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AN EFFECTIVE MEDIUM APPROXIMATION FOR THE ELECTRONIC STRUCTURE OF LIQUID
METALS
L. Roth
To cite this version:
L. Roth. AN EFFECTIVE MEDIUM APPROXIMATION FOR THE ELECTRONIC STRUC- TURE OF LIQUID METALS. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-317-C4-323.
�10.1051/jphyscol:1974461�. �jpa-00215652�
JOURNAL DE PHYSIQUE Colloque C4, suppldment au no 5, Tome 35, Mai 1974, page C4-317
AN EFFECTIVE MEDIUM APPROXIMATION FOR THE ELECTRONIC STRUCTURE OF LIQUID METALS
L. M. ROTH
State University of New York a t Albany Albany, N. Y., U. S. A.
RBsumB. - On traite la structure Blectronique d'un metal liquide par une approximation de milieu effectif d'un site simple. La description du milieu contient plus d'informations qu'une analyse unique de la self-knergie ou de matrice-T, c'est-a-dire qu'il y a des renseignements sur la dispersion entre les sites. Les caracteristiques du milieu sont dkterminkes dans une facon self-consistante en examinant la dispersion par un certain ion d'un electron du milieu qui consiste en les ions qui restent. Dans une expansion perturbative de la dispersion, notre theorie rend compte, dans le sens de I'approximation superpositive de Kirkwood, de la correlation entre les ions successifs dans une chaine de dispersions. 11 y a deux versions de la theorie, I'une derivant de I'expansion matrice-T bien connue de la theorie de la dispersion multiple, et une version de liaisons fortes que l'on traite par une methode ((locator D. Les deux versions se rkduisent en l'approximation du potentiel cohCrent pour le cas d'un alliage desordonne et en resultats connus dans le cas d'un metal liquide aleatoire, mais on croit que notre theorie est un perfectionnement des resultats precedents du genre de potentiel coherent, quand la correlation ionique est importante. La version a liaisons fortes de la theorie peut s'appliquer aux mktaux liquides a bande Ctroiteet se prete aux calculs numeriques.
Abstract. - The electronic structure of liquid metals is treated by a single site effective medium approximation. The description of the medium contains more detail than would be provided by a self energy or T-matrix alone, i, e. it contains some information about scattering between sites.
The properties of the medium are self-consistently determined by examining the scattering by a given ion of an electron from the medium consisting of the remaining ions. In a perturbation expansion of the scattering, our theory takes into account, in the sense of the Kirwood superposition approximation, correlation between successive ions in a chain of scatterings. There are two versions of the theory, one arising from the well known T-matrix expansion of multiple scattering theory, and a tight binding version which is treated by a locator method. Both versions reduce to the coherent potential approximation for the case of a disordered alloy, and to known results for the case of a random liquid metal, but are believed to be an improvement over previous coherent potential type results when ionic correlation is important. The tight binding version of the theory has application to narrow band liquid metals and is amenable to numerical calculation.
The problem I address myself t o in this work is the generalization of the coherent potential approxima- tion [l] (CPA) which has stimulated so much interest in the theory of alloys, t o apply t o systems with positional disorder, and specifically t o liquid metals.
The CPA is a self consistent single site approxima- tion [Z], and we briefly review it. An alloy consisting of components A and B is represented by a sum of potentials V,(r - R,) where V i is either V A or VB.
I n the CPA each potential is replaced by a coherent potential Vc(r - R,), which is the same for each site, but which is complex and energy dependent and so describes an average alloy. T o determine Vc self consistently we remove one coherent potential and replace it by a real atom, as indicated in figure la, and consider the scattering by the perturbation
Vi(r - R,) - Vc(r - R,). We then require that the total scattering from this perturbation, represented by the T-matrix, vanish, giving the well known condition
It has recently been recognized that this is not an altogether unique criterion, as the manner in which the total coherent potential is divided into contri- butions from each site is not unique. Much of the work has been done with a simple tight binding model [l, 21 in which the coherent potential is repre- sented simply by a complex frequency dependent but wave vector independent self-energy C(o).
The first attempt t o generalize this result t o the liquid metal case was by Faulkner [3] who considered
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974461
C4-3 18 L. M. ROTH the random liquid metal. The Hamiltonian for a liquid metal is
Faulkner considered the configurationally averaged one electron Green's function
where we adopt the notation of using script letters for a given configuration and Roman for averaged quantities. In a momentum representation
where Zk is the self-energy, here dependent on k.
