Valence density of states of TiS2 and ZrS2

255 VALENCE DENSITY OF STATES OF TiS2 AND ZrS2 H. I S O M A K I Department o]: General Sciences, Helsinki University of Technology, SF-02150 Espoo 15, Finland

and J. v. B O E H M Nordita, DK-2100 Copenhagen ¢J, Denmark T h e valence densities of states (DOS) of the IV A transition metal disulphides TiS2 and ZrS2 are calculated and analysed in detail. T h e D O S are based on the energy eigenvalues evaluated in the final self-consistent symmetrized O P W (SCSOPW) potential at 131 s y m m e t r y i n d e p e n d e n t k-points. A close a g r e e m e n t is found to exist between the present DOS, XPS m e a s u r e m e n t s and recent L C A O - D O S . Four pairs of valence bands (1 + 2, 3 + 4, 5 + 6 and 7 + 8) are found to give rise to four main peaks of the valence DOS. In addition, an interpretation of the extra features present in the D O S based on L C A O interpolation is given.

In this paper we present our study of the valence density of states (DOS) of the IV A transition metal disulphides TiS2 and ZrS2. The K K R - D O S of TiS2 by Myron and Freeman [1], the L C A O - D O S of TiS2 and ZrSz by Bullett [2] and the previous DOS of TiS2 and ZrS2 by Isom~iki et al. [3] based on the L C A O interpolation [4] of the self-consistent symmetrized OPW (SCSOPW) energies show considerable differences. The S C S O P W - L C A O - D O S differ also from the X-ray photoemission (XPS) measurements for TiS2 and ZrS2 by Wertheim et al. [5] and for ZrS2 by Jellinek et al. [6]. This is probably due to the fact that the least squares fit on the SCSOPW energies is not completely successful. For these reasons we have performed new DOS calculations which are more directly based on the SCSOPW results [3, 7, 8]. We will also present a detailed analysis of the structure of DOS in terms of contributions from individual bands. The S C S O P W - L C A O - D O S were calculated by using the Gilat-Raubenheimer (GR) method [9]. The bands for the G R calculation were obtained with 22 two-center energy and overlap parameters in the L C A O interpolation. Our present DOS are based on the valence eigenenergies evaluated in the final SCSOPW potential at a grid consisting of the following 131 symmetry independent k-vectors: 19 k-vectors in each of the ks = 0 and ks = b3/2 planes (corresponding to the filled circles in fig. 1) and 31 k-vectors in each of the ks = b3/8, kz = b3/4 and kz = 3b3/8 planes (corresponding to the filled and open circles in fig. 1). All DOS histograms

Physica 99B (1980) 255-258 (~ North-Holland

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presented in this paper are calculated from these valence energies (with proper weights). The more refined DOS of TiS2 is calculated by using the quadratic Lagrangian interpolation (QLI) and the G R methods. For this purpose the irreducible wedge of the first Brillouin zone (BZ) (one twelfth of B Z between the kz = 0 and kz = b3/2 planes) is divided into small identical cells (height b3/16) as shown in fig. 1 (i.e. 9216 cells in BZ). The energies and gradients for the G R calculation are evaluated from the QLI equations at the midpoints of the cells (indicated by dots in fig. 1). The SCSOPW-DOS histogram of ZrS2 is shown in fig. 2. This DOS compares quite well with the XPS-measurement [5] (broken line in fig. 2). (Because the XPS-results of [5] and [6]

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Fig. 2. DOS and XPS of ZrS2. Histogram is our present DOS. Capital letters are labels o[ DOS peaks. Broken curve is XPS result [5]. SCSOPW-LCAO-DOS [3] is placed in the lower part of the figure. Integers are indices of those bands which give the main contribution to the peaks. Arrows denote peak positions of LCAO-DOS [2]. Energy position of the central peaks at VB2 is the same in all DOS or XPS. Both DOS are given in the same arbitrary units.

