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Optical properties of non-ideal interface boundaries
214
Science 111 (1981) 2144226 North-Holland Publishing Company
Surface
OPTICAL PROPERTIES OF NON-IDEAL INTERFACE
BOUNDARIES
M.A. KRYKIN and S.F. TIMASHEV Karpov Institute of Physical Chemistry, Ul. Obukha IQ, 107120
Moscow B-120, USSR
Received 12 September 1980; accepted for publication 11 June 1981
Optical properties of an interface boundary containing thin inhomogeneous transitional tayers have been investigated within the framework of the classical Maxwell electrodynamics with the use of generalized functions. As a p~ticular case, expressions for reflection coefficients have been obtained for S- and P-polarized light.
1. Introduction
Optical methods of ~vestigation of interface boundaries (ellipsometry, electroreflectance, surface reflectance spectroscopy) may give unique information both on the state of interfaces and on the kinetics of physico-chemical processes taking place on them. In this case we mean the state of thin transition layers with characteristic thicknesses of -I--IO 8. These can be adsorption layers, layers produced by external fields or roughness of the surface, inversion layers in semiconductors and Mets-dielectric-semiconductor structures, thii island films on surfaces of semiconductors and dielectrics, etc. A phenomenological theory of light reflection from an interface with an arbitrary transition layer at this interface has been developed in ref. [l--4]. The optical parameters of the transition layer introduced in refs. [I-4] had a sense of response functions to the action of ele~troma~etic field of light and were associated with high-frequency conductivity and moments of surface high-frequency current (these parameters can be calculated in the microscopic approximation if a specific model is chosen, see, e.g., ref. [4]). Refs. [l-4] consider the case where the layer is homogeneous along the interface. It should be noted, however, that in some problems consideration of inhomogeneity of the transition layer at the interface is necessary and may result in qu~itative~y new effects absent in refs. [l-4]: the appearance of diffuse light scattering, interaction of bulk and surface modes (ii particular, absorption of light on surface polaritons), and so on. Such effects were investigated in a number of works (for instance refs. [5-71) in considering the problem of interaction of light with rough surfaces. However, the calculation methods used in those works are restricted to a large degree by the model itself and we believe that for 0039~028~81/0000-0000~~02.50
0 1981 North-Holland
M.A. Krykin, SF.
Timashev / Optical properties of interface boundaries
215
many problems of surface science these methods are not applicable. The purpose of this paper is to construct a theory of interaction between electromagnetic waves and interface boundary when the latter contains an arbitrary inhomogeneous layer (including the layer caused by the interface roughness). We consider the case where the characteristic thickness of a translational layer is small as compared to other characteristic lengths: skin depth or wave-length. Our task is to find a solution of the Maxwell equations in the corresponding classes of generalized functions with the use (in the general case) of the non-local matter equation. Expressions for the energy coefficient of light reflection from a non-ideal interface boundary are obtained, as a particular case.
2. Constitutent
equations
Let the interface between two non-magnetic isotropic media I and II coincide with plane (XY) of a rectangular coordinate system, and let a plane-polarized electromagnetic wave (with electric field intensity E0 and wave vector /co) be incident from transparent medium I (z < 0). The Maxwell equations for macroscopic electromagnetic field yield V’E-V(VE)+=o,
(1)
where E is the electric field intensity, D is the electrical displacement, and w and c are the frequency and velocity of light, respectively. To complete the system of field equations, let us add to it the matter equation which, according to ref. [3], can be written in the form D=I&E=ii,,Et(ii-l$,)E.
(2)
Here fi, is the linear dielectric operator dependent on the external parameters F, co is the linear operator chosen so that calculation of the light reflection coefficients by eqs. (1) and (2) should give ordinary Fresnel formulas, if I?, = ^no, and if spatial dispersion is neglected. For instance, ref. [l] gives for fro the expression corresponding to the so called dielectric approximation ~E=~e~(k,,,z-z’,o)E(k,,,z’,~)~‘+j~~(kl,lZ:~)~(k11,z’,~)~‘,(3) 90 0 where kll is the two-dimensional wave vector in the plane (Xu) and elcZ)(z) are the one-dimensional Fourier transforms of the local dielectric functions ~~(~~(4, w) for media I and II (4 is a three-dimensional wave vector). Representation of the matter equation in the form of eq. (1) is convenient when the transition layer thickness is small, as compared to other characteristic lengths of the problem - skin depth and wave-length. In this case, with a macroscopical
MA. Krykin, S.F. Titnashev / Optical properties of interface boundaries
216
description of the field, the second term in (2) (corresponding to the transition layer contribution into the displacement 0) is a generalized function with the null support which can be represented unambiguously in the form of expansion in 6function and its derivatives [S] L)j(r, W) = (IlQE)i + p
Cp&, W, F)
6’)(Z)*
(4)
Here r = (x, y, z),p = (x, y), S@)(z) is the pth order derivative of 6(z). The problem is thus reduced to finding solutions of eqs. (1) and (4) in the corresponding classes of generalized functions. Next, we shah confme ourselves, for simplicity, to the case where the spatial dispersion can be neglected; that is, we assume ir, = El(W) 6(-z)+
E*(W) Q(Z).
Let us note, however, that spatial dispersion can easily be accounted framework of the approach presented in ref. [ 11.
