D T, Khaihing

P K, Palra

Pachuau, Zaithanzauva

R K, Thappa

Zadeng, Lalthakimi

Zoliana, B

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Indian J. Phys. 77A (4). 393-396 (2003)a \O  f o t e5 U PPhotoemission studies of metals by using Mathieu potentialB Zoliana', Zaithanzauva Pachuau', Lalthakinii|Zadeng. P K Palra=, D T Khaihing' and R K Thapa»Condensed Matter Theory Research Group. OepanmcntLf Physics, Pachhiinga University College. Mi/oiam Umvei-sKy,Aizawl, Mia|>ram-7% 001, India^Department of Physics, Govt ZiiUri Residci^al Science College, Ai/awl, Mi/oram-7yo 001, India ^Science Education Centre, North-Eastern llill University, Shillong-7^3 00^, Meghalay, India ^Department of Physics, North-Eastern Hill University, Shillong-793 022, Meghalay, IndiaE-mail rktt (Q’sancharnetinReceived 25 Fehruarv 2003, auepied 7 Manh 2003Ab,stract : Photocurrent results from ferromagnetic materials F c and Ni and d-band metals Pd and Cr is discussed which is calculated hy using the Mathieu potential for the crystal A local dielectric functions have been used to calculate the electromagnetic field lor the bulk, surface and vacuum regions of the metals.Keywords : Photoemission, metals, Mathieu potentialPACS Nos. ; 79.60.Bm, 77.22.ChMathieu potential had been applied to photoemission calculations by Pachuau et al [1] in the case of empty potential and a strong potential [2]. These cases had been applied to free-elcctron type metals like Al and Be for empty potentials and d-band type metals Cu, W, Mo and also wSi. It has been seen that surface photocffcct had been exhibited in these metals for values of incident photon energies below and above the plasmon energy but within low photon energy range. With this view in mind» wc present in this Note, the extension of Mathieu potential model developed to the calculations of photocurrent from ferromagnetic metals Ni ^hd Fe and transition metals Pd and Cr.The photocurrent density formula [ 11 from golden rule approximation can be written asdj{E)d ax S ( E j - E i ^ h < o ) f o ( E - m U - M E ) \ , ( l )where ) refer to the initial ■(final) slate wavefunctions,perturbation H' can be written asIriLX{A .p + p.A).where m,. is the mass of the electron, p  the onc-elcctron momentum operator and A is the vector potential of the incident photon field. With simple modifications, the photon field used in our calculations for three regions can be written asAa(0),x) =Ai.C(Q)).d[1-£(<«)] A,.e(ca),x < - d  (bulk), - d < x ^ Q  (surface), (2) (vacuum),where /4| is a  constant depending on the dielectric function e(co), photon energy h(0 and angle of incidence d, e((0) is defined ItKally and interpolates linearly between the bulkCotrespooding A uthor © 2003 lACS394 R Zr)liana et aland ihc vacuum values. Considering onc-dimcnsional crystal, its potential is represented by V(a) = cos2a:x:/u , where a is the period of the crystal potential and maximum (repulsive) value is at jc = 0. The one dimensional SchroedingerVs equation can now be written as-f («2t/cos2z)*P = 0 ,/•/r ^ (3)where 2<7 =an_  ;cv j  _a * \  aEY ■ Here, q is themagnitude of the potential which measured its strength. Eq. (3) can be solved for obtaining initial state wavefunction for the bulk and surface regions {x< 0 )  and also for the vacuum regions. These wavcfunctions as derived are given by (in atomic units) :1/21 +  V y - / z ( .X ( > - A )  X(\16 640  ^ /  ^ (4)X > X o .The derivation of H^iix.q) had been discussed in detail by Pachuau [2,31The step potential at the surface encountered by an electron is defined by V(a) = -K o^ (.y), where 6>(jc) is unit function such that 6(x) -  1(0) for .y > 0 (a < 0). The final state wavefunction which i$ the solution of this step potential and is given by (in atomic units) :Vf ( x )  = \lltq ,2nq,where fc? = 2 £2q,q f +k,•I f 'JT<0(bulk and surface).q f - k* k ,L l - ^ rq j = 2 { E f-V o ) a n d E ,j f > 0  (v a c u u m ) ,(5)Ei + hcoobserved that maximum in photocurrent is obtained at tiO)- 15 eV for d = 10 a.u. For further increase in photon  energy, photocurrent decreases and becomes minimum at fiCO-\9 eV. A second peak of small height in photocurrent is obtained at ho) == 24 eV. Similarly in the case of Ni also.FeFigure 1. Vanation of photocurrent against photon energy tor surface width d -  10.0 a.u and narrow surface width (d = 0. 0 a.u.) in the case of Fewe have plotted (Figure 2) photocurrent as a function ot photon energy for = 10 a.u. For Ni, we find a maxirnuni in photocurrent at photon energy ^ m = 1 5 e V , hut it decreases towards zero as photon energy is further increasedNi9.ME40&In eq. (5), the factor is included on the surfaceand bulk side to take into account the inelastic scattering of the electrons.Photocurrent was calculated from metals Fe, Ni, Pd and Cr as a function of incident photon energy (fta>) for two different values of surface width d = iO a,u. and d  = 0.0. The real and imaginary parts of dielectric constants were ca lcu la ted  for each o f these m etals by using the experimentally determined values [4] of refractive indices and absorption coefficients (extinction constant). However, we have used the same surface parameters £ ,,  0 , for all the metals as was used in the case of Cu.Referring to Figure 1, we have plotted photocurrent as a function of photon energy (tico) in the case of Fe. It ist2 IS 90 34Photon energy (eV)Figure 2. Same as in Figure 1 in the case of Ni.Besides these two magnetic metals, photocurrent was also calculated