Shastri, S K

Shrivastava, S B

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Indian J. Phys. 73A (5). 717-723 (1999) N O T EU P  A- an international jountalBand structure calculation of momentum density in rutheniumS B Shrivastava and S K ShastriSchool of Studies in Physics, Vikram University,Ujjain-456 010, IndiaReceived 2 December 1998, accepted 24 March 1999Abstract : Band structure calculation of two photon momentum density distribution (TPMD), the angular correlations for positron annihilation radiations (ACAR) and the Compton profile (CP) have been performed in the/cc structure of the Ad transition metal ruthenium. The calculated results have been plotted along the three directions of high symmetry and discussed in terms of the observed energy band structure Strong discontinuities in the TPMD have been observed along the [100] direction a tp  = 4.8 mrad and along [111] direction a t/? = 3 7 mrad A comparison of ACAR in Ru has been made with that in Rh It has been observed that the ACAR in Ru gives a much broader distribution as compared to that in Rh.Keywords : Band structure of Ru, electron states, positron annihilationPACS Nos, : 71 10 -w, 71 60 +z, 78 70 BjThe electronic structure o f metals is of particular interest due to the unfilled d states and their peculiar properties. In these metal the Fermi level intersects the d bands leading to a com plex Ferm i surface topology. The strong p  dependence of the momentum density distributions corresponding to various Fermi surface sheets combined with the existence of im portant U m klapp com ponents, severely complicates the interpretation of experimental data. The only way to understand the electronic structure of these metals is to perform a d eta iled  band s tructu re  calculation  of electron momentum densities (EMD). Such calculations also provide a basis to u dersund the results of angular correlation ann ih ila tion  rad ia tions and the Com pton profile studies in these meta s * calculations have been reported in many of the Ad metals by a number o aut o s  How ever, no theoretical calculation o f EMD has been done so far in rut enium Ferm i su rface  study reported  was by Coleridge 16], for the De-Hass van Alphen oscillations. The result was interpreted on the basis of a ngid band model and using©  1999 lACS718 S B Shrivastava and S K Shastriband structure of Re. Recently, Sharma et al [7] have performed an experimental study of C om pton profile in hep Ru. However, they have given only the difference between experim ental and theoretical results. In the present work, we report the results o f our calculations o f the energy band structure, the positron wave function, the two photon momentum density (TPMD), the angular correlations for positron annihilation (ACAR) and the Compton profile (CP) in/cc* ruthenium. The aim is to understand the HMD in Ru and to compare the results with those in Rh sjo as to understand the effect o f change in d  band on EM D and ACAR.The m om entum  density  distribution  o f  the photon pairs (TPM D ) from  the annihilation of ihermaliscd positrons with electrons of wave vector k in the j-ih energy band is given by [ 11|2( 1)where f ( k j )  describes the occupation of the electron state having wave vector k  and band index j , A j ( k , p )  is the overlap matrix element and /ify is the reciprocal lattice vector responsible for the Umkiapp process. The overlap matrix element is related to the electron wave function j{r)  and the wave function o f the thermalised positron yf^{r) in the ground state {k = 0) through the relationAj (k , p)  = jexp (-ip .r)y /^  ^{r)\i/J r)d r . cell(2)Eq. (1) also represents the electron momentum density derived from Com pton scattering (EM D) if V^+(r) in eq. (2) is replaced by unity. The calculation o f pip) from equation (1) requires the computation o f the energy band structure the electron wave function V 't , / ' ')  and the positron wave function. In the present work, the calculation o f energy band structure and electron wave function has been perform ed em ploying the approxim ate technique of band structure calculation o f Hubbard [8] and as developed by M ijnarends 19). The crystal potential employed in the numerical calculation is the M uffin-tin potential as given by M oruzzi et al f 10], In this the charge densities were spherically sysmmetric inside, touching, but non-overlapping, spheres and constant in the region between the spheres. M oruzzi et al [10] had chosen the reference energy such that the potentials as zero in the interstitial