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Surface Brillouin scattering from acoustic phonons. I. General theory

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1980 J. Phys. C: Solid State Phys. 13 299

(http://iopscience.iop.org/0022-3719/13/2/017)

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J. Phys. C : Solid St. Phys.. 13 (1980) 299-317. Printed in Great Britain.

Surface Brillouin scattering from acoustic phonons : I. General theory

A M Marvin?, V. Bortolani $ and F Nizzolil t Istituto di Fisica Teorica, Universita di Trieste, Miramare-Grignano, 34014 Trieste, Italy $ Istituto di Fisica and GNSM del CNR, Universita di Modena, Italy

Abstract. The theory of Brillouin scattering from the surface of a semi-infinite medium is presented. The electromagnetic field is calculated to first order in the phonon displacements using the Rayleigh method. Both s and p polarisations are considered and the scattering geometry is not limited to the plane of incidence. The cross-section formulae can be inter- preted in terms of the ripple and elasto-optic mechanisms. The effects on the displacement field due to the surface are investigated in both the discrete and the continuous part of the frequency spectrum. In particular, a detailed analysis of the so-called ‘mixed modes’ belong- ing to the continuum is analytically given for the isotropic medium. The application of the theory to GaAs and the comparison with experimental data (including A1 coated surfaces) will be discussed in paper I1 .

1. Introduction

Brillouin scattering in bulk media has been used in the past to study solids and liquids (Fleury 1970, Sandercock 1975). This scattering is due to the fluctuating dielectric inhomogeneities associated with the thermal agitation of the medium (elasto-optic effect). From a classical viewpoint, the scattering can be interpreted as a result of re- flections from the thermal elastic waves. Because of the effect of interference, for scatter- ing in a given direction, only the elastic waves of a certain wavenumber are operative. Moreover, the light reflected by the elastic waves of frequency R suffers a shift in fre- quency of +IR as a result of the Doppler effect.

From the quantum-mechanical point of view, the process can be visualised as first- order Raman scattering by acoustic phonons. Although this effect is usually thought to be significant only in a transparent bulk material, with the use of intense laser sources and high-resolution detectors, scattered light can be measured even in semiconductors and metals. Because of the small penetration depth of the light in these materials, the inter- action with phonons is limited within a small region below the surface. This makes Brillouin scattering an important tool in investigating surface phonons and also surface resonances (Rayleigh waves, Love waves etc). In this context, the Brillouin scattering techniques (Sandercock 1978) appear competitive with the recent methods of atom diffractive studies on crystalline surfaces (Horne and Miller 1978).

The first requirement of any theory treating this interaction is to take into account the presence of the surface. The most important effect is to distort the bulk excitations through which the scattering takes place and to introduce new ones, called surface modes, which are intrinsically due to the existence of the surface and represent additional channels for the scattering. The distortion of bulk modes created by the surface was

0022-3719/80/020299+ 19 $01.00 @ 1980 The Institute of Physics 299

300 A M Marvin, V Bortolani and F Nizzoli

supposed to have a small effect on the final cross section and therefore to be negligible in a first approximation (Bennett et a1 1972). However it has recently been pointed out that this distortion affects some measurable quantities, such as lineshape resonances by bulk phonons (Dervisch and Loudon 1976) and introduces new structure in the cross section (Loudon 1978a, Bortolani et a1 1978a).

In addition there is a second surface contribution to the cross section due to the ripple mechanism. The phonon vibrations cause the surface to appear as a moving grating, i.e. a ripple, capable by itself of producing diffraction and changing the frequency of the incoming light without invoking any modulation of the dielectric constant in the interior of the crystal as happens in the ordinary ‘bulk Brillouin scattering’ previously discussed. When the light has a small penetration depth one would expect the ripple contribution to be the dominant process for the cross section. In fact, the modulation of the dielectric function caused by phonon displacements is felt by the incident beam only in a very narrow region below the surface because of the strong absorption of the electric field. The ripple contribution becomes more important as the skin depth diminishes. Recent experimental data on metals (Sandercock 1978) were explained in terms of ripples by Loudon (1978b), for polycrystalline aluminium and nickel, and by Bortolani et a1 (1978b), for the (111) face of chromium. However, when the light does penetrate appreci- ably into the medium one would expect the elasto-optic contribution to be important and even dominant, In fact, Bortolani et a1 (1978a) qualitatively explained the experi- mental data on GaAs (Sandercock 1978) by considering this effect.

