lc/81/16

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

DIELECTRIC BAUD STRUCTURE OF CRYSTALS:

GENERAL PROPERTIES, AND CALCULATIONS FOR SILICON

INTERNATIONALATOMIC ENERGY

AGENCY

R. Car

E. Tosatti

S. Baroni

and

S. Leelaprute

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1981 MIRAMARE-TRIESTE

IC/81/16

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

IBTERNATIOHAL CENTRE FOR THEORETICAL PHYSICS

DIELECTRIC BAND STC""1UBE OF CRYSTALS:

GENERAL PROPERTIES, AWD CALCULATIOHS FOR SILICON*

R. Car**

Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italyand

Laboratoire de Physique Experimentale, Ecole Polytechnique Federale,Lausanne, Switzerland***

E. Tosatti

Gruppo Nazionale di Stnittura della Materia, Istituto di Fisica Teorica,Universita di Trieste, Italy;

Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italyand

International Centre for Theoretical Physics, Trieste, Italy

S. Baroni

Laboratoire de Physique Theorique, Ecole Polytechnique Tederale,Lausanne, Switzerland

and

S. Leelaprute

International Centre for Theoretical Physics, Trieste, Italyand

Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy

MIRAMARE - TRIESTE

February 1981

* To be submitted for publication.

•* Address during 1981: IBM Research Laboratories, Yorfctovn Heights,Hev York 10598, USA.

••• Permanent address.

SUMMARY

We develop the dielectric band structure method, originally proposed

by Baldereschi and. Tosatti for the description of microscopic electronic

screening in crystals. Some general properties are examined first, including

the requirements of causality and stability. The specific test case of

silicon is then considered- Dielectric bands are calculated, according to

several different prescriptions for the construction of the dielectric matrix.

It is shown that the results allow a very direct appraisal of the screening

properties of the system, as well as of the quality of the dielectric model

adopted. The electronic charge displacement induced by V^ and X 3

phonon-lilce displacements of the atoms is also calculated and compared vith

the results of existent full self-consistent calculations. Conclusions are

drawn on the relative accuracies of the dielectric band structures.

-1-

I. INTRODUCTION

The study of electronic dielectric properties of crystals including

local fields implies the use of dielectric matrices (DM)}-y

}-*2' The calculationof these matrices start ing from rea l i s t ic band structures has proven to be

extremely elaborate, even i f a great simplieation can be introduced by using

e .g . , the mean-value point technique and i t s extensions , or other techniques

Such accurate DM calculations are available for fev materials only .

Furthermore, most numerical results are only available in the q-» 0 l imit .

Various models have been prop rea to simplify the task of calculating

the full DM at a l l q. - vectors. These models are generally based on sim-

plif ied electronic band structures and/or on other more or less

reasonable assumptions on the structure of the MT . The relalat ive

accuracy of the different models has not yet been established. On the other

hand, the development of reliable simple s ta t ic DM models i s absolutely

v i ta l to future important applications such as phonon calculations,

screening of defects, local field calculations in real space, e tc . If this

i s true for those fev materials vhere a great deal i s already known at

4-* 0, the need i s even more evident for the infini te variety of other crystals

vhere one hardly knows anything else than the s ta t ic 5 = 0 dielectric

constant.

Balderesehi and Tosatti have recently proposed a "dielectric band

structure" (DBS) scheme , whereby the s ta t ic (u = 0) DM i s diagonal!zed,

yielding screening eigenvalues and eigenvectors for each q in the Brillouin

Zone (BZ). They have shown that in this way symmetry emerges naturally in thescreening problem, and that the information contents of the fun DM, a verylarge matrix, i s cast into very few eigenvalues and eigenvectors for eachsymmetry.

In this paper, ve a) clarify further some general properties of thedielectric band structure approach, b) present explicit calculations for theDBS of Si throughout the Brillouin Zone and c) compare qualities anddefects of different available DM models, by direct calculation of the i rDBS, as vei l as of the screening charge induced by some high-symmetryphonon-llke displacements of the atoms.

The main point of general nature which apparently has not received

proper attention thus far concerns the restr ict ions which causality on one

hand, and thermodynamlc s tab i l i ty on the other hand, impose on any given DM.

-2-

We show that the dielectric band structure formulation is a very natural one

for stating clearly these requirements, thus extending all the very well-known

properties of diagonal response functions c(q , a) to the more complicated

non-local response matrix c(q + 3, q. *• 2' , 14) typical of all inhomogeneous

systems. This line is developed in Sect. II.

Calculating the dielectric band structure out of a given DM model -

there is a variety of such models on the market - turns out to be a rather

stringent and instructive test. In Sect. Ill we describe some of these

models, including one very newly developed, based on the density functional

formulation. A brief derivation of this model is sketched in the Appendix

Al. The dielectric band structures for all these models are then presented

and discussed in Sect. IV.

A certain class of models, namely those based on the"factorization

sosatz", which has been proposed for extreme tight-binding situations, turns

out to have surprisingly fev dielectric bands. As is shown in Appendix A2,

this is a general consequence of the factor!zable form of these dielectric

matrices. The obvious consequence is that only problems which involve the

limited set of symmetries represented by these fev bands can be handled

with such matrices.

As a realistic application, which allows a further comparison between

various screening models, ve present in Sect. V a calculation of the electron

charge density induced by a phonon-like displacement of the Si atoms. This

calculation is particularly informative, since the same quantity has also

been computed by an accurate self-consistent pseudopotential method by

Baldereschi and Haschke . The general conclusions that this type of com-

parison allows are very close to those drawn Tjy just looking at the DBS

of the different models. This lends further support to the validity of the

dielectric band structure method.

II. GEHERAL PROPERTIES

The exact Inverse longitudinal DM of a general many-body systewith crystalline translational properties i s given by

- 3 -

'

with

r

where v(Q) = !«e2 / Q2£l, X runs over a l l electrons, |0>, Bo andfn>, En denote exact ground state and excited states eigenvectors andenergies, a denotes a reciprocal l a t t i ce vector and q is the wavevectorinside the BZ. The existence of nonzero off-diagonal elements,

G ^ G' is connected with spatial inhomogeneity of the electrons, caused bythe periodic l a t t i ce potential .

At iii = 0, the matrix x(q + &\ q + 0*') becomes hermitian, but theDM e(q + G, q + 3') does not, due to the presence of the prefactor

v(q + G). Yet, the DM can be diagonalized, with real eigenvalues. Thisi s seen by f i r s t defining a hermitian %~l in the form

(2.2)

Once th is E" is diagonalized with eigenvalues t 1 ( q ) , and Bloch-likeeigenvectors

Gri t is easily verified that

and

(2.1.)

(2.5)

G. - ' " G

- I t -

are eigenvalues and eigenvectors of tne DM itself. This ia the dielectric22)band structure {DIS) scheme

22)Some properties of the DBS of crystals have already been cUscuased

In this section we extend this discussion, mainly to include causalityproperties.

