Gurung, Shivraj

Khaling, D T

Pachuau, Zaithanzauva

IACS Institutional Repository

Indian J. Phys. 73A (2), 231-243 (1999) NOTEU P  Aan international journalPhotocurrent calculations in semiconductors using Kronig-Penney modelZaithanzauva Pachuau", Shivraj Gurung, R K Tliapa, D  T  Khaling* and N Kar''Depaximeni of Physics, Pachhunga University College, North-Eastern Hill University, Ajzawl-796 001, Mizoram, IndiaReceived 16 December 1997, accepted 21 Oct ether 1Q9HAbstract : Photoemission calculations have been done usjng the Kronig-Penney model from band state (Fermi level) of semiconductors silicon and gallium arsenide. For the evaluation of pilotocurrent the initial state wavefunction u.sed is tlie one deduced by lhapa and Kar [Indian J PureAppl. Phys. 26 620(1988)] [1].Keywords : Photoemission, semiconductors. Kronig-Penney modelPACS No. : 79.60.BmThe existence of surface states on semiconductor surfaces was experimentaJly verified by using the angle integrated photoem ission [2], M oreover their existence is obvious through the pinning o f the Fermi level at the surface. However, the semiconductor surfaces are more complex than metal surfaces for the reason that semiconductor surfaces reconstruct 13], The presence o f these reconstructions and atom ic displacem ents on sem iconductor surfaces makes the studies of electronic structure a very interesting topic. O f the various tools and techniques, angle  reso lved  photoem ission  studies has also  been w idely used in understanding the surface states o f semiconductors. Various type of structural m odels of sem iconductors have been proposed [4], But in this short report, we will be m ainly concerned w ith the photoem ission studies by adopting a  sim ple calculational procedure which will be applied to  the case o f silicon and gallium arsenide.* Department of Physics, Government Aizawl College, Aizawl-796 001, Mizoram, India* Department of Physics, North-Eastern Hill University, Shillong-793 022, India Department of Physics, North Bengal University, Darjeeling-734 430, India® 1999 lACS238 Zaithanzauva Pachuau et alFor photoemission calculations the current density may be written with the help o f the golden rule expression [5] asdj(E)dQ X  S  S i E - E f ) 5 { E f  - E i  - fi(0 )(1)x / o  ( £ - « © ) [ !  - / o ( £ ) ] .where | ) (| <^/ ))  refer to the initial (final state), A = (e /2  mc){A.p + p.A), a n d p  beingthe one-electron m om entum  operator, A  the vector po ten tia l o f the photon  field . A lthough the one-electron states are treated quite accurately in m any photoem ission calculations, the variation o f  the photon fields in the surface region is usually neglected. The m odel dielectric function which takes into account the bulk, surface and vacuum  regions is given [6] bye(co)  3  e{(o,  z)  =f  1 + ie2 .1 + [1 -C (0 ))]- |,1,z < - a- a ^ z ^ Oz^O .(2)We have used the Drude-Lorentz model for calculating the frequency dependent dielectric constants which is given by£(a) = e„ 1 -0)‘0)((0 + iyi )(Cq -e ^ )Q } l(3)In eq. (3) above, fo  and e„ are the static and high frequency dielectric constants. By using tlic appropriate values o f constants Co, , X i, etc respectively for silicon and galliumftfsenide. the real and imaginary parts o f e(co) were calculated by using eq. (3) which were then applied to eq. (2) for computing the fields.Using the electrom agnetic field for p-polarized  radiation, we calcu late  the phoUiemission cross section by evaluating the matrix e lem en t:( i/T /|a | v^ , ) = J  yf*f ( z )A\ i / i ( z )dzr y / / ( z ) A ^ ( z ) ^ ^  + ^ ^ Y , ( z ) \ d z ,  (4)where |6]4 ; ,A^(Z)  =ae(co)\ \ - £ ( ( i ) ) ] z  + a 'A ie(© ),z < - a- a < z < 0z > 0 .(5)At is a constant whose value depends on the angle o f the incident radiation (0,) and its frequency (m) and dielectric constants e((o). In the surface region (-0  £  z £  0), the dielectricPhotocurrent calculations in semiconduaors etc 239function is linearly interpolated between the vacuum and tbe bulk values, in eq. (4) is the tree-electron final-state wavefunctitm given by2gq + k f  *i d - k f  ^e"’' + Jz < 0z > 0 (6)where k j  = 2Ef ,  = 2(Ey - V o ) a n d £ /  = E, +k ( 0 .To evaluate the initial state w avefunction v^,(z), one solves tbe one-dim ensional Schroedinger’s equationd-  y/(z)dz-+ k"-(z) = -2V(z)\f /(z), (7)where k f  = 2E,  and  V(z) is the 5-function potential o f  the K ronig-Penney (K -P ) model.Let 0(z) denote the Bloch wavefunction deep inside the metal and ^*(z) the