Optical measurement of the refractive index, layerthickness, 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 complex indexof refraction, the layer thickness, and induced volume changes of thin layers (due to a phase change, forexample). The principle of the calculation is fitting a curve in the reflectance-transmittance plane measuredon 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, inwhich a laser-induced amorphous-to-crystalline transition can be used in optical recording. The informationessential 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 calculatedunambiguously.

1. Introduction

For erasable and write-once optical recording, im-portant disk media properties are: the complex indexof refraction, the layer thickness, the thermal conduc-tivity, the heat capacity, and the stability of the thinfilm materials used. In general, a high reflectance, ahigh contrast between marks and unwritten areas (or,more precisely, a high modulation), and a reasonableabsorption in the active layer to obtain a low thresholdpower are sought.1 In the process of developing newmaterials for optical recording, the empirical adjust-ment of the desired properties is time consuming. Of-ten many disks with different layer thicknesses andcompositions are tested to obtain the optimum config-uration. In this paper a straightforward method isdiscussed to obtain the optical properties needed foroptical recording application. Reflectance R, trans-mittance T, and absorption A = 1 - R - T can beobtained by means of explicit relations if all indices ofrefraction, il = n- ikl, and layer thicknesses, dl, of themedia used are known [media are denoted by means ofan index 1 = 0 (air), 1, 2,. . .]. These properties have tobe known generally only at the wavelength Xo of thesemiconductor laser used.

The authors are with Philips Research Laboratories, P.O. Box80.000, 5600 JA Eindhoven, The Netherlands.

Received 13 January 1989.0003-6935/89/235095-10$02.00/0.© 1989 Optical Society of America.

The opposite case is not as straightforward: From ameasurement of the reflectance and transmittance of atransparent substrate (medium 1) supporting a thinabsorbing layer (medium 2), two relations should besolved to obtain the unknown complex index of refrac-tion of the layer.

For a nonabsorbing film, the real index of refractionand the layer thickness can be obtained by means ofexplicit relations.2 The calculated layer thickness is,however, multivalued with steps of Xo/(2n2)- Usually,the layer thickness is measured independently, forexample, by means of a mechanical profilometer, themethod of Tolansky, or with Rutherford backscatter-ing (at known densities). The calculated multivaluedlayer thickness is only then determined unambiguous-ly. The ambiguity in the layer thickness can also beavoided if spectrophotometric measurements aremade over a range of wavelengths. 3

For an absorbing film with known layer thickness d2 ,it is not possible to obtain explicit relations for n2 andk2 as functions of the measured properties.24 Twoequations with two unknowns of the form

Rth(n 2,k 2) - Rexp = 0, Tth(n 2,k2) - Texp = 0, (1)

2 = n2- ik2

can be solved numerically, to obtain the root (or roots)of these equations. In these formulas (Tth,Rth) denotethe theoretical expressions and (TexpRexp) denote theexperimental values for a certain layer thickness.This procedure can be repeated for 1 i < I filmthicknesses to obtain the common root of Eq. (1) and toreduce the error. The final index of refraction can beobtained by averaging such results. Because of errorsin the measurement of the film thicknesses, reflec-tance, and transmittance, or due to poorly chosen layer

1 December 1989 / Vol. 28, No. 23 / APPLIED OPTICS 5095

thicknesses in the range of interest, often a maximumrandom difference n2/n 2 = 0.1 and k2 /k 2 = 0.1 isobserved.

Instead of the above-mentioned methods we willpresent a straightforward (automated) measurementand 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 thecontrast function, all as a function of sputter time, forexample. On the other hand, the complex refractiveindex, the layer thicknesses, and volume changes (aftera phase change) can be calculated afterward unambig-uously. It is not necessary at all to measure the layerthicknesses. Before discussing the measurement andcalculation method in detail in the next sections, anexample of a measurement and a calculation will begiven first.

Part of the results of a typical measurement (at afixed wavelength of X0 = 820 nm) is given in Fig. 1(a)and the same data of the measurement are given in Fig.1(b). In the (T,R) plane the measured reflectance,Rp,, is plotted (points) as a function of the corre-sponding measured T', at 256 positions i of a thin filmof a-GaSb deposited on a transparent substrate. In-stead of a constant film thickness a layer thicknessdistribution has been prepared. In both Figs. 1(a) and(b) the arrow denotes the direction of increasing layerthickness (the latter, up to now unknown), while thesolid line represents calculated curves to be discussedbelow.

If a nonabsorbing medium were measured (k2 = 0), astraight line, R = 1 - Tp, would be observed forincreasing layer thickness, while the data run alongthis line up and down. From the minimum transmit-tance or maximum reflectance, the real part of theindex of refraction could be obtained directly. If k2 isgreater than zero, the transmittance finally becomeszero for large layer thicknesses and the reflectancereaches its bulk value. The curve shows fewer visibleoscillations if k2 increases.

