA DOS, coming from the correlated scattering by pairs,

for each of the four atomic planes in the intermediate region, are presented in Fig.5.12.

Although the amplitude of the oscillations is higher than the corresponding bulk one, the

corrections are found again to be very small. The differences in the amplitudes of different

planes come from the different values of concentration in each plane. The correction for

the deepest plane in the intermediate region approaches its bulk counterpart. However,

it should be mentioned that the concentarion in this plane is still different from the one

in the bulk.

The GEM corrections to the layer resolved SCPA DOS due to the presence of the

short range order are found to be of the same order of magnitude as the corresponding

corrections for the bulk alloy, i.e., the SRO produces, on each plane, a correction in the

percent range. The on-plane GEM corrections to the DOS are presented in Fig.5.13(a -

d) and compared with the bulk result presented in Fig.5.13(e).

The SRO parameters for on-planes nearest-neighbors are calculated according to eq.(4.39),

while the pair interaction Vi is calculated within the GPM.


25 () \\j \\

?ss

:«*

s

8

5

-0.6 -0.3

«Ry)

25 (c) 1

i.g

5 I

25 0>) A /

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IrST

S

5 \ /

-0.6 -0.3

E(Ry) E(Ry)

(•> y l 1 |?n«I

i

%

i8

5 I-0.6 -0.3

E(Ry)

Figure 5.11: Layer resolved CPA DOS for each offour surface atomic planes,

or PL, for the inhomogeneous concentration profile (a) - (d). DOS for the

bulk alloy, (e). The Fermi level is given by the thin vertical line.


-o.e -0.3

E<ny)

0

0.04

iI 0.02

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7

io a/I/V-

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CO

"Ha -0.02 1

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f -0.02

(b)

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1

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E(By)

Figure 5.12: On-plane GEM corrections to the SCPA DOS due to correlated

scattering by pairs of sites for each of four surface atomic planes, or PL, for

the inhomogeneous concentration profile (a) - (d). GEM corrections to the

DOS for the bulk alloy, (e). The Fermi level is given by the thin vertical

line.


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1 V /tat ~\ / 1/"-~ V /S /* Jo 1B-o.1

V-0.8 -0.3

E(Ry)

1\.

VAJ wn

0.4

of

o 0.2

!IS

0

J U /N^s °

s»

so

-0.2

~

v (jlr-0.6 -0.3

E(Hy)

-0.6 -0.3

E(Ry)

Figure 5.13: On-plane GEM corrections to the SCPA DOS due to short range

order (calculated on-plane SRO parameters for nearest neighbors) for each

of four surface atomic planes, or PL, for the inhomogeneous concentration

profile (a) - (d). GEM corrections to the DOS for the bulk alloy, (e). The

Fermi level is given by the thin vertical line.

Bibliography

[1] P.C.Hohenberg, W.Kohn, Phys. Rev. 136, B864, (1964)

[2] W.Kohn, L.J.Sham, Phys. Rev. 140, A1133, (1965)

[3] P. Soven, Phys. Rev. 156, 809 (1967)

[4] B. Velicky, S. Kirkpatrick, and H. Ehrenreich, Phys. Rev 175, 747 (1968).

[5] J. M. Cowley, J. Appl. Phys.21, 24 (1950).

[6] J. B. Staunton, D. D. Johnson, and F. J. Pinski, Phys. Rev. B 50, 1450 (1994).

[7] J. B. Staunton, D. D. Johnson, and F. J. Pinski, Phys. Rev. B 50, 1473 (1994).

[8] A. G. Khachaturyan, Theory of Structural Transformations in Solids (Wiley, New

York), (1983)

[9] R. Mills, Phys. Rev. B 8, 3650 (1973).

[10] I. V. Masanskii and V. I. Tokar, Theor. and Math. Phys. 76, 118 (1988).

[11] V. I. Tokar Phys. Lett. 110, 453 (1985)

[12] J. Kudrnovsky and V. Drchal, Phys. Rev. B 41, 7515 (1990).

[13] V. Drchal, J. Kudrnovsky, and P. Weinberger, Phys. Rev. B 50, 7903 (1994).

[14] J. S. Faulkner and G. M. Stocks, Phys. Rev. B 21, 3222 (1980).