The first theoretical approach to surface Brillouin scattering for opaque materials was given by Bennett et a1 (1972). They analysed the elasto-optic effect only, by using the displacement field of the bulk and neglecting any surface contribution.

The aim of this paper is to provide a general theory of surface Brillouin scattering by including both effects in terms of the displacement phonon field of the semi-infinite medium. Both the elasto-optic and the ripple mechanisms have already been considered by Mills (1977) who treated surface polaritons interacting with Rayleigh waves and matched the electromagnetic field at the surface. The same two effects are considered in the treatment of surface Brillouin scattering by Rowel1 and Stegeman (1978) but are valid for isotropic media and in-plane scattering geometry. In our approach, we refer to a generic scattering geometry and to anisotropic materials within a matching scheme formulated in a simple and advantageous manner.

A paper by Subbaswamy and Maradudin (1978) dealing with Brillouin scattering theory, appeared in print when this paper was completed. This work is based on the dynamical Green tensor for a planar stress-free surface. Our approach provides more transparent ways of investigating the contribution of each normal mode to the cross section. In the case of resonances in the spectrum, we can easily find the mode which becomes resonant. This type of analysis will be particularly developed in paper 11, where realistic calculations for various faces of GaAs and Si will be presented, together with the experimental data. Furthermore, with respect to Subbaswamy and Maradudin (1978), we also consider depolarisation effects due to the surface scattering. Our theory will be extended to coated surfaces in paper 11.

The plan of the paper is as follows. In $ 2 we relate the scattering light intensities to the modulated dielectric function and to the ripple distortion of the surface. The quan- tum cross section is derived in $3. In the last section we evaluate, as an example, the normal modes for a semi-infinite isotropic medium and give cross-section formulae for various polarisations of the incident light.

Surface Brillouin scattering : I 301

2. Calculation of the scattered light intensities

We assume that the amplitude of the electric field inside the material is attenuated with depth at such a rate that we may neglect any reflection from the back surface. It is sufficient, therefore, to treat the material as filling the half space z < 0, with the surface in the xy plane. The presence of phonons modifies the optical properties of this semi- infinite medium. As a good approximation, the relation between electric field E and displacement D can be represented through the modified dielectric function defined as

E $ @ , t ) = E$ + 6Eua(Y, t )

D,(Y, t ) = E$(Y, t ) EII(Y, t) .

The quantity E$ is the dielectric function of the infinite crystal in the absence of phonons. Strictly speaking, equation (2.2) has meaning only in Fourier space, since E$ is usually a function of k and w. In the following, we shall confine the discussion to non-dispersive media, such that equation (2.2) becomes perfectly meaningful for a monocromatic field offrequency w. The second term on the RHS in equation (2.1) represents the modulated dielectric function due to the presence of phonons and can be expanded as

Each term SE'&(Z; Ql , ) in equation (2.3) represents the optical effect on the crystal due to the single phonon state (Ql , , n) i.e. the state of parallel momentum Ql l and energy an. In this notation the label n includes the possibility of a degeneracy. Equation (2.3) represents the elasto-optic contribution to the scattering of light (Brillouin 1922) arising from the strain induced in the medium by vibrational modes. The relation between this modulated dielectric function (2.3) and the strain is linear for small strains and can be expressed as (Bennett et a1 1972)

(2 .4~)

(2.4b)

where kaar, are the elasto-optic coefficients and u(v, t) represents the phonon displace- ment field.

Equations (2.1t(2.3) define the model we shall use for the crystal interior. The use of the phenomenological model implies neglect of local field corrections. A derivation of equation (2.2), which is exact in the long-wavelength limit, has been given by Mills et a1 (1970). The ripple contribution is easily included once we define the medium as confined in the half space z < uZ(R, z = 0, t).

The two effects can be included in the simple formula below, which defines the di- electric function over all space as:

(2.5) E&> t ) = 6,,0 [z - 0,t)I + E$(Y, t ) @$(R, 0 , t ) - ZI

where 6(x) is the usual Heaviside step-function

1 x > o O(X) =

i o x < o and E$(Y, t ) is defined in equations (2.1) and (2.3).

