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Ellipsometry and the theory of photon scattering from surfaces
SURFACE
SCIENCE 22 (1970) 433458 o North-Holland
ELLIPSOMETRY
AND THE THEORY
SCATTERING P. W. ATKINS Physical
Chemistry
Laboratory,
FROM
Publishing Co.
OF PHOTON
SURFACES
and A. D. WILSON South Parks Road, Oxford,
England
Received 8 June 1970 The theory of photon scattering is applied to a discussion of ellipsometry. An operator that reproduces the effect of a reflecting and absorbing substrate is constructed phenomenologically by a consideration of the classical equations. An isolated monolayer is treated fully quantum mechanically, and its effect on the polarization of both a reflected beam and a transmitted beam is calculated and expressed in terms of a “monolayer operator” and a “transmission operator”, the parameters of which include the polarizability of the molecules of the monolayer. The monolayer and substrate calculations are then combined, and an expression for the ellipsometric parameters 6Y and &I obtained. An example of several adsorbates on silicon is described.
1. Introduction The fundamental process of ellipsometry may be considered to be the scattering of a polarized photon from the array of molecules that constitute the surface: the polarization change induced by the process is observed and used to determine the optical and geometrical characteristics of the surface, which may be either pure substrate or a substrate totally or partially covered by an adsorbate. A useful review of the type of information that may be obtained from the experiment and a description of the classical theory involved has been given by Bootsma and Meyerl) and a discussion of nomenclature and conventions was given at a recent conferencel). The theories that have been used to interpret ellipsometric data are classical, being based on the Maxwell equations and using the properties of Fresnel coefficients39 4), but as the basic process involved is quantum mechanical it is desirable to attempt to treat it in its fundamental form; that is, as a quantum mechanical scattering process. In a recent series of papers 5, 6,7) it has been shown how to calculate the change incurred in the polarization of a beam of photons of arbitrary initial polarization when they are scattered through any angle by a fluid solution of molecules: this approach may be adapted to the ellipsometric experiment as we shall show in this paper. In section 2 the model is described: the adsorbate and its interaction with 433
434
P. W. ATKINS
AND
A. D. WILSON
the incident light is described quantum mechanically, the effect of the substrate is incorporated classically by introducing, in section 3, a mirror operator whose parameters include the complex refractive index of the substrate. In section 4 the reflections from an isolated monolayer are described quantum mechanically, and in section 5 a monolayer is applied to a surface and the polarization changes are calculated in terms of the polarizability of the adsorbate molecules and the complex refractive index of the substrate. An example is described in section 6, and the results obtained are discussed in section 7. 2. The model The ellipsometric experiment may be imagined as a beam of plane wave photons initially of some polarization &) and wave vector k(‘) impinging at an angle of incidence cpi on a surface that consists of an adsorbate of a large number of identical molecules and a substrate whose molecular structure need be specified only in so far as it determines the density of the adsorbate. Each molecule of the adsorbate is the source of a spherical scattered wave which is superimposed at the detector with the spherical waves from all the other adsorbate molecules and the plane wave reflected from the substrate: the superimposition interferes constructively only in the specular direction. In the scattering processs the polarization of the beam is changed from A(~) to aCS) and the change is used to provide information concerning the extent of coverage and the type of adsorption. The polarization of the initial and final beams may be expressed in terms of two plane polarized basis states, corresponding to the electric vector lying along the unit vectors E( 1) and s(2) which are perpendicular to the wave vector k. Since we are concerned only with elastic scattering, the initial and final wave vectors may be written k&“‘(3) and k@(3) respectively, where e(3) is a unit vector. The vectors s(c) are chosen so that &(i’(1) A FP (2) = &(i)(3) ) E(f) ( 1) A E(J) (2) = &(J-)(3) .
(2.1)
When relations apply equally to initial and final states the distinguishing superscripts will be omitted. The vectors s’“(3) and ~‘~‘(3) determine the scattering plane (fig. 1). The initial state of the photons in the beam may be state In> may be written as a written Iks(‘)(3). , d’)), and as the polarization coherent superposition of the plane polarized basis states Is(a)), one may write 17&i)) = b(j) 1$)(g)) )
In’f’) = by’ I&‘f’(cT))
(2.2)
)
where the b, are the expansion
coefficients
(E(O)(Z) and cr= 1, 2. (We adopt
ELLIPSOMETRY
AND
THE THEORY
OF PHOTON
SCATTERING
435
the rule of implied summation over repeated greek indices, but not summation, unless it is written explicity, over repeated roman indices.) When the beam contains n photons of the same polarization, its state may be written 7, I$$(n))
= T [;I by’“-’ b:“’ Ike”‘(3); $+I
- r] #[r]),
(2.3)
where the notation implies that n - r photons occupy the polarization mode e(‘)(l) and r photons occupy e”‘(2), the total state being symmetrized. [f] is
Fig. 1.
The scattering plane, polarization vectors, and propagation from a surface.
vectors for reflection
the square root of the binomial coefficient (i). For single photon interactions it is unnecessary to use a beam of such complex structure because only one photon operators are involved: therefore we use a one photon state I$) = b, J,‘&‘(3); e”)(a)).
(2.4)
For the reflection of a beam of great intensity (when two OSmore photons may be scattered simultaneously from the same molecule or region of the substrate) the complex beam of eq. (2.3) must be useds). The coefficients b, may be related to the ellipticity and azimuth of the beam through either the polarization density matrix69 g, or the Stokes parameters59 lo, 11). In the latter case the Stokes parameters S, for the beam may be determined by evaluating the expectation values of the Stokes operators Z, for the state I$): S, = ($1 q4 w> = ($I(4
qP> Iti> 3
(2.5)
where the a&are Pauli matrices and a = (aI, u2) is composed of the annihilation operators a, for photons in the polarization mode [e(a)) ; u: are the
436
P. W. ATKINS
AND
A. D. WILSON
corresponding creation operators. It is easy to deduce I$) in eq. (2.3) the Stokes parameters are
that
for the state
s, = n(bFb, + b&), s, = n(bTb, + bTb,), S, = - in (bzb, - b:b,), s, = n(bp,
(2.6)
- b:b,),
and for the single photon state in eq. (2.4) the Stokes parameters are the same but n= 1. The connection with the normal polarization parameters of the beam is made by realising that the phase difference, 6, and the amplitude ratio, tan IX,are related to the Stokes parameters through tan6 = - S,/S,, tana Furthermore,
(2.7)
= (S: + SZ)+/(& + S,) = ((S, the azimuth,
i, and the ellipticity,
- S&(S,
(2.8)
q, of the beam are given by
tan21 = S,/S,, tan 21 = S,/(Sf
+ S,)}+.
(2.9) + S3)‘.
(2.10)
A helpful discussion of the properties of Stokes parameters has been provided by Schneiderli). The initial state vector of the beam of arbitrary polarization may therefore be constructed in the manner of eq. (2.3) or eq. (2.4) by determining the coefficients from eqs. (2.6)-(2.10) in terms of the angles tl and 6 or c and v]; for example 6) b, = cosq dos[ + i sinq sin<, b,=cosqsin[--isinycosc,
(2.11)
are the coefficients for an elliptically polarized beam of azimuth 5 to a chosen direction. If the manner in which the bc’ change when the beam is reflected from a characteristics of surface is determined, from the new b,U) the polarization the reflected beam may be deduced through the new Stokes parameters SLf’. The ba) may be determined by calculating the state vector of the reflected beam, (I,@)): this may be done by introducing the S-matrix of scattering theory, for (2.12) h+P’) = s ll+G”) . The details of the interaction between the beam and the surface are contained within the operator S whose elements constitute the S-matrix. In the quantum mechanical part of the calculation we use time reversed state vectors (2.13)
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
SCATI’ERING
437
in order to conform with the time dependent solution to Maxwell’s equations that has the form exp{ i(ot + S)>: this convention is that suggested for use in ellipsometric calculationss). It is a consequence of the theorem of detailed balance that the decomposition of the S-matrix S,,i, = 6,, - 2rci6(E/,i,) R,i
(2.14)
for normal states implies S,i = 6,i - 2rci 6 (EJi) R,i
(2.15)
for time reversed states12). In this decomposition the &function ensures that scattering is constrained to the energy surface (E/, z Ef - Ei =O). The reaction matrix R now contains the details of the interaction: the polarization changes on reflection may be expressed in terms of its elements, and the latter may be expressed in terms of, for example, the polarizability of the atoms that constitute the surface. 3. The mirror operator The operator that incorporates the effect of the reflection from the substrate will be termed the mirror operator, M. In principle, the effect of A4 may be deduced from the S-matrix itself, for the scattering from the components of the substrate must be included in R. It is simpler, however, to construct the A4 operator phenomenologically by investigating how it can express the changes in the polarization of reflected light which have been evaluated from classical electromagnetic theory: the r6le of the R-matrix is then confined to the description of the interactions with the adsorbate. The classical description of light incident on a plane surface of infinite extent and complex refractive index E is well knowns). The situation is illustrated in fig. 2: a plane wave of arbitrary polarization impinges at an
Fig. 2.
Light incident on a surface of complex refractive index ii.
438
P. W. ATKINS
AND
A. D. WILSON
angle or incidence
(pi; a reflected plane wave is observed at the specular angle; a transmitted wave is propagated at the angle cp: which is the real part of the complex angle F1= qP:+irp;’ obtained from the generalised form of Snell’s law: n, sin ‘pi = A sin &. (3.1) Henceforth the refractive index of the medium above the surface, n,, will be taken as unity and the complex refractive index will be written in the form A=n(l The polarization and A, where
changes
tan Y expid
-iK).
on reflection
(3.2)
may be described
= {Itanol(‘)l/ltanol(/)l)
expi(@)
by the angles
- @)).
Y
(3.3)
In later sections F and A will refer to reflection from the substrate alone and Y and A to reflection from the total system. The last equation may be written in terms of the Fresnel reflection coefficients tan(cp, - @,) A COS pi”’ = _____ tan(cp, + @,) Z COS
Cpi -
COS Qt
‘pi + COS & ’
(3.4) 8:M’=_~~C:11=~~~i~~,,s:,, 1
as
f
tan Y exp id = fi\“‘/fliM’ = -cos
The subscripts 1, 2 refer to the (s(l), Thus one may see that Y = arctan{(tana(“(/(tanatr)l>
f
i
(Vi+
Pt)lcos
(qi-
@t>.
a(3)), (e(2), ~(3)) planes
= arctan{(COS(qi
respectively.
+ @,)/cos(Cpi - &)I),
A = 6(f) - 8’) = arctan [arg { - cos (Cpi+ &)/cOS (Cpi- pJ}] .
(3.5)
(3.0) (3.7)
On writing /$/) =/IL”)’ + ipk”)” one may deduce that (W’ = (n4K: p2)/{n4rc$ + p2 + 2pn2 (ic- cos v - 2rc sin v)}, ;;W = _ 2pn2 (rc_ sin v -I- 2ii COsv)/
(3.8a)
{n4rc: + p2 + 2pn2(K-_ cosv - 2K sinv)},
(3.8b)
/!I$M”= (1 - $)/( 00” B2
where
=
-
(3.8~)
1 + /L2 + 2,u cos v),
2~ sin v/( 1 + p2 + 2p COSV),
(3.8d)
K* = 1+ic2, fi2 c0S2 ‘pi = ((n’ic_
(3.9)
- sin’ Cpi)’ + 4K2f14}‘,
v = - + arctan {2m2/(n2ti_
- sin’ Cpi)}.
(3.10) (3.11)
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
SCA’ITERING
439
The effect of the mirror operator on the state I@‘)) may be expressed in terms of the Fresnel coefficients; we propose that A4 has the following effect: M (k(i); .Ci)) = Mbz)Ik(i); e(i)(o)) = (M@) jk(J); e(/)(0)) = b’/’ 0 1k’f’; I’m’) where
= l/&f’, .(I))
,
Mb(i) = jj(M)bV) s s s .
