J. SLECHTA: Self-Consistent Continued-Fraction and Other Methods

phys. stat. sol. (b) 120, 329 (1983)

Subject classification: 2 and 13

Department of Physics, University of Leedsl)

329

Comparison between the SeIf-Consistent Continued-Fraction and Other Continued-Fraction Methods2) BY J. SLRCHTA

An alternative romparison of SCCP and other CP methods to that oue prtblished recerltly is given. It is shown that both, concrptnally and on numerical merits SCCF is the best method, in existence, of calculation of the density of states in disordered materials, in principle, as i t is the optimal, that is minimal yet fully tractfable, approximation of the exact problem formulated.

B 3 T O m CTaTLe OTHaHO aJLbTe1)HaTIlBHOe CpaBHelIEle MeToAa C06eCOrJIaCOBaHHbIX Henpe- PhIBHbIX npo6eZi (CCHa) M ApyrtIX MeTOAOB Hl?IIpe])bIBHEdX npo6eii ny6JIklKOBaHHbIX HeHaBHO. nOHa3aH0, 'IT0 Kali no KOHIleIILW~, TaK 110 WICJIeHIibIX HOCTMWeHHRX CCHD HBJIHeTCR CaMbln Hafi6OJIee Xa1lolllHfi MeTOU, I l O Cg4eCTBy, RJIR BbIqRCneHHR TIJIOTHOCTA

M I I H I l M ~ ~ L l I ~ H 110 B I lOJlbHe MCIlOJIb3YeMaR aIIpOHCUMa~IlH OpElrllHaJIbHOB TOqHOfi C O C T O R H ~ R B 6emopmorHHx ivmepanax, no ~ ~ E I I I U I I I I ~ , TaK Iialc OnmMamHaJx, T. 3.

aanasn.

1. Introduction Recently there was published a review paper by Wu et al. [l] in which there was presented a comparison of various methods of calculation of the density of states in disordered materials over a definite period of time. The focus there was upon the methods using the technique of continued fractions (CF), in one way or another. However, the methods included there were not exposed in a truely comparable and sufficiently coniprehensive way as not all of them were included up to the date of the issue of [l]. Actually sonie of them, like self-consistent continued fractions (SCCF), were exposed till much earlier date only. Especially [2] and [3], which were not in- cluded into [l] , had contained a major improvement of SCCF.

Though not much more radically new on calculation of the density of states has been published since [l] the points summarized briefly above make it necessary to publish more on that topic. This paper is not intended to be a new comprehensive review paper on the topic in question. I ts purpose is to put the interballance between SCCF and other CF methods into D, better proportion.

A similar comparison between SCCF and the methods based upon CPA had been given in [4]. It is to be shown here conclusively that SCCF is the best method of cal- culation of density of states in disordered materials, both conceptually and on numerical merits, in existence, as it is the optimal, that is minimal yet tractable, approximation of the core problem involved.

2. Brief Outline of the Methods Compared 2.2 SCCF

A t first we give a brief integrated expose of SCCF method which had been developed and presented, partially, in several works [2 to 101.

l) Leeds, Great Britain. 2, This paper was presented preliminary nt the 2nd EPS CMD conference in Mnnchester 1982

330 J. SLECHTA

The basic task of determination of the density of states of a solid, or a liquid, in

(1)

the tight binding approximation is to find roots of the secular equation

If we denote

where Hij is the Hamiltonian of the system and E is the energy, then it is equivalent t o finding poles of the Green function

(2)

I n solids we can write a = (R, x ) where R is the position vector of an atom and x denotes the set of the quantuni numbers of the atom. Then ( 2 ) has the form

H.;j - E6i.j = 0 .

K . . - H... - Ed.. 73 - u 7j 9

defined by the equation C Ki,jGjl = 6a . 1

C KabGbc = Sac . b

In the case of crystals there exists an elementary cell that is the minimal part of the solid in question from which the complete solid can be generated by its repetition. For that rea8on one can calculate the electron spectrum of the crystal by applying the Fourier transformation to (2 ) . The elementary cell forrns the kernel of that Fourier transformation.

