Chatterjee, S

Nandi, Manika

Sen, Mina

IACS Institutional Repository

Electronic surface states of beryllium, magnesium, zinc and cadmiumM a n ik a  N a n d i , M in a  S e n  a n d  S . C h a t t e e j e e  Indian A>s-so(yiation for the Cultivation of i^cience, Calcutta-100032 (liecewed 1 June 1977)Indian J .  Phys. 52A, 420-426 (1078)Wn liavo pi'ojocied tho dimoriHiomil oneigy bands ol‘ Bo, Mg,Zn and Cd on to tho two dimensional BriJlouin zone of the 001 surfaces oi' these hexagonal close packed mclals. The pi’ujectjons show fairly large band gaps at F foi Be and Mg closing at. points in the F - /i  and V -K  directions inside the two dimensional Biilloum zone. Zn and Cd, however, do not have any band gaxi at F, they have veiy small band gaps inside the two dimensional Biilloum zone From tho Ht.udy ol those baud gajis it is concluded that although Bo and Mg have good possibility of having surface state s, the jnobabihly o f surface stab's in Zn and C(^  is very small.\1. IntuoductionThe study of tho ehictronic struetiiri; of smlai'c's dalt-s btuh to 19110s with thi' Xiionoering work of Tamm (1932) and Shockley (1939). Smoii then it was neg­lected presumably IxJcaiiso of the tremendous onijihaMS jmt on the electronie energy stains oi bulk materials, flowevor, lecently the pioblem i.s taken up and has since proceeded in two deiinitc directions. The first a i^jiroacli is tJirough the analysis of electron gas (Lang 1973) and the second axijHoach is through the study of bulk band structures (Forostman Heine 1970, Alldredgi' i\: Kloinman 1972, Cai'uthors et al 1973, 1974; Caruthers & Kleiniiian 1974, rjemiisey et al 1975, 1970; Sohu et al 1970). The second ajuuoach have generally been con­centrated on the study of the states whoso energies tall in the gaps of the bullv: material and whose wave functions arc localized in the suiface region. All these calculations lack self-consistency and is mainly of paraineti izcd type (CanitlK'rs Kleiiiman 1974, Bemiisey et al 1975, 1970; 8olm el al 1970) whose imrametos are found out from tho bulk calculations Howevers, only recimtly some first principle mothody have been put forward (Kohn 1975, Kar & Sovon 1975) but. detail self-consistent calculation with these methods ai(' y(;t to be performed. Meanwhile a good insight in the surface states could be obtained by the studies ot the band gaps of bulk materials. In tact Pendry & Ournian (1975) have recently obtained a new condition for the existence- of suiface slates. They obtained that if the bulk crystal has a mirror plants parallel to the surface and a centre of symmetry, narrow gajis at centre and edges of the Biilloum zone may contain surface states. Another Avay of looking into the iiroblem is to420tJ lectron ic su r fa ce  sta tes 421j)rojcd/ tho throe dimensional energy bands oiit.o the Iavo diinensiomd BrilUmin zone. Unless one is spo(’Jally inlorc'sti^d in veiy Him films, the privjt'diou gives all the inlormalions about the OMsteiice of snriaeo statey in tb<‘ gai>s. As the oxistonce ol energy gaps m the pjojeeled bands is the piimary eondilion of tho cxistenee of siiifaco states, the study of the (Mieigy gaps lu diflerent. symmetry points and directions giv''(‘s a fairly goc>d insight in the study (vf their surface staucs. Uptil now the sLiidj'^  of the suifaee states ol metals wen* mainly eon- eentrated on the metals Li. Al, On and F(‘ in films ot dilh'icnt. cryslallogrnphie directions (Alldri'dge iV'. Klemman 1072, Oa.riilhers c4 al 1073, 107t; Curuthers & Kleinman 1974, Diunpsey ct al 1075. 