Jul 15, 1974 - of.charge transfer in alloys to electronegativity, to relative pure-constituent. Fermi energies, and to atomicvolume is discussed. The importance.
VOLUME
PHYSICAL REVIEW B
Charge transfer
10,
NUMBER
15 JULY 1974
2
in alloys: AgAn~
C. D. Gelatt, Jr. and H. Ehrenreich Division
of Engineering and
Physics, Harvard University, Cambridge, (Received 30 November 1973)
Applied
Massachusetts
02138
The relationship of. charge transfer in alloys to electronegativity, to relative pure-constituent Fermi energies, and to atomic volume is discussed. The importance of using comparable potentials for alloy calculations calculations is emphasized. The results are illustrated by coherent-potential-approximation for AgAu alloys using an s-d model Hamiltonian, Charge transfer as a function of concentration and the assumed pure-metal Fermimergy difference has been calculated self-consistently including the shift of the d-band energies resulting from the modified electron-electron Coulomb interaction. Renormalized-atom estimates are used for the Coulomb integrals which relate the shift in one-electron energies to the charge transfer. The result is in reasonable agreement with the charge transfer inferred core-level-shift measurements. Using the proper from isomer-shift and x-ray-photoelectron-spectroscopy charge transfer, the calculated optical-absorption edge agrees with e, experiments, and the self-consistently determined Fermi energy has the same qualitative concentration dependence as measured work-function
differences
for the alloy system.
I. INTRODUCTION
Calculations for a real, disordered, substitutional alloy system provide a quantitative test not only of the adequacy of the solution of the alloy problem, but also the potentials or parameters chosen to represent the constituents. This second ingredient has received less attention than the first. The construction of an effective one-electron potential is more difficult for an alloy than for a pure metal because results calculated from the potential are sensitive to the difference in potential between the constituents. The different techniques currently in use can lead to potentials for the same element mhich differ by several electron volts. This problem has not received adequate attention in previous calculations of alloy proper-
ties. A nonzero difference potential, or a difference in electron-per-atom ratio for the constituents, will lead to a transfer of charge in the alloy. This obviously is related to the concept of eleetronegativity. As mill be seen, the appropriate measure of electronegativity' for a metal is the negative of the Fermi energy, 2 and this definition corresponds to the Mullikens definition of electronegativity for an atom. The "bulk Fermi energy" is not experimenOne might think of choosing the tally measurable. difference in Fermi energy equal to the negative of the difference in work function. For AgAu alloys, which are of principal concern in this paper, this gives a self-consistently calculated charge transfer in good agreement with experiment; for metals with greatly different surface-dipole contributions to the mork function this choice mould clearly not be appropriate. Mott's argument that the Madelung contribution may be a significant part of the ordering energy of
10
an ordered alloy provides an estimate of the scale of charge transfer to be expected in an alloy; for P-brass the ordering energy corresponds to a net
transfer of 0. 075 electrons per atom.
This small
magnitude points out the importance of constructing the potential with a technique, such as the renormalized atom, in which the amount of charge transfer can be explicitly fixed rather than by overlapping atoms, where it is uncontrolled on the
scale of interest.
'
Given the alloy potential the properties of the alloy can be calculated either from an extension of the methods used to calculate the band structure of a pure metal, or from a model Hamiltonian with parameters determined from the potential. The band-theoretic techniques such as the muffin-tin coherent-potential approximation (CPA)' or the average-t-matrix approximation (ATA) allow one to calculate momentum-dependent properties such as the Fermi-surface topology or complex energy bands, but are difficult to implement numerically. The calculation of the alloy density of states mithin the muffin-tin framework has not yet been imple-
mented. A model Hamiltonian solved by the CPA, while it does not provide information on k-dependent properties, does allow efficient calculation of the density of states and its decomposition into ba, nd and constituent contributions. ' For the illustrative calculations reported here a two-band model Hamiltonian will be used to calculate properties of AgAu alloys. This alloy system is ideal for studying charge transfer: The atomic volumes differ by less the electronegativity difference on the than 1/0, Pauling scale is large for metals, ' and extensive experimental work has been done to measure the charge transfer. Levin and Ehrenreich" (hereafter LE) discussed
"
10
CHARGE TRANSFER IN
the effect of charge transfer on the concentration dependence of the AgAu optical-absorption edge. Their exploratory calculations, which included only the effect of conduction charge transfer, indicated a transfer of 0. 3 electrons per atom from a Au impurity in Ag. This result conflicted both with isomer-shift measurements, '3 which indicated the opposite sign, and with the lack of evidence of ordered pha, ses of AgAu, which would be expected from so large a charge transfer. The situation was clarified by the work of Watson, Hudis, and Perlman, ' who combined their measurements of core-level shifts with the isomer-shift results to demonstrate that there is significant d-band charge transfe r c ompe nsating the conduction-band charge transfer. The present calculations indicate that this picture of charge transfer is also consistent with the concentration dependence of the opticalabsorption edge. The calculations reported here are superior to those reported by LE in the following ways: (i) The model Hamiltonia, n and its CPA solution have been generalized to treat the effects of different site-diagonal conduction-band and hybridization parameters for the constituents. As a result it is possible to handle alloys in which the constituents have different d-band widths due to hybridization alone. (ii) The effects of d band as well as conduction-band charge transfer are included. (iii) The Coulomb integrals which relate the shift of energy levels to the charge transfer have been refined (LE's integrals contained a numerical error). (iv) The relative positions of the Ag and Au energy levels have been fixed by use of the contact-potential difference, which we regard as more reliable, although not necessarily correct, than positioning by use of currently available Ag and Au potentials. (v) The charge transfer has been calculated self-
eonsistently. No extensive effort has been made to optimize the alloy results by varying the pure-metal parameters in the model Hamiltonian. Rather, they were determined from band-structure calculations and photoemission results for the pure metals. Two sets of parameters are included: In the first model the d-band width arises solely from s-d as in LE, resulting in a resonancehybridization, like density of states; in the second model the d bands are given a nonzero intrinsic width, giving a qualitatively more realistic density of states. The results of the calculations compare well with experiments determining the isomer and core-level shifts, photoemission measurements for different alloy compositions, and the concentration dependence of both the optical-absorption edge and the work-function difference. Section II introduces the two-band model Hamiltonian and presents a brief discussion of how it is
A
LLOYS: AgAu
solved within the CPA to give the averaged and component densities of states for the alloy. Section III presents a discussion of charge transfer, the parameters which determine its sign and magni. tude, and the technique of calculating it self-consistently. Also included is a comparison of the various techniques of calculating the relevant Coulomb integrals within the framework of the renormalized atom. Section IV presents the choice of parameters used to describe pure Ag and Au, and the results of the alloy ealeulations and their comparison with experiment. II.
TYCHO-BAND
MODEL HAMILTONIAN 2X 2 CPA
AND
The effects of charge transfer on the electronic density of states of a substitutional disordered metal alloy, such as AgAu, are examined most readily by use of an appropriately chosen model Hamiltonian and the CPA. The Hamiltonian must be sufficiently simple as to allow efficient calculation of the alloy density of states and the density of states associated with each component, but yet realistic enough that it contains some of the essential properties of conduction and d bands. It would be computationally possible to choose a muffin-tinpotential Hamiltonian and treat the configuration average in the ATA, or in the CPA, but the computational complexity is prohibitive. Charge transfer is sensitive to the density of states in the neighborhood of the Fermi energy. In noble metals the density of states in that region is quite featureless and a, rises from a nearly-free-electron conduction band and the hybridization tails of the d bands which lie several eV below the Fermi level. Thus, for noble (in contrast to transition) metals, charge transfer is insensitive to the detailed structure of the d-band density of states. LE, ' in their paper on AgAu alloys, introduced a two-band (s and d) tight-binding model Hamiltonian which includes hybridization between the bands and is easily treated within the CPA. The Hamiltonian used for the present calculations is a generalization of that used by LE and includes nonzero dband hopping matrix elements.
A 2 &&2 matrix generalization of the CPA is used to perform the configuration average. The scattering due to different s-band energies is included. Furthermore, it is possible to include different hybridization widths for the two d bands. By contrast LE applied the virtual-crystal approximation to the s-s and s-d parts of the Hamiltonian, and used the CPA to treat only the d-d block. The Hamiltonian for an A„B, alloy is
1'f,
»„=
g e'(x) ns) (ns p f '„„. ns) I
I
+
~
&n's ~
D. GEI ATT,
400
JR.
