THE ELECTRONIC BAND STRUCTURE OF V3Ga AND V3Si

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THE ELECTRONIC BAND STRUCTURE OF V3Ga AND V3Si

I. Goldberg, M. Weger

To cite this version:

I. Goldberg, M. Weger. THE ELECTRONIC BAND STRUCTURE OF V3Ga AND V3Si. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-223-C3-233. �10.1051/jphyscol:1972333�. �jpa-00215067�

JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-223

THE ELECTRONIC BAND STRUCTURE OF V,Ga AND V,Si

I. B. GOLDBERG and M. WEGER

The Racah Institute of Physics, The Hebrew University of Jerusalem, Israel

RBsumB. - La structure klectronique des bandes de conduction des composks V3X (X = Ga, Si, Ge, Al, As) est calculke en s'interessant particulikrement a la bande 3 d du vanadiwh. &a struc- ture de cette bande est calculke par l'approximation des liaisons fortes ; les diverses intkgrales sont determinees directement a partir des fonctions atomiques et aussi par ajustement avec les calculs APW de Mattheiss aux points de haute symktrie de la zone de Brillouin. L'accord entre ces deux fa~ons de prockder est assez bon. La bande 4 s du vanadium est calcul6e dans l'approximation OPW en ajustant avec les niveaux APW, et la bande p de l'atome X a la fois dans les approximations des liaisons fortes et OPW. I1 apparait que dans les limites de prkcision du calcul APW (environ 0,05 Ry) la bande 3 d du vanadium peut dtre considerke comme composee de quatresous-bandes indkpendantes : a, n, 62 et 61-(X) p. Fonctions propres, knergies, densitks d'ktat et surfaces d'knergie constante sont calculkes pour chacune des sous-bandes. Un pic tr6s pointu dans la densitk d'etats (4 a 8 mRy de largeur) est dc aux Btats r f z s , X3, M5 de la bande 62, rendant compte decertains caracteres du mod6le Labbe-Friedel-Barisic. Des corrections au calcul APW, dues a des kcarts par rapport a la symetrie sphkrique dans les sphhres c< muffin-tin >> sont estimkes. La surface de Fermi de la bande 4 s du vanadium possirde des cols dans la direction (loo), ce qui rend compte de certaines donnQs obtenues par annihilation de positron dans V3Si. On ne suppose a pviori aucune proprikte unidimensionnelle, mais on trouve qu'elles dkcoulent sans ambiguM des caI- culs APW.

Abstract. -The electronic band structure of the conduction bands of the cqmpounds V3X (X ⁼ Ga, Si, Ge, Al, As) is calculated with special emphasis on the vanadium 3 d band.

The structure of this band is calculated by the tight-binding approximation ; the various integrals are determined directly, from atomic functions, and also by fitting to Mattheiss' APW calculation at high symmetry points of the BriIlouin zone. The agreement between these two procedures is quite good. The vanadium 4 s band is calculated in the OPW approximation by fitting to the APW levels, and the X atom p band, in both the tight-binding and OPW approximations. It turns out that within the accuracy of the APW calculation (about 0.05 Ry) the vanadium 3 d band cdn be described as composed of four independent sub-bands : ^{5 ,} n, 62 and 61-(X) p. Eigenfunctions, energies, density of states and constant-energy surfaces are calculated for each of the individual sub-bands. A very sharp peak in the density of states (4-8 mRy wide) is found to be due to the T.5,

X3, M5 states of the 62-band, accounting for some features of the LabbeFriedel-Barisic model.

Corrections to the APW calculation, due to deviations from spherical symmetry inside the muffin- tin spheres are estimated. The vanadium 4 s band Fermi surface is found to possess necks in the (100) directions, accounting for some of the data from positron annihilation in VSi. No one- dimensional properties are assumed npriori, but are found to follow unambiguously from the APW calculation.

1. Introduction. - This talk is not about new computational techniques, but rather about the relationship of an old (1965) APW calculation of Mattheiss [I] to some experimental features of a parti- cular system, namely V,Si, V3Ga, etc., and in parti- cular about the relationship to a theoretical model, namely the linear chain model [2], proposed t o des- cribe this system. The system in question is interesting because of the high superconducting transition tempe- rature (-- 18 OK), and possesses other peculiarities associated with the band structure, and the question arises how, if a t all, can c< standard ^)> computational techniques be related, and account for, the properties of such intermetallic compounds. This question is of particular relevance, since extensive work on the linear chain model by the Orsay group [3] did not make use of the APW calculation.