In the homogeneous case, i t i s well-known that 1/E(q , w) i s a

causal response function for a l l values of q, while the same is not true of

e(q. , in) i tself2 > 2 5 ' . The reason is that while l/e(q, , in) describes

the response to an external field, s(q , tu) is instead the response to the

internal f ield, which cannot be varied at wi l l , but depends Itself on the

response of the medium. Kirahnits has shown that only in the macroscopic

l imit , q •# 0 j the dielectric function e(q , u) becomes i t se l f a causal

response function. Thus, the Kramers-Kronig relat ions, which hold for

i~ (q , ID) at a l l wavelengths, become valid also for s(q , m) only in the

q-> 0 limitj

• (2.6b)

Introducing the important fact that a system at thermodynamic

equilibrium can only absorb energy at any given positive frequency, which

implies Im e (q , a) < 0, Im E(0 , o>) > 0, and specializing (2.6) for

ID = 0, one gets that for a stable system

(2.T)

(2.6)

most also hold. Therefore, a system i s unstable if either t (q , 0) i 1

0 for i£-» 0. However, at f ini te q, only c~l(i , 0) ) 1

for a l l q, while for q-> 0

E" (q , 0)

involves instability, while , 0) t 0 is perfectly allowed

Systems with negative ionic dielectric function e(q , 0) can be found

experimentally

g26} Mo system with negative electronic e(q , 0) appears

-5-

to have been explicitly discussed^but some must surely exist . Good candidatesshould be systems at temperatures Just above electronic superlattice phasetransi t ions, l ike for example charge-density waves or Wigner transitiona. Infact, a transition of th is type, if continuous, must be signalled by£ (q , 0) •• ^ —«^as discussed la ter in this section. Just above Tc,E (q , 0) should rise continuously from - « towards zero, to becomeeventually positive only vhen, sufficiently away from Te, the c r i t i ca lfluctuations are dying out.

We note, in passing, that the well-known one-electron, or RPA f27)approximation for the electronic response of a oryatal alwaya yields a

causal e(q , ») rather than a causal e~ (q , a). Specifically

,

(2.9)

which ia larger than one for all q, since by definition conduction states

E lie above valence states E . Therefore, the occurrence of a negative

electronic e(q , 0), which is in general possible, must be necessarily due

to so-called exchange and correlation effects, not contained in the RFA.

The natural question at this point is: how do these properties

change in passing from homogeneous systems, characterized by a dielectric

function e~ (q , u), to an inhomogeneous system, particularly a crystal,

characterized by a DM e (q + 2, q + 3', »)? As it turns out, the DBS

scheme provides a very natural way to answer this question. The equivalent

of (2.7) is that all dielectric eigenvalues must be smaller than unity,

(2.10}

The proof is as follows. The dielectric eigenvalues t~ (q) can be

seen as expectation values of an operator e" , between eigenpotentials

«sw,r.-**';>- 6 - (2.11)

ffo, '1 J

where use has been made of (2.3). The second terra in brackets can also be

written as

ft TO

and ia always positive, since E n > Bo* Hence (2.10) iB proved.

Having found the extension of (2.7) in (2.10), we can now look for

the extension of (2.8) to the crystalline case. The macroscopic dielectric2)

function is well-known to be

(2.13)

.25)whence, by repeating Kirzhnits1 arguments " , (2.8) is here replaced by

- I ^ ^C2.ll*)

This is not an obvious condition to be verified on any single eigenvalue oflthe DM, for q-»0. This is because in the crystal, e

glq-"r is not an

eigenpotential, but can only be expressed aa a sum of eigenpotentials

£ rZ ( r ) • (2.15)

Then, the inequality (2.11*), which is the crystal generalization of (2.8), 1B

only a weak condition on the eigenvalues

[2 >(2.16)

- 7 -

t*

The two inequalities (2.10) and (2.16) must be satisfied toy any DBSfor a 4 0, while for a general q in the BZ only (2.10) holds.

Again, in passing, we note that in the EPA the exact (2.10) is replaced by

which coincides with (2.10) only so long as e~ (q) * 0 .

We can now extend the condition for thennodynamic ins tabi l i ty

discussed previously to the inhomogeneous crystalline case. By virtue of

(2.10), any value of E (5) be vtt.i 1 and -« is compatible with

s tab i l i ty . At <j-*0, (2.16) must also be obeyed. A negative dielectr ic

eigenvalue simply means that a particular perturbation equal to the corres-

ponding eigenpotential is overscreened. Overscreening at some special s i tes

is possible even in the RPA. For instance, in the case of Si , an external

uniform field (which i s , of course, not an eigenpotential) i s oversereened

in a small volume centred at the bond s i tes

A crystal ia unstable when any of the E^Hq) i s > 1. I t i s clear

that i f a system ia to approach ins tabi l i ty in a continuous fashion taiB can

only occur with at least one of the eigenvalues crossing over to - »:

__ oo (£.17)

Inspection of (2.11) shows that this will occur when at least one excited

state of the crystal becomes degenerate vith the ground state

(2.16)

A well-known example of this behaviour is the occurrence of soft lattice29)

oe<tes in ferroelectric and antiferroeleetric phase transitions . Its

electronic equivalent, which is so far hypothetical however, is a soft

plasmon Or exciton mode in charge-density wave transitions. The

obvious implication of the equivalence between (2.17) and (2.18) is that

the order parameter for a continuous electronic phase transition in a

crystal mast be a dielectric eigenstate.

Armed vith this summary of the requisites that any DM must satisfy

if it is to describe a causal and stable crystal, we can now move on to

consider specific DM models.

- o -

I I I . MODELS FOE STATIC DIELECTRIC SCREENING IN S i

We give in this section a brief description or the various DM

models whose DBS's will be discussed in th is paper. Among several

poss ib i l i t i es , we have chosen as i l lus t ra t ive examples the models of2 1 'Sinha1 ' , of Johnson20' of Car and Selloni2 and a newly developed one

32)based ou the local-density functional formulation

(i! Sinha ModelWe consider only a particular form of the Sinha model which gives

rise to the so-called generalized shell model , when applied to latticedynamics calculations . In this model the DM is assumed of the followingfactorizable form:

13.1)

where eo(q + 5) Is an ad hoc function introduced in order to adjust the33)diagonal part of the DM (3.1) to the calculated values of Walter and Cohea

the sum Is over the s i tes in the unit c e l l , a i s an adjustable parameter,

f (q + 5) are spherical font factors defined in refs. 18 and 19 > and

v ' (Q) i s an effective electron-electron Interaction given by

(3.2)

where v(3) i s the Coulomb potential and f ($) allows approximately forexchange and correlation effects . In our application we use eg.. (3.1)with values of the parameter as appropriate for Si, given la ref. 18,19.

I t should be noted that the form (3. l) ia s t r i c t ly Juatifiad onlyin the extreme tight-Mnding limit with flat bands 3 6>3T'. Clearly a broad-band covalent crystal l ike Si i s very far from th is l imit . BeverthelesBaccording to Sinha et a l 1 °'^' this model works fairly well, at least in l a t t i cedynamics. A feature that makes th is model at tract ive in spite of i t s

crudeness i s obviously i t s factorizable form, that allows the DM to be

inverted analytically to obtain c~X(q + 5", q + S*'), the quantity of physicalimportance.