(3.12) (3.13)
Thus M is diagonal in the basis bj”, and its eigenvalues are the Fresnel coefficients; the effect of M on the wave vector kc’) is to rotate it into the wave vector of the reflected wave, kcf’, and to rotate e(‘)(a) into @(o). For a beam of n photons one obtains
=
n /j\M’“-r /?s”” b(li)“-’ bt)‘l k&“‘(3); &if) [n - r] t$) [r]) . r ic[l (3.14)
The Stokes parameters of the final beam relative to the vectors s(J)(o) may be calculated and expressed in terms of the Stokes parameters of the initial beam relative to the vectors J?(~)(O): SC/) = BilvS(i) (3.15) Y. P When M as previously defined is used and the initial wave is taken to be plane polarized (~5~~) =0) but of arbitrary azimuth the matrix B is found to have the following non-zero elements: B,, = B,, = 3 {IP:“‘12 + IB:“‘12), Ku = B,o = !I {IBi”‘12 - IB:“‘1219 B, 1 = 2 Re /3~“‘/3$“‘*, B,, = 2 Im /$“‘/I$“)*.
(3.16)
Using eqs. (2.7), (2.8), (3.15), and (3.16) one finds that ~5~~) = - arctan (B,,/B, Itan@ Thus
= {(B,,
- J&)/(~o~
b = - arctan (B,,/B,
(3.17)
J, + Bd>*
Itan aci)l ,
(3.19)
1),
p = arctan {(B,, + B,,)/(&_-,
(3.18)
- B&}*.
(3.20)
On introducing the values of /?L”“’given by eqs. (3.8) one obtains !P = arctan {lp~“‘l/I~$M’I) = arctan {Icos(cOi + Pt)lcos(Vi $$>I>, ii = arctan { - Im j?i”’ /3$“‘*/Re/3i”’ /I:“‘*> = arctan {arg [ - COs(Cpi+ $C~J/cos (vi - @,)I}.
(3.21) (3.22)
440
P. W. ATKINS
AND
A. D. WILSON
These values are the same as those obtained in the classical calculation, and so we conclude that, at least for an initially plane polarized wave, the mirror operator defined in eqs. (3.12) and (3.13) reproduces correctly the classical calculation. 4. Reflection of light from an isolated monolayer In the next step of the derivation the quantum mechanical part of the problem is considered: one considers the situation in which a beam of light of frequency w and arbitrary initial polarization impinges at an angle of incidence cpi on a plane surface of infinite extent but zero thickness, with no backing substrate, and composed of molecules at a density of c per unit area, each having the polarizability tensor CC(O).Each molecule is the source of spherical waves which bear a definite phase relation both to one another and to the transmitted plane wave. The asymptotic state after scattering consists of two parts: below the monolayer the outgoing spherical waves interfere with the transmitted plane wave to form a refracted plane wave, and above the monolayer the spherical waves interfere constructively to form a reflected wave in the specular direction. Therefore, the final state may be written (4.1)
If> = If>, + If&
where If), is the state “above” the plane, and If)b the state “below” the plane: this distinction has significance in the position representation, for if the point Y is above the plane (which we write YC~) then (r If>b=O; similarly
l~)=S/i)=li)-2~i~eXP(-i~j)lfi>,
(4.2)
where lfj> = & 6(E, - E) jkcf’; ~~~){~)) (k’?
~;~)((r)l
Rj ii>
(4.3)
is the wave that may be considered as emerging from the molecule j of the monolayer, 9 j is the phase difference between the plane waves arriving at the molecule j and the molecule at the origin of the set of coordinates, and the Ebb) are unit vectors based on the wave vector joining the molecule j to the point of observation (there will be two sets of such unit vectors depending upon whether the transmitted or reflected beam is considered). Rj is the part of the R-matrix involved with the interaction between the beam and molecule j. The coordinates used are illustrated in fig. 3; the direction of the E:‘)(O) will be specified later [eq. (4.28)]. In order to determine the amplitude of the wave at the points of observa-
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
tion rj (rjc a or 6) the position representation (rj
1 fj>
=
(g/2n)3
1
d/t(‘)
kcfj2
dQ(kcf))
6(E,
441
SCATTERING
may be employed to write -
E)
(rj 1 ken; tf)(o))
x (k (/I; E;~‘(cJ)[Rj Ii) = (dp3/8n3kc) 1 dkCf) kCs)’dQ (k(l)) 6 ( kCJ)- k) exp ( - ikCf’- rj) X
lS~“(~))
(k”‘;
(4.4)
&~~‘(~)IRj Ii))
where we employ the normalization (r
1k) = Z-*exp(-
(4.5)
ikar).
I b
Fig. 3.
Scattering from a monolayer: the basis vectors and propagation regions (a) above and (b) below the layer.
directions in the
59 is the quantization volume and the sum over wave vector states has been replaced by the integration & -, (~p/27~)~1 dkCs’ = &Y/279” s dk’l’ kCf)*dQ(@/‘).
(4.6)
Integrating over Q(ktf)) by parts and using kcs)rj+ 1 enables one to extract the outgoing component as a spherical wavels) (rj ) fj)
=
i(9’*/4n2/icA) [exp(- ikrj)/rj] ]&B)(C)> x (k.@‘(3). R.I Ii) ’ ’ ~(.~‘(a)] 3 .I
where A= l/k is the reduced vacuum wavelength of the incident wave.
(4.7)
442
P. W. ATKINS
AND
A. D. WlLSON
The appearance of a variety of vectors referred to different directions may be removed by the following transformations:
pay’(d);
e;yo>> = Ike’f’(v’); df’(y)) (;, ,,y
i
= where the if-coefficients m ( m’
propagation
Ike”‘($);&yp))
were introduced
n if = (e(‘)(m’)l P(n’)) n’ >
(;, ;,)i’(;, ;,)I’,
elsewheres)
(e(‘)(m)[ d.‘)(n))
(4.8)
and are defined by
(1 - 6,,,)
(1 - d,,), (4.9)
and the @coefficients n ( n’
are a convenient
s f* = (fP(n’)l s’> j
generalisation:
12$~‘(d)) (P(n)l
&Y)(S)) (1 - a,,,) (1 - S,,).
(4.10)
The if- andfisymbols disappear if the entries in any column are the same, and for plane polarized basis states are real. On using these coefficients the R-matrix may be expressed solely in terms of the basis vectors of the initial wave, and one may write (rj 1fj)
= i(P/4rPAcil) X ($
Since all molecules introduce
[exp( - ikrj)/rj]
;)i’(;,
1);
I&(/)(r)) (P(r)
(ke”‘(~);
in the monolayer
Isis’)
E’~‘(~L)IRj ( i).
are identical,
(4.11)
R j = R for all j, and we
the abbreviations R$,,,;ncn,j = (/d)(p’); X:,,(j)
=
s(‘)(p)] R Ikdi’(I’); s;“(a)>
(z,
e”‘(1)),
(4.12) (4.13)
1):) J
y =
For a one-photon phenomena)
8’/2nhcit.
(4.14)
state (which is all that is necessary for intensity
independent
one sets Ii> = l/C&“‘(3); e”‘(A)) b!” )
and the scattered wave at the point of observation is given by (rj ) jj) = - (1/27ci) [exp(ikrj)/rj] where Yj (7) = rX,‘,, (j) (z,
I,)‘i
emerging
(4.15) from molecule j
l&‘J’(r)) Yj(Z),
R:‘;‘,,,; ~(3) b:” 9
(4.16)
(4.17)
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
443
SCATTERING
and where the symmetry relation6) (4.18) has been used. The ~plitude be written :
of the superposition at the point of observation P may now
(a) for P c a, (r
1.0
=2~iCexP(-i?j)
= - Cil,~j)
Crj
exp(-
If>
iqj - ikrj) [e”‘(z)> Yj(r);
(4.19)
j
(b) for P c b,
= + [ 9 + hi C exd-
iv,) Qj I f>
= (V f k) jdi’(.A;) bf’ + 2ni x exp( - iqj)
(4.20)
j since only in the latter case (observation
of the transmitted
beam) does
Since the origin fixed in the monolayer is at the point r’ relative to the source and the molecule j is at ri one may write r(i- r’ = Ii for an infinitely distant source. The direction of the vectors r’ and I may be written P*&+ca,
(4.21)
c=cosf$ji+sin95jJ’,
(4.22)
where C=cos qi, S=sin cpi, and the unit vectors;, j, k are shown in fig_ 3. Hence, in the limit r’+co, the phase difference qj becomes vi = (rj - r’)/iz = (lj/A) S COS ~j.
(4.23)
Similarly one may write r = rj + Zj = r (Si^- CL),
so that
rj N r - Sij COS&j +(lj/2r)(l - S2
(4.24) COS”(pj),
where r is the position vector of the point of observation Consequently (l/ri> exp(-
(4.25) relative to 0.
iqj - ikrj) =
= (l/r) exp(-
ikr) {I -I- (El/r) S COS#jjzexp{-
i(Pl2rA) (1 - Sz COS"f$j)]. (4.26)
The next step is to define the eij’(a): s:j’(3) is chosen to lie in the direction
444
P. W. ATKINS
AND
A. D. WILSON
to the (lj, r) plane,
of rj, @(l) to be perpendicular this plane so that
and e:!)(2)
to lie in
&;f’(l) A &$J’(2) = &jf’(3).
(4.27)
This implies that the vectors are a:f)( 1) = (Ej A r) (llj A rl)-’
,
(4.28a)
~~“(2) = (Zj(r2 - lj.r) + r(Zf - liar)} (jlj A rJ )lj - rJ}-' aif)(3) = (r - lj) (lZj - rj)-’
,
.
(4.28b) (4.28~)
These definitions apply to both regions a and b. When expressed in terms of the angle of incidence Cpi and to terms first order in lJr= A one may write 6:f’(l)=gj{~Csin~j2~Ccos~jj”-Ssinf$jf}, ej:“(2) = gj {[C” cos +j + AS(sin’ +[Scos~j-n(l EY)(3)
=
[S - AC2
(4.29a) bj + C2 cos2 f#~~)]P+ sin +jj
-s2cos2~j)]ck}, COsbj]
;-
ti
where the upper signs correspond gy
(4.29b) C(l
Sh4jif
+ AS
COS4j)
(4.29~)
k,
to P c a, the lower to P c b, and
” = C2 cos2 4j + sin’ $j.
The E(~)(D) may be written
in a similar s”‘(1)
manner
= + cc+
(4.30)
as (4.31a)
Sk
e’f’(2) = -;
(4.31b)
&(f)(3) = S;T
(4.31c)
CL.
The transformation coefficients may now be deduced very easily, and are recorded to first order in A in table 1. The jkoefficients that are required are given in table 2. TABLE
The transformation
1
coefficients
a
1 2 3
1
2
3
- 9j sin 4j F 91 cos $j 0
c 91 cos #j - 9j sin 41 A91-’
F AC cos +j A sin #j 1
gj-2 = C2 + S2 sin2 ~$1
ELLIPSOMETRY
TABLE The
( > 3
mff 35
P
=
( 2 1
35> ff = 1
( 31
35 1 I=ff
(2 1
2ff 35 >
1 3
2ff
(
ficoefficients.
Upper
signs
refer
2
to region
0
-
g5A
sin2 &
95 sin $5
= & g5,4C cos $5
sin 45
= * g5c cos $5
35 )
The sum overj
445
AND THE THEORY OF PHOTON SCATTERING
a, lower
signs
to region
3 ( fit
1 ff d >5
=
0
( 21
35 lff >
=
g5AC2
( 23
35 lff>
=
( 21
2ff 35 >
= i
g5AC
2 ( 3
2ff =
95 sin45
3>5
in eq. (4.2) may now be replaced
zk g5ccos
-
z(l/rj)
sin 45 cos
45
0
of molecules.
Then one seeks the asymptotic
exp[-i(qj + krj)]
’
$5
(4.32)
0
I(z) =
$5
by an integral
4-.~dljljjd$j,
where (T is the surface density limit as r-bco of
cos
b
q(T)
(z, $)‘iR~l,r,;,c,,by),
= yo(e -jk’/r) x:,.