302 A M Marvin, V Bortolani and F Nizzoli

Equation (2.5) completely defines our model. The problem is thus reduced to cal- culating the electromagnetic field through Maxwell’s equations:

V x E = - ( l / ~ ) aB/at

V x B = ( l / c ) d ( ~ E ) / a t

( 2 . 6 ~ )

(2.6b)

where E is the tensor defined in equation (2.5).

frequencies R are in the far infrared, the inequality Since we are interested in light scattering in the optical region and the phonon

w S R (2.7)

is largely satisfied. Therefore equation (2.6b) can be rewritten to a good approximation as

( 2 . 6 ~ )

Equations ( 2 . 6 ~ ) and (2 .6~) can be solved using straightforward perturbation theory to first order in and U,(& z = 0, t), applying the standard boundary conditions for the field across the crystal-vacuum interface

V x B = ( l / c ) ~ a E / i t .

put = B’”; Dpl.t = D r , k y t = E,” (2.8) where the subscripts ‘n’ and ‘t’ denote the normal and parallel components to the actual surface and ‘out’ and ‘in’ refer to the vacuum and crystal, and all the quantities have to be evaluated at the surface (Marvin et al 1975).

In order to get more insight into the physics it is convenient to rewrite (2 .6~) using (2.1) as

V x B = (47c/c) jeff + (l/c) E,,(z) ;E/& ( 2 . 9 ~ )

E;,(z) = + E z 0 e ( - z ) (2.9b)

where jeff is an effective current present in the medium and defined by

(4n/c) jeff = (l/c) [ E - E ~ ( Z ) ] ?Z?/i;t. ( 2 .9~)

Since we are interested in the linear response in d~~~ we can substitute in the RHS of (2.9~) the zeroth-order solution for the electric field so that the effective current has to be regarded as a perfectly defined quantity.

The proper solution for the problem will be a general solution of the inhomogeneous set of Maxwell equations ( 2 . 6 ~ ) and (2.9a), plus a particular solution of the homogeneous set (i.e. with jeff = 0) to satisfy the boundary conditions (2.8). Physically, the inhomo- geneous solution is the field carried by the effective current inside the medium and repre- sents the bulk solution. The presence of the surface induces a homogeneous field propa- gating away from the surface, on both sides. It is precisely the part of this field in vacuum which is detected in scattering experiments.

Let us call KO and wo the parallel momentum and frequency of the incident light. Using equations (2.3) and (2 .9~) in ( 2 . 9 ~ ) one can see that the current induces a field of momentum and energy

KO + Q l , WO + On (2.10)

where Ql, and R,, are the parallel momentum and energy of the phonon involved in the scattering. Equation (2.10) defines the allowed channels for the scattering and is particu- larly important for surface excitation studies.

We shall now describe how the calculation is carried out. Multiplying (2 .6~) by E - ’ ,

Surjace Brillouin scattering : I 303

evaluating the curl and using (2 .6~) we find the explicit equations for the magnetic field

{rot[€-' rot] + (1/c2) (d2/o"t2)} B = 0 (2.114

V . B = 0. (2.11b)

The inverse dielectric function E-' can be found by taking the derivative of the identity

( E - l I a p EDY = day . To the lowest order we get

@ E - l Ip7 = - ( E o -

Ea;' = ( E o - ' )@ + ( 6 € - l ) J D .

E o a0 = € O S J P .

(Way

In the following we shall assume €$ to be diagonal, i.e.

In this case the equations (2.12) give

(2.1 2a)

(2.12b)

€@;I = (1/EO) d@ - [b€JB/(E0)2]. (2.13)

The solution for the magnetic field can be found perturbatively by using equations (2.11) and (2.13).

The equations for the electric field can be found in a similar way from equations (2 .6~) and (2.6c), giving

{ - V 2 + (E/c') (d2/dt2)} E + V(V . E ) = 0

v . ( € E ) = 0.

(2.14~)

(2.14b)

We shall now consider the incident light polarised either in the plane of incidence (p polarisation or TM wave) or normal to it (s polarisation or TE wave). In order to write the boundary conditions explicitly, it is convenient to work with the electric field (equa- tions 2.14) for s incident waves and the magnetic field (equations 2.1 l) for p waves. In fact in the first case the relevant equations (2.14) become simply scalar for in-plane scattering and only bidimensional for out-of-plane scattering. The same holds for equations (2.1 1) in the case of p waves. The boundary condition (2.8) written in terms of the electric field becomes simply

(V x put)su,f = (V x W s u r f ; D:'Jt = DF, ,;Ut = gtll (2.15)

To express the boundary conditions in terms of the magnetic field, we multiply equation (2.6~) on the left by E - ' and get

E - E - ~ ( V x B).