in the case of Pd and Cr with the same initial parameters as was used in the case of Fe and Ni. The behaviour of photocurrent is shown in Figure 3. For surk«ee width d =  10 a.u., we find that a maximum in photocurrent occurred  at Afi)s=8eV, decreased  to minimumPhotoemission studies o f metals by using Mathieu potential 395cV. It showed a small hump in photocurrenl for eV and decreased to minimum again for further inLicasc of photon energy. Photocurrent data of Pd showed behaviour exactly similar to that obtained by Thapa [5] but hy using the Kronig-Penney potential model. For narrow sijiface width, photocurrent data showed a constant val^e also for Pd when plotted against photon energy. \Pd 'fWe show in Figure 4, the plot of photocurrent as a liinclion of photon energy in the case of Cr. F o r d -  10 a.u., there is a peak in photocurrenl obtained at h o ) - \4  eV. Again, photocurrent decreased to minimum at h (0 ^ \9  cV  and showed a second hump of lower magnitude at hco = 27 eV. Cr also showed similar behaviour for narrow surface width as seen in the case of Fe, Ni and Pd.CrPhoton energy (eV)Pigurc 4. Same as in Figure I in the case of Cr.We find from our results that in the low photon energy range all the metals showed similar trends in the behaviour photocurrent for both the surface widths. The origin of first peak at = 8 eV in the case of Pd is attrtibuted tosurface photoeffect as explained by Thapa [5] This had been evidenced by the plot of field for photon energy below and above 8 eV. The decrease in photocurrent to minimum after the first peak in all the metals is due to loss of photon energy by excitations of bulk plasmons. Plasmons in metals and semiconductors play a significant role in transport and optical properties. Surface plasmons and in general, the dynamical screen properties at the surface attracted therefore considerable attention [6,7]. Surface plasmons in particular, arc of importance as it is linked to the position of the centroid of the screening charge [8] at the surface plasmon frequency. Wc assume the surface plasmon frequency {(Os) to be the value of photon frequency at which photocurrenl is maximum and this is related to bulk plasmon frequency (cOg) by (0^ ss cogyfl. Further, it had been already shown from electrodynamics of surface phenomena [9] that at frequency co^ (O/ ,^ the component of electromagnetic field (vector potential) tends towards minimum which causes the photocurrent to be also minimum. We find that the metals under study, satisfy this condition although bulk plasmon frequency (energy) is not a well-defined quantity. Here, we have considered it to be equal to the value at which C| —> 0.We have also plotted photocurrent as a function of hco for narrow surface width Le. /^ = 0 a.u. by including Fresnel fields for all the metals under study. 7’he behaviour of photocurrent in this case, was found to he quite different. There is no peak in photocurrent found as in the case for d = 10 a.u. The photocurrent in this case, is constant and remains parallel with the x-axis. This indicates that inclusion of field is important in photoemission calculations.Wc have tried in this report to see the effect of Mathieu type of potential on photoemission calculations. Wc find that photocurrent results from these metals showed similar behaviour as had been obtained by 7bapa et al [10| in the case of other metals and semiconductors [11]. However, the main drawback of the model presented here is that the same initial state wavefunction is used both for the surface and the bulk regions of the metal to calculate the photocurrent. We are working to represent the initial state wavefunction accurately for these regions by applying the projection operator method of group theory (12].AciuiowledginentsRKT and PKP thanks University Grants Commission, New Delhi for a research grant. Prof, S G Davison and Pruc Davison, Department of Applied Mathematics and Physics, University of Waterloo, Ontario, Canada, have helped with many relevant literature connected with the works reported here.m BZolianaeialBefernMH(1] Z l^duuiu, B Zoiiana, D T Khatmg, P K PBtta ami R K Thtpa, Pkys. Un. KXn 459 (2000)(2] Zoithanzauva Paciwan, B Zoliana, P K Palta, D T Khatiag and R K Thapa U tt A294 52 (2002)(3] Zaithanzauva Pachuau PhD Thesis (unpublished) (North-Eastern Hill University, Shillong, India){4] } Weaver Hmibook of Chemistry and Physics (Boca Raton ; CRC Press) (1987)[5] R K Thapa Phys. Stat. M  B179 391 (1993)(6) P J Feibelman Prog. Surf. Sci. 12 287 (1982)P*] P Apell, A Ljongbeit and S Lundqvist Phys. Scripta 30 367 (1984)(8] M Rocca and F Moresco Prog. Sutf. ScL S3 331 (19%)[9] Zaithanzniva Pachuau, B Zoliana, P K Patra and R K Thapa 3rd Regional Conference on Physics Research in Narti,. East (Dibrugarh University, Dibrugath, India. 9th November 2002)[10] R K Thapa and N Kar Phys. Rev. B51 17980 (1995)[11] R K Thapa and N Kar Mod. Phys. Lett. B8 361 (1994)[12] B Zoliana, Zaithanzauva Pachuau, P K Patra, D T Khotini; and R K Thapa 3rd Regional Cot^ rence on Physics Restart h tn North-East (Dibrugarh University, Dibrugarh, India, 9ih November 2002); B Zoliana, Z Pachuau, R K Thapa, Laltliakimi Zadeng, P K Patra and D T Khathing liul. J .  Mod. Phys n (accepted)

Photoemission Studies of Metals by Using Mathieu Potential

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Photoemission Studies of Metals by Using Mathieu Potential

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