(ii) Johnson Model

This model was constructed start ing from a very different approach,i . e . from an off-diagonal generalization of the popular diagonal Perm model3

where an effectively one-dimensional gap i s superposed to an otherwise freeelectron model. In this sense Johnson's model is much more free-electron-like than Sinha's. Johnson's DM has the form:

where i s

n33\in the HPA.%,o) » as competed, e-6-

is the valence charge density that can be taken39)

eEPAby Walter and Cohen

either from experiment or from empirical pseudopotential band calculations

and K is a cutoff, fixed by Johnson as

= o i C3.ii)

)where k , is the Fermi momentum and m = (kme /m) , This cutoff, whichis introduced rather a rb i t ra r i ly , has the effect of seriously overestimatingsome off-diagonal matrix elements. Tor example for small q andJG - G'| > 2ky the cutoff may give r ise to important non-zero off-diagonalDM elements even for very large | * | and I <*'|, which i s unphyaical since

e(q + G, q + 5') should go to zero at least of 0(G~ ) for large Z and

G' = G+ const. Johnson's model was originally devised for diamond, giving

quite good results for the DM elements in the limit q-> 0, when compared

with the accurate calculations of Van Vechten and Martin . I t is also

rather simple and parameter-free, both the charge density f[G) and e

being normally known. We did however, get very poor results when we t r ied

to apply this model to other semiconductors like Si or Ge1(0)

(iii) Car-Selloni Model

A modification of Johnson's model was introduced by Car and Selloni,21)who applied i t to screening of point-charge impurities in Si

model the DM takes the form:

In their

(3.5)

Here the function f(q + S) i s f i t ted to the diagonal dielectr ic function

as calculated by Walter and Cohen , and the constants A and B are

adjusted to reproduce the off-diagonal EM elements as calculated in the

limit q-»0 by Baldereschi and Tosat t i1 1 .

The dielectric model (3.5) does not have the unphysical cutoff K

of the Johnson model. I t s accuracy has been found to be nearly the same in^0)various sejniconductors . However, unlike Johnson's model, this formulation

depends upon the availabil i ty of accurate calculations of at least some off-

diagonal DM elements (at q = 0, or elsewhere) so that the parameters A

and B can be adjusted to f i t them.

(iv) Local Penalty Model

A very simple dielectric matrix without adjustable parameters can be

obtained from the local density approximation of the density functional

theory. The form we use here is

(3-6)

where the matrix R i s given by:

(3.7)

-10- -11-

Here c 0 = • (3ir )3 , ^ = - , P i s the valence charge ueasity, and

[f(r)]* denotes the 0 Fourier transform of the function f(r). The most

important steps in the derivation of this model are sketched in Appendix Al.32)

For more details ve refer to a future publication

Due to the Thomas-Fermi form of the approximate functional (Al.l),

the DM (3.6) has an undesired "metallic" behaviour in the limit q-> 0,

that is, a 1/q divergence in e(q , q) and e(q_ , q, + G) . We

expect however that, except for these elements at q-»-0, and generally for

q. / 0, this model should constitute quite a reasonable approximation for a

semiconductor DM. This can already be seen in the homogeneous approximation

where the frey electron Iindhard or Thomas-Fermi dielectric function approaches

rapidly the Penn dielectric function of a semiconductor of same average

electron-density, as soon as |i( is larger than a fraction of kpp.

IV. DIELECTRIC BAND STRUCTURES FOE SILICON

Ve present in this section the results of the diagonalization of the

different model DM's discussed in the previous section. To provide a fixed22)

reference point, we first reproduce from the work of Baldereschi and Tosatti ,

the "ewpty lattice" DBS obtained by siaple folding of the inverse diagonal

dielectric function E (i) without any off-diagonal elements. This is

shown in Fig.l. Also shown there is the result of an accurate diagonalization22)

of the full DM calculated in the RPA and in the 4 -» 0 limit only .

The shifts and splittings of these q = 0 dielectric eigenvalues with respect

to the empty lattice DBS are indicative of the effect of crystal inhomogeneity

on screening, in the same way that the shifts and splittings in the electronic

band structure with respect to the empty lattice energy bands are indicative

of the effect of the crystal potential.

It will soon become apparent that shifts and splittings in the DBS,

though quite noticeable, are not such as to make it look totally unlike the

empty lattice dielectric band structure of Fig.l. This explains why for

many non-critical applications the usual diagonal e(q) is sufficient.

The symmetries of the most screened eigenpotentials (i.e., those

corresponding to the lowest cy (q)) reflect indirectly important

characteristics of the underlying electronic structure, that is of the

chemical bond in the crystal, as explained in ref. 22. In Si and at i» 0

the most screened symmetries are, in order of increasing magnitude of E V ,

YYV ' ri5 ' r25'' ri2/' V ThB 1 u a l l t a t i v e reBaon

perturbation is screened best,is Because it excites transitions from bonding

(F ') to antibonding (T^.)' states most efficiently.-

Another general fact worth noticing is that the (0, 0, 0) plane wave

gives rise to a r eigenpotential which is the lowest of all, and is also

uncoupled to all the higher r1'

s" ** i s i n a t e a d coupled to all higher T^

eigenvalues, by the so-called "wings" of the DM, f ^ q , q + 3 ) . For q-» 0,

these elements take the non analytic form _£_ • C (S). The coupling 1^ - V^

\his at the origin of the "longitudinal transverse spl i t t ing" of a l l r ^eigenvalues (r -> + A ), for q*0 . Physically,what happens is that alongitudinal perturbation, even i f cell-periodic, gives rise at long wave-length to a macroscopic field paral lel to q. This f ield, in turn, causes aresponse which i s cell-periodic, and acts to reinforce the original perturbation,thus yielding a to ta l reduction of screening power, i . e . , an increase ofE"1 ( 0 ^ 0 ) . The same does not happen for a transverse perturbation, whence

41

the L-T splitting, which is a completely analogous in nature to that well-

known for phonons.We can now proceed to illustrate the DBS obtained with each of the

models chosen.

(i) Slnha Model

The DBS obtained by diagonalizing (3.1) i s shown in Fig.2. The

dielectric bands are very few and very f l a t . Remarkably, only perturbations

of symmetries r ^ - and V^ ( - ^ + i ? ) at q, • 0 are screened at a l l ,

while any other perturbation is left unscreened, including T^, which i s

known22' to be in reali ty si l icon'3 best screened. Hawover, there are no

other q -»0 phonons in Si than r . q and r 'which shows that the modell8 19)

is practically custom-made for use in la t t ice dynamics ' . A generalconnection between DM factorizability and selective screening of a fewsymmetries only is elucidated in Appendix A2.

The flatness of the dielectric bands of Fig.2 indicates that Sinha'smodel leads to expect exceedingly large local field effects, with strongdeviations from purely diagonal screening. Comparison with either the

-12--13-

accurate q. => 0 eigenvalues of Fig.1,or v M k

in this section indicates that local field effects are indeed overestimated

in this model. This la not surprising, silicon being very far from a tight-

"binding two-band insulator, which is the ideal case where Sinha's model

applies exactly.

(ii) Johnson Model

Not being factorizable, Johnson's DM screens strongly all symmetries,

and not just some of them. However, a glance at the DBS of Fig.3, is

sufficient to conclude that this is not a very good model either. A large

number of bands are strongly compressed towards the lower values of £

This means that the local field effects, instead of becoming less and less

important for higher bands (that is for large I SI values)fhave an unphysical

tendency to remain constant. For instance, the L-T splittings at q -> 0

remain of the same order for higher as for lower bands, while they decrease

strongly in the realistic calculation of ref. 22. Furthermore, there are

suspicious oscillations in the bands in the middle of the Brillouin Zone,

which do not seem to have a proper physical motivation,

A closer inspection shows that the unphysical features just described

are due to the cutoff K introduced in the denominator for )G G'/ > 21^.