(4.33)
where m X&*
2n
dlj Zj
= s 0
d~j (1 + nS
COS ~j)
X:,,(j)
s 0 x
exp { - i(Z3/2rJ) (1 - S2 cos2 +j)>.
The combination of transformation coefficients X&,) may be deduced from tables 1 and 2: X,‘,, = 6,,6,.,
+ A sin 4j 6,,&,
Xy”,’= Bv28”‘3 T AC
COS ~j
X,“, = &- AC cos 4j 6,,6,,,
6,26,‘,
that occurs in xi,. (that is
,
(4.34a) ,
- A sin +j 6,,6,,,
(4.34b) ,
(4.34c)
where, as usual, the upper signs refer to region a and the lower to region b.
446
P. W. ATKINS
Upon integrating
s
dl lk exp (-
AND
A. D. WILSON
first over &j and then over Zj using the integral14)
iZ2/2riZ)
0
= +(2rA)*(k+1) T’(+k + 4) exp { - +i(k + 1) rr}
(k > - 1,2rit. > 0),
(4.35) one obtains x:,. = - irA(l + 3S2) (6,,6,1&f3
+ 42h2&p3),
(4.36)
from which may be deduced
Consequently
(r
1f)
= (r 1k) le”‘(A))
b$’ 6(r, b)
- iyoA( 1 + 4s’) eeikr { Wl))
(; (4.38)
R$,,,); A(3)b:“,
where S(r, b) = 0 unless r c b, when 6(r, b) = 1. The polarization is still confined to ~‘~‘(1) and ~‘~‘(2); that is, to directions perpendicular to r, so that rk=r*k. Therefore exp(-i k-r) is the plane wave LZ’p-t(r 1k), and so we may write (r 1f)
= (r
I k) {It’s)>
b:” S(r, b) + (d’)(A))
8,))
(4.39)
b(i) r *
(4.40)
where 8, = - i(Z30/tic)
(1 + 4s’)
fi
R(i) lw):f(3)
In region b one has
(4.41) and if the cancellation of _F3 by a like factor in the denominator pated one may deduce that
of Ris antici-
5, = - i(a/tzc) (1 + +S2) {a,, Ry()3j; T(3j + S,, R:ir3); r(3j) b’,“,
(4.42)
and (r I f)
NOW suppose
= (r
that the monolayer
I k) l.di)(l)) is composed
(by’ + 6,).
of molecules
(4.43) with an axially
ELLIPSOMETRY
AND THE THEORY OF PHOTON SCATTERING
447
symmetrical polarizability tensor with a component aI perpendicular to the surface and components all in the surface, then one may deduce from the rules for calculating the elements of the R-matrix6) that for single photon interactions (4.44) R$,,,; l(l,j = (~c$c~) ?a(W)rl m,,,,, y where cc, is the vacuum permittivity, yeis a Lorentz correction term, and m is a matrix with all elements mp,lf = 1: this result is derived in the Appendix. Since the only non-zero elements of the polarizability tensor are aI1 = Pa,
+ C2c(,, ,
(4.45a) (4.45b)
a22 = all 7 al3 =
cl31
=
SC(a,
-
(4.4%)
q),
a33 = C’a, + S2all,
(4.45d)
one may deduce that 6,, which in this region refers to the transmitted and so is written br’, is related to the initial bj” by
wave
@) = 8;) b;i)
(4.46)
By’ = - i(rc~a/s,1) (1 + 3s’) (a,, + (aI - a,,) S”}
(4.47a)
with Jlf’ = - i(rcya/sJ)
(1 + 3s’) a,, .
(4.47b)
In region a the sole contribution to the amplitude is due to the scattered spherical waves, which superimpose to produce asymptotically the plane wave = W’(~))
6,.
(4.48)
The effect of the monolayer resembles in this region the effect of a mirror, and so 6, will be written b:““. Four if-symbols are required: they may be constructed from the values of (@(rr) 1#)(a’)) recorded in table 3, and found to be (4.49a) (4.49b) Consequently, one finds that b:“’ = - i(a/hc) {6,, [(S’ - c2)2 jp l(3),
+
6~2
16’
-
C”>
@3):
r(3)
r(3)
-
4S2C2 R$); rt3J]
- XC R$,,; r(3J} b!“.
(4.50)
448
P. W. ATKINS
AND
TABLE
The transformation
A. D. WILSON
3
coefficients CT’
rs
__~
-~
1
2
3
1
s2 _ C”
0
2sc
2
0
1
0
3
- 2sc
0
p _ C”
Using the same model for the monolayer as in the former calculation that bj”’ = p;“’ b!” )
one finds (4.51)
with
fi\") = - i(qa/&,)
(1 + $P)
{LY,I (S2 - C’) + S’[(S’
- C2)2 -
x (@I- "II)L pi”) = - i(rrfIo/le,)
4Sc31 (4.52a)
(1 + 4s’) (S2 - C2 - 2SC) Ml1.
(4.52b)
In each of the regions above and below the monolayer the new b, coefficients, from which the polarization will be determined, are proportional to the initial coefficients by’, and the scattering process does not mix together the by’. Therefore we may define a monolayer operator, m, and a transmission operator, t, which are diagonal in the basis b(” and are such that (4.53) (4.54) These operators obviously resemble the mirror operator M as defined in eq. (3.13). It should also be noted that m has an additional function, for it replaces e(“)(a) by E(~)(O), just like M, and so the full definition of m and t is mb~)Is(‘)(o))
= (mb!)) m Idi)(
tbb”IE@)(o)) = (tbb”) t\@(a))
= C fl~“)b~‘)lg(f)(s)), = C ;:“bj”($+)). s
(4.55) (4.56)
The approach to the calculation of Y and A is now straight forward: the initial polarization determines the by’; above the monolayer the new polarization is determined for the bB)=j?!“‘)by) and below the monolayer it is calculated from the b? which now are a!” bc’.
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
449
SCA’ITERING
5. Reflection of light from an adsorbed monolayer In order to treat the problem of a monolayer on a reflecting substrate the results of the two preceding sections must be combined. Let us suppose that the monolayer is at a height d above the surface: this entails that a phase difference x is introduced between the ray reflected from the monolayer and that reflected from the substrate mirror: x =
If the initial state of the direction will consist of a component transmitted transmitted through the reflected back from the transmitted through the
2dCliZ.
(5.1)
ray is written 1i) the ray observed in the specular a component reflected from the monolayer m (i) ; through the layer, reflected by the mirror, and then layer again to give tMt 1i); a component that was underside of the layer on to the mirror, and then layer tMmMt )i); and so on. These contributions
t d mMmh
1
,
Fig. 4.
Multiple reflections from a monolayer and a substrate surface contributing the total reflected and transmitted rays.
are illustrated
in fig. 4. The total amplitude
(r 1f)
at the detector
= (r I m + eeiX tMt + eeZiX tMmMt
therefore
reflection
mirror
operator
will be
+... Ii)
= (r I m + eTiX tM 1 eminx(mM)” t Ii) ” = (r I m + eeiX tM(1 - eCiX mM)-’ t 1i), and so the total multiple
to
which applied
(5.2) to the
450
initial
P. W. ATKINS
AND
A. D. WILSON
state yields the final state at the detector (5.3)
is given by (5.4) When the monolayer in absent m =O, t = 1 and so a= M, the mirror operator for the pure substrate. Since the operators m, t, and A4 are all diagonal in the basis b(‘), the polarization change induced in the reflected beam is contained in the eigenvalue equation (5.5) with eigenvalues pp
= @)
+
e-i~
py”
p!“)(l
e-iw film)
_
pp)-1.
(5.6)
From this equation one may calculate the Stokes parameters of the reflected beam in terms of the properties of the adsorbate and the substrate: whereas 1 and ?? for the pure substrate were expressed in terms of /3:“’ in eqs. (3.21) and (3.22), when an adsorbate is present A and Y are given by the same expressions, but with pi”) in place of /?:““‘. This section may be elucidated by noticing that the operators m, t, and M, commute: this is not quite trivial, for although the b(‘) are simultaneous eigenbases of all three, m and A4 affect the e(‘)(o) but t leaves them unchanged. Nevertheless we observe that mt Ii) = mtb~)ld’)(A))
= c m @I bi’)js(‘)(r))
= T a!” /$“” bl” ,e’f;(r)> = T t j3:“” bli)lecf)(r))
= tmby’(di’(,l))
= tmli),
(5.7)
and so [m, t] =O.
(5.8)
Likewise [m,M]=[M,t]=O. This enables
eq. (5.4) to be written
(5.9)
as
m + M( tt - mm) eCiX
JJf=
~ i
1+ mMe-iX
I ’
(5.10)
which is the operator form of the Fresnel equationsr). The eigenvalues of A? constitute the Fresnel equations themselves, and so we may write, for
ELLIPSOMETRY
r=
AND THE THEORY
OF PHOTON
SCATTERING
451
1, 2, fl(M)
=
r
B!“’+ *B!“‘@?” + B(m)8(M)
i
r
. I”“’e-iX
I
- I.7
,
(5.11)
For completeness one may add that a transmission operator for the substrate, T, may be constructed phenomenologically, and it leads to an operator form of the classical result for the polarization of a beam that penetrates a region of complex refractive index ii. Thus one may write T b(i) = p(r) b(i) r r I
(5.12)
T(&“‘(A)) = l#)(A)))
(5.13)
with and @’ = 2K/(fiC
+ cos @,),
(5.14a)
/I$” = ZAC/(C + n”cos fj+)) which are the Fresnel treansmission components /3zT” and j$“” are
n4K+
+ p2 + 2/d(K_
- 2n2 [~pcosv
_ --
-
-
sin v] 2K
sinv)’
sinv]
+ picosv
-
2K
sin v)’
(5.15a)
(5.15b)
+pcosv-rcpsinv] 1+ $
(‘0”
2~p
cosv
+ ,u2 + 2pqK_
dK+
p$r” = 2n[l
P2
Their real and imaginary
2n2 [n”K”++ p2~- cos v -
/$r” =
CT)”= BI
coefficients.
(5.14b)
+ 2p cos v
(5.15c)
’
2n[rc+rc~cosv+~sinv] (5.15d)
’
1 +j?+2/fcosv
where the same notation as in eqs. (3.8) for the j?:“’ has been used. From fig. 4 it is simple to deduce that the total transmitted beam is (r
1f) = (rlTt
+ TmMt e-” + TmMmMt
e-2ix +...li)
= (4 Tli), where
(5.16)
T = T(l - mM edi”)-’ t,
(5.17)
which is the operator equivalent of the classical result3). Since all the operators are diagonal in b,(‘), the polarization of the transmitted beam may be expressed in terms of the eigenvalues of T which are jj:T) = a!‘) p!‘)(l _ ,y!W ,$W
,-ix)-1
.
(5.18)
452
P. W.ATKINS
AND
A.D.
WILSON
In conclusion we note that the same results may be obtained by a manner that conforms more closely to the classical discussion of light reflection by ignoring the vectorial superposition of the scattered waves but introducing instead the Kirchoff damping factor that arises from the Kirchoff-Helmholtz integral will be
theorem3).
The superposition
of waves observed
at the detector
= d(r, b) -t 7 (@/hA)
(eeikr/r) eMirpjlP(c7))
(i
t,)‘i
Rtl,,j;ACjj by’,
(5.19)
where ‘pj is the total phase difference between the waves that took a path via molecule j and molecule 0. The damping factor that must be introduced according to the Kirchhoff-Helmholtz theorem reduces the contribution from points j very distant from 0 (in the vectorial addition procedure the vector addition of the polarization vectors achieved this in a different way), and in the asymptotic limit the factor is 1+ S cos 4 j. Consequently
lime-i.l=j”(“_I’.):idm,j.dl,ii(l+Scos~,) r-m
x exp(=Therefore, (r ( f)
i(r~/L-Q(l”-
iail( 1 + $3’) lim eeikr. r-m
the total amplitude
fY
J fi
3
v’
( >
@‘)
.