In the case of the disordered niaterials i t i s not possible to apply the Fourier trans- formation for the lack of periodicity. Then the question is this: can one find a proce- dure analogous to that for a crystal described above? If yes what is a generalized concept of the elementary cell in the case of disordered niaterials and what replaces the Fourier transformation there ! If one finds an answer to these questions one can start to see what kind of concept replaces the concept of k-space in disordered materi- als. I t will be shown that in disordered materia.1~ the elementary cell is replaced by the concept of M-testing cluster and that the type of the averaging connected with it takes place instead of the Fourier transformation.

At first let us partition the material in question into the chosen cluster a and the rest of the material b. Because (2) is a system of linear algebraic eqnations we can always rename the positions of the atoms in such a way that its part corresponding to the cluster a is in the left-hand side corner of the matrix I( (the corner approxirna- tion of SCCF [9]). Then one can write

The partitioning procedure can be recurrently continued by dividing the cluster b into its subclusters c and d. So one can write

Kaa K a c Kac~ K = K,, ( 4 )

(Kd. i:: z)’ where d is the rest of the mat,erial minus the cluster a and c. Let iis apply the partitioning one more step further dividing the cluster d into clusters f and h. One can write then

Kaa K a c Knf Kah

Km Kw Kcf Kc11 (5) ( K f , K f , K f f Km)’

Klrn K I ~ Klit Khh where h is the rest of the material minus the clusters a , c, and f .

331 Self-Comistent Continued-Fraction and Other Continued-Fraction Methods

h -- - - - - - - - - - - - - I Fig. 1. DiviRion of the s e m h i n g into clusters in t,he corner

, - - C - - - - - - - - - - - - - - - d form of SCCF (see text) T--e 1 ; -,-------- 0 - ------ I l l I ! ! I ' q ' c ' f

I n the following text we shall make two assumptions, according to which the ma-

a) the cluster a neighbours the cluster c and the cluster c neighbours the cluster f ,

b) while the cluster a is of an arbitrary size the clusters c and f, etc., have their

The division of the semistring which models the alloy in question into clusters in

Then (4) and (5) have the forms

trix K can always be reorganized [6] :

etc.;

linear dimensions equal to the Penetration depth of the interatomic potential.

the way described above is given in Fig. 1.

We can partition the Green function G siniilarly to the matrix Zi and write

It is easy to calculate G,, that is the block of the Green function corresponding to a-cluster. For it holds

(8) I a n

~

Kaa - K a c Kc, C,, == Ia,(K,, - K a t ~ i ; i ~ b a ) - l I , ,

K c c - K,dKi:Kd, It is possible to see easily that the determinant of the denominator of the fraction (8) , which yields the poles of G,,, is equal to the determinant of equation (2). That nieans that the poles of G,, give the energy spectrum of the whole material.

TO this point the basic problem leading to (1) has been only reformulated and ex- pressed in the recursive form of a continued-fraction (8). That, however, has not rendered the problem of determination of the spectriim of the related system more tractable. To achieve that it is necessary to introduce additional ideas.

Let us start with the idea of M-testing cluster [Z, 61. This means to take a virtual copy of the a-cluster, a t first, which is denoted as M-testing cluster in analogy with the concept of a testing particle in the ordinary Green function theory [ll]. Here M is related to the size of the a-cluster. I n the one-dimensional case it is equal to the number of atoms in that cluster. Then the M-testing cluster is put into an arbitrary position replacing the local structure, there, of the size of the M-testing cluster. By that replacenient we get a sample of the material in question slightly changed rela- tively to the original one; a virtual sample. The spectrum of that virtual sample differs relatively to the correct spectrum of the material in question. However, we show now that for nearly any size of M-testing cluster the spectrum of the virtual sample represents the spectruni of the original sample nearly precisely. Namely when M-testing cluster is very small relatively to the size of the original sarnple then the spectrum of the virtual saniple is obviously nearly same as the spectrum of that