The continuity of the electric field is therefore

(V x B y t s u r f = (ECIV x (2.16)

The continuity of the magnetic field across the boundary and equation (2.16) are sufficient to solve the problem. Projecting equation (2.6~) on the normal to the surface it can be seen that the remaining condition in (2.8), namely D;ut = DF, is identically satisfied, as it must be.

The geometry of the scattering is specified in figure 1. The incident light of frequency

304 A M Marvin, V Bortolani and F Nizzoli

o, is characterised in vacuum by the wavevector

k, E ( K O 2 - P o ) (2.17)

where the 2 axis is chosen to be parallel to the K , direction and

p o = [(oo/c)2 - K p 2 = (o&) cos $,, K , = (o,/c) sin 9,. (2.18)

The subscript 0 refers to the incoming beam.

i

Figure 1. Geometry of the scattering. The x axis is chosen so that the total wavevector k , of the incident light is in the x-z plane; 9 and cp are the polar and azimuthal angles of the scattered light with momentum k. The phonon which contributes to the scattering has parallel momentum Qll = K - KO where KO and K are the projections of the two momenta k , and k in the x-y plane.

In the medium, the incident light has a total wavevector *

k o ( K O , -go); go = (oo/c) ( E , - sin2 $,)1'2 Im 4 , > 0. (2.19)

The scattered light of frequency o is characterised in vacuum by the wavevector

k = ( K , P ) K = + Q i , K x = (w/c) sin 3. cos cp K , = ( o / c ) sin 3. sin rp

p = (o /c ) cos 3. 0 = W O + Rn and in the medium by -

k E ( K , - q ) : g = ( o / c ) (eo - sin2 $)112 I m g > 0.

Since (2.7) is valid we need not distinguish between o and wo, except in the time-depend- ent phase factors. Using equations (2.14) and (2.15) or (2.11) and (2.16), the following expression is found for the scattered field in vacuum:

(2.20)

Surface Brillouin scattering : I 305

where 3 and B are the unit vectors for s and p polarisation:

3(K, CO) = [ (K /K) x 41; B(K, CO) = (l/k) [G - p(K,K)].

We may summarise our results as follows:

where

(2.22)

- 2iPo (jo dz exp[ -i(q + qo) z] %Y: n, - xo K) -m

for a TE incident field Efnc and

g p - 5 = 2ipo 0 (Jo dz exp[ -i(q + qo) z] (Q 1 1 ; ” ) E o P o + qo ( p + 4) -

\

+ - € 0 ) (qqo ‘Os - (‘1) (2.25)

for a TM incident field Ern,. In these expressions U“ (0) is defined in analogy to (2.3) by zQ I I

u(v, t ) = 1 (z) exp(iQlI . R) exp( -innt); Y E (R,z) (2.26)

where u(u, t ) is the amplitude of the vibrating phonon field. All the other quantities have been previously defined. We recall that o and n are continuous variables, so that the summation indicated in equations (2.3), and (2.20) to (2.26) actually has the meaning of an integration.

Neglecting the elasto-optic effect (2.3) in the above formulae, we find for the ripple

I1

Q l l

K 2

306 A M Marvin, V Bortolani and F Nizzoli

contribution the same expressions as were found by Marvin et a1 (1975) for scattering of light from a rough surface. We note further that for in-plane scattering the ripple con- tribution is zero for ‘exchange polarisation’ s + p and p + s. Using the time reversal symmetry, it can be shown that these ‘exchange polarisations’ are interrelated quantities. In particular, the time reversal changes the geometry of the scattering, while equations (2.23) and (2.24) refer to the same geometry as figure 1. It follows therefore that they are related through a rotation and a reflection in the xy plane. This is sketched in figure 2, where the relation between the two sets of axes is given by

xi = tl. .x. ( i , j = 1,2) 13 J

-cos40 -sincp

-sincp cos cp a = (

In the new axes {Xi}, the tensor quantities

(2.27)

which appear in equation (2.23) are easily transformed to

- [ J E ~ ~ ( z ; Q ~ ~ ) c o s v - J ~ ~ ~ ( ~ ; Q l l ) s i n ~ ] , - 2(z ; Q 1 1 1 COS v - 1 ( Z ; Q 1 1 1 sin VI

which appear in equations (2.24). Adding the ripple contribution and exchanging the ‘0’ subscript, it can be seen that the expressions in large brackets in equations (2.23) and (2.24) transform into each other (apart from sign), that is, the two equations fulfil time-reversal symmetry as is required.