tim e reversal o f 0  (z). The eigenfunction in the semi-inflnite solid (z < 0) was chosen [7] to have the form  v^, (z) = 0 (z ) -  P 0 * (z ) , where P is the reflection  coefficient obtained by matching the wavefunction and its derivative at z = 0. One can then show [1] that tbe initial state wavefunction for z < 0  m ay be written asVr,(z) = s in 5 )c * '*  -  ( P - i e ' *  sin5 )e~ * '* . (8)where cot 5  = ^  being the strength o f the potential. The initial-state w avefunctionSoutside the metal (z > 0) isVr, (9)T being the uaiism ission coefficient across the boundary plane and = 2(Vo -  £ ,), where V,) is the surface step potential. From  tbe m atching conditions a t z 0, one can easily deduce the values o f P  and T  in eqs. (8) and (9) and write tbe m ost explicit form o f  initial state w avefunction I v^ >. Tbe photoemission cross section was obtained viad ad ai i0) (10)The m atrix elem ent / = ( | A | v r , ) in eq. (10) can be expanded asr C  • A , • A ^ ¥ i  j  1 f° .  dAatlz) .I -  i L * ' / - * - * ' " *(11)73A(2)-15240 Zaithanzauva Pachuau et alIn calculating the pbotocurrent, these integrals were evaluated analytically w herever possib le , o r num erically  by developing a  FO R TR A N  program m e. To ensu re  the convergence, a  factor o f (oris the .scattering factor) was introduced for z < 0 which represents the effect o f inelastic collisions.Since it is strictly a model calculation, we have used the same set o f data (in atomic units) both for silicon and gallium arsenide. The data used are ;Fermi level (Ep) = 0.463,W ork function (0) = 0.198,Surface width (a) = 10,Height o f the potential barrier (Vb) = 0.662,Strength o f the potential barrier (g) = 0.60,Phase factor (5) = -0.5776,Scattering factor (cO = 0.35.W e have chosen the initial state energy (£,) to lie at the Ferm i level. The values o f the parametCTS g, 5 and a  are chosen arbitrarily to fit in such a  way that it can reproduce qualitatively the nature o f  the pnoton energy dependence o f the square o f the field. The reason for this being that the matrix element for photoemission cross section is a quadratic function o f the photon field.In Figure 1, we have shown the plot o f photocurrent as a  function o f photon energy {h(o). For the surface width a = 10 a.u., we find that at fioi = 12 eV, photocurrent peak isSIHconflwlon Energy (eV)Figure 1. Plots of log (photocurrent) (in arb, units) as a funcuon of photon energy for silicon for surface widths a = lO.O a.u. and 0.0 using the Drude- Lorentz dielectric model.Photocurrent calculations in semiconductors etc 241maximum. It .shows a  minimum at ^ m = 16 eV followed by a small hump at 17 eV. Further increase o f ^o) cau.ses the photocurrent to decay towards a  minimum value. We have taken the plasmon energy (hcOp) of silicon to be 16 eV. Interesting feature that is seen here in the case o f silicon is that photocurrent data .showed behaviour similar at least qualitatively with the earlier results [8] when fields were calculated by using thp experimentally m easured dielectric constants. For example, the maxima in photocurrent was obtained at fi0 <i.e.. at 12 eV photon energy followed by a minima occurring 3kfico= ti0 p. F'or the case of narrow surface width (a = 0.0) the behaviour in photocurrent Is completely different. We find that a peak o f low height in photocurrent occurred but af plasmon energy of silicon. This is quite different behaviour obtained than with the one 18] calculated by using the experimentally determined dielectric constants.Gallium Arsenide610 12 14 16 16Photon Energy (eV)20Figure 2. Plots of log (pholocurrent) (in arb. units) as a funcuon of photon energy for galhuin arsenide for surface widths a = 10.0 a.u. and 0.0 using the Drude-Lorcntz dielectric model.In Figure 2, the plot o f photocurrent as a  function of photon frequency in the case o f gallium arsenide is shown for two different surface widths a = 10.0 a.u. and a =  0.0 respectively, bu t with the same values o f Kronig-Penney parameters as were used in the case o f silicon. T he photocurrent structures o f  gallium  arsenide data show ed three peaks at = 10, 12 and 15 eV respectively but showing a  m inim a at hm = 16 eV (the a.ssumed plasmon energy o f gallium arsenide). Another bump in the pholocurrent is seen at 17 eV beyond w hich the photocurrent decreases gradually. The photocurrent data for242 Zaithamauva Pachuau et alnarrow surface widtb (a »  0.0) is found to be similar as in tbe case o f  