Faulkner now sought to remove one atom from the medium and to consider the scattering from one real atom. However, here there is a difficulty, as Zk des- cribes a medium in which positional information about the atoms has been lost. He used as a pertur- bation V(r - R,) - Z/N, i. e. he assigned 1/N of the self-energy to each atom, where N is the number of atoms. He then applied the criterion of eq. (1) to determine Z. Faulkner's result agreed with the earlier
@re-CPA) perturbation expansion summation result of Klauder [4], and is given by
fjIc. 1. - Scattering problems in a) coherent potential approxi- mation and b) liquid metal problem.
where tkk is the scattering amplitude for scattering by an extra ionic potential V(r - R,) added to the medium described by Z. Gyorffy [5] then attempted to gene- ralize this result to the more realistic case of a liquid with ionic correlation, and Korringa and Mills [6]
put his work into a more useful form. However in this case the use of Zk/N to describe the removal of an ion from the medium turned out to be inadequate and as we shall see below, their result is unsatisfactory.
In the limit as N -t co this procedure does not really remove the scattering ion from the medium. This did not matter in the random liquid case because the medium is not affected by the location of a particular ion, but for the general case there is trouble.
Schwartz and Ehrenreich [7] attempted to improve upon this situation by assigning a part of Z to each scattering ion. They used a generalization of a result
derived by Lax [8] called the quasicrystalline approxi- mation (QCA), a version of which we shall discuss below. Here let me point out that Lax's work formed an important part of the multiple scattering structure upon which the CPA rests. The Schwartz-Ehrenreich result called the self consistent approximation (SCA) is rather complicated to derive and does not, as we shall see below, completely solve the correlation problem.
Now I should like to go into what really should be done here. We need a more detailed description of the medium than the average Green's function or self energy. That is, we need to include sufficient positional information that we can remove an ion from the medium and describe the medium consisting of the rest of the ions in some average way. Clearly this does not matter in the case of a random liquid but if there is correlation between ionic positions, we expect there to be a hole in the medium, as sketched in figure lb. We then put the ion back and consider the resulting scattering problem. Finally we require that on the average this system describe the original effective medium self-consistently. This procedure is clearly appropriate for a self-consistent single site approximation.
There are two versions of the theory with which I have been working and I shall only derive here the result for one of these, the simpler. Let me first outline the general multiple scattering theory for the Hamiltonian of ea. (2). Here we follow a well travelled A . , path and define the scattering or T-matrix by relation
where Go is the free electron Green's function.
% we have the well known T-matrix expansion to Foldy [9]
the (6) For due 6 = ti + x' ti Go tj + x' ti Go tj Go ti + ... (7)
i i j i j t
where the prime means that no two successive indices are equal, and where ti is the T-matrix for scattering of an otherwise free election by the ith ion. A more, compact statement of this expansion is obtained by writing
Zij, as indicated in figure 2a, is the sum of all scattering events beginning on ion i and ending on ion j. Now we can define a medium quantity [lo] based on Gij by writing
T(R, R') = (x pi(R) gij pj(Rf)) . (10)
ij
Here pi = 6(R - Ri) is the density of the ifth ion at R.
T(R, R) describes scatterings which begin at R and end at R' as indicated by figure 2b and is called the
C4-319
FIG. 2. - a) Qij or 'I;$, ; b) G(R, R') or T(R, R') ; c) Typical terms in expansion of gza or 5 2 2 ; d ) Typical terms in expansion
of gar or %zr
medium site-site scattering matrix [ l o ] . This quantity which is configurationally averaged but which contains positional information is just what we need to obtain the above mentioned more detailed description of the medium. The average T-matrix is given by
(11) and leads through eq. (3) to the average Green's function and self-energy.
Let me now discuss the other version of the theory, from which I shall actually derive the EMA. As I mentioned earlier, in the alloy problem a simple tight binding model has received a great deal of attention. Several authors [ll-131 have applied this model to the liquid metal problem, and it is expected to be useful for narrow bands such as occur in liquid transition metals and in dilute fluids such as super- critical fluid mercury. We assume that the energy spectrum is dominated by an atomic s-level on each ionic site and that the Green's function is given by
Then G i j is found to obey the equation
z ( wSir - X a ) Q l j = aij
1 (13)
where
and
( 1 5)
In the second equalities in eq. (14) and (15) we make the simplifying assumptions that the hopping integral depends only on the distance between ions, and that the orbitals are orthogonal. The latter approximation is readily relaxed [ l l ] . With these simplifications we can write for Sij
where Lo = ( w - H0)-l is a locator [14]. Eq. (16) is analogous to eq. (9), our compact version of the T-matrix expansion. Sij corresponds to Z i j , Lo to a one site T-matrix, and H(Rij) to Go. The analogy can be made more precise [lo, 1 1 1 , but eq. (16) is sufficient for our present purposes.