n'$n
Q(r. (x)l«&&ssl+H
')
10
Z's are the components of the 2&2 matrix self-
+Q»„'(x)~«)&«~+Q f„'.~«&&n'd~ +
AND H. EHRENREICH
(2
~
1)
Here ins&, l«& are Wannier kets associated respectively with s and d electrons located on site n. The subscript n in E„and y„denotes that the parameter in question depends on whether the site n is occupied by an A or a B atom. The composition dependence of the parameters is the result of charge transfer and will be discussed in Sec. III. The &'s correspond to the center of gravity of the unhybridized bands in cubic lattices, but not the atomic energy levels as sometimes claimed. y is the hybridization parameter which determines the strength of the s-d mixing. The t„'~" are the hopping matrix elements which give nonzero bandwidths. In the original LE model, the absence of the d-band hopping term (f„„,=0) gave rise to an unphysical hybridization gap. Brouers and Vedyayev'6 pointed out that by giving the unhybridized d band a finite
"
the gap could be closed. For one of the models considered in the present paper t~„, will be assigned values consistent mith their values in the It is assumed that the hopping pure constituents. matrix elements are independent of the configuration of the alloy and thus periodic, and further that the g and g hopping matrix elements are proportional:
energy. The derivation of the CPA self-consistency equation for the matrix seU-energy is in exact correspondence with the single-band calculation. The multiple-scattering expansion for the Green's function can be written in terms of the site-scattering (f) matrices. If one makes a single-site decoupling of the configuration average, one finds the selfconsistency requirement &f„(Z)& =0. Expressing t„ in terms of the self-energy, the two-band generalization of Soven's' form of the CPA equation results:
Z(z)
=-
—[» -Z(z)]Z(z, Z(z)) [»' -Z(z)]. (2. 3)
»
This is a matrix equation in mhich
. ") *
(2. 4)
+(1 —x)»
(2. 5)
yA
f»A ~A
» = x»
width,
is the diagonal matrix element of the configurationaveraged Green's function G in a%annier basis. Here &0 =(&Os [, &Od(]. E can be expressed in terms of the self-energy and the unhybridized p-band densIty of states determined by s(k). In a Bloch basis l
Q
P(z, Z)=&olg~o&=X This assumption assures the k independence of the self-energy at little cost in terms of the physical reasonability of the results. The application of the single-site CPA to this two-band model Hamiltonian is a. straightforward generalization of the calculation for a single-band model. Since the scattering potential in the disordered alloy is site diagonal in a Rannier representation, the self-energy operator defined in terms Green's function is of the configuration-averaged Transforming to a Bloch basis, k independent.
"
(kI ={&k,I, &(z
(k, l],
e.„.„)-'&„=G,
—
z
—s(k)
- Z„(z)
—Z~(z)
z —o.s(k)
—Z „(z) (2. 2)
where the brackets ( average,
s(k)
g
e'lk
~ ~ ~
) denote
a,
configuration
Iis fs
the Fourier transform of the hopping matrix elements, is the g-band dispersion relation, and the
(2. 7)
&klclk&
or, more explicitly,
E(z, Z) =Q, (2x) (z
"B.Z.
-s(k)
X
d k
-Z„-Z„ z-ns(k)-Z„~
7
(2. 8)
where Q, is the unit-cell volume, and the integral extends over the Brillouin zone (B. Z. ). The only k-dependent term of the integrand is s(k). Defining the unhybridized s-band density of states po, (E) = 0 ( s2)z~
—Z, „(z)
(2. 6)
Z(z, Z) =&OiG(z, Z)io&
J
d k5(E —s(k)),
(2. 9)
and its Hilbert transform
.
f
Z, (z) = „p„(E)(z—Z) 'dZ,
(2. 10)
it can be shown by performing a partial-fraction expansion of the denominator of the matrix inverse in Eq. (2. 8), and using Eqs. (2. 9) and (2. 10) to carry out the k integration, that the components of 5' are given by
&-= [n(E
-E)] '[(z-Z -«)&
(E,)
CHARGE TRANSFER IN
10
A
LLOYS: AgAu
—(z —Z~ „— nz2)FO~(zz)],
III. CHARGE TRANSFER
F„=z„[n(z,-z, )]-'[F„(z,)-F„(z,)], F„= [n(Z, —E, )]-' [(z —Z„- Z, )F„(Z,) - (z - Z„-Z, )F„(Z,)],
(2. »)
where
s~
2 12
In the limit n-0, Eqs. (2. 11) reduce to Eqs. (2. 24) and (2. 31) of LE. The solution of the CPA for the two-band model Hamiltonian is obtained by iterating Eq. (2. 3) for the matrix self-energy to self-consistency' with the site-diagonal matrix element of the configuration-averaged Green's function calculated from Eqs. (2. 11). The densities of states associated with d and s electrons are
p~(z, Z)
=
—10(X&) ' Im T r~ G(z+ jo)
=-Io(xx)-'im
2
()tdlclfd&
kt=B, Z ~
—low ' ImF~~(E+ fo& Z), ' p, (E, Z) = —2w ImF„(z+ io, Z),
(2. 13)
=
(2. 14)
where the factor of 10 is included for the degeneracy of the d band and 2 for that of the s band. The component density of states can be defined in terms of the averaged Green's function corresponding to configurations restricted to have an A or 8 atom at the site @=0. This restricted average Green's function can be shown to have the form
G~'s=cp- [lo)(e"'-z)(ol]cj The component
are given p",
s-
(2. iS)
and d-band densities
of states
by
(E) = —Io~ 'Im(n=0, dl G"
"'(z) =-2~ 'Im(n=o,
p,
l
n
=0, d), (2. 16)
s G"'ln=o, s). (2. I '7)
%ith this definition, the consistency equations, xp, ~(z)+(1 —x)p, ~(z) =p, ~(z), are satisfied. The two-band generalization of the single-site CPA shares the virtues of its single-band sibling': It is symmetrical in rand 1 —g, it is exact to order x in the low-density limits (x- 0 and x- 1); and it reduces to the virtual-crystal result in the weakscattering limit. It also provides a simple model for describing alloys of metals with different dband hybridization widths since it is not assumed
"
.
that
y" = y~.
401
%hen dissimilar atoms are brought together to form a solid, the electronic charge distribution about a nucleus will almost always be altered from that in the corresponding pure solid. This change can be detected experimentally, by isomer-shift and core-level-shift measurements for example. The description of charge transfer in transition metals is difficult because (i) the magnitude of the transfer is expected to be small, of order 0. 1 electrons per atom or less; (ii) it is necessary to consider d-band and conduction-band charge transfer separately; and (iii) the lack of well-defined atomic radii in the alloy makes the separation of the charge density into its components uncertain. AgAu alloys are attractive for the study of charge transfer in alloys because they have relatively large electronegativity difference, d bands which do not intersect the Fermi level, and nearly equal%ignerSeitz radii. The-fact that many transition- and noble-metal alloys have substitutional disordered phases indicates that the net charge transferred is quite small, probably less than a tenth of an electron per atom for equiconcentration alloys. As pointed out by Mott, if the charges on the atoms are too large, the Madelung energy gained by forming an ordered solid overcomes the entropic term which favors disorder, and the alloy will order at sufficiently low temperature. If the net charge is smallenough, the ordering temperature might be so low that kinetic effects make it unlikely that an ordered phase would ever be observed. ' A useful picture of the formation of an alloy is that of the renormalized atom. ' The valence wave functions of an atom are cut off at some effective Wigner-Seitz radius r„s and renormalized within that radius to preserve the total electronic charge of the atom. By choosing the charge of the starting atom, it is possible to control the amount of charge transferred, but at the expense of having to pick a value of y~s for the atom in an alloy, a procedure which is not well defined. In fact, the "radius" of an atom in an alloy should depend on the amount of charge which has been transferred. The advantages of using the renormalized-atom approach are several. The Chodorow' potential for Cu, the prototype renormalized-atom potential, is still the standard of comparison for Cu potentials. The renormalized atom allows the use of the Hartree-Fock Wigner-Seitz exchange and correlation potential, which is probably more accurate for tightly bound d shells than an averaged [p(x)]'~3 exchange potential derived from a free-electron gas. It also provides a convenient method of calculating the average d-band energy and the on-site electronelectron Coulomb and exchange integrals. ' It is,
"
C. D. GELATT) JR.
402
however, probably no more accurate than other methods of constructing potentials in determining the magnitude of the difference in potential of two different atoms, and it is from this that the charge transfer results. The case of 4d and 5d elements like AgAu is further complicated because of the greater importance of relativistic effects for the heavier atoms. Previous attempts at determining the correct relative potentials for CPA calculations have either arbitrarily equated the I', states of the alloys constituents, ' or, as in the case of LE, accepted the results of band structures calculated from potentials which were not really comparable. As demonstrated below, the sign and magnitude of the charge transfer depend on the difference in the calculated pure-metal Fermi energy of the constituents. Accordingly, if an error is made in the sign of the relative Fermi energies (as was the case in the calculation of LZ), the sign of the charge transfer will also be wrong. It is possible to calculate the charge density and charge transfer self-consistently in an ordered~~ or substitutional-disordered alloy. Considering an A„B, transition-metal alloy, one could define component d- and conduction-band densities of states p", ' (E, x), p", 's(E, x). They will be composition dependent not only because of the composition dependence of the coherent potential, but also because of the modification of the component potentials produced by charge transfer. The result of such a calculation could be characterized by the five composition-dependent quantities: the Fermi energy values of the amount of charge transand the tr(x), ferred, bn,"'s(x) and dn~~'s(x). They are related through the equations
[p, (E, x)+p, (E, x)]dE= xX" +(1 —x)X', (3 1) d,
n"'(x) = f
b, n", '
(x)=
f
p"'(E p", '
x)
dE-X"'
(E„x)dE —N", '
AND H. in a
EHRENREICH
8 crystal
is given
10
by
B
~H(0)
=
f „"p, (E) dE,
(3. 4)
S~
which has the sign of czB —Hz and whose magnitude is a nondecreasing function of ) P~ —E~ I. For small differences in Fermi energy the charge transfer is of order he+ p, (ez). For a noble or free-electronlike metal, the density of states at the Fermi level is several tenths per atom per eV, and Fermilevel differences are probably of order 1 eV or less, implying a charge transfer = 0. 1 electrons
per atom. The This estimate is obviously oversimplified. potenti. al at an A site will be modified by the charge transfer. If the A site gains electrons, the extra Coulomb repulsion will raise the one-electron energy levels at that site, and, by raising some previously filled levels above the Fermi level, the Coulomb effect opposes charge transfer. Since a typical electron-electron Coulomb repulsion energy U on a given site is 1 Ry, a charge transfer of 0. 1 electrons per atom can cause level shifts of U'bn =1 eV. The renormalized-atom picture' provides a technique for calculating the Coulomb integrals which relate a one-electron level shift to the amount of Consider the shift of the cencharge transferred.