Let us look a t the crystal structure of V,Si (Fig. 1).

We see immediately that the vanadium atoms are arranged in families of chains, such that the distance between vanadiums of different chains is about 20 %

(4%)

In this approximation, the constant energy surfaces are planes perpendicular t o the respective chains (Fig. lb). This model was proposed to account for some anomalies in the electronic properties of these alloys, and to suggest new experiments.

I n 1965, Mattheiss carried out an APW calcula- tion [I] (Fig. 2) ; the band structure seemed t o differ from the nearest neighbour, 2-center approximation, and therefore it was argued that the linear chain model

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972333

I. B. GOLDBERG AND M. WEGER

( ~ 0 0 ) (loo) much more effective and the system will not display

, . ^{1 0}

one dimensional properties.

In addition to this << topological >> factor, the sym- metry of the lattice and of the wavefunctions has to be considered, for some states, scattering between different families of chains is forbidden by symmetry (in the one-electron approximation) since the transfer integrals from an atom on one chain to different atoms on a neighbouring chain are equal and oppo- site and cancel each other (Fig. 4b). This property holds over whole planes of the Brillouin zone.

3. Positron annihilation experiments. - In order to perform a critical test of the linear chain model, an experiment was performed, following suggestions by W. Kohn and J. Friedel, with S . Berko at Brandeis using the positron annihilation method [5]. This method probes the momentum distribution of the electrons, and in a one dimensional structure should

( c ) show up as a step function of the distribution (inte-

( b )

grated over the [010] and [OOl] directions), in the [loo]

FIG. 1. - a) The crystal structure of V3X. The large spheres are direction, while in other directions, nothing particu- vanadium atoms and the small ones are X atoms. b) The Fermi larly should happen. ~~~~~i~~~~ indeed surface of a 3 d sub-band in the nearest-neighbour tight-binding

approximation. c) The first Brillouin zone. bears out this strong anisotropy [5]. The sharp, small discontinuities in the momentum distribution should be particularly strong when we look at the derivative was all wrong. But, in particular, the APW calcula-

tion looked so complicated, that it appeared difficult to interpret and utilise it.

Since 1966, Friedel, Labbe and Barisic did extensive work on the linear chain model, ignoring the APW calculation. This work describes quite well the elastic anomalies and crystalline phase transition of this system, and also attempts to account for the high superconducting transition temperature. In parti- cular, LFB account for the observed sharp peak in the density of states at the Fermi level in terms of the k = 0 van Hove I/& singularity characteristic of one-dimensional systems.

2. The coupled chain model. - The main argument against the linear chain model was that the inter- action between different chains is not very weak compared with the intra-chain interaction. (A factor of order 1000 is required by the Labbe model.) So, an attempt was made to consider a model of coupled chains [4]. It was found that the constant-energy sur- faces consist of planar sections, ⁽⁽ torn out planes )>, even for rather strong coupling (Fig. 3). This is a rigorous mathematical feature of this model, but a handwaving argument to account for it is as follows : (Fig. 4). When an electron is scattered out of a chain, successive scatterings are into different states ; conse- quently they are incoherent and transition probabilities (rather than amplitudes) have to be added ; while scattering inside the chain is coherent and amplitudes add, therefore it is much more effective. Thus, when different chains are parallel (or approach each other at several points), scattering between chains may be

(Fig. 5). However, when we look at the derivative in the [ I l l ] direction, we see a structure (maxima at 4.5 and 7.5 mrad) much stronger than that in the [loo]

direction ; and although this structure is not as sharp as that in the [loo] direction, it nevertheless is odd to attept to account for cc small >> structure, while a

(( big ^>) structure remains unexplained ; and the linear

chain model does not appear to be able to account for this structure.

Phenomenologically, the [ I l l ] peaks can be describ- bed as due to planar sections of the Fermi surface perpendicular to the [I 111 direction (Fig. 6). The simplest Fermi surfaces possessing such planar sec- tions are a large cr electron ⁾⁾ structure with necks in the [loo] directions, centered around the zones (2 nla)

<

>

<

>

<

>,

> .