In both Figs. 1(a) and (b) the same curve obtained bythe numerical calculation method is shown in the mid-dle (solid line: 2 = 4.88 - il.466). In Fig. 1(a)calculations are also plotted where the value of k2 iskept constant and n2 is changed by ±10% (outer solidlines). In Fig. 1(b) the value of n2 is kept constantwhile k2 is changed by 10% (outer solid lines). Thisexample shows that at constant k2 the flank at F re-mains almost fixed while at constant n2 the height atthe extreme at E remains almost fixed. Using thisobservation, an approximate value for the index ofrefraction can be obtained quickly by trial and errorcalculating a theoretical curve for an assumed complexindex of refraction and comparing it to the measureddata (optimize k2 first and then optimize n2). Thisvalue of the complex index of refraction can be used asa start value in the numerical method and once it isknown, the layer thicknesses can also be calculated.As shown, the value of the complex index of refractioncan be determined without knowing the layer thick-

0.7

0.6

0.5

0 0.4I:

0.3

0.2

0.1

a

b

Fig. 1. Measured reflectance as a function of the measured trans-mittance (points) at certain positions of the prepared sample a-GaSb,N.A. = 0.2, Xo = 820 nm. Three theoretical curves (solid lines)are plotted with, in the middle, the value obtained from the least-squares method, h2 = 4.88 - il.466, and for (a) n2 = +0.1n2 and (b)bk2= 0.1k2. Notethatn 2 determines the height of the extremumEand k2 determines the position of the flank F. The arrow denotes anincreasing layer thickness along the curve. The measured reflec-tance and transmittance as a function of the calculated layer thick-

nesses are shown in Fig. 5(b)

nesses at all. Note that at a thin film thickness equalto zero, when representing a measurement on a trans-parent nonabsorbing substrate, the curves in Fig. (1)start at (T,R) (0.92,0.04), instead of at the values(0.92,0.08). In general, the reflectance is 0.04 lowerwhen a focused beam is used (as in optical recording)instead of a parallel beam to measure the transmit-tance and reflectance.

In Sec. II a quick preparation method of the samplesand a measurement method is discussed first. Expres-sions for the reflectance and transmittance of a stack ofthin films on a thick transparent substrate depend onthe geometry of the beam of light used (focused/paral-lel beam, collecting optics, substrate-incident/air-inci-dent light) and are discussed in Sec. III. In Sec. IVtheoretical expressions are given to explain the nu-merical calculation method used. In Sec. V the resultsof this general method when applied to the amorphousand the crystalline state of the phase change materialsGaSb and InSb are discussed.

II. Experimental

The samples were prepared by means of magnetronsputtering. A shield with a small slit of 2 mm X 50 mm

5096 APPLIED OPTICS / Vol. 28, No. 23 / 1 December 1989

was placed above the sputter target (100 mm in diame-ter) with the long side of the slit in the radial directionof the target. A 1.2-mm thick clean transparent glasssubstrate (B270 glass from Schott), was displacedstepwise behind the slit during sputtering. Duringthese displacements the sputtering conditions weremaintained, while the time between successive dis-placements was increased. The final sample con-tained stripes with variable layer thicknesses d2.

The reflectance and transmittance were measuredon home-built equipment.5 The relevant part of theexperimental setup is shown in Fig. 2 and is discussedin more detail in Sec. III. The sample displacement,parallel to the focus plane, and the (TR) measure-ments have been automated. The transmittance andreflectance arm were calibrated by means of a set ofsamples with a known transmissivity and reflectivity(air, and 200 nm thick Au and Al layers on glass sub-strates measured air incident) and by making use of anadditional detector to normalize all detector currents.A part of the incident light beam was directed towardthis detector by means of a neutral beam splitter. Alsothe focusing on the glass-stack interfaces was auto-mated by means of a standard CD focus motor andFoucault knife-edge method.5 Measurements onsamples of GaSb and InSb were made with light inci-dent from the substrate side.

The measurement strategy for the GaSb sample wasas follows: First, two equal parts were prepared bycutting the original sample perpendicular to the de-posited stripes. Second, one part of the original sam-ple was heated in an oven to obtain the crystallinestate. During heating, the transmittance of one stripewas monitored to determine the temperature of theamorphous-to-crystalline transition (GaSb: Ttr =2890C, InSb: Ttr = 21000). In addition, to preventoxidation during the heating process, argon gas waspassed through the oven. The applied heating ratewas 200C/min.

Finally, the reflectance and transmittance of theamorphous and crystalline parts were measured atequidistant positions. The measurements at thoseparts were taken along a line parallel to the cut, at adistance of 2 mm from the cut. One may also expectlayer thickness variations due to inhomogeneous sput-tering. The procedure in the case of the InSb samplehas been changed to avoid this minor problem. The a-InSb sample was first measured at equidistant posi-tions along a line perpendicular to the depositedstripes, then it was converted to the crystalline stateand measured along the same line again.

