Thapa, R K

Zoliana, B

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Indian J. Phys. 82 (6), 729-734 (2008) 111 J Pi Application of projection operator method to define basis functions for use in photoemission calculations B Zoliana1* and R K Thapa2 •Department of Electronics, Govt Zirtin Residential Science College, Aizawl, M,zoram-796 001, India department of Physics, Mizoram University, Aizawl, Mizoram- 796 001, India E-mail bzohana(« yahoo com Abstract : We have applied projection operator method of group theory in deriving the symmetry adapted wavefunctions defined by ¥l which represents the surface electronic states in the case of a strong periodic potential. The method involves the defining of basis function pertaining to a particular symmetry point of Cu(110), from which v, is formulated by the LCAO method This has been applied to calculate photocurrent from Cu (110) surface state. We have also used this method of calculation for a metal like Al Keywords : Periodic potential, projection operator, wavefunctions, photocunent PACSNo. : 79.60.Bm 1. Introduction For the use of Projection Operator for the case of strongly bonded metals, we define the crystal potential by a Kronig-Penney 8 -potential which is periodic with the periodicity of i 10 . ..AJ mcfal 1 1 I' L ,1 i-i i va< r V(y) • - » FJgurt 1. Schematic representation of Kronig-Penney 3 -potential model. * Corresponding Author ©2008IACS 730 B Zoliana and R K Thapa the lattice as shown in Figure 1. In our calculation, the Schrodinger equation will then be solved to obtain the solution in terms of Green function [1]. The method involves the projection operator operating on linear combination of atomic orbitals (LCAO) representation of the surface states giving the basis function and this basis function has been incorporated in defining y/t. This had been applied to calculate photocurrent from the Cu (110) surface state. The calculation has been carried out in one-dimension and it was found that the basis function derived in one dimensional form can be extended to a metal like Al. 2. Formalism 2.1. Derivation of basis function by projection operator method: For Cu (110), surface state occurs [2] in L? - L, band gap. The LCAO representation for L2- point is -r=r(x+ y + z) . The projection operator formula [3] is given by v3 y T Here lp is the dimension of the unitary irreducible representation of the group g, g is order of g and 2 ^ extends over all the transformation T of g. T Eq.(1) operating on the LCAO representation for L2, gives a set of basis functions —f=\x + y + z). This basis function in one dimensional form is incorporated in the derivation of initial state wave functions. 2.2. Derivation of initial state wavefunction: The solution of Schrodinger equation in terms of Green function, in one dimensional form may be written by the integral equation V{y)\i/(y)G(z,y)dy m (2) Let the Kronig-Penney potential be represented by a linear Dirac 5 -function i.e V{y)>*[2£] X*(y-na) (3) where P = Km ^:Xt * b with Xo : = 2 ( v i ~ Ei). Here, b is the width of Kronig-Penny potential b-~*0 £ moad and V1 is the height of the potential, p is the strength of the s -function barrier Application of projection operator method to define basis functions etc 731 and it is assumed to be positive. The final electronic wavefunction after substitution of the above value (eq.3) in eq. (2) gives ^ (z )= -^Lre / / c a s in /c / z~s in /c / ( z -a ) l iM sin/c,aL ' /V 'J (4) 2C/psin2fc,a where r(0) = —— - —_ - . (5) kla(e,ka-e~,H,a)(ooska-coskta) Introducing the atomic orbital <j>(z) which includes the basis function derived by P.O method as discussed above the initial state wavefunction can be represented by , . IvWQWe*'^ k, cos k,z- xs\n k,z), z<0 (bulk& surface) [if/l(0)klexz, z>0( vacuum) v ' 2/sin/ca r(0) kt cos k,a - k,e~lk a - x sin k,a where * , « » - , --_,z _._,;.' ..._.._• < 7 > 2.3. Photocurrent calculations: The photocurrent density formula from golden rule approximation can be written as ffi.^ry^^iH'I^H2 5(E-Ef)S(Ef - E, -hw)10(E -»<«>)[1- f0{E)] (8) d£2 ti ^ where ^ ( fF , ) refer to the initial (final) state wavefunctions and perturbation H ' can be written as mc "*.«5+i!*M (9) where > \ , ( z ) = ^ d ^ , / ^ ( z ) is the component of vector potential along z-axis. The final state wavefunction |V,) used is the scattering state of the step potential[4] which is encountered by the electron. The matrix element / - (v, | H' |y/,) in eq. (8) can be written in one dimensional form as : The integrals in Eq. (10) are evaluated numerically by writing FORTRAN programmes. 