332 J. SLECHTA

original sample. Also when M-testing cluster is so large that it is nearly of the size of the original sample, the spectrum of the virtual sample represents the spectrum of that original sample nearly precisely. Generally the latter case holds for any M - testing cluster big enough so that its average properties are already representative for the properties of the original sample, as then substitution of it into any position of that original sample causes only small disturbances of its spectral properties. For example in the one-dimensional case and the nearest neighbour approximation the minimal size of M-testing cluster for which the statement above is already true is several tens of atoms. However, that number of atoms is still very small relatively to the number of the atoms in a sample of a macroscopic size and so for such a M - testing cluster there holds also simultaneously the consideration for the very small M-testing cluster, formulated above. I n the case of higher dimensions than one similar holds for M-testing clusters which have their linear dimensions larger than several tens of atoms [2]. From the reasoning above follows the statement preceding it. That statement is still improved when the average (. . .)b of G,, is taken all over the possihle positions of M-testing cluster in the original sample. Then we get the quan- t i t y

Here Zaa can be called, in a cdose analogy with the ordinary Green function theory, the self-energy of M-testing cluster imbedded into the rest of the material in question. I t s energy spectrum yields the density of states of the material nearly precisely, as discussed above. However, the straight determination of Za8 is not a smaller task than that set by the original formidation by (2) and it is necessary to find an approxi- mative way of doing so. For that purpose let us now repeat the procedure described above for the initial cluster a + c and M + 1-testing cluster associated with it [2]. The resulting quantity is

1

Siniilarly for the initial cluster a + c + f holds

etc.

sizes of the initial cluster, above there hold axact relations It is easy to see that among the different levels of the averaging, that is different

(12) < G a a ) f ( < G a a ) d c

Self-Consistent Continued-Fraction and Other Continued-Fraction Methods

Moreover there also holds “2, 61)

<Gaa)d = <Gaa)b + <GaR)b G!a<Qaa>, =

= <Gaa)b + <Gna)b TL(Qan)b , where

333

From (14) follows immediately

<T&)b = 0 (17) etc.

All the relations presented so far are exact and nearly precise. They involve these basic procedures :

a) development of the Green function of a very large system into the forin of a continued fraction (8) ; it is an exact relation.

b) The idea of M-testing cluster used within (9) to (11). Those equations hold nearly precisely for any size of that M-testing cluster with negligibly small difference from the precise results;

c) a generalization of the Dyson equation (14); d) a generalization of CPA idea within (14) to (17); it holds exactly there. For the features ad b) and ad d) the method had been called also CSEM [4, 6, 91

At thid stage, to make the scheme practically tractable, it is necessary to introduce or MCP, [ 6 ] , a t first.

an -Rpproximation. Nainely that ([el)

etc. which is the condition of self-consistency of the method. This feature together with the features a) to c) above gave the method its present calling SCCF [2, 101. If we take into account that relations (9) to (17) hold nearly exactly, to a negligibly small error, relation (18) is the only approximation within the whole SCCF scheme. Equation (17) is a nonlinear algehraic equation for X. Relation (18) is valid the more precisely the bigger the initial cluster is, as the more this takes place the smaller is the relative difference between the sizes of M-testing and M + 1 testing clueters and also the difference between ZM and Z M + ~ . In the limit when the size of M-testing cluster converges to the size of the whole material relation (18) holds exactly. The reasoning above contains the proof that with increasing M the density of states calculated according to SCCF converges towards the true density of states of the material in question and that, nearly always, each consecutive approximation yields improved result relatively to that of the preceding one, in that direction. However, it is difficult to estimate the speed of convergency within the whole range of E other than nume- rically and such studies were presented in [Z , 8, lo]. There was shown that the speed of convergency of SCCF is really high even in the one-dimensional case where the conver- gency is ii~ually the slowest 1121.

