Polarization properties of surface plasma radiation excited by electrons

Solid State Communications, Vol. 15, pp. 1387—1390, 1974.

Pergamon Press.

Printed in Great Britain

POLARIZATION PROPERTIES OF SURFACE PLASMA RADIATION EXCITED BY ELECTRONS* K.L. Ngai Naval Research Laboratory, Washington, D.C. 20375, U.S.A. and E.N. Economou Department of Physics, University of Virginia, Charlottesville, Virginia 22901, U.S.A. (Received 9 October 1973; in revised form 14 May 1974 by E. Burstein)

Some interesting features of the radiation associated with non.radiative surface plasmons, excited by electrons have not been explained previously. We give here an explanation of the origins of these features, based on recent work by Elson and Ritchie.

RADIATION associated with the usually non-radiative surface plasmon oscillations (SPO) excited by electrons have been observed and studied extensively in the past. Boersch and coworkers~4have measured the radiation emitted from thick silver foils irradiated by electrons at grazing incidence. They found an intense peak at 3500 A which corresponds to the non-radiative SPO energy for semi-infinite Ag bounded by vacuum. The position of the peak was shifted as expected for the SPO energy when the surface was covered by a thin dielectric layer. Jones, Cram and Arakawa5 observed the same effect. The experiment was subsequently repeated with unoxidized Al by Burker and Steinmann,6 and by Braundmeier, er aL7 They found a pronounced peak of emitted radiation at 10 eV, the Al SPO energy. More recent works~onAg are by Dobberstein and Sauerbrey,8 and Dobberstein.9 While surface roughness can be invoked to ~ the coupling of supposely non-radiative SPO with the radiation field, certam features of the phenomenon have not been explained previously. It is the object of this paper to explain these features. Presence of random roughness of the surface breaks *

Work supported in part by the National Science Foundation: NSF-GH-34404.

the translational invariance on a smooth plane surface. Momentum parallel to the surface is no longer required to be conserved in any physical interaction process. This has a profound influence on the optical interactions of both radiative and non-radiative SPO. The relevance of surface roughness was proved by an experiment of Teng and Stern11 by using an Al coated optical reflection grating. Thereby the surface roughness is known and its effects verified. The calculation of the intensity, angular dependence and polarization properties of surface plasmon radiation excited by electrons (SPREE) can be divided into two parts. The first part involves the inelastic excitation of SPO by fast electrons. Methods for the treatment of this effect have been discussed.10 The second part concerns the SPO-photon conversion process made possible by surface roughness. This is the inverse of the photon—SPO conversion that causes the reflectance drop. In this paper we shall employ a recent formalism by Elson and Ritchie12 to demonstrate that some formal solution of the second part can explain several features of SPREE as were observed experimentally.9”1 Dobberstein9 recognized that for any calculation of the properties of the radiation field of SPREE, the

1387

1388

SURFACE PLASMA RADIATION EXCITED BY ELECTRONS

surface structural factor g(ic) [Dobberstein called it S(g)] must be known. He devised and interesting method of measuring g(ic) of an evaporated 1 j.tm thick Ag film on a 0.2 pm rough CaF2 layer ofi a glass substrate He found that g(ic) has a prominent peak at K0 ~ 7 X iO~A-’. The Ag film was bombarded with a beam of 40 keV electrons at an angle of 50 from the surface. Angular dependence of the intensity of SPREE with wavelength X = 3850 A, observed in the plane defmed by the incident electron direction and the surface normal, shows that the main intensity is emitted in the backward direction against the obliquely incident

indicates that whatever the emission plane, the electric field of the 5400 A radiation is always in the plane determined by the normal to the grating surface and g. This rather remarkable polarization property is depicted in Fig. 1. On the other hand, for a flat surface, the electric field for the SPO is in the plane determined by the surface normal and k11. This difference in behavior was, so far, left unexplained. We shall resolve this difference.

-

parallel to electron beam. the plane This occurs of observation. for radiation For polarized the perpen-

VACUO

The incident electrons couple most strongly to those surface plasmons which have a phase velocity

METAL

withmatches that face. tomponent For this a surface with k11the plasmon in electron the plane which velocity ofhas incidence a along wave for the the vector sur electron, condition’°is =

(1)

.

