1. Optical measurement of the refractive index, layer
thickness, and volume changes of thin films
A. H. M. Holtslag and P. M. L. 0. Scholte
Preparation, measurement, and calculation methods are discussed for the determination of the complexindex
of refraction, the layer thickness, and induced volume changes of thin layers (due to a phase change, for
example). The principle of the calculation is fitting a curve in the reflectance-transmittance plane measured
on a range of layer thicknesses, instead of fitting the reflectance and transmittance as a function of
independently measured layer thicknesses. This general method is applied to thin films of GaSb and InSb, in
which a laser-induced amorphous-to-crystalline transition can be used in optical recording. The information
essential for optical recording applications is measured quickly by making use of a stepwise prepared layer
thickness distribution, while the complex refractive index and the layer thicknesses can also be calculated
unambiguously.
1. Introduction The opposite case is not as straightforward: From a
For erasable and write-once optical recording, im- measurement of the reflectance and transmittance of a
portant disk media properties are: the complex index transparent substrate (medium 1) supporting a thin
of refraction, the layer thickness, the thermal conduc- absorbing layer (medium 2), two relations should be
tivity, the heat capacity, and the stability of the thin solved to obtain the unknown complex index of refrac-
film materials used. In general, a high reflectance, a tion of the layer.
high contrast between marks and unwritten areas (or, For a nonabsorbing film, the real index of refraction
more precisely, a high modulation), and a reasonable and the layer thickness can be obtained by means of
absorption in the active layer to obtain a low threshold explicit relations.2 The calculated layer thickness is,
power are sought.1 In the process of developing new however, multivalued with steps of Xo/(2n)- Usually,
2
materials for optical recording, the empirical adjust- the layer thickness is measured independently, for
ment of the desired properties is time consuming. Of- example, by means of a mechanical profilometer, the
ten many disks with different layer thicknesses and method of Tolansky, or with Rutherford backscatter-
compositions are tested to obtain the optimum config- ing (at known densities). The calculated multivalued
uration. In this paper a straightforward method is layer thickness is only then determined unambiguous-
discussed to obtain the optical properties needed for ly. The ambiguity in the layer thickness can also be
optical recording application. Reflectance R, trans- avoided if spectrophotometric measurements are
3
mittance T, and absorption A = 1 - R - T can be made over a range of wavelengths.
obtained by means of explicit relations if all indices of For an absorbing film with known layer thickness d2 ,
refraction, il = n- ikl, and layer thicknesses, dl,of the it is not possible to obtain explicit relations for n2 and
2 4 Two
media used are known [media are denoted by means of k2 as functions of the measured properties.
an index 1= 0 (air), 1, 2,. . .]. These properties have to equations with two unknowns of the form
be known generally only at the wavelength Xoof the Rth(n 2,k 2) - Rexp= 0, Tth(n 2,k2) - Texp = 0, (1)
semiconductor laser used. 2 = n2- ik2
can be solved numerically, to obtain the root (or roots)
of these equations. In these formulas (Tth,Rth)denote
the theoretical expressions and (TexpRexp)denote the
experimental values for a certain layer thickness.
This procedure can be repeated for 1 i < I film
The authors are with Philips Research Laboratories, P.O. Box
thicknesses to obtain the common root of Eq. (1) and to
80.000, 5600 JA Eindhoven, The Netherlands.
reduce the error. The final index of refraction can be
Received 13 January 1989. obtained by averaging such results. Because of errors
0003-6935/89/235095-10$02.00/0. in the measurement of the film thicknesses, reflec-
© 1989 Optical Society of America. tance, and transmittance, or due to poorly chosen layer
1 December 1989 / Vol. 28, No. 23 / APPLIEDOPTICS 5095

2. thicknesses in the range of interest, often a maximum a
random difference n2/n 2 = 0.1 and k 2 /k 2 = 0.1 is
observed.