V(V’),1(3)
(5.20)
is
= (r 1 k) {d(r, b) b:“je”‘(A))
*
S2 co~‘4~)}
b(i) a
- i(,/&)
)
(1 + +S2) IS(‘)(~)) (5.21)
1
which is identical to the result obtained by the vector method; see eq. (4.39). The advantage of this less direct method seems to be that it is more easily extended to a surface with depth: the integrals over a volume of adsorbate may be performed more easily than in the other method. 6. An example We shall consider as an example a silicon substrate covered by a nonabsorbing adsorbate. The polarization changes of the light on reflection from the substrate may be calculated on the basis of the formalism in section 3. The optical parameters for silicon are knownr5) to be n=4.05 and
ELLIPSOMETRY
AND ‘IHE THEORY
OF PHOTON
SCAlTERING
453
~=0.006914. Using eqs. (3.8)-(3.11) the p(M) coefficients for an initially plane polarized beam incident at ‘pi= 70” are found to be /I(M) 1 = 5.7149 + 0.1038 i = c1 + id 19
(6.la)
pi”’ = - 1.1901 - 0.0015 i = c1 + id,.
(6.lb)
From eqs. (3.21) and (3.22) one then deduces that F = 78.24*,
2 = 179.03”,
in accord with the experimental valuesr5) p = 78.23” + 0.05”,
a = 179.05” + 0.19”.
[The !P used in this paper corresponds to 90°-$ in the work of Archerr5) and Bootsma and Meyerr).] At an angle of incidence ‘pi= 45”, the /?:“’ have the values /?1 = 2.0476 + 0.0107 i3
PC;“’ = - 1.4310 - 0.0037 i,
and so !P = 55.07”,
a = 179.85”.
For silver, n=0.20 and K= 17.2; at 4oi=45” fii”) = 0.7329 + 0.7434i,
ficM) Z = - 0 .9426 - 0 .3944 1,
and so p = 55.62”,
a = 165.27”.
When an adsorbate is present the p!“’ are changes in to 8:” ‘) in accord with eqs. (5.6) and (5.11), and the values of Y and A may be calculated accordingly. A convenient simplification, which could be removed easily, is to suppose that the monolayer molecules are isotropically polarizable so that ~1~=a,, =c(. Then the coefficients fi!m)and /I;) become j3’;“’= - i(rc~a/e,l) (1 + +S’) (S2 - C2)2 cL= - ia, ,
(6.2a)
/I$“’ = - i(n~o/s,,~) (1 + 3s’) (S2 - C2 - 2SC) CY = - ia,,
(6.2b)
/I$‘)= 1 - i (rcvcr/s,1) (1 + 4s’) CY = 1 - ib,;
r=1,2.
(6.3)
On writing fli”)=flIM” + iDi”)” and e-iX N 1 -ix, and inserting the expressions for a!“‘, j?lf), and /?z”‘)given by eqs. (6.1), (6.2), and (6.3), one obtains
x41 Cl+ a,(4 - x41 - t-(4- x4 (1 - b,2+ d> - 2b(cr+ x4 - 4 Car Cc,+ x4)1)3
BZ”” = sr {C(c, + Xd,) (1 - b: + a;) + 2b,(d, -
(6.4)
454 B ?“’
P. W. =
~r{C(cr +
[Cd,
x
Cl
+ +
x4 a,(4
xd.)(l (1 -
-
ATKINS
b,2 b,2
+
+
a;)
AND
d)
+ -
A. D. WILSON
2b,(&
%(c,
+
XG.)]
x4)
-
[a,(~,
+
x4)]
4
x41>9
(6.5)
where BP-’ = (1 + a,(&
- XCJ}” + (a,(c,
+ Xd,)}2.
(6.6)
The values of 8,(M) obtained from these equations may be used to determine Y and A, and hence SF=!?--Y and SA=ii-A. The values of SY and SA were calculated for an initially plane plarized beam of I= 546.1 nm incident at 70” on a (111) plane of silicon covered by a monolayer. The values of LX,the polarizability, were taken from table 2 of Bootsma and Meyer l). The values of 0, the fractional surface coverage, was obtained from the same source and used to calculate 0 on the assumption that the surface (111) plane has the same structure as the (111) plane of the bulk crystal. The value of d, the distance between the surface and the monolayer was taken to be the sum of the Van der Waals radius of the adsorbed atomi) or, in H,A, the radius of A, and the covalent radius for Si. Fortunately SY and SA are not very sensitive to the value of d and these rather crude choices do not affect the calculation significantly. The values of SA so calculated are shown in table 4 and compared there with the experimental values and values calculated by classical methodsl). It is far from clear from the equations involved that SA is proportional to the extent of surface coverage, as is observed experimentally: that in fact the dependence of SA on 0 is indeed linear is illustrated in fig. 5 where SA for H, calculated as a function of 0 is plotted. 7. Discussion The values of SY were found to be much smaller than those of SA and in the same direction, in accord with the predictions of the classical theory. It can be seen from table 4 that the fractional change in the imaginary parts of the p are more affected by the adsorption than their real parts; furthermore, the real parts of fi are much larger than the imaginary parts. Consequently, thevalues of 1/3!“‘1and Re(fli”’ pi”*) are not affected significantly by changes in pl”“’ wherease the values of Im(/3i”) B\“‘*) are altered markedly even by small changes in prW” . Thus one may account for the relative magnitudes of SA and SY. For a monolayer or submonolayer, A is much more sensitive to changes in c1and G than to changes in d. Of course, for coverage greater than just a monolayer, one should consider a three dimensional array of scatterers rather than a two dimensional array, and the value of d will be more significant. Our model implies that the absorbate molecules scatter coherently even
0.240
5.7152 1.1901 0.0675 0.0104 0.21
0.137
0.81 0.5 0.04 5.7150 - 1.1901 0.0810 0.0076 0.21 0.20 0.26 0.25
HZ
Cl2
4.50 0.5 0.04 5.7138 - 1.1896 0.0313 0.0441 1.47 1.00 0.94 1.39
0.297
Values of 6A are in degrees. 8 Calculated according to method of the present paper. b Experimental values’). c Drudels), Sivukhinsa) calculationsl). * Strachanl*) calculationsl) .
6AS SAb 6AC 6Ad
Thickness d (nm) (~(x 10-30m-3) 6 u ( x lOlsm-a) /&WY &W’ /&SW ~&W”
Adsorbate
5.7142 1.1895 0.0231 0.0458 1.47
0.360 3.78 0.25 0.02 5.7151 - 1.1900 0.0525 0.0198 0.51 0.43 0.46 0.59
0.302
H2S
5.7153 1.1900 0.0436 0.0216 0.51
0.370 2.65 0.5 0.04 5.7148 - 1.1900 0.0467 0.0259 0.76 0.77 0.64 0.82
0.297
HCl
5.7150 1.1900 0.0393 0.0276 0.76
0.360
HBr
3.64 0.5 0.04 5.7144 - 1.1898 0.0380 0.0349 1.09 0.77 0.83 1.13
0.312
on clean silicon (111) plane for qt = 70”, L = 546.1 nm
TABLET
Calculated and experimental parameter for adsorption
5.7148 1.1897 0.0278 0.0370 1.09
0.390
456
P. W. ATKINS
AND
A. D. WILSON
for small values of 0: this is reflected in the direct proportionality between P(M) or P(t) and 0. The plot of 6A against 0 shows that 6A is directly proportional to 0 even for very small surface coverage. Bennett et a1.16) have shown that the classical ellipsometric equations predict correctly the average
0.;
0
bb
0.
0.t
Fig. 5.
The dependence
of 6A on the extent of surface coverage 0 for hydrogen silicon (111) plane.
on
thickness of a thin discontinuous absorbing film by studying the growth of silver sulphide tarnish on silver. There is also ample experimental evidence’) for the proportionality of 6A and 0 for films adsorbed on silicon and germanium substrates. This proportionality between ellipsometric effect and surface coverage is not surprising, since even when 0 - 0.01 the monolayer comprises a large number of scatterers; for example for H, on Si( 11I), IS- 5 + 10” mm-‘, which is equivalent to 1.5+ lo4 molecules in the first Fresnel zone. We have chosen the Lorentz field correction term appropriate to an isotropic dielectric medium for the effects due both to the monolayer and the
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
SCATTERING
457
substrate. In neither case is this very satisfactory for a variety of reasons but until a better correction term can be obtained this point cannot be improved. The theory can quite easily be extended to include both the effect of absorption by the monolayer molecules and the effect of strong static fields at the interface on the values of the R-matrix elementsIT). The model presented here bears a strong resemblance to the Strachan quasi-microscopic theory 1s) which treats the monolayer as an array of two dimensional Hertzian oscillators but does not interpret the scattering indices so obtained in terms of molecular structure and polarizability. His results are included in table 4 and compared with ours. Acknowledgement
We are grateful to the Science Research Council for financial support of A.D.W. Appendix THE R-MATRIX ELEMENTS
The procedure for the calculation of the R-matrix elements in terms of diagrammatic perturbation theory has been described elsewheres). Only two diagrams are required, and these are illustrated in fig. 6. The application of
Fig. 6. Diagrams contributing to the matrix element RfQ,(,,,; p(p,).
the standard rules5) and the imposition of the limitation that only electric dipole transitions are considered, leads to
for all A’, $. In order to preserve the indices I’, $, yet obtain the same values for all I’, q’, we introduce the matrix m, I pI all elements of which are unity. Introducing the polarizability tensor +&a
= 22E~~~~~~~~/(E~~ - A2mZ),
(A?J
the matrix element becomes Rf’) nf~‘t;r(r’)= (~u~~~32&~) a (mjPi. mPps1
(A3)
as used in eq. (4.47).
f
458
P. W. ATKINS
We include
a correction
AND A. D. WILSON
term r
=
pp
in the R-matrix to allow for the enhancement of the field experienced by a molecule in the monolayer owing to the presence of the other adsorbed molecules (q”‘) and the atoms comprising the substrate (q”‘). The expressions for q(l) and q(‘) are derived from the Lorentz-Lorenz formula for the polarizability of an isotropic dielectric medium :
__3dmlam_ 3d,la, - 4na ’
(1) =
? where d,,, is the diameter a complete monolayer;
of an adsorbed
(2)_’ ?
molecule
n2 ~3
-,j
645)
and cm the value of u for
+2
(
)
where n is the refractive index of the substrate the fact that the adsorbed molecule interacts substrate molecules.
and the factor of + reflects with only a hemisphere of
References 1) G. A. Bootsma and F. Meyer, Surface Sci. 14 (1969) 52. 2) R. H. Muller, Surface Sci. 16 (1969) 14. 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965). A. VaSiZek, Optics of Thin Films (North-Holland, Amsterdam, 1960). P. W. Atkins and L. D. Barron, Proc. Roy. Sot. (London) A 304 (1968) 303. P. W. Atkins and L. D. Barron, Mol. Phys. 16 (1969) 453. P. W. Atkins and L. D. Barron, Mol. Phys. 18 (1970) 721. P. W. Atkins and L. D. Barron, Mol. Phys. 18 (1970) 729. U. Fano, Rev. Mod. Phys. 29 (1957) 74. J. M. Jauch and F. Rohrlich, Theory of Photons and Electrons (Addison-Wesley, Reading, Mass., 1955). R. W. Schneider, J. Opt. Sot. Am. 59 (1969) 297. P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965) p. 539. M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964). W. Grijbner and N. Hofreiter, Zntegraltafel, Vol. 2 (Springer-Verlag, Vienna, 1966) p. 67. R. J. Archer, J. Opt. Sot. Am. 52 (1962) 970. H. E. Bennett, D. K. Burge, R. L. Peck and J. M. Bennett, J. Opt. Sot. Am. 59 (1969) 657. P. W. Atkins and L. D. Barron, Proc. Roy. Sot. (London) A 306 (1968) 119. C. S. Strachan, Proc. Cambridge Phil. Sot. 29 (1933) 116. P. Drude, Wiedemann’s Ann. 36 (1889) 532, 865; 39 (1890) 481. D. V. Sivukhin, Soviet Phys.-JETP 3 (1956) 269.
