ANALYSIS OF ELECTRONIC STRUCTURE USING EMPIRICAL PSEUDOPOTENTIALS

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ANALYSIS OF ELECTRONIC STRUCTURE USING EMPIRICAL PSEUDOPOTENTIALS

W. van Haeringen, M. Schuurmans, H.-G. Junginger

To cite this version:

W. van Haeringen, M. Schuurmans, H.-G. Junginger. ANALYSIS OF ELECTRONIC STRUCTURE USING EMPIRICAL PSEUDOPOTENTIALS. Journal de Physique Colloques, 1972, 33 (C3), pp.C3- 185-C3-190. �10.1051/jphyscol:1972327�. �jpa-00215061�

JOURNAL DE PHYSIQUE Colloque C3, suppldment au no 5-6, Tome 33, Mai-Juin 1972, page C3-185

ANALYSIS OF ELECTRONIC STRUCTURE USING EMPIRICAL PSEUDOPOTENTIALS

W. VAN HAERINGEN, M. F. H. SCHUURMANS Philips Research Laboratories Eindhoven-The Netherlands

and H.-G. JUNGINGER

Philips Forschungslaboratorium G.m.b.H., Aachen-Germany

Rksumk. - La structure de bandes d'knergie de plusieurs polytypes de Sic a ete calculk par une m6thode utilisant un pseudo-potentiel empirique (EPM). L'application de cette methode A Sic est donnee en exemple d'illustration des possibilitCs futures d'obtenir les bandes pour des structures complexes.

On rapporte des tentatives de representation des bandes d'energie, calcul6es a l'aide d'EPM pour Al, Si et GaAs, par un dkveloppement en ondes planes symetriskes dans toute la zone de Brillouin. Des ajustements obtenus par la methode des moindres carres, avec ou sans contraintes additionnelles, sont discutes. Les deviations moyennes tombent a 0,01 Ry lorsqu'on utilise 30 termes. On n'a pas trouve de procede pour Bliminer les oscillations parasites introduites par cette m6thode.

Abstract. - Electronic band structures of several polytypes of Sic have been calculated using an empirical pseudopotential method (EPM). The application of the method to Sic is taken as an example to illustrate the future possibilities of obtaining band structures for complicated structures.

Attempts are described to fit EPM computed energy bands in Al, Si and GaAs to an approxima- tion consisting of an expansion of symmetrized plane waves throughout the whole Brillouin zone.

Least squares fits, with and without extra constraints, are discussed. Average deviations go down to 0.01 Ry at 30 terms in such an expansion. No way has been found to eliminate the unwanted oscillations introduced in the representations by this particular method.

I. Introduction. - The use of empirical pseudopo- tentials in determining energy band structures has become widespread since its introduction in 1961 [I].

I t is known that satisfactory agreement with experi- mental crystalline energy levels can be obtained by employing local atomic pseudopotentials in a large number of semiconductors 121-[6]. Also in the case of the NFE metals [2] it is possible t o fit energy bands or Fermi surfaces with no more than two or three form factors.

In this paper we want to discuss two different topics, in the treatment of which we make use of the exis- tence of empirical pseudopotentials.

a) An important feature of some pseudopotentials is their apparent transferability [2] : An example is the use of the same pseudopotentials in cubic and hexagonal ZnS [4]. Two of us (W. v. H. and H.-G. J.) took up this idea in an attempt to obtain energy bands for some of the polytypes of S i c [7]. The increasing com- plexity of the different polytypes (3 C, 2 H, 4 H, 6 H, 8 H, 1 0 H , 15 R... with 2, 4, 8, 12, 16, 20, 30

...

We therefore considered it worthwhile to try out the EP method. In section 11 we briefly describe the application of the EPM to Sic, diamond and graphite

and we place emphasis on the possibilities today and in the near future of computing energy bands in complicated structures.

b) A topic of general importance is whether or not energy bands can be represented by expansions in complete sets of properly chosen functions. If such functions could be constructed from energy values at only a small number of k points throughout the Bril- Iouin zone (BZ), the practical gain would be considera- ble because not only the proper energy function, but also local and total densities of state could be much more easily determined. In our present studies of BZ effects on the critical temperature of some (alloyed) NFE superconductors 181, it would be very convenient if the local derivatives of the energy bands were avai- lable in functional form. In section 111 we report on investigations concerning the fitting of energy bands using symmetrized plane waves, since to our knowledge this has not hitherto been systematically investigated.