band in an alloy when both sTo lowest order and d-band charge is transferred. the shift in one-electron energy will be linear in the amount of charge transferred,
ter of gravity of the d
~~a =
4~«~+ ~~&&E
(3. 5)
where b E~ is the change in the average one-electron d energy, «„and 4z, are the amounts of charge
transferred,
and
(3. 6)
(3. 2) (3. 3)
in which p, and p~ are the configuration-averaged densities of states in the alloy, @"' =+,"'B++"„'B is the number of valence electrons per atom in the pure metals expressed as a sum of conduction- and d-band contributions, and p, 's and p,"'s are the component densities of states. It is apparent from Eqs. (3. 1)-(3.3) that the difference in Fermi energy of A and 8 has an important influence on the self-consistent charge transfer. This may be understood by considering a simple rigid-band model in which A, and 8 are assumed to have the same density of states but a different number of valence electrons per atom. If level shifts are neglected, charge-transfer-induced the A. -site conduction charge transfer for dilute A
is the effective Coulomb integral. Numerical calculations have shown that for transition and noble metals, &a= ~u
=—
(3. 7)
((alH
provides an accurate estimate of the center of The renormalized-atom gravity of the d bands. wave function for a given configuration, (r)g, ) — = P, (r), is defined in terms of the corresponding self-consistent Hartree-Fock wave function for the free atom of that configuration, P, (r), as N(&f)((r),
r& Rws
0, R~ is
the Wigner-Seitz and N, is chosen to satisfy
Here
(3. Sa) radius (4xRvg3 = V/Ã)
CHARGE TRANSFER IN ALLOYS:
10
(3. 8b) The effective Coulomb integrals defined by Eq. by generating renormalizedatom estimates of &„ for configurations with different numbers of s and d electrons and evaluating the For example, U« partial derivatives numerically. for Ag can be estimated from
(3. 6) can be evaluated
karen(d
10s)
sren(d
(3. 9)
&'„"(d"s ) is given by Eq. (3. 7) using a renormalized-atom wave function defined by Eqs. (3. 8) in terms of the Hartree-Fock wave function for a free A better estimate atom in the configuration d"s of the partial derivative in Eq. (3. 6) might be given by taking the difference between the one-electron energies for two configurations differing by only a fraction of an electron. Wave functions for Ag in were calculated using the configuration d reSlater's average-of-configuration equations, one-elecnormalized, and the renormalized-atom tron energies were generated. Evaluating U« from the one-electron energies for the configurations d' s and d s led to an estimate different from that of Eq. (3. 9) by less than 1/~. Similar calculations were carried out for other Ag Coulomb integrals with the same result. We conclude that the finite difference approximation used in Eq. (3. 9) is an adequate evaluation of the partial derivative. The definition of the Coulomb integrals used here is closely related to the polar-energy terms discussed by Van Vleck24 and Herring. 2' Van Vleek considered the energy U required to transfer a d electron from one Ni atom in a d s configuration For this transto another, without s compensation. fer, d s+d s-d s+d' s, the total energy difference
.
"s
'
s) —E(d's)]- [E(d s) —Z(d's)]
can be related to a difference in one-electron energies by Koopman's theorem,
—E(d' s) —E(d's), e,(d' s) = e,(d's) = Z(d's) E(dn s)-,
ables in the present calculations, the Herring definition is not relevant to Eq. (3. 5). In previous charge-transfer calculations several other approximate techniques have been used to oneevaluate Eq. (3.6). The renormalized-atom electron energy can be written as OCC
&['"=f«9+ g((ij
I
&lij
&
-&ijl Vlji&),
(3. 10)
where h&; is the one-electron kinetic-energy and nuclear potential-energy integral, and the terms in brackets are the two-electron Coulomb and exBecause the largest effect of change integrals. charge transfer is to modify the valence-electron Coulomb integrals, it is useful to separate these and rewrite Eq. (3. 10) for en in the form
en" = (nn —1)Fnn+ n, F,n+ &9,
s) —fn(d s)
This is analogous to the definition of U«given in Eq. (3. 9). Van Vleck used atomic spectra to estimate this energy. Because of the compression of charge in the solid, U is underestimated when atomic spectra are used. Herring's discussion assumed complete compensation of s and d charge transfer, i. e. , d s+d s-d s +d', leading to a U analogous to U«+ U„-2U, ~ in the present notation. Since b, g, and b, yg„are regarded as independent vari-
(3. 11)
where g„and n, are the number of d and s electrons per atom in the assumed configuration, the
f
F, q = J' R((r, )(1/r&)Rq(99)9', dr,
rqdr9
(3. 12)
3
are Slater integrals for the renormalized atom, all of the other terms of Eq. &~ represents (3. 10). In Eq. (3. 12) the R(9.) are renormalizedatom one-electron radial functions. The earliest papers assumed the 5' s and &~ to be unchanged when the configuration of the renormalized atom was varied. ' ' Then Eq. (3. 6) for U«becomes
and
0
Udd
=
1. 37 Ry for Ag(d" s)
.
(3. 13)
An improvement of this method, which has been used in recent calculations, ~~ ~' includes some of the change in the E 's due to wave-function relaxation but still assumes no change in E~. In this case
Eq. (3. 9) is evaluated using Eq. (3. 11) for en" to yield U~ = F„',(d"s
)+(n —2)[F'„(d"s ) —F,'„(d" 's )]
', (d" s") —F,', (d" 's")]
+ m[F,
=0. 98 —
which implies
U~ fn(d
403
9
=1. 10 Ry,'
U= [E(d'
AgAu
Ry for Ag(dms)
.
(3. 14)
The differences between the various techniques for evaluating the effective Coulomb integrals can be explained by the extent to which the approximations include the relaxation of the wave functions when the configuration of the atom is varied. The use of the bare Slater integral, as in Eq. (3. 13), does not include any effect of this relaxation. The use of Eq. (3. 14) to calculate U, , includes the modification of F„., which is the largest contribution to the Coulomb repulsion between orbitals i and Consider the ionization of Ag: d' s-d s. In the final state there is one less d electron. The screening of all the other electrons will be reduced and the
j.
404
C. D. GE LATT, JR.
charge density will be pulled in closer to the nucleus. This will cause an increase in all of the Slater integrals since they are a measure of the inverse of the electron-electron distance. This implies that SF'/Sn& 0, which explains why the U's calculated from Eq. (3. 14) are smaller than the corresponding Slater integral. Equation (3. 9) provides the best estimate of the effective Coulomb integrals because by taking the difference of oneelectron energies directly the effects of relaxation on all contributions to the one-electron energy, not just the valence-valence Coulomb repulsion term, are taken into account. A discussion of charge transfer in metals parallels the consideration of the equivalent problem for molecules and semiconductors, for which the con"3 has provided qualitacept of electronegativity tive insight. In this connection, consider the construction of an AI3 alloy composed of equal-sized atoms. One models' is to cut up pure A and 8 crystals into atomic cells, freeze the charge density, then form the alloy by stacking the A and 8 cells together and allowing the charge to redistribute itself. Which way will the charge flow? If it is possible to define a local Fermi energy for each of the frozen cells, charge will flow from the cells with higher local Fermi energy until the Fermi levels have become equal. The problem is more complicated if the atomic volumes in A and B crystals are not equal. In that case a viewpoint on the change in local Fermi energy as a function of atomic volume is needed. Since the atomic volumes of Ag and Au are so nearly equal, this point is mentioned only as a. warning and will not be discussed further. This viewpoint is quite similar to that introduced ' who established a scale of electroby Pauling, negativities X by considering the binding energies of diatomic molecules. It is assumed that when a molecule is formed, charge will flow from the atom with the smaller electronegativity to the one with the larger. The relationship between electronegativity and local Fermi energy is more apparent when. one considers the Mulliken~ scale, for which the electronegativity is defined as X= -', (I+A), where I is the atomic ionization potential and A the electron affinity. Let E(n) be the total energy of the atom with n electrons; then I=E(n —1) —E(n), and A=E(n) -E(n+1). The Mulliken definition is a finite difference expression for —dE/dn, or, more precisely, —(SE/Bn, )n;, the change in total energy when the occupation of the jth shell is varied, holding the occupation of all others fixed. dE/dn is the usual definition of the Fermi energy e~. (Admittedly, the derivative has no precise meaning for an atom which must have an integral number of electrons. ) This relationship between an "atomic Fermi energy" and electronegativity has been dis-
AND H. EHRENREICH
cussed previously
by
Slater.