The electron structure accounts better for the large observed structure, since it occupies a much larger volume in momentum space.

Thus, from experiment, we are faced with the follow- ing questions, which the APW calculation should answer :

(i) Are the constant energy surfaces planar ? (ii) Is there a peak in the density of states, about 2 mRy wide, containing about 0.15 electronlatom, at the Fermi level [6] ?

(iii) Does this peak tend to ((stick ⁾⁾ to the Fermi level (as appears to be observed from doping experi- ments) ?

(iv) What causes the double-peaked structure in the [ I l l ] positron annihilation derivative curve ?

THE ELECTRONIC BAND STRUCTURE OF V3Ga AND V3Si C3-225

FIG. 2. - The band structure of V3Ga calculated by Mattheiss by the APW method.

C3-226 I. B. GOLDBERG AND M. WEGER

band A _{band B}

FIG. 3. - The band structure of a two-dimensional system of coupled chains, when coupling between the chains is as strong as the coupling between atoms inside the chain. a) The constant energy surfaces.

b) The Fermi surface for an occupation of one electron per atom.

FIG. 4. - a) Sketch indicating why scattering between chains is relatively inefficient for crossing chains, compared with parallel ones, b). c) Sketch indicating the vanishing of interchain coupl-

ing due to symmetry, for some states.

THE ELECTRONIC BAND STRUCTURE OF V3Ga AND V3Si C3-227

8 in rnrad

8 in rnrad

RG. 5. -The derivative of the positron annihilation angular correlation curve for [I001 and [I 11 I planes for V3Si, according to

Berko et al. [ 5 ] .

FIG. 6. - Sketch indicating a simple Fermi surface that can give rise to peaks in the positron annihilation derivative curve like

those observed.

4. Mattheiss' APW calculation. - As mentioned before, an APW calculation for several V,X compounds (X ⁼ Ga, Si, Ge, AI, As) was carried out by Mattheiss in 1965, but the resulting band structure appears to be very complicated, and no details, such as energy levels away from symmetry points, shapes of constant-energy surfaces, wavefunctions (except at very few symmetry points), or fine structure in the density of states were drawn from it.

In trying to analyse the APW data, we are faced with the problem that the crystal structure is relatively

complicated, with 6 transition-metal atoms and 2 non- transition-metal atoms per unit cell. Thus there are about 80 energy-levels values at symmetry points that have to be fitted. If we use effective tight-binding integrals for the 3 d band, and consider only nearest and next-nearest neighbour interactions, there are about 25 independent integrals ; and since the X-atom p-band admixes strongly with the 3 d band, it must be described too at the same time, requiring 5 integrals for itself, and a few more to describe the 3 d-p admix- ture ; the vanadiun 4 s band requires about 7 parame- ters to describe it in the OPW approximation. Thus, about 40-50 parameters are required for a proper description of the conduction band ; and the assign- ment of the various energy levels to the various bands is not a priori unique (for example, a

r ,

However, with the aid of group theory, and a method of successive approximations, this fitting can be done. The method is similar to the diagonal sum rule employed in atomic spectroscopy to determine the Slater integrals from the term values.

The basis functions for the 3 d-band correspond to the irreducible representations of the point group of the vanadium atom, 42 m (D,,) ; they are A , , yielding o-bonds between nearest neighbours (with dzz functions for atoms having nearest neighbours in the z-direc- tion), B1(6,, dX2-,,2) ; B2(J2, dxy) ; and E(n, d,,, dyz)- Looking at the energy levels at the high-symmetry points of the BZ

and

we find that a T i z state occurs only for the 6, sub- band ; thus we have

where K is the crystal-field integral, J is the nearest- neighbour transfer integral, and I is the next-nearest neighbour transfer integral. These integrals are ⁽⁽ effec- tive >> and already include the corrections due to over- lap (non-orthogonality of the atomic functions), those due to 3-center integrals, and more important still, they include to some extent the effect of admixture of conduction-band states into the atomic d-states, which apriorimay be the most important factor casting doubt on the tight-binding approximation [7]. A similar equation can be written for the state

r;.