334

K =

J. SLECKTA

Kcc Kco 0 1 Koc K o o Kob

K b o Kbb Kbo'

K,yb KO*,* KO,,,

<0 Kco' Kcc, ,

and

(23)

(24)

335 Self-Consistent Continued-Fraction and Other Continued-Fraction Methods

where P represents the kind of matrices involved in his method, is structurally equi- valent to (6) to (8), that is before the idea of M-testing cluster is applied. Also its application to a linear string, namely that

where Gc is the Green function of the central cluster and the parameters involved are defined by the expression for the related Hainiltonian of the form

and z is coniplex variable, does not go beyond those equations either. It is a mere application of (24) to that case.

I t is shown by the author of ([13]) himself that SCBS is, in one-dimensional case, equivalent to CPA. For that reason there holds for it the same comment as for CCPS above.

2.3 Wu at al. [ I , 151

This method develops the element of the Green function G,, into a continued-fraction form by applying, recursively, a perturbation method ([El). I n the latter work there was considered a system within the nearest neighbour approximation defined by the equation

c t7;;dlnf = 0 ,

GZAr

(27)

(28)

12.'

where

Here u,. are amplitudes of the eigenstate with its eigenvalue En$. There was found ([ 151) that

(29)

( E - Van*) 6,,* - Vnn+l6n+ln' I Vna-16n-lnt *

Considering the relation between the electronic and vibrational states of disordered solids presented in [2] we shall now follow the application of the general theory above to vibrations of a linear disordered chain modelled by the equation [15]

--mzW2% = --2yuz -I- yuz+1+ yl'z-1 >

where mz is the mass of the atom at the position 1 and y the elastic constant in har- monic approximation. Then after introducing the auxiliary quantities

336

we can write

1

A N - 1 - I

A N - X N and

1 1

which are algebraic equations of the second order in XN and yl. It is easy to see that (33) and (34) are structurally similar to (8) before the idea of M-testing cluster is applied. Thus those principle core equations (33) and (34) of this method are not more, in principle, than the opening of SCCF even when developed substantially “deeper” along the “staircase” than in [13]. Moreover the cyclic conditions imposed upon (33) and (34) lead to polynomial equations of second order only, whereas SCCF gives, for one-dimensional alloys, a polynomial equation of fifth order. That yields the richer structure of SCCF than that of the method by Wu et al. fl, 151 as a higher-order equa- tion is bound to have, in principle, except in special cases, a richer interdependence of its more plentiful coefficients and roots than a lower-order one. Actually the cyclic conditions are generalized Born-Karman conditions which are rather valid for crystals.

A s for more recent developments of the method by Wu et al. [l] it can be reasoned by a similar way that it is not more, in principle, than the opening of the central formulation of SCCF till equation (22).

2.4 Kuplun und Gray [16,17]

Even when in that work there is treated the off-diagonal disorder, contrary to most methods other than SCCF, the plain “staircase”

G, 1 (35)

z - ... where an, Pn are defined by

Gn = ( X -an) G ~ + I - BnGn+z (36)

is again not more than the initial part of SCCF up to equation (8). Though it is devel- oped up to the level 10 down that staircase it does not give better results than SCCF for M = 0 [ 2 , 71. The merits of comparison are given below and in [7]. They hold also for the recent work by Gray and Kaplan [18].

337 Self-Consistent) Continued-Fraction and Other Continued-Fraction Methods

3. Further Compwison of Those Exposed CP Methods and SCCF

3.2 Advantages of SCCF

1) Though it has a realistic crystalline limit it does not need any information about the crystalline density of states of any reference lattice, as most the other methods do. It requires information about the spectrum of isolated atoms in vacuIIIn, of which the given alloy consists, their interactions and the spatial structure of that alloy, like whether it is amorphous, or purely random binary alloy or with statistical short-range order, etc.. only.