Instead of the above-mentioned methods we will
present a straightforward (automated) measurement
and calculation method. On the one hand, the mea-
surement method directly gives the information need-
ed in optical recording applications, e.g., the reflec-
tance, the transmittance, the absorption, and the
contrast function, all as a function of sputter time, for
example. On the other hand, the complex refractive
index, the layer thicknesses, and volume changes (after
a phase change) can be calculated afterward unambig-
uously. It is not necessary at all to measure the layer b
thicknesses. Before discussing the measurement and 0.7
calculation method in detail in the next sections, an
0.6
example of a measurement and a calculation will be
given first. 0.5
Part of the results of a typical measurement (at a
0
0.4
fixed wavelength of X0= 820 nm) is given in Fig. 1(a) I:
and the same data of the measurement are given in Fig. 0.3
1(b). In the (T,R) plane the measured reflectance, 0.2
Rp,, is plotted (points) as a function of the corre-
sponding measured T', at 256 positions i of a thin film 0.1
of a-GaSb deposited on a transparent substrate. In-
stead of a constant film thickness a layer thickness
distribution has been prepared. In both Figs. 1(a) and Fig. 1. Measured reflectance as a function of the measured trans-
(b) the arrow denotes the direction of increasing layer mittance (points) at certain positions of the prepared sample a-
thickness (the latter, up to now unknown), while the GaSb,N.A. = 0.2, Xo 820 nm. Three theoretical curves (solidlines)
=
solid line represents calculated curves to be discussed are plotted with, in the middle, the value obtained from the least-
below. squaresmethod,h2 = 4.88 - il.466, and for (a) n2= +0.1n2and (b)
If a nonabsorbing medium were measured (k2 = 0), a bk2= 0.1k2 . Notethatn 2 determines the height of the extremumE
straight line, R = 1 - Tp, would be observed for and k2 determines the position of the flank F. The arrow denotes an
increasing layer thickness, while the data run along increasing layer thickness along the curve. The measured reflec-
tance and transmittance as a function of the calculated layer thick-
this line up and down. From the minimum transmit- nesses are shown in Fig. 5(b)
tance or maximum reflectance, the real part of the
index of refraction could be obtained directly. If k2 is
greater than zero, the transmittance finally becomes nesses at all. Note that at a thin film thickness equal
zero for large layer thicknesses and the reflectance to zero, when representing a measurement on a trans-
reaches its bulk value. The curve shows fewer visible parent nonabsorbing substrate, the curves in Fig. (1)
oscillations if k2 increases. start at (T,R) (0.92,0.04), instead of at the values
In both Figs. 1(a) and (b) the same curve obtained by (0.92,0.08). In general, the reflectance is 0.04 lower
the numerical calculation method is shown in the mid- when a focused beam is used (as in optical recording)
dle (solid line: 2 = 4.88 - il.466). In Fig. 1(a) instead of a parallel beam to measure the transmit-
calculations are also plotted where the value of k2 is tance and reflectance.
kept constant and n2 is changed by ±10%(outer solid In Sec. II a quick preparation method of the samples
lines). In Fig. 1(b) the value of n2 is kept constant and a measurement method is discussed first. Expres-
while k 2 is changed by 10% (outer solid lines). This sions for the reflectance and transmittance of a stack of
example shows that at constant k 2 the flank at F re- thin films on a thick transparent substrate depend on
mains almost fixed while at constant n2 the height at the geometry of the beam of light used (focused/paral-
the extreme at E remains almost fixed. Using this lel beam, collecting optics, substrate-incident/air-inci-
observation, an approximate value for the index of dent light) and are discussed in Sec. III. In Sec. IV
refraction can be obtained quickly by trial and error theoretical expressions are given to explain the nu-
calculating a theoretical curve for an assumed complex merical calculation method used. In Sec. V the results
index of refraction and comparing it to the measured of this general method when applied to the amorphous
data (optimize k2 first and then optimize n2 ). This and the crystalline state of the phase change materials
value of the complex index of refraction can be used as GaSb and InSb are discussed.
a start value in the numerical method and once it is
known, the layer thicknesses can also be calculated. II. Experimental
As shown, the value of the complex index of refraction The samples were prepared by means of magnetron
can be determined without knowing the layer thick- sputtering. A shield with a small slit of 2 mm X 50mm
5096 APPLIEDOPTICS / Vol. 28, No. 23 / 1 December 1989

3. was placed above the sputter target (100mm in diame-
ter) with the long side of the slit in the radial direction
of the target. A 1.2-mm thick clean transparent glass
substrate (B270 glass from Schott), was displaced
stepwise behind the slit during sputtering. During
these displacements the sputtering conditions were
maintained, while the time between successive dis-
placements was increased. The final sample con-
tained stripes with variable layer thicknesses d2. a | LW a
- *t
The reflectance and transmittance were measured
Iz
arm
Transmissivity
on home-built equipment.5 The relevant part of the
arm
Reflectivity
experimental setup is shown in Fig. 2 and is discussed Fig. 2. Part of the experimental setup used for measuring the
reflectance and transmittance of substrate-incident light. Depend-
in more detail in Sec. III. The sample displacement, ing on the positions and distances used, the light reflected from the
parallel to the focus plane, and the (TR) measure- air-glass interface is collected partially or completely at the detec-
ments have been automated. The transmittance and tor.
reflectance arm were calibrated by means of a set of (2) This procedure allows a large number of layer
samples with a known transmissivity and reflectivity thicknesses to be measured efficiently, reducing er-
(air, and 200 nm thick Au and Al layers on glass sub- rors.
strates measured air incident) and by making use of an (3) There is no need to measure the layer thick-
additional detector to normalize all detector currents. nesses independently, because they are obtained un-
A part of the incident light beam was directed toward ambiguously from the calculation, as will be shown in
this detector by means of a neutral beam splitter. Also Sec. IV.
the focusing on the glass-stack interfaces was auto- (4) For lateral reference it is obviously useful to
mated by means of a standard CD focus motor and have some sharp features (a peak, valley, or step) in the
Foucault knife-edge method.5 Measurements on layer thickness distribution. In this way lateral ad-
samples of GaSb and InSb were made with light inci- justment errors are eliminated and it becomes possible
dent from the substrate side. to compare the layer thicknesses of the amorphous and
The measurement strategy for the GaSb sample was crystalline states afterward. A relative change in the
as follows: First, two equal parts were prepared by layer thickness may be expected if a layer is converted
cutting the original sample perpendicular to the de- from the amorphous to the crystalline state.5 7
posited stripes. Second, one part of the original sam- (5) By making use of a focused beam, measurements
ple was heated in an oven to obtain the crystalline obtained can be used directly in evaluating optical disk
state. During heating, the transmittance of one stripe properties.