M,, R1 _R,, and R, must belong to the n-band ; thus the energies can be expressed unambiguously in terms

C3-228 I. B. GOLDBERG AND M. WEGER of the n-band integrals K(n), J(n), IIl(n), I12(n),

IZl(n), and J,(n). For this band, there are 3 indepen- dent next-nearest-neighbour integrals ; (since there are 2 basis n-functions, there are in principle 4 next- nearest-neighbour integrals, but two of them are equal by symmetry). J,(n) is a 3-center integral due to the potential of the X-atoms (mainly) and to func- tions on nearest-neighbour vanadiums. Its effect if to split the n-band at its center, and (in the indepen- dent linear chain approximation), to produce there sharp peaks in the density of states. Therefore, it is quite important [4]. Thus, we have 6 equations in 9 unknowns, and 3 more equations are needed. These are provided by the states X,, and M,, which occur several times, but are due to the 6, and n bands only, thus equations can be written for the sum of energies of these states, which depend on the values of the above-mentioned integrals only ; for example,

(The individual energies, unlike their sum, depend also on the integrals connecting the 6, and n sub-bands, and the dependence is non-linear.) Thus we have 9 linear equations for 9 unknowns, which yield the integrals for the 6, and n sub-bands. The procedure is easily extended to the 6, sub-band. As for the o sub-band, it has the same symmetry as the 4 s band, therefore its levels cannot be identified by symmetry alone ; however, the TI levels can be identified from the wavefunction analysis of Mattheiss, and the states

rZ5 and M I of the 4 s band are much higher than the 3 d band, and thus the T,, and MI states of the o band can be identified, and its integrals determined.

Actually, the admixture of the 4 s band into the 3 d band was found to be very small ; the reason for this small hybridisation is not yet understood. On the other hand, admixture of the t< X xx atom p-band is large, in particular into the 6 , sub-band which consists of functions with lobes in the direction of the X-atom.

Adding the p-band to this scheme, and considering its admixture with the 6, sub-band, is straightforward and it is found that the p-band too can be described very well by the TB approximation with nearest- neighbour X-V and n. n. and n. n. n. X-X integrals only.

5. Use of the intra-sub-band intregrals. - Once the intra-sub-band integrals are determined, we have two alternatives :

(i) We can use them to determine the inter-sub- band integrals 181, or ;

(ii) We can try to ignore the inter-sub-band inte- grals.

As for (i), the determination of the << inters >> requi- res the decomposition of the {< intras >> into 2-center and 3-center components ; the inter Zcenter compo- nents are then determined by the standard Koster- Slater method, while the 3-center components are

determined by an interpolation procedure. The decomposition depends on the observation that the main contribution to the 2-center integral

comes from regions of small r, and @(r

+

The interpolation for the 3-center integrals

assumes that V(r) can be replaced by 6-functions for the vanadium and X atoms. Thus 2 additional para- meters determine all 3-center integrals. A direct calculation of the 2-center integrals using tabulated (Herman-Skillman) wavefunctions and potentials yields excellent agreement (about 15 %-20 % standard deviation) with the values determined from the APW calculation. The reason for this excellent agreement (indicating negligible 3 d-4 s hybridisation), is still not understood. However, it should be noted that the accuracy of both the APW and the TB approximations is actually lower, since the exact configuration is not known (the difference between the values for the 3 d3 4 s2 and 3 d4 4 s configurations is of the order of 20 %), and both are based upon a <<full Slater xx exchange, which may introduce even somewhat larger errors, and both neglect the possibility of charge transfer between the V and X atoms.

Once these integrals are determined, the band structure and density of states can be determined by diagonalising the resulting 36 x 36 matrix. (30 for the 3 d, 6 for the p band.) For an accuracy of about 2-3 % at a resolution of 2 mRy (and a total bandwidth of about 0.9 Ry), 40 mn on a CDC 6 400 computer are required, using the Gilat algorithm 191. This high resolution is essential, since the density of states dis- plays peaks of this width.

