Kar, N

Thapa, R K

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Indian]. Phys. 6 4 A  (4 ) , 3 2 1 -3 2 4  (1 9 9 0 )Electromagnetic fields near surfaces in a simple model : the case of aluminiumR K Thapa and*N Kar -Department of P h y s ics , Pachhunga U niversity  College, North Eastern H illU n iversity , A iza w l-7 9 6  001 , M izoram , IndiaReceived  9  November 1 9 8 9 , accepted 21 February 1 9 9 0In angle resolved photoemission, the photon field responsible for photoemission may play an important role especially when the spatial symmetries of the initial and final states are known. If the photoemission cross section is measured as a function of photon energy then the behaviour of the photon field may become crucial for explaining the experimental data. One example is the photoemission from the surface state of the tungsten (1(X)) surface. The data of Weng et al (1977) for p-polarised light showed a pronounced dip near the plasmon energy which was interpreted in terms of refraction effects by Bagchi and Kar (1978) using a simple model. The calculation of electromagnetic fields in the presence of real metal surfaces is a formidable problem. However, ab initio calculations have been done for jellium (Feibelman 1975a, b, Mukhopadhyay and Lundqvist 1977). Levinson et a/(1979) used these results for jellium for comparison with the experimental data for aluminium and showed that the variation of the photon field in the surface region is extremly Important for the explanation of the variation of photo-current as a function of photon energy. However, the ab initio calculations have been performed only for jellium and even this is an extremly complex computational procedure. The standard methods for calculation of photo-current, for example, that of Pendry and Hopkinson (1978) neglected the variation of the photon field in the surface region. The model of Bagchi and Kar (1978) which involves a linear interpolation in the surface region between the experimentally measured frequency-dependent bulk value of the dielectric constant and its vacuum value is relatively easy and can be used for both metals and semiconductors. It was shown to be successful in explaining the qualitative behaviour of the photo- emission cross section for the surface states of tungsten. We propose to use the model for bulk states also. A s a first step, we show that for aluminium the use of this model Indicates that the photon field near the surface shows qualitatively the same behaviour as that obtained from the photo-current measure­ments.'Department of Physics, North Bengal University, Oarieeling-734 430, West Bengal, India321322 R K Thapa and N KarThe model used by Bagchi and Kar (1978) is a fairly simple one. The z-direction is taken to be the normal to the nominal surface plane which is chosen to b e th e z = 0  plane. The metal is assumed to occupy all the space to the left of the z - 0  plane. It is assumed that the response of the metal to an electro­magnetic fields is bulk-like every where except in a surface region which extends over - 0/2 <  z <  a/2. In this region, the model dielectric constant is chosen to be a local function which interpolates linearly between the bulk value inside the metal and the vacuum value (unity) outside. The model frequency-dependent dielectric function is, therefore the following :e(a>) =  € i(a ))  +  ieB(w ), Z < - 0 / 2e ( w ,  Z ) — H I +«(« ')]+  [ ! “ «(<")]• -/ -0 /2  <  Z < 0 / 2Q1, Z ^  0/2For the complex dielectric function of aluminium, we use the experimental values given by Weaver (1987-88). For p-polarised light, the magnetic field B(z)=B(Q , 0);  z), (where Q=c<j/c sin is small) is in the /-direction and obeys the following equation (Landau and Lifshitz 1960) with €=«(w, z).The electric field components can be obtained from the magnetic field bycjLf^ . 0 c/BE -H Q , 01 ; z ) = r ~luie dz£*(Q, : z)• -Sm  0i B.The solution of the above equation leads to the following long wave length result (Bagchi and Kar 1978) (wo/c 0),sin Zdit*(co)-sin« 6 ',P '*4 -€(io) cos a, sin 2 at, z ^ - a / 2F (Q -> 0 ,  a., Z), Z/0 +  K 1 + ‘(‘«>)]/[1-«(a>)]e(cu)-sin* a,]^'®+c(w) cos a, sin 26i({(o)[e((o) — sin* a<]^'*+«(a») cos a,—  , -fl/ 2< z< o /2, z^o/2We are interested in the z-component of the electric field because in normal photo­emission is the component which w ill give a non-vanishing matrix element if the initial and final states are symmetric. We have computedz) gg g function of cd for various values of z using the experi­mental values of e(a>). The plots for z/o=0.5, 0.0 and + 0 .5  are shown in the Figure 1. The curves for z/o=0.0 and z/a=0.5 both exhibit a minimum around 15 eV (The plasmon energy for aluminium is 15.75 eV). Further, the plot of A„(z)E l e c t r o m a g n e t i c  f i e l d s  n e a r  surfaces etc 3 2 3F ig u re  I .  V ariation  of vector potential A,„(z) in a lum inium  for z / o = - 0 .5 ,  0 .0  and  0 .5  w ith  respect to th e change in incident photon energy u (e V ).for z/d 0.0 show s a strong peak at around 12 eV. These facts lead us to correlate the experimentally observed minimum at 15 eV and a peak at 12 eV for both the state at Fermi level and the surface state at r  (Levinson et al 1979) with refraction effects. Preliminary calculations for the photoemission cross section with a simple model seem to confirm this. We note that these conclusions were reached by Levinson et al (1979) using a much more sophisticated calculation of the photon field but their calculation was restricted to jellium. We have used this much simpler model for tungsten w ith some success and now see that the results for aluminium also agree qualitatively w ith experiments. We are going to incorporate this model for photoemission cross section calculations and w ill be using it for cases where results obtained for iellium would not be applicable.AcknowledgmentOne of the authors (RKT) Is very much grateful to R V  Rao, N IC  unit, Aizawl, Mizoram, for sharing some of his time in computer works.324 R K  Thapa and N  K a rReferencesBagchi A and Kar N 1978 Phys. Rev. BIS 8240 Feibelman P J 1975a Phys. Rev. Lett. 341092 - 1975b Piiys. Rev. B i l l 319Levinson H J, Plummer E W and Feibelman P J 1979 Phys. Rev. Lett. 43 952 Landau L D and Lifshitz E M I960 Electrodynamics o f Continuous Media (New York; Pergamon) Sec 68Mukhopadhyay G and Lundqvist S 1977 Solid State Commun. 2 | 629 Pendry J B and Hopkinson J F L1978J. Phys. F81009 Weng S L, Gustafsson T and Plummer E W 1977 Phys. Rev. Lett. 39 822 Weaver J H 1987-88 Hondbook of Chemistry and Physics (Boca Raton: Florida CRC Press) 68-th edn, E-377

Electromagnetic Fields Near Surfaces in a Simple Model : The Case of Aluminium

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Electromagnetic Fields Near Surfaces in a Simple Model : The Case of Aluminium

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