- Email: [email protected]
Implications of three parameter solutions to the three-layer model
SurfaceScience56 (1976)354-372 0 North-Holland Publishing Company
IMPLICATIONS OF THREE PARAMETER THE THREE-LAYER MODEL
SOLUTIONS TO
B.D. CAHAN ~epartr~ent of Chemistry, Ohio 44105. USA
Case Western Re.wrve
University,
Cleveland,
The use of relative reflectivity changes as an independent third ellipsometric parameter provide insight into shortcomings of the three layer model in thin film systems. An analysis of the successes and failures of this approach in the study of thin films produced electrochemically is presented. An algorithm for the rapid computer evaluation of two (A and $) and three (A, $I, and &R/R) parameter value for n, k and t of an absorbing fiim is discussed.
1. Introduction Modern automatic ellipsometers [ 11 which make possible the collection of relative reflectivity data * together with the conventional ellipsometric parameters have opened new avenues in the study of thin surface films. It is now possible to obtain unique solutions for the complex index of refraction and the thickness of absorbing films which approximate an “ideal” three-layer system [2]. However, recent measurements on real systems have shown that many systems are far from ideal in that the data will not fit any set of parameters chosen for the simple model. This paper will explain the principles involved, give some examples, and attempt to pinpoint some of the causes for the discrepancies observed.
2.
Theoretical
implications
of the three-parameter
solution
The interpretation of eliipsometric data requires a model for the optical properties of an interface. A three-layer model is usually used, where the three layers are the substrate, a film and the ambient. Many ellipsonletr~c investigations are concerned with the characterization of the optical constants and thickness of this intermediate * The ellipsometer (see section 7) used in this study was designed as a non-nulling instrument, in order to preserve the information contained in the reflectivity. On the Poincare sphere, the reflectivity is the radius, which varies independently from the more usual trigonometric relationships on the surface of the sphere.
B.D. Cahan /Implications
of three parameter solutions to three-layer model
355
film layer. Using only the “classic” ellipsometric parameters A and J/, there is an infinity of values for n rilm, k film, and thickness, t, which together with the n and k of the substrate and the n of the ambient will satisfy the Fresnel equations. The problem is to decide which of these solutions, if any, corresponds to the “real” situation using an independent third parameter. It has been shown [2-41 that in many cases the change of relative reflectivity &R/Rcan be used successfully for this purpose. and can be used to provide insight into those cases that deviate from ideality. The methods used for the solution of the relevant equations has often been cumbersome, generating large volumes of computer printout and large computer bills. At our laboratory [3] we have developed a set of programs in Fortran * suitable for implementation on a time-share system or on a minicomputer, which generates these solutions in a short time and in an easy-to-interpret format. While the procedure was originally devised for the solution of the three-parameter problem, it is equally suited to solving the more usual two-parameter equations and indeed uses that approach as a first step.
3. Numerical (computer) (2P) systems The equations
solution of the three-parameter
(3P) and two-parameter
to be satisfied are, following the Nebraska conventions,
(1) R = ipfP*
sin2
where the reflection
(y. -I- isps*
coefficients
cos2
01=
(2)
a tan2 $ + cos2 cz) ,
isis*(sin2
of the surface are given by
iv = fy2 + i;3 exp(-4nti2t
cos c#J~/X)
1 t i’;2i53exp(-4niri2t
v=p,s,
cos ~$~/h) ’
(3)
or iv = F1;2 + 43T
1 + i’;2?53T T = exp(S)
(34
’ and
s = -47+?,
COS
&t/h
,
(3b)
where v is either p or s polarization,
and the subscript
* A more recent version of this program,
written for use on our PDP-11, is available from the for use in an interactive mode, but could be readily
author. adapted
In its present form, it is written to a batch-mode computation.
1 denotes the ambient,
2 the
356
3.D. Cahan /Implications
of three parameter solutions to three-layer model
film and 3 the substrate, 12 the interface between 1 and 2, and 23 the interface between 2 and 3. Further, the individual interfacial reflection coefficients rij where i and j are either 1 and 2 or 2 and 3, are given with complex notation by: 'iiCOS~i-iziCOS
~j
'f=
# = ~j COS~i+
AiCOS~j
12,23 ,
(4)
'
although the n’s and Cp’smay of course be real. Inversion of these equations analytically is not possible, but solutions can be found using an extension to two or three dimensions of Newton’s approximation, -dX=X,+,
- Xm = -flXM(X)
(6)
>
and dX is the where X, denotes a trial root and Xm + 1 is the next approx~ation, correction to be applied to the trial value. XN1+ I then becomes the new trial value and the procedure is repeated until convergence is obtained. This algorithm is ideally suited to computer iteration and usually converges in less than 4-5 steps. Assuming that A, and J/u for the surface and nI for the ambient are known, we wish to find values for n2, k, and r2 which will make the functions F(A)=SAC-6A,
=A,-A,,
Pa)
all equal to zero (or less than the experimental the zero (or minimum) of the function F = F(A)* + F($)2 In three dimensions
c(X,,+, i
error). This is equivalent
to finding
(8)
+ F@R)* .
it is then necessary to solve the set of three equations
- Xj,m)
72
1.m
in Vi:
Xj=n2,k2,t =F(Vj)
Vi = A, ~, 6R I
to refine the trial root set Xj,na to the Xj,m+l set. The partial derivatives W( r3,aXj are obtained from eqs. (1) and (2). Differen-
(9)
B.D. Cahan /Implications
of three parameter
solutions
to three-layer
model
351
tiating (1) and rearranging terms gives - 1 sip -fp ax
___ - 1 _ais is ax
LL!L?! sin2GaX
+i?!!, aX
(10)
Since aF( VI/ax = a( V)/aX and the a( V)/aX must be real, one can separate: an
_!.ayp_Lar^”
ax=‘”
;f
_
sin?+ Re
(11)
is ax 1 ’
( ?P ax^
arp__1 ar^’ (ipax^ 1
(12)
ax )
is
From (2), since --1 aw +Lawb=~~eG!!! w* ax w ax ( w ax 1 3 a(&R/R)_ 2 sin2 cytan2 $ Re [( 1/iP)/(aiP/aX)j + 2 cos2 (YRe [( l/is)/(W/&Y)j ____ _________. ax sin2 (Ytan2 $ + cos2 (Y (13) From (3), since aA/ak = -i(M/an) 1
apv
--= FV at2
(aiy,/atg(i
= -i,
-(i%q2)+(i
p(r^‘;2)2)r(ai”,/at2) ____ (1 + i’;2y^y)(il;2 + i53T)
---i au^v =___ i ar^v f” ak iv at1 3 i ai” v^ at ”
tfg(ar/an) ___
3 (14)
(15)
tsr^53(1 - (fy2)2) __-1 + i’;2i;3 t(
T)(“;2
(16)
+ F;3 7J
From (4) and (5) +-f2
2n,
cos uj, cos $,(l
- tan2 i2)
YG=
(A2 cos $1 t nl cos ljJ2)2
a93
X2,
anai;, ----=_ an
(17)
’
cos & cos i2( 1 - tan2 $2) (fz3 cos $2 + A* cos &)2
’
(18)
2n, cos qbl cos G2(1 + tan2 G2) (n1 cos $1 + A2 cos G2)2
’
(19)
(31)
While a hunt- and search method of finding the trial roots (X,,) for eq. (9) cat1 be used, it is both tedious and frustating, since all initial guesses either converge to the same few sets of roots or diverge, and possible roots can be readily missed. A sitnplcr and more elegant solution is to use a subset of two of the eqs. (7a. b. c), find the locus of all possible 2P roots, and locate those solutions which give a rcasonable match for the third parameter, using the full iteration scheme to refine only those roots. For example. the minimization l of FR = F(A)2 + F($)2
(13)
)
can be used to find those values of II and k which satisfy eq. ( 1) for a series of thickTable 1 _
-__
Trial value ___~_
_ ______~~_
-_
_._..
“Exact”
solution
(1) 2-Parameter (% k)ambient~ (rrl-kl,
'3Zj
(2) Choose
t;R(“I.k,.r,),F(~.)=F(~)=O
‘1
t2)
,,kj
,,
i--t F(6R)
fj’
“I
E‘R(Q,
_i
b'Rhj.