Here v is the incident electron velocity, a is the grazing angle, and w is the SPREE frequency. For 40 keV electrons and A = 3850 A, this occurs with k1~~ 5 X l0~ cm~.Dobberstein noticed that the equation k11 K0 = (w/c) sin 0 for the conservation of overall momentum the along the intensity surface, implies sin 0should —1.beThis means that main of SPREE emitted in the backward direction. However, the difference in behavior of both polarizations has remained unexplained, An explanation of this property wifi be given below. —

SPREE at optical frequencies of 5400 A from Al grating surfaces has been measured by Teng and Stern.” This radiation comes from SF0 near the w = ck~ 1line, with k11 ~ 1 ~5 cm’. The wavelength positions ofofemission peaks, their dependences on the the emitted radiation and the angle of the emission plane relative

U3

PLANE EMIssIO~,.~\J

intensity is quite symmetric about the surface dicular polarization the angular dependence ofnormal. the

v cos a

Vol. 15, No. 8

POLARIZATION

PLANE

—

u~

— __________

______ -________

—

I. Coordinate system and incident electron beam for Dobberstein’s SPREE. Shown also are the fields of a surface plasmon, traveling along the u, axis. FIG.

Theoretical considerations of the effect of surface roughness on the optical properties of non.radiative 12 surface plasmon been carried Fedders and by Elson andhave Ritchie.’3 Elsonout andby Ritchie started out with a perfectly smooth surface bounding a semiinfinite plasma (z >0) from vacuum (z <0), and described the rough surface by a funtion z = ~(x,y). The hamiltonian of the system of a photon plus a semi-infmite electron plasma takes the form H = (8irc2)~ d3r{A2 + 0 [z ~(x,y)] w~A2

$

—

2 (V X A)2). (2) + cFedders,’3 they transform coordinates to Following the non-orthogonal system u 1 = x, u2 = y, u3 = z ~(x,y). The vector potentials are expressed in terms of the unit vectors (1,, 12, 13) tangential to the coordinate curves (u, u2, u3) as A = 11A1 + 12A2 + 13A3. H can then be expanded in powers of ~ —

to the grating rulings have been accounted for by Teng and Stem as SF0—photon conversion assisted by the grating which supplies a momentum ±g, the reciprocal grating vector. They were able to construct, for the first time, the low energy portion of the dispersion curveforthe SN) inAl. Teng and Stem noted that the SPREE emitted

,

H

=

H0 +JJ~+H2

+...

.

(3)

Only thepower terms H0 which contain terms up to the first in ~and wereH, kept.

Vol. 15, No.8

SURFACE PLASMA RADIATION EXCITED BY ELECTRONS

2)’

H0

SPO’s are shown in Fig. 2. Let A

(8irc

=

H,

=

2H? 1

[A?+ 0(u

3 3u E

X

1389

Jd 3)~A?+ c J d3u (D A3 + O(u3 )w~,DA

(4)

i~1

5 be the vector potential SF0—radiation for the SF0 fields and A~ theElson—Ritchie vector potential for the photon. The SF0— photon interaction hamiltonian given by equations

—~-~

(5)—(9) will lead us to the nature of As the has SF0only radiation photQn observed in the u,u3 plane. the

4irc

components A5, and A33 in the 11 and 13 directions,

_c2[D(~~!_~~+(—~--i~H3 GaAS’) ~a~2 au,j \au3 / (5)

I~

—

where D=~A,+~A 2

(6)

au2

respectively. We found 1 a~(AA+AA+ H5_~ =—jfdu— 4~rc~ au 2 A~, +0(u3)~~(A5jA~3 +Ap,As3)+c aui aHP2M~H _~4~Hs21J. (11)

{

—~—

—

F

=

G

=

(7)

au, 2(?i~/au,) au3 p2 As, ôu3 (aH~,/au Terms like —c 2)vanish be-

H,

(8)

OAiIOuk

(9)

cause for photons in the ~/au2 u, u3 plane, /au2since is zero. Also terms that contain can beaH~, ignored momentum conservation requires that for initial state SF0 with k11 = k111, and fmal state photon in the u,u3 plane, the k that appears in ~k must also be in the 1, direction. On examining the expresseion for H3 ~, in equation (11), we observe that a SPO with k11 = k111, can be converted into a photon polarized parallel to the u, u3 plane only.

-~-A2—-~-A1 au, au2 —~-

au,

H2

—

-~-

au2

and H~ = oAk IOu,

—

-

with cycliupermutations of the coordinate indices. H, is treated as a perturbation on H0. H0 describes mutually orthogonal e.m. modes that propagate along the actual rough surface. These zero order fields are stifi orthogonal to each other and no physical effects emerge. H, is non-diagonal in these zero order modes, hence it causes coupling of modes to occur and induces new effects. We shall apply this theory to explain the anomalous properties of SPREE observed by 9 and by Teng and Stem.” Dobberstein The surface lies on thexy oru 1u2 plane in Fig. 2 and the Ag plasma is confmed to the region u3 = [z ~(xy)] >0. The surface normal pointing out to vacuum is —u3. The plane of incidence of the electrons and the plane of observation of the radiation in Dobberstein’s experiment is the u,u3 plane. The surface roughness function ~(u, u2) has Fourier transform ~ defmed by