As for (ii), we can write down the expressions for all the energy levels ignoring the << inters ⁾⁾ and then determine the (( intras >) by least squares. The agree- ment turns out to be to a standard deviation of 0.03 Ry, if admixture of the X atom p-band into the 6, sub-band is included. Thus, to within this accu- racy the band structure may be described as consist- ing of independent o, n, 6, and 6,-p sub-bands. This description is rather simple and physically illuminat- ing. The density of states curve obtained by the super- position of the densities of states of the individual sub-bands is almost identical to that obtained from the

THE ELECTRONIC BAND STRUCTURE OF V3Ga AND V3Si C3-229

36 x 36 matrix ; the peaks are slightly shifted by the Thus we shall regard the description by independent (tinter >> integrals, but not broadened significantly. sub-bands as adequate, at present (Fig. 7, 8).

FIG. 7. - The assignment of the energy levels at symmetry points obtained by the APW method (right) by the TB approximation (left).

6. The peak in the 6, band. - Perhaps the most striking feature in the density of states curve is the sharp peak in the 6, band, in the Ti5-X3-M5 plane. The width of this peak is approximately 4 mRy ; it contains about 0.15 electron per vanadium atom, and it is triply degenerate (i. e. there are dege- nerate states corresponding to the k, = 0, k, = 0, and k , = 0 planes). These features are just right to account for the anomalies observed in V3Ga, V3Si, and Nb3Sn6. This peak is not the only one found, but it appears to be the biggest and sharpest. In Matt- heiss' calculation, it is roughly 1 eV below the Fermi level. Generally, APW calculations may be expected to be more accurate (the convergence here is to 1 mRy) ;

however, we should note that the APW calculation employed a muffin-tin potential with spherical symme- try inside the atomic spheres, while here the local symmetry is very low, and the spherical averaging may cause a large error. This is because each vana- dium has two nearest neighbours at a distance of 2.36

a

The sharpness of this peak is not an accidential

C3-230 I. B. GOLDBERG AND M. WEGER

FIG. 8. - The density of states of the Ga 4 p and V 3 d bands, obtained by the TB interpolation method, compared with

Mattheiss' histogram, b).

electron theory (and in experiment) may serve as a starting point for more elaborate theories.

Present Labbe

Cubic Mattheiss Model Axial Friedel FIG. 9. -The centers of gravity of the individual sub-bands

(crystal field integrals) in the various approximations.

feature of the chosen potential, since it occurs through- out the series V3Ga(3 d3 4 s2)-V3Ga(3 d4 4 s)-V3Si- V,Ge-V,Al-V,As studied (Fig. 11). The width of the peak is roughly given by Min ( E(&)-E(X3) ; E(X3)-E(M,)) which is seen to be of order 4 mRy (within a factor of 2) throughout the series.

This peak is in effect the peak suggested by Labbe and Friedel. Its sharpness is due to the vanishing of interchain coupling due to symmetry (Fig. 4b) and the vanishing of hybridisation with the V 4 s, X s and p bands, also due to symmetry (at least at Tk).

It should be noted that for peaks that are so sharp, one-electron theory cannot be expected to be adequate, since electron-phonon and electron-electron interac- tions may be expected to affect the energy bymore than 4 mRy. However, the existence of the peak in one-

7. The vanadium 4 s band. - After the energy levels of the 3 d band have been identified, those of the 4 s band at the symmetry points are determined by elimination. (In a few cases, namely symmetry points X I , M,, and R4, and perhaps a few others, there is strong hybridisation. But at most symmetry points, the dominant state may be determined uniquely.) These energy levels can serve as a basis for the deter- mination of the energy levels of the 4 s band by the OPW method by interpolation. Actually, it was found necessary to include the X atom s and p bands in this scheme. Thus, a total bandwidth of about 1.5 Ry is fitted by this method with 7 parameters to a stan- dard deviation of about 0.2 Ry. In the least-squares fitting, more weight is given to <<critical >> levels near the Fermi level, so the accuracy there is consi- derably better. This behaviour is typical of what can be expected from OPW's with a local, energy-indepen- dent potential. This way the Fermi surface is deter- mined and found to possess necks in the ( 100 ) directions and to be doubly degenerate. Also, it is found that the main contribution to the wavefunctions comes from the Brillouin zones around points

<

>

<

>

(The Fermi surface is centered around a T , , state, which has no component with momentum zero.)