-+
(rrf. kf. ‘f)
3
F,(t,;.
k2. r2), F(h)
ki, ?$,
= F(G) = 0
p(A) = F(+)
= 0
r 0 (if any)
3-parameter (np kb ti) (3) Repeat
(‘*, %ubstratc
etc.
(F = 0)
for
___..___-
k;. t;’
.-_
* The symbol FR is used in the figure to denote those solutions obtained by minimizing E’(A) and F(i). Similarly F, denotes minimization of F(G) and F(&R), and Fti denotes k’(A) and F(sR).
B.D. Cahan /Implications
of three parameter solutions to three-layer model
359
nesses tn. This procedure is equivalent to finding the roots of a “classical” ellipsometric problem, with the added feature that F(6R) (eq. 7c) is also calculated. The procedure is outlined in table 1. Intuitive reasoning (with the help of experience) provides a procedure for finding all of the roots. It is obvious that for a given 6A and S$ the 2P equations can be satisfied by a thick film whose optical constants are close to those of the ambient or close to those of the substrate. Choosing some value, t, much thicker than that expected for the real film, the n and k for the ambient are used as trial guesses, and an nl and k, are calculated. These values are then used as trial values at a lesser thickness r2 and a new set (tz2, k2) are calculated. This algorithm is repeated (calculating F(6R) at each step) by suitably decrementing * t until (1) a thickness is reached well below the anticiapted film thickness; (2) the solutions diverge, or (3) the new values of n and k become impossibly high or n and/or k goes negative (a physically impossible condition). The procedure is then repeated using the n and k for the substrate. In practice, for thin films (t < 100 a) we have found that almost any initial trial values with thicknesses at least ten times greater than the expected value lead to the same two branches (which we label the “water” branch and the “metal” branch) or else diverge and give no solution at all. Occasionally we can find additional branches with physically absurd optical constants for large trial values of thickness but these have never produced viable alternate solutions. Other symmetric branches with negative II, negative k or both negative can be obtained, but these can be eliminated on physical grounds. If, for any set (ni, ki, ti), F(SR) z 0 (i.e., 6R/Rc.culated _- 6R/R ,,,easured) or F(FR) changes sign from one ti value to the next, those values can be used as trial values of a full three-dimensional iteration, and the “exact” root determined.
4. Examples of three-parameter
solutions
Fig. la illustrates the two branches as developed for an oxide covered Pt surface, with two possible roots on the “metal” branch, and none on the “water” branch. The corresponding values of n and k are also shown, labeled 1 and 2. Fig. 1b illustrates a similar case for an oxide covered Au surface, with one root on each branch. Fig. Ic shows a case where no roots exist. In this case, there is no single film of any optical constants which will satisfy the experimentally measured values of A, $, and 6R/R. The commonly followed practice of choosing a thickness from an independent measurement (such as from coulometry in electrochemistry) will of course give two solutions for n and k, but they will be invalid, as would a reflectivity cal-
l
The size of the step taken is usually not too important, since the curves are in general smooth and continuous, but sometimes too large a step causes a “jump” to the other branch. Occasionally, one branch terminates (as will be shown later), giving only a single branch below a given thickness. The solutions will then “jump” to that branch or diverge.
360
B.D. Cahan J Implications
of three parameter
solutions
to three-layer
model
Cc)
3 2 2
ixl_l OO Fig. 1. Plots of AR/R,,lc (top), n ing several types of solution of the solid line represents the measured = -0.84%. (b) Gold oxide on gold, 6R/Rmeas = -2.28%.
IO T- &
00
IO
(middle), and k (bottom) for three different systems, illustratthree parameter equations with plots versus thickness. The 6R/R value. (a) Platinum oxide on platinum, 5820 A. SR/R,,e,,s (c)Gold oxide on gold, 5064A. 7439 A. SR/R,,,,, = -0.945.
culated on the basis of assuming a calculated R,, from the A, ti measurements alone [5,6]. (It need hardly be pointed out that the two branches have no physical significance whatsoever, but are merely the loci of all possible 2P solutions. Only the points of intersection with the third parameter are solutions for the complete 3P set.) In figs. la, b, c it can be seen that even at 18 A the values of n and k are tending toward those of “water” and the “metal”. Although the calculated reflectivity change in fig. lc misses the measured value by almost 0.4%, far too much to be attributed to experimental error, it is clear that in fig. la a small error in &R/Rof even 0.02% would have caused the curve to miss completely and give no solutions at all. In fig. lb the roots are clear, but the t values are so close that it would be difficult to choose between them on the basis of coulometry alone *. While the preceding section illustrated the two-parameter solution by the minimization of F(A) and F(G) to find values of n and k as a function of f, it is just a feasible to minimize F(A) and (6R)or F(JI) and F(6R), or to solve for any two of the three * Other
criteria
will be discussed
later.
(b)
I
OO
I
IO
I
1
I
T (A)
Fig. 2. (a) Data of fig. la replotted to show two intersections of FA, F$, FR curves in k-t (top) and n-t (bottom) space on same branch. (Only relevant branches drawn, for clarity.) (b) Data for platinum oxide on platinum, 7329 A. Plotted as in (a), but with no convergence (caused by small error in data). Lower FA branch terminates at X.
(a)
0
.
’
’
’
’ ’ IO
(b)
3
’
’
0 ’ c 3 c c 8 3 8 IO T(;;(: Fig. 3. (a) Data of fig. lb replotted as in fig. 2a, to show two intersections on different branches. (b) Data of fig. lc replotted as to show no 3P intersections. Only one FA branch exists at these thicknesses. 0
362
B.D. Cahan /Implications
of three parameter
solutions
to three-layer
model
film parameters (n, k, and r) as a function of the third. Figs. 2a, b and 3a, b show typical plots of the three sets of (n, k) versus r that can result from this procedure. Intersection of all three curves at a point in both n-t and k-t space is necessary for a three-parameter solution to exist. Such is the case for fig. 2a. corresponding to the two roots of fig. la. Note that the intersection at rr is much clearer than that at r2, reflecting the approach to quasi-homogeneity and consequent lack of resolution for that root. Fig. 2b has no roots. The inset in the n-t plane shows that what appears to be a root in the upper F, branch is not, as is evident in the k -t plane. The lower FA branch in fig. 2a terminates before it reaches the curve, and only one F, branch exists below -5 A. Figs. 3a, b show similar plots for the data of figs. 1b, c, where there is one root on each branch and no roots, respectively.
5. Effects of errors in experimental
data
It is often informative in an analysis of this type of data to consider the effects of small errors in the experimentally determined values. The method of two-parameter locus tracing allows a convenient visualization of these effects. As one example l , consider the results of small variations in S$ as shown in fig. 4. An experimentally determined value of -0.224” converges to a 3P root at a thickness of -4.5 A. Varia-
Fig. 4. Gold oxide on gold, 6570 A, showing Left: &R/R; lower right: n; upper right: k.
l
All of the experimental for purposes of brevity
errors resulting
from small deviations
variables are subject to errors, and can be analyzed and clarity only variations in 6$ are shown here.
in J, meaS
in this fashion,
but
B.D. Cahan /Implications
Fig. 5. Spectrum Kruger [ 71.
for an Fe surface,
of three parameter solutions to three-layer model
passivated
at +0.7 V in borate
solution.
363
After McBee and
tions of -0.01“ and +O.Ol” change the calculated thickness to 5.2 a and 2.1 a while +0.02” makes the 3P system divergent with no toots at all. The same type of figtue can be used to estimate the error range in II and k when calculated from a 2P (e.g. A and $) measurement using an independently determined thickness as a third parameter. Especially in the case of thin films of unknown stoichiometry, thickness can be difficult to estimate precisely, but the range of possible n and k values are clear.
6. Use of
6R/Rmeasurements
to eliminate false roots by locus tracing
The problem of a choice between the two roots can often be minimized by even a rough measurement of &R/R.In ths illustrations of fig. la (or 4) the expected reflectivity difference between the two branches is of the order of 0.5-l%. A reflectivity measuring system with a resolution of -0.1% might not be sufficiently accurate for a full 3P solution but would suffice to differentiate between the roots. As an example, consider the analysis of data similar to that obtained by McBee and Kruger [7] for the passive film on iron (fig. 5). Using values of A and $J recalculated from their data or those measured in our laboratory (see e.g. figs. lOa, b) with the method of locus tracing. one obtains dual values of n and k for all wavelengths. It becomes clear that those multiple values in fig. 5 represent cases where the discrete root procedure [8] used by McBee and Kruger only found some of the roots, missing others. The two sets of roots then correspond to the two branches obtained by the 2P solu-
364
B.D.