10 2
By comparing Pauling and Mulliken electronegativities for halide atoms where the affinity is known, it is possible to relate the two scales~ as X„ = —3. 1X~. Those atoms with more negative "atomic Fermi level" will have larger electronegativity, and it is apparent that if two atoms of differing atomic Fermi energy are brought together, charge will flow to the atom with the lower Fermi level (greater electronegativity). Caution should be exercised when using atomic electronegativities to discuss charge flow in solids. Often the configuration appropriate to the solid is not the most stable configuration of the free atom. Transition-metal atoms are usually assigned a. d" ~ s configuration but band calculations indicate that d" 's' is more appropriate in the solid. ~ When atoms are brought together to form a solid, there is a compression of the electronic charge density. This can cause a significant increase in the local Fermi energy, particularly for polyvalent atoms for which the compression of charge is substantial. The local Fermi energy of a metal atom at some assumed Wigner-Seitz radius would provide a useful electronegativity scale for metals. Note, however, that the electronegativity of an atom depends on the screening of the atom, so it is not obvious, and probably not even true, that a unique electronegativity could be assigned to all metals which would correctly describe alloying behavior with other metals of greatly different electron concentration. Unfortunately the local Fermi energy is not a directly measurable quantity. One might hope to use the work function Q of the metal as a measure of the local Fermi energy. ' However, the work function depends not only on the local Fermi energy, but also on the surface-dipole layer. While surface dipoles have been calculated successfully for nearly-free-electron metals, 3 no reliable calculations have been made for metals where delectron effects are likely to be important. With that warning in mind, we will now discuss why it might still be reasonable to use the work function as a measure of local Fermi energy at least for the AgAu system. The surface dipole results from the relaxation of the electron charge density into vacuum. For nearly-free-electron metals the surface dipole is determined primarily by the electron density. The pseudopotential terms for metals with s- and p-electron densities si. milar to noble metals only make small modifications. However, the pseudopotentials in Ag and Au are not small, even though the electron densities are nearly the same. An estimate of the sign of the difference of dipole layers can be obtained by considering the s and d energies of the free atoms. The Ag atom with a
CHARGE TRANSFER IN
10
tightly bound d shell has a relatively loosely bound g shell; the Au atom with a loose d shell has a more tightly bound s shell. The s charge density will extend farther out from Ag than Au atoms. To the extent that a surface atom approximates a free atom one would expect Ag to have more loosely bound surface s electrons and thus a larger surface dipole. This implies that the difference in work functions would underestimate the difference in local Fermi energies since the Au work function (5. 1 eV) is greater than that of Ag (4. 0 eV). 3~ In spite of these reservations, it nevertheless is the case that for AgAu the assumption that h&~ = —hP
leads to self-consistently
fer in semiquantitative (See Sec. IV. )
calculated charge transagreement with experiment.
That this procedure may well not work in general is demonstrated by CuNi alloys, the prototype minimum-polarity system. ~ ~ Published work = functions (P„, 5. 15 eV, Pc„= 4. 65 eV) would suggest an s charge transfer from Cu to Ni of order 0. 1 electron per atom for dilute Ni in Cu, in mild disagreement with the isomer-shift measurements of Love eg gl. , ' which indicate essentially zero s charge transfer at a Ni site. However, because of uncertainties in the correction for the second-order Doppler shift, a charge transfer of 0. 1 electron per atom cannot be ruled out. Wenger ef, al. have measured the ratio of Ni L, to L, x-ray emission intensity in CuNi alloys. They claim that this ratio can be related almost directly to the number of d electrons at a Ni site. The results indicate that the change in the number of d electrons going from pure Ni to the alloy is zero within experimental error, which the authors indicate to be about + 0. 1 electrons per atom. Apparently the presently available experimental techniques do not have sufficient precision to measure reliably the charge transfers that might be characteristic of CuNi. The use of published band-structure calculations to determine relative Fermi energies of metals is reliable only if (i) the metals are in the same row of the Periodic Table, and (ii) the same method has been used to generate the crystal potential. Comparisions down a column are clouded by large changes in relativistic and correlation effects. Different techniques of generating crystal potentials, even if they start from the same atomic wave functions, can lead to pure-metal Fermi energies which differ by more than 0. 5 Ry. For example, Christen and Seraphin find Ez = —0. 6 Ry for Au in a relativistic calculation while Ballinger and Marshall, using a Gaspar potential, obtained ez = —0.01 Ry. Since differences in w'ork functions and local Fermi energies are usually about 1 eV, these systematic differences in crystal potentials can completely swamp the differences related to electroansfer. negativity and charge
t.
A
LLOYS: AgAu
405
At present the best one can do in determining relative Fermi energies is to consider relative work functions and electronegativities keeping in mind the pitfalls. If they give answers in agreement in sign and are of nearly the same magnitude (recall that Pauling electronegativities should be multiplied by a factor of about 3 to give results in electron volts), the result is probably reliable. For example, the difference in Au and Ag work functions is 1. 1 eV, while the difference in Pauling electronegativity is 0. 5, implying a difference in Fermi energy of about 1. 5 eV and of the same sign. IV. AuAg ALLOY CALCULATIONS
In this section model-Hamiltonian calculations of several experimentally measurable properties of Au„Ag, „alloys will be presented. They are extensions of the earlier work of LE and will illustrate some of the ideas of Sec. III concerning charge transfer as applied to this system, for which the problem of different atomic volumes is minimized. As in any calculation using a simplified model Hamiltonian, the choice of the form of the model is motivated by the features of the system which are of particular interest. In these calculations the effects of charge transfer on various properties associated with the density of states are of primary importance. Comparison will be made with isomer-shift and core-level-energy-shift measurements of charge transfer, and the agreement between the calculated density of states and opticalabsorption edge and photoemission results will be discussed. These goals motivated the model Hamiltonian described in Sec. II and used in these calculations. In particular, the model must include the composition dependence of the s-d hybridization to give a reasonable account of the d-band Fermi-
level density of states. The model Hamiltonian described in Sec. II imposes the requirement that the shape of the unhybridized d- and s-band densities of states be the same. For computational simplicity we have chosen a semicircular band,
Because of the requirement of the similarity of sof states, it is not possible to choose a d-band density of states fitted to the results of a band-structure calculation as was done ' for CuNi and Stocks by Kirkpatrick et a/. et al. ' for CuNi and Agpd. As a result the density of states calculated here is not in as good agreement with photoemission experiments as would have been the case if such a single-band model with a more intricate d-band density of states had been used. The previous calculations have not, however, been capable of presenting an accurate picture of:he hyand d-band densities
406
C. D.
GE LATT,
JR.
bridization of the s and d bands which leads to a nonzero d-band density of states at the Fermi level in noble metals. It would have been possible to use a "steeple-model" density of states as was used by Hasegawa and Kanamori~ to discuss the magnetic properties of NiFe alloys. The unphysical peak (steeple) at the high-energy end of the s ba. nd would have been several eV above the Fermi energy and would not have affected the class of results to be considered here, but the extra computational time was not considered warranted. Two different sets of calculations were performed. In the first the d-band width resulted primarily from the s-d hybridization, as was the case in the calculations of I E. The unhybridized dband width resulting from d-d hoppi. ng matrix elements was chosen to be only as large as necessary to close the unphysical hybridization gap. In the second d-d hopping provided most of the d-band width, and hybridization was included to give reasonable d-band density of states at the Fermi level. Both calculations are included to provide a link with the I E results, and to demonstrate the degree of sensitivity of the results to the parameters chosen to describe the alloy system. In neither case has the choice of parameters been extensively optimized. No doubt it would have been possible to obtain closer agreement with experiment if a different set of parameters had been chosen, but the model itseU is simplified to the point where detailed agreement is more suspect than satisfying. Section IVA will describe the choice of parameters used to model pure Ag and Au. In Sec. IV B the modification of parameters due to charge transfer and the calculation of the self-consistent charge transfer are discussed. The results of the alloy calculations are presented in Sec. IVC.
AND H. EHRENREICH edge are much more sensitive to the Au than the Ag d band except for extremely dilute AgAu alloys because the Au d band is both closer to the Fermi energy and wider. In choosing the parameters for the model more care is taken in fitting the Au d band. In the first set of calculations the d-band width results primarily from hybridization, as mentioned before. The value of the hybridization parameter is chosen to give the same hybridization splitting as that given by band-structure calculations along the b, direction. The Au d resonance energy E~" is chosen to give reasonable agreement of the position of the d band relative to the Fermi level. The unhybridized d-band width 20, w, is given the smallest value that will close the unphysical hybridization gap. The choice of parameters for Ag is less straightforward. It is necessary to raise the Ag d band higher above the bottom of the conduction band than is indicated by band-structure calculations in order to obtain the experimentally observed optical-absorption edge of 4 eV. The values of y and 0( are chosen as for Au. The relative placement of the Au and Ag bands is accomplished by requiring that the difference in Fermi energy is equal to the difference in work functions, z~~ —Ez =1 eV. In previous single-band CPA calculations for real systems, the problem of relative energies was either ignored, or the F, conduction-band minima of the constituents were arbitrarily equated. ' The total density of states for pure Ag and Au calculated using this set of parameters is presented in Fig. 1. The Ag d-band peak is resonancelike with relatively small hybridization tails on top of the conduction-band density. The Au peak is less
A. Pure metals
For both sets of calculations the half-bandwidth of the unhybridized s band, m„ is 7 eV. This value, which is that used by LE, yields occupied conduction bandwidths in reasonable agreement with band calculations, an effective mass of 0.84mo at the bottom of the unhybridized s band, and an adequate s-band density of states at the Fermi level. The other parameters used in sets of calculations I and II are given in Table I. In this table F, refers to the bottom of the unhybridized s band. The model Hamiltonian does not include symmetry restrictions on the hybridization of s and d states, and as a result the F, state at the bottom of the conduction band hybridizes with the d states. Because of this effect, the bottom of the conduction band is pushed down below the energy I', when the hybridization is turned on. Both charge transfer and the optical-absorption
18—
(o) Ag
4—
d toP
O Icl
6F
2
V) LLI
cf
I
-10
-8
CA
t -4 8Rg
-2 E
d
(ev)
0
UJ
-10
-8
g dAU
-2 E (eY)
FIG. 1. Density of states for (a) Ag and P) Au calcuusing the parameters of model I (see Table I).
l. ated
CHARGE TRANSFER IN
10
TABLE I. Pure-Au and -Ag parameters used in the model Hamiltonian for the two sets {I and II) of calculations. Note that in the table I'& refers to the bottom of the unhybridized condution band. Also listed are results calculated with the model Hamiltonian, and the experimentally measured Fermi-level density of states and work function. {All energies in eV. ) Au
Parameters
fp
a~ —I ed
1
—r, 'Y
—l. 0 9. 35 4. 50 2. 00
—1. 0 8. 10 3. 00
11.7
11. 7 14. 1
1.25
l. 40
2oi los
Uu.
