Ghosh, S K

Laskar, A K

English

IACS Institutional Repository

E N E R G Y  ST A T ES A T  TH E IN T ER FA CE OF TW O  SEMI-INFINITE O N E-D IM ENSIO NAL C R Y ST A L SH X. (mOSH AND A. K. LASKAKIJiCPAllTMENT OF P u Y S lO S , iNOTAN INSTITUTE OF TECHNOLOGY,K ilA R A G rU H[H t c n v e d  J u l y  10, 1963)ADSIRACT. Loonlizod onergy levels are computed, by the sfatterin g  iiu itnx m ethod 1)1 Saxon and B u tu ei (1949), a t tho m terlaeo of two sim ilar Hemi-iiifiriito crystals with adsoibed gas at tho iiitoifaoe, F irst, tho calculations liiivo been made for a clean iiitorfacp imd tlieii foi an impunity atom  at the m torface Tho results show the existence of locahzofl s lates a t cortaui potential strength of tho adsorbed gas Some experim ental ovideiice ol the theorel ical lesnlts are given.I  N J l  O D U ( ' T  I O NTamm (U)32) showed the exintenuc ol‘ localised ciierg}  ^ levt'ls at the h‘ec sur­face of a one dmiciisional scmi-infimic crystal. iSmet' tlien sevcial autluTrs have calculated surface energy levels by various metliods. iShockley (1939), and Aorts (1900) ealeulated surface energy levels on the assumjition that the potential of the boundary atom shows a symmetne behaviour. Aerts fiu'ther extended tho case to a surface contaminated by an adsorbed im])urity atom. On assuming this kind of potential, the results obtained by Aerts conform to the expermiental observationsIn this paper, the electronic energy slates localised at the interface of two semi- infinite crystals with adsorbed gas layt r^ at the interface, are computed Aerts (I960) has shoAvn that the interface boLween two crystals is a kind of imperfection, which gives rise to energy state and thus plays a great role in its electneal behaviour, ft is v^ cll established that the electrical conductivity ol both scmic-onducting ciystals and hlins are strongly dependent on the adsorficd gas. When films of a substancic are deposited on a substrate, we have adsorbed gas at the interface of two crystallites of identical nature. The presence ol an impurity may also he possible at the interface of identical crystallites. We propose to investigate the energy-states in the above mentioned systems A simple method based on the work of Saxon and Hiitncr (1949) has been followed.T H E  C R Y S T A L  M O D E L  A N D  T H E O R YSince the essential features of the problem would remain same, a simplified model of two semi-infinite crystals, of identical nature in contact is considered. The semi-infinite monatomic lattice has tho same spacing on both sides. Lei59534U be the atomic potential Due to the adsorbed gas at the interface the surface atoiUvS may be assumed to have the potential strength Let the interfacepotenlial be U f  A graphical I'epresentation of the crystal model is given inrig 1.Energy State at the Interface of two Semi-Infinite, etc. 635acII1 (liMj l^iiCLil roprosoiitatum  o f the cryKial inoclol w ith adsorbed im purity (I) at Ihc interfaco.Choosing the origin of the co-ordinate system at tlu^  interface, the atoms of the It H iS are localized at the iiomts with abscissa x — (2?t-|-l)a and those ofL.ILS. are localised at the points .r (2a-f l)u, n being a positive integer.On the K, H ,»S the potential emu’gy iR ef the form\ m / 0 m ( 1)and on the L H S. the fiotential energy is of the formI (t\ mS I {2n-\ l)«,]-| «) mThe v aves projiagatmg from eithei side will undergo scattering by the coinbhied disturbances, first at the absorbed imimrity atom, then at the interlace and again at the adsoihed impurity atom rollowing kSaxon and Hutner’s method, the Tlf-matrix associated Avith the three combined distiirliances may b(‘ given byM .. (3)The matrix is associated ii’ith the adsorbed impurity situated at x ^  a. It is I 1 -f- i tan 0 i tan 0, (+)- 1 tan 0 1 -  i tan 0 iwhere,tail e ““iX sin 2/^ tt2 mEi )^Ai^ here fi is the wave vector and x^  — ^  , E  brung the electron energy in the crystal.536 S. K. Ofiosh and A. K . Laskar(^2) is i-iie translation matrix for a; ^  a to =  0 and is of the form' 0. . .  (6)TIui matrix (-/^ o)^  associated with the disturbance al the interface potential, is given byj _  iau  ^sin ya 2x sin /Iddii(2 Sin yr/ sill /4<ilaM^  sm 2y sin fiurm^sm yr/ 2y sm /mIS the translation matrix for .« — 0 to a; ——a and is of the formI 0 ,(I(h (8)The matrix (R )^ is associated ivitJi atlsorhed impurity situated at x = —a and it is I l-l-?ltan^y t tan tVl i , ~  • \ , ... (9)'I tan 0 1 —1, tan 6 IWriting tan 0 ~  A, -= Fi and ^2y sill jitaand then substituting the values of (ft) ^ 1  equation (3),we have\\-\ 1/4 I i ' 0 I -10 ~ ic  B 0 l-\-iA iAJlf =  i ' , ... (10)—1/4 1 lA ' i) /? , ] IV 14-«r 0 R ~?A 1 —iAThe localized energy levels al. tlu‘ mteii’ace are determined by the conditions-  0; /, -  , I > 0 (11)Using second and third conditions of (‘(piation (11), the appropriate Kronig and Penny relation can be writLon as/ an \(—1)" ch 2 a  ^ — cos 2ay+\ Aj sin 2 a%The matrix element j4fo2 as obtained from equation (10) is given by1^ 2^2 — tan^ ?