Line shapes and threshold resonances for the scattering of waves from surfaces

L657

Surface Science 122 (1982) L657-L662 North-Holland Publishing Company

SURFACE

SCIENCE

LETTERS

LINE SHAPES AND THRESHOLD RESONANCES SCATTERING OF WAVES FROM SURFACES N. GARCIA IBM

Zurlrh

Received

* and W.A. SCHLUP

Research Laboratory,

14 April

FOR THE

1982; accepted

CH- 8803 Riischlrkon,

for publication

Switzerland

25 August

1982

An analytical study of threshold resonances in scattering of scalar waves from periodic surfaces is presented. Resonances are found with a variety of line shapes which provide characteristic information on the surface potential. The analytical predictions on line shapes are supported numerically. These resonances should be observed in atom-surface scattering and in scattering of light from large-amplitude gratings. We think that these resonances may be important for the field enhancement on surfaces.

In elastic scattering from periodic surfaces, threshold resonances occur when a diffracted beam merges with or emerges from, the surface. Let the incident radiation have a wave vector ki( K, - k,J whose components parallel and perpendicular to the surface are K and k,,, respectively. The wave vectors of the diffracted beams k,( K,, k,,)are given by

K,=K+G, k&=+(K+

(1) G)',

(2)

where G are the surface reciprocal vectors. For particular incident conditions, the beam G whose perpendicular component vanishes. Associated structures appear in the other outgoing beams, and these are called threshold resonances. Cabrera and Solana [I] suggested that these resonances contain valuable information for the atom-surface interaction potential. They discussed the problem in a simple but physical way by assuming certain properties of matrix elements. Garcia [2] studied the resonances numerically using the hard corrugated-surface (HCS) model with a corrugation function that only contains the first cosine Fourier component. Calculations showed that: (a) the scattered beam intensity merges or emerges with infinite slope, and (b) the other outgoing beams have an associated structure consisting basically of an infinite

kg_, = 0, T denoting

* Permanent address: Departamento Canto Blanco, Madrid 34, Spain.

Fisica

Fundamental,

Universidad

0039~6028/82/0000-0000/$02.75 0 1982North-Holland

Autonoma

de Madrid,

L65X

N. Garcia, W.A. Schlup / Line shapes and threshold resonances

slope in the region k:, > 0 but “finite” slope in ki-Z < 0. Recently, Armand, Lapujoulade and Lejay [3] studied the problem in connection with experimental data using a soft-exponential corrugated potential. They assert that precise experiments around the resonance may give final information on the potential, They used only one cosine for the corrugation function, and reach basically the same conclusions as in ref. [2], but claim on the basis of the computations that for soft potentials the slope of the resonance for k:, > 0 also remains finite. In this Letter, we present for the first time an analytical study of threshold resonances by using the series expansion of Lopez, Yndurain and Garcia [4] on the HCS model, and the distorted-wave Born approximation for the soft-corrugated exponential potential. It is shown that by introducing several Fourier components in the corrugation function, a large variety of resonance line shapes is obtained. This new feature can provide systematic information on the number and strength of Fourier components of the corrugation function. The analytical predictions are confirmed by computation for a HCS. For a given corrugation function D(x) (for simplicity one-dimensional but without loss of generality), D(x)

= c DG e’“‘“.

(3)

G

it can be shown from ref. [4] that the diffracted in power series Pc=t3G,o+

z

intensities

PG can be expanded

Pg’,

(4)

n=2

where P&“’ (for n > 2) are of nth order in the coefficients order, with G * 0, these are

D,. Up to the fourth

PA2’ = 4k,,,k,,lDG12. P$)=

8k,;k,,

(5)

Re -iD;;*~DG-G.kGszDGs i

C’

,

(6)

!