(5) for in the
3. Solution of field equations Solution of eqs. (1) and (4) will be sought in the form [3] Ej = EF •t Cd&p) P
6@(z),
(6)
where Ep is the regular part of the field. Let us separate in expansion (4) the transition layer parameters as functions of the linear response to the action of electromagnetic radiation. To this end, expand the coefficients CM in a series in the field EF and quantities d+ retaining only the first terms of the expansion: C&r,
0, F) = E o$ E(p) EFv + C a$ d&p) a ” P
(7)
Here oii and iii$ are transition layer parameters independent of the high-frequency field; the supercript Y (v = C) takes two values corresponding to the electric field intensity on both sides of the layer; t(p) is a function characterizing the change of optical properties in the plane (XI’). This description covers a wide class of surface irregularities, the explicit form of the funfction E(p) being dependent on the specific model of the transition layer. For instance, if the layer is due to the surface roughness, then according to the simplest model t(p) can be considered as the form factor of the interface boundary (i.e. the surface specified by the equation g(p) = 2); for a homogeneous layer we can assume f(p) E 1. Below we shall consider the case of a stochastically inhomogeneous surface and show which statistical characteristics of t(p) determine the field of scattered electromagnetic waves. In particular,
M.A. Krykin, S.F. Timashev / Optical properties of interface boundaries
217
for a surface with a transition layer caused by roughness, we can assume in the simplest case for the mean value (t(p)) = 0. If, moreover, the transition layer at the interface boundary is related to some other factors, e.g. external fields, then (C;(p)) # 0. In the case of adsorption layers, t(p) is determined by the dependence of surface concentration of adsorbed particles on the vector p and so on. To obtain final expressions for the field of scattered electromagnetic waves, one has to specify a certain number of singular terms in (4) and (6); in other words, to determine classes of generalized functions in which solutions of the Maxwell equations are sought. In the present paper we shall retain only one term in the expansion (4). Moreover, only the field E, will be assumed singular, according to ref. [31 (such an approximation is sufficient to prove the generality of the approach developed and to analyse results obtained previously in papers [5-71 concerning the problem in question). Substitution of (6) into (1) with due account for (4) and (7) yields the equation for the regular parts of the field ER and displacement DR: V2ER - V(VER) + $ boundary
DR = 0;
conditions: =-$(L$E;+
(9)
t qE,R-),
(10)
=-$(,$E;+ta;E;-),
a&
ax
and relation for the quantity
g(x, y)+
(cx;E,R++ a;ER,-)v,
r$(x,y)+
(a;E;++cx;E:-)
(11) y,
dez :
doz =(a$E,R+ +a,-E;-)$(x,y). The following notations 6 = a;,
= c&,
and
(12)
(13)
are introduced o; = -a&/&
in the above relations: .
The solution of (8)-( 13) is sought in the form ER-- = E,, exp(ik’
*r) + $
ER+ = 2
exp(ik2 * r) ,
C+(Q)
C-(&l)
exp(ikr * r),
(14) (15)
218
M.A. Krykin, S.F. Tirnashev / Optical properties
of interface boundaries
where ER‘ are the field intensities in media I and II, and the wave vectors kl = (kll, k;) and k2 = (kll, kz) satisfy ordinary dispersion relations kf =?$,
k;
=$.
(16)
Summation in (14) and (15) is perfo~ed only over waves whose solitudes do not grow into the depth of both the media. Substitution of (14) and (15) into eqs. (8)(13) yields the system of equations for determining the quantities C’(kll) = cc”,, c’y, Gj:
C; -C,-
=~&+(k,,
-kp)+&
k,C_;
+k,C,-
+k;C,
=O,
k,C;
+k,C;
+k;C;
=O,
k;C;
- k,C;
-
k,fC:
- k,Ct-
ky~(C;+C;jE(k;
c* (2rr)2
(1’W
k;C,
+ k,C,
cd’ ioL, =-CC: + Gj c* (27rj2 s
k;C;
+k,C;
@I2 =-+,,,k!$j(k,,
S
+
s
=~E,,x&(k,i-k;j+ia, s
(c;
+c,je(k;,
-kiij&i
(27rj2 kx
I
Hk;l - kid %>
(174
-kf)
j(C;+C,)W;l -k,,jdkrc;,, 1
-Eoyt(kj-k,,j+ C
for S-polarized incident light (E, has the component C;-C,
(17b) (17c)
~2 iol, @I*
+
-klljdkb,
(17f)
EoY only along the y-axis), and
[@.f.&,~(k~_kilj s
>
Wa)
k,C;
+k,C;
+k,C,
=O,
(I8c)
k,C;
+
+ k;Ct
= 0,
Wdj
k;C;
- k,C,+ - k,C,
k,C;
+ k,C,-
=-(;)*
(k,“Eo, - k,oEozj 6(k,, - k:)
MA. Krykin, S.F. Timashev /Optical properties of interface boundaries
219
ioL, 0’ -+ (27r)2 c* k;C;
-
k,C;
-
k;C,
+
W2
=‘OLTj-(C; +C,) (27r)2 c*
k,C,
((k;, - k\\) dk;,,
W-I
for P-polarized wave (E, has two components Eox and Eoz along the x- and z-axis, respectively). In systems (17) and (18) the transition to continuous values of k 11is performed by means of known relations
c =,s j-d&, (W2
-9
akllki,
W,, - kid>
(19)
4
and the notation
is introduced
Here Akllki, is the Kronecker symbol, S is the interface area (S -+ -). In deriving (17) and (18) we also assumed (II, = a: = q and (Y, = CY~ = ai. The system of integral equations (17) or (18) will be solved by the iterative method, that is, we shall assume c’-(ki) = Ci t C’: + c”, t ... . Let us note that c’(kll) are generalized functions of k11, so the problem is reduced to solving the corresponding systems of linear equations for generalized functions. Next, we have to define the “zero” problem, that is, to choose the zero approximation for the solution of systems (17) and (18). This choice depends on the particular situation. Let us consider system (17) in detail, namely the case where E is small and the general solution of (17) for t(p) G 0 is taken as the zero approximation. System (18) is slightly more complicated, but it can be solved similarly. The general solution of system (17) is of the form (see the Appendix): 0
0
k,k, 2kll
~1
kxk,
~2
kf
k;
cw* =- -EOy S
f
k;
k,+ - k, 2k, k;
- k,
;
k,k,
W,, - k;)+C
~2
k:
~1
k,k,
~2
WI1 - k,), k;
~1 E2
0
El
0
e2
-1
(21)
M.A. Krykin, S.F. Tirnashev / Optical properties of interface boundaries
220
where the first term in (21) is a particular solution of the inhomogeneous system (the Fresnel solution), and the second term containing an arbitrary constant C is the general solution of the homogeneous system and corresponds to waves propagating along the surface and decaying exponentially into the bulk of both the media (surface polaritons). If there are no sources of surfaces waves, the zero approximation contains only the Fresnel part of (21) i.e. C= 0. Substitution of (21) into (17) results then in the system of equations for determining c(kil). Solving this system, we obtain (22a)
Eoy 2k: TF = 2iol,-- S k”++ko L
k”+ z
=k;
z
ml
- kll)
>
(k,, =kfl).