SCIENCE 22 (1970) 433458 o North-Holland
ELLIPSOMETRY
AND THE THEORY
SCATTERING P. W. ATKINS Physical
Chemistry
Laboratory,
FROM
Publishing Co.
OF PHOTON
SURFACES
and A. D. WILSON South Parks Road, Oxford,
England
Received 8 June 1970 The theory of photon scattering is applied to a discussion of ellipsometry. An operator that reproduces the effect of a reflecting and absorbing substrate is constructed phenomenologically by a consideration of the classical equations. An isolated monolayer is treated fully quantum mechanically, and its effect on the polarization of both a reflected beam and a transmitted beam is calculated and expressed in terms of a “monolayer operator” and a “transmission operator”, the parameters of which include the polarizability of the molecules of the monolayer. The monolayer and substrate calculations are then combined, and an expression for the ellipsometric parameters 6Y and &I obtained. An example of several adsorbates on silicon is described.
1. Introduction The fundamental process of ellipsometry may be considered to be the scattering of a polarized photon from the array of molecules that constitute the surface: the polarization change induced by the process is observed and used to determine the optical and geometrical characteristics of the surface, which may be either pure substrate or a substrate totally or partially covered by an adsorbate. A useful review of the type of information that may be obtained from the experiment and a description of the classical theory involved has been given by Bootsma and Meyerl) and a discussion of nomenclature and conventions was given at a recent conferencel). The theories that have been used to interpret ellipsometric data are classical, being based on the Maxwell equations and using the properties of Fresnel coefficients39 4), but as the basic process involved is quantum mechanical it is desirable to attempt to treat it in its fundamental form; that is, as a quantum mechanical scattering process. In a recent series of papers 5, 6,7) it has been shown how to calculate the change incurred in the polarization of a beam of photons of arbitrary initial polarization when they are scattered through any angle by a fluid solution of molecules: this approach may be adapted to the ellipsometric experiment as we shall show in this paper. In section 2 the model is described: the adsorbate and its interaction with 433
434
P. W. ATKINS
AND
A. D. WILSON
the incident light is described quantum mechanically, the effect of the substrate is incorporated classically by introducing, in section 3, a mirror operator whose parameters include the complex refractive index of the substrate. In section 4 the reflections from an isolated monolayer are described quantum mechanically, and in section 5 a monolayer is applied to a surface and the polarization changes are calculated in terms of the polarizability of the adsorbate molecules and the complex refractive index of the substrate. An example is described in section 6, and the results obtained are discussed in section 7. 2. The model The ellipsometric experiment may be imagined as a beam of plane wave photons initially of some polarization &) and wave vector k(‘) impinging at an angle of incidence cpi on a surface that consists of an adsorbate of a large number of identical molecules and a substrate whose molecular structure need be specified only in so far as it determines the density of the adsorbate. Each molecule of the adsorbate is the source of a spherical scattered wave which is superimposed at the detector with the spherical waves from all the other adsorbate molecules and the plane wave reflected from the substrate: the superimposition interferes constructively only in the specular direction. In the scattering processs the polarization of the beam is changed from A(~) to aCS) and the change is used to provide information concerning the extent of coverage and the type of adsorption. The polarization of the initial and final beams may be expressed in terms of two plane polarized basis states, corresponding to the electric vector lying along the unit vectors E( 1) and s(2) which are perpendicular to the wave vector k. Since we are concerned only with elastic scattering, the initial and final wave vectors may be written k&“‘(3) and k@(3) respectively, where e(3) is a unit vector. The vectors s(c) are chosen so that &(i’(1) A FP (2) = &(i)(3) ) E(f) ( 1) A E(J) (2) = &(J-)(3) .
(2.1)
When relations apply equally to initial and final states the distinguishing superscripts will be omitted. The vectors s’“(3) and ~‘~‘(3) determine the scattering plane (fig. 1). The initial state of the photons in the beam may be state In> may be written as a written Iks(‘)(3). , d’)), and as the polarization coherent superposition of the plane polarized basis states Is(a)), one may write 17&i)) = b(j) 1$)(g)) )
In’f’) = by’ I&‘f’(cT))
(2.2)
)
where the b, are the expansion
coefficients
(E(O)(Z) and cr= 1, 2. (We adopt
ELLIPSOMETRY
AND
THE THEORY
OF PHOTON
SCATTERING
435
the rule of implied summation over repeated greek indices, but not summation, unless it is written explicity, over repeated roman indices.) When the beam contains n photons of the same polarization, its state may be written 7, I$$(n))
= T [;I by’“-’ b:“’ Ike”‘(3); $+I
- r] #[r]),
(2.3)
where the notation implies that n - r photons occupy the polarization mode e(‘)(l) and r photons occupy e”‘(2), the total state being symmetrized. [f] is
Fig. 1.
The scattering plane, polarization vectors, and propagation from a surface.
vectors for reflection
the square root of the binomial coefficient (i). For single photon interactions it is unnecessary to use a beam of such complex structure because only one photon operators are involved: therefore we use a one photon state I$) = b, J,‘&‘(3); e”)(a)).
(2.4)
For the reflection of a beam of great intensity (when two OSmore photons may be scattered simultaneously from the same molecule or region of the substrate) the complex beam of eq. (2.3) must be useds). The coefficients b, may be related to the ellipticity and azimuth of the beam through either the polarization density matrix69 g, or the Stokes parameters59 lo, 11). In the latter case the Stokes parameters S, for the beam may be determined by evaluating the expectation values of the Stokes operators Z, for the state I$): S, = ($1 q4 w> = ($I(4
qP> Iti> 3
(2.5)
where the a&are Pauli matrices and a = (aI, u2) is composed of the annihilation operators a, for photons in the polarization mode [e(a)) ; u: are the
436
P. W. ATKINS
AND
A. D. WILSON
corresponding creation operators. It is easy to deduce I$) in eq. (2.3) the Stokes parameters are
that
for the state
s, = n(bFb, + b&), s, = n(bTb, + bTb,), S, = - in (bzb, - b:b,), s, = n(bp,
(2.6)
- b:b,),
and for the single photon state in eq. (2.4) the Stokes parameters are the same but n= 1. The connection with the normal polarization parameters of the beam is made by realising that the phase difference, 6, and the amplitude ratio, tan IX,are related to the Stokes parameters through tan6 = - S,/S,, tana Furthermore,
(2.7)
= (S: + SZ)+/(& + S,) = ((S, the azimuth,
i, and the ellipticity,
- S&(S,
(2.8)
q, of the beam are given by
tan21 = S,/S,, tan 21 = S,/(Sf
+ S,)}+.
(2.9) + S3)‘.
(2.10)
A helpful discussion of the properties of Stokes parameters has been provided by Schneiderli). The initial state vector of the beam of arbitrary polarization may therefore be constructed in the manner of eq. (2.3) or eq. (2.4) by determining the coefficients from eqs. (2.6)-(2.10) in terms of the angles tl and 6 or c and v]; for example 6) b, = cosq dos[ + i sinq sin<, b,=cosqsin[--isinycosc,
(2.11)
are the coefficients for an elliptically polarized beam of azimuth 5 to a chosen direction. If the manner in which the bc’ change when the beam is reflected from a characteristics of surface is determined, from the new b,U) the polarization the reflected beam may be deduced through the new Stokes parameters SLf’. The ba) may be determined by calculating the state vector of the reflected beam, (I,@)): this may be done by introducing the S-matrix of scattering theory, for (2.12) h+P’) = s ll+G”) . The details of the interaction between the beam and the surface are contained within the operator S whose elements constitute the S-matrix. In the quantum mechanical part of the calculation we use time reversed state vectors (2.13)
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
SCATI’ERING
437
in order to conform with the time dependent solution to Maxwell’s equations that has the form exp{ i(ot + S)>: this convention is that suggested for use in ellipsometric calculationss). It is a consequence of the theorem of detailed balance that the decomposition of the S-matrix S,,i, = 6,, - 2rci6(E/,i,) R,i
(2.14)
for normal states implies S,i = 6,i - 2rci 6 (EJi) R,i
(2.15)
for time reversed states12). In this decomposition the &function ensures that scattering is constrained to the energy surface (E/, z Ef - Ei =O). The reaction matrix R now contains the details of the interaction: the polarization changes on reflection may be expressed in terms of its elements, and the latter may be expressed in terms of, for example, the polarizability of the atoms that constitute the surface. 3. The mirror operator The operator that incorporates the effect of the reflection from the substrate will be termed the mirror operator, M. In principle, the effect of A4 may be deduced from the S-matrix itself, for the scattering from the components of the substrate must be included in R. It is simpler, however, to construct the A4 operator phenomenologically by investigating how it can express the changes in the polarization of reflected light which have been evaluated from classical electromagnetic theory: the r6le of the R-matrix is then confined to the description of the interactions with the adsorbate. The classical description of light incident on a plane surface of infinite extent and complex refractive index E is well knowns). The situation is illustrated in fig. 2: a plane wave of arbitrary polarization impinges at an
Fig. 2.
Light incident on a surface of complex refractive index ii.
438
P. W. ATKINS
AND
A. D. WILSON
angle or incidence
(pi; a reflected plane wave is observed at the specular angle; a transmitted wave is propagated at the angle cp: which is the real part of the complex angle F1= qP:+irp;’ obtained from the generalised form of Snell’s law: n, sin ‘pi = A sin &. (3.1) Henceforth the refractive index of the medium above the surface, n,, will be taken as unity and the complex refractive index will be written in the form A=n(l The polarization and A, where
changes
tan Y expid
-iK).
on reflection
(3.2)
may be described
= {Itanol(‘)l/ltanol(/)l)
expi(@)
by the angles
- @)).
Y
(3.3)
In later sections F and A will refer to reflection from the substrate alone and Y and A to reflection from the total system. The last equation may be written in terms of the Fresnel reflection coefficients tan(cp, - @,) A COS pi”’ = _____ tan(cp, + @,) Z COS
Cpi -
COS Qt
‘pi + COS & ’
(3.4) 8:M’=_~~C:11=~~~i~~,,s:,, 1
as
f
tan Y exp id = fi\“‘/fliM’ = -cos
The subscripts 1, 2 refer to the (s(l), Thus one may see that Y = arctan{(tana(“(/(tanatr)l>
f
i
(Vi+
Pt)lcos
(qi-
@t>.
a(3)), (e(2), ~(3)) planes
= arctan{(COS(qi
respectively.
+ @,)/cos(Cpi - &)I),
A = 6(f) - 8’) = arctan [arg { - cos (Cpi+ &)/cOS (Cpi- pJ}] .
(3.5)
(3.0) (3.7)
On writing /$/) =/IL”)’ + ipk”)” one may deduce that (W’ = (n4K: p2)/{n4rc$ + p2 + 2pn2 (ic- cos v - 2rc sin v)}, ;;W = _ 2pn2 (rc_ sin v -I- 2ii COsv)/
(3.8a)
{n4rc: + p2 + 2pn2(K-_ cosv - 2K sinv)},
(3.8b)
/!I$M”= (1 - $)/( 00” B2
where
=
-
(3.8~)
1 + /L2 + 2,u cos v),
2~ sin v/( 1 + p2 + 2p COSV),
(3.8d)
K* = 1+ic2, fi2 c0S2 ‘pi = ((n’ic_
(3.9)
- sin’ Cpi)’ + 4K2f14}‘,
v = - + arctan {2m2/(n2ti_
- sin’ Cpi)}.
(3.10) (3.11)
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
SCA’ITERING
439
The effect of the mirror operator on the state I@‘)) may be expressed in terms of the Fresnel coefficients; we propose that A4 has the following effect: M (k(i); .Ci)) = Mbz)Ik(i); e(i)(o)) = (M@) jk(J); e(/)(0)) = b’/’ 0 1k’f’; I’m’) where
= l/&f’, .(I))
,
Mb(i) = jj(M)bV) s s s .