Fig. 3. DOS and XPS of TiS2. Upper and lower curves (continuous line) represent SCSOPW-QLI-DOS and SCSOPW-LCAO-DOS [3], respectively. Otherwise see the caption of fig. 2.

are practically the same only the result of [5] is redrawn in fig. 2.) Only p e a k C lies ~0.5 eV below the lower energy shoulder. The S C S O P W - L C A O - D O S [3] (lower part of fig. 2) agrees well with our present D O S at the lower valence band group (VB1) but differs considerably at the u p p e r valence band group (VB2). The L C A O - D O S peaks [2] at VB2 (positions indicated by arrows in fig. 2) compare within ~ 0 . 5 e V with peaks A, B and C but the two L C A O - D O S peaks at VB1 are placed ~1 eV below peaks D and D'. The S C S O P W - Q L I - D O S of TiS2 (see fig. 3) has again the sharp peak D at VB1 and the distinct p e a k C at VB2 in agreement with the XPS result [5] (broken curve). H o w e v e r p e a k A is exceedingly high at the cost of the almost disappearing central p e a k B. T h e S C S O P W LCAO-DOS (lower part in fig. 3) is in agreement with the S C S O P W - Q L I - D O S at VB1 but at first sight there does not seem to be any agreement at VB2. Also the agreement of the S C S O P W - L C A O - D O S with the XPS result at VB2 turns out to be somewhat misleading as will be analysed in detail below. Again the L C A O -

D O S peaks [2] at VB2 (positions indicated by arrows in fig. 3) match accurately peaks A, B and C but the two L C A O - D O S peaks at VB1 lie ~ l . 5 e V below p e a k D. The K K R - D O S [1] seems to be even m o r e concentrated towards the Fermi energy than our D O S (see fig. 5 below). To deepen our understanding of the structure of the D O S of TiS2 and ZrS2 we have divided our D O S into partial D O S from individual bands. This division of ZrS2 is shown in fig. 4 (columns at peaks of the total D O S as well as the corresponding contributing columns of partial D O S are filled). Peak D at VB1 consists almost entirely of the p e a k of band 2. The adjacent peak D' originates mainly from the p e a k of band 1 but has also a clear contribution from band 2. Peak C is due to the overlap between the steep upper energy and lower energy edges of bands 3 and 4, respectively. The high central p e a k B originates similarly from the overlap between the edges of the sharp peaks of bands 5 and 6. Peak A is mainly formed by the overlap between the slow slope at upper energy of band 7 and the steeper lower energy edge of band 8. However, band 6 also gives a small contribution. The over-

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Fig. 5. DOS division of TiS2. All histograms including KKRDOS redrawn from [1] are given in the same arbitrary units. Integers are band indices. Capital letters A-D are labels of DOS peaks.

lap between the slow slopes of bands 4 and 5 has tendency to cause an extra p e a k between peaks B and C. It is now straightforward to understand the structure of the S C S O P W - L C A O - D O S of ZrS2 at VB2 within this D O S division f r a m e w o r k (see fig. 2 where the integers at peaks are indices of the main contributing bands). T h e deep valleys between the pairs of peaks from bands 3 and 4, 5 and 6 as well as 7 and 8 are artifacts of the LCAO-interpolation. In case of a perfect L C A O fit we would expect both peaks of each pair to m o v e together to form peaks C, B and A in place of the valleys. The division of the D O S of TiS2 is shown in fig. 5. Peak D is again due to the sharp p e a k of band 2. The adjacent p e a k has practically disappeared. The overlap between the edges of bands 3 and 4 is again responsible for p e a k C. However, the formation of peaks B and A differs here considerably from that of ZrS> This difference is due to the spread of bands 5 and 6. Thus although p e a k A is again formed mainly from the contributions of bands 7, 8 and 6 this time band 5 gives a noticeable contribution too. This exaggerates p e a k A. On the other hand the contributions from bands 5 and 6 to the central p e a k B are correspondingly diminished causing the shrinkage of this peak. The reason for this difference between TiS2 and ZrS2 is probably due to the fact that the S C S O P W bands containing d-type admixture are expected to converge m o r e