was monitored to determine the temperature of the
amorphous-to-crystalline transition (GaSb: Ttr = 111. Theory: The Calculation of Reflectance and
2890C, InSb: Ttr = 21000). In addition, to prevent Transmittance
oxidation during the heating process, argon gas was In Fig. 3, three situations of interest are shown.
passed through the oven. The applied heating rate Figure 3(a) demonstrates that at an interface, the Fres-
was 200C/min. nel equations and Snell's law must be fulfilled (Ref. 8,
Finally, the reflectance and transmittance of the pp. 40 and 615). For the p- and s-components of the
amorphous and crystalline parts were measured at electric field vector E, the Fresnel coefficients T,t are
equidistant positions. The measurements at those
-9sl l cosl - 12COS 2
parts were taken along a line parallel to the cut, at a
distance of 2 mm from the cut. One may also expect =E+ 11 coSnl + 12 coSn2
layer thickness variations due to inhomogeneous sput- (2)
tering. The procedure in the case of the InSb sample Ep 12 coSn
1 - hl coS 2
has been changed to avoid this minor problem. The a- Tp12 l= 112coSn + h1 coSn
1 2
InSb sample was first measured at equidistant posi-
tions along a line perpendicular to the deposited s 2nk cosl
stripes, then it was converted to the crystalline state 1 h1 conS + 12 coSn2
1
and measured along the same line again. (3)
The above preparation and measurement method Ep2 2h1 cos4
1
has the following advantages: =pp21h2 coSn1 + hl CSn 2
(1) Instead of preparing I samples with I different
constant layer thicknesses, only one sample using the nOsint0 0 = 11h
sinl = 2 sin0 2 = *- - - (4)
same sputter conditions is prepared with a layer thick-
ness distribution in the range of interest. When mea- The plus and minus indices refer to the direction of
suring the layer thicknesses and refractive indices in propagation of the plane waves, the tilde denotes a
both phases it is important to have a single sample complex quantity, the media are represented by means
because alignment errors in the reflectance-transmit- of indices 0,1,2, ... , and 00 represents the angle of
tance equipment are neutralized. incidence.
1 December 1989 / Vol. 28, No. 23 / APPLIEDOPTICS 5097

4. For a thin layer positioned between two half-infinite a b C
layers, see Fig. 3(b), the next equations are valid for the
p- as well as the s-components of the electric field T1 3
vector (see Ref. 8, pp. 62 and 628):
2 3
EE.
- r + 23 exp(-2it)
12
E 2
1+ 12r23 exp(-2ig)
0
L 3
-3 = t12t23 exp(-it) Fig. 3. Electromagnetic fields at (a) a boundary, (b) a thin layer;
1+ 12P23 exp(-2iX) arrows denote plane waves,and (c) a thin layer or stack of thin layers
on a thick substrate; the values at the boundary of the dashed-line
27rd2 -
box represents the calculated transmittance and reflectance by
£x= 119 conk2 .
means of Eq. (6) of a stack of thin layers or a single thin layer. Now
In this equation, x denotes a phase delay of interfering the influence of the substrate should be evaluated.
plane waves. Stacks of thin layers give analogous
expressions, which can be obtained easily by substitut-
ing for r23and 2 3 in Eq. (5) the reflection and transmis-
sion coefficients of the underlying layers. The reflec-
tance and transmittance of a thin stack of layers can be area is canceled due to the oscillating phase factor.
obtained from (Ref. 8, pp. 41 and 630) Therefore, the measured intensity due to the interfer-
ence of the spherical waves of the air-glass interface
R= Iiirst/Iifrst I?12,
= and the resulting glass-stack-air wave often results in
las | 2irs cosk) ,t
an addition of the intensities of those waves, even at a
= (n (6)
irt
1
(n COSk)frst high coherence length of the light beam used4 :
valid for the intensities I of the p- and s-components of R =R1 + R1 3 T, T= T T Do, < Ddet- (8)
the electric field vector. In these equations the indices 1 R 13 R0 1 1-R133 01
1 RoI
first and last refer to the first and last dielectric medi- For investigations with parallel substrate-incident
um. beams Eq. (8) should be used.
Finally, care should be taken using the above equa- If Do, >>Ddet,the substrate-incident light reflected
tions in the case of a stack of thin layers on a thick at the air-glass interface collected by the detector is
substrate. A thick substrate (and thick protection negligible (as in optical recording). For a pencil of
layer) should be treated separately. In Fig. 2 a rele- light the next equation can be applied (we take medi-
vant part of our experimental setup is shown. For a um 3 equivalent to medium 0):
thin layer (medium 2) or a stack of thin layers on a
thick transparent substrate (medium 1), see Fig. 3(c). R 1 3To2, T= T1 3T61, Doi >>Ddet, (9)
The appropriate equations to be applied depend on the while an averaging of all incident angles 00and p- and
numerical apertures N.A. and focal lengths f of the s-components is required in the experimental situa-
objective and collector used and on the positions and tion of Fig. 2. In the interesting case of incident circu-
the diameters Ddet of the detectors. Using a detector larly polarized light (as in optical recording in phase
at a distance a from the objective or collector, the beam change materials or for the compact disk) of a plane
size of the substrate-incident light (beam diameter wave,the final reflectance and transmittance are given
Dinc = 2f N.A.beam), reflected once at the air-glass by R = (Rp + R)/2 and T = (Tp + T)/2. These
interface, in the geometric approximation (see Fig. 2) functions are, up to a N.A. of 0.5, almost independent
becomes of the angle of incidence. Thus the averaging of all
incident angles will give almost the same result as at 00
Do = Dinc ( a _1 +1 (7) = 0 (see, for example, Ref. 8, p. 44).