732 B Zoliana and R K Thapa 3. Results and discussions 3 1 Copper The plot of photocurrent against the photon energy ho) for Cu is shown in Figure 2 For the same value of surface state [5] energy (2.72 eV below Fermi level) and potential barrier height (11.4 eV) , photocurrent was calculated for two values of surface widths We find that for d= 10 a.u., a maximum in the value of photocurrent occurs at fico = 14 eV. With further increase of photon energy, photocurrent decreases to a minimum value at h(D = 19 eV and the next hump occurs again at ttw = 23 eV But for a narrow surface width (of = 0), the behaviour of photocurrent is quite different as shown in Figure 2. We do not find any peak for values of photon energy below and above 19 eV The behaviour of photocurrent shows a qualitative agreement with those shown by other metalslike W, Si, etc. in which Kronig-Penney potential model was also used [5] its) «£, | R 710E+00 7 00E+00 6 90E+00 6 80E+00 6 70E+00 6 60E+00 6 50E+00 6 40E+00 6 30E+00 6 20E+00 10 15 20 25 Photon Energy (eV) 30 Figure 2. Plot of variation of photocurrent with photon energy for surface widths d = 10 a u and d = 0 in copper using Kronig-Penney 5 -potential where y/t is derived by projection operator method of Group Theory 3.2. Aluminium: Figure 3 shows the plot of photocurrent versus photon energy (hco ) for the surface width d= 10 a.u and narrow surface width d= 0 a.u. We have chosen the initial state energy £, for a high lying surface state occurring at 0.41 eV below Fermi level, the location of the Fermi level and the work function 0 for AI are taken as 11.7 eV and 4.25 eV, Application of projection operator method to define basis functions etc 733 respectively. The photocurrent profile for d = 10 a.u. showed a strong photoemission at photon energy too = 9 eV, which was followed by a decrease in photocurrent and the minimum occurring at too =15 eV (the plasmon energy of Al is lnop = 15.3 eV). There is another hump in the photocurrent data at hco = 20 eV. The behaviour of photocurrent in the case of narrow surface width is quite different. For example, almost a flat peak in a photon energy range 10 eV-14 eV was obtained. Although the minima in photocurrent was seen at hco = 15 eV, contribution to photoemission was far less important. However the photoemission results for d = 10 a.u. seems to be in qualitative agreement with the experimental results of Levison et al [7] (see Figure 3 (inset)). For example, the experimental data of Levinson etal [7] showed a maxima in photocurrent at hw = 13 eV with occurrence of a minima at the plasmon energy (^ 15.3 eV). Also, it showed similar behaviour with the calculated results of Thapa et al [8,9]. 5.00E+01 4.50E + 01 -4.00E+01 — 3.50E + 01 to E £ 3.00E + 01 S 2.50E + 01 8 o 2.00E + 01 CL 1.50E+01 1.00E + 01 5.00E+00 0.00E + 00 10 15 20 25 30 Photon Energy (eV) Figure 3. Plot of photocurrent variation against photon energy In the case of aluminium, tor surface width d = 10 a.u. and d = 0 by employing Kronig-Penney tf-potential where v, is derived by projection operator method of Group Theory. Inset • Experimental results for aluminium and beryllium as obtained by Levinson et al [7]. 4. Conclusion In our calculations of photocurrent, we have seen that photoemission was considered to take place in metal, for location of surface state below the Fermi level. Symmetry state or direction to the existence of surface state in a particular band gap was ignored. Therefore, to include band structure effect, we have applied projection operator method of T — i 1 — i 1 r \^^Jb^o 06 0? 0J 0V tO M 12 * * • > V -7 N \ k»-x 734 B Zoliana and R K Thapa group theory to derive initial state wavefunction ^, pertaining to a particular band gap When compared to experimental data and other previous theoretical results like Kronig* Penney potential model of Thapa et al [6], we find that calculated data with this approach of calculations seemed to agree in a better way. This is quite evident from the photoemission result from Al especially for which also experimental result is available. However, there are still certain drawbacks remaining for further corrections. For example, we have not considered the nature and type of potentials which defines the bulk and surface regions This plays equally important role in the definition of dielectric response function too. From all the above facts, the model employed in our calculations seems to be highly simplified However, as evidenced from the comparison with other approach of calculations, as well as with experimental results [7], our method seems to be quite appropriate. Acknowledgment B Z acknowledges Minor Research Project from UGC and RKT a grant from CSIR (Delhi) References [1] S G Davison and M Steslicka Basic Theory of Surface State (Oxford : Clarendon) (1992) [2] E Bertel Phys. Rev. B50 4925 (1995) [3] J F Cornwall Group Theory and Electronic Energy Band in Solids (Amsterdam, Holland North Holland Publishing Company) (1969) [4] B Zoliana A Theoretical Formulation of initial Statewave functions for photoemission calculations, Ph.D. Thesis, NEHU (2004) [5] E W Plummer and E Eberherdt Advance in Chemical Physics 49 533 (1982) [6] R K Thapa and N Kar Phys. Rev B51 17980 (1995) [7] H T Levlnson, E W Plummer and P J Feibelman Phys. Rev. Lett 43 952 (1979) [8] R K Thapa, P Das and N Kar Mod Phys. Letts. B8 361 (1994) [9] Zaithanzauva Pachuau, B Zoliana, P K Patra, D T Khathing and R K Thapa Phys. Letts A294 52 (2002). 

Application of Projection Operator Method to Define Basis Functions for Use in Photoemission Calculations

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Application of Projection Operator Method to Define Basis Functions for Use in Photoemission Calculations

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