THE ELECTRONIC BAND STRUCTURE OF V3Ga AND V3Si C3-231

Pzs? 3?&(0)

r;s,&,MS(bz)

$80

B

1 ' I f I

% 70,

5 ^{M,, (n)}

60k -

- - -

734 -30 726 722 -.I8 -.I4 -10 :06 -.02 02 .06 .I0 14 .I8 .2Z .26 .30 34 38 42 46 50 54 E N E R G Y (RYDBERGS)

FIG. 10 a. -The density of states when corrections for non- spherical symmetry in the atomic spheres are made. The model density of states following Labbe and Friedel is also shown, b).

The FS determined this way, ignores 4 s-3 d level

- W crossings, which affect the FS locally (i. e. along lines

-

dHvA, these local perturbations alter the electron orbits altogether, since an orbit cut at only two points will have entirely different periods. But for most

-.2 -.l ⁰ .1 ^.2 .3 . .5 Ry

<t important >> properties, such as density of states, FIG. 10 b. superconductivity, cohesive energy, electronic specific heat, and susceptibility, elastic constants, average momentum distribution, these local crossings are

rn

E ~ Y

8. Application of the Electron Network model. - Up to now, the treatment of the 3 d band was within the framework of the tight-binding approximation, which assumes, more of less, that the electrons are localised on the atoms. The question arises, whether due to some fine features of the itinerant nature of the electrons, the one-dimensional features of the band structure may be altered. To investigate this, a model FIG. 11. - The position of the energy levels responsible for the was constructed following Coulson's Electron Net- peak in the density of states of the & band, for the various work [101, in which are assumed be materials studied. (Following Mattheiss.) on infinitely narrow ^<( bonds ))between the atoms, but

In the { 11 1 } directions, the FS possesses rather plane sections, in excellent agreement with the data from positron annihiIation (Fig. 6) ; The FS deter- mined this way predicts peaks at 4.5 and 7.5 mrad, in good agreement with experiment. Small discre- pancies might b e expected due to admixture of the 3 d3 4 s2 configuration into the 3 d4 4 s configuration studied (thus increasing the population of the 4 s band), or perhaps due to charge transfer between the silicons and vanadiums. Anyway, the agreement is considerably better than could have been expected for a system of this complication.

entirely free to move along the bonds. With S . Alexan- der and G. Della Riccia [ll], a model for a system of coupled families of chains was constructed, and the band structure determined. It is found that as the energy is varied, the << one dimensional D nature of the bands displays ((resonances D (in which the electrons move practically along the chains, and the coupling is ineffective), and << anti-resonances ⁾⁾ (in which the effective coupling between the chains is very strong, and no one-dimensionalfeatures persist). The density of states prossesses I / ~ E divergences associated with peaks in the ^<( one dimensional >> character of the band.

Scattering of the electrons at tht vertices of the net- work (the << atoms >>) does not remove these diver-

C3-232 I. B. GOLDBERG AND M. WEGER gences ; only cr direct ⁾⁾ interaction between bonds

(i. e. interaction not through vertices, but due to the finite width of real bonds) removes them. This feature is not restricted to the coupled chain model ; for a model simulating the diamond lattice, the density of states also diverges at the top of the valence band, as it does in the tight-binding, nearest neighbour approxima- tion. However, there, for real systems, there is a direct interaction between bonds which removes this diver- gence, while for the P-tungstens, direct interaction between parallel bonds is apparently weak, due to the large distance between them and the intervening X atoms, and this divergence persists and is present experimentaIIy.

Thus the one-dimensional nature of the band struc- ture, and the divergences in the density of states, are probably not artifacts of the tight-binding approxima- tion, but are present under more general conditions.

9. Deviations from the rigid band approximation.

- Experimentally, it appears that when materials like V,Ga are doped, the peak in the density of states seems to stick to the Fermi level, though it may be somewhat broadened. Also, it would be surprising to find, for several substances, a peak 4 mRy wide just happening to fall and coincide with the Fermi level.

This raises the question as to whether the rigid-band approximation, in which the band structure is assum- ed to be independent of the occupations of the bands, is valid for systems like the P-W. To investigate this, interaction between electrons was treated in the atomic Hartree-Fock approximation, with interactions bet- ween electrons on different atoms being neglected [12].