[15] P. Weinberger, in Electron Scattering Theory for Ordered and disordered Matter

69

BIBLIOGRAPHY 70

[16] A. Messiah, Quantum Mechanics, North-Holland, Amsterdam. (1969)

[17] J. S. Faulkner Prog.in Mat. Sci. 27 1 (1982)

[18] B. L. Gyorffy, Phys. Rev. B 5 2382 (1972)

[19] A. Gonis, X.-G. Zhang, and D. M. Nicholson Phys. Rev. B 38, 3564 (1989)

[20] R. Monnier Phil. Mag. B 75 67 (1997)

[21] H. Ehrenreich, and L. M. Schwartz, Solid State Physics, 31 150 (1976)

[22] N. D. Mermin, Phys. Rev 137, A1441 (1965)

[23] U. Gupta and A. K. Rajagopal, Phys. Rev. A 20, 2972 (1980)

[24] F. Perrot and M. W. C. Dharma-Wardana, Phys. Rev. A 30, 2972 (1984)

[25] D. G. Kanhere, P. V. Panat, A. K. Rajagopal, and J. Callaway Phys. Rev. A 33, 490

(1986)

[26] R. M. Dreizler and E. K. U. Gross Density Functional Theory (Berlin: Springer)

[27] D. D. Johnson, D. M. Nicholson, F. J. Pinski, B. L. Gyorffy, and G. M. Stocks, Phys.

Rev. B 41, 9701 (1990).

[28] B. L. Gyorffy, and G. M. Stocks, J. Phys. (Paris) 35, C4-75 (1974).

[29] P. Lloyd and P. R. Best, J. Phys. C 8, 3752 (1975)

[30] V. Drchal, J. Kudrnovsky, A.Pasturel, I. Turek, and P.Weinberger, Phys. Rev. B 54,

2028 (1996)

[31] H.L. Skriver and N.M. Rosengaard, Phys. Rev. B 43, 9538 (1991)

[32] O. K. Andersen, Phys. Rev. Lett. 27, 1211 (197273).

[33] O. K. Andersen and O. Jepsen, M. Sob, in Electronic Structure and its Applications,

edited by M. Yussouff (Springer, Berlin, 1987)

BIBLIOGRAPHY 71

[34] O. K. Andersen and O. Jepsen, G. Krier, Lectures in Methods of Electronic Structure

Calculations (Singapore: World Scientific)

[35] O. K. Andersen and 0. Jepsen, Phys. Rev. Lett. 53, 2571 (1984).

[361 F. Garci'a-Moliner and R. Velasco, Prog. Surf. Sci. 64, 96 (1986)

[37] J. Kudrnovsky, I. Turek, V. Drchal, and M. Sob Stability of Materials, Edited by A.

Gonis et al, Plenum Press, New York, (1996)

[38] B. Wenzien, J. Kudrnovsky, V. Drchal, and M. Sob J. Phys. Condens. Matter 1,

9893 (1989)

[39] L. Szunyogh, B. Ujfalussy, P. Weinberger, and J. Kollar Phys. Rev. B 49, 2721 (1994)

[40] H. Wright, P. Weightman, P. T. Andrews, W. Folkerts, C. F. Flipse, G. A. Sawatzky,

D. Norman, and H. Padmore, Phys. Rev. B 35, 519 (1987).

[41] V. Drchal et al, Phys. Rev. B 45, 14328 (1992).

[42] F. Ducastelle and F. Gautier, J. Phys. F 6, 2039 (1976)

[43] I. Z. Fisher, Statistical Theory of Liquid, (Chicago: University Press) (1964)

[44] J. P. Perdew and A. Zunger Phys. Rev. 23, 5048 (1981)

[45] D. M. Ceperly and B. J. Alder Phys. Rev. Lett. 45, 566 (1980)

[46] S. Y. Yu, B. Schonfeld, and G. Kostorz, Phys. Rev. B 56, 6626 (1997).

[4~] W. B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys,

(Pergamon Press, Oxford 1964)

[48] L. U. Hsuen-Shan, S. S. Lu, and L. Ching-Kwei Chi. J. Phys. 22, 505 (1966)

[49] E. Raub and G. Falkenburg Z. Metallkde, 55, 392 (1964)

BIBLIOGRAPHY 72

[50;

[51

[53

[54

[55;

[56

[57

[58

[59

[60;

[61

[62

[63

[64

[65

[66

Landolt-Bornstein, New Series, edited by K.-H. Hellwege (Springer-Verlag, Berlin

1971) 6; 11 (1979)

See Binary Alloy Phase Diagrams, edited by T. B. Massalski (1990)

Z. W. Lu, S. H. Wei, and A. Zunger Phys. Rev. B 44, 10470 (1990)

T. Saha, I. Dasgupta, and A. Mookerjee, J. Phys.: Condens. Matter 6, L245 (1994).

T. Saha, I. Dasgupta, and A. Mookerjee, Phys. Rev. B 50, 13267 (1994).

C. Wolverton and A. Zunger Phys. Rev. Lett. 75, 3162 (1995)

Binary Alloy Phase Diagrams, edited by J. L. Murray, L. H. Bennet, and H. Baker,

(American Society of Metals, Metals Park, Ohio, 1986), 1, pp. 5 and 55.