In the alloy problem in the tight binding model, Ducastelle [15] and Shiba [16] have shown that the CPA can be derived using a locator approach as well as by using the T-matrix formalism. This is fortunate because in the tight binding model of a liquid metal, an unperturbed Hamiltonian cannot be defined to use with the T-matrix approach, so that the locator version of the theory is appropriate. We shall develop the effective medium approximation in this model - the more general multiple scattering result is derived in a forthcoming article [ l o ] .
For the tight binding-locator version of our theory the medium quantity analagous to T(R, R') is
which has been termed the continuum Green's func- tion [l 11. We can iterate eq. (16) to obtain the analogue of the T-matrix expansion
Then the expansion for the continuum Green's function can be put in the form [ l l ]
G(R, R') = Lo ( z pi@) ) 6(R - R') +
i
+L% J ( pi(R) pj(Rrl) PLR') ) x
x H(R - R") H(Rt' - R') dR" + .-. . (19)
Here the positional information is all in the correlation functions. We have
where n is the density, and
where g(R - R') is the pair distribution function
C4-320 L. M. ROTH
giving the relative density of different atoms at R', of (( scatterers )) but neglect any effect on M. Then we given one at R. In the third term we must distinguish have
between the cases i = I and i f 1 :
G R 0 ( ~ , R') = nLg(R - R,) S(R - R') +
C3 = (
z'
pi@) ~j(R'0 pi@')) (+ M(R - R1') GRO(~", Rt) dR" ) . (28)
= n"(~ - R") S(R - R') + n3 g(R, R", R') (22) which involves the three body distribution function.
This quickly becomes complicated and we must resort to diagrams [12] or to approximate summation methods [l I].
Consider the approximation in which we include correlation only between ions having neighboring indices, i. e. in the last term above between i and j, j and I, but ignoring the correlation between ions i
and I. Eq. (22) is then approximated by
C3 E n3 g(R - R") g(R" - R') . (23) In this case we find that H(R) always appears with a factor g(R), i. e. in the form H ( R ) = g(R) H(R).
Eq. (19) becomes
= nLo ( 6(R - R') + 5 f i ( ~ - R") GQC*(R1', R') dR" )
or, Fourier transforming, noting that CQcA depends on (R - R'),
This is the locator analogue of the QCA [8, 111, which we call QCA'. It has a simple interpretation in that the energy in eq. (25-6) is an obvious generaliza- tion of the Bloch sum. G(R, R') has a local part nL,, which we can think of as due to scattering by a single site, and then a multiple scattering intersite contribu- tion, the second term of eq. (24). Having introduced the QCA' we now wish to improve upon it. Let us replace Lo by a medium locator L, and g(R) by an intersite interactor M(R), where L and Mare unknown, and M is regular at R = 0. Then
G(R, R') =
= nL S(R 1 - R') + 1 M(R - R") G(RU, R') dR" ) . (27)
We now wish to remove an ion at R = R,. Let G ~ O ( R , R') be the continuum Green's function for all other ions, given that there is one at R,. Let us assume that only the local part L is affected, and that L -+ Lg(R - R,). That is, we alter the distribution
We thus achieve our goal of removing a ((medium ion >> at R,, and in fact it is done in a very similar way to what is done the alloy case [16].
We now wish to consider the scattering problem.
To do this we generalize Shiba's [16] work on the alloy problem, and make use of an exact relation for the diagonal matrix element of the Green's function
Here Sf, is the Green's function for the system in which atom i is omitted. This is derived by considering scattering trips in which the electron goes out from site i and back any number of times, as indicated in figure 2c. This is the same equation obtained by Shiba [I61 for the alloy case, except that for the alloy case Lo depends on the site. Here we need also an expression for the off diagonal matrix elements gij which is readily obtained by considering one last scattering trip from i to j, as indicated in figure 2d.