(224
Using (22), we can find the final expressions for the energy flux of electromagnetic waves scattered at the interface boundary. In fact, the relation for the Poynting vector P in the Fraunhofer zone is of the form
S2 p=cZ._ exp[i(kr 8nw (27r)4 s
- k:) * r] (C{ X kl X Cl) dk11dk\~,
(23)
where kl = k,(kli), k; = k,(k;l), Cl = C,(kll), C\ = C,(k;l). As for experimental study of surface phenomena by optical methods, however (ellipsometry, electroreflectance, surface reflectance spectroscopy), a specularreflected wave (kll = kl) is of greater interest. But it follows from (22) that the quantities Ci(kll) become zero at kll = k\ (the case of (l(p)) = 0 is considered), so the second approximation for C’ has to be sought. In this paper we shall confine ourselves to calculating the energy coefficient R of light reflection, that is the energy flux P, in the direction of the specular reflected ray (the wave vector kb).
4. Coefficient
of light reflection
According to (14), the energy flux P, (the second order in <) in the Fraunhofer zone [9] is of the form P, = &
kb[ IC, I2 t 2 Re(C2e, CT,*)],
(24)
221
M.A. Krykin, S.F. Timashev /Optical properties of interface boundaries
where C, = C;(ki), i.e. C; = C,Aklrkfl; CZO = C,(@). Substitution of (21) and (22) into (17) yields
4@i2 w2
=-----
k,+k;
2k:
(27r)Z c2 (k,o+ + k,0)2
c20y
%I-
k;E,
- k;
g(lkll
- k;E2
(25)
- kB 1) %,
where familiar notations [5] 62g(lktl ])=S-’ k&)1’ are introduced, and 6 is the mean square deviaton of t(p) from zero. Combining, then, (24) and (21) we obtain, with due account for (25), the expression for the change in the light reflection coefficient MS = Rs - RFs (RFSis the Fresnel reflection coefficient): 16 o2 ---kk,062Re aRS = (2n)2 c2
Q&+/$)2
0: e; -e:
(“‘-““)‘I, k;+k,o+
(26)
where (27) Similarly, we can obtain the expression ARp of P-polarized light:
t CY;E~(E~ t c2)2k:2Z;
+ q(Ynk:k,O+
for the change in the reflection coefficient
[(f~ +
~2)Zn
+
'&~2I:1),
(28)
where 1, =
s
.#I1
-km)
k;+k, - k2 k+f _ k_EY 4 z 1 z 2
1: ‘Sg(lk,,
kx(k; - kll I) k+e z
+k,+) _ k-E 1
z
ill, 2
(29a)
5. Results and discussion First of all, let us consider the case where one of the media is vacuum (el = l), and assume that absorption in the second medium can be neglected (i.e. E$’--t 0 and E; < -1, where e2 = ei t kg). Let us note that it is juest the case which is considered in the literature [S-7] most frequently and is of interest for comparison of results obtained with these data. Taking the limit r$ + 0 in (26)-(25), we obtain in the particular case Im CQ= Irn (Y, = 0: AR S(P)= Af#p,
+ A%+
(30)
222
M.A. Krykin, SF.
Timashev / Optical properties of interface boundaries
where
cos 0
= -f$
ARgc
=-pcoseh2a2
(1 + le;l)2(le;l-
(1 t lf;l)(lE;P
X k,,;-
ki)ln
277
482 IE; I2
AR?
,I&(;
oJ-
l)=
g(lk,kll
-k~I)sin2~d~,
(31)
- 1) - kjj
(32) 2n
m;P
h2 w4 If; I2 cos e = _-s 77 c4 (1 t I& l)2( I& I - 1)5n sin28 t I& I cos2e o
X [2c+(sin2B + le~l)lR co~~-~~le~l~‘~(l
A@
=Ew
X [4$(sin2e
- Ie;I)sin0]2dq,
cos e
7r2c (1 t le;l)(~in~e
t le;lc0s2e)o
g(lkskii -@I)
?
(33)
*/c dlkii - kfll> hdo2/c2- k;t dv o
s
(le;]’ - l)(ki
t l&l) (kj t IE~I(w~/c~ - k2Ii sm . 2cp)) t o; l&l’(l
- k,2)
- IE~I)~ sin20 ki
(34) are the changes in the light reflection coefficient due to absorption of incident wave energy by surface polaritons (AR!&) and diffuse scattering (A&&). In (3 l)-(34) a polar coordinates are used and the following notations are introduced: 8 is the incidence angle of a light wave, k,, = k 111 YcltI is the unit vector along the direction ktl. It can be seen from (31) and (33) that absorption of light by surface polaritons is possible if the interface boundary contains irregularities of any kind (roughness, cluster adsorption, nucleation of a new phase, field effects and so on). In particular, for a simple (and fairly wide-spread) model of rough surface where the interface can be considered as an absolutely sharp surface at which the dielectric permittivity has a jump (in this case g( Ik 11 I) is the form factor), relations (3 1) and (33) coincide with the corresponding results of ref. [7]. In this case (Y, = (1 + le; l)/2, CY,= (le; I t l)/ (I& I - 1). Note that the latter equalities also directly follow from comparing boundary condition (9)-(12) and corresponding formula’s (9) of ref. [9]. Refs. [5-71 do not present expressions for aRg. Of great interest is, in our opinion, the case where the transition layer at the interface is caused by several factors, e.g. roughness of the surface and external electric field. Such a situation arises, in particular, in the study of electroreflectance
MA. Krykin, S.F. Timashev/Optical properties of interface boundaries
223
of light from metal-electrolyte interface (see, for instance, ref. [lo]). In this case it is convenient to represent the quantity l(p) as a sum of two terms, t(p) = &, t n(p), and take the exact solution of system (17) or (18), corresponding to a homogeneous transition layer at the surface (n(p) z 0 [l]), as the zero approximation. In this case &, = lo(q) where cp is the electrode potential of a metal (that is potential of the metal electrode in question relative to that of some standard reference electrode). If the surface is smooth (n(p) = 0), surface polaritons (surface plasmons in that case) do not interact with bulk electromagnetic waves and do not contribute to the elctroreflectance signal. If the Interface is non-ideal (n(p) # 0), surface modes become radiative. Let us note, however, the fact that these modes contribute to the electroreflectance signal due to a change in the to value under modulation of cp (since in the framework of the model concepts considered n(p) does not depend on cp). Moreover, the dependence of &, on cp results in the rearrangement of the surface polariton spectrum (chane in the dispersion law, appearance of new branches and so on). Therefore, the “resonance” frequency of interaction of light with surface polaritons must depend on the electrode potential cp,that was in fact observed in ref. [lo]. The explanation of this effect presented in ref. [lo] and based on the assumption that the metal dielectric permittivity changes with variation of electrode potential is, in our opinion, incorrect because the screening length of dc field in a metal (-1 a) is small, as compared to the skin depth (-100 A). The approach developed enables one to solve a number of problems associated with the transformation of surface waves into bulk ones (in the presence of surface wave sources). In this case the second term in (21) corresponds to the zero approximation. Therefore, the formalism suggested permits the solution of a wide range of problems related to the analysis of the interface structure (roughness, superlattices, dispersed layers, etc.) and to the study of kinetics of physico-chemical processes taking place on this interface (adsorption, formation of a new surface phase, chemical reactions, initial stages of corrosion damage and so on).