The above preparation and measurement methodhas the following advantages:

(1) Instead of preparing I samples with I differentconstant layer thicknesses, only one sample using thesame sputter conditions is prepared with a layer thick-ness distribution in the range of interest. When mea-suring the layer thicknesses and refractive indices inboth phases it is important to have a single samplebecause alignment errors in the reflectance-transmit-tance equipment are neutralized.

a L | W a

Reflectivity arm Transmissivity arm

Fig. 2. Part of the experimental setup used for measuring thereflectance and transmittance of substrate-incident light. Depend-ing on the positions and distances used, the light reflected from theair-glass interface is collected partially or completely at the detec-

tor.(2) This procedure allows a large number of layer

thicknesses to be measured efficiently, reducing er-rors.

(3) There is no need to measure the layer thick-nesses independently, because they are obtained un-ambiguously from the calculation, as will be shown inSec. IV.

(4) For lateral reference it is obviously useful tohave some sharp features (a peak, valley, or step) in thelayer thickness distribution. In this way lateral ad-justment errors are eliminated and it becomes possibleto compare the layer thicknesses of the amorphous andcrystalline states afterward. A relative change in thelayer thickness may be expected if a layer is convertedfrom the amorphous to the crystalline state.57

(5) By making use of a focused beam, measurementsobtained can be used directly in evaluating optical diskproperties.

111. Theory: The Calculation of Reflectance and

Transmittance

In Fig. 3, three situations of interest are shown.Figure 3(a) demonstrates that at an interface, the Fres-nel equations and Snell's law must be fulfilled (Ref. 8,pp. 40 and 615). For the p- and s-components of theelectric field vector E, the Fresnel coefficients T,t are

-9sl l cosl - 12 COS2

=E+ 11 coSnl + 12 coSn2(2)

Ep 12 coSn1 - hl coS 2

Tp12 l= 112 coSn1 + h1 coSn2

s 2nk cosl1 h1 conS1 + 12 coSn2

(3)

Ep2 2h1 cos41=pp2 1h 2 coSn1 + hl CSn 2

nO sint0 0 = 11h sinl = 2 sin02 = * - - - (4)

The plus and minus indices refer to the direction ofpropagation of the plane waves, the tilde denotes acomplex quantity, the media are represented by meansof indices 0,1,2, ... , and 00 represents the angle ofincidence.

1 December 1989 / Vol. 28, No. 23 / APPLIED OPTICS 5097

Iz - *t

For a thin layer positioned between two half-infinitelayers, see Fig. 3(b), the next equations are valid for thep- as well as the s-components of the electric fieldvector (see Ref. 8, pp. 62 and 628):

- r12 + 23 exp(-2it)1 + 12r23 exp(-2ig)

-3 = t12t23exp(-it)1 + 12P23 exp(-2iX)

27rd2 -£x= 119 conk2 .

In this equation, x denotes a phase delay of interferingplane waves. Stacks of thin layers give analogousexpressions, which can be obtained easily by substitut-ing for r23 and 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 beobtained from (Ref. 8, pp. 41 and 630)

R = Iiirst/Iifrst = I?12,

las 2irs = | (n cosk)1 ,t (6)irt (n COSk)frst

valid for the intensities I of the p- and s-components ofthe electric field vector. In these equations the indicesfirst and last refer to the first and last dielectric medi-um.

Finally, care should be taken using the above equa-tions in the case of a stack of thin layers on a thicksubstrate. A thick substrate (and thick protectionlayer) should be treated separately. In Fig. 2 a rele-vant part of our experimental setup is shown. For athin layer (medium 2) or a stack of thin layers on athick transparent substrate (medium 1), see Fig. 3(c).The appropriate equations to be applied depend on thenumerical apertures N.A. and focal lengths f of theobjective and collector used and on the positions andthe diameters Ddet of the detectors. Using a detectorat a distance a from the objective or collector, the beamsize of the substrate-incident light (beam diameterDinc = 2f N.A.beam), reflected once at the air-glassinterface, in the geometric approximation (see Fig. 2)becomes

Do = Dinc a ( _1 +1 (7)

if the beam is focused on the interfaces of the thinstack. In this equation the positive sign should beused for the beam directly reflected at the air-glassinterface toward the reflectance detector arm, whilethe negative sign should be used for the beam indirect-ly reflected toward the transmittance detector arm.

For the situation, Do < Ddet, almost all light reflect-ed at the air-glass interface is collected at the detector.At a high coherence length of the light beam, fringesarise due to the interference of the spherical wavesreflected at the air-substrate interface and the mainwave reflected at the glass-stack-air position, all witha different origin. For many fringes the integral of thecross-term of the electric field vectors over the detector

a

EE.