The large L"T splittings are due to exceedingly large DM wings £"(q, q + 0 ) ,

while the oscillations arise from the sudden onset of the cutoff K in

q - space.

There is alao another problem introduced by the cutoff, which is not

directly shown on Fig,3 in order not to make the figure too cumbersome, but

is nevertheless serious. At large I q.1 , where the diagonal DM elements are

already very close to one, the excessive magnitude of the off-diagonal elements

leads to high-lying eigenvalues e^ (a.) larger than 1. This la in contra-

diction with the general requirements of causality and thermodynamic

stability (£.10). Thus, Johnson's model describes a system that has a short-

wavelength instability, unless care ia taken to drop all the unphysical bands,

now quite reasonable magnitudes «* *ft» P*

a sensible behaviour over all the BZ, The L-T splitting also have the right

order of magnitude, which is not surprising since the parameters in eq. (3-5)

are given the values that best fit the off-diagonal elements of the DM

of Baldereschi and Tosatti . However, although the roost screened eigenvalues

at r are the same as in ED, the ordering Is not the same. We have found

that it is not possible to reproduce this correct ordering with a formula like

eq, (3.5), neither by changing the values of the parameters A and B, nor by

slightly modifying the formula itself. Vfe conclude that the simple local

dependence on the valence charge density of eq. (3.5) while quite good for a

semiquantitative description of screening in Si - and, ve would expect, other

crystals as well - is not suitable when a high accuracy is required.

(iv) Local Density Model

The DBS obtained with the LD model Is presented in Tig-5. The overall

behaviour is not very different from that of the Car-Selloni model, the

deviations from the empty lattice bands being generally somewhat smaller. This

indicates that a general tendency of the LD model may be to underestimate the

local field effects. All best screened symmetries at r are represented,

though not always in the same order as in Ref. 22. However, the lowest 1^

eigenvalue i3 identically zero, and no L-T splittings are found. This is a

consequence of the fact, already mentioned in the previous section, that the

LD formulation is not capable of giving a noij-metalllc behaviour In the

dielectric response. Hence, e"(q, i)/|^J * . =(a» 1 + °> * el1 + °« a)"T4

in this model, Irrespective of the existence of a gap in the electronic

spectrum.

In Bpite of this defect, this model seemB to us quite an attractive •

one, both because it has a more firm theoretical basis than the others, and

because it is parameter-free, requiring the cryBtal density as its sole input.

It can be of direct use to study screening in all those crystals where our

Knowledge has been so far very limited.

(iii) Car Selloni Model

Clearly, once the cutoff is dropped many of the undesirable features

described above should disappear. The DBS corresponding to tbi3 model is

shown in Fig.lt. By comparison with Fig.l, we see that the eigenvalues have

-llt-

V. AH APPLICATION: CHARGE INDUCED BY HIGH-SYMMETRY LATTICE VIBRATIONS

Though the dielectric band structure itself is a rather Instructive

object, it does of course not contain all the information of the full DM.

-15-

The remaining part of Information ia contained in its eigeopotentials, vhicn

we have not discussed so far. A proper appraisal of the differences between

DM models should Include a compariaon of their eigenpotentials. This ia

however both very cumbersome to do, and ratter devoid of immediate physical

appeal. A more natural choice can be to compare, for example, real-spade

distributions of induced electronic charge vhen the crystal is subject to a

given perturbation of practical occurrence. Examples of such perturbations

, or23)s

p p

could be a uniform external field , or point charge impurity * , or a23)

phonon-llke displacement of the atoms . One extra bonus is that, in the

lack of any direct experimental me^"*urements, there exist accurate self-

consistent calculations of these induced charges against which the various

models can be tested.

In this section we apply the different screening models discuBsed

in the previous sections to the calculation of the induced charge caused by

a high-symmetry lattice vibration in Si crystal. The local charge dis-

tribution is, incidentally, a very relevant quantity to the microscopic theory

of lattice vibrations, because the point-like ion core feels the local field *

rather than the mean field, such as one would calculate with a diagonal e(q).

We nave chosen the following two perturbations:

a) the unit cell of the distorted crystal contains two atoms, one at (000)

the other at *. (ill) (1 + * ) . This corresponds for x ^ o to a roly

phonon displacement, b) the distorted unit cell consists of four atoms, at

(0,0,0); J:(lll); ( + x,x,2) and £(3 + X l 1 + X j 3). This corresponds to1

X, phonon displacement.23)

The reason for this choice ia because Baldereschi and Maachie^

have provided a self-consistent pseudopotential calculation of the charge-

density change for these displacements of the atoms where x" is small.

Fig.6 and 7 show their results, plotted as constant density contours in the

Si unit cell. These authors have shown, furthermore, that the charge

induced via linear screening by a ^05' phonon displacement defined in terms

of the bare atomic pseudopotential at . as

where

- 1 6 -(5.2)

is, if e"1 is the DM of ref. 11, almost indistinguishable from that obtaineddirectly by difference between two full self-consistent pseudopotential cal-culations. Ho such comparison had been possible for the X, phonon, as there

was no DM calculation at X. However, since the diagonal dielectric functione(q) is well-known at all q in Si33 , i t is possible to separate thediagonal and the off-diagonal contributions to the total screening charge,both at r and at X. The separate contributions to the screening chargefound in this way by Baldereschi and Maachke are plotted in Fig.6(b),(c) and7(b) and (c).

We have calculated the induced charge (5.1) tor the r£ ' phonon,and that, defined similarly, for the X, phonon,by using the DM models des-cribed in the previous sections. The bare pseudopotential va tr) has been

1,1,1chosen to be the same as that of Baldereschi and MsJchke ' , and also thedistortion, magnitudes x are the same. The results of the Sinha, Car-Selloniand loeal^Iensity models are presented, for comparison with the full calculations,in Figs. 8, 9, 10, 11, 12 and 13. (Johnson's model has been dropped by now, onaccount of £ts many unphysical features).

The following comments can be made. It is known that, while thediagonal screening gives rise to dlpoles centred on the atoms, the off-diagonalscreening in the diamond structure gives rise to additional dipoles centredon the bonds ^ * * £ . For a T^y perturbation, which is itself parallel to

the bond, the atom and bond dipoles are lined up, that i s , the off-diagonalscreening merely reinforces the diagonal effect. In a X, perturbation,instead, the atom and the bond screening keep their individuality. Therelative importance of the bond contribution is larger, with the bond screen-ing charge oriented differently from the atomic screening charge. This con-trasting behaviour can be seen by comparing Pig.6 and 7.

The results for the Sinha model, Fig.8 and Fig,9 indicate a tendencyto overestimate the off-diagonal screening_,that is. the magnitude of the bonddipole, which has become even more important than the diagonal. A secondfact is that the bond dipole lines up with the atom dipole not only in the

- 1 7 -

r 'case, but al3o In the X case^contrary to the self-consistent results.

The Car-Selloni model gives of course good results for a r ^

perturbation (Fig.10), with off-diagonal effects practically indistinguishable

.from those of Fig.6. Hovever, the bond dipoles are somevhat underestimated

for the X, perturbation (Fig.ll), and their direction is intermediate between

parallel and transverse to the bond direction, while the full calculation

indicates a rather more transverse direction.