We discuss least squares fits as well as fits with extra constraints. The results turn out t o be not particu- larly encouraging.

11. Energy band structures of polytypes of Sic. - The band structure of S i c in the sphalerite structure (3 CSiC) has been calculated by the OPW method [9], as well as by an EP method 171, [lo] [ll]. Apart from

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972327

C3-186 W. VAN HAERINGEN, M. F. H. SCHUURMANS AND H.-G. JUNGINGER a preliminary band structure calculation of 2 H

S i c [9] by the OPW method, no first principles calcu- lation of any polytype has been published. EPM band structure results on 2 H, 4 H and 6 HSiC have been reported by two of the present authors [lo].

Essential in this EPM approach is the use of transfer- able local atomic pseudopotentials for the elements Si and C. For Si the use of local potentials is less doubtful than for C. The carbon core contains no p electrons. Hence there is no repulsive p potential keeping the valence electrons out of the core region.

This could in principle make an EPM approach impossible. We have shown [7], however, that possibly apart from the location of the

r,

In the eigenvalue equation to be solved in the EP method form factors vi(K) occur, where K is a reci- procal lattice vector and i indicates the respective atom.

The form factors are needed for different K values when going from one crystal to another. This requires a careful interpolation procedure, such that new adjust- ments in new structures do not spoil the earlier ones in other structures. If, however, new data, become available, for instance in diamond or in some poly- type of S i c the whole interpolation scheme has to be reconsidered. For that reason our EPM band struc- ture results on 3 C, 2 H, 4 H and 6 HSiC are essen- tially provisional.

We note in passing an application of our C potential in the case of graphite [13], where we obtained some interesting results concerning the energy band struc- ture along the KH axis : Positions of electrons and holes on the Fermi surface were found at K and H respectively, contrary to earlier beliefs [14], but in accordance with recent investigations 1151.

We considered it worthwile to exploit in the compu- ter program symmetry properties of the crystals, leading to separate secular systems of lower dimen- sion in the case of symmetric k points. We first arrange the sequence of plane waves exp i(k

+

asterisk means complex conjugation. The eigenvalue equation is then transformed into a similar equation with symmetrized plane waves as basis functions.

The procedure is described in more detail in reference [lo]. In addition to the quantities defining the eigenvalue problem (i. e. the bais vectors in the reciprocal space, the number and positions of the atoms, and the form factors) we need to specify all point group operations and non-primitive transla- tions for the group of the wave vector k as well as the matrices representing the irreducible representa- tions.

We did not use the Lowdin method [16], which seems to yield equally good convergence of the energy eigenvalues with matrices which are a factor 2 to 3 smaller in size [3]. This additional feature would make it possible to handle structures with about twice as many atoms per unit cell. Our ALGOL program was run on an EL-X8 computer which is about 5 to 10 times slower than an IBM 360175 compu- ter. Because of storage limitations we could handle no more than 380 plane waves (if in less than about 135 shells). The maximum dimension of matrices to be diagonalized was 103. We observed that conver- gence within 0.1 eV was obtained for Si, diamond, 3 CSiC, 4 HSiC at about 60, 130, 100, 320 plane waves respectively. The applied Housholder/QR procedure took about 7.5 x 1 0 - ~ . n ~ seconds for the diagonalization of complex Hermitian n x n matrices on the EL-X8 computer.

In table I an outline is given of the possibilities of our EPM program on the EGX8 computer, on the middle size IBM 360175 computer and on the advanced

<< STAR D (CD) computer. The table contains rough estimates of the maximum number of atoms per unit cell to be handled, the maximum number of plane waves and the maximum duration of the eigenvalues determination per k point. It seems that a 15 RSiC energy band structure comes into sight, which is interesting in itself. The more complicated S i c polytypes, however, are not within the reach of pre- sent-day or near-future computers, nor are any crys- tals with more than 40 atoms per unit cell. It will be clear that similar estimates for other methods, which are necessarily more complicated and time- consuming, will greatly reduce this number of 40.