E 2

b

2 3

C

T13

0 L 3

Fig. 3. Electromagnetic fields at (a) a boundary, (b) a thin layer;arrows denote plane waves, and (c) a thin layer or stack of thin layerson a thick substrate; the values at the boundary of the dashed-linebox represents the calculated transmittance and reflectance bymeans of Eq. (6) of a stack of thin layers or a single thin layer. Now

the influence of the substrate should be evaluated.

area is canceled due to the oscillating phase factor.Therefore, the measured intensity due to the interfer-ence of the spherical waves of the air-glass interfaceand the resulting glass-stack-air wave often results inan addition of the intensities of those waves, even at ahigh coherence length of the light beam used4:

R = R 1 + R1 3 T,1 R 13 R0 1

T= T13T011-R 13 RoIDo, < Ddet- (8)

For investigations with parallel substrate-incidentbeams Eq. (8) should be used.

If Do, >> Ddet, the substrate-incident light reflectedat the air-glass interface collected by the detector isnegligible (as in optical recording). For a pencil oflight the next equation can be applied (we take medi-um 3 equivalent to medium 0):

R 1 3To2, T= T1 3T61, Doi >>Ddet, (9)

while an averaging of all incident angles 00 and p- ands-components is required in the experimental situa-tion of Fig. 2. In the interesting case of incident circu-larly polarized light (as in optical recording in phasechange materials or for the compact disk) of a planewave, the final reflectance and transmittance are givenby R = (Rp + R)/2 and T = (Tp + T)/2. Thesefunctions are, up to a N.A. of 0.5, almost independentof the angle of incidence. Thus the averaging of allincident angles will give almost the same result as at 00= 0 (see, for example, Ref. 8, p. 44).

In the experimental setup the following conditionshave been used (see Fig. 2): a p-polarized beam of asemiconductor laser pen (with beam shaping optics inthe collimator pen) at a wavelength of Xo = 820 nm, anobjective numerical aperture of 0.45 (reflectance arm),a collector numerical aperture of 0.6 (transmittancearm), a = 400 mm, f = 4.5 mm, d = 1.2 mm, n = 1.523,and N.A.beam = 0.2. From Eq. (7) it follows that Do, -55 2 mm >> Ddet = 3 mm. The fraction of theincident light collected by the detector in the reflec-tance arm from the air-glass interface with Rol 0.04is, therefore, only (3/55)2 0.04 104 a negligibleamount. Therefore, in our calculations we use Eq. (9).

As stated, explicit relations for the reflectance andtransmittance are obtained. It is not possible to ob-

5098 APPLIED OPTICS / Vol. 28, No. 23 / 1 December 1989

tain an explicit relation for the index of refraction f 2 asa function of the measured reflectance and transmit-tance in a situation as in Fig. 3(c), because this index ofrefraction arises in both terms of a product composedof exp(-ix) and r23 or t2 3 . It is possible, however, foran absorbing layer studied using plane waves, to re-duce the numerical problem to other attractive analyt-ical expressions.2 The final calculation, however,must still be done numerically, 2 while such a method isnot attractive at all if an averaging over angles ofincidence and polarization states is required. There-fore, a straightforward numerical method is preferred,as discussed in the next section.

IV. Theory: Calculation of the Index of Refraction and

Layer Thicknesses

Consider the measured reflection and transmissionas a function of the (up to now unknown) layer thick-nesses d2. For a least-squares fit [as in Figs. 5(b) and6(b), discussed below] it is required that the next sum,

lS = gT E [Tth(n2,k 2,dl)-

j=1

+ gR E [Rth(n2 ,k2,di)- s (10)

is a minimum. The theoretical values (Tth,Rth) areexplicitly given as a function of the properties n2, k2,

and d2 of the film under study, while the substratevalues are assumed to be obtained from a previousmeasurement. The letter i denotes that the properlayer thickness of measurement i should be substitut-ed. If required, the theoretical expressions include anaveraging over the angles of incidence and over the p-and s-components of the electric field.9 In this formu-la the values gT and gR denote weight factors which willbe taken finally equal to 1. If the weight factors arechosen equal, the contribution of a certain measure-ment i to this sum S can be interpreted in the (TR)plane as the square of the distance, connecting a pointon the theoretical curve and the measured value at(T xpRxp). To find an extremum (minimum) of thesum S, the following equations should be solved:

-= O -an2 ak2

as 2 gT(Tah- i dT= d T(T - Ttxd)

+ 2gR(R -R Kp) th = ,th ex d'2 1 i sI. (12)

If the following vectors are defined:

Pi =-(Ti - Rth p)

(13)

1 (T'h . Rlthd' 1•i d adth Ad2) lSiSIa asi 2 2e =)

and assuming, for example, g9T = R= 1, Eq. (12) can be

b

theoretical curve

during calculation

Fig. 4. Some examples illustrating the geometric relation pi * li = 0in the (TR) plane. (a) Expression corresponds to looking for aglobal minimum distance. (b) Desired solution during the calcula-tion is positioned in the part of the curve between the two smallest

local minimum distances.