2) With variations of the concentration parameter 0 5 x =( 1 and the affinity parameter E of the component A in A-B binary alloy the density of states e ( E ) reflects all the natural features of it for all values of x, E [2? 81. Namely:

a) The main bands are centred around the levels of atoms A and B related and have the correct width according to the strength of A-A and B-B interactions.

b) With changing values of x, E the shape of e ( E ) undergoes changes expected froin the physical reasoning, as discussed in details in [2], more realistically than all the other methods. For example it gives, a8 the only method, the proper kind of the van Hove singularities for one-dimensional solids for x + 0 and x --t 1, a t the edges of the main bands [2, 81.

c) For 2 = 0.5 (50% concentration) and those parameters selected, like VAA = VBB, e ( E ) is, within the tolerance of the rounding-off errors (numerical noise), symmetrical around h’ = 3.5 as it should be. This aspect is violated by most other methods (see below and in [2, 81).

d) The side bands have a rich enough structure to calculate most properties of materials of interest from e ( E ) like conductivity in the Hartree approximation de- coupling scheme for the two-particle Green function involved [3] even when, as hinted to in [lo], SCCF is capable of treating that two-particle Green function straight. Because SCCF has many useful scaling properties in the parameters involved which characterize the material in question, there can be tabulated a “prototype” e ( E ) for parameters of choice which is valid for a whole group of materials and then scaled for each member of the group accordingly as required. As shown and reasoned in [2] and this paper SCCF p(E) resembles well, if not best, the true density of states in binary alloys.

3.2 Disadvantages of the other methods

Butler [13] : As discussed in the previous section it caters for side bands only though in that energy region i t is, for the same size of the initial cluster, one or two steps richer than SCCF as the extra self-consistency built into SCCF has an overdamping effect of a numerical character (noise) onto the sharp line-like structure of the side bands.

Wuet al. [l]: It does not give naturally physical results comparable to the SCCF ones as it is much simpIer in its structure. The criteria of comparison are given in [2] and throughout this paper. Here we still add that none of the results published from their method show the true physical features of the binary alloys with statistical order as discussed above.

Kaplan and Gray [IS, 171: It gives consistently less rich and realistic p(E) to both Dean’s reference [19] and SCCF results, though more realistic ones than those by Wu et al. [I]. 22 physics (b) 120/1

338 J. SLECHTA 3.3 Further commeiits

a) Another different scheme which leads to a sort of continued fractions is that by Moorkerjee [20]. Here the structure of the continued fractions i s not more sophisti- cated than that of [13]. However, more comprehensive coinparison of that method with those included is not given here as the concept of that method is rather different to those derived from a sort of CPA scheme, SCCF inclusive. Moreover not nluch numerical results comparable to those lesults for one-dimensional systellls dealt with in this paper and also in that by Wu et al. ([l]) have been publishecl.

b) One more difference between SCCF and other C P methods presented in this paper is that its mathematical core, namely (15) there, consists of subtraction of two continued fractions, rather than one only. They are truncated a t different depths. I t is interesting to notice that that form resembles a basic formula from the theory of in- formation [21] namely that

Ipxssrd Ibackground - Imessage , (37) where Ipaxsec, is the amount of information passed during the transmission of a message to a receiver, is the measure of the size of the informational background of the receiver in the direction of the message and ImeHsagc defines the informational size of the message. Comparing (15), or similar forniulae in [ 2 ] and [lo], and (37) one can conclude that that M + 1-testing cluster defines the background information and M-testing cluster provides the message. It has been shown in [lo] that in most physical situations i t is sufficient to develop SCCF up to M = 3 only. However, if we develop its structure for larger M than 3, then the structure of the SCCF self- energy for the levels below that corresponding to M = 3 down the staircase ([lo]) can serve for structural modelling of the “hidden” parameters.