3. The scattering cross section

In the previous section both the electromagnetic field and the phonon field were treated as classical quantities. This clearly appears in equations (2.3) and (2.26) where the time dependence is explicitly included. However, in calculating the Poynting vector for the EM field (i.e. the cross section) a thermal average over phonons must be performed. In general, this should be done quantum-mechanically. A full quantum calculation allows a better comparison of om formulae with those existing in the literature. It will be found, however, that the results of the previous section still remain valid with proper modifica- tions and that all the formal machinery is not really needed. In this case, the general solution for the displacement field u(v, t ) can be written as

where the normal coordinates A, ,(t) are expressed in terms of the second-quantisation phonon-field operators a, a+ as f8lows

In equation (3.1) the normal modes wI(z; Qll) satisfy the orthonormality condition r n

Surface Brillouin scattering: I 307

i f + - i i Y

Figure 2. Direct and time-reversed scattering process for 'exchange polarisation' s + p.

In these equations p is the mass density, L is the side of the surface and the summation over n means as usual an integration.

As in equations (3.1) and (2.4), the modulated dielectric function can now be written as:

8E&, t) = 1/LP1'' c exP(iQI, R)f igzp(Z ; Qll)AQ,, ,n( t ) (3.4) Q l l , n

where

(3.5)

In the quantum treatment of the field, equations (3.1) and (3.4) are used instead of equations (2.26) and (2.23) respectively, so that the final equations (2.21) to (2.25) are still valid with the substitution

In compact notation the scattered field can be written

From (2.22) and (3.6) we have for instance for s -+ s scattering

- &;,(z; Q l l ) sin 'p] dz - (1 - eo) wZ(0; Ql l ) cos 'p . (3.8) I The same can be found for the other polarisations (equations 2.23 to 2.25).

The Poynting vector in vacuum is

S(v, t ) = ((c/8n) Re(E(v, t) x B*(u, t))) (3.9)

308 A M Marvin, V Bortolani and F Nizzoli

where the symbol ( ) means thermal average. From equations (2.6a), (2.20) and (3.7) and using the relation below,

( A Q ,,(t) , n ( t )> = ' Q , l , Q J(n - n') ( A Q n ( t - "1 ,,('I>; (3'10) I \ I 1 1 .

the Poynting vector can be expressed as

(3.1 1)

(AK-K, ,n( t )AKf-Ko, , , (0)) ' 1 Since the modes of the crystal are perfectly harmonic, the integration in the large brackets in (3.1 1) gives

(h/2Qfl) LW-4 - (m - WO)) + mfl + (0 - q-J) (NQ,) + 111 (3.12)

which includes anti-Stokes and Stokes processes. N(Q) is the usual Bose-Einstein thermal factor.

At this point, our treatment differs from that by Mills et a1 (1970) who expanded the phonon displacement u(r. t) in the bulk normal modes and took the surface into account through a phenomenological damping introduced in the A's.

We have used instead the correct normal modes of the seminfinite medium. To proceed further we use the identity

(3.13)

where the index j labels the various normal modes (except surface modes) having energy Q, and parallel momentum Qll . Inserting (3.12) in (3.11) and using the above prescription, the Poynting vector becomes

with k = N , ( N + 1) for anti-Stokes and Stokes processes. By using

(3.15)

where dQa is the differential solid angle, projecting the expression (3.14) on the z axis (Marvin et a1 1975), and dividing by the projected Poynting vector of the incident field, we get the differential cross section

(3.16)

For scattering by surface states where the energy Q,, and the parallel momentum satisfy the dispersion relation Qj = Q,(Q,,), equation (3.13) is no longer valid, but istead

(3.17)

Formally, in all the previous equations the index n has been replaced by the discrete index j labelling surface modes with momentum Qll . In this case, equation (3.16) is

c w-4 - Q) = C'Pj(Q1,) - QI. n i

Surface Brillouin scattering : I

substituted by

309

(3.18)

The surface states contribute with an infinitely narrow line in the cross section. This is a consequence of our approximation in assuming these states as perfectly harmonic. The ripple effect and the impurities present in the neighbourhood of the surface will, along with other processes, add a damping to the energy (Maradudin and Mills 1976) so that these infinitely narrow lines will be spread out in Lorentzians.