interpreted in the (TR) plane by means of a geometricrelation:

(14)

This means that in the (T,R) plane the vector pi, di-rected from the theoretical curve at (Tth,Rth) towardthe measured value (TexpRt ) should be perpendicu-lar to the vector li denoting the increase (OT,6R) alongthe theoretical curve at increasing layer thicknesses.A few examples are illustrated in Fig. 4. Most timesEq. (14) and the condition to look for a minimummeans that the shortest distance from the measuredvalue (TxpnRexp) to the theoretical curve gives thecorrect layer thickness, [see Fig. 4(a)]. That might beexpected intuitively from Eq. (10) (gT = R = 1) be-cause, as noted, squares of distances in the (TR) planeare added, while a minimum sum should be obtained.At extrema, as in Figs. 1(a) and (b) at point E, however(also during the calculation), this is not correct [seeFig. 4(b)]. At such extrema there are often two localminimum distances. The squares of these two small-est minimum distances are denoted by SA and SB (SA S

SB), arising at two almost equal layer thicknesses, dA

and dB. In such cases the following equation is appliedto give a first-order result for the actual layer thick-ness:

(SA + AS) (Sg - As) - d = dA,

(SA + AS) > (Sg - As) -d' =dASB + dBSA(sA+As)>(sn~s2 d - SA +SB

(15)

(16)

The value As = S/I, is determined from the error sum Sin the previous iteration step.

Because each of the I equations (12) is connected toone measured set (Te.PRe.P) only, the following itera-tive method is possible:

(1) Calculate for a certain chosen h2 the theoreticalcurve in the (TR) plane, using, for example, steps of 1nm increase for the layer thickness. Those values aredenoted as (Tth,Rth).

(2) For the chosen f 2 and a certain i, all distancesfrom the theoretical values to the experimental pointare calculated by means of

sj'= (Tj - Tp) 2 + (Rjth -ReR) 2. (17)

The two smallest distances are selected, called SA andSB (SA < SB). If, after applying Eqs. (15) and (16), Eq.

1 December 1989 / Vol. 28, No. 23 / APPLIED OPTICS 5099

a

pi -1 = 0, I < i < L

(15) is selected, a more accurate value can be obtained:In the triangle with base from (TItj',RIjl) to (Tih j,Rijtj)and top (Ti R the height (/s) and the intersec-tion of the perpendicular with the theoretical line(=d') are calculated. If Eq. (16) is selected, sii iscalculated by substituting the obtained d' in Eq. (17).This procedure is repeated for all experimental points,1 < i < I. All first results for the layer thicknesses areobtained and the sum, S = sii, is calculated.

(3) By first making a calculation with different k2 atconstant n2 and then by taking different n2 at constantk2, this procedure can be repeated a few times. Eachtime an interval with the value of n2 or k2 of the lastminimum sum is taken in the middle. Also at thisstage the random search method4 can be applied.

(4) Finally the standard deviation, as = ISminimum/Ican be used to decide if it is necessary to use a stackmodel to represent the single thin film. Of course,when applying a stack model (assuming that the layeris not homogeneous), the previously obtained a, shouldbe greater than the standard deviation exp of the mea-sured reflectance and transmittance obtained from arepeated measurement at a certain layer thickness.This direct comparison between these standard devi-ations is also a reason to prefer this straightforwardnumerical calculation method rather than a mixedanalytical and numerical method.

Up to now it is assumed that the index of refractionof the substrate n is known. However, if the sum

IS = gT 3 [T'h(nl,n2,k2,di) - Tp]

i=1

+ gR E [R-(nn2,k2,d) -Rexp]2 (18)

is defined and if it is also required that

as= O. (19)

the substrate value can also be calculated. If the sub-strate is slightly absorbing, the substrate values can beobtained in the same way. In Eq. (9), for example,instead of To,, the value To, exp(-47rk 1 d1 /Xo) is substi-tuted. Instead of k and d, only the product k1d, canbe obtained in this case.

V. Results for GaSb and InSb

In Figs. 5 and 6 the measurements of the GaSb andInSb samples are shown in detail. In Figs. 5(a) and6(a), the measured results are shown (points) in the(T,R) plane as already discussed for a-GaSb. In thecase of c-InSb, at layer thicknesses d2 < 15 nm, thecrystallized layer clusters into islands on the substrate,as was observed with a microscope. Therefore, for thecalculation of 2, c-InSb values obtained at T > 0.45are disregarded. In Figs. 5(b) and 6(b) the measuredtransmittance and reflectance are shown in the T(d)and R(d) planes, as a function of the calculated layerthicknesses d. Also shown in Figs. 5(c) and 6(c) arethe calculated layer thicknesses as a function of the

substrate position for the amorphous and crystallinestates. It only serves as an illustration of the layerthickness distributions prepared. To eliminate ad-justment errors, only a shift of <0.02 mm of the com-plete curve for the crystalline state is needed to givecoincidence of the peaks and valleys of both states.Finally, in Figs. 5(d) and 6(d) the crystalline layerthickness obtained is plotted as a function of the initialamorphous layer thickness. The values derived fromthese measurements are collected in Table I.