In the experimental setup the following conditions
if the beam is focused on the interfaces of the thin have been used (see Fig. 2): a p-polarized beam of a
stack. In this equation the positive sign should be semiconductor laser pen (with beam shaping optics in
used for the beam directly reflected at the air-glass the collimator pen) at a wavelength of Xo= 820 nm, an
interface toward the reflectance detector arm, while objective numerical aperture of 0.45 (reflectance arm),
the negative sign should be used for the beam indirect- a collector numerical aperture of 0.6 (transmittance
ly reflected toward the transmittance detector arm. arm), a = 400 mm, f = 4.5 mm, d = 1.2mm, n = 1.523,
For the situation, Do < Ddet,almost all light reflect- and N.A.beam 0.2. From Eq. (7) it followsthat Do, -
=
ed at the air-glass interface is collected at the detector. 55 2 mm >> Ddet = 3 mm. The fraction of the
At a high coherence length of the light beam, fringes incident light collected by the detector in the reflec-
arise due to the interference of the spherical waves tance arm from the air-glass interface with Rol 0.04
reflected at the air-substrate interface and the main is, therefore, only (3/55)2 0.04 104 a negligible
wave reflected at the glass-stack-air position, all with amount. Therefore, in our calculations we use Eq. (9).
a different origin. For many fringes the integral of the As stated, explicit relations for the reflectance and
cross-term of the electric field vectors over the detector transmittance are obtained. It is not possible to ob-
5098 APPLIEDOPTICS / Vol. 28, No. 23 / 1 December 1989

5. a b
tain an explicit relation for the index of refraction f 2 as
a function of the measured reflectance and transmit-
tance in a situation as in Fig. 3(c), because this index of theoretical curve
refraction arises in both terms of a product composed during calculation
of exp(-ix) and r23 or t2 3 . It is possible, however, for
an absorbing layer studied using plane waves, to re-
duce the numerical problem to other attractive analyt-
ical expressions.2 The final calculation, however, Fig. 4. Some examples illustrating the geometric relation pi *li = 0
2
must still be done numerically, while such a method is in the (TR) plane. (a) Expression corresponds to looking for a
not attractive at all if an averaging over angles of global minimum distance. (b) Desired solution during the calcula-
incidence and polarization states is required. There- tion is positioned in the part of the curve between the two smallest
fore, a straightforward numerical method is preferred, local minimum distances.
as discussed in the next section.
IV. Theory: Calculation of the Index of Refraction and
Layer Thicknesses
interpreted in the (TR) plane by means of a geometric
relation:
Consider the measured reflection and transmission
(14)
as a function of the (up to now unknown) layer thick- pi - 1 = 0, I <i<L
nesses d2. For a least-squares fit [as in Figs. 5(b) and This means that in the (T,R) plane the vector pi, di-
6(b), discussed below] it is required that the next sum, rected from the theoretical curve at (Tth,Rth)toward
l the measured value (TexpRt ) should be perpendicu-
S = gT E [Tth(n2,k 2,dl)- lar to the vector li denoting the increase (OT,6R)along
j=1
the theoretical curve at increasing layer thicknesses.
A few examples are illustrated in Fig. 4. Most times
+ gR E [Rth(n2 ,k2,di)- s (10) Eq. (14) and the condition to look for a minimum
means that the shortest distance from the measured
is a minimum. The theoretical values (Tth,Rth) are value (TxpnRexp)to the theoretical curve gives the
explicitly given as a function of the properties n2, k 2 , correct layer thickness, [see Fig. 4(a)]. That might be
and d2 of the film under study, while the substrate expected intuitively from Eq. (10) (gT = R = 1) be-
values are assumed to be obtained from a previous cause, as noted, squares of distances in the (TR) plane
measurement. The letter i denotes that the proper are added, while a minimum sum should be obtained.
layer thickness of measurement i should be substitut- At extrema, as in Figs. 1(a) and (b) at point E, however
ed. If required, the theoretical expressions include an (also during the calculation), this is not correct [see
averaging over the angles of incidence and over the p- Fig. 4(b)]. At such extrema there are often two local
9
and s-components ofthe electric field. In this formu- minimum distances. The squares of these two small-
la the values gT and gR denote weight factors which will est minimum distances are denoted by SA and SB (SA S
SB), arising at two almost equal layer thicknesses, dA
be taken finally equal to 1. If the weight factors are
chosen equal, the contribution of a certain measure- and dB. In such cases the followingequation is applied
ment i to this sum S can be interpreted in the (TR) to give a first-order result for the actual layer thick-
plane as the square of the distance, connecting a point ness:
on the theoretical curve and the measured value at (SA + AS) (Sg - As) - d = dA, (15)
(T xpRxp). To find an extremum (minimum) of the
sum S, the following equations should be solved: (SA+ AS) > (Sg - As) -d' -=dASB+ dBSA
(sA+As)>(sn~s2 d SA+SB (16)
-= -
Oak The value As = S/I, is determined from the error sum S
an2 2
in the previous iteration step.
as 2
gT(Tah- dTi Because each of the I equations (12) is connected to
d
= T(T - Ttxd) one measured set (Te.PRe.P) only, the following itera-
tive method is possible:
(1) Calculate for a certain chosen h 2 the theoretical
+ 2gR(R -RKp) th 1 i sI. (12)
curve in the (TR) plane, using, for example, steps of 1
th
ex d'2 = ,
nm increase for the layer thickness. Those values are
If the following vectors are defined: denoted as (Tth,Rth).