Both situations are included in the exact expression
where G i i is given by eq. (29). To make a single site approximation we replace sLj by a medium quantity.
Actually we construct and make use of the continuum Green's function to obtain
G(R, R') =
= nGd 6(R [ - R') + i H(R - R") GR(R", R') dRY] (3 1)
These two equations together with eq. (27) and (28) are sufficient to determine L and M self-consistently.
First, let us compare eq. (27) and (31). Since L and Gd both appear in the singular part of G, they must be identical. Equating the second terms and writing GRO(R, R') = G1(R - R,, R' - R,), we Fourier transform to obtain
The second equality comes about because we assume
AN EFFECTIVE MEDIUM APPROXIMATION FOR THE ELECTRONIC STRUCTURE G1 to be symmetric. The Fourier transform of eq. (28)
can be written (L- - nMk) G:,, =
d,, + h(k - k') + I h(k - k") Mku G:,,, dk"/8 n3
I
where h(k) is the Fourier transform of h(R) = g(R) - 1.
The bracket on the left hand side is just Gkl. Multi- plying by Hk, integrating over k', and using eq. (33) we have
M, = ng, -t n S h(k - k") M$, Gkt, dk1'/8 n3 . (35) Eq. (32) and (33) also give
L-' = - I H, M, G, dk/8 n3 (36) which completes the derivation.
We can rewrite G, in the form
%
G , = n [ o - H o - n H , - Z k ] - ' (37) where Zk is thus a conection to the QCA' energy.
We can divide C, into a constant term 2, and a term C,, which vanishes for k -t co. Then
L = [ o - Ho - c,]-' (38) nMk = nHk + Clk. (39) The equations for Ern and Z,, can then be written, after some manipulation of the former, in the form
Z,, = n-I S h(k- k') (ng, + z , , ~ ) ~ Gw dkr/8 n3 .
(41) Before examining these results in detail, let me quote the analagous result for the multiple scattering version of the theory. The medium site-site scattering matrix T(R, R') obeys the self-consistent set of eq. [lo]
T(R, R') =
= ntc(R) 6(R { - R') + 1 G(R, R") T(R", R1) dR" ) (42)
t,(R) = t, + tR S Go g(R - R') nt,(R ') x
G(R', R") T(R" , R'") Z(R'", R) dR" dR"'
%
G(R, R1) = Go g(R - R') + h(R - R') 1 G(R, R'') x
x T(Rfl, R"') Z ( R , R') dR" dR"' . (44)
These are clearly more complicated than the tight binding results because the single site scattering matrix tc(R) and the intersite propagator E(R, R) as well as T(R, R'), are operators on the electron coordi- nates.
Let us now discuss the results. First it is quite easy to show that our multiple scattering result (eq. 6, 11, 42-4) reduces to the Faulkner-Klauder result for the case of a random liquid. The locator version of the theory gives the equally simple result for the random liquid case
2 = n 5 H: G, dk/8 n3 . (45)
Next we remark that the method must reduce to CPA for the alloy case, because we have used a gene- ralization of Shiba's locator version of CPA. In fact, Shiba's development is the alloy version of the theory.
The alloy case for the multiple scattering version of the theory is discussed in another article [lo], where it is shown that the CPA again appears.
Let us now compare this theory with other work, namely that of Gyorffy [ 5 ] , and Korringa and Mills [6], and that of Schwartz and Ehrenreich 17, 1 I], which results have been translated into the tight binding version of the theory ; and the more recent result of Ishida and Yonezawa [12], which was obtained for the tight binding model. All of these theories can be put in the form of eq. (31'1, with Z, given by different results, namely
GKM' : 2, = S [I + nh(k- k')] G,, Hir dk1/8 n3 (46) SCA' : 8, = n 5 fi, H , G , P(kf , k) dkJ/8 n3 (47) nF(k, k') = 1 + n h(k - k') +
The last result can be obtained from the present theory by neglecting Z,, in eq. (40).
An analysis of the merits of the various results can be made by examining the moments of the spectral function - (nn)-' Im G, about Ho. These correspond exactly to the coefficients in the expansion of G in eq. (19). Thus
W'O' = 1 (50)
These two moments are given correctly by all of the
C4-322 L. M. ROTH
theories considered. In the second moment, however, we find differences. We have
d2) = ( n a 2 + y; (52) where yk is a measure of the width of the quasi- particle peak in the spectral function. The various theories give
QCA' : y; = 0 (53)
GKM' : y: = n J (1 + nh(k- k')) HZ, dk1/8 7 c 3 (54) SCA' : y; = n (1 + nh(k- kt)) H,. gk. dk1/8 n3 (55)
EMA' : y; = n I (H,. +- nh(k- k') G,) I?,, dk1/8 7c3 .