1, 75 0. 90
0, 94
3. 50 11.7
0. 42 11, 7 15. 0
5, 60
.
14, 1
0. 0 7. 45
0. 0 7. 60 3. 15
15. 0
Calculated Results
p{cz) (electrons/atom
eV)
Occupied conduction-
0. 333 10. 20
0. 236
0. 222
0. 204
9, 25
7. 89
8. 35
9. 66
9, 83
9. 91
9. 93
band width
Integrated d character
Experimental
{electrons/atom from specific heat p(& z)
0. 296
e V)
0. 275 4. 0
5. 1
Work function P
aUncorrected for phonon enhancement ~Reference 35.
{Ref. 52).
symmetrical and the larger y gives longer and higher hybridization tails. The small dip near 5 eV is a remnant of the hybridization gap. The method of determining the top of the d band is explained later. These results bear little resemblance to the results of band calculations or photoemission experiments in the region of the d band, but they do provide reasonable charge transfer be-
-4.
— Ag
A
LLOYS: AgAu
407
cause the density of states near the Fermi energy is given adequately by the model. A second set of calculations (1I) was performed using a much larger unhybridized d-band width to give somewhat better agreement with the photoemission results as demonstrated in Figs. 2(a) and 2(b). The parameters for Au and Ag were determined primarily by comparison with vacuum-uv photoemission experiments. 3 The unhybridized d-band width is 2ezg, = 5. 6 and 3. 5 eV for Au and The position of c~ relative to the Ag, respectively. unhybridized s-band minimum was chosen to give the correct value for the optical-absorption edge with the top of the d band determined by the sharp drop in the d density of states. The value of y used for Ag is essentially the same as that used in I. Using y"" = 2 eV, as in 1, resulted in a density of states at the Fermi level much larger than indicated by experiment. The increased unhybridized d-band width results in unhybridized d states only 2. 5 eV below the Fermi level which contribute to p(ez) when hybridization is turned on. In I, e~ is about 4 eV below cr, giving a smaller p~(ez). Since most of the d-band width comes from d-d hopping in model II, y can be chosen by some other criterion. A reasonable alternative is to choose y to give a Fermi-level d-band density of states in agreement with interpolation fits of calculated band structures. This choice indicates a y for Au of about 1. 25 eV, the value used in the second set of calculations. There is a trade off between y and the unhybridized width determined by z. With a smaller value of a, it would be possible to use a larger y yet maintain a constant d-band width. The value of y used is smaller than would be calculated using the d-resonance models of Heine and Hubbard, 4 but those theories, developed for a narrow d band, are of questionable validity for Au, where the d-band width is comparable with the d-to I'1 separation.
Model E X PS (Hufner,
O
et
Ol)
2—
Shirley )
0 I-
(A LLI
I-
l
Cl
I-
(A
'I
M
4J
1
(fJ
l
I-
d tap
pl I
-8
-6
I
fA 9
&a
0
-2
E(e~)
-6 Au
E (eY)
d
FIG. 2. Density of states for {a) Ag and (b) Au calculated using the parameters of model II {see Table I). XPS measurements normalized to the same area are presented for comparison. The Ag curve(shifted by 0. 25 eV as explained in the text) is from Ref. 46, the Au data from Ref. 47.
C. D. GELATT, JR. The pure-metal density of states with this modified set of parameters is shown in Figs. 2(a) and 2(b). X-ray photoemission results normalized to the same occupied area are given for comparison. The model density of states does not include the 7& peak at the top of the d band characteristic of fcc metals. Figure 2(a) compares the model results for Ag with the data of Hufner ef gl. 6 The top of the d band is 4 eV below the Fermi energy, and the d-band width is a.bout right. In Fig. 2(b) comparison is made with the Au measurements of Shirley. (The experimental curve has been shifted to lower energy by 0. 25 eV to give a d-band-toFermi-level separation of 2. 5 eV, in agreement 8) The "ramp" at the botwith e~ measurements. tom of the d band is an effect of hybridization noted previously in Cu and is a result of the center of gravity of the s band lying above that of the d band and pushing the d states to lower energy. The pronounced two-humped structure in the x-ray-photoelectron-spectroscopy (XPS) measurements is a result of the spin-orbit splitting of the d bands and It is is not found in nonrelativistie calculations. not correct to interpret the 3-eV separation of the humps as a measure of the spin-orbit splitti. ng parameter. Relativistic band-structure calculations which give a density of states in good agreeindicate that the spinment with experiment orbit splitting parameter for the solid is about 1 eV, somewhat smaller than that of the free atom. It is a puzzling feature of the experimental XPS results that it is necessary to shift the curves, even for the pure metals, to obtain an optical-absorption One edge in agreement with c~ measurements. would expect that since both the rapid rise in the XPS signal and the increase in a2 at the opticalabsorption edge are due to transitions from the toy of the d band to the Fermi level, the energies should agree. Since comparison will be made later it was with optical-absorption edge measurements, decided to treat the experimental determination of the optical edge as the more basic, and shift the XPS curves accordingly. Different techniques are used to determine the
"''
"
upper edge of the d band for the two sets of calculations. In model I there is no structure to use as the top edge as shown in Figs. 1(a) and 1(b). The same problem was faced by LE, who suggested that the integrated d-band density of states could be used to determine the top of the d band, E'~, They obtained E~, from No= J (E) dE, where No is chosen to give the correct optical-absorption edge for Au. This method is used in connection with model I. Using a value of 2. 5 eV for the Au optical-absorption edge, the present calculations with model I requixe No = 8. 925 electrons yer atom. In model II there is a sharp drop-off of the d-band
„'p,
.
AND H.
EHRENREICH
10
density of states [Figs. 2(a.) and 2(b)] which can be used to fix the position of the top of the d band. A suitable choice, also discussed by LE, is to define the top ot the d band as that energy where p~(E)/ pd'" = R. Here pd is the maximum of the d-band density of states and R is chosen to fit the Au optical-absorption. edge. The edge i.s determined by this method in model II with R=0. 16. The definition of the upper d-band edge would have been less amibguous if the d-band density of states had included the T& peak characteristic of fcc structures instead of going to zero as (Eo Eo) ~ -. The square root is easily washed out by alloy scattering while the peak would
persist.
The input parameters for the calculations pertaining to the pure metals as well as their alloys are summarized in Table I. This also contains several quantities derived from the model-Hamiltonian calculations for the pure metals. The experimental specific heat~~ (uncorrected for phonon enhancement) and work function 5 (used in the model to place relative Fermi energies) are given for comparison. Band calculations give total occupied conduction bandwidths of 8. 3 to 10. 4 eV for Au, and 6. 8 to 7. 8 eV for Ag. The integrated d character is not measurable, but it can be calculated from an interpolation fit of a calculated band structure4&'3 which makes a somewhat arbitrary separa, tion of an eigenstate into d and plane-wave components. Such fits of published band structures indicate an integrated d character of 9. 68 to 9. 87 for Au, and 9. 87 to 9. 98 for Ag. 8. Effects of alloying
The parameters which describe a pure metal will be modified when the metal is alloyed. The most important of these effects is the shift of the energy of the localized d electrons due to charge transfer discussed in Sec. III. The shift in d energy, Aa„ is given by Eq. (3. 5), B g@, d
pA, Bg ndA, B+ pA, Bg dd d
A, B
He re b nd and hn,
are the change in the number of d-band a. nd conduction-band electrons at the site, and the U's are the effective Coulomb integrals discussed previously. The subscripts c for conduction band and s for s band will be used almost The effective Coulomb integrals interchangeably. are calculated by changing the s configuration of the renormalized atom (not a weighted s and p configuration) and this is denoted by the s subscript. LE included the effects of s charge transfer only and used the value of bare renormalized-atom Coulomb integrals, F~, for U", „' Due to a numerical error, the values of E,~ they used were incorrect: Both the Au and Ag integrals used were
.