Because,.when measuring the reflectance and trans-mittance repeatedly at a certain layer thickness, thestandard deviations equal AT- AlR 5 X 10-3 (crexp -

7 X 10-3), we did not apply a stack model. Therefore,it can be concluded that the layers can be regarded ashomogeneous, except at layer thicknesses d2 5 15 nm.

For the accuracy of the layer thicknesses obtained,the following analytical expression is illustrative. Bytaking a measurement error into account, denoted as(ATAR), implicit differentiation of Eq. (12) (withgT=gR = 1) gives

(Adi)2=

T 2th ATi\ d2

2 S 8~~2( th \22 + -A-7-

'd h2 { th 22[\aTh2 (haR th \21

Lad,I ad'° (aexp)2 * (20)

If the partial derivatives are small, for example, at d2 -Xo/(4n2), the error is large. An example of Eq. (20),normalized to d2, and taking AT = AR = AT, isillustrated in Fig. 7 for the values obtained in Fig. 5.From this figure or from Eq. (20) we conclude that it isimportant to prepare also layer thicknesses differingfrom -N o/(4n2), with N an integer. At the layerthicknesses - NXO/(4n2) the accuracy is small, becausethe partial derivatives R/0d2 and T/d 2 are small oreven vanish. Because those values are not knownbeforehand, it is preferred to prepare a large number oflayer thicknesses. At layer thicknesses -(1 + 2N)Xo/(8n2) the partial derivatives and the accuracy are high-er, while with thick layers the accuracy and the partialderivatives become smaller, see Eq. (20), and Figs. 5(b)and 6(b). In addition, with thick layers the accuracyof the layer thickness becomes lower at higher k (seeFig. 7).

The accuracy of the index of refraction obtained canbe expressed by means of equivalent expressions. Im-plicit differentiation of Eq. (11) results in

A

k-.12 -

(k 2 )2 =

I E E-ARi)

[I '(Th)2 1 (R)22

±A(nih ATi) + E (a A)

(exp) ) 2,

(2

/((aexp) P2

I ' (a 22 - #k )

5100 APPLIED OPTICS / Vol. 28, No. 23 / 1 December 1989

(22)

2' taTtih AT i1 � �a2

A N - if-1co

- I i 2taT th

Yi=1

b

toesI0

200

C

200GaSb

150

Ca

100

50 - r -.c I /\I ' ._

0 50 100z/dz

f

0

150

100

50

150 200

d

I I I

GaSb

l~~~~~~~~~~ :

'i',1

, I' , I I I I '

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, with nl = 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 significantdigit are given in parentheses in Table I and in thecaptions of Figs. 5 and 6.

The calculated layer thicknesses d2 are now denotedas da and dc and refer to the amorphous and crystallinestate of medium 2. With a least-squares method thefunction dc = ada is fitted. The following abbrevia-tions are introduced:

I I I

Saa = 3 (di)2, Sac = 3 d, = (di)

2, (23)

jil j~1 i= 1

to obtain expressions for slope a, correlation coeffi-cient c, standard deviation Urd:

Sac Saca n = f o t e i : I

and for the error in af:

ad = V I- I (24)

(At)2 =3 -( dz Ad' + -Ad'i ~ O W 0 / -d \ d ' c ,

- [(di - 2ad)2 (Ada)2 + (da)2(Ad 2] Sa i=3 ( a )

(25)

For GaSb the values are a = 0.999(G2), Ud = 5.41 nm,and a correlation coefficient of c = 0.998. Almost nosignificant difference in layer thickness is observed.Of course, the complex indices of refraction do differ(see Table I). This result also gives an idea of theaccuracy of the method, if the obtained deviation isexplained completely within the accuracy in the deter-mination of layer thicknesses. For InSb the resultindicates a small relative increase of the layer thick-ness: a = 1.015(+5), Ud = 6.76 nm, with a correlationcoefficient of c = 0.995.

1 December 1989 / Vol. 28, No. 23 / APPLIED OPTICS 5101

I

150d(nm)

300

a

zv ouv in11

U

a b

I-

C 200

zldz -

t

150

100

50

d(nm) -

d

I , I . IInSb

, I-

... - .

) ' s

0 "~. 50 1 00 1 0 20

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 afunction 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 layerthicknesses obtained from the least-squares fit (c = 0.995). It results in d, = 1.015(45)da with ad = 6.76 nm. This indicates a small relativelayer thickness increase after the crystallization.