111. On the Fourier representation of energy bands. - Fourier fitting procedures have been successfully applied to the Fermi surfaces in the noble metal series [17], 1181. In the opinion of the authors the possibility of fitting energy band structures throughout the whole BZ has not been sufficiently investigated.

An attempt made by Schliiter [I91 to fit the lower Co, rp(oi>j*l oi (1) energy bands of A1 kith only 10 Fourier terms was claimed by him to be fairly successful. Preliminary to lone of the members in each shell. Oi is an element of attempts by the authors to fit the second A1 energy the group of k, l"p(Oi)j, is the (il) element of the pth band yielded average deviations decreasing to 0.01 Ry irreducible representation of the group of k. The at about 20 Fourier terms, but the maximum devia-

ANALYSIS OF ELECTRONIC STRUCTURE USING EMPIRICAL PSEUDOPOTENTIALS C3-187

Comparison of EPM on three dzyerent computers. Numbers are given, yielding estimated maxima in : 1) the number of atoms per unit cell

2) the number of plane waves 3) the computing time per k point.

In each of the nine categories possible applications are given.

Arbitrary accuracy atoms per unit cell k point 0.1 eV plane waves

time in minutes examples

Symmetric accuracy atoms per unit cell k point 0.1 eV plane waves

time in minutes examples

Symmetric accuracy atoms per unit cell k point 0.3 eV plane waves

time in minutes examples

X8 - 2 100 10 111-V, 3 CSiC

4 200 5 Diamond, 2 HSiC

12 380

10 Graphite, 6 HSiC

tions turned out to be 0.06 Ry. The fit was obtained from 100 randomly chosen k points only. We consider- ed it worth studying the Fourier fitting procedures in more detail in the case of Si and GaAs valence and conduction bands. There are three reasons for this choice. Firstly, our EPM program quickly yields energy eigenvalues throughout the BZ. Secondly, the mentioned energy bands are smoother and more

(( cosinelike ^>) than in the A1 case. Thirdly, the energy bands have complicating features such as degenera- cies, giving rise to cusps in energy bands.

111.1. EXPANSION IN SYMMETRIZED PLANE WAVES. -

An fcc lattice of a given crystal consists of the points

where a is the lattice constant, 1, m, and n are integers

A A A

and x, y, z are unit vectors in the x, y, z direction. An exact representation of the ith energy band of the crystal can be given by

Ei(k) = ai(Rj) exp[ik.Rj] ,

I

where k is restricted to the first BZ and for shortness j = (I, m, n). As Ei(k) is invariant under all point group operations 0 of the crystal, we have ai(ORj) = ai(Rj) for all 0 and j. Two coefficients ai(Rl.,m.,n.) and u,(R,,,,~) are equal if and only if R,r,m,,n, is one of the 48 permutations of

In this way the R vectors are collected in shells. Writ-

IBM 360175

-

4 200 10 Diamond, 2 HSiC

8 400 5 Graphite, 6 HSiC

20 600 10 8, 10 HSiC

STAR (CD) -

20 1 000 10 Graphite, 6 HSiC

40 2 000 5 15 RSiC

100 4 000 10 Boron, 33 RSiC

ing k ⁼ (k,, ky, k,), we then find the alternative expression :

x cos

[

⁺

a?]

where bi is invariant under the six permutations of the arguments. Let the function S [I, m, n ; k] be constructed from the product

ma ky

cos

[%I

^[--I--]

_[%I

by adding the 6 cosine products obtained by permut- ing 1, m, and n in the three cosine functions. If the 6 terms are equal we divide the sum by 6 ; if only three terms are different we divide by 2. With the help of these symmetrized plane waves (spw's) we arrive at

l + m f n even

In an attempt to fit an actual band structure func- tion Ei(k) it is not a priori clear which terms in (3) are to be retained. If Ei(k) is sufficiently smooth such that we have a reasonable hope that spw's with high (I, m, n) will be relatively unimportant, an easy way to proceed is first to order the terms in (3) with increas- ing shell radius Rj. Note that different shells can have equal radii. Next we define S [p ; k] = S [I, m, n ; k]

where p

<

+

<

+

+

13

C3-188 W. VAN HAERJNGEN, M. F. H. SCHUURMANS AND H.-G. JUNGlNGER

and p is an integer. Similarly ci [p] can be introduced. and B is a symmetrical N x N matrix. CT denotes Eq. (3) then reads the transposed matrix. It may be readily verified that

As a matter of course similar expressions for other structures can be obtained.