CHARGE TRANSFER IN
10
too small by a factor of 3 to 4 (0. 29 Ry wa, s used instead of 0. 9 Ry for Ag, 0. 23 Ry insteadof 0. 9Ry for Au). The calculations of LE determined the concentration dependence of c, and, through Eq. (4. 2), the magnitude of An,"' F, d" If correct Coulomb integrals had been used, the charge transfer which gave values of e~(x) in agreement with optical-absorption measurements would have been about a third as large as that quoted by LE, that is, = —0. 1, a more reasonalbe magnitude hn~ "(x =0) — for the total charge transfer, but still with the wrong sign because LE assumed that the Au Fermi level was 2. 6 eV above that of Ag. We have assumed that charge transfer does not shift F, The conduction-band wave function is more diffuse than the d-band functions and samples the average potential in the crystal. Thus the equivalent of Eq. (4. 2) for the shift of I', is
.
.
hEr,
= AU, ~An~+
U„bn, ) + (1 —x)(U, ~dn„+ U„An, )
One f inds that UA
sd
~Ugss ~UQsd
ss
This fact combined with the requirement of charge neutrality implies that the shift of F, is small. Band calculations for n-brass mhich include charge-transfer effects confirm this conclusion, The effect of the Madelung potential, which is important in ordered ionic solids, is not included. In a completely disordered alloy the average of the Madelung potential is zero since the average charge on every site is zero. In the spirit of the singlesite CPA, me have replaced the actual Madelung potential by its mean value. The Heine-Hubbard ' theory of transition-metal band structures is used to determine the change in when Ed is shifted y, the hybridization parameter, jas pointed out above, y for the pure metals is determined by comparison with band calculations). the energy separation y scales linearly with Ed — between the center of gravity of the d band and the conduction-band minimum. Thus, when charge is transferred, y(x) = y + y Ae, (x). Here y is the hybridization parameter for the pure metal, d, s~(x) is the shift in d-band energy given by Eq. (4. 2), and y' = y'l(~', —r', ). The unhybridized d-band width 2m~, is not the same for pure Ag and Au. If this difference is treated accurately, one is faced with the off-diagonal disorder problem discussed recently by Schwartz et al. and Blackman et al. ' The added complication is sufficiently prohibitive that the dband width is treated instead in the virtual-crystal
A
LLOYS: AgAu
409
parameters, it is now possible to implement the calculation of the self-consistent charge transfer outlined in Sec. III. Written explicitly for Au+g, alloys the self-consistent equations are
f
eF (x) I
p, (Z, x)+ p, (Z, x)]dZ=
which determines
xA
.„
""+(I-x)A "',
the Fermi energy,
(4 4) and the set of
four equations e~ (x)
f p, "'~(Z, x)dZ-N„""'"'=an,"" "'(x), f „p, "'(Z, x)dZ —X,"" "' ~n,"" "'(x),
(4. 5)
ep (x)
=
(4. 6)
which determine the charge transfer. Here cz(x) is the Fermi energy for an alloy of concentration .x, p, (Z, x) =xp~(Z, x)+(I-x)p~(Z, x) and p, (Z, x) = xp~~" (Z, x) + (1 —x) p~(Z, x) are the total conductionand d-band densities of states given by Eqs. (2. 13) and (2. 14) expressed in terms of the component densities given by Eqs. (2. 16) and (2. 17), and ~Au, Ag ~Au, Ag + ~Au, Ag c d Self-consistency is imposed on this set of five equations, (4. 4)-(4. 6), by including the shift of the Au and Ag d levels given by Eq. (4. 2). They can be iterated to self-consistency as in a self-consistent band-structure calculation {Brouers and Vedyayev were the first to report a calculation of this type for a disordered alloy56). The d-level shifts will modif y the density-of -states functions and thus give a modified charge transfer. The self -consistent charge transfer yields level shifts which reproduce the assumed charge transfer. Care must be taken in the iterative procedure since simple iteration is
unstable. The model in its present form does not necessarily satisfy the Friedel sum rule
F„
approximation,
a(x) = xn(l) + (1 —x) n(0)
.
(4 3)
Having obtained prescriptions for treating the char ge -transfer modif ications of the pure-metal
If d&z/dx were calculated from Eqs. (4. 4)-(4. 6) the result would depend implicitly on the effective Coulomb integrals. It would be possible to satisfy the sum rule since it imposes only two restrictions, while there are four Coulomb integrals. The &=0' limit for Au„Ag, „mould place a restriction on the ratio of U« to Uds for Au. If the Coulomb integrals were assumed to be independent of concentration, then they would not have the correct value for pure Au. A more natural assumption would be to let the Coulomb integrals change with concentration, taking on the pure-metal value in one concentration. limit, and the value imposed by the sum rule in the other. This added complexity was considered unwarranted. The relevance of the sum rule, a result derived for a vanishing concentration of impurities, to calculations of alloy properties for the full concentration range is questionable.
C. D.
410 Au BRND
GE LATT,
Auog Ago g N00FL
MODEL
NODAL
JR.
Ag BAND
CALC.
CALC.
E(eV)
-2-
-8-10— FIG. 3. Comparison of the pure-metal parameters of model II with the results of band-structure calculations for Au (Bef. 39) and Ag (Bef. 59). The energy levels indicated are &z, the Fermi energy; ~&, the center of gravity of the d 'band; and I'&, the bottom of the conduction band. The two central sets of levels are those of a Au and a Ag site in a Auo &Ago alloy resulting from a self-
AND H. EHRENREICH
from predominantly Au states while the maximum comes from states with large Ag character. For compa, rison, Siegbahn's XPS measurement of AuAg~ normalized to the same area is shown (the experimental curve has been shifted to lower energy by 1.0 eV to give a d-band-to-Fermi-level separation in better agreement with the 3. 1 eV implied by optical measurements of e, ). 4' The overall width is in good agreement but more structure, which is not included in the model, is observed experimentally. Nilsson's+ and Siegbahn's~ unshifted photoemission measurements are plotted in the upper part of Fig. 4. The photon energy used in the uv-photoemission-spectroscopy (UPS) experiment48 (10 eV) is not sufficient to reveal the full d-band width. Both curves imply that the top of the d band is closer to the Fermi level than indicated by &2 measurements. As noted for the pure metals, this discrepancy cannot be explained by the stated errors in placement of the Fermi energy or by instrumental broadening of the structures. C. Results
&
consistent charge-transfer
calculation.
Figure 3 illustrates the parameters used in model II and how they change when charge is transferred in the alloy. Included in the figure are &z, the Fermi energy, c~, the center of gravity of the d band (e, =-', I'25+&I', 2 for the band calculations), and I'„ the conduction-band minimum. The outermost columns are the results of band calculations for Au+ and Ag shifted to give a Fermi-energy difference in agreement with that assumed for the calculations. The next set of model-Hamiltonian levels toward the center of the figure are the parameters used in the model Hamiltonian for the pure metals. The occupied conduction band width (&z —I', ) is in good agreement with the band calculation for Au, but the Ag width is a bit too large. The value of t. „used in the model Hamiltonian is about 0. '?-5 eV low for both Au and Ag. The center set of levels are those self-consistently calculated for the equtconcentration alloy Auo 5Ago 5. The Coulomb repulsion of the charge transferred onto Au has shifted the d band upward by 0. '? eV. The reduced Coulomb repulsion. on Ag has shifted the d band downward by about the same amount. The 42 eV is slightly above the Fermi energy of arithmetic mean of the Au and Ag values. The density of states of AuAg is plotted in Fig. 4. The d-band peak is asymmetrical with a shoulder on the high-energy side. The dotted and the dashed lines give the concentration-weighted Ag and Au This decomcomponent densities, respectively. position demonstrates that the shoulder results
-0.
10
SeU-consistent configuration calculations have been made over the full concentration range for Au+g, alloys using the parameters listed in Table I. Comparison w ith experimental measurements of charge transfer, optical-absorption edge, and alloy work function will be presented in this sec-
,
tion.
—
AUp
5 Agp 5 Toto{ {Model
{{)
---- . Ag component
—- Au component . ——XPS CSiegbahn, ——XPS CNilsson)
/'
—
shifted)
—"—X PS CSiegbahn)
O
l-
2-
UJ
IV) LLJ
1—
'&Ag Gd
~~ Cd
4
-2 ECeV)
FIG. 4. Density of states of Auo &Ago z calculated with model II. Also shown are the decomposition of the density of states into its Au and Ag components, and XPS measurements for this alloy (Refs. 48 and 60). The experimental curve in. the lower half of the figure has been shifted by 1.0 ev.
CHARGE TRANSFER IN 0.2
0.2
(a)
(b)
l
SIte
Ag
Au
Site
---Model I Model E
0E ~o
Au
~ inc
Ol
Ol
C
0
Net
O 0) 0&
I
LLJ
I
I
6
.8
I
4
.2
.
.
p
~
X
Z'
I
.2
4, I
.