Appendix
Solution
of field equations in the class of generalized functions of a slow growth
Let us represent, as usual, the general solution of system (17) or (18) as the sum of a partial solution of an inhomogeneous system and the general solution of a homogeneous one (the case .$= 0 is considered). An ordinary “Fresnel” solution Ce(kll) can be chosen as a partial solution of the inhomogeneous system. For sys-
224
M.A. Krykin, S.F. Timashev 1 Optical properties of’interface
boundaries
tern (18) we have, for example,
k,O(k:+Q - k,Oez) 2k”ko+e zz
1
0 0
6(kll -kfl>,
(A-1)
k;(k,O+e, - k,oc2) -2k0 ZXlk” E where E. is the incident wave amplitude (EoX = E. cos 8, I?,, = -E. sin O), 19is the incidence angle. To obtain the general solution of system (17) or (18), we have, therefore, to find general solution of the following type system:
00
-1
10
0
c*
k,
0
k,
0
k,
0
C;
0
0 0 0
k, 0 0
0 0 k,
k; 0 -k,
C$
‘-1 -k,
k, 1 k;
“c;
Lo
0
k;
k,
-k,
J ,cj
=o.
-k;
(A.21
I C,-
Introducting new variables Cl1 and cp(the polar coordinates): C, = C; = G= Cl1 cos cp, C, = CT = C$ = Cl, sin cp, k, = kll cos cp, k, = kll sin p, we can lower the order of system (A.2). Then we have instead of (A.2) Ac=o,
(A.3)
where
++
f:
j-j,
+j.
The determinant of this system, A equal to A = (kg - ki)(klk; - kj) may turn to zero ifka=ki =( w2 / c2 ) er e2 /(er t e2), that is at the frequency of a surface polariton. The general solution (A.3) is, therefore, a generalized function with the null support. This generalized function is uniquely represented in the form [8]: C(kII) = c b,+‘)(k,, P
- k,) .
(A.4)
M.A. Krykin, SF.
Substitution &be
225
Timashev /Optical properties of interface boundaries
of (A.4) into (A.3) yields the system of equalities
- A&
+A&
t . .. t (-l)%@‘b,
= 0,
\
1 GW A&,_~ &b,,
^I - p*Aobp = 0, = 0,
/
where the notation A$‘) = (#/%c~)~ (ktl = k,) is introduced. that the system of matrix equation (AS) has the only solution bl = b2 = .., = b, = 0, ‘G -k, bo =C
(A.6a)
ez ’ et
-e2 e1 ,-1
Let us demonstrate
,
(A.6b)
,
where C is an arbitrary constant (an arbitrary function of kit =klt/lktlI, to be exact), $ = kz(kll = k,). In fact, substituting the fundamental solution of the system AobM = 0 into the equation for bp_l, we obtain
(A.7)
-
1
i El System (A.7) is compatible if the rank of the system matrix (matrix 2,) is equal to the rank of the extended matrix. It can be easily shown that the rank of the extended matrix is equal to the rank of the system matrix at no value of w. The case e1 = -e2 might be the exception, but then the determinant 2 is not zero at either kll. Therefore, eq. (A.6) is the only solution of system (A.5). This solution corresponds to freely propagating surface waves exponentially decaying into the depth of both media.
References [l] M.A. Krykin and S.F. Tamashev, Fiz. Metallov Metalloved. 40 (1975) 958. [2Jvl.A. Krykin and S.F. Timashev, in: Proc. 11th Mendeleev Symp. on General and Applied Chemistry, Alma-Ata, 1975, No. 3, p. 279 (in Russian). [3] S.F. Timashev and M.A. Krykin, Phys. Status Solidi (b) 76 (1976) 67.
226
MA. Krykin, S.F. Timashev / Optical properties
of interface boundaries
[4] A.G. Grivtsov, R.M. Yergunova, Z.M. Zorin, M.A. Krykin, Yu.N. Mikhaylovsky, A.A. Nechaev, SF. Timashev and A.E. Chalykh, Opt. i Spektroskopiya 45 (1978) 738. (51 J.M. Elson and R.H. Ritchie, Phys. Status Solidi (b) 62 (1974) 461. f6] A.A. Maradudin and D.L. Mills, Phys. Rev. Bll (197.5) 1392; B. Laks and D.L. Mills, Phys. Rev. B20 (1979) 4962. [7] G.S. Agarwal, Phys. Rev. B15 (1977) 2371. [ 81 I.M. Gelfand and GE. Shilov, Spaces of Fundamental and Generalized Functions @Ioscow, 1958) p. 149 (in Russian). [9] F.G. Bass and I.M. Fuks, Scattering of Waves at Statistically Rough Surfaces (Nauka, Moscow, 1972) (in Russian). [lo] D.M. Kolb, J. Physique 38,Colloque C-5, Suppl. 11 (1977) 167.