Cahan /Implications
of three parameter
solutions
to three-layer
model
tion method. The calculated values of 6R/R for the two branches are different by several percent and are even of opposite sign at some of the wavelengths in this range. Preliminary measurements [9] using 6R/R as the third parameter show that the correct set of roots is probably the lower k branch.
7. Practical implications
of the three-parameter
solution
In the preceding sections, it was pointed out that not all sets of experimental data will yield a 3P root (e.g. fig. 2b) and that the discrepancy was often far larger than could be accounted for by experimental inaccuracies. Some of the causes of this dilemma and a few explanations for them will be discussed in this section. Some are obvious, some are less obvious but demonstrate, and some are still only speculative, but are the subject of further theoretical and experimental scrutiny. For purposes of this section, “real” data will be used as examples when possible, but “thought” experiments with synthetic data are sometimes invoked. A typical set of electrochemical-ellipsometric data as collected by the computer 3.87
93.9 -1
22.01,
-l&O0
3e.441-
’
’
’
I
’
’
E VS PD-H
E VS PD-H
Fig. 6. Raw data as collected from triangular sweep ot’ Au in 1 N HCl04. ~1 = 65”; A = 53 12 A; sweep speed = 100 mV/sec; time = 19:04. Lower left: i versus V; upper left: A versus V; lower right: $ versus V; upper right: --6R/R versus V. (Note: relative reflectivity plot is inverted.)
B.D. Cahan / Implications
of three parameter
solutions
t---+
to three-layer
model
365
t--+
Fig. 7. (a) Waveform used in scanning potential for fig. 6. (b) &me but for figs. 8 and 9. Note that while potential “jumps” to 1.35, dotted line indicates “apparent” potential for recording data.
from the automatic ellipsometer l is shown in fig. 6. The lower left hand curve is a current-potential plot as the potential of an Au electrode in 1 N HClO, is scanned from 0 to 1.8 V and back according to the waveform in fig. 7a. Currents above the line are anodic. representing charging of the double layer from 0.0 V to -1.2 V, and formation of an oxide film of a few monolayers to 1.8 V, where the sweep reverses direction. A small anodic current continues to pass until the potential drops to 1.6 V, and starting at about 1.2 V, the film is reduced back to metallic Au. By 0.9 V, the film has been completely reduced and the small residual current is cathodic charging of the double layer. Corresponding points of the curves of A, $J, and &R/R versus potential, measured at 5321 8, near the absorption edge (-5000 A) of gold, can be compared to the E-i plots. 7.1. Choice of “bare substrate” values. A0 and Go It is clear from fig. 6 that in the anodic curve from 0 to 1.2 V and from 0.9 to 0 V in the cathodic branch, where no surface oxide films are present on the surface, there are relatively large (several tenths of a degree and several tenths of a percent) changes in A, $, and FR/R. This effect is not new and has been reported from many reflectometric and ellipsometric studies of Au [ 10-13, etc.] but it points up the problem of which values to choose (if any) for A, and J/o, since the surface is “bare”, i.e. free of films, over the entire potential range. To be sure, the surface is covered by the electrolyte ambient, but its optical properties are known, and it is easily demonstrable [ 141 that no conceivable change in the solution could account for more than a small fraction of the observed effect. It is widely recognized that this effect is caused by some electromodulation of the surface layers, but there the agreement
l
A Rudloph Research RR2000 [ 11 Automatic Ellipsometer was used, with the data collected, analyzed and plotted by a PDP- I l/45 minicomputer. The computer generates the electrochemical waveforms, and synchronizes the data collection to the experiment. Each curve consists of 5 12 data points. This corresponds to a voltage resolution of - 7 mV/point. With a sweep speed of 100 mV/scc. and a measurement rate of 54 sets of (I, A, $J and 6R) points per set, -3.8 data sets are averaged per point. The r.m.s. data deviations are typically less than 0.001 degrees for A and @ and O.Ol%, for aR/R.
366
B.D. Cahan /implications
of three parameter solutions to three-layer model
stops. The depth of this layer and its “optical constants” are still a matter of controversy. Using a 3P approach, no simple thin film of any optical constants and thickness will satisfy the experimental data. Attempts to calculate the problem in reverse. i.e. using optical constants calculated from McIntyre’s [ 151 approximate equations and measured R, and R, data to ‘*predict” aA/?l V and a$/a I’ fared no better. In the case of Au (and other metals like Ag and Cu) the interaction with the ambient, and the electric field at the surface, including the effects of truncation even in UHV studies, can preclude an unambiguous definition of the “bare” substrate optical constants. 7.2. Substrate-film
interaction
Just as there are modulation effects caused by electric fields at the surface of Au in the absence of a definite “film”, so there are similar effects induced in the metal by the presence of a film on its surface. In fig. 6, both $ and 6R/R begin to change immediately after the passage of the first anodic current at 1.25 V while changes of A are delayed by 50 mV to 1.30 V. At 1.35 V l there is an abrupt change of slope of aA/aJ/ (and an/N), which then continues constant to 1.8 V. A similar effect is evident in the cathodic (reduction) branch where A returns to the double layer line at 0.95 V, 50 mV before G or 6R/R. For wavelengths any shorter than -6500 8, we have not been able to find any 3P convergences for either the initial monolayer or the thicker film at 1.8 V l * , regardless of how we choose A,, and Go or 6A, SJ/, and 6RfR. 7.3. Electromodulation
of the surface film
Anodic current does not stop immediately after reversal of the potential sweep. Additional oxide continues to form down to 1.7 V as evidenced by the continued increase in A. From 1.7 V down to 1.3 V, no significant current passes, and the A value stays essentially constant. The J/ and 6R/R values show a definite decrease in this region. To decide whether or not this was an effect of “aging” [ 131 or dissolution, the potential sweep regime of fig. 7b was used. Instead of reversing after reaching 1.8 V, the sweep was abruptly stepped down to 1.35 V, at which potential no reduction occurs, and the potential maintained steady for the 4.5 set the normal(dotted line) sweep would have required, and then the sweep resumed. In this way, aging could proceed without the complication of further film growth or parallel 0, evolution. Fig. 8 shows the results plotted versus time rather than potential as in fig. 6. The anodic portion of the curve is almost identical. The negative current spike in
* At this point, just past the first anodic peak, the coverage by oxide or oxygen is one or onehalf depending on how one defines the valency of the adsorbing or depositing species. ** In contrast to the case for platinum [ 21 where reasonable convergences are obtained at all thicknesses and wavelengths.
B.D. Cahan / implications of three parameter solutions to three-layer model
361
DEL
E VS PD-H
Fig. 8. Data similar to fig. 6, but including “jump”. Time = 23:56.
the cathodic branch at 1.8 V corresponds to the charging of the double layer. Both $J and 6R/R show a distinct jump at the transition while A changes less than O.OOl”! Then all three parameters (A, I/Jand 6R/R) remain constant until the sweep is resumed, after which the curves are indistinguishable from those of fig. 6. There is thus a voltage sensitive component of r+!~ and &R/R for an oxide covered Au surface which is not visible from A modulation measurements alone. This is in direct contrast to the bare surface electromodulation where A shows the largest effect. Indeed, as can be seen from fig. 9, the effect is visible in aJ//a Voxide even at wavelengths (-4000 A) where 3$/a VAu is zero or slightly negative. To date, we have not yet learned how to account for this effect, which appears to occur with several metal oxide-metal systems, nor have we decided whether the seat of this modulation is at the oxidesolution or the oxide-metal interface, or both. 7.4. Problems of surface roughness Figs. 6 and 8 illustrate yet another problem. These measurements were part of a long series of measurements on the same sample, changing many variables including wavelength. Fig. 8 was obtained almost 5 h after fig. 6. Although the curves are al-
B.D. Cahan /Implications
368
of three parameter solutions to three-layer model 42.21
t
E VS PD-H
E VS PD-H
2.0
Fig. 9. ri, XISUS t plots similar to fig. 8. upper left: h = 4172 A; lower left: 5312 A; upper right: 6070 1$; lower right: 7430 A. most superimposable,
the A values have been displaced
by -0.2 1” and the $ values
by -0.09”. (The absolute reflectivity was not monitored.) Blondeau and Cahan [ 161 have shown that these changes are due to a roughening of the surface during cycling,
and appear to follow the predictions l
* of the Ohlidal-LukeS
[ 171 equations.