Ph4’= 4ko,k,;(CD,_,.k,,:D,,12 G’

-

8kozkcz

c D,_.,k,.ZD,._,..k,..;D,.. Re D$ I i G’G”

II Our goal is to study the above expression G = T. The coefficients Pp’ can themselves

with one beam around be expanded as

(7) threshold

n-2 pP)=

c

/,.l=O

Pp’(I,,

l) k&+‘k;‘,

(8)

N, Garcia, W.A. Schlup / Line shapes and threshold resonances

L65Y

where the values for P$“)(I,, /) follow from eqs. (4)-(7). to From the definition of ksZ, it is clear that the largest contribution d P,/dyi comes from the linear term in k,. Here, y, is any polar coordinate of k,( ki, (pi, 8;). That contribution is divergent while the higher terms remain finite. Here, we conclude analytically that the beams appear or disappear with infinite slope for HCS. Analogously, the diffracted intensities G * 0 go to zero linearly for 8; = 90” because the expansion in k,= starts with the linear term. Now, the interesting point of this work is to analyze the line shape and structure of the other G f T outgoing beams at the emerging points kg, = 0 (analogous results are obtained for merging beams.). For this, we rewrite the series (4) in powers of k,: n-2

Pg’ = k,,i,

i c

P&“‘( T, I, I’) kLzk;;‘.

I./‘=0

Again, by the same argument as above, one concludes that in general an infinite slope d P,/dy, appears on both sides of the singularity (i.e., kg, < 0 and kf, > 0). This result was not obtained in previous numerical work [2,3]. We proceed by studying the line shape of this “extra” structure depending on the Fourier components of the corrugation. We take the “general” case of three Fourier coefficients 0; = DL, (D, real without loss of generality) and D, = DE, = 0; + iD;‘. In fig. 1, we present the different line shapes that are then obtained from eq. (9). The upper part of fig. la shows the line shapes for the lowest diffracted beams for D, f 0. We notice that up to fourth-order expansion only the specular beam shows the typical singularity calculated up to now [2,31. The lower part of fig. la indicates the behavior of k,, and where the threshold in connection with the upper part appears. The cases in which more than one Fourier component is different from zero are presented in figs. 1b- le for the specular and the first few diffracted beams. In these plots, one can see different behaviors in the different beams that account very distinctly for the number, value and sign of the Fourier coefficients in eq. (3). The details are given in the figure. To check the analytical information up to fourth order, we performed calculations with the expansion in ref. [4] taking a n = 50 and 40 G-vectors. We also took kia = 20 with the unit cell a = 2.84 A. Under these conditions, beam 1 disappears at angle Bi = 43.302”. The increments used were Aei = 0.1”. In fig. 2, we show the results for the specular beam considering the same examples as before for the Fourier coefficients, and these are indicated for each case in the figure. They confirm all the analytical predictions of fig. 1, but also give us quantitative information on the corrugation by looking at the line shape, as well as at the strength of the resonance. For example, compare figs. 2c and 2d with 2b. We should comment that the slope is infinite on both sides of the

I.660

N. Gurcra,

W.A.

Schlup

/

Lme

shapes

und threshold

re.~onunc~ec

D; =O D;
Fig. 1. Various types of singularities exhibited by the first fe\s intensities PC, near the angle of disappearance of beams 0: > 0,’ > 8,’ z 0. for a corrugation IX u) = ZlI; cos(27u/tr)+2D~ x coa(4n_x/a)-2D;’ sin(4w/u) with 0; > 0. The lowrer part of (a) shows I\,, for 0, < 6”’ and K(,, for 0, z 0: versus the angle of Incidence; the upper part LSfor 0; = 0;’ = 0. (b)-(e) present results for the Fourier coefficients indicated, exhibiting either cusps at P,, m (d), “semicusps” at I’, , tn (c), or “anticusps” at P, , in (e) with infinite slope in the point of inflection.

resonance. The calculation also seems to show this is not the case for only one Fourier component, fig. 2a, even with a finer grid. For the case of a soft-exponential corrugated potential, we have used the distorted-wave Born approximation equivalent to the second order in our expansion. It is straightforward to see that the intensity behaves like [5,6] PC?) = ak G

G,

_)

( 10)

where a is independent of G when G + T. This again proves that the beams for this exponential potential appear and disappear with infinite slope in agreement with exact calculations by Armand [3]. The same happens for the Morse potential. As seen in the case of HCS, the infinite slope dP,/dv, is due to the behavior of k, and not to the potential (corrugation shape). so that this

N. Garcia,

W.A.