111.2. THEORY OF FITTING. - We restrict ourselves to N terms on the right-hand side of eq. (4). The resul- ting function will be called Ei(k). The function Ei(k) is determined at M randomly chosen k points k,,

...,

In our fitting attempt in Si and GaAs, N varied from 1 to 60 and the maximum value of M was 398. The fit of Ei(k) to Ei(k) will be performed under a least squares condition. It may sometimes be desirable to demand exact coincidence of Ei(k) and Ei(k) at some chosen points kM+ ,, ..., k,,,. This calls for L Lagran- gian multipliers. Finally, in order to be able to smooth the zi(k) function obtained, we investigate the possi- bility of using a roughness functional (cf. Shank- land 1201). It consists of a sum of integrals over the entire BZ of I (grad)"E",(k) 12, where s = 0, 1,2,

...

Summarizing, we try to minimize the functional

M

Fi =

2

i = 1

with respect to ci [p] and ,Ij. The roughness coefficients r, can still be chosen freely. Using (4) it follows that

I

^I

1'

BZ

where Rp is the length of the lattice vector in the pth shell and where fp is the proportionality constant between S[p ; k] and the sum of all plane waves in the pth shell, i. e.

The sum over h runs over g, elements. The ^fp as well as g p can easily be calculated.

Differentiating Fi with respect to ci [p] and Aj results in N -I- L linear equations with N

+

unknowns ci[l],

. ..,

,, .

Yp =Ei(kp-,+~)(N

+

+

111.3. APPLICATION TO THE ENERGY BANDS OF Si

AND GaAs. - We define the average and maximum deviations in a set of M' k points by respectively

M D = max

1

1

j= 1, ..., M' (10) In referring to the primary set we shall mean the set of M k points used to obtain the actual fit, and the secon- dary set refers to the set of k points to be used as an extra test for the representation obtained. A sub- script ~(primary), ~(secondary) or p

+

We first discuss pure least squares fits. Figure 1 gives the dependence on N for the third energy band of Si. M is chosen equal to 100. It is seen that

AD, > AD, and of course the AD, and AD, decrease

with increasing number of spw's. The MD,+? shows a more irregular behaviour. For M = 100 and increas- ing N we can expect the maximum error in the fitting of the s set to become large. The representations gi(k) thus obtained reveal that the coefficients ci[pr]

do not decrease monotonically with p'. For that reason we tried to make alternative adjustments to the obviously most important N spw's. The results of these adjustments are also shown in figure 1. We find slightly smaller deviations at equally large numbers of spw's. The gain of such a procedure is hardly significant.

Figure 2 gives the dependence of the deviations on M for both the third and fifth energy bands of Si.

When going from 75 to 400 points the AD,'s do not increase or decrease significantly, which means that these deviations are almost equal to their saturation value for M -, co. The AD, which are not shown in the figure tend, as they should, to the same satura- tion value as AD,. For M = 200 the AD' s in p and s set were found to be almost equal. In GaAs the devia- tions in comparable energy bands are slightly smaller than in Si, but show the same M and N dependence.

Therefore, when restricting ourselves to 30 spw's,

ANALYSIS O F ELECTRONIC STRUCTURE USING EMPIRICAL PSEUDOPOTENTIALS C3-189

there seems to be no reason to make an adjustment using more than 200 k points in Si and GaAs. However, the MD is then still quite large (about 0.01 and 0.025 Ry for the third and fifth Si band respectively). The study of the decrease of the AD and MD upon a further simultaneous increase of M and N is of great impor- tance. But what the optimum NIM ratio in such an investigation should be is not a priori clear. We suggest in view of the above observations that the ratio NIM % 301200 is a good starting.point in such an investigation. The diminishing rate of the respec- tive deviations for increasing M and N is of course of vital importance in a final judgement concerning the possibility of fitting with spw's. Because of computer limitations we were not able to investigate the situa- tion for large M and N values sufficiently.