silicon. The reason for this being that for a »  0.0, both silicon and gallium  arsenide bad alm ost sam e values c i  dielectric constants as calculated by using tbe Drude*Lorentz formula.Photoemission is considered basically to be a  surface effect. It is  dependent m ainly on the spatial variation o f the photon field vector at tbe surface region. W e find therefore that in the case o f  silicon, tbe nuxlel dielectric response function o f Drude-Lorentz seem s to  work quite fittingly for values o f photon energies below and above tbe plasm on energy. However, su ik ingly  different behaviour is seen in pbotocurrent in tbe case o f  gallium  arsenide. For example, we fm d three peaks in pbotocurrrat iotiim<h(Op& result seen quite different frc»n that o f  silicon [8J and other m etals like alum inium  [9], palladium  [10], tungsten [8] etc. Tbe reason for this m ay be attributed to tbe fact that tbe E)rude-Lorentz m odel for calculating the dielectric constants is not applicable to the case o f gallium  arsenide. The other reason for tbe occurrence o f such peaks may be that for h(o < fitOp the photon field vector A Jx)  as deduced by using tbe m odel o f  Bagcbi and Kar [6] is not applicable to tbe case o f compound semiconductcnr. W e cannot rule out tbe weakness o f tbe K -P  m odel potential as used by Thapa and K ar [1] for deducing tbe in itita l state w avefunction \ff,. The effect o f the surface was no t taken into consideration  w hile formulating yr, for a  semi-infinite solid. However, tbe occurrence of peaks dXh(o<ha>p both for silicon and gallium arsenide can be attributed to the spatial variation o f vectw  potential. Lapeyre and Anderson [11] bad also experimentally found the existence o f surface state in gallium arsenide from tbe constant initial state spectroscopy. Tbe com plicated line shapes in their m easurem ent for the surface states w ere no t fully understood. H ow ever, the conclusion found in their measurement was that tbe photoemission intensity w as strohgly a polarization dependent.Though we have not taken into consideration the effect o f tbe type o f  semiconductor, density etc, however, we find that in the case o f sem iconduaors, tbe spatial dependence o f  vector potential is an essential ingredient in photoem ission calculations. Instead o f  using the sim ple type o f dielectric form ula like that o f Drude-Lorentz type, it w ould be m ore realistic if  one can em ploy tbe m ethod as developed by Cappellini [12] et al which is specifically defined tally for tbe semiconductors. Further, tbe inclusion o f  structure into this type o f  calculations w ill enable one to  com pare the data  w ith experim ent in a  m ore appropriate way. For exam ple, a detailed  study o f  pbotoem ission from  sem iconductor gallium  arsenide by using the one-step model o f pbotoemission had been done by Schattke[13]. He had used the Green function-form alism  to the valence states in LC A O  basis by taking photon field vector as constant in dipole approximation. Tbe pbotoem ission data for the ideal gallium arsenide surface agreed well with the experimental data.AcknowkdginentsRK T adcnow ledges die sanction o f a  research grant and SG the award o f a  fellowship from D qiartm en t o f  Science and Technology, New Delhi.Photocurrent calculations in semiconductors etc 243References(1] R K Thapa and N Kar Indian I  Pure Appl. Phys, 26 620 (1988)(2] F J H infsel Appld. Phys. A38 2QS (1985); L F Wagner and W E Spicer Phys. Rev. Lett. 2 S 1381 (1972)(3] G P Srivastava Rep. Prog. Phys. 60 561 (1996); E W Plummer and W Ebertiardt Adv. Chem. Phys. 40 533(1982)(4] K C Pandey and J C Phillips Phys. Rev. Lett. 32 1433 (1974); D Haneigu Phys. Rev. 121 1093 (1961)(5] D R Penn Phys. Rev. Lett. 28 1041 (1972)|6] A Bagchi and N Kar Phys. Rev. B18 5248 (1978)[7] N W Ashcroft and N D Manran Solid State Phys. (Japan: CBS PuWising) p 146 (1976)[8] R K Thapa and N Kar Surf. Sci. 338 138 (1995)[91 P Das, R K Thapa and N Kar Mod Phys. Utt. B35 65 (1991)[lOj R K Thapa Phys. Slat Sol. (b) 179 621 (1993)111) G J Lapyere and P J Anderson Phys. Rev. Lett. 35 117 (1975)[12] G Cappelhni, R Del Sole. L Reimng and F Bechstedl Phys. Rev. B47 9892 (1993)(13) W Schattke Prog. Surf. Sci. 54 211 (1997)

Photocurrent Calculations in Semiconductors using Kronig-Penney Model

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Photocurrent Calculations in Semiconductors using Kronig-Penney Model

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