Science 111 (1981) 2144226 North-Holland Publishing Company
Surface
OPTICAL PROPERTIES OF NON-IDEAL INTERFACE
BOUNDARIES
M.A. KRYKIN and S.F. TIMASHEV Karpov Institute of Physical Chemistry, Ul. Obukha IQ, 107120
Moscow B-120, USSR
Received 12 September 1980; accepted for publication 11 June 1981
Optical properties of an interface boundary containing thin inhomogeneous transitional tayers have been investigated within the framework of the classical Maxwell electrodynamics with the use of generalized functions. As a p~ticular case, expressions for reflection coefficients have been obtained for S- and P-polarized light.
1. Introduction
Optical methods of ~vestigation of interface boundaries (ellipsometry, electroreflectance, surface reflectance spectroscopy) may give unique information both on the state of interfaces and on the kinetics of physico-chemical processes taking place on them. In this case we mean the state of thin transition layers with characteristic thicknesses of -I--IO 8. These can be adsorption layers, layers produced by external fields or roughness of the surface, inversion layers in semiconductors and Mets-dielectric-semiconductor structures, thii island films on surfaces of semiconductors and dielectrics, etc. A phenomenological theory of light reflection from an interface with an arbitrary transition layer at this interface has been developed in ref. [l--4]. The optical parameters of the transition layer introduced in refs. [I-4] had a sense of response functions to the action of ele~troma~etic field of light and were associated with high-frequency conductivity and moments of surface high-frequency current (these parameters can be calculated in the microscopic approximation if a specific model is chosen, see, e.g., ref. [4]). Refs. [l-4] consider the case where the layer is homogeneous along the interface. It should be noted, however, that in some problems consideration of inhomogeneity of the transition layer at the interface is necessary and may result in qu~itative~y new effects absent in refs. [l-4]: the appearance of diffuse light scattering, interaction of bulk and surface modes (ii particular, absorption of light on surface polaritons), and so on. Such effects were investigated in a number of works (for instance refs. [5-71) in considering the problem of interaction of light with rough surfaces. However, the calculation methods used in those works are restricted to a large degree by the model itself and we believe that for 0039~028~81/0000-0000~~02.50
0 1981 North-Holland
M.A. Krykin, SF.
Timashev / Optical properties of interface boundaries
215
many problems of surface science these methods are not applicable. The purpose of this paper is to construct a theory of interaction between electromagnetic waves and interface boundary when the latter contains an arbitrary inhomogeneous layer (including the layer caused by the interface roughness). We consider the case where the characteristic thickness of a translational layer is small as compared to other characteristic lengths: skin depth or wave-length. Our task is to find a solution of the Maxwell equations in the corresponding classes of generalized functions with the use (in the general case) of the non-local matter equation. Expressions for the energy coefficient of light reflection from a non-ideal interface boundary are obtained, as a particular case.
2. Constitutent
equations
Let the interface between two non-magnetic isotropic media I and II coincide with plane (XY) of a rectangular coordinate system, and let a plane-polarized electromagnetic wave (with electric field intensity E0 and wave vector /co) be incident from transparent medium I (z < 0). The Maxwell equations for macroscopic electromagnetic field yield V’E-V(VE)+=o,
(1)
where E is the electric field intensity, D is the electrical displacement, and w and c are the frequency and velocity of light, respectively. To complete the system of field equations, let us add to it the matter equation which, according to ref. [3], can be written in the form D=I&E=ii,,Et(ii-l$,)E.
(2)
Here fi, is the linear dielectric operator dependent on the external parameters F, co is the linear operator chosen so that calculation of the light reflection coefficients by eqs. (1) and (2) should give ordinary Fresnel formulas, if I?, = ^no, and if spatial dispersion is neglected. For instance, ref. [l] gives for fro the expression corresponding to the so called dielectric approximation ~E=~e~(k,,,z-z’,o)E(k,,,z’,~)~‘+j~~(kl,lZ:~)~(k11,z’,~)~‘,(3) 90 0 where kll is the two-dimensional wave vector in the plane (Xu) and elcZ)(z) are the one-dimensional Fourier transforms of the local dielectric functions ~~(~~(4, w) for media I and II (4 is a three-dimensional wave vector). Representation of the matter equation in the form of eq. (1) is convenient when the transition layer thickness is small, as compared to other characteristic lengths of the problem - skin depth and wave-length. In this case, with a macroscopical
MA. Krykin, S.F. Titnashev / Optical properties of interface boundaries
216
description of the field, the second term in (2) (corresponding to the transition layer contribution into the displacement 0) is a generalized function with the null support which can be represented unambiguously in the form of expansion in 6function and its derivatives [S] L)j(r, W) = (IlQE)i + p
Cp&, W, F)
6’)(Z)*
(4)
Here r = (x, y, z),p = (x, y), S@)(z) is the pth order derivative of 6(z). The problem is thus reduced to finding solutions of eqs. (1) and (4) in the corresponding classes of generalized functions. Next, we shah confme ourselves, for simplicity, to the case where the spatial dispersion can be neglected; that is, we assume ir, = El(W) 6(-z)+
E*(W) Q(Z).
Let us note, however, that spatial dispersion can easily be accounted framework of the approach presented in ref. [ 11.