_~3

—

=

~~~exp(_iK.P)

Esi

U1

(10)

peaked at K = K0. For a SF0 with wave vector k I along the u, -axis, the conservation of momentum i~f the SPO—photon conversion process implies k11 ± K0 = (w/c) sin 0. Real solutions for 0 only exst for the minus sign case, and as Dobberstein pointed out 0 = ir/2. The electromagnetic fields of these k1~i, —

~

FIG. 2. Teng and Stem’s SPREE has the property that whatever the angle 0 the emission plane makes with the grating rulings (assumed to be parallel to the u, direction), the radiation is always polarized in a plane parallel to u2 X u3.

1390

SURFACE PLASMA RADIATION EXCITED BY ELECTRONS

~1extlet us turn to the SPREE observation by Teng and Stern.~Referring to the coordinate system as in Fig. 1, we assume that the grating ruling is along the u, -axis. The roughness function ~is then a function of u 2 only ~(u

)

~ pg0u~

=

(12)

where g~= (2~n/d)12 andd is the ruling spacing. The SF0—photon coupling is again describable by the hamiltonian of equation (13). 3u ~ A. 1 H3...~ = d 82A~3+ A~2A83+ ~

J

,

2

+ o’~ ~U3JWpk ‘ 2(~52 A p3 + Ap2 A 8~ ~/ 2 A ~I2~1\+A ~! c 32 \0u 2 Ou, 0u2

I

—

)

—~~H~3 +II~,~+Hp~

—

~

~3})

Ou, .

as observed by Teng and Stern, o. = ck can be considered to be small compared with c~,.In equation (13), the group of terms inside the square bracket and also the first two terms are of the order of (w/w~,)2cornpared with the third and fourth terms. Hence they can be neglected and H5~= ~

f

3u 0(u d

3)w~(As2Aps+ A~2A53) (14)

l’his expression implies for an arbitrary emission plane, that the low frequency SP couples to a photon that has non-zerocomponents of the vector potential in either SPREEthe is polarized 12 or 13 direction. in the u2u3 Hence, plane theor,low equivalently, frequency the plane defmed by the surface normal and the recip-

—

—

Vol. 15, No. 8

)

rocal grating vector. This is in accord with Teng and Stern’s observation. (13) Acknowledgement The author wishes to thank E.D. Palik for a critical reading of the manuscript and helpful suggestions. —

We first notice that for SF0 radiation with wave vector k and frequency ~ that corresponds to near 5400 A

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SAUERBREY G., Proc. 5th mt. Congr. ElectronMicroscopy, Vol. 1, p. AA-l3, Academic Press, New York (1961).

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BLANCKENHAGEN P.V., BOERSCH H., FRITZSCHE D., SAUERBREY G and SEIFERT H.G.,Phys. Lett. 11,298 (1964).

4.

BOERSCH H., DOBBERSTEIN P., FRITZSCHE D. and SAUERBREY G.,Ann. Phys. 187,97(1965). See also BOERSCH H. and SAUERBREY G., Optical Properties and Elecbvnic Structure ofMetals and Alloys (edited by ABELES F.) p. 386, North Holland, Amsterdam (1966).

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JONES G.E., CRAM L.S. and ARAKAWA E.T.,Thys. Rev. 147,515(1966). BURKER U. and STEINMANN W., Phys. Status Solidi 12, 853 (1965).

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BRAUNDMEIER A.J.,.Jr. and ARAKAWA E.T., Opt. Commun. 2,257(1970); BRAUNDMEIER A.J. and ARAKAWA E.T., Z. Phys. 239,337(1970); BRAUNDMEIER AJ., Jr., WILLIAMS M.W., ARAKAWA E.T. and RITCHIE R.H.,Phys. Rev. B5, 2754 (1972).

8.

DOBBERSTEIN F. and SAUERBREY G., Phys. Lett. 31A, 328 (1970).

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DOBBERSTEIN P., Phys. Lett. 31A, 307 (1970).

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STERN E.A., comment in Optical Properties and Electronic Structure ofMetals and Alloys (edited by ABELES F.), p. 397, North Holland, Amsterdam (1966); RITCHIE R.H., Phys. Status Solidi 39,297 (1970). TENG Y.Y. and STERN E.A., Phys. Rev. Lett. 19,511(1967). FEDDERS P.A., Phys. Rev. 165, 580 (1968).

13.

ELSON J.M. and RITCHIE R.H.,Phys. Rev. B4, 4129 (1971).

10.