A. Bieber and F. Gautier, Acta. Metall. 34, 2291 (1986)

Z. W. Lu, D. B. Laks, S.-H. Wei, and Alex Zunger, Phys. Rev. B 50, 6642 (1994).

Dilip Kumar Saha, Kenji Koga and Ken-ichi Oshima, J. Phys. Condens. Matter 4,

1093 (1992)

B. Ginatempo et al Phys. Rev. 42, 2761 (1990)

N. Takano, S. Imai, and M. Fukuchi, J. Phys. Soc. Japan 60, 4218 (1991)

B. D. Butler and J. B. Cohen J. Appl. Phys. 65, 2214 (1989)

M. Cyrot and F. Cyrot-Lackmann J. Phys. F 6, 2257 (1976)

M. Sluiter, P. Turchi and D de Fontaine, J. Phys. F 17, 2163 (1987)

E. Raub, J. Less-Common Metals 1, 3 (1959)

B. M. Klein, Z. Wu, and A. Zunger, Stability of materials Edited by A. Gonis et al,

(plenum Press, Ney York, 1996)

BIBLIOGRAPHY 73

[67] V. I. Tokar and R. Monnier (unpublished)

[68] A. Mookerjee, J. Phys. C 6, 1340 (1973).

[69] A. Mookerjee, J. Phys. F 17, 1511 (1987).

[70] Yang Wang et al., Phys. Rev. Lett. 75, 2867 (1995)

[71] I. A. Abrikosov et al., Phys. Rev. Lett. 76, 4203 (1996)

[72] F. Gautier, F. Ducastelle and J. Giner, Phil. Mag. 31, 1373 (1975)

[73] Z. W. Lu and Alex Zunger, Phys. Rev. B 50, 6626 (1994).

[74] C. Wolverton and Alex Zunger, Comput. Mater. Sci. 8, 107 (1997).

[75] H.Sato and R.S.Toth, Phys. Rev 127, 469 (1962).

[76] P.Weinberger, V.Drchal, L.Szunyogh, J.Fritscher, and B.I.Bennett, Phys. Rev. B 49,

13366 (1993).

[77] A.Gonis, X.G.Zhang, A.Freeman, P.Turchi, G.M.Stocks, and D.M.Nicholson Phys.

Rev. B 36, 4630 (1987).

[78] J.Florencio, D.M.Ren, and T.T.Tsong, Surf. Sci. 345, L29 (1996)

[79] D.D.Johnson and F.J.Pinski, Phys. Rev. B 48, 11553 (1993).

[80] J. Kudrnovsky, S. K. Bose, and O. K. Andersen, Phys. Rev. B 43, 4613 (1991).

[81] J. Kudrnovsky, I. Turek, V. Drchal, P. Weinberger, S.K. Bose, and A. Pasturel, Phys.

Rev. B 47, 16525 (1993)

[82] V. Drchal, J. Kudrnovsky, and P. Weinberger, Phys. Rev. B 50, 7903 (1994)

[83] J. Kudrnovsky, P. Weinberger, and V. Drchal, Phys. Rev. B 44, 6410 (1991)

[84] LA. Abrikosov and H.L. Skriver, Phys. Rev. B 47, 16532 (1993)

BIBLIOGRAPHY 74

[85] LA. Abrikosov, A.V. Ruban, H.L. Skriver, and B. Johansson, Phys. Rev. B 50, 2039

(1994)

[86] A.V. Ruban, LA. Abrikosov, D.Y. Kats, D. Gorelikov, K.W. Jacobsen, and H.L.

Skriver, Phys. Rev. B 49, 11383 (1994)

[87] Z. W. Lu, S.-H. Wei, and Alex Zunger, Phys. Rev. B 45, 10314 (1992).

CURRICULUM VITAE

Surname: BORICI

First name: Mirela

Nationality: Albanian

Birth date: 11 September 1965

Birth place: Tirana, Albania

Marital status: Married

1971-1979: Primary school, "Skender Cagi" 8-year school, Tirana.

1979-1983: Middle school, "Partizani" Gymnasium, Tirana

1983-1988: Diploma in Physics, University of Tirana: 5-year Physics Branch.

Diploma Work: "An algorithm for determination of

point symmetry group of some materials".

Supervisor: PD. Dr. Bardhyl Guda

1988-1992: Research and Teaching Assistant, Materials Physics Chair,

University of Tirana

1992-1993: Research Assistant, Laboratory of Powder Technology, EPF

Lausanne. Supervisor: Dr. Paul Bowen

1993-1994: Auditor student, Swiss Federal Institute of Technology:

ETH Zurich.

1995-1998: Ph.D. work in ETH Zurich,

Influence of Short Range Order on the Electronic

Structure of Alloys and their Surfaces".

Supervisors: Prof. Dr. Danilo Pescia and PD Dr. Rene Monnier.

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