(5) for in the
3. Solution of field equations Solution of eqs. (1) and (4) will be sought in the form [3] Ej = EF •t Cd&p) P
6@(z),
(6)
where Ep is the regular part of the field. Let us separate in expansion (4) the transition layer parameters as functions of the linear response to the action of electromagnetic radiation. To this end, expand the coefficients CM in a series in the field EF and quantities d+ retaining only the first terms of the expansion: C&r,
0, F) = E o$ E(p) EFv + C a$ d&p) a ” P
(7)
Here oii and iii$ are transition layer parameters independent of the high-frequency field; the supercript Y (v = C) takes two values corresponding to the electric field intensity on both sides of the layer; t(p) is a function characterizing the change of optical properties in the plane (XI’). This description covers a wide class of surface irregularities, the explicit form of the funfction E(p) being dependent on the specific model of the transition layer. For instance, if the layer is due to the surface roughness, then according to the simplest model t(p) can be considered as the form factor of the interface boundary (i.e. the surface specified by the equation g(p) = 2); for a homogeneous layer we can assume f(p) E 1. Below we shall consider the case of a stochastically inhomogeneous surface and show which statistical characteristics of t(p) determine the field of scattered electromagnetic waves. In particular,
M.A. Krykin, S.F. Timashev / Optical properties of interface boundaries
217
for a surface with a transition layer caused by roughness, we can assume in the simplest case for the mean value (t(p)) = 0. If, moreover, the transition layer at the interface boundary is related to some other factors, e.g. external fields, then (C;(p)) # 0. In the case of adsorption layers, t(p) is determined by the dependence of surface concentration of adsorbed particles on the vector p and so on. To obtain final expressions for the field of scattered electromagnetic waves, one has to specify a certain number of singular terms in (4) and (6); in other words, to determine classes of generalized functions in which solutions of the Maxwell equations are sought. In the present paper we shall retain only one term in the expansion (4). Moreover, only the field E, will be assumed singular, according to ref. [31 (such an approximation is sufficient to prove the generality of the approach developed and to analyse results obtained previously in papers [5-71 concerning the problem in question). Substitution of (6) into (1) with due account for (4) and (7) yields the equation for the regular parts of the field ER and displacement DR: V2ER - V(VER) + $ boundary
DR = 0;
conditions: =-$(L$E;+
(9)
t qE,R-),
(10)
=-$(,$E;+ta;E;-),
a&
ax
and relation for the quantity
g(x, y)+
(cx;E,R++ a;ER,-)v,
r$(x,y)+
(a;E;++cx;E:-)
(11) y,
dez :
doz =(a$E,R+ +a,-E;-)$(x,y). The following notations 6 = a;,
= c&,
and
(12)
(13)
are introduced o; = -a&/&
in the above relations: .
The solution of (8)-( 13) is sought in the form ER-- = E,, exp(ik’
*r) + $
ER+ = 2
exp(ik2 * r) ,
C+(Q)
C-(&l)
exp(ikr * r),
(14) (15)
218
M.A. Krykin, S.F. Tirnashev / Optical properties
of interface boundaries
where ER‘ are the field intensities in media I and II, and the wave vectors kl = (kll, k;) and k2 = (kll, kz) satisfy ordinary dispersion relations kf =?$,
k;
=$.
(16)
Summation in (14) and (15) is perfo~ed only over waves whose solitudes do not grow into the depth of both the media. Substitution of (14) and (15) into eqs. (8)(13) yields the system of equations for determining the quantities C’(kll) = cc”,, c’y, Gj:
C; -C,-
=~&+(k,,
-kp)+&
k,C_;
+k,C,-
+k;C,
=O,
k,C;
+k,C;
+k;C;
=O,
k;C;
- k,C;
-
k,fC:
- k,Ct-
ky~(C;+C;jE(k;
c* (2rr)2
(1’W
k;C,
+ k,C,
cd’ ioL, =-CC: + Gj c* (27rj2 s
k;C;
+k,C;
@I2 =-+,,,k!$j(k,,
S
+
s
=~E,,x&(k,i-k;j+ia, s
(c;
+c,je(k;,
-kiij&i
(27rj2 kx
I
Hk;l - kid %>
(174
-kf)
j(C;+C,)W;l -k,,jdkrc;,, 1
-Eoyt(kj-k,,j+ C
for S-polarized incident light (E, has the component C;-C,
(17b) (17c)
~2 iol, @I*
+
-klljdkb,
(17f)
EoY only along the y-axis), and
[@.f.&,~(k~_kilj s
>
Wa)
k,C;
+k,C;
+k,C,
=O,
(I8c)
k,C;
+
+ k;Ct
= 0,
Wdj
k;C;
- k,C,+ - k,C,
k,C;
+ k,C,-
=-(;)*
(k,“Eo, - k,oEozj 6(k,, - k:)
MA. Krykin, S.F. Timashev /Optical properties of interface boundaries
219
ioL, 0’ -+ (27r)2 c* k;C;
-
k,C;
-
k;C,
+
W2
=‘OLTj-(C; +C,) (27r)2 c*
k,C,
((k;, - k\\) dk;,,
W-I
for P-polarized wave (E, has two components Eox and Eoz along the x- and z-axis, respectively). In systems (17) and (18) the transition to continuous values of k 11is performed by means of known relations
c =,s j-d&, (W2
-9
akllki,
W,, - kid>
(19)
4
and the notation
is introduced
Here Akllki, is the Kronecker symbol, S is the interface area (S -+ -). In deriving (17) and (18) we also assumed (II, = a: = q and (Y, = CY~ = ai. The system of integral equations (17) or (18) will be solved by the iterative method, that is, we shall assume c’-(ki) = Ci t C’: + c”, t ... . Let us note that c’(kll) are generalized functions of k11, so the problem is reduced to solving the corresponding systems of linear equations for generalized functions. Next, we have to define the “zero” problem, that is, to choose the zero approximation for the solution of systems (17) and (18). This choice depends on the particular situation. Let us consider system (17) in detail, namely the case where E is small and the general solution of (17) for t(p) G 0 is taken as the zero approximation. System (18) is slightly more complicated, but it can be solved similarly. The general solution of system (17) is of the form (see the Appendix): 0
0
k,k, 2kll
~1
kxk,
~2
kf
k;
cw* =- -EOy S
f
k;
k,+ - k, 2k, k;
- k,
;
k,k,
W,, - k;)+C
~2
k:
~1
k,k,
~2
WI1 - k,), k;
~1 E2
0
El
0
e2
-1
(21)
M.A. Krykin, S.F. Tirnashev / Optical properties of interface boundaries
220
where the first term in (21) is a particular solution of the inhomogeneous system (the Fresnel solution), and the second term containing an arbitrary constant C is the general solution of the homogeneous system and corresponds to waves propagating along the surface and decaying exponentially into the bulk of both the media (surface polaritons). If there are no sources of surfaces waves, the zero approximation contains only the Fresnel part of (21) i.e. C= 0. Substitution of (21) into (17) results then in the system of equations for determining c(kil). Solving this system, we obtain (22a)
Eoy 2k: TF = 2iol,-- S k”++ko L
k”+ z
=k;
z
ml
- kll)
>
(k,, =kfl).