Equations (3.16) and (3.18) complete the calculation of the cross section. The quan- tities 13Z:~(z; Q, l ) appearing in the 3 ' s are related to normal modes through the elasto- optic coefficients as in (3.5). All the other quantities have been previously defined.

Equations (3.16) or (3.18) differs from that given by Mills et a1 (1970) by a factor cos 9/ cos 9, which was missed in their treatment. The same discrepancy can be found in sub- sequent work by Maradudin (Bennett et a1 1972). The factor (cos9,)-' was still not included in a recent paper by Loudon (1978a). These factors arise from the projection of the Poynting vector along the z axis, for both the diffracted and incident fields. If the elasto-optic contribution is set to zero, our formulae agree with those of Agarwal(l977) who used a quite different technique to calculate the cross section.

To conclude this section we will show how equations (3.16) and (3.18) can be obtained by a classical approach. The total cross section is expressed in this case as

Using (3.15) the differential cross section becomes

r, t = s, p. (3.19)

(3.20)

where (IEJ;,,12/IE:n,12) is given in (2.22H2.25) and where we use (2.3), (2.4) and (2.26). The symbol (

Let us confine the discussion to scattering by distorted bulk modes. Since the scattered field can be expressed through the phonon displacements by equations (2.3) and (2.4), the thermal average can be easily found with the aid of the relation

) denotes the thermal average.

(3.21)

To connect the U'S of equation (2.26) to the w's of equation (3.3) we use the energy equi- partition theorem

s Q l , . j

Ekin = i p dr(lu(r, t)I2> = +E,,,, = dSZ (k,T). (3.22)

Inserting (2.26) in (3.22) above, we find the relation

(3.23)

310 A M Marvin, V Bortolani and F Nizzoli

It is not difficult to show, using equations (3.21) and (3.23), that the cross section (equation 3.20) coincides with the quantum treatment in equation (3.16), provided that the correct factors R(Q) are used instead of their classical limits. The same can be done for scattering by surface modes.

At first sight equation (3.16) or (3.18) appears very complicated to handle. Some particular cases in which it simplifies are given in paper 11. A general property, however, can be outlined here. Suppose we fix the geometry of the scattering. The only phonon modes which are involved in the scattering process are those satisfying the following relation between Q l l and a,: Ql, = (oo/c) {[l + (Q,,/oO)] sin 9 cos cp - sin 9,, [I + (Q,/mO)] sin 9 sin cp}. (3.24)

To a good approximation, by using (2.7) we get

Ql l = (oo/c) [sin 9 cos cp - sin go, sin 9 sin cp]. (3.25)

This relation shows that all the modes, and only those, with Q,l given in equation (3.25), are active. This result is important for experimental studies since it permits detection of surface modes one by one.

4. Normal modes of vibration in an isotropic medium

In this section, we solve the problem of normal modes of vibrations in a semi-infinite isotropic medium, starting from the equation of elasticity for the bulk medium and apply- ing the free stress boundary conditions at the surface.

In real crystals the use of the elastic theory is justified as long as the wavelength of the vibrating modes is long compared with the atomic distance in the bulk (Born and Huang 1966) and with the distance between neighbouring layers below the surface. In this limit, the vibrating mode is indeed insensitive to any microscopic detail (relaxation of planes, reconstruction of the surface), as a full microscopic calculation has shown (Bortolani et a1 1978~).

The equation of elasticity can be written as (Farnell 1970)

where p is the density of the medium already defined in (3.1) and czBo, are the elastic coefficients. Assuming the solution of the form

U - exp[i(Ql, . R + q,z - Qt)] (4.2)

and choosing the reference frame such that the P axis is parallel to Q,, one has from (4.1) the equivalent

U = 0. (4.3)

Surface Brillouin scattering : 1

Defining the longitudinal and transverse velocities by

c: = C , , / P > CT = C44/P

equation (4.3) can be rewritten as

311

(4.5)

(4 .8~)

(4.8b)

The first above is doubly degenerate and gives the two independent solutions

u(l) t = (ct/Q) (qt) 0, - Q 1 1 1 exp[i(Q 1 1 * R + qtz - at)] u ( 2 ) t = (0,1)0) ex~[i(Qil * R + qtz - Qt)]