(2) For the chosen f 2 and a certain i, all distances
Pi =-(Ti - Rth p) from the theoretical values to the experimental point
(13) are calculated by means of
1• sj'= (Tj - Tp)
2
-ReR) 2.
+ (Rjth (17)
1 (T'h . Rlthd'
i d adth Ad2) lSiSI
=)
a asi2 2e The two smallest distances are selected, called SA and
and assuming, for example, g9T
= R= 1, Eq. (12) can be SB (SA < SB). If, after applying Eqs. (15) and (16), Eq.
1 December 1989 / Vol. 28, No. 23 / APPLIEDOPTICS 5099

6. (15) is selected, a more accurate value can be obtained: substrate position for the amorphous and crystalline
In the triangle with base from (TItj',RIjl) to (Tih j,Rijtj) states. It only serves as an illustration of the layer
and top (Ti R the height (/s) and the intersec- thickness distributions prepared. To eliminate ad-
tion of the perpendicular with the theoretical line justment errors, only a shift of <0.02 mm of the com-
(=d') are calculated. If Eq. (16) is selected, sii is plete curve for the crystalline state is needed to give
calculated by substituting the obtained d' in Eq. (17). coincidence of the peaks and valleys of both states.
This procedure is repeated for all experimental points, Finally, in Figs. 5(d) and 6(d) the crystalline layer
1 < i < I. All first results for the layer thicknesses are thickness obtained is plotted as a function of the initial
obtained and the sum, S = sii, is calculated. amorphous layer thickness. The values derived from
(3) By first making a calculation with different k2 at these measurements are collected in Table I.
constant n 2 and then by taking different n 2 at constant Because,.when measuring the reflectance and trans-
k 2, this procedure can be repeated a few times. Each mittance repeatedly at a certain layer thickness, the
time an interval with the value of n 2 or k 2 of the last standard deviations equal AT- AlR 5 X 10-3 (crexp -
minimum sum is taken in the middle. Also at this 7 X 10-3), we did not apply a stack model. Therefore,
stage the random search method4 can be applied. it can be concluded that the layers can be regarded as
-
(4) Finally the standard deviation, as= ISminimum/I homogeneous, except at layer thicknesses d2 5 15nm.
th
can be used to decide if it is necessary to use a stack For the accuracy of the layer thicknesses obtained,
model to represent the single thin film. Of course, the following analytical expression is illustrative. By
when applying a stack model (assuming that the layer taking a measurement error into account, denoted as
is not homogeneous), the previously obtained a, should (ATAR), implicit differentiation of Eq. (12) (withgT=
be greater than the standard deviation exp of the mea- gR = 1) gives
sured reflectance and transmittance obtained from a T 2 2S 8~~2
repeated measurement at a certain layer thickness. th ATi (
A-7-
22
+
This direct comparison between these standard devi- d2
(Adi) 2 = 2
° (aexp)*
ations is also a reason to prefer this straightforward 'd 2
[aTh(haR 22
h2 { th
th (20)
21
numerical calculation method rather than a mixed
analytical and numerical method. Lad,I ad'
Up to now it is assumed that the index of refraction If the partial derivatives are small, for example, at d 2 -
of the substrate n is known. However, if the sum Xo/(4n ), the error is large. An example of Eq. (20),
2
I normalized to d 2, and taking AT = AR = AT, is
S = gT 3 [T'h(nl,n ,k ,di)
2 2 - Tp] illustrated in Fig. 7 for the values obtained in Fig. 5.
From this figure or from Eq. (20) we conclude that it is
i=1
important to prepare also layer thicknesses differing
+ gR [R-(nn2,k2,d)-Rexp]2 from -N o/(4n ), with N an integer. At the layer
2
E (18)
thicknesses - NXO/(4n) the accuracy is small, because
2
the partial derivatives R/0d2 and T/d 2 are small or
is defined and if it is also required that even vanish. Because those values are not known
beforehand, it is preferred to prepare a large number of
as layer thicknesses. At layer thicknesses -(1 + 2N)Xo/
= O. (19)
(8n2 ) the partial derivatives and the accuracy are high-
the substrate value can also be calculated. If the sub- er, while with thick layers the accuracy and the partial
strate is slightly absorbing, the substrate values can be derivatives become smaller, see Eq. (20), and Figs. 5(b)
obtained in the same way. In Eq. (9), for example, and 6(b). In addition, with thick layers the accuracy
instead of To,, the value To, exp(-47rk 1 d1 /Xo) is substi- of the layer thickness becomes lower at higher k (see
tuted. Instead of k and d, only the product k1d, can Fig. 7).