region. These potentials are interpolated on a Herm ann-skilim an mesh. In the calculation o f energy eigen values partial waves up to A^ax = 3 were used in the expansion o f  wave function inside the inscribed sphere while 137 reciprocal lattice vectors including 15 in the preferred set were used in the expansion of the wave function outside the inscribed sphere.The positron is know n to be therm alised before annihilation and is, therefore, assumed to be in the ground state (Jfc = 0). The positron wave function is periodic accordingBand structure calculation afnmmentum density in ruthenium 719to B loch theorem  and m ay be expanded as a Fourier series in te rn s  o f the reciprocal lattice vectors following G ould e ta l { \ \ \V^+(r) =  1/ ( t )*'2 A k exp l iK .r )is written in sym m etrized form and solved easily rising the Fourier coefficient.K-K' =  (1 /  T )V (r)exp  [ i( ir  -  K 'tr ]d r(3)(4)The first coeffic ien t was adjusted so that \ff^(r = 0) = = 0 subject to thenorm alization condition |i4j^ | =  1. The potm tial V(r) in which the positron moves is assum ed to be the electrostatic  potential used ih the calculation o f the electronic wave function. Thus, no correlation between electron m d  the positron is taken into account. The first ten shells (ten representative vectors o f the shells) and 137 plane waves in the expansion o f  th e /c c  structure were included in the calculations of yt+{r). The band profile for the long slit geometry angular correlation has been obtained fromN i P , )  =  r  p,pi,ip)pJo (5)where the spherically averaged psphiP) was obtained averaging p,oo, p m  and p ,n  according to Betts eta l [12].PsphiP) =  (l/3 5 )[l0 p io o (p ) + 16p ,,o (p ) + 9 p ,i ,(p ) ]  (6)The band profile for Com pton scattering /(P^) has been obtained using eq. (6) but by taking p^ph(P) pertaining to EM D Le. without the positron wave function.The energy band structure as obtained from the present calculation are shown in Figure 1 along some im portant symmetry directions. The Fermi energy is shown by dottedF ig u r e  I .  E n e rg y  b an d s  fo r  R u  a lo n g  so m e  sym m etry  d irections.720 5  B Shrivastava and S K Shasthlines. A look at the band structure (Figure 1) shows that along A direction the A |(ti) band lies well below the Fermi level and the A\{b) band crosses the Fermi level. A long S  direction the X i(c) band lies above the Fermi level while 2T|(a) and Z^ib) bands arc below the Fermi level. Along A direction A, (a) band lies below the Fermi level while A, (b) band crosses the Fermi level. These features have been found to give prominent contribution to corresponding momentum density distributions as discussed in Section 4.The results of the two photon momentum density distributions (TPM D) in fee  Ru as obtained from the present calculations for 4d and conduction electrons arc plotted in Figure 2 along the three symmetry directions [100], [110] and [ Ml ] .  According to a selection rule pointed out by Mijnarends [9] only the respective bands A |, Z i, A| belongingFigure 2, Calculated two photon inoincntuin density distnbution in Ku along (a) (1001. (b) 1110] and (c) {III ] directionsto the totally symmetric irreducible representation can contribute to the total TPM D in the first Brillioulin zone. The contribution to total TPM D from the individual bands is also shown in above figure. The TPM D along [100] direction shows significant structures between p  » 4.8 to 9 mrad. The band structure along this direction shows that the A |(a) band lies well below the Fermi level and is fully occupied. Hence, this band contributesBand structure calculation of momentum density in ruthenium 721fully to the total TPM D and gives 5-like contribution at low momenta. However^ the ^^(b) band is partially  occupied  and crosses the Fermi level. This gives rise to a strong discontinuity in the total TPM D  a tp  »  4.8 mrad. This is because of the unoccupied part of the Ai(b) band show n by the dotted line. The -contribution due to the A,(ib) band rises steeply at low m om enta due to the s-d hybridiz^ion. As a result of this hybridization the upper band contributes m ainly as 5-character near the Fermi energy. From the position of the discontinuity, the Fermi m om enta can be determined. The effect of Umklapp processes has been observed in the higher zones, thus, giving a peak at p  = 18 mrad.Along [110] direction, both the I i ( a )  ai|d Ii(fe ) bands lie much below the Fermi level and are fully occupied w hile the S i (c) .band lies above the Fermi level and