Finally, the local-density model exhibits a screening charge of

qualitative features similar to those of the full calculation, both for ^25'

(Fig.12) and X, perturbations (Fig.13). Quantitatively, however, there is

a general tendency to underestimate local field effects (that i s , the bond

dipoles are smaller than they should be) both with respect to the full cal-

culation and with respect to the Car-Selloni model.

Summing up, the above comparison of induced charge-densities leads

to fairly similar conclusions to those based on comparing dielectric band

structures: local field effects in Si are too strong and lack symmetry in

Sinha's model, and are substantially better represented but somewhat weaker

than they should be in the Car-Selloni and local-density models.

leads to the conclusion that local field effects are generally overestimated by

Sinha and Johnson's models, while they are underestimated, but generally

better represented, by the Car-Selloni and the local-density model. Also, the

important general fact that factor!zable models necessarily screen only per-

turbations of a small incomplete set of symmetries is brought out. It is alao

found that certain dielectric matrices, such as Johnson's, violate either

causality or stability, and must be dropped.

la compensate for the fact that no direct comparison with experiment

is possible at this stage, we have calculated with linear response the induced

charge densities for periodic phonon-llke T_c and X, displacement of the

atoms, and compared the results with, the full self consistant pseudopotential

calculations of Baldereschi and Maschfce . The extent to which the two

approaches agree varies substantially from one model to another. Conclusions

that can be drawn this way support strongly those obtained by simple

comparison of dielectric bands.

ACKHOWLEDGMENTS

VI. CONCLUSIONS

In this work, we have addressed the general problem of electronic

screening in a crystal, and further developed the dielectric band structure

approach to handle i t .

The extension of the general properties of causality and thermo-

dynamic stability to this case shows that all inverse dielectric eigenvalues

must be smaller than one, but not necessarily positive. Continuous electronic

phase transitions occur, in particular, when one inverse dielectric eigen-

value goes to - ™.

The problem of a full dielectric band structure calculation through-

out the Brillouin Zone is then considered, for the test case of silicon, which

is the crystal whose dielectric properties are best known. Four different

dielectric matrix models axe considered, among those existing in literature.

They are the models of Sinha, Johnson, Car and Selloni, and a newly developed

one based on a local density functional. Comparison among their dielectric

bands, and with those calculated by Baldereschi and Tosatti at q, -^ 0

The authors would ltke to thank Professor Ahdus Salam and

Professor Paolo Budini, tbe International Atomic Energy- Ageoey and UNESCO for

hospitality at the International Centre for Theoretical Physics, Trieste and

at the Scuola Interaazionale Superiore di Studi Avanzatl, Trieste. They also

wish to thank K. Maschke for ertlmulating discussions, and for his help with

the phonon charge calculation, and A. Baldereschi and A. Selloni for useful

discussions at various stages.

- 1 8 - - 1 9 -

APPENDIX Al - DENSITY FUNCTIONAL DIELECTRIC MATRIX

k'6)According to the theorem of Hohenberg and Kohn ' the ground 3tate

energy of a system of If interacting electrons in the external (local) field

V(r) is a unique functional of the electron charge density, vhich can be

written in the form

(Al.l)

where Q\f^\ is a universal functional of P(r) , that includes kinetic andexchange-correlation contribution to the total energy.

From the minimum property of (Al.l) subject to the conditionfP<?)d3r = H we get

— O

i.e.,

Here is the chemical potential and

(A1.2)

(A1.3)

a 3r'. The left-hand

side term of eq. (A1.3) has the meaning of the Belf consistent field V (r)

seen by a test charge on the system.

By taking the functional derivative of eq. (A1.3) with respect to

^(r) we get the response function

lf>Cr)or, in Fourier space:

CA1.5)

20

where fi-i are Fourier componenta of the charge dsaslty. The DM is given by:

in terms of the density response

We get therefore Eq. (.3.6) with R given by *i. (Al.7).

This formulation is exact and completely general; from the knowledge

of the functional G[p] and of the equilibrium charge density which satisfies

eq. (A1.3) we can calculate R # whose inverse giws the proper polarization

part \. In practice, hovever, we must content ourselves with an approximate

form of G[p]. In the present paper we have chosen the Thomas-Fermi local

form corrected with first gradients < •.

(Al.8)(X?)

from which eq.. (3.7) follows directly. The form (Al.8) does not includeexchange and correlation effects,and i » appropriate to the comparison withthe RPA results of Baldereschi and Tosatti ' .

In our application we do not use in eq. (3.7) the equilibrium chargedensity obtained with eq. (A1.3) with the functional (Al.8), but simply usein eq. (3.7) the empirical pseudopotential charge density. This i s a commonpractice in Thomas-Fermi methods. Indeed i t is known that the Thomas' Farm!approximation with gradient corrections can fa i l badly in some applicationswhen u 3 e d as a variationa.1 method, but gives in any case a fairly accurateapproximation of the to ta l energy i f an accurate charge-density (Hartree-Focli,

pseudopotential, experimental...) is used as an input to the Thomas-Fermi

formulas.

A more complete account of this density-functional based method will32)be given in a future publication

-21-

It should be pointed out, in closing, that a complete self-consistentformulation of the local-density linear screening theory in real space has

48)been given previously by Ying, Smith and KJohn , vho applied i t to theproblem of c&emisorption from jellium surfaces.

- 2 2 -

APPENDEC A2 - SYMMETRY ANALYSIS OF FACT0RI2ABLE DM MODELS

The fact that only a fev specifia symmetries are Bcreened in Sinha's

model is a direct consequence of the faetorizable nature of that model. To

see this more clearly let us set EQ(Q) = 1 in eq. (3.1) and look in

detail to the part of the Sinha DM that is responsible for LF effects, i.e. to

where

We can restrict our analysis to the r —point and consider only the matrix

(A2.1) at q = o 'in order to avoid the complication of the DM non-anali-

in the limit q->0. This is equivalent to consider the DM without

the wings, i.e., without the first row and the first column. Considering

the complete matrix for q -> 0 will only cause L-T splittings of the ri(.

eigenvalues found for q. = 0

We- note first, by inspection of eq. (A2.l), that the symmetry of the

eigenvalues of TtS, 3') is the same as that of the eigenvalues of

\{Z, Z'), Therefore in the following ve will concentrate on the

"polarization matrix" x- w* * 1 B O switch to a real space representation,

which is more appropriate to this discussion. The matrix50)

given in the RPA by :

» r'J *•'

(A2.3)

where the sums run over all the states i in the crystal and the fys arc

Fermi occupation numbers.

We expand t h e wave f u n c t i o n s i > , ( r ) i n terms o f l o c a l i z e d Wannier

f u n c t i o n s , <i i ( r ) >

- 2 3 -

.1* - »

tA2.lt)

Here n is a band index, k is the reduced momentum, X is a site in the

cell S , and N is the number of cells. Substituting (A2.k) into (A2.3)

we find11*);

2

"X,

(A2.5)

If we now res t r i c t to a tvo-band model, v i th t ight ly bound f lat bands,theproduct of two Wannier functions is different from zero only i f both belongto the same unit cell and the same s i t e . We obtain, like Ortuno and Inkson '

r-(2(A2.6)

where

CA2.7)

One recovers Sinha's model (A2.l) from (A2.6) i f the s i tes X are the atomics i tes s and the conduction and valenee Wannier functions are p- ands-like respectively . On the other hand if the s i tes \ are the bond s i tes

^ ^ '"'val ^cond a r e feoni'-n6 a n d antibonding combinations of sip hybrids,one obtains another popular form of the extreme-tight-binding model, namelythe model used by Ortuno and Inkson , Turner and Inkson and Arya and

17)Jha . From (A2.T) and the assumption of tvo flat bands , i t followsthat:

-2k-

This means that the matrix (A2.6) is a real-space representation of the

operator

A „ _

i.e., a sum of projection operators over the states

of (A2.9) having Bloch character are, for q. =• 0,;

CA2.9)

*• 'Hl* eigenvectors

(A2.10)

The number of different s i t e s in the unit ce l l times the possible degeneracyof state A gives therefore the tota l number of eigenvalues of theoperator (A2.9). Of course some of these eigenvalues can be degenerate.

