Surface photoeffect with non specular surface scattering

of electrons

P. De Andres, R. Monreal, F. Flores, F. Garcıa-Moliner

To cite this version:

P. De Andres, R. Monreal, F. Flores, F. Garcıa-Moliner. Surface photoeffect with nonspecular surface scattering of electrons. Journal de Physique, 1982, 43 (4), pp.685-689.<10.1051/jphys:01982004304068500>. <jpa-00209440>

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Submitted on 1 Jan 1982

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685

Surface photoeffect with non specular surface scattering of electrons

P. de Andrés, R. Monreal, F. Flores and F. García-Moliner

Instituto de Fisica del Estado Sólido(C.S.I.C.) and Departamento de Física de Sólidos (U.A.M.), C-XII, UniversidadAutónoma de Madrid, Cantoblanco, Madrid, Spain

(Rep le 31 juillet 1981, révisé le 9 décembre, accepté le 10 d6cembre 1981 )

Résumé. 2014 On argumente (a) que la diffusion non spéculaire des électrons à la surface augmente le rendementphotoélectrique et (b) que, en désaccord avec la notion généralement acceptée, il y a des diverses situations où ilest aussi important de retenir une description non locale de la réponse diélectrique transversale.

Abstract. 2014 It is argued that (a) diffuse surface scattering of electrons enhances the internal photoyield and (b)contrary to the commonly held view, non locality in the transverse dielectric response also plays an importantrole in many situations.

J. Physique 43 (1982) 685-689 AVRIL 1982,

ClassificationPhysics Abstracts79 . 60

Photoemission and the electronic properties ofsurfaces are receiving a great deal of current interest [1].Effects associated with the electromagnetic field nearthe surface [2, 3], surface potential [4] and surfacewavefunctions [5] have been discussed by variousauthors. Others [6-8] have hinted that the non specularscattering of electrons from the surface might have anappreciable influence on the photoyield. The purposeof this note is to provide an explicit discussion of theeffect of non specular scattering on the internal pho-toyield.For this we follow Kliewer’s approach [7], based on a

semiclassical infinite barrier model. This amounts to aloss of accuracy as compared with a more detailedquantum mechanical analysis [4], but allows us toisolate the problem of non specularity and to focus onits effect. The calculation of the photoyield, avoidingthe complicated problem of the escape probability forelectrons with different energies, is based on theformula

where 6 is a representative average escape length anddA/dz is the differential absorptance per unit length.This is only the internal photoyield, the passage throughthe surface barrier is not discussed here. The purposeof this paper is to discuss surface effects in the theoryof the internal photoyield. These effects are manifestedin dA/dz which is determined by the EM field insidethe material. The escape factor weights more strongly

positions closer to the surface and thus the field dis-tribution in this range is important. If light is incidentat an angle 0 through vacuum, expressing the diffe-rential power absorption as the Joule term per unitincident (time-averaged) energy flux yields

As in Kliewer’s calculation [7] we study P-polarizedincident light. The analysis is based on the methodof extended pseudomedia [9, 11], which has beenused [11] to study in the same way the total absorp-tance, the limiting case 6 --+ oo of the problem studiedhere. As in [9, l lJ we use simplified dispersive dielectricfunctions 8L (k, m) and 8T(k, (J). These model dielectricfunctions are chosen so that they incorporate the mainfeatures of spatial dispersion while being sufficientlysimple that the problem is amenable to analytical study.This avoids the mathematical complications whicharise when one uses elaborate dielectric functions [12,13], in which case one must soon resort to heavynumerical computation. Previous experience hasshown that the use of these simplified dielectricfunctions gives very satisfactory results in the calcu-lation of reflectivity [11] and surface plasmon disper- .sion relation [9]. It will be seen here that they also pro-duce results for the photoyield which compare quitewell with the results obtained with more elaborate cal-culations in the cases in which such calculations havebeen performed. The form of 8L is like that of the

hydrodynamic model, but the approximation pro-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004304068500

686

ducing SL and ET is different [10] in one importantrespect, namely, that spatial dispersion is also includedin the transverse response. We stress that the analysisof [7] was based on a very elaborate (essentially RPA)’6L and a local ST. The point of this paper is that inclu-ding dispersion in ST is important for non specularsurfaces.