I
I
6
.8
X
Au
hand Ag
LLI
(3
X
X
C
& Net
-Ol—
-Ol—
C3
a -0.2—
FIG. 5. Concentration dependence of the conduction band, d band, and net charge transfer in A~gt „ for (a) a Ag site and {b) a Au site. 1. Charge transfer
Figures 5(a) and 5(b) show the concentration dependence of the Ag and Au conduction band, d band, and net charge transfer. In this and following figures results of calculations using the parameters of set I are given by the dashed line, those of set II by the dotted line. Conduction charge is transferred onto the Au sites (bn~ & 0) and there is a compensating but smaller d charge transfer off the Au sites. The situation is reversed at Ag sites, which lose conduction electrons but gain a smaller number of d electrons. The sign of the total charge transfer is that expected from the relative Fermi energies (ez~& z~~) as explained in Sec. III. The magnitude of the total charge transfer in the lowconcentration limits is somewhat smaller than the —0. 25 electron per atom that hn„, = p&, &(fr)h&z = the simple rigid-band model predicts. This is reasonable because the d-level shifts included in the present calculations, but neglected in the rigidband model, oppose charge transfer. The qualitative concentration dependence of the charge transfer is right: The charge transfer is zero for the pure metal and increases monotonically as the concentration of the constituent decreases. The combination of isomer-shift and core-levelshift measurements gives an experimental value for both Au charge transfers. The ' ~Au Mossbauer isomer shift has been measured for AuAg alloys over the full concentration range. '3 For dilute Au in Ag it is +2. 1 mm/sec relative to pure Au. The measured shift is linear with concentration within I
I
I
I
A
LLOYS: AgAu
experimental error (+ 0. 01 mm/sec for the latest measurements}. This experiment measures changes in the electronic charge density at the nucleus. As a result it is only sensitive to s (and relativistic Watson, Hudis, and Perlman'4 p&&2) charge density. (WHP) estimate that the quoted result corresponds to a transfer of 0. 26 s electrons onto the Au sites in the x- 0 limit. This interpretation takes into account the difference between the atomic 6s wave function and the wave function of a Fermi-level electron in the metal and allows for the contraction of the conduction-band charge density when the s-d screening is reduced by compensating d transfer away from the Au sites. The value quoted for the experimental s charge transfer could be in error by as much as a factor of 2, owing to uncertainties in the value used for the change in nuclear radius during the Mossbauer transition and in the corrections used to relate the nuclear contact charge density to the total number of conduction electrons at similar isomer-shift the Au site. Unfortunately, measurements have not been made, and indeed may not be possible for the Ag sites. The experimental determination of the other three constituent component charge transfers is less direct. %'HP point out that ha~" can be estimated using an equation analogous to Eq. (4. 2), b (CF —f4f) 4 Ep U4'f ghn~ —U4f g4n, , (4 7) where h(ez —c«) is the Au 4f core shift, he+ the difference between the alloy and pure-metal Fermi energies, and Ape, is the conduction-band charge transfer inferred from the isomer shift. The charge transfers at the Ag sites are given by the solution of anequation similar to Eq. (4. 7) for the Ag 3d core shiftand the charge neutrality requirement
x(~n,"" + ~n~) + (1 — )(~x~,"'+ ~
n)
=0
.
(4. 8)
For the case &=0. 5, the appropriate inputs are (i) the core shifts' for Au 4f, +0. 1 eV, and Ag 3d, —0. 25 eV; (ii) the value of An,""(0 5) =+0..13 electron per atom'"'4; (iii) the effective Coulomb
'
U4f g 1. 19 Ry, U4~, =0. 95 Ry, U3gg =1.24 Ry, UAf, =0. 94 Ry; and (iv) the assumption that az, (x) = xcp" + (1 —x)tp' with ez' —cp" = 1 eV. The resulting "experimental" values for the charge transfers are Ape~~= —0. 08, 6n,"'= —0. 15, and
integrals
+0. 10 electrons per atom. These values could be in error by as much as 0. 1 electrons per atom because of uncertainty in the relative Fermi energies and in the evaluation of b, n", " from the isob, n~~ =
mer-shift measurements. Comparing these experimental values with the calculated charge transfer in Figs. 5(a) and 5(b) it is apparent that the calculated Au s charge transfer is less linear with concentration than the isomershift data imply. It is also somewhat smaller in magnitude than the %HP analysis indicates. The
412
C. D.
GE LATT,
JR.
ratio of d charge transfer off Au sites to s charge transfer on is smaller for the calculated results than for the %HP core-level-shift measurements. The calculated charge transfers imply an Au 4f core-level shift of -0. 16 eV (-0. 17 eV) for model I (II) instead of the +0. 1 eV actually measured. Note that the core-level shift in the absence of d compensation would be 1.2 eV, so the error in sign is not so significant, although the error is outside of the experimental uncertainty of about + 0. 1 e V. Similarly the calculated Ag charge transfers imply a 3d core-level shift of +0. 42 eV (+0. 30 eV) for model I (II) instead of the —0. 25 eV measured. One further test of the reasonableness of the calculated result is provided by Mott's argument that the Madelung energy due to charged atoms will stabilize ordered alloys. The Madelung energy E„ for an equiconcentration alloy is given by
(4. 9) where n is the Madelung constant for the structure of the ordered crystal, a the lattice constant, and b, Q the net charge on the atoms. For P-brass Mott estimated that A@=0.076e from resistivity measurements (in good agreement with band-structure estimates~) and calculated an ordering energy equivalent to a temperature of about 300'K for Pbrass. By contrast, no ordered phases have been observed for AuAg alloys. The activation energy for volume diffusion of about 2 eV makes it unlikely that any ordered phases could be observed at a temperature below about 200 'C because the order' Thus the value ing time would become too long. of b, g must be smaller than 0. 1 electrons per atom. The calculated net charge transfer for Aup, Agp g of 0. 072 (0. 069) electrons per atom for model I (II) satisfies this requirement as does the net charge transfer of 0. 05 electrons per atom inferred from
isomer-shift and XPS experiments. Aside from the shortcomings of the model Hamiltonian itself, the least certain of the parameters entering the calculation is the choice of the relative Fermi energies of Ag and Au. Several experiments measure work functions or relative work functions but we are not aware of any which can measure the "bulk" Fermi energy or the surface It is therefore interesting to dipole separately. examine the effects of using a different value for the relative Fermi energy than that implied by the relative work functions. Figures 6(a) and 6(b) show how the dilute-limit charge transfers vary when the relative Fermi energy, or relative electronegativity following the discussion of Sec. III, is changed holding the other parameters fixed. The charge transfer is a monotonically varying function of the Fermi-level difference. For zero difference it is small; the residual transfer is due to band-shape changes, an
AND H. EHRENREICH
10
effect not directly related to Fermi-level differences. For &~ —E~" &0 the net charge transfer is from Ag to Au. The more positive the charge on the Ag sites, the more tightly the Ag d level is bound. As seen in Fig. 6(a), the Ag d band continues to gain electrons as the Fermi-level difference is increased until the d charge transfer saturates when the band is completely full. The qualitative features of the self-consistently calculated charge transfer are in accord with the discussion of Sec. III. The details, such as the magnitude of the charge transfer or the ratio of d to s charge transfer, are sensitive to the values and ratios of the effective Coulomb integrals, the value of the hybridization parameter and how it changes when the d levels are shifted, and, of course, the assumed relative Fermi energy. 2 Optical absorption edge
The concentration dependence of the opticalabsorption edge has played a central role in pointing out the importance of charge transfer in alloys. LE found that they obtained a qualitatively wrong optical edge in AgAu alloys unless they included the shift of the d levels which results from charge transfer. In a recent ATA calculation of the complex energy-band structure of n-brass Bansil et a/. found that the optical-absorption edge increased too rapidly as Zn is added to Cu unless the reduction in Cu d-level binding due to conduction charge transfer from Zn to Cu is taken into account. Figure '7 displays the comparison with experiment of the optical-absorption edge calculated in the self-consistent charge-transfer model for the parameters listed in Table I. The calculated opti-
Ag
E
0
ate
AU
———Model I
E
Q
c
02— "" ' Model
o
ste
02—
C
0I O
0.1—
Q rid
Ag
— 0.1—
Net Ag
&F
0
0.5
1.5
1.
CAg
~~
. -'
'~ +5
Au
-BF
(eV)
1.0
1.5
E Au( v)
-01— x UJ
AU
Cf)
,
Net
0, 2
f -0.2—
4J
X w
K
~-0.&-
-0.5—
43
g~
Ag
-0.4— FIG. 6. Dependence of the self-consistently calculated impurity charge transfer on the assumed pure-metal Fermi-energy difference. (a) The charge transfer at a Ag impurity in Au, and |b) the charge transfer at a Au impurity in Ag. All parameters but the relative puremetal Fermi energy are those of Table I. -0.4—
CHARGE TRANSFER IN ALLOYS: I
I
I
I
I
I
I
the differential techniques are most sensitive. The rapid increase of the optical edge near x=1 for model II is an artifact of the technique used to determine the top of the d band from the calculated density of states. The sharp cutoff in the density of states at the top of the d band seen in pure Au [Fig. 2(b)] is washed out by the disorder scattering in the alloy. As a result the "top of the d band" comes at a lower energy than it did before the cutoff was rounded out, and the optical edge increases.
I
———MODEL I ~
LOP-E=-2. 5
C3
"-.- MODEL
AgAu
E
i' ..