be obtained in this case. The accuracy of the index of refraction obtained can
be expressed by means of equivalent expressions. Im-
V. Results for GaSb and InSb plicit differentiation of Eq. (11) results in
In Figs. 5 and 6 the measurements of the GaSb and ' taTtih AT i
2
InSb samples are shown in detail. In Figs. 5(a) and A E E-ARi)
6(a), the measured results are shown (points) in the A
k-.12
N -
-
1 a2
if-1
[I
I
'(Th)2 1(R)22
co
(exp) ) 2,
(T,R) plane as already discussed for a-GaSb. In the
case of c-InSb, at layer thicknesses d2 < 15 nm, the
crystallized layer clusters into islands on the substrate,
(2
as was observed with a microscope. Therefore, for the
calculation of 2, c-InSb values obtained at T > 0.45
are disregarded. In Figs. 5(b) and 6(b) the measured ±A(nih
ATi) + E (a A)
/((aexp)
P2
2
transmittance and reflectance are shown in the T(d) (k 2) =
and R(d) planes, as a function of the calculated layer
- I
taT
i 2
th I (a ' 22 - #k )
thicknesses d. Also shown in Figs. 5(c) and 6(c) are Y
i=1
the calculated layer thicknesses as a function of the (22)
5100 APPLIEDOPTICS / Vol. 28, No. 23 / 1 December 1989

7. a b
toes
I0
I
150 300
d(nm)
d
200 I I I
GaSb
C
200 150
GaSb
f l~~~~~~~~~~ :
150
Ca 0 100
100
'i'
50 ,1
50 - r -.
I c /I ' ._
, I' , I I I I '
U zv ouv
11
in
0 50 100 150 200
z/dz da(nm)-
Fig.5. Results for a-GaSb and c-GaSb, N.A.be, = 0.2, XO 820 nm. (a) Least-squares fit (solid lines) in the (T,R) plane yields na = 4.88(1)
=
-1.466(+2)i, withnl = 1.523(±2)and i, = 4.23(+1)-0.510(+5)i, with n1 = 1.523(+2). (b) Measured reflectance and transmittance (points)
as a function of the calculated layer thicknesses obtained from the least-squares fit, asa = 5 X 10-3 and aSc 9 x
= 10-3, respectively. (c) Calcu-
lated layer thicknesses as a function of the substrate position, dz = 0.15 mm. (d) Correlation of the amorphous and crystalline layer
thicknesses obtained from the least-squares fit with a correlation coefficient of c = 0.998. It results in d, = 0.999(+2)da with rdd 5.41 nm.
=
This indicates the same layer thickness after crystallization.
Errors obtained corresponding to the last significant
digit are given in parentheses in Table I and in the
captions of Figs. 5 and 6.
The calculated layer thicknesses d2 are now denoted
as da and dc and refer to the amorphous and crystalline
state of medium 2. With a least-squares method the
function dc = ada is fitted. The following abbrevia-
(At)2
d
'c =3 ( -~ Ad' + /-d -Ad'
Sa
dz
O
-
W i
i=3
0
2 2
[(di - 2ad)2 (Ada) + (da)2(Ad ]
,
( a )
(25)
tions are introduced: For GaSb the values are a = 0.999(G2), Ud = 5.41 nm,
I I I and a correlation coefficient of c = 0.998. Almost no
Saa = 3(di) ,
2
Sac = 3 d, =
2
(di) , (23) significant difference in layer thickness is observed.
jil j~1 i=1 Of course, the complex indices of refraction do differ
(see Table I). This result also gives an idea of the
to obtain expressions for slope a, correlation coeffi- accuracy of the method, if the obtained deviation is
cient c, standard deviation Urd:
explained completely within the accuracy in the deter-
Sac Sac
mination of layer thicknesses. For InSb the result
a nt
= fo
e i : I ad = V I- I (24)
indicates a small relative increase of the layer thick-
ness: a = 1.015(+5), Ud = 6.76 nm, with a correlation
and for the errorin af: coefficient of c = 0.995.
1 December 1989 / Vol. 28, No. 23 / APPLIEDOPTICS 5101

8. a b
I-
300
d(nm) -
d
C 200 I , I . I
InSb
kiSb
150
150
t I,
100
100
-
-
.. ..
50 50
' ; 0c 00 150 20(
) ' s
50 - 0 10 10 0
0 "~. 50 100 1 0 20
)O _0
zldz - Wm) -)
Fig.6. Results for a-InSb and c-InSb, N.A.b,a, = 0.2, Ao 820 nm. (a) Least-squares fit (solid lines) in the (TR) plane
= yields ha = 4.82(:1)-
1.95(+1)i, with hl = 1.523(+2)and h, = 4.07(1)-0.755(+5)i, with hn = 1.526(+2). (b) Measured reflectance and transmittance
(points) as a
function of the calculated layer thicknesses obtained from the least-squares fit, asa = 1.9 X 10-2 and as = 1.3 X 10-2, respectively. (c)
Calculated layer thicknesses as a function of the substrate position, dz = 0.15 mm. (d) Correlation of the amorphous
and crystalline layer
thicknesses obtained from the least-squares fit (c = 0.995). It results in d, = 1.015(45)dawith ad= 6.76nm. This indicates
a small relative
layer thickness increase after the crystallization.