Schlup

/ Line shapes

and threshold

resonances

Lb6 I

Fig. 2. Numerical results of the specular beam intensity versus 8, for a three-component corrugation function close to the point of disappearance of the first beam 8,’ = 43.302” for various representative values of the Fourier coefficients. (b) and (d) exhibit cusps in the maximum, whereas (c) shows S-like behavior (“anticusp”) with infinite slope in 0,‘. Notice that (e) and (f) have the same structure as those in (b) and (c), but now the resonance is larger in relative value by a factor of 20. The corrugattons have been changed by a factor of 2 only.

is a geometric characteristic of the threshold beams. We cannot discuss these slopes in terms of the distorted-wave Born approximation for G * 0, but for G = 0, we have Po=l-

c

PG.

(11)

G-0

If, say, for a G = T in (1 l), the slope dP,/du; + 00, then also the slope dP,,/dy, + co at the side ks2 > 0. To know what happens at k& < 0, we have to consider higher-order terms in the Born series. Although the slopes are infinite as for the HCS, the width of the threshold resonance decreases with increasing softness of the potential as found in the exact treatment of the exponential corrugated potential by Armand [5], and therefore it is hard to observe them either numerically or experimentally [7]. Figs. 2e and 2f are the resonances calculated for cases b and c, respectively, but with Fourier coefficients which are doubled. It is noteworthy that the resonance strength is augmented by an order of magnitude. In this case, the merging beam corresponds to a rainbow peak and then its intensity has to be relocated in the other beams. This effect of resonant strength depending on corrugation could provide very valuable information on the characterization of random surfaces and optical gratings [8-141. The same case of resonant

L662

N. Grrrcia, W.A. Schlup / Line shapes and threshold resonances

scattering can arise by coupling light with the dispersion relation of plasmas in a corrugated metal surface [ 13,141. Here, the control of the diffracted intensity is important for light spectroscopy. In conclusion, our results suggest that very specific information on the shape of the surface corrugation can be gained by studying threshold resonances. The line shapes predicted by the series expansion up to fourth order have been confirmed by precise calculations. The fact that a systematic knowledge of the line shapes can be obtained analytically for different diffracted beams could enable direct relation of the experimental results to the corrugated function. Here, we have treated the problem of a non-penetrating radiation. However, it should be interesting to consider the case in which part of the radiation is transmitted and part diffracted using the same perturbation expansion [4]. Furthermore, the approach appears promising for the calculation of evanescent waves that might enhance the field near the surface and thus lead to enhanced Raman scattering. It is a pleasure to thank E. Courtens for critical and N. Cabrera and K.H. Rieder for discussions.

reading

of the manuscript.

References [l] N. Cabrera and S. Solana, in: Proc. Intern. School of Physics “Enrico Fermi”. Ed. F.O. Goodman (Compositori, Bologna, 1974) p. 530. [2] N. Garcia, Surface Sci. 71 (1978) 220. [3] G. Armand, J. Lapujoulade and Y. Lejay, in: Proc. 4th Intern. Conf. on Solid Surfaces. Cannes. 1980, Eds. D.A. Degras and M. Costa [Suppl. to Le Vide. les Couches Minces (1980) 8571. [4] C. Lopez, F.J. Yndurain and N. Garcia, Phys. Rev. B18 (1978) 970; T. Engel and K.H. Rieder, Structural Studies of Surfaces with Atomic and Molecular Beam Diffraction, in: Springer Tracts in Modern Physics, Vol. 91, Ch. 3.5 (Springer. Heidelberg, 1982). [5] G. Armand, J. Physique 41 (1980) 1475. [6] N. Garcia, to be published. [7] G. Armand, private communication. [S] N. Garcia, V. Celli and M. Nieto-Vesperinas, Opt. Commun. 30 (1979) 279. [9] J. Shen and A.A. Maradudin, Phys. Rev. B22 (1980) 4234. [lo] B. Laks and D.C. Mills, Bull. Am. Phys. Sot. 26 (1981) 358. [l l] N.E. Glass and A.A. Maradudin, Phys. Rev. B24 (1981) 595. [12] F.U. Hillebrecht, J. Phys. D13 (1980) 1625. [13] H. Raether, in: Physics of Thin Films, Vol. 9 (Academic Press, New York, 1977) pp. 145-261. [14] H. Raether, Excitations of Plasmons and Interband Transitions by Electrons, in: Springer Tracts in Modern Physics, Vol. 88, Ch. 10 (Springer, Heidelberg, 1980) p. 155.