300

0

Different densities of amorphous and crystallinematerials using different preparation and annealingmethods have been previously observed in other sys-tems as well. In an experiment on the laser-inducedamorphous-to-crystalline transformation in a flash-evaporated thin film of a-InSb, an increase of 12% ofthe layer thickness has been observed.5 For the tran-sition of an a-Te alloy to a c-Te alloy, by means of lowtemperature annealing, a layer thickness decrease of

Table I. Results Obtained for Magnetron Sputtered Amorphous andFurnace Annealed Crystalline GaSb and InSb Films, A = 820 nm,

15 nm d 170 nm

GaSb InSb

Ttr (at 201C/min) 2890 C 2100Ch1 4.88(+1) - 1.466(+2)i 4.82() - 1.95(+1)iasa 5 X 10-3 1.9 X 10-2hc 4.23(11) - 0.510(+5)i 4.07(11) - 0.755(45)iGsc 9 X 10-3 1.3 X 10-2c 0.998 0.995

= dc/da 0.999(+2) 1.015(:5)CR -0.707 -0.543dRC 97.5 nm 101.4 nmdm 97.6 nm 101.5 nm

-4% (Ref. 6) occurs. For the difference in volume of a-Si with respect to c-Si, a volume increase of 5-10%(Ref. 7) is reported after ion bombardment of c-Si.For the III-V compounds investigated (InSb, InAs,InP, GaSb, GaAs, and GaP), a lower density is reportedfor the flash-evaporated thin films with respect to thebulk crystalline materials (Ref. 10, p. 515). It shouldbe noted that in Refs. 7 and 10 the density of theamorphous thin film is compared with the density ofthe bulk crystalline material, while in Refs. 5 and 6 andour data it is compared with the crystalline state of thethin film. These crystalline and amorphous statesmay differ due to voids, substrate-induced stresses,different compositions, etc.

A comparison of the index of refraction obtainedfrom literature data is rather difficult. Only a fewvalues have been reported and such values also dependon the preparation method. For the amorphous stateof flash-evaporated GaSb and InSb, values Of a = 5.04- il.60 and a = 4.91 - i2.10, respectively, are report-ed' 0 at Xo = 820 nm. In both cases our values fromsputtered samples are a few percent lower indicating,for example, a smaller density for sputtered materialcompared to flash evaporation, or a different composi-

5102 APPLIED OPTICS / Vol. 28, No. 23 / 1 December 1989

kiSb

150

100

50 ' ; 0 c 00 150 20(

50 - 0 10 10 0

)O

Wm) -)

_

a0.1ia

100d(nm)

Fig. 7. Accuracy of the calculated layer thickness, using for the indices of refraction, i" = 4.88- il.466 and hc =

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-

creasing layer thicknesses the accuracy is small because the partial derivatives as given in Eq. (20) become small.

tion, etc. For the crystalline state a comparison withthe crystalline bulk material can be made: n, = 4.388- iO.344 for GaSb (Ref. 11) and h, = 4.418 - iO.643 forInSb." In these cases again a lower n is obtained, buta higher k. Also in Refs. 1 and 5 some data of the indexof refraction of magnetron-sputtered GaSb and flash-evaporated InSb are reported; however, these are mea-sured at Xo = 780 nm (Ref. 12) and Xo = 800 nm,13

respectively.A quantity often used is the contrast function (as a

function of sputter time or layer thickness), defined inreflection as

CR = R+ R. (26)

This function can be obtained from the measurementsdirectly without calculating the index of refraction orlayer thicknesses. The maximum contrast I CRI andthe corresponding layer thickness d can be obtainedfrom Figs. 5(b) and 6(b) (see Table I). The directmeasurement of the contrast function gives a firstimpression of the optimum layer thickness wanted inoptical recording.

For optical recording, however, the modulationfunction 4 should be calculated to obtain the optimumlayer thickness. This requires the knowledge of theindex of refraction and of the layer thickness. Inaddition, the geometry of the marks on the disk shouldalso be known. The analyses of the modulation func-tion are generally complex. Therefore, only one spe-cific disk structure will be considered. Suppose thatthe amplitude reflection function of the disk can bedescribed by means of

P = 12(C + P) + 2(PC - P.) cos(2lrx/p)

= O + P, [exp(2irix/p) + exp(-27rix/p)]. (27)

coefficients To = (P, + ra)/2 and i: = ( - Ta)/4. Thedisk is assumed to move in the positive x direction witha speed v. The total intensity on the detector Idet inthe reflectance arm is, due to the interfering orders,explained in detail in Ref. 14, Chap. 2:

Idet Dinc[ ?012 + 2MTF(M)IY'j2

+ 2MTF(6)(W0 i + FiF,1) cos(2irvt/p)I

+ 7rD?,,[2 MTF(26)1i'l 2 cos(47rvt/p)]. (28)