Now, when the excitations near the surface areimportant one might expect that the dominant rolethat diffuse surface scattering plays in determiningthe decay of collective excitations [9, 11] may enhanceincoherent excitations and therefore the photoyield.In order to formulate this qualitative idea we use theanalysis of [9-11] in which the strength of diffuse surfacescattering is represented by the phenomenologicalparameter p (0 , p 1). This parameter has a statisti-cal meaning. It represents the fraction of incomingelectrons which are specularly reflected from thesurface. This phenomenological model has been

widely used in the theory of surface transport and radiofrequency size effects in metals [14] and also in theprevious studies of electrodynamic [9, 11, 15] andoptical properties of non ideal surfaces performed onthe basis of the approach used here to study the pho-toyield.The outline of the calculation is as follows : we put

k = (K, 0, q), so that K is the component of k parallelto the surface and the Ox axis is chosen as the directionof K. For given (K ; m) we consider the root L(q ; co) of

and the two roots R(q ; m) and r(q ; m) of

L, R and r are determined by the model dielectricfunctions

The field inside the metal (z &#x3E; 0) is obtained in theform

As regards the transverse field, the mode q = + iRcan be approximately obtained by taking

and represents the normal EM mode propagatingin the metal, while the mode q = ir is a polariton-likevibration associated to the electronic transverse mode

given by ST -+ oo.

Fig. 1. 2013 Differential absorptance as a function of z for0 = 45°. Simplified non local CL and ET. Full line : p = 1,dashed line : p = 0. a) 0 = 0.33, b) Sl = 1.1.

The method of extended pseudomedia [9, 10] yieldsthe longitudinal (EL) and transverse (ER, E,) amplitudesas functions of p. The current is likewise obtained in , the form

and with this one can evaluate (2) carrying throughthe effects of p. We shall concentrate on the extremecases p = 0 and p = 1. In the specular limit (p = 1)we should make contact with [7]. All the calculationshave been done with same input parameters as in [7] :cop 1: = 101 and the sodium density.

In order to see the effect of accounting for spatialdispersion in ET, figure la shows the differential

absorptance dA/dz as a function of z for 0 = 450 andreduced frequency Q = w/wp = 0.33 for the extremecases p = 0 (dashed line) and p = 1 (full line). Thesuite appreciable differences in the photoyield calcu-lated with p = 0 and p = 1 for St 1, to be seenlater in figures 2 and 3, are due to the large differenceshown in figure la for the cases p = 0 and p = 1.For p = 0 the oscillations in dA/dz have a periodicityof about 10 A, the value of 2 x/Im r. Thus at thesefrequencies and p = 0 energy absorption is dominatedby the excitation of the transverse mode r, which onlyexists if &#x26;r non local. This effect can be understood in

687

Fig. 2. - Calculated photoyield vs. Sl for different models,3 = 2 A and 0 = 45°. Full line : Present. Simplified 8vDispersive 8T’ p = 1. Dot-dashed : Kliewer [1]. Elaborate a,.Local e,. p = 1. Dashed : Present. Simplified &#x26;T. DispersiveET p = 0.

physical terms by noting that surface scatteringstrongly affects the transverse current density fluc-tuations near the surface [17]. In our case, the pola-riton-like transverse mode has an energy essentiallyassociated with transverse electron vibrations, whilethe normal EM mode is associated to a propagationof an EM energy. This explains why electron surfacescattering predominantly affects to the polariton-likemode.When the plasmon threshold is reached, the lon-

gitudinal mode L begins to play an increasingly impor-tant role. Figure I b (Q = 1.1) shows that this modedominates if p = 1. In this case the periodicity in theoscillations of dA/dz is 2 Tr/Im L N 13 A. However,for p = 0 the situation is still dominated by the

Fig. 3. - Same as figure 1, with 5 = 20 A.

transverse mode r. At this frequency 2 n/Im r =- 3 A,which is the periodicity of the oscillations of dA/dz.As co increases the situation changes and is increa-singly dominated by the longitudinal mode. Thedifference between the results for p = 0 and p = 1decreases and this is why the resulting photoyield forQ = 1.5 is almost the same, as is seen in figures 2and 3. Notice that large differences in dA/dz result inlarge differences in the photoyield only for smallvalues of 5. As 5 increases the details of,dA/dz tendto wash out in the integral (1).

It is also interesting to remark that if one assumes aliteral hydrodynamic model for EL(k, co), and conse-quently a local &#x26;T(co), then no assumption aboutreflection of the free electrons at the surface is neces-

sary. It turns out that in this case standard electroma-

gnetic matching conditions and current conservationat z = + 0 suffice to determine the problem [16].In this model the fields and currents near the surfacedo not depend on p. The situation changes entirelyif By is non local. In this case [15] the field inside con-tains a non vanishing amplitude of the r mode and thefull determination of the solution requires an extracondition at the surface. As discussed in [9, 10] thisamounts to a model for the scattering of the quasi-particles at the inner surface of the medium. It is thismodel that is represented by the phenomenologicalparameter p in the case of metals.