(4 .9~)

(4.9b)

with U!') polarised in the plane of incidence and with U:') perpendicular to it. The eigen- value in (4.8b) gives a unique in-plane polarised solution

U I = (c, /R) ( Q , , , 0, qJ ex~[i(QIl * R + q,z - at)]. (4.9c)

From the definition of isotropy of the medium, the result does not depend on the direction of Q,, ; namely, (4.8) and (4.9) have to be regarded as general solutions for the bulk. For a fixed Q , , , the bulk solutions are confined to the region

2 c,Qli, (4.10)

since qt, qI have to be real. In the range R < clQ,, only the two solutions (equation 4.9a) b) exist, while for R 2 clQ,, (4.9~) is also present.

When the surface is introduced, the free stress conditions

[ c 1 3 p " ( ~ / ~ x y ) ~ p l z = O = 0 (4.10)

have to be added to (4.1). For the geometry above, the last can be rewritten

+ d Z U x ) z = 0 = o (4.1 1)

The problem is easily solved once we express the U'S as a linear combination of (4.9) in a complex extension where Q , , and R are real but Imq,,t < 0. From the structures of equations (4.9) and (4.1 1) we see immediately that the transverse mode is decoupled from the longitudinal ones and gives by itself the solution

(4.12)

(4.13)

wT,*(z; Q l l ) = ( 2 / ~ ) " ~ co~(q,z)(^Z x Q , ; / Q , , ) 0

[wT,*'(z; Q,,)]* [wT**(z; Q,,)1 dz = &q, - 4) - m

312 A M Marvin, V Bortolani and I; Nizzoli

Equation (4.12) is a 'bulk type' solution which exists only for Q 2 ctQiI and is not particu- larly interesting in surface problems.

All the other modes will arise from a combination of U!'), U, in complex extension and are therefore longitudinal in character. In the region Q < ctQ,, we find the well-known localised states: Rayleigh or Rayleigh-type waves (Farnell 1970).

Above the transverse threshold

we have a continuum of solutions arising from bulk modes. It is convenient to distinguish the two regions SZ < Q, and R > Q, where

defines the longitudinal threshold. The first reason is connected with the actual degeneracy of longitudinal modes above

Q, and not below. The second is related to the different nature of the solutions depending on whether q, is imaginary or real.

In the first case, which happens for R < a,, a decaying wave which resembles a surface mode of vibration is present in the solution. These modes are called 'mixed (Dennis and Huber 1972) since they appear as a mixture of sinusoidally varying waves and exponentially decaying ones.

For real q, (i.e. l2 2 Q,), however, the solution appears with a typical bulk character, since it is simply a combination of allowed bulk solutions. Those are called 'combination modes' and are less interesting than the previous ones as far as the surface scattering is concerned. Using the same procedure as above, we find the mixed modes to be

(4.17~)

These modes are defined for Q < Q, where qf is purely imaginary

q, = -iQll[l - (SZ/R,)2]'i2 = - 1Y ' 1' (4.18)

while qt is given by (4 .8~) with (4.17c), the modes satisfying the orthonormality condition:

(4.19)

Formally, one can obtain the degenerate modes above the threshold R, by separating

0

dz[wL,*'(z; Q,,)]* [wLSR(z; Q l l ) ] = 6(q, - 4:). s_,

Surface Brillouin scattering: I 313

(4.22~)

(4.22b)

Using the last two equations, the modes in equation (4.20) satisfy the orthonormality condition

[W?'*(Z; Qll)]* , [wjL,*(z; Ql,)] dz = hjj,6(q, - 4:). Jr m

(4.23)

As an application of the formulae above let us calculate the (s + s) scattering cross section by bulk phonons (3.16). For simplicity we shall consider only in-plane scattering (cp = 0). Using (3.8) and (3.5) in (3.16) gives:

cos2 3 cos 3,

(4.24~) for Q, 6 R 6 R, where cr = (E, - sin2 3)'12 and

for SZ 3 R,. The formulae above are still valid for cp = 7-c but with the substitution 3 - -3.

314 A M Marvin, V Bortolani and F Nizzoli

Neglecting the elasto-optic contribution, we find the simple expression (Loudon 1978 b) :

(4.25)

Equation (4.25) can be used for metals where the ripple effect is supposed to dominate. In the opposite limit, i.e. neglecting the ripple contribution, we recover equation (5 1) of Loudon (1978a). Formulae (4.24) contains both effects.