(224
Using (22), we can find the final expressions for the energy flux of electromagnetic waves scattered at the interface boundary. In fact, the relation for the Poynting vector P in the Fraunhofer zone is of the form
S2 p=cZ._ exp[i(kr 8nw (27r)4 s
- k:) * r] (C{ X kl X Cl) dk11dk\~,
(23)
where kl = k,(kli), k; = k,(k;l), Cl = C,(kll), C\ = C,(k;l). As for experimental study of surface phenomena by optical methods, however (ellipsometry, electroreflectance, surface reflectance spectroscopy), a specularreflected wave (kll = kl) is of greater interest. But it follows from (22) that the quantities Ci(kll) become zero at kll = k\ (the case of (l(p)) = 0 is considered), so the second approximation for C’ has to be sought. In this paper we shall confine ourselves to calculating the energy coefficient R of light reflection, that is the energy flux P, in the direction of the specular reflected ray (the wave vector kb).
4. Coefficient
of light reflection
According to (14), the energy flux P, (the second order in <) in the Fraunhofer zone [9] is of the form P, = &
kb[ IC, I2 t 2 Re(C2e, CT,*)],
(24)
221
M.A. Krykin, S.F. Timashev /Optical properties of interface boundaries
where C, = C;(ki), i.e. C; = C,Aklrkfl; CZO = C,(@). Substitution of (21) and (22) into (17) yields
4@i2 w2
=-----
k,+k;
2k:
(27r)Z c2 (k,o+ + k,0)2
c20y
%I-
k;E,
- k;
g(lkll
- k;E2
(25)
- kB 1) %,
where familiar notations [5] 62g(lktl ])=S-’ k&)1’ are introduced, and 6 is the mean square deviaton of t(p) from zero. Combining, then, (24) and (21) we obtain, with due account for (25), the expression for the change in the light reflection coefficient MS = Rs - RFs (RFSis the Fresnel reflection coefficient): 16 o2 ---kk,062Re aRS = (2n)2 c2
Q&+/$)2
0: e; -e:
(“‘-““)‘I, k;+k,o+
(26)
where (27) Similarly, we can obtain the expression ARp of P-polarized light:
t CY;E~(E~ t c2)2k:2Z;
+ q(Ynk:k,O+
for the change in the reflection coefficient
[(f~ +
~2)Zn
+
'&~2I:1),
(28)
where 1, =
s
.#I1
-km)
k;+k, - k2 k+f _ k_EY 4 z 1 z 2
1: ‘Sg(lk,,
kx(k; - kll I) k+e z
+k,+) _ k-E 1
z
ill, 2
(29a)
5. Results and discussion First of all, let us consider the case where one of the media is vacuum (el = l), and assume that absorption in the second medium can be neglected (i.e. E$’--t 0 and E; < -1, where e2 = ei t kg). Let us note that it is juest the case which is considered in the literature [S-7] most frequently and is of interest for comparison of results obtained with these data. Taking the limit r$ + 0 in (26)-(25), we obtain in the particular case Im CQ= Irn (Y, = 0: AR S(P)= Af#p,
+ A%+
(30)
222
M.A. Krykin, SF.
Timashev / Optical properties of interface boundaries
where
cos 0
= -f$
ARgc
=-pcoseh2a2
(1 + le;l)2(le;l-
(1 t lf;l)(lE;P
X k,,;-
ki)ln
277
482 IE; I2
AR?
,I&(;
oJ-
l)=
g(lk,kll
-k~I)sin2~d~,
(31)
- 1) - kjj
(32) 2n
m;P
h2 w4 If; I2 cos e = _-s 77 c4 (1 t I& l)2( I& I - 1)5n sin28 t I& I cos2e o
X [2c+(sin2B + le~l)lR co~~-~~le~l~‘~(l
A@
=Ew
X [4$(sin2e
- Ie;I)sin0]2dq,
cos e
7r2c (1 t le;l)(~in~e
t le;lc0s2e)o
g(lkskii -@I)
?
(33)
*/c dlkii - kfll> hdo2/c2- k;t dv o
s
(le;]’ - l)(ki
t l&l) (kj t IE~I(w~/c~ - k2Ii sm . 2cp)) t o; l&l’(l
- k,2)
- IE~I)~ sin20 ki
(34) are the changes in the light reflection coefficient due to absorption of incident wave energy by surface polaritons (AR!&) and diffuse scattering (A&&). In (3 l)-(34) a polar coordinates are used and the following notations are introduced: 8 is the incidence angle of a light wave, k,, = k 111 YcltI is the unit vector along the direction ktl. It can be seen from (31) and (33) that absorption of light by surface polaritons is possible if the interface boundary contains irregularities of any kind (roughness, cluster adsorption, nucleation of a new phase, field effects and so on). In particular, for a simple (and fairly wide-spread) model of rough surface where the interface can be considered as an absolutely sharp surface at which the dielectric permittivity has a jump (in this case g( Ik 11 I) is the form factor), relations (3 1) and (33) coincide with the corresponding results of ref. [7]. In this case (Y, = (1 + le; l)/2, CY,= (le; I t l)/ (I& I - 1). Note that the latter equalities also directly follow from comparing boundary condition (9)-(12) and corresponding formula’s (9) of ref. [9]. Refs. [5-71 do not present expressions for aRg. Of great interest is, in our opinion, the case where the transition layer at the interface is caused by several factors, e.g. roughness of the surface and external electric field. Such a situation arises, in particular, in the study of electroreflectance
MA. Krykin, S.F. Timashev/Optical properties of interface boundaries
223
of light from metal-electrolyte interface (see, for instance, ref. [lo]). In this case it is convenient to represent the quantity l(p) as a sum of two terms, t(p) = &, t n(p), and take the exact solution of system (17) or (18), corresponding to a homogeneous transition layer at the surface (n(p) z 0 [l]), as the zero approximation. In this case &, = lo(q) where cp is the electrode potential of a metal (that is potential of the metal electrode in question relative to that of some standard reference electrode). If the surface is smooth (n(p) = 0), surface polaritons (surface plasmons in that case) do not interact with bulk electromagnetic waves and do not contribute to the elctroreflectance signal. If the Interface is non-ideal (n(p) # 0), surface modes become radiative. Let us note, however, the fact that these modes contribute to the electroreflectance signal due to a change in the to value under modulation of cp (since in the framework of the model concepts considered n(p) does not depend on cp). Moreover, the dependence of &, on cp results in the rearrangement of the surface polariton spectrum (chane in the dispersion law, appearance of new branches and so on). Therefore, the “resonance” frequency of interaction of light with surface polaritons must depend on the electrode potential cp,that was in fact observed in ref. [lo]. The explanation of this effect presented in ref. [lo] and based on the assumption that the metal dielectric permittivity changes with variation of electrode potential is, in our opinion, incorrect because the screening length of dc field in a metal (-1 a) is small, as compared to the skin depth (-100 A). The approach developed enables one to solve a number of problems associated with the transformation of surface waves into bulk ones (in the presence of surface wave sources). In this case the second term in (21) corresponds to the zero approximation. Therefore, the formalism suggested permits the solution of a wide range of problems related to the analysis of the interface structure (roughness, superlattices, dispersed layers, etc.) and to the study of kinetics of physico-chemical processes taking place on this interface (adsorption, formation of a new surface phase, chemical reactions, initial stages of corrosion damage and so on).