4. Conclusioii

As exposed above all those CP methods other than SCCF correspond to not more than the structure of SCCF up to equation (8) before the idea of M-testing cluster is applied. They are basically only different ways of how to reduce the eigenvalue of a large system into the form of truncated continued fractions. They all include only very rudimental form of averaging fo,r less comprehensive than that of SCCF [2].

In comparison with all the works reviewed in [l], and in this paper, SCCF comes out to be the best method, in principle. I t is least approximative, actually minimally possible, yet incorporating all the features of the complete initial theoretical formula- tion of the problem, as discussed in Section 2, and fully tractable with a good con-

vergency. Though all the studies of the properties of SCCF have been performed for rather

siniple model systems [2, 31, so far, there are no principle restrictions, as the type of disorder, the type of atoms, the strength of interactions, etc., as stiessed above, which would limit the usage of SCCF to any type of real material.

All the rest of the methods reviewed here do not have all those properties. Their performance, like convergency of the density of states, they yield, to the true density of states of the materials in question, is difficult to prove within themselves. I t must be carried out by their conlparison with works like that by Dean [19]. However, though his work has been widely taken for an absolute standard so far, this is not justi- fied in principle as Dean’s work is mostly computational method [ l , 151. In general all the Green function methods as a group, SCCF inclusive, are theoretically superior to mere number given schemes, like that by Alben et al. [22], as stressed in [I].

Self -Consistent Continued-Fraction and Other Continued-Fraction Methods 339

Acknowledgements

The author of this work would like to thank the Institute of Physics in London for their support of himself both morally and financially and the Brotherton library of the university of Leeds, namely Mr. D. Cox, its librarian, for providing him with a privilege access to its possessions.

References [I] S. W. Wu, J. P. BOWEN, and K. S. DY, CRC Crit. Rev. Solid State Sci. 10,43 (1980). [21 J. SLECHTA, J. Phys. C 10, 2047 (1977). [3] J. SLECHTA, J. Phys. C 12, 1819 (1979). 141 J. ~LECHTA, J. Phys. P 6, 2081 (1976). [5] J. SLECHTA and R. V. ALDRIDGE, J. Phys. F 2, L132 (1972). [6] J. ~LECHTA, J. Phys. F 4, 1148 (1974). [7J J. SLECHTA, J. Phys. F 5, 1285 (1975). [S] J. SLECHTA, phys. stat. sol. (b) 71, K165 (1975). [9] J. SLECIITA, J. Phys. F 6, L29 (1976).

[lo] J. SLECHTA, phys. stat. sol. (b) 104, K143 (1951). [Ill V. 2. BONCH-BRWEVICH and S. V. TYABLIKOV, Green Function Methods in Statistical Physics,

[12] E. H. LIEB, Mathematical Physics in one Dimension, Academic Press, Inc., New York 1966. [13] W. BUTLER, Phys. Rev. 8, 4499 (1971). [14] M. TSUKADA, Phys. SOC. Japan 32, 1475 (1972). [15] Y. S. Wu, C. C. TUNG, and nip. J. SWARTZY, J. Math. Phys. (New York) 16, 938 (1974). [16] T. KAPLAN and L. J. GRAY, Phys. Rev. 14, 3462 (1976). [17] T. KAPLAN and 1,. J. GRAY, Phys. Rev. B 15, 3260 (1977). [18] L. J. GRAY and T. KAPLAN, Phys. Rev. B 24, 1862 (1981). [19] P. DEAN, Rev. mod. Phys. 44, 127 (1972). [20] A. MOORKERJEE, J. Phys. C 6, 1340 (1973). [all W. R. ASHBY, An Introduction to Cybernetics, Chapman & Hall, London 1956. [22] R. ALBEN, M. BLUME, H. KRAKUS, and L. SCHWARTS, Phys. Rev. B 12,490 (1975).

Fizmatgiz, Moscow 1961.

(Received July 22, 1953)

22*