197(>, Sohn v1 al 1970) But divalent, hep metals Be, Mg, Zii and Ctl hav(, got litth^  attention calthough a largt^  number of exporiniental and theoietieal studies of the bulk eleclronie projKuties ot'these solids are available (l^eddeinieyor 1970, Holdei and Alonso 197(i, iSlu'vehick et al 1973, Tejida et al 1973, Townsend 1953, FalieoA’' 1902, Boueks & Cutler 1904, Torre] 1906, Chatterjoe & Sinha 1975, SJnha & Chattei-jeo 1977) Amongmtr those metals tho most experimentully studied metal is Mg Although the eailiost. experiment of Townsend (1953) could not show any surface stutes, the- experiments o f Cardona (1973) did show a surface |lasmon states just out,side the valence band spoctia of Mg In view (d those experiments one would be ml-erest(!d t,o kmnv tho surface' ,st,at,es of these metal,s from the theoietieal point of view. In trying to do so one would have to take loeourse to the lieavy (‘al- culations by either tho parametrized tight binding iiu'thod of (^aruthers el al (1976) or fir.st principal calculation of Kohn or Soven (B375, 1975) Before (‘mbarkmg on such a huge' calculation one can easily study the general features of the surlaco-states by stndymg the baud gaps in the projected two dimensional energy bands of the,ye metals. In thi,s ivoik we have obtained the 001 proji'cted energy bands ol these metals liom bulk ealeulation.s of Chatterjeo Sinha (1976) Tho various oiuu’gy gaps obtained m diffeient, symmetry points and axes of the two dimensional Biilloum zone aio then discussed in t,he light of possible existence o f surface states.2. Tiiii P r o j e c t e d  B a n d sAs the metals Be, Mg, Zii and Cd have hep slructare, botli real and roeipro- cal lattice have tlie same hexagonal symmetry The unit, cells in l,be real space and the reciprocal space have the sliajx' o f Jiexagonal prism aw sliown in figure 1 differing only in orientation and magnitude. Tlievofoie, il wo consider the 001 surface o f  those metals i-he two dimensional Brilloiiin zone o f those metals will simply be the VKM  plane o f the three dimensional Biillouin zone and tho two dimensional Brillouin zone would tlu'ii look like as sliou'ii ni figure 2. Tho .symmetry jioints and axes would t lioii bo describ(*d by the triangle VKM.  Tho projected bands along these triangle ai*e sliown in figures 3, 4, 5' and 6 for Be,ie S  Manika Nandit Mina Sen and 8- ChatterjeeF.g. 1. Bnllaum ^ono for tho hexagonal Drava« lattice with eymmetry pomtc and axe, following the notations of HerringElectronic etirfaee aUxtesOr4 2 SP TFig. 3 (001) onorgy banciH of Be,- 0 - 5- l O-1 -5AQ- 2 0 K K T' M M T rk - * -F ig. 4. (001) energy banda o f  M g.424 M a n ik a  N a n d i,  M in a  S en  a n d  S . C h a t te r je cZn and ('d rcspootjvt'ly Smcn those are projeetcd from bulk states, surface states can t‘xist at a value of K  wlittjo there is an energy gax) in the projected band Lf eoiu e^nt-ratc our attention to the origin of these gai>s we nolieo that the V eigen v^ alu(‘s are projected from the (OOf)A direction oi tlie three dimen­sional bamts. Similarly at the i>oint K, the eigenvalues are projected from the *3JJ liaufls along the lino I*. Jiocaiise of Iho inherent degiuioraoy of the bulk states at A, II and at syminoLry linos S then  ^ is no gap in the T-K  dneetii n Ixitween tlu» first two valence bands. Similar situation i x^ist in F-ilf direction where the points A, L and tlui symmetry line It has the same natuie. The pro- jiicted bands nie thus conneetod in xiairs and gaps occur only fiotween two such Xiairs of judjeeted bands. We now deserib(‘ th(^  projected bands and possible oxisteiu'c of surface stales of those metals one by one From figui’o 3 it is quite evident tliat tbero is a large band gaj) at f  and hence the possibility of siirfaia] statics occurring in this gap is qiiii-e high The gax> then decreases along tlu- 2-0t’lg. r>. (001) oiievgy bands of Zn.r~K  lino cuiting the