is unoccupied. Thus, the total TPM D is a sum of the contributions from the two bands in the first Brillouin zone. The contribution from the S i(« )  band falls rapidly at low momenta and is mainly due to 5-p character. W ith the increase in A:-values the wave fuention hybridizes with the d  bands. The contribution from the d bands increases and give a large contribution to the total TPM D fo rp  between 4 to 13 mrad. Thus, giving a broad distribution to the total TPM D. The contribu tion  o f this band is ^ -likc  throughout the Brillouin zone. No discontinuity in the first Brillouin zone could be observed along this direction. The effect of Umklapp processes has been observed in the higher zone.A prom inent feature along [111] direction is the occurrence of a strong discontinuity a tp  ~ 3.7 mrad arising from the crossing of the Ai(b) band with the Fermi level in the first B-zone, The total TPM D  shows a strong dip between p  ~ 3.7 to 8.3 mrad and a strong step a tp  »  8.3 mrad. The contribution from A\(b) band rises rapidly and becomes unoccupied at p ^  3.1 mrad. The contribution from A\(a) band is more or less similar in nature to that along other directions. A broader contribution to the total TPMD has been observed for p  between 8.3 to 14.7 mrad along this direction. The effect of U-processes has been observed in the form o f a small discontinuity a tp  = 16.5 mrad and a step at p  = 21.3 mrad.Besides the calculation o f TPM D, we have also performed the computation of EMD pertaining to Compton profile in ruthenium. The EMD for 4d and conduction electrons inFigure 3.722 S B Shrivastava and S K Shastriruthenium along the three symmetry directions are presented in Figure 3. Looking to these curves one observes that the nature of the TM PD and HMD are the same. The main difference occurs in the higher zones where the HMD curves give a larger contribution to p(p). Thus the effect o f the absence o f is found to be important in case o f the high momentum region only. Along [100] direction the p(p) extends up to p  »  23.5 mrad, while the same in case o f TPM D terminates a tp  «  21 mrad. Similar effects have been observed along other directions also.The calculated band profile for ACAR (N{p^)) and the band electron Compton profile (J (p^)), are plotted in Figure 4 for the comparison o f band ACAR and band CP. The curves are identical at low momenta Le. up to p  «  6 mrad. But the high momenta the J (pj) curve gives a higher and broader contribution. W hile the band ACAR curve ends at 20 mrad, the band CP curve extends up to 23.5 mrad. Thus, the amplitude o f the Compton profile is higher at high p  values. This is consistent as in case of the Compton profile the absence of positron wave function increases the contribution to the p(p). In the same figure the band ACAR for Rh also has been plotted for comparison.Figure 4. Comparison of the band CP J {p2 ) with ACAR (N^2) 1^ “-band (ACAR) in Rh is also plotted for comparison.The ACAR in Ru has been observed to be much broader as compared to in Rh. This could be due to large d  band width in Ru, suggesting strong hybridization in case o f  Ru. Experimental studies of EMD in Ru are, however, needed to verify the above facts.References[1] P E  Mijnarends in Positron Solid State Physics eds. W Brandt and A Dnpasquier (Amsterdem : North Holland) (1983)[2] A Kshirsagar, D G Kanhere and R M Singru in Positron Annihilation eds. L Dorikens-Vanpraet. M Dorikens and D Segers (Singapore: Worid Scientific) p 236 (1988)[3] S B Shrivastava and H P Bonde Phys. Siast, Sol, (b) 139 155 (1987)[4] D O Kanhheie and R M Stngm Phys. U u , 53A 67 (1975)Band structure cakuladon of momentum density in ruthenium 723[5] S M Sharma, P ChidanAiaram and S K Sikka in Paiitron AnnihUation eds.PC Jain, R M Singni and K P Ciopinathan (Singapoie: Worid Scientific) p S2 (1985)[6] P T Coleridge J. Low Temp. Phys. 1 1254 (1%9)[7] B K Sharma, R K Kothaii, K B Joshi, M D Sharma and B L Ahuja Solid Slate Phys. (India) 37C 108(1994)[8] J Hubbard J. Phys. C2 1222 (1969)(9| P E Mijnarends Physica 63 235 (1973)[10] V L Moruzzi, J P Janak and A R Williams Calculuted Electronics Properties of Metals (New York ; Pergamon) (1978)[11] AG Gould, R N West and B G Hogg Con. J. Phys. H  2294 (1972)[12] D D Betts, A B Bhatia and M Wyman Phys. Rev. 104 37 (1956)

Band Structure Calculation of Momentum Density in Ruthenium

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Band Structure Calculation of Momentum Density in Ruthenium

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