slowly in the case of TiS2 (no d core states in Ti) than in the case of ZrS2 (3d core states in Zr). Indeed, by using group theoretical analysis (see e.g. table V in [10]) we find smaller probability for d-type admixture in the u p p e r part of VB2 of ZrS2 [8] than in that of TiS2 [7]. This supports accordingly expectations of slower convergence for TiS> As to the interpretation of the S C S O P W - L C A O - D O S of TiS2 at VB2 (see fig. 3) the D O S division above indicates that the first three peaks (3 + 3 + 4) represent p e a k C, the next two (5 + 6) p e a k B and the last three (7 + 8 + 8) p e a k A. So far we have completely neglected the effects of transition probabilities (matrix elements) in the comparison between the D O S and the XPS measurements. These effects may really be large in some cases (see for example Jarlborg and Nilsson [11] in the case of A15 compounds). However, in m a n y cases the effect would essentially be just a relatively slight rescaling of the existing D O S peaks (as for example in the case of the trigonal Te semiconductor [12-15]). We would expect some minor rescaling effects to occur also in the D O S of ZrS: and TiS2 but we would not expect this to explain the relative sizes of peaks B and A of the D O S of TiS> In conclusion we have found our present D O S of ZrS: and TiS2 to agree well with both the XPS results and the L C A O - D O S except for the forms of peaks B and A in the case of TiS2, For both

Fig. 4. DOS division of ZrS=. All histograms are given in the

258

materials the three peaks C, B and A correspond to the lower energy shoulder, central peak and upper energy shoulder at VB2 of the XPS-curves without ambiguity and are formed from the pairs of bands 3 + 4, 5 + 6 and 7 + 8, respectively. The lack of this grouping into pairs is responsible for the extra valleys present in the SCSOPWLCAO-DOS. References [1] H.W. Myron and A.J. Freeman, Phys. Rev. B9 (1974) 481. [2] D.W. Bullett, J. Phys. C: Solid St. Phys. 11 (1978) 4501. [3] H. Isom~iki, J. v. Boehm and P. Krusius, J. Phys. C: Solid St. Phys. 12 (1979) 3239. [4] J.C. Slater and G.F. Koster, Phys. Rev. 94 (1954) 1498. [5] G.K. Wertheim, F.J. DiSalvo and D.N.E. Buchanan, Solid St. Commun. 13 (1973) 1225.

[6] F. Jellinek, R.A. Pollak and M.W. Sharer, Mat. Res. Bull. 9 (1974) 845. [7] P. Krusius, J. v. Boehm and H. Isom~iki, J. Phys. C: Solid St. Phys. 8 (1975) 3788. [8] H. Isom~iki, J. v. Boehm and P. Krusius, Nuovo Cimento 38B (1977) 168. [9] G. Gilat and L. J. Raubenheimer, Phys. Rev. 144 (1966) 390. [10] L.F. Mattheiss, Phys. Rev. B8 (1973) 3719. [11] T. Jarlborg and P.O. Nilsson, J. Phys. C: Solid St. Phys. 12 (1979) 265. [12] J.D. Joannopoulos, M. Schliiter and M.L. Cohen, Phys. Rev. B 11 (1975) 2186. [13] T. Starkloff and J.D. Joannopoulos, Phys. Rev. B 19 (1979) 1077. [14] J. v. Boehm, H. Isom~iki, P. Krusius and T. Stubb, Proc. of the Intern. Conf. on the Physics of Selenium and Tellurium, 28-31 May, K6nigstein (Springer, Berlin-New York, 1979). [15] H. lsom~iki, J. v. Boehm, P. Krusius and T. Stubb (1979, to be published).