There are some typographical and mathematical errors in ref. [ 171. Their eq. (16) should read i?‘;,,
=
ii’no {(h2Bz
a2 G2E + noa)
_ (2a2B1 - n2d IB)(G*niA Ri’=
=
no (Y2(noB + a)3
+ nzd IB, - nZ&2B)&2B
,B/2a
+ no(y)
+ 2&i*B1 + $ n&4 1)}.
201~6~ + r~&4~B~or - r1&4~Ba)(noB
+a)
r
It appears that Ohlidal and LukeS may have used the incorrect equations in refs. [ 17,181.
B.D. Cahan /Implications
of three parameter solutions to three-layer model
7.6
iiP
III
M
369
7qi
(L d B
-0.4
-0.4 I31
q ‘I
d 0
-0.5v
c
III -$50
it VS
PD-H
E VS
PD-H
25.6
(a) 1204r
1
l.9r
I.OV
PSI
29.6
1
_2jfy-j 2jly,I‘jyq -0.50 E VS PD-H
1.50
-0.50
1.50 E VS PD-H
29 .40
2140 PSI
(b) Fig. 10. (a) Data for triangular sweep of Fe in buffered borate between -0.5 V and +l.O V (versus a-Pd-H): $ = 65’; A = 4172 .k; sweep speed = 20 mV/sec. Lower left: i versus V; upper left: A versus V; lower center: \L versus V; upper center: -6R/R versus V; lower right: A versus Q ; upper right: -6R/R versus Q. (b) Same as (a) but for h = 7430 A.
7.5. The problem of the anomalous reflectivity Figs. lOa, b show preliminary results [9] of electrochemical-ellipsometric measurements for the passivation of Fe in buffered-borate solution at 4171 A and 6570 A. Although the A-$ plots appear well behaved and give 2P solutions similar to those
370
B.D. Cahan /Implications
of three parameter solutions to three-layer model
in the literature for the passive film, we are currently at a loss as to how to ascribe the wild gyrations obtained in the &R/R versus rj curves to the growth of a three layer system. 7.6. The problem of “equivalent” surfaces As long as only the two parameters, A and $ were measured ellipsometrically, it was often tacitly assumed that it was possible to assign a set of “pseudo-optical constants”, n* and k*, to describe the surface from the measured parameters and an idealized two-medium model. Indeed, since it is very difficult if not &possible to measure the absolute reflectivity to an accuracy equivalent to that calculable from A and J/ (or n and k) values, absolute reflectivities are often determined in this indirect fashion. At the basis of the 3P (A, +, and 6R/R) method is the assumption that while only relative reflectivity changes (AR/R) are measured, Ro, the base surface reflectivity is calculable from A0 and Go. If this calculation is not valid, the results of the 3P approach will be in error. Specifically, reflectivities calculated from A and $ measurements on a rough surface [ 161 or a film-covered surface by assuming equivalent pseudo-optical constants are not, in general, equal to those calculated from the more rigorous equations. Reflectivities calculated on the basis of the two models differ by several percent even for very thin (
Fs WI
x=n,k, ’
(22)
* A similar conclusion was reached by Schueler [ 191 about the use of multiple-angle of incidence measurements on thin (<200 A) fiis. l * E.g., if nI = 1.33, nz = 1.7 an angle of incidence 41 = 65” gives a 02 of 45.16”. Of course, k is usually complex, but similar considerations hold.
B.D. Cahan /Implications of three parameter solutions to three-layer model
311
and
WRIR) ax
or
WV0 ax
aJ/ =k3=,
(24)
which is just the condition for homogeneity and therefore indeterminacy! On the other hand, it is possible to calculate from eqs. (1 l)-( 13) values of G2 which will optimize the sensitivity of any (or all) of the three parameters to a desired physical constant, within the limitations imposed by the experimental conditions *.
8. Conclusions Although the three parameter (including &R/R)approach is not always the answer to an ellipsometrists prayer, it does often lead to solutions, and can at least help decide between alternate solutions. While the existence of a three parameter solution for a given substrate-film-ambient system does not guarantee that the numbers so obtained are a real solution (i.e., the three layer model is adequate) the lack of a convergence, as in fig. lc), guarantees that something is wrong with the model. In this fashion we often have a check on the validity of the conchrsions drawn from ellipsometric data.
Acknowledgements The author is pleased to acknowledge the support of this research by the National Science Foundation and by the U.S. Office of Naval Research. The computer facility used in these studies was acquired under a grant by the Chemistry Research Instrumentation program of the National Science Foundation. Thanks are due to Drs. G. Blondeau and T. Grcev for their help and to Mr. C.T. Chen for his invaluable assistance.
l
Once approximate values of the n, k, and t are determined, a simple computer calculation of the nine derivatives as a function of e.g. the angle of incidence can establish the optimum $1.
312
B.D. Cahan /implications
of three parameter solutions to three-layer model
References B.D. Cahan and R.F. Spanier, Surface Sci. 16 (1969) 166. J. Horkans, B.D. Cahan and E. Yeager, Surface Sci. 46 (1974) 1. J. Horkans, Ph.D. Thesis, Case Western Reserve University (1973). W.-K. Paik and J. O’M. Bockris, Surface Sci. 28 (1971) 61. R.C. O’Handley, Surface Sci. 46 (1974) 24. S. Gottesfeld and B. Reichman, Surface Sci 44 (1974) 377. CL. McBee and J. Kruger, Surface Sci. 16 (1969) 340. Program by F.L. McCracken, Natl. Bur. Std., Tech. Note No. 49 (1969). [ 91 B.D. Cahan, C.T. Chen and T. Grcev, unpublished data. [ 101 W.N. Hansen, Surface Sci. 16 (1969) 205. [ 111 J.D.E. McIntyre, in: Advances in Electrochemistry and Electrochemical Engineering, Vol. 9 (Wiley, New York, 1973). [12] B.D. Cahan, J. Horkans and E. Yeager, Surface Sci. 37 (1973) 559. [ 131 B.E. Conway, H. Angerstein-Kozlowska and L.H. Laliberte’, J. Elec. Sot. 121 (1974) 1596. [ 141 M. Stedman, Symp. Faraday Sot. 4 (1970) 64. [ 1.51 J.D.E. McIntyre and D.M. Kolb, Symp. Faraday Sot. 4 (1970) 99. [16] G. Blondeau and B.D. Cahan, Paper presented at the May 197.5 Meeting of the E.C.S. (Toronto). 1171 1. Ohlidal and F. Luke;, Opt. Acta 19 (1972) 817. [ 181 I. Ohlidal, F. LukeS and K. Navratil, Surface Sci. 45 (1974) 91. [ 191 D.G. Schueler, Surface Sci. 16 (1969) 104. [l] [ 21 [3] [4] [S] [6] [ 71 IS]
Critique
A.B. Buckman: I just wanted to point out another example of the failure of the three-parameter model you mentioned. This happened with thicker films and multiple-angle-of-incidence ellipsometry. The program discussed by lbrahim and Bashara that we used to minimize the sum of the squares for differences between experimental and calculated values would always find the minimum but in the case of some work we did on lead iodide on glass, these were thick lead iodide films better than a 1000 A thick, we found that unless we included in this model the effect of a surface layer on glass which had a different refractive index than the bulk glass itself, the sum of the square on conversions would come out to be from 100 to a 1000 square degrees. The program would find the minimum which converged very poorly. The total sum of the squares could be much greater than we would expect from experimental error. This is just another example of the failure of this three-layer model that you mentioned. It also happened in static situations with multiple angle of incidence ellipsometry and with thicker films. B.D. Cahan: The failure, may I point out, is not in the three-parameter-solution techniques but in the three-layer model. One can start from any crazy starting set of n’s and k’s and the program will converge in fractions of a second and then proceed to give you the entire locus. Even on our minicomputer we can piot out the entire locus in a short time. 0. Hunderi, I have difficulty understanding how you can get any model to work for these thicknesses where you have monolayers or submonoldyers. What does a layer mean at this level. B.D. Cahan: That’s a question often asked by physicists. I’m not sure. The only defense is, there are some systems where it does work. For example, with careful measurements on the platinum oxide film we get the same thickness over a wavelength range from 4000 A to 7000 A. Yet the optical constants in both substrate and film vary considerably over this range.
