Research Collection
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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ETH Library
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)
1.1.The Coherent-Potential Approximation 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)
1.1.The Coherent-Potential Approximation 5
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)
1.1.The Coherent-Potential Approximation 7
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)
1.1.The Coherent-Potential Approximation 8
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
1.1.The Coherent-Potential Approximation 9
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
1.2.Elements of Density Functional Theory 11
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
1.2.Elements of Density Functional Theory 12
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
1.2.Elements of Density Functional Theory 13
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^
1.2.Elements of Density Functional Theory 14
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
2.2.TB-LMTO Surface Green's function 21
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)
2.2.TB-LMTO Surface Green's function 22
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^,i^o,
(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'fcM01^oM >
= M10[(MvV)-1]00M01^i^i, (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)
2.2.TB-LMTO Surface Green's function 24
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
4.1.GEM corrections to the CPA DOS for bulk alloys and their surfaces 38
(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
4.1.GEM corrections to the CPA DOS for bulk alloys and their surfaces 39
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-cJalpAtpg^Atp,
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+Jalp^Atpg^Atp+xg^p
+ (1 - Cp-Jal^Atpg^Atp^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)Atpg^Atpg^Atpg^Atp,
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.
4.2.Short Range Order 41
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)
4.2.Short Range Order 42
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
4.2.Short Range Order 43
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
4.2.Short Range Order 44
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.
5.1.Results for bulk alloys 47
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
5.1.Results for bulk alloys 48
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
5.1.Results for bulk alloys 49
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-
site coherent potential approximation as the concentration-weighted average
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
5.1.Results for bulk alloys 50
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-
site coherent potential approximation as the concentration-weighted average
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.
5.1.Results for bulk alloys 52
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-
site coherent potential approximation as the concentration-weighted average
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.Results for bulk alloys 54
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
5.3.Results for alloy surfaces 60
I 0.1(«) l
i /3
^ /W* A/^V Ia7
s
to.1tti
o
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-0.6 -0.3 0
E(Ry)
.,DC 0.03
om
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«o
-0.03
£ 0.04
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(b)
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-0.6 -0.3
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V-0.03
1-o.e -o.3 -0.« -0.3 0
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i 1
|0.« i•
i
1i
* ji
*
{
M i*%'I sy Mw \r 1/ '
i
S
\\t,"*
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
5.3.Results for alloy surfaces 61
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.
5.3.Results for alloy surfaces 62
£
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1(c) 1
a/1a.1I»
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£• I
i
1•
*0 A -n 1
1 f^-vs
°
I Vooa
v
,-0.02
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10.5 -0.3
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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.
5.3.Results for alloy surfaces 63
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.
5.3.Results for alloy surfaces 64
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.Results for alloy surfaces 65
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.
5.3.Results for alloy surfaces 66
25 () \\j \\
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s
8
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«Ry)
25 (c) 1
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5 I
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5 \ /
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E(Ry) E(Ry)
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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.
5.3.Results for alloy surfaces 67
-o.e -0.3
E<ny)
0
0.04
iI 0.02
.
7
io a/I/V-
8 "
CO
"Ha -0.02 1
I
f -0.02
(b)
i Ar—v
,
" "
V
1
o.e -o.3
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.
5.3.Results for alloy surfaces 68
0.1
-u
£ \E \ r>« N \\ /*
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.
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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.