fk-rini energy hwel at a K  xioint 10/17th of the V-K  line away from the F xjniut, finally closing at a K  ])oint 10/15tb of the T~K line away from the F point. iSimilarly along the F-ili direction the gap is largest at F and then closes at lO /iilh  of the T~M line away from F. The Fermi level alsoElectronic surface states 425intorosects the gap almost at. the same point. In the K -M  direction, however, there is no gap and hence thoio is no possihility of surface slates Next if we go on to figure 4 for Mg, we find that hero also the gap is laigesl. at V although the magnitude is much less than that ot B(‘ The gap deei’enses from T to zi'ro at the point 10/I5th of the T-K  lino and 10/J5tli of F-M  line UAvay from F. Here the Fermi line unlike Be do not tail m the gap near 1^ point Thi' b'ermi line crosses the gap at the points J/3tli and l/2tli of F-7f line and 10/271 h and 10/171h6. (001) 1‘norgy bands of C!d,of F-M  line and has only limited region of I\ A^ alues The nature of the gaps in the next two metals Zn and Od is tinile difleient. from that of Be and Mg Zn and Cd have no gap at F and only very small gaps exist in the F~K and F-M  direction. The gap in the F~M diiuctioii of Cd is almost negligible. Although the Forrai lino crosses these small gaps, there is only very small possibility of finding surface states experimentally considering the smallness oi magnitude of the gaps.4 Disonssroi^Tf wo go through the figures 3, 4, 5 and G, w'o find that the gap at F for all those metals decreases from Bo to Mg, tlnai overlapping lor Zn and Cd, the over­lapping being largest for Cd. Since the criteria of Gurman & Pendry (1975)hold good for these metals one would generally expect surfooe statefl for all these metals inspite of the non-existence of bond gaps at F point for Zn and Cd. How­ever, considering the nature of the band gaps one may conclude that there is a high possibility of surface states for Bo and Mg while the possibility of surface states for Zn and Cd being quite small426 Manika Nandi, Mina Sen and S. ChatterjeeAcknowledgementsThe authors arc grateful to Dr. D. Basil for his keen interest in the problem.B efeeencesAlldmlgo Q. P. fvlomman L. 1972 Phys. Jtev. Lett. 2Q, 1264Cfirutbors E. E , Klcinman L. Alldrodgo G. P. 1973 Phya. Pev. B8, 4070; 1974 Phya. Bev.B9, 3320; 1974 Phya. Bev. B9, 3330.Camihers Ed. & Khiinitian L. 1974 Phya. Bev. BIO, 370. jOhaUorjyd iS & iSmha Parsathi 1975 J . /*%«. F5, 2089, 'DornpHoy 1) G , Kloinman L. & CoruthnrB E. 1975 Phya, Bev. B12, 2932; 1976 Phya. Bev. BIB,148$.Falicov L M. 19G2 Phil. Trana, B Soc. A255, 663. \Forsiinan & Ht'iiio V. 1970 Phya. Bev. Lett. 24, 1419. \Holder W. C & Alonso J. 1976 Jr, Phya Bev, BIS, 2418.Kohn Walter 1976 Phya, Bev. Bll, 3766.Kar KikhilOHh & Soven P 1976 Phya, Bev Bll Louoks T. L. &, Cutler P. H. 1964 Phya. Bev. 133A, 819 Lang Norton D 1973 /Solid State Phya. 28, 223,Neddormoyor N. 1970 Phya. Rev BIS, 2411.Pendry J. 11 &, Gummn S. .1. 1976 Surf. Sci. 49, 87,Shookly W. 1939 Phya. Bev. 56, 377.iSolin K. y., DempHoy D. G., Kleinmnn L. & Caruthors Ed 1076 Phya. Bev. BIS, 1616. Shovchiek N. .1 , Tejida J , Langer J). W. & Cardona M. 1973 Phya Statua Solidi B57, 245. Sinha Parsathi & Chattorjecj R. 1977 J, Phya. F7, 105.Tamm 1. 1932 E. Phya. 70, 849; 1932 Plvya. Z Sowjitunion 1, 733.Tejida .1 , Cardona M , Shevehick N. J., Langer B W. & Sohonherr E. S. 1973 Phya, StatuaSolidi B58, 189.Terrel .1. H. 1906 Phya. Bev 149, 526.Townsend J U 1953 Phya. Rev, 92, 656.

Electronic Surface States of Beryllium, Magnesium, Zinc and Cadmium

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Electronic Surface States of Beryllium, Magnesium, Zinc and Cadmium

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