This approximation involves ⁽⁽ unrestricting w the standard Hartree-Fock-Slater (or HF-Kohn-Sham) approximation, so that different spin states, and diffe- rent orbital states, 'see different HF potentials. In this case, the energy of each individual sub-band depends on the populations of all other sub-bands. (But Koopman's theorem holds and we can talk of one- electron energies.) The interaction between electrons on the same atom is expressed in terms of the Slater integrals F,, F2, F,. For F,, a value much lower than the free-atom value must be taken, due to screening by the conduction electrons, while F, and F, are probably less screened (since they correspond to multipole interactions decaying much faster with distance) and perhaps may be roughly approximated by the free-atom values. It is found that the interaction between the 6 , and o bands (in particular) is such, that when the 6, band is relatively empty, then as an electron moves from the o to the 6, band, the energy of the 6, band falls (and vice versa when the 6, band is nearly full). Let this interaction energy be given by AE = an, n,, and let us neglect the other bands, so that n,

+

tendency for the 6, band not to be completely full, or completely empty, but partially occupied. When the peak in the 6, band contains a sizable fraction of the states of the band, this argument applies to this peak too ; and in order for it to be partially full, it has to be at the Fermi level. When a > 0, the peak tends to be repelled from the Fermi level, and to be either completely full or completely empty. (This is for example the case in Anderson's model for magnetic moments, where U is positive, and in the magnetised state, the peak for one spin direction is almost full, and for the other it is almost empty.)

FIG. 12. - Sketch illustrating breakdown of the rigid band approximation ; when the peak is lowered by cr An when An electrons occupy it, and anmax > AE, where nmax is the maximum occupation of the peak, the peak will be pulled towards the

Fermi level and << stick >> to it.

The sign of a depends on whether there is a relative attraction or repulsion between electrons of different sub-bands. In the simple case of a non-degenerate (s-like) band, there is effective repulsion of electrons of opposite spins, due to the Pauli principle ; while for different orbital sub-bands, there may be an effec- tive attraction, because electrons of different orbital states may be on the average further apart than electrons of the same orbital state. This effective inter- action gives the right sign for a to cause sticking of the peak to E,. Numerically, this effect depends on F, being relatively large, and Fo being relatively small.

Reliable values for the effective values of these Slater integrals for metals are unfortunately not available.

If we assume that U = 1 eV, and F, and F, have their atomic values, there may be sticking if the peak hap- pens to fall within a distance of 0.2 eV from the Fermi level, roughly.

This effect is obviously absent in the muffin-tin approximation, in which the H F potential is assumed to be spherically symmetric inside the atom. Thus, the muffin-tin approximation is probably not adequate here, and an orbitally-unrestricted Hartree-Fock ealculation is necessary.

Charge transfer between the V and X atoms, or between the V 3 d and 4 s bands, and lattice expansion, may have similar effects.

10. Conclusion. - The APW calculation of Matt- heiss is apparently a good calculation, and contains a considerable amount of information about the band

THE ELECTRONIC BAND STRUCTURE OF V3Ga AND V3Si C3-233

structure of materials of considerable basic and prac- tical interest. Nevertheless, apparently the calculation did not have the impact that it should have had. Pro- bably, the reason for this is that the band structure is rather complicated, and the graph of the energy levels along symmetry lines (Fig. 2) is not by itself very intelligible. This seems to indicate the need of models for complicated systems, which can serve to suggest what quantities should be calculated (in the present case, momentum distributions, shapes of constant energy surfaces, types of van Hove singularities, etc.) ; what approximations to make, and to avoid (here, for example, the muffin-tin approximation should be avoided, or at least corrected for, while 3 d-4 s hybridi-

sation, and even hybridisation between different 3 d sub-bands, may be ignored in the zeroth approxima- tion) ; what experiments to make, since complicated systems often call for specialised experimental techni- ques (in the present case, the method of positron annihi- lation ; while dHvA is not extremely useful, both for experimental reasons - the collision time is short in available samples, and for theoretical reasons -there are so many orbits that it is virtually impossible to identify the observed ones unless the band structure is already known to an extremely high accuracy).

Such models for physically important systems are usually available, but frequently ignored because they are not regarded as sufficiently ⁽⁽ basic ⁾⁾ or exact D.

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