(3.12) (3.13)
Thus M is diagonal in the basis bj”, and its eigenvalues are the Fresnel coefficients; the effect of M on the wave vector kc’) is to rotate it into the wave vector of the reflected wave, kcf’, and to rotate e(‘)(a) into @(o). For a beam of n photons one obtains
=
n /j\M’“-r /?s”” b(li)“-’ bt)‘l k&“‘(3); &if) [n - r] t$) [r]) . r ic[l (3.14)
The Stokes parameters of the final beam relative to the vectors s(J)(o) may be calculated and expressed in terms of the Stokes parameters of the initial beam relative to the vectors J?(~)(O): SC/) = BilvS(i) (3.15) Y. P When M as previously defined is used and the initial wave is taken to be plane polarized (~5~~) =0) but of arbitrary azimuth the matrix B is found to have the following non-zero elements: B,, = B,, = 3 {IP:“‘12 + IB:“‘12), Ku = B,o = !I {IBi”‘12 - IB:“‘1219 B, 1 = 2 Re /3~“‘/3$“‘*, B,, = 2 Im /$“‘/I$“)*.
(3.16)
Using eqs. (2.7), (2.8), (3.15), and (3.16) one finds that ~5~~) = - arctan (B,,/B, Itan@ Thus
= {(B,,
- J&)/(~o~
b = - arctan (B,,/B,
(3.17)
J, + Bd>*
Itan aci)l ,
(3.19)
1),
p = arctan {(B,, + B,,)/(&_-,
(3.18)
- B&}*.
(3.20)
On introducing the values of /?L”“’given by eqs. (3.8) one obtains !P = arctan {lp~“‘l/I~$M’I) = arctan {Icos(cOi + Pt)lcos(Vi $$>I>, ii = arctan { - Im j?i”’ /3$“‘*/Re/3i”’ /I:“‘*> = arctan {arg [ - COs(Cpi+ $C~J/cos (vi - @,)I}.
(3.21) (3.22)
440
P. W. ATKINS
AND
A. D. WILSON
These values are the same as those obtained in the classical calculation, and so we conclude that, at least for an initially plane polarized wave, the mirror operator defined in eqs. (3.12) and (3.13) reproduces correctly the classical calculation. 4. Reflection of light from an isolated monolayer In the next step of the derivation the quantum mechanical part of the problem is considered: one considers the situation in which a beam of light of frequency w and arbitrary initial polarization impinges at an angle of incidence cpi on a plane surface of infinite extent but zero thickness, with no backing substrate, and composed of molecules at a density of c per unit area, each having the polarizability tensor CC(O).Each molecule is the source of spherical waves which bear a definite phase relation both to one another and to the transmitted plane wave. The asymptotic state after scattering consists of two parts: below the monolayer the outgoing spherical waves interfere with the transmitted plane wave to form a refracted plane wave, and above the monolayer the spherical waves interfere constructively to form a reflected wave in the specular direction. Therefore, the final state may be written (4.1)
If> = If>, + If&
where If), is the state “above” the plane, and If)b the state “below” the plane: this distinction has significance in the position representation, for if the point Y is above the plane (which we write YC~) then (r If>b=O; similarly
l~)=S/i)=li)-2~i~eXP(-i~j)lfi>,
(4.2)
where lfj> = & 6(E, - E) jkcf’; ~~~){~)) (k’?
~;~)((r)l
Rj ii>
(4.3)
is the wave that may be considered as emerging from the molecule j of the monolayer, 9 j is the phase difference between the plane waves arriving at the molecule j and the molecule at the origin of the set of coordinates, and the Ebb) are unit vectors based on the wave vector joining the molecule j to the point of observation (there will be two sets of such unit vectors depending upon whether the transmitted or reflected beam is considered). Rj is the part of the R-matrix involved with the interaction between the beam and molecule j. The coordinates used are illustrated in fig. 3; the direction of the E:‘)(O) will be specified later [eq. (4.28)]. In order to determine the amplitude of the wave at the points of observa-
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
tion rj (rjc a or 6) the position representation (rj
1 fj>
=
(g/2n)3
1
d/t(‘)
kcfj2
dQ(kcf))
6(E,
441
SCATTERING
may be employed to write -
E)
(rj 1 ken; tf)(o))
x (k (/I; E;~‘(cJ)[Rj Ii) = (dp3/8n3kc) 1 dkCf) kCs)’dQ (k(l)) 6 ( kCJ)- k) exp ( - ikCf’- rj) X
lS~“(~))
(k”‘;
(4.4)
&~~‘(~)IRj Ii))
where we employ the normalization (r
1k) = Z-*exp(-
(4.5)
ikar).
I b
Fig. 3.
Scattering from a monolayer: the basis vectors and propagation regions (a) above and (b) below the layer.
directions in the
59 is the quantization volume and the sum over wave vector states has been replaced by the integration & -, (~p/27~)~1 dkCs’ = &Y/279” s dk’l’ kCf)*dQ(@/‘).
(4.6)
Integrating over Q(ktf)) by parts and using kcs)rj+ 1 enables one to extract the outgoing component as a spherical wavels) (rj ) fj)
=
i(9’*/4n2/icA) [exp(- ikrj)/rj] ]&B)(C)> x (k.@‘(3). R.I Ii) ’ ’ ~(.~‘(a)] 3 .I
where A= l/k is the reduced vacuum wavelength of the incident wave.
(4.7)
442
P. W. ATKINS
AND
A. D. WlLSON
The appearance of a variety of vectors referred to different directions may be removed by the following transformations:
pay’(d);
e;yo>> = Ike’f’(v’); df’(y)) (;, ,,y
i
= where the if-coefficients m ( m’
propagation
Ike”‘($);&yp))
were introduced
n if = (e(‘)(m’)l P(n’)) n’ >
(;, ;,)i’(;, ;,)I’,
elsewheres)
(e(‘)(m)[ d.‘)(n))
(4.8)
and are defined by
(1 - 6,,,)
(1 - d,,), (4.9)
and the @coefficients n ( n’
are a convenient
s f* = (fP(n’)l s’> j
generalisation:
12$~‘(d)) (P(n)l
&Y)(S)) (1 - a,,,) (1 - S,,).
(4.10)
The if- andfisymbols disappear if the entries in any column are the same, and for plane polarized basis states are real. On using these coefficients the R-matrix may be expressed solely in terms of the basis vectors of the initial wave, and one may write (rj 1fj)
= i(P/4rPAcil) X ($
Since all molecules introduce
[exp( - ikrj)/rj]
;)i’(;,
1);
I&(/)(r)) (P(r)
(ke”‘(~);
in the monolayer
Isis’)
E’~‘(~L)IRj ( i).
are identical,
(4.11)
R j = R for all j, and we
the abbreviations R$,,,;ncn,j = (/d)(p’); X:,,(j)
=
s(‘)(p)] R Ikdi’(I’); s;“(a)>
(z,
e”‘(1)),
(4.12) (4.13)
1):) J
y =
For a one-photon phenomena)
8’/2nhcit.
(4.14)
state (which is all that is necessary for intensity
independent
one sets Ii> = l/C&“‘(3); e”‘(A)) b!” )
and the scattered wave at the point of observation is given by (rj ) jj) = - (1/27ci) [exp(ikrj)/rj] where Yj (7) = rX,‘,, (j) (z,
I,)‘i
emerging
(4.15) from molecule j
l&‘J’(r)) Yj(Z),
R:‘;‘,,,; ~(3) b:” 9
(4.16)
(4.17)
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
443
SCATTERING
and where the symmetry relation6) (4.18) has been used. The ~plitude be written :
of the superposition at the point of observation P may now
(a) for P c a, (r
1.0
=2~iCexP(-i?j)
= - Cil,~j)
Crj
exp(-
If>
iqj - ikrj) [e”‘(z)> Yj(r);
(4.19)
j
(b) for P c b,
iv,) Qj I f>
= (V f k) jdi’(.A;) bf’ + 2ni x exp( - iqj)
(4.20)
j since only in the latter case (observation
of the transmitted
beam) does
Since the origin fixed in the monolayer is at the point r’ relative to the source and the molecule j is at ri one may write r(i- r’ = Ii for an infinitely distant source. The direction of the vectors r’ and I may be written P*&+ca,
(4.21)
c=cosf$ji+sin95jJ’,
(4.22)
where C=cos qi, S=sin cpi, and the unit vectors;, j, k are shown in fig_ 3. Hence, in the limit r’+co, the phase difference qj becomes vi = (rj - r’)/iz = (lj/A) S COS ~j.
(4.23)
Similarly one may write r = rj + Zj = r (Si^- CL),
so that
rj N r - Sij COS&j +(lj/2r)(l - S2
(4.24) COS”(pj),
where r is the position vector of the point of observation Consequently (l/ri> exp(-
(4.25) relative to 0.
iqj - ikrj) =
= (l/r) exp(-
ikr) {I -I- (El/r) S COS#jjzexp{-
i(Pl2rA) (1 - Sz COS"f$j)]. (4.26)
The next step is to define the eij’(a): s:j’(3) is chosen to lie in the direction
444
P. W. ATKINS
AND
A. D. WILSON
to the (lj, r) plane,
of rj, @(l) to be perpendicular this plane so that
and e:!)(2)
to lie in
&;f’(l) A &$J’(2) = &jf’(3).
(4.27)
This implies that the vectors are a:f)( 1) = (Ej A r) (llj A rl)-’
,
(4.28a)
~~“(2) = (Zj(r2 - lj.r) + r(Zf - liar)} (jlj A rJ )lj - rJ}-' aif)(3) = (r - lj) (lZj - rj)-’
,
.
(4.28b) (4.28~)
These definitions apply to both regions a and b. When expressed in terms of the angle of incidence Cpi and to terms first order in lJr= A one may write 6:f’(l)=gj{~Csin~j2~Ccos~jj”-Ssinf$jf}, ej:“(2) = gj {[C” cos +j + AS(sin’ +[Scos~j-n(l EY)(3)
=
[S - AC2
(4.29a) bj + C2 cos2 f#~~)]P+ sin +jj
-s2cos2~j)]ck}, COsbj]
;-
ti
where the upper signs correspond gy
(4.29b) C(l
Sh4jif
+ AS
COS4j)
(4.29~)
k,
to P c a, the lower to P c b, and
” = C2 cos2 4j + sin’ $j.
The E(~)(D) may be written
in a similar s”‘(1)
manner
= + cc+
(4.30)
as (4.31a)
Sk
e’f’(2) = -;
(4.31b)
&(f)(3) = S;T
(4.31c)
CL.
The transformation coefficients may now be deduced very easily, and are recorded to first order in A in table 1. The jkoefficients that are required are given in table 2. TABLE
The transformation
1
coefficients
a
1 2 3
1
2
3
- 9j sin 4j F 91 cos $j 0
c 91 cos #j - 9j sin 41 A91-’
F AC cos +j A sin #j 1
gj-2 = C2 + S2 sin2 ~$1
ELLIPSOMETRY
TABLE The
( > 3
mff 35
P
=
( 2 1
35> ff = 1
( 31
35 1 I=ff
(2 1
2ff 35 >
1 3
2ff
(
ficoefficients.
Upper
signs
refer
2
to region
0
-
g5A
sin2 &
95 sin $5
= & g5,4C cos $5
sin 45
= * g5c cos $5
35 )
The sum overj
445
AND THE THEORY OF PHOTON SCATTERING
a, lower
signs
to region
3 ( fit
1 ff d >5
=
0
( 21
35 lff >
=
g5AC2
( 23
35 lff>
=
( 21
2ff 35 >
= i
g5AC
2 ( 3
2ff =
95 sin45
3>5
in eq. (4.2) may now be replaced
zk g5ccos
-
z(l/rj)
sin 45 cos
45
0
of molecules.
Then one seeks the asymptotic
exp[-i(qj + krj)]
’
$5
(4.32)
0
I(z) =
$5
by an integral
4-.~dljljjd$j,
where (T is the surface density limit as r-bco of
cos
b
q(T)
(z, $)‘iR~l,r,;,c,,by),
= yo(e -jk’/r) x:,.