IMPLICATIONS OF THREE PARAMETER THE THREE-LAYER MODEL
SOLUTIONS TO
B.D. CAHAN ~epartr~ent of Chemistry, Ohio 44105. USA
Case Western Re.wrve
University,
Cleveland,
The use of relative reflectivity changes as an independent third ellipsometric parameter provide insight into shortcomings of the three layer model in thin film systems. An analysis of the successes and failures of this approach in the study of thin films produced electrochemically is presented. An algorithm for the rapid computer evaluation of two (A and $) and three (A, $I, and &R/R) parameter value for n, k and t of an absorbing fiim is discussed.
1. Introduction Modern automatic ellipsometers [ 11 which make possible the collection of relative reflectivity data * together with the conventional ellipsometric parameters have opened new avenues in the study of thin surface films. It is now possible to obtain unique solutions for the complex index of refraction and the thickness of absorbing films which approximate an “ideal” three-layer system [2]. However, recent measurements on real systems have shown that many systems are far from ideal in that the data will not fit any set of parameters chosen for the simple model. This paper will explain the principles involved, give some examples, and attempt to pinpoint some of the causes for the discrepancies observed.
2.
Theoretical
implications
of the three-parameter
solution
The interpretation of eliipsometric data requires a model for the optical properties of an interface. A three-layer model is usually used, where the three layers are the substrate, a film and the ambient. Many ellipsonletr~c investigations are concerned with the characterization of the optical constants and thickness of this intermediate * The ellipsometer (see section 7) used in this study was designed as a non-nulling instrument, in order to preserve the information contained in the reflectivity. On the Poincare sphere, the reflectivity is the radius, which varies independently from the more usual trigonometric relationships on the surface of the sphere.
B.D. Cahan /Implications
of three parameter solutions to three-layer model
355
film layer. Using only the “classic” ellipsometric parameters A and J/, there is an infinity of values for n rilm, k film, and thickness, t, which together with the n and k of the substrate and the n of the ambient will satisfy the Fresnel equations. The problem is to decide which of these solutions, if any, corresponds to the “real” situation using an independent third parameter. It has been shown [2-41 that in many cases the change of relative reflectivity &R/Rcan be used successfully for this purpose. and can be used to provide insight into those cases that deviate from ideality. The methods used for the solution of the relevant equations has often been cumbersome, generating large volumes of computer printout and large computer bills. At our laboratory [3] we have developed a set of programs in Fortran * suitable for implementation on a time-share system or on a minicomputer, which generates these solutions in a short time and in an easy-to-interpret format. While the procedure was originally devised for the solution of the three-parameter problem, it is equally suited to solving the more usual two-parameter equations and indeed uses that approach as a first step.
3. Numerical (computer) (2P) systems The equations
solution of the three-parameter
(3P) and two-parameter
to be satisfied are, following the Nebraska conventions,
(1) R = ipfP*
sin2
where the reflection
(y. -I- isps*
coefficients
cos2
01=
(2)
a tan2 $ + cos2 cz) ,
isis*(sin2
of the surface are given by
iv = fy2 + i;3 exp(-4nti2t
cos c#J~/X)
1 t i’;2i53exp(-4niri2t
v=p,s,
cos ~$~/h) ’
(3)
or iv = F1;2 + 43T
1 + i’;2?53T T = exp(S)
(34
’ and
s = -47+?,
COS
&t/h
,
(3b)
where v is either p or s polarization,
and the subscript
* A more recent version of this program,
written for use on our PDP-11, is available from the for use in an interactive mode, but could be readily
author. adapted
In its present form, it is written to a batch-mode computation.
1 denotes the ambient,
2 the
356
3.D. Cahan /Implications
of three parameter solutions to three-layer model
film and 3 the substrate, 12 the interface between 1 and 2, and 23 the interface between 2 and 3. Further, the individual interfacial reflection coefficients rij where i and j are either 1 and 2 or 2 and 3, are given with complex notation by: 'iiCOS~i-iziCOS
~j
'f=
# = ~j COS~i+
AiCOS~j
12,23 ,
(4)
'
although the n’s and Cp’smay of course be real. Inversion of these equations analytically is not possible, but solutions can be found using an extension to two or three dimensions of Newton’s approximation, -dX=X,+,
- Xm = -flXM(X)
(6)
>
and dX is the where X, denotes a trial root and Xm + 1 is the next approx~ation, correction to be applied to the trial value. XN1+ I then becomes the new trial value and the procedure is repeated until convergence is obtained. This algorithm is ideally suited to computer iteration and usually converges in less than 4-5 steps. Assuming that A, and J/u for the surface and nI for the ambient are known, we wish to find values for n2, k, and r2 which will make the functions F(A)=SAC-6A,
=A,-A,,
Pa)
all equal to zero (or less than the experimental the zero (or minimum) of the function F = F(A)* + F($)2 In three dimensions
c(X,,+, i
error). This is equivalent
to finding
(8)
+ F@R)* .
it is then necessary to solve the set of three equations
- Xj,m)
72
1.m
in Vi:
Xj=n2,k2,t =F(Vj)
Vi = A, ~, 6R I
to refine the trial root set Xj,na to the Xj,m+l set. The partial derivatives W( r3,aXj are obtained from eqs. (1) and (2). Differen-
(9)
B.D. Cahan /Implications
of three parameter
solutions
to three-layer
model
351
tiating (1) and rearranging terms gives - 1 sip -fp ax
___ - 1 _ais is ax
LL!L?! sin2GaX
+i?!!, aX
(10)
Since aF( VI/ax = a( V)/aX and the a( V)/aX must be real, one can separate: an
_!.ayp_Lar^”
ax=‘”
;f
_
sin?+ Re
(11)
is ax 1 ’
( ?P ax^
arp__1 ar^’ (ipax^ 1
(12)
ax )
is
From (2), since --1 aw +Lawb=~~eG!!! w* ax w ax ( w ax 1 3 a(&R/R)_ 2 sin2 cytan2 $ Re [( 1/iP)/(aiP/aX)j + 2 cos2 (YRe [( l/is)/(W/&Y)j ____ _________. ax sin2 (Ytan2 $ + cos2 (Y (13) From (3), since aA/ak = -i(M/an) 1
apv
--= FV at2
(aiy,/atg(i
= -i,
-(i%q2)+(i
p(r^‘;2)2)r(ai”,/at2) ____ (1 + i’;2y^y)(il;2 + i53T)
---i au^v =___ i ar^v f” ak iv at1 3 i ai” v^ at ”
tfg(ar/an) ___
3 (14)
(15)
tsr^53(1 - (fy2)2) __-1 + i’;2i;3 t(
T)(“;2
(16)
+ F;3 7J
From (4) and (5) +-f2
2n,
cos uj, cos $,(l
- tan2 i2)
YG=
(A2 cos $1 t nl cos ljJ2)2
a93
X2,
anai;, ----=_ an
(17)
’
cos & cos i2( 1 - tan2 $2) (fz3 cos $2 + A* cos &)2
’
(18)
2n, cos qbl cos G2(1 + tan2 G2) (n1 cos $1 + A2 cos G2)2
’
(19)
(31)
While a hunt- and search method of finding the trial roots (X,,) for eq. (9) cat1 be used, it is both tedious and frustating, since all initial guesses either converge to the same few sets of roots or diverge, and possible roots can be readily missed. A sitnplcr and more elegant solution is to use a subset of two of the eqs. (7a. b. c), find the locus of all possible 2P roots, and locate those solutions which give a rcasonable match for the third parameter, using the full iteration scheme to refine only those roots. For example. the minimization l of FR = F(A)2 + F($)2
(13)
)
can be used to find those values of II and k which satisfy eq. ( 1) for a series of thickTable 1 _
-__
Trial value ___~_
_ ______~~_
-_
_._..
“Exact”
solution
(1) 2-Parameter (% k)ambient~ (rrl-kl,
'3Zj
(2) Choose
t;R(“I.k,.r,),F(~.)=F(~)=O
‘1
t2)
,,kj
,,
i--t F(6R)
fj’
“I
E‘R(Q,
_i
b'Rhj.
-+
(rrf. kf. ‘f)
3
F,(t,;.
k2. r2), F(h)
ki, ?$,
= F(G) = 0
p(A) = F(+)
= 0
r 0 (if any)
3-parameter (np kb ti) (3) Repeat
(‘*, %ubstratc
etc.