In the case of Sinha's model we have two atomic s i tes in the unitce l l and the localized functions A * have p- character. The correspondingsymmetrized combinations of Bloch sums can have T _ and T^y symmetries;hence Sinha's DM model can have only two eigenvalues different from 1,

corresponding to these two symmetries, T and V'. The addition of the

DM wings in the case q -^ 0 introduces the L-T spl i t t ing of the P^eigenvalue into A and Ar. as can be seen in Fig.2. In the same figurethere are also other eigenvalues close to 1 but not exactly 1 due to thea r t i f i c i a l addition of the homogeneous screening term EO{Q).

The same symmetry analysis can be applied to the extreme tight-binding*model based on Wannier functions localised at the bonds

for this model only eigenvalues of symmetry T£ and

One obtains

at q. = 0, which

becojae T£ and Lapplied to phonons

+ 4,. in the limit <i-?0. This model haa been extensively5 ' , giving rise to reasonable dispersion curves. However,

since it does not contain the symmetry r ' vhich is the symmetry of the

optical phonon at V in covalent semiconductors with the diamond structure,

- 2 5 -

the induced electronic charge due to a r ^ lattice distortion is exactly-

zero in this model. We conclude therefore that even if the phonon

frequencies obtained at r with this model are numerically good, the

microscopic field behind them is certainly misrepresented by this type of DM.

-26-

FIGUHE CAPTIONS

Fig.l

rig. 9

"Empty lattice* dielectric hand structure of Silicon obtained by

simple folding in the £Z of the diagonal e J4). The dots at

r are the q.-> 0 accurate HPA results of Baldereschl and

Tosatti (from ref. 22).

Dielectric hand structure of Si obtained with Sinha's model

(eq. 3.1).

Dielectric band structure of Si obtained with Johnson's model

(eq. 3.3). Vertical units are the same as in Fig.2.

Dielectric band structure of Si obtained with the Car-Selloni

model (eq.. 3.5).

Dielectric hand structure of Si obtained with the local density

model (eq.. 3.6).

Accurate results (from ref. 23) for the induced electron

charge density (in units of 0.01 electrons/cell) produced in the

(110) plane by the r - distortion described in the text,with'JFO.OOl*.

)total induced charge; (b) diagonal contribution; (c) off-diagonal

contribution.

Accurate results (from ref. £3) for the induced electron charge

density (in units of 0.1 electrons/cell) produced in the (110)

plane by the X distortion described in the text .with. xstl.l. (a) total

induced charge; (b) diagonal contribution; (c) off-diagonal con-

tribution.

Induced electron charge density in Si (in units of 0.01 electron/

cell) obtained with Sinha's EM model in the (110) plane by the

r25' distortion described in the text, with x = 0.004. (a) Total

induced charge; (b) diagonal contribution; (c) off-diagonal con-

tribution ,

Induced electron charge density in Si (in units of 0.1 electrons/

cell) obtained with Sinha's DM model in the (110) plane by the X

distortion described in the text, with x « 0.1. (a) Total induced

charge; (b) diagonal contribution; (c) off-diagonal contribution.

Same as fig.8 vith the Car-Selloni DM-

Same as fig. 9, vith the Car-Selloni DM.

Same as fig.8 with the local density model.

Same as fig.9 with the local density model.

- 2 T -

REFERENCES

1) P.C. Martin and J. Schvinger, Phys. Rev. 115_, 13^2 (1959).

2) S.L. Adler, Phys. Rev. Ig6_, 1*13 (1962); H. Wiser, Phys. Rev. 129,

62 (1963).

3) A. Baldereschi, Phys. Rev. B£, 5212 (1973); D.J. Chadi and M.L. Cohen,

Phys. Rev. B12., 57^7 (1973).

k) J.T. Devreese, P.E. Van Camp and V.E. Van Doren, Phys. Rev. Letters

1>2, 122U (1979).

5) J.A. Van Vechten and E.M. Martin, Phys. Rev. Letters £8,, U 6 (1972)-

6) W. Hanke and L.J. Sham, Phys. Rev. Letters 33., 582 (l97lt).

7) H.E. Brener, Phys-. Rev. B12, llt87 (1975).

8} S.P. Singhal, Phys. Rev. B12_, 561* (1975).

9) S.G. Louie, J.R. Chelikovsky and M.L. Cohen, Phys. Rev. Letters 3jj_,

155 (1975).

10) J.T. Devreese, P.E. Van Camp and V.E. Doren, Bull. Am. Phys. Soc. gg,

22U (1978).

11) A. Baldereschi and E. Tosatti, Phys. Rev. jOJ, 1*710 (1978).12) CM. Bertoni, V. Bortolani, C. Calandra and E. Tosatti, P h y 3. Hev_ £ a >

1710 (I97tt).

13) J.C. Inkson, J. Phys. C: Proc. Phys. S o c , London 7, 1571 (197^).lit) M. Ortuno and J.C. Inkson, J. Phys. CJ£j IO65 (1979)-15) R.D. Turner and J.C. Inkson, J. Phys. CJ1, 2875 (1978)

16) L.E. Oliveira and M. Ortuno, Solid State Commun. _3JL 821 (1980).

17) K. Aiya and S,S. Jha, Phys. Rev. J 2 , 1*1*85 (1971*).

18) S.K. Slnha, Phys. Bev. JJ7., 1256 (1969).19) S.K. Sinha, K.P. Gupta and D.L. Price, Phys. Rev. B2, 2561* and 2573

(191k).

20) D.L. Johnson, Phya. Rev. B2, 1*1*75 (l97i*).

21) R. Car and A. Selloni, Phys. Rev. Letters J*H, 1365 (1979).

22) A. Baldereschi and E. Tosatti, Solid State Commun..£2, 131 (1979).

23) A. Baldereschi and K. Maschke, Proc. Int. Conf. Lattice Dynamics,

M. Balkanski ed,, Flammarion, Paris 36 (1978) and Proc. Int. Conf.

Phys. Semiconductors, Edinburgh, 673 (1978).

21*) P.C. Martin, Phys. Rev. 1&\, ll*3 (1967).

25) D.A. Kirzhnita, Usp. Fiz. Mauk 11£, 357 (1976); Engl. transl. Sov.

Phys. Usp. _19_, 530 (1976); O.V. Dolgov, D.A. Kirzhnits, E.G. Maksimov,

Revs. Mod. Fhys. £3., (1981).

-28-

£6) A. Fasolino, M. Parrinello and M.P. TOBI, Phys. Letters SfiL 119 U9T8).

27) M.H. Cohen and H. Ehrenreict, Phya. Rev. 015, 786 (1959)-

28) A. Baldereschi, R. Car and E. Tosatt i , Solid State Caranmn. J £ , 757

(1979).