The procedure can be repeated for (p + p) scattering. Limiting ourselves to the same geometry (cp = 0) and using equations (4.16) and (4.17) we find for the cross section (3.16) by bulk phonons the expression

cos2 9 cos 9, / E o cos 9 + d 12 IEo cos 9, + do12

(4.26~) /1 + i(1 - E , ) c:y,(ooo - E , sin 9 sin 9,) + k44(R,/Q)2 C

for R, < R < R, and using (4.21) and (4.22),

/1 - (1 - E , ) c:(ao, - E , sin 9 sin 9,) + ~,,(R,/R)~ D

for R 3 R,. C, D and E are rather complicated expressions related by

C = (2Qic: - Q2)/(2Q,,e:) E - iy,D, (4.27~) where

(4.27b)

Surface Brillouin scattering: I 315

X , = 4c:[(q + q0)/QiI (Qiao, - 412 sin9 sing,) - qf(sin90, - sin$,o)]

5 = 2(2Qic: - Q2) {[m, - (q:/Qi) sin 9 sin So] - (q + qc)/Q,l(~in 90, - sin 9,~))

2Qicp - R2 X , = -( ) [(2Qfjc: - R2)(sin9cr, - sin9,cr)

Q;C:

+ 2c:Q (q + q,,) (m, + sin 9 sin 9,)]

= 2[(2Qficf - Q 2 ) ( q + LI,)/Q,~ (sin90, - sin9,o) +2q;c:(o0, + sin9 sin 9,)].

In the limit ] E , / B 1 we have from equations (4.26) and (4.27) the simpler expressions (4.27~)

for R, < R < R, and

R q, (1 - sin 9 sin 9,,) (2Qfc: - Q2)’ + 4Q;c:G ( 1 X

(4.28b)

for R 2 0,.

gously to (4.25); On the other hand, neglecting the elasto-optic contribution in (4.26), we find analo-

(4.29)

which is the ripple contribution to the cross section for (p -, p) scattering. In the same way, using (4.12) one finds the contribution for the in-plane scattering

3 16 A M Marvin, V Bortolani and F Nizzoli

cross section (3.16) due to ‘exchanged polarisations’ s -+ p and p -+ s to be

cos2 9 cos 9,

(4.31)

in agreement with equation (58) of Loudon (1978a), since the ripples do not contribute for this geometry.

looQll(4 + 61,) + sin 9,q:I2 X

1(4 + 4,12 - q:I2

5. Conclusions

We have presented the Brillouin scattering theory for a semi-infinite medium bonded by a phonon-corrugated surface. Compared with previous treatments, the scattered field has been derived in a more simple manner by performing the matching in terms of either the electric or the magnetic field, according to the type of polarisation of the incident light.

The main results of the paper are the expressions (2.22H2.25) for the scattered field, through which the Brillouin cross sections (3.16 and 3.18) are expressed. Both the elasto- optic and the ripple effects are considered at the same time. For highly reflecting materials like metals, the corrugation scattering is dominant and the elasto-optic terms vanish when E -+ CO. For semiconductors, however, the elasto-optic contribution becomes com- parable with the ripple term and in some frequency range is even larger.

As an application, we presented in 54 explicit results for the cross section in an isotropic medium and for different polarisations of the light, generalising the results of Loudon (1978a, b). The analytic expressions which can be obtained for the isotropic

medium are a very useful guide for understanding the nature of the modes in the various frequency ranges. We have found that surface features arise not only from the well-known localised modes but also from the ‘mixed modes’ of the continuous spectrum between the transverse and longitudinal thresholds.

The calculations for realistic anisotropic systems will be discussed in paper 11, where numerical techniques are used to evaluate the displacement field. However, it is still possible to perform an analysis in terms of the relevant modes entering the dis- placement field, in order to detect the single contribution to the scattering. This would be a difficult task within the Green function formalism. In our approach we can treat realistic systems equally well. Calculations of the cross section for various faces of GaAs and Si will be presented together with experimental data in paper 11, where a definite proof of the importance of the elasto-optic mechanism in semiconductors will be also given.

Acknowledgments

The authors would like to express their gratitude to V Celli and E Tosatti for useful comments and discussions.

Surface Brillouin scattering : I 317

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