C) LLI
3
3. Work function CL
SLOPE = —. 9
I
0
0.4
I
0.6
I
0.8
1.0
X AU FIG. 7. Concentration dependence of the optical absorption edge in Au„Ag& „alloys. The experimental points are derived from measurements of &2(cu) (Ref. 48). The error bars represent our estimate of the error made inferring the energy of the optical edge from the experimental curves. The short lines at the low-concentration limits show the slope determined from differential reflectivity me as ure ments (Ref. 62). Ag
cal-absorption edge is the dirierence in energy between the Fermi level given by Eq. (4. 4) and the top of the d band determined by the techniques discussed above. The experimental points are defor the alrived from Nilsson's c2 measurements loys. The optical edge is taken to be the energy at which E~"' achieves half of its peak value in the rapid rise as the energy becomes sufficient to excite electrons from the top of the d band to the Fermi level. Note that LE compared their- results with Wessel's ' ref lectivity data, not the edge determined from e~. The error bars on the experimental points allow for the estimated error in assigning an energy to the optical-absorption edge from the plot of ez(u). A line is drawn at the two low-concentration limits to indicate the rate of change of the optical-absorption edge with concentration measured by Beaglehole and Erlbach+ using a differential reflectance technique. The over-all concentration dependence of the calculated optical edge is in good agreement with experiment. The calculated slope near x=0 is greater than the slope near x=1, but the agreement with the slope determined by differential reflectance is only fair. This is not surprising because the simple model used in the calculations does not contain any of the fine structure to which
In Fig. 8 the difference between the Fermi energy of pure Ag and the self-consistently determined alloy Fermi energy from Eq. (4. 4) is plotted versus concentration. For this calculation the difference between the Ag and Au pure-metal Fermi energies is taken to be 0. 9 eV. All other parameters are those of model II in Table I. Also plotted is the alloy Fermi energy calculated with the effective Coulomb integrals in Eq. (4. 2) arbitrarily set to zero (labeled "no level shifts" in the figure). The dashed curve shows the behavior of the contactpotential difference with respect to pure Au of AgAu alloys evaporated on heated mica substrates as measured by Fain and McDavid. The results of the self-consistent charge-transfer calculation are in qualitative agreement with experiment. The calculated concentration dependence is nonlinear with a larger slope at the Aurich end, but the nonlinearity of the experimental results is more pronounced than that calculated. The "no level shift" curve bows the wrong way. This is most readily understood by considering a simplified solution of the system of self-consistent equations for the charge transfer [Eqs. (4. 4)-(4.6)]. The equations can be linearized by assuming that
QQ
-Q2
l
Q6 Ag
I
1
AU
FIG. 8. Concentration dependence of the I'ermi energy of Au+g& „alloys. The self-consistent charge-transfer calculation was performed using the parameters of model II (with the exception that the pure-metal Fermi-energy difference is taken as 0. 9 eV for better comparison with the experimental data). The long-dashed curve was calculated assuming no d-band shift upon charge transfer. The short-dashed curve presents the behavior of the contact potential difference measured by Fain and Mcoavid (Ref. 63).
C. D.
414
GE LATT,
JR.
the density of states in the vicinity of the Fermi level is independent of energy and concentration. When level shifts are excluded, the solution of these linearized, approximate equations for the Fermi level in an A„B, alloy is'4
s~(x)
=
8 - x) xc~+(I - x)c~+x(l A,
-
(er'- eg)(p" p') xp
+(1-x)
(4. 10) are the pure constituent Fermi energies, and p"' are the pure constituent total densities of states. In the dilute-A limit Eq. (4. 10) reduces to Kp(x) = Ep+ x(ep —Eg)p /p (4. 11)
where
e~+
The slope is proportional to the ratio of the Fermilevel state densities. Since the Au density of states is greater than that of Ag (see Table I), the s1olx.' curve is of the Fermi-level-versus-concentration greater in the dilute-Au (x 0) limit when level shifts are excluded. It can be shown~ that within the linearized, approximate solution of Eqs. (4. 4)-(4. 6), the effect of including level shifts is to replace p" in Eqs. (4. 10) and (4. 11) by p" =—d&" s/de+, the "effective total densities of states" (here N" is the total number of electrons at an A (B) site for a given Fermi energy). The ratio p ' /p s is less than 1,
~Work supported in part by Grant No. GH-32774 of the National Science Foundation. 'L. Pauling, The Nature of the Chemical Bond {Cornell U. P. , Ithaca, N. Y. , 1960).
J. C.
B. S.
Slater,
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L.
F,
Mott, Hodges,
Proc. Phys. Soc. Lond. 49, 258 (1937).
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P. G.
(1972). N. Sen and M. H. Cohen used an equivalent formalism for a CPA treatment of a two-band model of a dis-
P.
AND H. EHRENREICH
10
and it decreases as the ratio of d band to conduction-band Fermi-level density of states increases. In the case of AgAu alloys, while p~ & p~, p"" ( p~. ~ Referring to Eq. (4. 10), when level shifts are included the curvature of the Fermi-level-
versus-concentration curve changes sign, as is seen in Fig. 8. It is remarkable that the relationship between bulk Fermi energy and work function that was assumed when establishing the relative energy scale for pure Ag and Au apparently also holds for the alloy. One can think of several possible explanations for this result, but the resolution of the question is impossible in the absence of reliable ab Agitio calculations of the work function of transition or noble metals. The existence of reliable work-function data for alloys will hopefully provide both insight and motivation for such calculations. ACKNOWLEDGMENTS
The authors are indebted to R. E. Watson for providing the renormalized-atom programs used to calculate the effective Coulomb integrals, and for several helpful discussions. S. C. Fain, Jr. kindly provided us with results prior to publication. We have benefited from discussions with P. de Chatel, L. Schwartz, and A. Bansil.
ordered semiconducting alloy, in Amorphous and Liquid Semiconductors, edited by M. H. Cohen and G. Lucovsky (North-Holland, Amsterdam, 1972), p. 147. The Soven form of the self-consistency equation is not well suited for iterative numerical solution. The calculation reported in Sec. IV used the matrix generalization of the form used by Velickg et al. {Ref. 10), Z =& +x(l -x)b[F + Z —e-(1 —2x)6] 6, which can be solved iteratively starting from Z = ~. 'SD. Turnbull (private communication). M. I. Chodorow, Phys, Hev. 55, 675 (1939). 'G. M. Stocks, R. W. Williams, and J. S. Faulkner, Phys. Hev. 8 4, 4390 (1971); J. Phys. F 3, 1688 (1973). F. J. Arlinghaus, Phys. Rev. 186, 609 (1969); A. R. Williams (private communication). ~~J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill, New York, 1960), Vol. I. 4J. H. Van Vleck, Rev. Mod. Phys. 25, 220 (1953). -5C. Herring, in Magnetism, edited by G. T. Rado and H. Suhl (Academic, New York, 1966}, Vol. . IV. ~ T. Koopmans, Physica 1, 104 {1933). P. Fulde, A. Luther, and R. E. Watson, Phys. Rev. 8 8, 440 (1973). B. M. Friedman, J. Hudis, M. L, Perlman, and R. E. Watson, Phys. Rev. 8 8, 2433 {1973). H. Pritchard and H. Skinner, Chem. Rev. 55, 745 3
3
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J. C.
CHARGE TRANSFER IN
10
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J.
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(unpublished).
V. Heine, Phys. Rev. 153, 673 (1967); J. Hubbard, Proc. Phys, Soc. Lond. 92, 921 (1967). 46S. Hiifner, G. K. Wertheim, J. H. Wernick, and A. Melera, Solid State Commun. 11, 259 (1972). 'D. A. Shirley, Phys. Rev. B 5, 4709 (1972). P. O. Nilsson, Phys. Kondens. Mater. 11, 1 (1970). N. V. Smith and M. M. Traum, in Electron Spectroscopy, edited by D. A. Shirley (North-Holland, Amsterdam, 1972), p. 541. J. W. D. Connolly and K. H. Johnson, M. I. T. Solid 4
A
LLOYS: AgAu
State Molecular Theory Group 72, p. 19, 1970 (unpublished). ~'Atomic Energy Levels, edited Bur. Stand. Circ. 467) (U. S. 1952), Vol. 3. T. H. Davis and J. A. Rayne,
415 Semiannual
Report, Vol.
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Phys. Rev.
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(1972).
F. M. Mueller, Phys. Rev. 153, 659 (1967). 4L. Schwartz, H. Karkauer, and H. Fukuyama, Phys. Rev. Lett. 30, 746 (1973}; Phys. Rev. B {to bepublished}. 5~J. A. Blackman, D. M. Esterling, and N. F. Berk, Phys. Rev. B 4, 2412 (1971). F. Brouers and A. V. Vedyayev, in Band Structure Spectroscopy of Metals and Alloys, edited by D. J. Fabian and L. M. Watson {Academic, New York, 1973). "In the present calculations the iteration was stabilized by a combination of averaging the charge transfer from
~
successive iterations, and using the Aitken & procedure. This instability is also a problem for self-consistent band calculations. J. Friedel, Adv. Phys. 3, 446 (1954). B. R. Cooper, E. L. Kreiger, and B. Segall, Phys. Rev. B 4, 1734 (1971). 0K. Siegbahn, ESCA: Atomic, Molecular, and Solid State Structure Studied by Electron Spectroscopy (Almqvist and Wiksells, Uppsala, 1967), Chap. 4. 'P. R. Wessel, Phys. Rev. 132, 2062 (1963). 62D. Beaglehole and E. Erlbach, Phys. Rev. B 6, 1209 (1972). S. C. Fain, Jr. and J. M. McDavid, Phys. Rev. B 9, 5099 (1974}. For details of the linearized charge-transfer model, and a more complete discussion of the comparison of the calculated results with experiment, see C. D. Gelatt, Jr. , Ph. D. thesis (Harvard University, 1974) (unpublished).