(F = 0)
for
___..___-
k;. t;’
.-_
* The symbol FR is used in the figure to denote those solutions obtained by minimizing E’(A) and F(i). Similarly F, denotes minimization of F(G) and F(&R), and Fti denotes k’(A) and F(sR).
B.D. Cahan /Implications
of three parameter solutions to three-layer model
359
nesses tn. This procedure is equivalent to finding the roots of a “classical” ellipsometric problem, with the added feature that F(6R) (eq. 7c) is also calculated. The procedure is outlined in table 1. Intuitive reasoning (with the help of experience) provides a procedure for finding all of the roots. It is obvious that for a given 6A and S$ the 2P equations can be satisfied by a thick film whose optical constants are close to those of the ambient or close to those of the substrate. Choosing some value, t, much thicker than that expected for the real film, the n and k for the ambient are used as trial guesses, and an nl and k, are calculated. These values are then used as trial values at a lesser thickness r2 and a new set (tz2, k2) are calculated. This algorithm is repeated (calculating F(6R) at each step) by suitably decrementing * t until (1) a thickness is reached well below the anticiapted film thickness; (2) the solutions diverge, or (3) the new values of n and k become impossibly high or n and/or k goes negative (a physically impossible condition). The procedure is then repeated using the n and k for the substrate. In practice, for thin films (t < 100 a) we have found that almost any initial trial values with thicknesses at least ten times greater than the expected value lead to the same two branches (which we label the “water” branch and the “metal” branch) or else diverge and give no solution at all. Occasionally we can find additional branches with physically absurd optical constants for large trial values of thickness but these have never produced viable alternate solutions. Other symmetric branches with negative II, negative k or both negative can be obtained, but these can be eliminated on physical grounds. If, for any set (ni, ki, ti), F(SR) z 0 (i.e., 6R/Rc.culated _- 6R/R ,,,easured) or F(FR) changes sign from one ti value to the next, those values can be used as trial values of a full three-dimensional iteration, and the “exact” root determined.
4. Examples of three-parameter
solutions
Fig. la illustrates the two branches as developed for an oxide covered Pt surface, with two possible roots on the “metal” branch, and none on the “water” branch. The corresponding values of n and k are also shown, labeled 1 and 2. Fig. 1b illustrates a similar case for an oxide covered Au surface, with one root on each branch. Fig. Ic shows a case where no roots exist. In this case, there is no single film of any optical constants which will satisfy the experimentally measured values of A, $, and 6R/R. The commonly followed practice of choosing a thickness from an independent measurement (such as from coulometry in electrochemistry) will of course give two solutions for n and k, but they will be invalid, as would a reflectivity cal-
l
The size of the step taken is usually not too important, since the curves are in general smooth and continuous, but sometimes too large a step causes a “jump” to the other branch. Occasionally, one branch terminates (as will be shown later), giving only a single branch below a given thickness. The solutions will then “jump” to that branch or diverge.
360
B.D. Cahan J Implications
of three parameter
solutions
to three-layer
model
Cc)
3 2 2
ixl_l OO Fig. 1. Plots of AR/R,,lc (top), n ing several types of solution of the solid line represents the measured = -0.84%. (b) Gold oxide on gold, 6R/Rmeas = -2.28%.
IO T- &
00
IO
(middle), and k (bottom) for three different systems, illustratthree parameter equations with plots versus thickness. The 6R/R value. (a) Platinum oxide on platinum, 5820 A. SR/R,,e,,s (c)Gold oxide on gold, 5064A. 7439 A. SR/R,,,,, = -0.945.
culated on the basis of assuming a calculated R,, from the A, ti measurements alone [5,6]. (It need hardly be pointed out that the two branches have no physical significance whatsoever, but are merely the loci of all possible 2P solutions. Only the points of intersection with the third parameter are solutions for the complete 3P set.) In figs. la, b, c it can be seen that even at 18 A the values of n and k are tending toward those of “water” and the “metal”. Although the calculated reflectivity change in fig. lc misses the measured value by almost 0.4%, far too much to be attributed to experimental error, it is clear that in fig. la a small error in &R/Rof even 0.02% would have caused the curve to miss completely and give no solutions at all. In fig. lb the roots are clear, but the t values are so close that it would be difficult to choose between them on the basis of coulometry alone *. While the preceding section illustrated the two-parameter solution by the minimization of F(A) and F(G) to find values of n and k as a function of f, it is just a feasible to minimize F(A) and (6R)or F(JI) and F(6R), or to solve for any two of the three * Other
criteria
will be discussed
later.
(b)
I
OO
I
IO
I
1
I
T (A)
Fig. 2. (a) Data of fig. la replotted to show two intersections of FA, F$, FR curves in k-t (top) and n-t (bottom) space on same branch. (Only relevant branches drawn, for clarity.) (b) Data for platinum oxide on platinum, 7329 A. Plotted as in (a), but with no convergence (caused by small error in data). Lower FA branch terminates at X.
(a)
0
.
’
’
’
’ ’ IO
(b)
3
’
’
0 ’ c 3 c c 8 3 8 IO T(;;(: Fig. 3. (a) Data of fig. lb replotted as in fig. 2a, to show two intersections on different branches. (b) Data of fig. lc replotted as to show no 3P intersections. Only one FA branch exists at these thicknesses. 0
362
B.D. Cahan /Implications
of three parameter
solutions
to three-layer
model
film parameters (n, k, and r) as a function of the third. Figs. 2a, b and 3a, b show typical plots of the three sets of (n, k) versus r that can result from this procedure. Intersection of all three curves at a point in both n-t and k-t space is necessary for a three-parameter solution to exist. Such is the case for fig. 2a. corresponding to the two roots of fig. la. Note that the intersection at rr is much clearer than that at r2, reflecting the approach to quasi-homogeneity and consequent lack of resolution for that root. Fig. 2b has no roots. The inset in the n-t plane shows that what appears to be a root in the upper F, branch is not, as is evident in the k -t plane. The lower FA branch in fig. 2a terminates before it reaches the curve, and only one F, branch exists below -5 A. Figs. 3a, b show similar plots for the data of figs. 1b, c, where there is one root on each branch and no roots, respectively.
5. Effects of errors in experimental
data
It is often informative in an analysis of this type of data to consider the effects of small errors in the experimentally determined values. The method of two-parameter locus tracing allows a convenient visualization of these effects. As one example l , consider the results of small variations in S$ as shown in fig. 4. An experimentally determined value of -0.224” converges to a 3P root at a thickness of -4.5 A. Varia-
Fig. 4. Gold oxide on gold, 6570 A, showing Left: &R/R; lower right: n; upper right: k.
l
All of the experimental for purposes of brevity
errors resulting
from small deviations
variables are subject to errors, and can be analyzed and clarity only variations in 6$ are shown here.
in J, meaS
in this fashion,
but
B.D. Cahan /Implications
Fig. 5. Spectrum Kruger [ 71.
for an Fe surface,
of three parameter solutions to three-layer model
passivated
at +0.7 V in borate
solution.
363
After McBee and
tions of -0.01“ and +O.Ol” change the calculated thickness to 5.2 a and 2.1 a while +0.02” makes the 3P system divergent with no toots at all. The same type of figtue can be used to estimate the error range in II and k when calculated from a 2P (e.g. A and $) measurement using an independently determined thickness as a third parameter. Especially in the case of thin films of unknown stoichiometry, thickness can be difficult to estimate precisely, but the range of possible n and k values are clear.
6. Use of
6R/Rmeasurements
to eliminate false roots by locus tracing
The problem of a choice between the two roots can often be minimized by even a rough measurement of &R/R.In ths illustrations of fig. la (or 4) the expected reflectivity difference between the two branches is of the order of 0.5-l%. A reflectivity measuring system with a resolution of -0.1% might not be sufficiently accurate for a full 3P solution but would suffice to differentiate between the roots. As an example, consider the analysis of data similar to that obtained by McBee and Kruger [7] for the passive film on iron (fig. 5). Using values of A and $J recalculated from their data or those measured in our laboratory (see e.g. figs. lOa, b) with the method of locus tracing. one obtains dual values of n and k for all wavelengths. It becomes clear that those multiple values in fig. 5 represent cases where the discrete root procedure [8] used by McBee and Kruger only found some of the roots, missing others. The two sets of roots then correspond to the two branches obtained by the 2P solu-
364
B.D.