Different densities of amorphous and crystalline -4% (Ref. 6) occurs. For the difference in volume of a-
materials using different preparation and annealing Si with respect to c-Si, a volume increase of 5-10%
methods have been previously observed in other sys- (Ref. 7) is reported after ion bombardment of c-Si.
tems as well. In an experiment on the laser-induced For the III-V compounds investigated (InSb, InAs,
amorphous-to-crystalline transformation in a flash- InP, GaSb, GaAs,and GaP), a lower density is reported
evaporated thin film of a-InSb, an increase of 12% of for the flash-evaporated thin films with respect to the
the layer thickness has been observed.5 For the tran- bulk crystalline materials (Ref. 10, p. 515). It should
sition of an a-Te alloy to a c-Te alloy, by means of low be noted that in Refs. 7 and 10 the density of the
temperature annealing, a layer thickness decrease of amorphous thin film is compared with the density of
the bulk crystalline material, while in Refs. 5 and 6 and
Table I. Results Obtained for Magnetron Sputtered Amorphousand
our data it is compared with the crystalline state of the
Furnace Annealed CrystallineGaSb and InSb Films, A = 820 nm, thin film. These crystalline and amorphous states
15 nm d 170 nm may differ due to voids, substrate-induced stresses,
different compositions, etc.
GaSb InSb A comparison of the index of refraction obtained
0
Ttr (at 201C/min) 289 C 210 C 0 from literature data is rather difficult. Only a few
h1 4.88(+1) - 1.466(+2)i 4.82() - 1.95(+1)i values have been reported and such values also depend
asa 5 X 10-3 1.9 X 10-2 on the preparation method. For the amorphous state
hc 4.23(11) - 0.510(+5)i 4.07(11) - 0.755(45)i
Gsc 9 X 10-3 1.3 X 10-2 of flash-evaporated GaSb and InSb, values Of a = 5.04
c 0.998 0.995 - il.60 and a = 4.91 - i2.10, respectively, are report-
= dc/da 0.999(+2) 1.015(:5) ed' 0 at Xo= 820 nm. In both cases our values from
CR -0.707 -0.543 sputtered samples are a few percent lower indicating,
dRC 97.5 nm 101.4nm
dm 97.6 nm 101.5 nm
for example, a smaller density for sputtered material
compared to flash evaporation, or a different composi-
5102 APPLIEDOPTICS / Vol. 28, No. 23 / 1 December 1989

9. a0.1
ia
100
d(nm)
and hc =
Fig. 7. Accuracy of the calculated layer thickness, using for the indices of refraction, i" = 4.88- il.466
4.23-iO.510, while AT = AR = 5 X 10-. Note that at layer thicknesses -NXo/(4n 2 ) (N is an integer) and at in-
small.
creasing layer thicknesses the accuracy is small because the partial derivatives as given in Eq. (20) become
tion, etc. For the crystalline state a comparison with coefficients To = (P, + ra)/2 and i: = ( - Ta)/4. The
the crystalline bulk material can be made: n, = 4.388 disk is assumed to move in the positive x direction with
- iO.344 for GaSb (Ref. 11) and h, = 4.418 - iO.643 for a speed v. The total intensity on the detector Idet in
InSb." In these cases again a lower n is obtained, but the reflectance arm is, due to the interfering orders,
a higher k. Also in Refs. 1 and 5 some data of the index explained in detail in Ref. 14, Chap. 2:
of refraction of magnetron-sputtered GaSb and flash- Idet
2
Dinc[ ?012 2MTF(M)IY'j
+
evaporated InSb are reported; however, these are mea-
13
sured at Xo = 780 nm (Ref. 12) and Xo = 800 nm, 0i
+ 2MTF(6)(W + FiF,1)cos(2irvt/p)I
respectively. + 7rD?,,[2MTF(26)1i'l
2
cos(47rvt/p)]. (28)
A quantity often used is the contrast function (as a
function of sputter time or layer thickness), defined in In this equation T*denotes the complex conjugate of P,
reflection as and the abbreviation MTF(b) denotes the modulation
CR=R+ R. (26) transfer function. The modulation transfer function
equals the relative overlapping region in the pupil, of
This function can be obtained from the measurements the first and the zeroth order, and can be given as a
directly without calculating the index of refraction or function of the reduced spatial frequency 6 = X/(p
layer thicknesses. The maximum contrast ICRI and N.A.),14
the corresponding layer thickness d can be obtained MTF(b) = -
2
arccos(6/2) - /7r1 - 62/4, 0 < 6 < 2,
from Figs. 5(b) and 6(b) (see Table I). The direct
(29)
measurement of the contrast function gives a first MTF(b) = 0, > 2.