From figure 3 where both the third and fifth energy bands of Si as well as their representation obtained from 100 k points and 30 coefficients are plotted along three symmetry axes, we see that the situation is in fact rather bad : The valence band representation along the A-axis and the conduction band representa-

--L N, the number of symmetrized plane waves

FIG. 1. - Fitting of the third energy band of Si. Average devia- tion in @ primary and @ secondary set of 100 random points as a function of N, the number of symmetrized plane waves.

@ and @ as in @ and @ respectively, but now adjustment is made to a selected set of N symmetrized plane waves, which were observed to be the most dominant. @ and @ indicate the maxi- mum observed deviations in @, @ and @, @ respectively.

FIG. 2. - Fitting of the third and fifth energy band of Si with 30 symmetrized plane waves. Average deviation in primary set of M points as a function of M @ for the third and @ for the H t h band. Maximum observed deviation @ for the third band

and @ for the fifth band.

FIG. 3. - Third and fifth energy band of Si along three symme- try axes. The crosses indicate fitting attempts with 30 symmetriz- ed plane waves. a) M = 100. Least squares fit. b) M = 398.

Least squares fit. Parts of bands 3 and 5 are shown. c) M = 100.

Least squares fit, but with additional constraints. Points r, L, X and

2

roughness parameter r l = 0.001.

C3-190 W. VAN HAERINGEN, M. F. H. SCHUURMANS AND H.43. JUNGINGER tion along a11 three symmetry axes (TLXT) have a

strongly oscillating character. The maximum devia- tions with respect to the EPM energy bands are about 0.04 Ry. One could think of double degeneracies along the symmetry axes A, A , and Z as the cause of these oscillations : Away from these axes these dege- neracies are lifted, yielding in general energy bands with cusps. The fact that the fifth GaAs energy band has no double degeneracy at all, but still shows the same oscillating behaviour (Fig. 4), excludes the double degeneracy as the only cause of oscillations.

In some regions of the BZ there are marked varia- tions over short distances in the slope of Ei(k). An example is the fifth energy band of GaAs along TLXT.

For a good fit in these regions we need cosines of small period. These in turn could then be responsible for oscillations elsewhere in the BZ. This is confirmed by our observation of the large systematic variation of I Ei(k) - Ei(k) I throughout the whole BZ. Although we did not have detailed information about the cause of the unwanted oscillations, we did try to eliminate or at least diminish them.

First we raised the number of k points for the adjust- ment in Si to 398. Parts of the results are shown in figure 3b. A slight improvement was obtained. The convergence is seen to be very slow, however, and computer limitations prevent us from going far beyond 398 k points. An almost trivial way to diminish the oscillations is of course to omit cosines of short period. But, as we have seen in figure 1, this gives rise to a larger AD and MD and is not a real improvement.

Next we tried to suppress the influence of short period cosines with the aid of the roughness functional defined in 111.2. In figure 3d a part of the result is shown for r , = 0, r , = 0.001 and all higher derivative roughness parameters equal to zero. Comparison with figure 3b reveals that no improvement is obtained.

Obviously, if the roughness functional is able to improve the adjustment at all, the use of higher deriva- tive terms will have to be considered. This point is still under consideration.

Finally we investigated the possibility of fitting Ei(k) exactly to Ei(k) in some ^<( special ⁾⁾ k points. This could be useful, for instance, if one wants &(k) to give values of some optical transition energies with high accuracy. An example of such an attempt using the method of Lagrangian parameters is shown in figure 3c. The AD, MD and the oscillations increase only slightly if the number of extra Lagrangian cons- traints is limited to a number much smaller than N.

FIG. 4. -Third and fifth energy band of GaAs along three symmetry axes. The crosses indicate a fitting attempt from 148 randomly chosen k points with 30 symmetrized plane waves.

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