Cahan /Implications
of three parameter
solutions
to three-layer
model
tion method. The calculated values of 6R/R for the two branches are different by several percent and are even of opposite sign at some of the wavelengths in this range. Preliminary measurements [9] using 6R/R as the third parameter show that the correct set of roots is probably the lower k branch.
7. Practical implications
of the three-parameter
solution
In the preceding sections, it was pointed out that not all sets of experimental data will yield a 3P root (e.g. fig. 2b) and that the discrepancy was often far larger than could be accounted for by experimental inaccuracies. Some of the causes of this dilemma and a few explanations for them will be discussed in this section. Some are obvious, some are less obvious but demonstrate, and some are still only speculative, but are the subject of further theoretical and experimental scrutiny. For purposes of this section, “real” data will be used as examples when possible, but “thought” experiments with synthetic data are sometimes invoked. A typical set of electrochemical-ellipsometric data as collected by the computer 3.87
93.9 -1
22.01,
-l&O0
3e.441-
’
’
’
I
’
’
E VS PD-H
E VS PD-H
Fig. 6. Raw data as collected from triangular sweep ot’ Au in 1 N HCl04. ~1 = 65”; A = 53 12 A; sweep speed = 100 mV/sec; time = 19:04. Lower left: i versus V; upper left: A versus V; lower right: $ versus V; upper right: --6R/R versus V. (Note: relative reflectivity plot is inverted.)
B.D. Cahan / Implications
of three parameter
solutions
t---+
to three-layer
model
365
t--+
Fig. 7. (a) Waveform used in scanning potential for fig. 6. (b) &me but for figs. 8 and 9. Note that while potential “jumps” to 1.35, dotted line indicates “apparent” potential for recording data.
from the automatic ellipsometer l is shown in fig. 6. The lower left hand curve is a current-potential plot as the potential of an Au electrode in 1 N HClO, is scanned from 0 to 1.8 V and back according to the waveform in fig. 7a. Currents above the line are anodic. representing charging of the double layer from 0.0 V to -1.2 V, and formation of an oxide film of a few monolayers to 1.8 V, where the sweep reverses direction. A small anodic current continues to pass until the potential drops to 1.6 V, and starting at about 1.2 V, the film is reduced back to metallic Au. By 0.9 V, the film has been completely reduced and the small residual current is cathodic charging of the double layer. Corresponding points of the curves of A, $J, and &R/R versus potential, measured at 5321 8, near the absorption edge (-5000 A) of gold, can be compared to the E-i plots. 7.1. Choice of “bare substrate” values. A0 and Go It is clear from fig. 6 that in the anodic curve from 0 to 1.2 V and from 0.9 to 0 V in the cathodic branch, where no surface oxide films are present on the surface, there are relatively large (several tenths of a degree and several tenths of a percent) changes in A, $, and FR/R. This effect is not new and has been reported from many reflectometric and ellipsometric studies of Au [ 10-13, etc.] but it points up the problem of which values to choose (if any) for A, and J/o, since the surface is “bare”, i.e. free of films, over the entire potential range. To be sure, the surface is covered by the electrolyte ambient, but its optical properties are known, and it is easily demonstrable [ 141 that no conceivable change in the solution could account for more than a small fraction of the observed effect. It is widely recognized that this effect is caused by some electromodulation of the surface layers, but there the agreement
l
A Rudloph Research RR2000 [ 11 Automatic Ellipsometer was used, with the data collected, analyzed and plotted by a PDP- I l/45 minicomputer. The computer generates the electrochemical waveforms, and synchronizes the data collection to the experiment. Each curve consists of 5 12 data points. This corresponds to a voltage resolution of - 7 mV/point. With a sweep speed of 100 mV/scc. and a measurement rate of 54 sets of (I, A, $J and 6R) points per set, -3.8 data sets are averaged per point. The r.m.s. data deviations are typically less than 0.001 degrees for A and @ and O.Ol%, for aR/R.
366
B.D. Cahan /implications
of three parameter solutions to three-layer model
stops. The depth of this layer and its “optical constants” are still a matter of controversy. Using a 3P approach, no simple thin film of any optical constants and thickness will satisfy the experimental data. Attempts to calculate the problem in reverse. i.e. using optical constants calculated from McIntyre’s [ 151 approximate equations and measured R, and R, data to ‘*predict” aA/?l V and a$/a I’ fared no better. In the case of Au (and other metals like Ag and Cu) the interaction with the ambient, and the electric field at the surface, including the effects of truncation even in UHV studies, can preclude an unambiguous definition of the “bare” substrate optical constants. 7.2. Substrate-film
interaction
Just as there are modulation effects caused by electric fields at the surface of Au in the absence of a definite “film”, so there are similar effects induced in the metal by the presence of a film on its surface. In fig. 6, both $ and 6R/R begin to change immediately after the passage of the first anodic current at 1.25 V while changes of A are delayed by 50 mV to 1.30 V. At 1.35 V l there is an abrupt change of slope of aA/aJ/ (and an/N), which then continues constant to 1.8 V. A similar effect is evident in the cathodic (reduction) branch where A returns to the double layer line at 0.95 V, 50 mV before G or 6R/R. For wavelengths any shorter than -6500 8, we have not been able to find any 3P convergences for either the initial monolayer or the thicker film at 1.8 V l * , regardless of how we choose A,, and Go or 6A, SJ/, and 6RfR. 7.3. Electromodulation
of the surface film
Anodic current does not stop immediately after reversal of the potential sweep. Additional oxide continues to form down to 1.7 V as evidenced by the continued increase in A. From 1.7 V down to 1.3 V, no significant current passes, and the A value stays essentially constant. The J/ and 6R/R values show a definite decrease in this region. To decide whether or not this was an effect of “aging” [ 131 or dissolution, the potential sweep regime of fig. 7b was used. Instead of reversing after reaching 1.8 V, the sweep was abruptly stepped down to 1.35 V, at which potential no reduction occurs, and the potential maintained steady for the 4.5 set the normal(dotted line) sweep would have required, and then the sweep resumed. In this way, aging could proceed without the complication of further film growth or parallel 0, evolution. Fig. 8 shows the results plotted versus time rather than potential as in fig. 6. The anodic portion of the curve is almost identical. The negative current spike in
* At this point, just past the first anodic peak, the coverage by oxide or oxygen is one or onehalf depending on how one defines the valency of the adsorbing or depositing species. ** In contrast to the case for platinum [ 21 where reasonable convergences are obtained at all thicknesses and wavelengths.
B.D. Cahan / implications of three parameter solutions to three-layer model
361
DEL
E VS PD-H
Fig. 8. Data similar to fig. 6, but including “jump”. Time = 23:56.
the cathodic branch at 1.8 V corresponds to the charging of the double layer. Both $J and 6R/R show a distinct jump at the transition while A changes less than O.OOl”! Then all three parameters (A, I/Jand 6R/R) remain constant until the sweep is resumed, after which the curves are indistinguishable from those of fig. 6. There is thus a voltage sensitive component of r+!~ and &R/R for an oxide covered Au surface which is not visible from A modulation measurements alone. This is in direct contrast to the bare surface electromodulation where A shows the largest effect. Indeed, as can be seen from fig. 9, the effect is visible in aJ//a Voxide even at wavelengths (-4000 A) where 3$/a VAu is zero or slightly negative. To date, we have not yet learned how to account for this effect, which appears to occur with several metal oxide-metal systems, nor have we decided whether the seat of this modulation is at the oxidesolution or the oxide-metal interface, or both. 7.4. Problems of surface roughness Figs. 6 and 8 illustrate yet another problem. These measurements were part of a long series of measurements on the same sample, changing many variables including wavelength. Fig. 8 was obtained almost 5 h after fig. 6. Although the curves are al-
B.D. Cahan /Implications
368
of three parameter solutions to three-layer model 42.21
t
E VS PD-H
E VS PD-H
2.0
Fig. 9. ri, XISUS t plots similar to fig. 8. upper left: h = 4172 A; lower left: 5312 A; upper right: 6070 1$; lower right: 7430 A. most superimposable,
the A values have been displaced
by -0.2 1” and the $ values
by -0.09”. (The absolute reflectivity was not monitored.) Blondeau and Cahan [ 161 have shown that these changes are due to a roughening of the surface during cycling,
and appear to follow the predictions l
* of the Ohlidal-LukeS
[ 171 equations.