(4.33)
where m X&*
2n
dlj Zj
= s 0
d~j (1 + nS
COS ~j)
X:,,(j)
s 0 x
exp { - i(Z3/2rJ) (1 - S2 cos2 +j)>.
The combination of transformation coefficients X&,) may be deduced from tables 1 and 2: X,‘,, = 6,,6,.,
+ A sin 4j 6,,&,
Xy”,’= Bv28”‘3 T AC
COS ~j
X,“, = &- AC cos 4j 6,,6,,,
6,26,‘,
that occurs in xi,. (that is
,
(4.34a) ,
- A sin +j 6,,6,,,
(4.34b) ,
(4.34c)
where, as usual, the upper signs refer to region a and the lower to region b.
446
P. W. ATKINS
Upon integrating
s
dl lk exp (-
AND
A. D. WILSON
first over &j and then over Zj using the integral14)
iZ2/2riZ)
0
= +(2rA)*(k+1) T’(+k + 4) exp { - +i(k + 1) rr}
(k > - 1,2rit. > 0),
(4.35) one obtains x:,. = - irA(l + 3S2) (6,,6,1&f3
+ 42h2&p3),
(4.36)
from which may be deduced
Consequently
(r
1f)
= (r 1k) le”‘(A))
b$’ 6(r, b)
- iyoA( 1 + 4s’) eeikr { Wl))
(; (4.38)
R$,,,); A(3)b:“,
where S(r, b) = 0 unless r c b, when 6(r, b) = 1. The polarization is still confined to ~‘~‘(1) and ~‘~‘(2); that is, to directions perpendicular to r, so that rk=r*k. Therefore exp(-i k-r) is the plane wave LZ’p-t(r 1k), and so we may write (r 1f)
= (r
I k) {It’s)>
b:” S(r, b) + (d’)(A))
8,))
(4.39)
b(i) r *
(4.40)
where 8, = - i(Z30/tic)
(1 + 4s’)
fi
R(i) lw):f(3)
In region b one has
(4.41) and if the cancellation of _F3 by a like factor in the denominator pated one may deduce that
of Ris antici-
5, = - i(a/tzc) (1 + +S2) {a,, Ry()3j; T(3j + S,, R:ir3); r(3j) b’,“,
(4.42)
and (r I f)
NOW suppose
= (r
that the monolayer
I k) l.di)(l)) is composed
(by’ + 6,).
of molecules
(4.43) with an axially
ELLIPSOMETRY
AND THE THEORY OF PHOTON SCATTERING
447
symmetrical polarizability tensor with a component aI perpendicular to the surface and components all in the surface, then one may deduce from the rules for calculating the elements of the R-matrix6) that for single photon interactions (4.44) R$,,,; l(l,j = (~c$c~) ?a(W)rl m,,,,, y where cc, is the vacuum permittivity, yeis a Lorentz correction term, and m is a matrix with all elements mp,lf = 1: this result is derived in the Appendix. Since the only non-zero elements of the polarizability tensor are aI1 = Pa,
+ C2c(,, ,
(4.45a) (4.45b)
a22 = all 7 al3 =
cl31
=
SC(a,
-
(4.4%)
q),
a33 = C’a, + S2all,
(4.45d)
one may deduce that 6,, which in this region refers to the transmitted and so is written br’, is related to the initial bj” by
wave
@) = 8;) b;i)
(4.46)
By’ = - i(rc~a/s,1) (1 + 3s’) (a,, + (aI - a,,) S”}
(4.47a)
with Jlf’ = - i(rcya/sJ)
(1 + 3s’) a,, .
(4.47b)
In region a the sole contribution to the amplitude is due to the scattered spherical waves, which superimpose to produce asymptotically the plane wave
6,.
(4.48)
The effect of the monolayer resembles in this region the effect of a mirror, and so 6, will be written b:““. Four if-symbols are required: they may be constructed from the values of (@(rr) 1#)(a’)) recorded in table 3, and found to be (4.49a) (4.49b) Consequently, one finds that b:“’ = - i(a/hc) {6,, [(S’ - c2)2 jp l(3),
+
6~2
16’
-
C”>
@3):
r(3)
r(3)
-
4S2C2 R$); rt3J]
- XC R$,,; r(3J} b!“.
(4.50)
448
P. W. ATKINS
AND
TABLE
The transformation
A. D. WILSON
3
coefficients
rs
__~
-~
1
2
3
1
s2 _ C”
0
2sc
2
0
1
0
3
- 2sc
0
p _ C”
Using the same model for the monolayer as in the former calculation that bj”’ = p;“’ b!” )
one finds (4.51)
with
fi\") = - i(qa/&,)
(1 + $P)
{LY,I (S2 - C’) + S’[(S’
- C2)2 -
x (@I- "II)L pi”) = - i(rrfIo/le,)
4Sc31 (4.52a)
(1 + 4s’) (S2 - C2 - 2SC) Ml1.
(4.52b)
In each of the regions above and below the monolayer the new b, coefficients, from which the polarization will be determined, are proportional to the initial coefficients by’, and the scattering process does not mix together the by’. Therefore we may define a monolayer operator, m, and a transmission operator, t, which are diagonal in the basis b(” and are such that (4.53) (4.54) These operators obviously resemble the mirror operator M as defined in eq. (3.13). It should also be noted that m has an additional function, for it replaces e(“)(a) by E(~)(O), just like M, and so the full definition of m and t is mb~)Is(‘)(o))
= (mb!)) m Idi)(
tbb”IE@)(o)) = (tbb”) t\@(a))
= C fl~“)b~‘)lg(f)(s)), = C ;:“bj”($+)). s
(4.55) (4.56)
The approach to the calculation of Y and A is now straight forward: the initial polarization determines the by’; above the monolayer the new polarization is determined for the bB)=j?!“‘)by) and below the monolayer it is calculated from the b? which now are a!” bc’.
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
449
SCA’ITERING
5. Reflection of light from an adsorbed monolayer In order to treat the problem of a monolayer on a reflecting substrate the results of the two preceding sections must be combined. Let us suppose that the monolayer is at a height d above the surface: this entails that a phase difference x is introduced between the ray reflected from the monolayer and that reflected from the substrate mirror: x =
If the initial state of the direction will consist of a component transmitted transmitted through the reflected back from the transmitted through the
2dCliZ.
(5.1)
ray is written 1i) the ray observed in the specular a component reflected from the monolayer m (i) ; through the layer, reflected by the mirror, and then layer again to give tMt 1i); a component that was underside of the layer on to the mirror, and then layer tMmMt )i); and so on. These contributions
t d mMmh
1
,
Fig. 4.
Multiple reflections from a monolayer and a substrate surface contributing the total reflected and transmitted rays.
are illustrated
in fig. 4. The total amplitude
(r 1f)
at the detector
= (r I m + eeiX tMt + eeZiX tMmMt
therefore
reflection
mirror
operator
will be
+... Ii)
= (r I m + eTiX tM 1 eminx(mM)” t Ii) ” = (r I m + eeiX tM(1 - eCiX mM)-’ t 1i), and so the total multiple
to
which applied
(5.2) to the
450
initial
P. W. ATKINS
AND
A. D. WILSON
state yields the final state at the detector (5.3)
is given by (5.4) When the monolayer in absent m =O, t = 1 and so a= M, the mirror operator for the pure substrate. Since the operators m, t, and A4 are all diagonal in the basis b(‘), the polarization change induced in the reflected beam is contained in the eigenvalue equation (5.5) with eigenvalues pp
= @)
+
e-i~
py”
p!“)(l
e-iw film)
_
pp)-1.
(5.6)
From this equation one may calculate the Stokes parameters of the reflected beam in terms of the properties of the adsorbate and the substrate: whereas 1 and ?? for the pure substrate were expressed in terms of /3:“’ in eqs. (3.21) and (3.22), when an adsorbate is present A and Y are given by the same expressions, but with pi”) in place of /?:““‘. This section may be elucidated by noticing that the operators m, t, and M, commute: this is not quite trivial, for although the b(‘) are simultaneous eigenbases of all three, m and A4 affect the e(‘)(o) but t leaves them unchanged. Nevertheless we observe that mt Ii) = mtb~)ld’)(A))
= c m @I bi’)js(‘)(r))
= T a!” /$“” bl” ,e’f;(r)> = T t j3:“” bli)lecf)(r))
= tmby’(di’(,l))
= tmli),
(5.7)
and so [m, t] =O.
(5.8)
Likewise [m,M]=[M,t]=O. This enables
eq. (5.4) to be written
(5.9)
as
m + M( tt - mm) eCiX
JJf=
~ i
1+ mMe-iX
I ’
(5.10)
which is the operator form of the Fresnel equationsr). The eigenvalues of A? constitute the Fresnel equations themselves, and so we may write, for
ELLIPSOMETRY
r=
AND THE THEORY
OF PHOTON
SCATTERING
451
1, 2, fl(M)
=
r
B!“’+ *B!“‘@?” + B(m)8(M)
i
r
. I”“’e-iX
I
- I.7
,
(5.11)
For completeness one may add that a transmission operator for the substrate, T, may be constructed phenomenologically, and it leads to an operator form of the classical result for the polarization of a beam that penetrates a region of complex refractive index ii. Thus one may write T b(i) = p(r) b(i) r r I
(5.12)
T(&“‘(A)) = l#)(A)))
(5.13)
with and @’ = 2K/(fiC
+ cos @,),
(5.14a)
/I$” = ZAC/(C + n”cos fj+)) which are the Fresnel treansmission components /3zT” and j$“” are
n4K+
+ p2 + 2/d(K_
- 2n2 [~pcosv
_ --
-
-
sin v] 2K
sinv)’
sinv]
+ picosv
-
2K
sin v)’
(5.15a)
(5.15b)
+pcosv-rcpsinv] 1+ $
(‘0”
2~p
cosv
+ ,u2 + 2pqK_
dK+
p$r” = 2n[l
P2
Their real and imaginary
2n2 [n”K”++ p2~- cos v -
/$r” =
CT)”= BI
coefficients.
(5.14b)
+ 2p cos v
(5.15c)
’
2n[rc+rc~cosv+~sinv] (5.15d)
’
1 +j?+2/fcosv
where the same notation as in eqs. (3.8) for the j?:“’ has been used. From fig. 4 it is simple to deduce that the total transmitted beam is (r
1f) = (rlTt
+ TmMt e-” + TmMmMt
e-2ix +...li)
= (4 Tli), where
(5.16)
T = T(l - mM edi”)-’ t,
(5.17)
which is the operator equivalent of the classical result3). Since all the operators are diagonal in b,(‘), the polarization of the transmitted beam may be expressed in terms of the eigenvalues of T which are jj:T) = a!‘) p!‘)(l _ ,y!W ,$W
,-ix)-1
.
(5.18)
452
P. W.ATKINS
AND
A.D.
WILSON
In conclusion we note that the same results may be obtained by a manner that conforms more closely to the classical discussion of light reflection by ignoring the vectorial superposition of the scattered waves but introducing instead the Kirchoff damping factor that arises from the Kirchoff-Helmholtz integral will be
theorem3).
The superposition
of waves observed
at the detector
(eeikr/r) eMirpjlP(c7))
(i
t,)‘i
Rtl,,j;ACjj by’,
(5.19)
where ‘pj is the total phase difference between the waves that took a path via molecule j and molecule 0. The damping factor that must be introduced according to the Kirchhoff-Helmholtz theorem reduces the contribution from points j very distant from 0 (in the vectorial addition procedure the vector addition of the polarization vectors achieved this in a different way), and in the asymptotic limit the factor is 1+ S cos 4 j. Consequently
lime-i.l=j”(“_I’.):idm,j.dl,ii(l+Scos~,) r-m
x exp(=Therefore, (r ( f)
i(r~/L-Q(l”-
iail( 1 + $3’) lim eeikr. r-m
the total amplitude
fY
J fi
3
v’
( >
@‘)
.