^[2i' (sin im ~ c  cos [w)-\-c\—2 tan 0 [i(l+?'c)c“2^ ®^H-c]+  (H-ic)e-2^''®(12).. (13)M ergy States at the Interface of two Sem i-Infinite, etc. 537Using the first condition of en M1  ^ iof the quadratic Eq. (13), arc given by ° °  ^ as a solution( . sin/ia- “3  tan cos/,«](l_2 cos>= fta- 2i sin fta cos /m )tan $~  tan xa cos /ia±i sm /m2 i[ & sin= eos / ,a -  tan ^a(\ - 2  s.,,^... (14)tan x(f' nos imr» X \ ■■■negative w rits ™ *  (14) and takingta n i? ,= l^ r ! tan x„,(l [(1 - r f  2 a | ) l + , ( ] 2 of)ll2 ( 1  tan y ajd  |_„A2 of) .. (ig.To obtain the energy states wii.hin the first ferbidden vn, w, i„l tion ( 1 2 ) and by ehnnnating f between (1 2 ) and (15) I  g l  ‘ "r/2 [ i (  o«^ tanay(cot^„y+ ‘'^^Iot «y)l ][(oot2 ay-|- e o s « ,y y + i ( l -  c o ta y j ‘ jtan 0  ^ =( I +  “^ 2  tan ay j ( J -  cet „,y j(16)X --/I yHubstitutmg the value of ton U, as in cquatnm (5 ) n, (16) we have2 2 , 2 Z ^JC,I ' l l c o t  u f ]*rS o tT K / 2 1 [ ’ - y  ](“ t - r 1 - 7tKK cot2 " '' 1 -  1  cot 7tF( F 2 (17)638where X , = —  V^, T =  ~  U ,^ Z71 7T2 a“8. K , Ghosh avd A, K . Laskar-C7 A T7and V =  — XIT(17)Now taking the positive sign in equation (14) for the second solution of tan 0 , we similarly obtainBin(7rF)H(18)Equations (17) and (IH) are rather involved in V, therefore, it is difficult to cajl- culato the energy values directly for different values of x, i.e. different types (^ fFig. 2Z-rX ----►Kip ;i Solutions of the equations (17) and (18)----------- for Z =  - i ,  Y -  -1forZ Y - ~ i—o— o— for Z -  —  Y =  0adsorbed gaa. Hence, a asfiumiiig reasonable values of V, X.^  and have been (calculated with the help ol equations (17) and (IS) It is thus possible to have same energy for different nature of adsorbed gas.K  E  S U  L T  S A  N  V) U I S  C  U  S H I  O  NThe plots in Figs. 2  and II represent the results of our icaleulationB. It  should be mentioned that onl)" the solutions that haAn^  a pliysical meaning are plotted Fig 2 represents the solutions foi' a definite value of atomic potential Z, and positiA’-e values of interlace potential Y ; Fig. 3 shows tlu' solutions for a definite value of atomic potential 7j with zero and negative values of interface potential Y. For positive values of Y  one finds the localized energy levels at the interface when there is only adsorbed gas at the clean interface ol two crystal­lites of identical nature and for negative values of Y  there are adsorbed gas and also an impurity at the interface The plots are interrupted at certain values of V because we have restricted the calculations to the first forbidden gap.The family of curves in Figs. 2  and 3 can in gtmeral be explained if we consider that when the potential strength of the adsorbed impurity is smaller than that of the hulk, an electron can escape more easily because it is more loosely bound and there is possibility of larger interaction with the neighbouring atomic potentials. On the contrary Avheii the potential strength of adsorbed impurity is deeper than that of the bulk, the electron is more tightly bound to the imjmrity and the possibility of interaction with the neighbouring atomic potential decreases. Tt is, however, not possible to explain every detail of the curves only on the above (jonsiderations. Tt is striking to note that the energy levels move towards the conduction, creating degenerate levels near the conduction band as the impurity potentials become deeper. This means the creation of n-type levels by highly p-type impurities Avhen they arc consecutively placed in the atomic lattice or are in close proximity. It was experimentally observed by (Ihosh (I960) that there IB a possible existence of rt-type regions in the films deposited from highly 2>-type tellurium The existence of w-type tcDurium as such is not found, but it appears from the above discussion that an »i-type region is possible in a film deposited from highly p-type materials at places where there is an accumulation of impurity atoms or a highly p-type gas adsorbed at the interface of two crystallites.Energy States at the Interface of two Semi-Infinite, etc. 539R E F E R E N C E S  Aerta, E ., 1960, Physica, 26, 1063-Ghosh, S. K .,  (1960) Thesis subnutted for tho Ph .D  degree a t the Indian  In stitu te  of Technology, Kharagpur.Saxon, D. S and H utner, R . A., 1949, Philips Res. Jtep. 4, 81.Shockley, W ., 1939, Phya. Rev., 66, 317 Tam m , I . L . ,  1932, Phya. Z. Bowjetuiuon, 1, 733.

Energy States at the Interface of Two Semi-Infinite One-Dimensional Crystals

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/3981/1/Energy%20States%20at%20the%20Interface%20of%20Two%20Semi-Infinite_By%20S%20K.%20Ghosh.pdf

Energy States at the Interface of Two Semi-Infinite One-Dimensional Crystals

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