Appendix
Solution
of field equations in the class of generalized functions of a slow growth
Let us represent, as usual, the general solution of system (17) or (18) as the sum of a partial solution of an inhomogeneous system and the general solution of a homogeneous one (the case .$= 0 is considered). An ordinary “Fresnel” solution Ce(kll) can be chosen as a partial solution of the inhomogeneous system. For sys-
224
M.A. Krykin, S.F. Timashev 1 Optical properties of’interface
boundaries
tern (18) we have, for example,
k,O(k:+Q - k,Oez) 2k”ko+e zz
1
0 0
6(kll -kfl>,
(A-1)
k;(k,O+e, - k,oc2) -2k0 ZXlk” E where E. is the incident wave amplitude (EoX = E. cos 8, I?,, = -E. sin O), 19is the incidence angle. To obtain the general solution of system (17) or (18), we have, therefore, to find general solution of the following type system:
00
-1
10
0
c*
k,
0
k,
0
k,
0
C;
0
0 0 0
k, 0 0
0 0 k,
k; 0 -k,
C$
‘-1 -k,
k, 1 k;
“c;
Lo
0
k;
k,
-k,
J ,cj
=o.
-k;
(A.21
I C,-
Introducting new variables Cl1 and cp(the polar coordinates): C, = C; = G= Cl1 cos cp, C, = CT = C$ = Cl, sin cp, k, = kll cos cp, k, = kll sin p, we can lower the order of system (A.2). Then we have instead of (A.2) Ac=o,
(A.3)
where
++
f:
j-j,
+j.
The determinant of this system, A equal to A = (kg - ki)(klk; - kj) may turn to zero ifka=ki =( w2 / c2 ) er e2 /(er t e2), that is at the frequency of a surface polariton. The general solution (A.3) is, therefore, a generalized function with the null support. This generalized function is uniquely represented in the form [8]: C(kII) = c b,+‘)(k,, P
- k,) .
(A.4)
M.A. Krykin, SF.
Substitution &be
225
Timashev /Optical properties of interface boundaries
of (A.4) into (A.3) yields the system of equalities
- A&
+A&
t . .. t (-l)%@‘b,
= 0,
\
1 GW A&,_~ &b,,
^I - p*Aobp = 0, = 0,
/
where the notation A$‘) = (#/%c~)~ (ktl = k,) is introduced. that the system of matrix equation (AS) has the only solution bl = b2 = .., = b, = 0, ‘G -k, bo =C
(A.6a)
ez ’ et
-e2 e1 ,-1
Let us demonstrate
,
(A.6b)
,
where C is an arbitrary constant (an arbitrary function of kit =klt/lktlI, to be exact), $ = kz(kll = k,). In fact, substituting the fundamental solution of the system AobM = 0 into the equation for bp_l, we obtain
(A.7)
-
1
i El System (A.7) is compatible if the rank of the system matrix (matrix 2,) is equal to the rank of the extended matrix. It can be easily shown that the rank of the extended matrix is equal to the rank of the system matrix at no value of w. The case e1 = -e2 might be the exception, but then the determinant 2 is not zero at either kll. Therefore, eq. (A.6) is the only solution of system (A.5). This solution corresponds to freely propagating surface waves exponentially decaying into the depth of both media.
References [l] M.A. Krykin and S.F. Tamashev, Fiz. Metallov Metalloved. 40 (1975) 958. [2Jvl.A. Krykin and S.F. Timashev, in: Proc. 11th Mendeleev Symp. on General and Applied Chemistry, Alma-Ata, 1975, No. 3, p. 279 (in Russian). [3] S.F. Timashev and M.A. Krykin, Phys. Status Solidi (b) 76 (1976) 67.
226
MA. Krykin, S.F. Timashev / Optical properties
of interface boundaries
[4] A.G. Grivtsov, R.M. Yergunova, Z.M. Zorin, M.A. Krykin, Yu.N. Mikhaylovsky, A.A. Nechaev, SF. Timashev and A.E. Chalykh, Opt. i Spektroskopiya 45 (1978) 738. (51 J.M. Elson and R.H. Ritchie, Phys. Status Solidi (b) 62 (1974) 461. f6] A.A. Maradudin and D.L. Mills, Phys. Rev. Bll (197.5) 1392; B. Laks and D.L. Mills, Phys. Rev. B20 (1979) 4962. [7] G.S. Agarwal, Phys. Rev. B15 (1977) 2371. [ 81 I.M. Gelfand and GE. Shilov, Spaces of Fundamental and Generalized Functions @Ioscow, 1958) p. 149 (in Russian). [9] F.G. Bass and I.M. Fuks, Scattering of Waves at Statistically Rough Surfaces (Nauka, Moscow, 1972) (in Russian). [lo] D.M. Kolb, J. Physique 38,Colloque C-5, Suppl. 11 (1977) 167.