impression of the optimum layer thickness wanted in
optical recording. The modulation function m, defined as the relative
For optical recording, however, the modulation intensity variation on the detector, Idet 1 + m
function 4 should be calculated to obtain the optimum cos(27rvt/p), neglecting the second harmonic, becomes
layer thickness. This requires the knowledge of the
) roFr=+ror± 2MTF(a) ror, + ror,
index of refraction and of the layer thickness. In m = 2MTF(
m1F + 2MTF(5)17pI
2
2
1 o12
addition, the geometry of the marks on the disk should
also be known. The analyses of the modulation func- (30)
tion are generally complex. Therefore, only one spe- 2
To obtain the second expression it is assumed that I 1d
cific disk structure will be considered. Suppose that >> 2MTF(b)rpJ2. Only then is the optimum layer
the amplitude reflection function of the disk can be thickness independent of the reduced spatial frequen-
described by means of cy of the marks. After substitution of Po and T± as
P= 12(C + P) + (PC
2 - P.) cos(2lrx/p) given in Eq. (27), the final result is, in this specific
= O+ P, [exp(2irix/p) + exp(-27rix/p)]. (27) example, given by
m 2MTF(6) (1,-R ) (31)
In this equation T, = /Rcexp(ic) and Ta= Raexp(ika) 0
Rc +Ra+ 2 Rcos(4,- 0 )
denote the crystalline and amorphous amplitude re-
flection values of the disk, and p denotes the periodici- This modulation function shows a similar behavior to
ty of the marks. In the second expression the disk the contrast function defined above. Due to the same
reflection function is given as a Fourier series with amount of amorphous and crystalline material on the
1 December 1989 / Vol. 28, No. 23 / APPLIEDOPTICS 5103

10. disk, the modulation function becomes zero at Ra = R, References
even while the phases rbaand 4c are not equal. The
contrast function and this modulation function be-
come zero at the same layer thicknesses, while in be-
tween extremum values are reached. Therefore, the 1. D. J. Gravesteijn, H. F. J. J. van Tongeren, M. M. Sens, T. C. J.
layer thickness at the extremum of the contrast func- M. Bertens, and C. J. van de Poel, "Phase-Change Optical Data
tion can be used as a first order result for the layer Storage in GaSb," Appl. Opt. 26,4772-4776 (1988).
thickness at the extremum of the modulation function. 2. J. C. Manifacier, J. Gasiot and J. P. Fillard, "A Simple Method
for the Determination of the Optical Constants n, k and the
The optimum layer thicknesses for GaSb and InSb, dR, thickness of a Weakly Absorbing Thin Film," J. Phys. E. 9,
as determined from the modulation function Eq. (31), 1002-1004 (1976).
are given in Table I as well. As expected, these values 3. W. E. Case, "Algebraic Method for Extracting Thin-Film Opti-
d' almost equal the values of d, as determined from cal Parameters from Spectrophotometer Measurements," Appl.
the contrast function. Opt. 22, 1832-1836 (1983).
4. M. Chang and U. J. Gibson, "Optical Constant Determinations
VI. Conclusions of Thin Films by a Random Search Method," Appl. Opt. 24,504-
507 (1985).
The measuring method described makes it possible 5. C. J. van de Poel, "Rapid Crystallization of Thin Solid Films," J.
to obtain the optical properties of optical disks quickly Mater. Res. 3, 126-132 (1988).
and straightforwardly. For automatic reflectance and 6. L. Vriens and W. Rippens, "Optical Constants of Absorbing
transmittance measurements, a single sample mini- Thin Solid Films on a Substrate," Appl. Opt. 22, 4105-4110
mizes the number of alignments and neutralizes the (1983).
calibration errors making a stepwise layer distribution 7. H. Hora, "Stresses in Silicon Crystals from Ion-Implanted
preferred. If a certain state should also be compared Amorphous Regions," Appl. Phys. A 32, 217-221 (1983).
with another state, the layer distribution should con- 8. M. Born and E. Wolf, Principles of Optics (Pergamon, New
tain a maximum or minimum to reduce the lateral York, 1980).
9. M. Mansuripur, "Distribution of Light at and Near the Focus of
adjustment error. With sputtering or flash evapora- High-Numerical-Aperture Objectives," J. Opt. Soc. Am. A 3,
tion, such samples are easily prepared. The procedure 2086-2093 (1986).
described allows the measurement of a large number of 10. J. Stuke and G. Zimmer, "Optical Properties of Amorphous 3-5
layer thicknesses and therefore the contrast function Compounds," Phys. Status Solidi B 49, 513-523 (1972).
can be measured directly. A first-order impression of 11. D. E. Aspnes and A. A. Studna, "Dielectric Functions and Opti-
the optimum layer thickness (or sputter time) can be cal Parameters of Si, Ge, GaP, GaAs,GaSb, InP, InAs, and InSb
easily obtained. It can be concluded in the range of from 1.5 to 6.0 eV," Phys. Rev. B 27, 985-1009 (1983).
selected layer thicknesses whether the layers can be 12. The value reported in Ref. 1 for ,, of a-GaSb at Xo= 780 nm
considered as homogeneous or if a stack model is need- equals 4.6 - 1.2i instead of the printed value 4.6 - 0.2i: a
ed to represent the thin film. Once the index of refrac- printing error.
13. Private communications to C. J. van de Poel, see Ref. 5.
tion is known, the setup can be used to measure layer 14. G. Bouwhuis,J. Braat, A.Huijser, J. Pasman, G. van Rosmalen,
thicknesses of samples prepared in the same condi- and K. Schouhamer Immink, Principles of Optical Disk Sys-
tions. tems (Hilger, Bristol, 1986).
We would like to thank P. van de Werf for preparing
the samples by magnetron sputtering.
5104 APPLIEDOPTICS / Vol. 28, No. 23 / 1 December 1989