There are some typographical and mathematical errors in ref. [ 171. Their eq. (16) should read i?‘;,,
=
ii’no {(h2Bz
a2 G2E + noa)
_ (2a2B1 - n2d IB)(G*niA Ri’=
=
no (Y2(noB + a)3
+ nzd IB, - nZ&2B)&2B
,B/2a
+ no(y)
+ 2&i*B1 + $ n&4 1)}.
201~6~ + r~&4~B~or - r1&4~Ba)(noB
+a)
r
It appears that Ohlidal and LukeS may have used the incorrect equations in refs. [ 17,181.
B.D. Cahan /Implications
of three parameter solutions to three-layer model
7.6
iiP
III
M
369
7qi
(L d B
-0.4
-0.4 I31
q ‘I
d 0
-0.5v
c
III -$50
it VS
PD-H
E VS
PD-H
25.6
(a) 1204r
1
l.9r
I.OV
PSI
29.6
1
_2jfy-j 2jly,I‘jyq -0.50 E VS PD-H
1.50
-0.50
1.50 E VS PD-H
29 .40
2140 PSI
(b) Fig. 10. (a) Data for triangular sweep of Fe in buffered borate between -0.5 V and +l.O V (versus a-Pd-H): $ = 65’; A = 4172 .k; sweep speed = 20 mV/sec. Lower left: i versus V; upper left: A versus V; lower center: \L versus V; upper center: -6R/R versus V; lower right: A versus Q ; upper right: -6R/R versus Q. (b) Same as (a) but for h = 7430 A.
7.5. The problem of the anomalous reflectivity Figs. lOa, b show preliminary results [9] of electrochemical-ellipsometric measurements for the passivation of Fe in buffered-borate solution at 4171 A and 6570 A. Although the A-$ plots appear well behaved and give 2P solutions similar to those
370
B.D. Cahan /Implications
of three parameter solutions to three-layer model
in the literature for the passive film, we are currently at a loss as to how to ascribe the wild gyrations obtained in the &R/R versus rj curves to the growth of a three layer system. 7.6. The problem of “equivalent” surfaces As long as only the two parameters, A and $ were measured ellipsometrically, it was often tacitly assumed that it was possible to assign a set of “pseudo-optical constants”, n* and k*, to describe the surface from the measured parameters and an idealized two-medium model. Indeed, since it is very difficult if not &possible to measure the absolute reflectivity to an accuracy equivalent to that calculable from A and J/ (or n and k) values, absolute reflectivities are often determined in this indirect fashion. At the basis of the 3P (A, +, and 6R/R) method is the assumption that while only relative reflectivity changes (AR/R) are measured, Ro, the base surface reflectivity is calculable from A0 and Go. If this calculation is not valid, the results of the 3P approach will be in error. Specifically, reflectivities calculated from A and $ measurements on a rough surface [ 161 or a film-covered surface by assuming equivalent pseudo-optical constants are not, in general, equal to those calculated from the more rigorous equations. Reflectivities calculated on the basis of the two models differ by several percent even for very thin (
Fs WI
x=n,k, ’
(22)
* A similar conclusion was reached by Schueler [ 191 about the use of multiple-angle of incidence measurements on thin (<200 A) fiis. l * E.g., if nI = 1.33, nz = 1.7 an angle of incidence 41 = 65” gives a 02 of 45.16”. Of course, k is usually complex, but similar considerations hold.
B.D. Cahan /Implications of three parameter solutions to three-layer model
311
and
WRIR) ax
or
WV0 ax
aJ/ =k3=,
(24)
which is just the condition for homogeneity and therefore indeterminacy! On the other hand, it is possible to calculate from eqs. (1 l)-( 13) values of G2 which will optimize the sensitivity of any (or all) of the three parameters to a desired physical constant, within the limitations imposed by the experimental conditions *.
8. Conclusions Although the three parameter (including &R/R)approach is not always the answer to an ellipsometrists prayer, it does often lead to solutions, and can at least help decide between alternate solutions. While the existence of a three parameter solution for a given substrate-film-ambient system does not guarantee that the numbers so obtained are a real solution (i.e., the three layer model is adequate) the lack of a convergence, as in fig. lc), guarantees that something is wrong with the model. In this fashion we often have a check on the validity of the conchrsions drawn from ellipsometric data.
Acknowledgements The author is pleased to acknowledge the support of this research by the National Science Foundation and by the U.S. Office of Naval Research. The computer facility used in these studies was acquired under a grant by the Chemistry Research Instrumentation program of the National Science Foundation. Thanks are due to Drs. G. Blondeau and T. Grcev for their help and to Mr. C.T. Chen for his invaluable assistance.
l
Once approximate values of the n, k, and t are determined, a simple computer calculation of the nine derivatives as a function of e.g. the angle of incidence can establish the optimum $1.
312
B.D. Cahan /implications
of three parameter solutions to three-layer model
References B.D. Cahan and R.F. Spanier, Surface Sci. 16 (1969) 166. J. Horkans, B.D. Cahan and E. Yeager, Surface Sci. 46 (1974) 1. J. Horkans, Ph.D. Thesis, Case Western Reserve University (1973). W.-K. Paik and J. O’M. Bockris, Surface Sci. 28 (1971) 61. R.C. O’Handley, Surface Sci. 46 (1974) 24. S. Gottesfeld and B. Reichman, Surface Sci 44 (1974) 377. CL. McBee and J. Kruger, Surface Sci. 16 (1969) 340. Program by F.L. McCracken, Natl. Bur. Std., Tech. Note No. 49 (1969). [ 91 B.D. Cahan, C.T. Chen and T. Grcev, unpublished data. [ 101 W.N. Hansen, Surface Sci. 16 (1969) 205. [ 111 J.D.E. McIntyre, in: Advances in Electrochemistry and Electrochemical Engineering, Vol. 9 (Wiley, New York, 1973). [12] B.D. Cahan, J. Horkans and E. Yeager, Surface Sci. 37 (1973) 559. [ 131 B.E. Conway, H. Angerstein-Kozlowska and L.H. Laliberte’, J. Elec. Sot. 121 (1974) 1596. [ 141 M. Stedman, Symp. Faraday Sot. 4 (1970) 64. [ 1.51 J.D.E. McIntyre and D.M. Kolb, Symp. Faraday Sot. 4 (1970) 99. [16] G. Blondeau and B.D. Cahan, Paper presented at the May 197.5 Meeting of the E.C.S. (Toronto). 1171 1. Ohlidal and F. Luke;, Opt. Acta 19 (1972) 817. [ 181 I. Ohlidal, F. LukeS and K. Navratil, Surface Sci. 45 (1974) 91. [ 191 D.G. Schueler, Surface Sci. 16 (1969) 104. [l] [ 21 [3] [4] [S] [6] [ 71 IS]
Critique
A.B. Buckman: I just wanted to point out another example of the failure of the three-parameter model you mentioned. This happened with thicker films and multiple-angle-of-incidence ellipsometry. The program discussed by lbrahim and Bashara that we used to minimize the sum of the squares for differences between experimental and calculated values would always find the minimum but in the case of some work we did on lead iodide on glass, these were thick lead iodide films better than a 1000 A thick, we found that unless we included in this model the effect of a surface layer on glass which had a different refractive index than the bulk glass itself, the sum of the square on conversions would come out to be from 100 to a 1000 square degrees. The program would find the minimum which converged very poorly. The total sum of the squares could be much greater than we would expect from experimental error. This is just another example of the failure of this three-layer model that you mentioned. It also happened in static situations with multiple angle of incidence ellipsometry and with thicker films. B.D. Cahan: The failure, may I point out, is not in the three-parameter-solution techniques but in the three-layer model. One can start from any crazy starting set of n’s and k’s and the program will converge in fractions of a second and then proceed to give you the entire locus. Even on our minicomputer we can piot out the entire locus in a short time. 0. Hunderi, I have difficulty understanding how you can get any model to work for these thicknesses where you have monolayers or submonoldyers. What does a layer mean at this level. B.D. Cahan: That’s a question often asked by physicists. I’m not sure. The only defense is, there are some systems where it does work. For example, with careful measurements on the platinum oxide film we get the same thickness over a wavelength range from 4000 A to 7000 A. Yet the optical constants in both substrate and film vary considerably over this range.