(57) Exact :
yk 2 = n S Hkt gk, dkr/8 7c3 +
+ n2 1 ekR ( g(R, R', 0) -g(R -R1) g(R1) 1
x H(R - R') H(R') dR dR' . (58)
It is readily shown that if in the exact result we make the Kirkwood superposition approximation [17]
for the 3-particle distribution function :
g(R, R', R 3 g(R - R') g(R - R") g(R' - R") (59) then the result reduces to that for the effective medium approximation, eq. (57). Comparing this with the other CPA-type results, eq. (54-6), it appears that the present approximation gives the closest to the correct second moment.
If we examine the third moment of the spectral function, the EMA' begins to show deviations typical of CPA, such as incorrect treatment of repeated scatterings between two ions, from a superposition version of the exact result. Similar observations have been made for successive terms in the T-matrix expansion for the multiple scattering version [lo]
of the theory.
Another aspect of the present theory is worth noting. We see that in the second part of eq. (57)
gk appears, which means that H(R) appears with a pair distribution function which takes into account correlation between the ions. The first term in eq. (57) involves H2(R) g(R), which also includes the ionic correlation. However, in eq. (54) and (55) the bare Hk appears, in which ionic correlation is not properly taken into account. This can be particularly damaging in the case of a hard core liquid, as the result depends on what goes on within the hard core. This fact in the
SCA' in fact led me into difficulties [ I l l as for an unfortunate choice of interaction Y; changed sign.
In eq. (40) and (41) in fact we see that except for the first term in eq. (40), which we have dealt with, the interaction always appears in the form s. This is
the reason for the particular form in which we have exhibited Z, - it is not evident from eq. (36). If we were to use a diagram expansion [lo] we would find that this means that intersite correlation is taken into account along a chain of scatterings, for all terms included in the approximation. This chain- correlated property also holds for the QCA' and IY theories, and would seem to be a minimum requirement for a theory including ionic correlation. An analagous condition holds for the multiple scattering version of the theory, eq. (42-4).
Concerning numerical results, we note first that some results for a random liquid metal have already been published [ll]. For the more general case the numerical results which I have obtained are rather preliminary showing mainly that it does seem to be possible to solve the equations. In figure 3 is shown the QCA cr band energy )> nEk and the width function y, for a hard sphere model of a liquid metal. The pair distribution function was taken as that for the
-4
FIG. 3. - The QCA' {(Band E unction )> nHk and the width function yk for a hard sp uid metal with packing fraction q = .45 and unit hard sphere diameter. Interaction e-are-a I r - R I dr, with ~ 1 = - - 1 anda=4.
Also, Ho = 0.
exact solution of the Percus-Yevick equation for the hard sphere liquid, which was shown by Ashcroft and Leckner [18] to be a good representation for liquid alkalis. H(R) was taken to be a simple overlap between exponential wave functions e-aR. In figure 4 is shown the real and imaginary parts of Z, in the effective medium approximation. These results are compared with those obtained from the Ishida-
FIG. 4. - a) Real and b) imaginary part of Za for effective medium approximation (EMA') and Ishida-Yonezawa Theory
(IY) for liquid metal of figure 3.
Yonezawa theory [12], i. e. by setting I,, = 0 in eq. (40).
The density of states was also calculated using the result for orthogonal orbitals [l 11
N ( o ) = - ~ - ~ I m ( o - H , - I,)-' (60)
which comes about from the identification of L with G,, the diagonal part of the Green's function.
These results are shown in figure 5 and we see that, while the overall band width is quite similar to the Ishida-Yonezawa result, the detailed band shape differs at the upper end. The parameters used are for a somewhat more extended wave function than were used by Ishida and Yonezawa [12] to simulate a liquid transition metal.
FIG. 5. - Density of states for liquid metal of figure 3.
In conclusion, an effective medium theory has been developed to extend the coherent potential approxi- mation to the case of positional disorder such as is found in liquid metals. The theory appears to take good account of ionic correlation and, in the tight binding-locator version, to be reasonably tractable.
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