V(V’),1(3)
(5.20)
is
= (r 1 k) {d(r, b) b:“je”‘(A))
*
S2 co~‘4~)}
b(i) a
- i(,/&)
)
(1 + +S2) IS(‘)(~)) (5.21)
1
which is identical to the result obtained by the vector method; see eq. (4.39). The advantage of this less direct method seems to be that it is more easily extended to a surface with depth: the integrals over a volume of adsorbate may be performed more easily than in the other method. 6. An example We shall consider as an example a silicon substrate covered by a nonabsorbing adsorbate. The polarization changes of the light on reflection from the substrate may be calculated on the basis of the formalism in section 3. The optical parameters for silicon are knownr5) to be n=4.05 and
ELLIPSOMETRY
AND ‘IHE THEORY
OF PHOTON
SCAlTERING
453
~=0.006914. Using eqs. (3.8)-(3.11) the p(M) coefficients for an initially plane polarized beam incident at ‘pi= 70” are found to be /I(M) 1 = 5.7149 + 0.1038 i = c1 + id 19
(6.la)
pi”’ = - 1.1901 - 0.0015 i = c1 + id,.
(6.lb)
From eqs. (3.21) and (3.22) one then deduces that F = 78.24*,
2 = 179.03”,
in accord with the experimental valuesr5) p = 78.23” + 0.05”,
a = 179.05” + 0.19”.
[The !P used in this paper corresponds to 90°-$ in the work of Archerr5) and Bootsma and Meyerr).] At an angle of incidence ‘pi= 45”, the /?:“’ have the values /?1 = 2.0476 + 0.0107 i3
PC;“’ = - 1.4310 - 0.0037 i,
and so !P = 55.07”,
a = 179.85”.
For silver, n=0.20 and K= 17.2; at 4oi=45” fii”) = 0.7329 + 0.7434i,
ficM) Z = - 0 .9426 - 0 .3944 1,
and so p = 55.62”,
a = 165.27”.
When an adsorbate is present the p!“’ are changes in to 8:” ‘) in accord with eqs. (5.6) and (5.11), and the values of Y and A may be calculated accordingly. A convenient simplification, which could be removed easily, is to suppose that the monolayer molecules are isotropically polarizable so that ~1~=a,, =c(. Then the coefficients fi!m)and /I;) become j3’;“’= - i(rc~a/e,l) (1 + +S’) (S2 - C2)2 cL= - ia, ,
(6.2a)
/I$“’ = - i(n~o/s,,~) (1 + 3s’) (S2 - C2 - 2SC) CY = - ia,,
(6.2b)
/I$‘)= 1 - i (rcvcr/s,1) (1 + 4s’) CY = 1 - ib,;
r=1,2.
(6.3)
On writing fli”)=flIM” + iDi”)” and e-iX N 1 -ix, and inserting the expressions for a!“‘, j?lf), and /?z”‘)given by eqs. (6.1), (6.2), and (6.3), one obtains
x41 Cl+ a,(4 - x41 - t-(4- x4 (1 - b,2+ d> - 2b(cr+ x4 - 4 Car Cc,+ x4)1)3
BZ”” = sr {C(c, + Xd,) (1 - b: + a;) + 2b,(d, -
(6.4)
454 B ?“’
P. W. =
~r{C(cr +
[Cd,
x
Cl
+ +
x4 a,(4
xd.)(l (1 -
-
ATKINS
b,2 b,2
+
+
a;)
AND
d)
+ -
A. D. WILSON
2b,(&
%(c,
+
XG.)]
x4)
-
[a,(~,
+
x4)]
4
x41>9
(6.5)
where BP-’ = (1 + a,(&
- XCJ}” + (a,(c,
+ Xd,)}2.
(6.6)
The values of 8,(M) obtained from these equations may be used to determine Y and A, and hence SF=!?--Y and SA=ii-A. The values of SY and SA were calculated for an initially plane plarized beam of I= 546.1 nm incident at 70” on a (111) plane of silicon covered by a monolayer. The values of LX,the polarizability, were taken from table 2 of Bootsma and Meyer l). The values of 0, the fractional surface coverage, was obtained from the same source and used to calculate 0 on the assumption that the surface (111) plane has the same structure as the (111) plane of the bulk crystal. The value of d, the distance between the surface and the monolayer was taken to be the sum of the Van der Waals radius of the adsorbed atomi) or, in H,A, the radius of A, and the covalent radius for Si. Fortunately SY and SA are not very sensitive to the value of d and these rather crude choices do not affect the calculation significantly. The values of SA so calculated are shown in table 4 and compared there with the experimental values and values calculated by classical methodsl). It is far from clear from the equations involved that SA is proportional to the extent of surface coverage, as is observed experimentally: that in fact the dependence of SA on 0 is indeed linear is illustrated in fig. 5 where SA for H, calculated as a function of 0 is plotted. 7. Discussion The values of SY were found to be much smaller than those of SA and in the same direction, in accord with the predictions of the classical theory. It can be seen from table 4 that the fractional change in the imaginary parts of the p are more affected by the adsorption than their real parts; furthermore, the real parts of fi are much larger than the imaginary parts. Consequently, thevalues of 1/3!“‘1and Re(fli”’ pi”*) are not affected significantly by changes in pl”“’ wherease the values of Im(/3i”) B\“‘*) are altered markedly even by small changes in prW” . Thus one may account for the relative magnitudes of SA and SY. For a monolayer or submonolayer, A is much more sensitive to changes in c1and G than to changes in d. Of course, for coverage greater than just a monolayer, one should consider a three dimensional array of scatterers rather than a two dimensional array, and the value of d will be more significant. Our model implies that the absorbate molecules scatter coherently even
0.240
5.7152 1.1901 0.0675 0.0104 0.21
0.137
0.81 0.5 0.04 5.7150 - 1.1901 0.0810 0.0076 0.21 0.20 0.26 0.25
HZ
Cl2
4.50 0.5 0.04 5.7138 - 1.1896 0.0313 0.0441 1.47 1.00 0.94 1.39
0.297
Values of 6A are in degrees. 8 Calculated according to method of the present paper. b Experimental values’). c Drudels), Sivukhinsa) calculationsl). * Strachanl*) calculationsl) .
6AS SAb 6AC 6Ad
Thickness d (nm) (~(x 10-30m-3) 6 u ( x lOlsm-a) /&WY &W’ /&SW ~&W”
Adsorbate
5.7142 1.1895 0.0231 0.0458 1.47
0.360 3.78 0.25 0.02 5.7151 - 1.1900 0.0525 0.0198 0.51 0.43 0.46 0.59
0.302
H2S
5.7153 1.1900 0.0436 0.0216 0.51
0.370 2.65 0.5 0.04 5.7148 - 1.1900 0.0467 0.0259 0.76 0.77 0.64 0.82
0.297
HCl
5.7150 1.1900 0.0393 0.0276 0.76
0.360
HBr
3.64 0.5 0.04 5.7144 - 1.1898 0.0380 0.0349 1.09 0.77 0.83 1.13
0.312
on clean silicon (111) plane for qt = 70”, L = 546.1 nm
TABLET
Calculated and experimental parameter for adsorption
5.7148 1.1897 0.0278 0.0370 1.09
0.390
456
P. W. ATKINS
AND
A. D. WILSON
for small values of 0: this is reflected in the direct proportionality between P(M) or P(t) and 0. The plot of 6A against 0 shows that 6A is directly proportional to 0 even for very small surface coverage. Bennett et a1.16) have shown that the classical ellipsometric equations predict correctly the average
0.;
0
bb
0.
0.t
Fig. 5.
The dependence
of 6A on the extent of surface coverage 0 for hydrogen silicon (111) plane.
on
thickness of a thin discontinuous absorbing film by studying the growth of silver sulphide tarnish on silver. There is also ample experimental evidence’) for the proportionality of 6A and 0 for films adsorbed on silicon and germanium substrates. This proportionality between ellipsometric effect and surface coverage is not surprising, since even when 0 - 0.01 the monolayer comprises a large number of scatterers; for example for H, on Si( 11I), IS- 5 + 10” mm-‘, which is equivalent to 1.5+ lo4 molecules in the first Fresnel zone. We have chosen the Lorentz field correction term appropriate to an isotropic dielectric medium for the effects due both to the monolayer and the
ELLIPSOMETRY
AND THE THEORY
OF PHOTON
SCATTERING
457
substrate. In neither case is this very satisfactory for a variety of reasons but until a better correction term can be obtained this point cannot be improved. The theory can quite easily be extended to include both the effect of absorption by the monolayer molecules and the effect of strong static fields at the interface on the values of the R-matrix elementsIT). The model presented here bears a strong resemblance to the Strachan quasi-microscopic theory 1s) which treats the monolayer as an array of two dimensional Hertzian oscillators but does not interpret the scattering indices so obtained in terms of molecular structure and polarizability. His results are included in table 4 and compared with ours. Acknowledgement
We are grateful to the Science Research Council for financial support of A.D.W. Appendix THE R-MATRIX ELEMENTS
The procedure for the calculation of the R-matrix elements in terms of diagrammatic perturbation theory has been described elsewheres). Only two diagrams are required, and these are illustrated in fig. 6. The application of
Fig. 6. Diagrams contributing to the matrix element RfQ,(,,,; p(p,).
the standard rules5) and the imposition of the limitation that only electric dipole transitions are considered, leads to
for all A’, $. In order to preserve the indices I’, $, yet obtain the same values for all I’, q’, we introduce the matrix m, I pI all elements of which are unity. Introducing the polarizability tensor +&a
= 22E~~~~~~~~/(E~~ - A2mZ),
(A?J
the matrix element becomes Rf’) nf~‘t;r(r’)= (~u~~~32&~) a (mjPi. mPps1
(A3)
as used in eq. (4.47).
f
458
P. W. ATKINS
We include
a correction
AND A. D. WILSON
term r
=
pp
in the R-matrix to allow for the enhancement of the field experienced by a molecule in the monolayer owing to the presence of the other adsorbed molecules (q”‘) and the atoms comprising the substrate (q”‘). The expressions for q(l) and q(‘) are derived from the Lorentz-Lorenz formula for the polarizability of an isotropic dielectric medium :
__3dmlam_ 3d,la, - 4na ’
(1) =
? where d,,, is the diameter a complete monolayer;
of an adsorbed
(2)_’ ?
molecule
n2 ~3
-,j
645)
and cm the value of u for
+2
(
)
where n is the refractive index of the substrate the fact that the adsorbed molecule interacts substrate molecules.
and the factor of + reflects with only a hemisphere of
References 1) G. A. Bootsma and F. Meyer, Surface Sci. 14 (1969) 52. 2) R. H. Muller, Surface Sci. 16 (1969) 14. 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965). A. VaSiZek, Optics of Thin Films (North-Holland, Amsterdam, 1960). P. W. Atkins and L. D. Barron, Proc. Roy. Sot. (London) A 304 (1968) 303. P. W. Atkins and L. D. Barron, Mol. Phys. 16 (1969) 453. P. W. Atkins and L. D. Barron, Mol. Phys. 18 (1970) 721. P. W. Atkins and L. D. Barron, Mol. Phys. 18 (1970) 729. U. Fano, Rev. Mod. Phys. 29 (1957) 74. J. M. Jauch and F. Rohrlich, Theory of Photons and Electrons (Addison-Wesley, Reading, Mass., 1955). R. W. Schneider, J. Opt. Sot. Am. 59 (1969) 297. P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965) p. 539. M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964). W. Grijbner and N. Hofreiter, Zntegraltafel, Vol. 2 (Springer-Verlag, Vienna, 1966) p. 67. R. J. Archer, J. Opt. Sot. Am. 52 (1962) 970. H. E. Bennett, D. K. Burge, R. L. Peck and J. M. Bennett, J. Opt. Sot. Am. 59 (1969) 657. P. W. Atkins and L. D. Barron, Proc. Roy. Sot. (London) A 306 (1968) 119. C. S. Strachan, Proc. Cambridge Phil. Sot. 29 (1933) 116. P. Drude, Wiedemann’s Ann. 36 (1889) 532, 865; 39 (1890) 481. D. V. Sivukhin, Soviet Phys.-JETP 3 (1956) 269.