Figure 2 shows the calculated photoyield vs.

reduced frequency Sl = (9/o)p. The results with a

simplified SL are in excellent agreement with thoseobtained [7] with an elaborate SL for Q &#x3E; 1, when theplasmon excitation dominates. Of course there is aloss of accuracy for Q 1, where single particleexcitations dominates and this is just the structurewhich is lacking in our simplified EL. This is not

important in the calculation of the total absorptance,even down to very low g [11], but figure 2, with6 = 2 A (a really extreme case), weights very heavilythe differential absorption for very short distances andthis is already sensitive to the details of SL. However,we can now see what happens when diffuse surfacescattering is switched on. The result (dashed line) is avery considerable enhancement of the photoyield for0 1. The case 6 = 2 A is only shown for illustrativepurposes. A more plausible value could be 6 rr 20 A(actually 21.2 A, to compare with [7]). The results areshown in figure 3. There is no doubt that the effectof diffuse surface scattering is far more important thanthat of single particle structure for Q 1. For Q &#x3E; 1the photoyield tends to be insensitive to the value ofp = 0, has we have just seen.A complementary view is obtained by fixing Q and

studying Y vs. 0. Figure 4 shows the results forQ = 1.01, when plasmon excitation is already suffi-ciently important, although at this frequency it is

appreciably damped. There is no need to comparewith reference [7], as the results for p = 1 would beindistinguishable. We can now see the effects of diffuse

688

Fig. 4. - Fixed Q = 1.01. Photoyield vs. 0 for 5 = 2 Aand 6 = 20 A. Full line : p = 1. Dashed : p = 0.

surface scattering for all angles and these are every-where significant even for 6 = 20 A. The minimumdifference between p = 1 and p = 0 appears (for any6) at the peak, which corresponds to the critical anglefor total external reflection, when the transmitted lightwave runs parallel to the surface. This residual, but notnegligible, difference is due to the dominant role of thetransverse part of the transmitted wave. A local &#x26;r and

p = 0 would give in this case a poor approximation.The role of ET, which is usually treated in a local

approximation, can be more explicitly discussed in thelight of figure 5, which shows the calculated photo-yield for normal incidence and 6 = 20 A. The diffe-rence between p = 1 and p = 0 is again quite conside-rable (in fact at all frequencies). In this case Ez = 0and there is no longitudinal excitation. Moreover, forp = 1 the analysis shows that the power absorptionassociated with transverse waves is negligible, andusing a local ST is a good approximation. However,this changes drastically for p = 0. The large enhance-ment of the photoyield is then due to the transversetransmitted waves and it is here that Ex is important.In this case the Fourier amplitude of Ex(z) inside themetal turns out be

where J M, T M, f2 and f3 are K and a) dependent para-meters which are eventually eliminated [9, 10]. The

Fig. 5. - Normal incidence. Photoyield vs. 0. Full line :

p = 1. Dashed : p = 0.

resulting E,,(z) is then very sensitive to the details of&#x26;r(k; a)). These features remain for small angles ofincidence. The case 0=0 serves also to make contactwith the case of S-polarization and shows that keepinga dispersive &#x26;p is important for the study of the photo-yield Y. with S-polarized light This is importantbecause the experimental information is usuallygiven as the ratio Yp j YS, which is very convenient forthe theoretical interpretation. Assuming that the

passage through the surface barrier is indifferent tothe polarization of the light which has produced theexcitations, one can compare directly experimentaldata with calculations of internal photoyield, but it isimportant to have equivalent theories for both Ypand YS. In conclusion we find : (i) The non specular surface

scattering of electrons enhances the photoyield. (ii)While a local ET is a good approximation for p = 1,inclusion of non specularity requires the use of adispersive &#x26;r. A detailed analysis shows that this canbe important in a variety of cases, contrary to thecommonly held view that it is only the use of a disper-sive CL that matters.

Furthermore, in connection with other related

phenomena it is worth stressing that [11] : (i) Thereflectivity, unlike the photoyield, is very insensitiveto the value of p. (ii) The total absorptance, corres-ponding to 6 -+ oo in (1), does depend on p a little morethan the reflectivity, but considerably less than thephotoyield for a typical representative escape lengthof 20 A.

689

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