29) P.W. Anderson, in Proceedings of Ail-Union Conference on the Physics

of Dielectrics. USSR Academy- of Sciences, Moscow (1958) p . 290;

¥. Cocnran, Adv. Phys._9_, 387 (i960).

30) W. Kohn: in "Many Body Physics", R. Balian ed. , (Gordon and Breach,

1967) p . 355-

31) G.F. Giuliani, E. Tosatti ana M.P. Tosi, J . Phys. Cia. 2769 (1979).

32) 5. Baroni, R. Car and E. Tosatti , to he published.

33) J.P. Walter and M.L. Cohen, Fhys. Rev. B2, 1821 (1970).31*) Actually this form applies only to the screening of the "electron-

nonelectron" interaction as discussed, e.g. hy L. Kleinman, Phya,

Rev. 160. 535 (1967) and Phys. Rev. 172. 383 (1968); by L. Ballentlne,Phys. Rev. 1?8, 670 (196T); and also by V. Heine and D. Weaire,

Solid State Phys. _gl+_. I t should be noted that a l l other models dis-

cussed here refer instead to screening of a "nonelectron-nonelectron"

interaction.

35) K.S. Singwi, A. SJolander, M.P. Tosi and R.H. Land, Fbya. Rev. £ k

10!*l» (1970).

36) W. Hanlte and L.J. Sham, Phys. Rev. B12. 1*501 (1975).

37) P-V. Gianuinta, M. Parrinello, E. Tosatti and M.P. Tosi, J . Phys.

£9., 2031 (1976).

38) D.R. Penn, Phys. Rev. igJL, 2093 (1962).

39) In our calculation we use the local pseudopotential form factors

defined for Si by M.L. Cohen and J . Chelikovsky, Fhys. Rev. Blft.

5095 (1971<).

1*0) R. Car and A. Selloni, unpublished work.

1*1) V.E. Godwin and E. Tosatti, Kuovo Cimento 5U5., 1*97 (1979).

1*2) L.J. Sham, Phys. Rev. 188_, ll*31 (1969).

1*3) R.M. Pick, M.H. Cohen and R.M. Martin, Phys. Rev. SL, 910 (1970).

1*!*) Baldereschi and Maschke used in their calculation the bare pseudo-

potential Si of J. Appelbaum and O.B. Hamann, Phys. Rev:. Bfi,

1777 (1973).

1*5) E. Tosatti, C M . Bertoni, V. Bortolani and C. Calandra, J. Phys.. C .,

L299 (1972).

-29-

k6) P. Hbhenberg and W. Kbhn, Fhys. Rev. 12&, B861* {1961*).

**7) See, for instance, N.H. March, Advan. Phya. 6_, 1 (1957).

U8) S.C. Ying, J.E. Smith and W. K3hn, Fhye. Rev. Bll, 1U83 (1975).

1*9) We recall that gauge invarisnce implies that E(0 , 0) = z[0 , Q) = 0

(see ref. 11,22).

50) See for instance, L. Hedin and S. Lundquist, Solid State Phys. 2£

(1969).

51) In this ease f\ Is a composite index standing for the site and

the x, y, z character of the conduction state.

-30- -31-

, - ( • > * C4 P» * i - — PI Pi "

Q

FIG. 6

-33-

WWMlla(a) (b)

FIG. 7

<aFIG. 8

ICt

FIG. 10

(Ct

FIG. 11

-35 -FIG.' 9 -34.".

IC/80/31* I.Zh. PETKOV and M.V. STOITSOV: On a generalization of the Thomas-Feral

IBT.EEP.* method to finite Fermi systems.

FIG. 12

FIQ. 13

IC/80/36

IC/80/37

IC/80/ltOINT.REP.*

ic/8oAlINT.REP.*

IC/80/51

IC/8O/83

IC/80/81.INT.REP.•

IC/60/93

lC/60/107

IC/80/108

IC/80/109INT.REP.*

I (ID/80/110

IC/8O/IUINT.HEP.*

IC/80/112

1 IC/80/113

INT.REP.*

IC/80/llU

IC/80/115

IC/80/116

S.K. SHARMA: On the suppression of magnetic octupole momenta in nucleiwith closed J-J shell plus or minus one nucleon.

A.N. ASTONOV, V.A. NIKOLAEV and I.Zh- PETKOV: Nucleon momentum and densitydistributions of nuclei.

W. KROLIKOWSKI: An integral transform of the Salpeter equation.

Z.A. KHAN: Elastic scattering of intermediate energy protons on He1 £

0. BUMITRESCU, L. FONDA and K. MANKOC-BORSTHIK: Alpha decay i>f coherentrotational states.

L.K. CHAVDA and B.B. DESAI: Self similar solutions for Implosion andreflection of strong and veak shocks in a plasma.

L. OLUMKKOR: Sutstrate temperature and resistivity of Hn and Mn/HgPgthin fila resistors.

S.P. MISRA and J.C. PATI: Is quantum chromodynamics effectivelyperturbative everywhere?

T. KUNIMASA: The Casimir effect aa a screening effect in quantizedfield theory.

E. ADESIYI BANOUDU: The method of moments and nested Hiltert spaces inquantum mechanics.

F. FURLAH'and R. RACZKA: On higher dynamical synmetriea in models ofrelativiatic field theories.

R. PAETHASARATHY - Composite quarks and their magnetic moments.

A. BREZINI ana G. OLIVIER: Localization on veakly disordered Cayley tree.

S.K. OH: The Bjorken-Paschos relation in the unified gauge theory.

A.A. FADIJiLLA: On a tioundary value problem in a strongly pseudo-convex

domain.

P. EAEKLER, F.W. HEHL and E.W. HIELKE - Vacuum solutions with double dualUfproperties of a quadratic Poincare' gauge field theory-

WITHDRAWM

M. SOCOLOVSKY - Spin-spin asymmetries with vpolarized nucleons.

and beams and

-36-• Internal Reports: Limited distribution.

THES^l PREPRINTS ARE AVAILABLE FROM THE PUBLICATIONS OFFICE, ICTP, P.O. BOX 586,1-3^100 TRIESTE, ITALY.

IC/80/117 ' P. FAZEKAS: Laser induced switching phenomena in amorphous GeSe2: Aphase transition model.

IC/8O/II8 P. BUDINI: On spinor geometry: A genesis of extended supersymmetry.

IC/80/119 G. CAMPAGHOLI and E. TO S ATT I: AsF^intercalated graphite: Self-consistent

band structure, optical properties and structural energy.

IC/80/120 Soe YIN and E. TOSATTI: Core level shifts in group IV semiconductors and

semimetals.

IC/80/121 D.K. CHATURVEDI, G. SENATORE and M.P. TOSI; Structure of liquid alkaliINT.REP.* • metals as electron-ion plasma.

IC/80/126 The Seventh Trieste Conference on Particle Physics, 30 June - h July l°80

INT.REP.* (Contributions) - Part I.

IC/80/l?7 The Seventh Trieste Conference on Particle Physics, 30 June - U July 1980

IHT.REP.* (Contributions)- Part II.

IC/8O/I28 FARID A. KHWAJA: Temperature dependence of the short-range order parameterand the concentration dependence of the order-disorder temperature for Ni-Ptand Hi-Fe systems in the improved statistical pseudopotential approximation.

IC/80/130 A. <yipiR: Dirac'3 large number hypothesis and the red shifts of distant

IHT.REP.* galaxies.

IC/80/131 S. ROBASZKIEWICZ, R. MICHAS and K.A. CHAO: Ground-state phase diagram ofIHT.REP.* extended attractive Hubbard model.

IC/8O/I33 K.K. SIKGK: Landau theory and Ginzburg criterion for interacting bosons.

R.P. HAZOUME: Reconstruction of the molecular.distribution functions from thesite-site distribution functions In classical molecular fluids at equilibrium.

R.P. HAZOUME: A theory of the nematic liquid crystals.

K.G. AKDENIZ and M. HORTAC.SU: Functional determinant for the Thirring model

vith instanton.

IHT.REP.*

IC/80/135INT.REP.*

IC/80/136IKT.REP.*

lC/80/137 M. SINGH: Phonon conductivity and relaxation rate in solids withIHT.REP.* disturbances by the Green function method.

IC/80/138 F.A. KATOUT; Mott excitons in ferrodielectrics.INT.REP.*

IC/80/139 F. BROUERS and O.L.T. de MENEZES: ELectron-phonon interaction inmixed valence systems.

IC/8o/l'tO T.C. CHOY: Some exact results for a degenerate Hubbard model in oneINT.REP.* dimension.

IC/8O/II4I AMOUZOU TEKOU - Multiple scattering in the nv.clear rearrangement

reactions at medium energy.

IC/8O/1J42 VIJAY KUMAR, A. MOOKERJEE and V.K. SBIVASTAVA - Electronic structure ofdisordered alloys - I: Self-consistent cluster CPA incorporating off-diagonal disorder and short-range order-.

-IC/8O/1U3 A. MOOKERJEE and V. CKAUDHRY - Electronic structure of disorderedalloys - I I : Self-consistent CCPA c.alucla,t£qns for I3J-Y semi-conducting al loys .

IC/80/lUl* F.A. KHWAJA and A.A. KATSNELSOH - The experimental study of establishinglocal order in binary metallic solid solutions.

. IC/8O/ll»5 H. DORN and E. WIECZOREK - Renormall station and short distance .propertiesof string type equations in QCD.

IC/8O/1I46 A. QADIR - A crit icism of Tivari 's paper on coupled zero mass andINT .REP.* electromagnetic f ie lds .

IC/8O/l!*7 -M.W. MORSY, A.A. HILAL and M.A. El-EABAGH -The scattering matrix elementof the three-body reactive col l is ion.

IC/8O/1I48 F. BALDRACCHINI, H.S. CRAIGIE.V. ROBEFTO and H. SOCOLOVSKY - A survey Ofof polarization asymmetries predicted "by QCD.

IC/8O/1I49 R. PARTHASARATHY and V.N. SRIDHAR - Effect of meson exchange correctionsINT.REP.••' on allowed union capture.

IC/80/150 N.H. MAECH and M.P. TOSI - Interpretation Of X-ray diffraction fromIHT.REP.* liquid a lkal i metals.

IC/8O/I5I ' -K.S. SINGWI and M.P. TOSI - Simple considerations on the surface tension.IHT.REP.* and the c r i t i c a l temperature, of. the electron-hole l iquids .

IC/80/152 Z. AKDENIZ, G. SEHATORE and M.P. TOSI - Concentration fluctuations andINT.REP.* ionic core polarization in molten sa l t mixtures.

IC/80/153 F.A. KHWAJA and ANIS ALAM - Concentration and temperature dependence ofshort-range order in Si-Ta solid solution using X-ray diffraction method.

IC/80/155 M, STESLICKA and L. PERKAL - Effect of the field penetration on surfaceINT.REP.* s ta tes .

IC/80/156 I.Y, YANCHEV, Z.G. K0IN0V and A.M. PETKOVA - Density of states in heavilyINT.REP.* doped strongly compensated ssaiconduetors with correlated impurity

distr ibut ion.

IC/80/157 CAO XUAH CHUAN - A theorem on the separation of a system of coupledINT.REP.* differential equations.

IC/8O/158 E.W. MIELKE and R. SCHERZER - Geon-type solutions of the non-linearHeisenberg-KLein-Gordon equation.

IC/80/159 A. RABIE, M.A. Ei-GAZZAE and A.Y. ABUL-MAGD - The Watanabe model for^Li-nucleus optical potent ia l .

IC/80/160 A. RABIE, M.A. El-GAZZAR and A.Y. ABUL-MAGD - A correction to theWatanabe potential .

IC/80/161 E.W. MIELKE - Snpirical verification of recently proposed hadron massINTiREP.* formulas.

IC/80/162 A. SADIQ, R.A. TAKnt-KHM,!, M. WORTIS and N.A. BHATTI - Percolation andspin glass t rans i t ion . ^ V

IC/80/163 WITHDRAWN

-38-

IC/80/165

IC/80/166

IC/6O/167

IC/80/168

IC/8O/169INT.HEP.*

IC/80/170IBT.REP.*

IC/80/171IHT.REP."

IC/eo/172IKT.REP.*

IC/80/173

IC/80/171*

IC/8O/1T5

.IC/80/176IBT.REP-*

IC/80/177IHT.EEP.*

S. NAHISOM - QCD sum rules for light mesons.

D.K. CHATURVEDI, 0, SEHATORE and M.P. TOSI - Structure of the stronglycoupled classical plasma in the self-consistent mean sphericalapproximation.

M. YUSSOUFF and H. ZELLER - An efficient Kbrringa-Kohn-Fostocker-method for"complex" lattices.

H.S- CRAIGIE and H. DORM - On the renormalization and short-distanceproperties of hadronic operators in QCD.

D.S. KULSHRESHTHA and R.S. KAUSHAL - Heavy mesons in a simple quark.-eonfining two-step potent111 model.

D.S. KULSHKESHTHA ond R.S. KAUSHAI, - Form factor of K-meson and the mesonradii in a quark-confining two-step potential model.

J. TAESKI - Unitary symmetries in Budini's theory of quarks.

H.H. MARCH and M.P. TOSI - Charge-charge liquid structure factor and thefreezing of alkali halides.

W. DEPPERT - Remarks on the "'beginning" and the "end" of the Universe.

M. YUSSOUFF - Generalized structural theory of freezing.

A. RABIE, M.A. EL-GAZZAR and A.Y. ABUL-MAGD - Diffraction model analysis ofvector polarized 6Li elastic scattering on 12c, l 6 0 , 28Si and 5%i nuclei,

T. JAROSZEWICZ - High energy multi-gluon exchange amplitudes..

M.R. MOHGA and K.N. PATHAK - Dispersion and damping ot plasmons in metals-

N.H. MARCH and M.P. TOSI - Interacting Prenkel defects at high concentrationan* the superiomc transition in fluorite crystals.

IC/8O/18O

IC/60/lSl

ic/ao/ite

J.C. PATI, ABDUS S4L&M and J. STOATHDEE - A preon aodel with biMenel»ctric <na

P. FURLAff and R. RACZKA - A new approach to unified f i e l d theor ies .

L. BEEB&Xa,.H.F. TOSLwd. 5 .B . MARCH - Exchange energy o f inhomogeneouselectxon a

IC/8O/1&3 J.C. PATI, ABDUS SALAM and J. 3TEand IT—n oecillBtiana*