In this equation T* denotes the complex conjugate of P,and the abbreviation MTF(b) denotes the modulationtransfer function. The modulation transfer functionequals the relative overlapping region in the pupil, ofthe first and the zeroth order, and can be given as afunction of the reduced spatial frequency 6 = X/(pN.A.),14

2MTF(b) = - arccos(6/2) - /7r1 - 62/4, 0 < 6 < 2,

MTF(b) = 0, > 2.(29)

The modulation function m, defined as the relativeintensity variation on the detector, Idet 1 + m

cos(27rvt/p), neglecting the second harmonic, becomes

m = 2MTF( ) roFr= + ror± 2MTF(a) ror, + ror,m1F 2 + 2MTF(5)17pI2 1 o12

(30)

To obtain the second expression it is assumed that I 1d 2>> 2MTF(b)rpJ2. Only then is the optimum layerthickness independent of the reduced spatial frequen-cy of the marks. After substitution of Po and T± asgiven in Eq. (27), the final result is, in this specificexample, given by

In this equation T, = /Rc exp(ic) and Ta = Raexp(ika)denote the crystalline and amorphous amplitude re-flection values of the disk, and p denotes the periodici-ty of the marks. In the second expression the diskreflection function is given as a Fourier series with

(31)m 2MTF(6) (1,-R 0 )Rc + Ra + 2 Rcos(4,- 0 )

This modulation function shows a similar behavior tothe contrast function defined above. Due to the sameamount of amorphous and crystalline material on the

1 December 1989 / Vol. 28, No. 23 / APPLIED OPTICS 5103

disk, the modulation function becomes zero at Ra = R,even while the phases rba and 4c are not equal. Thecontrast function and this modulation function be-come zero at the same layer thicknesses, while in be-tween extremum values are reached. Therefore, thelayer thickness at the extremum of the contrast func-tion can be used as a first order result for the layerthickness at the extremum of the modulation function.The optimum layer thicknesses for GaSb and InSb, dR,as determined from the modulation function Eq. (31),are given in Table I as well. As expected, these valuesd' almost equal the values of d, as determined fromthe contrast function.

VI. Conclusions

The measuring method described makes it possibleto obtain the optical properties of optical disks quicklyand straightforwardly. For automatic reflectance andtransmittance measurements, a single sample mini-mizes the number of alignments and neutralizes thecalibration errors making a stepwise layer distributionpreferred. If a certain state should also be comparedwith another state, the layer distribution should con-tain a maximum or minimum to reduce the lateraladjustment error. With sputtering or flash evapora-tion, such samples are easily prepared. The proceduredescribed allows the measurement of a large number oflayer thicknesses and therefore the contrast functioncan be measured directly. A first-order impression ofthe optimum layer thickness (or sputter time) can beeasily obtained. It can be concluded in the range ofselected layer thicknesses whether the layers can beconsidered as homogeneous or if a stack model is need-ed to represent the thin film. Once the index of refrac-tion is known, the setup can be used to measure layerthicknesses of samples prepared in the same condi-tions.

References

1. D. J. Gravesteijn, H. F. J. J. van Tongeren, M. M. Sens, T. C. J.M. Bertens, and C. J. van de Poel, "Phase-Change Optical DataStorage in GaSb," Appl. Opt. 26,4772-4776 (1988).

2. J. C. Manifacier, J. Gasiot and J. P. Fillard, "A Simple Methodfor the Determination of the Optical Constants n, k and thethickness of a Weakly Absorbing Thin Film," J. Phys. E. 9,1002-1004 (1976).

3. W. E. Case, "Algebraic Method for Extracting Thin-Film Opti-cal Parameters from Spectrophotometer Measurements," Appl.Opt. 22, 1832-1836 (1983).

4. M. Chang and U. J. Gibson, "Optical Constant Determinationsof Thin Films by a Random Search Method," Appl. Opt. 24,504-507 (1985).

5. C. J. van de Poel, "Rapid Crystallization of Thin Solid Films," J.Mater. Res. 3, 126-132 (1988).

6. L. Vriens and W. Rippens, "Optical Constants of AbsorbingThin Solid Films on a Substrate," Appl. Opt. 22, 4105-4110(1983).

7. H. Hora, "Stresses in Silicon Crystals from Ion-ImplantedAmorphous Regions," Appl. Phys. A 32, 217-221 (1983).

8. M. Born and E. Wolf, Principles of Optics (Pergamon, NewYork, 1980).

9. M. Mansuripur, "Distribution of Light at and Near the Focus ofHigh-Numerical-Aperture Objectives," J. Opt. Soc. Am. A 3,2086-2093 (1986).

10. J. Stuke and G. Zimmer, "Optical Properties of Amorphous 3-5Compounds," Phys. Status Solidi B 49, 513-523 (1972).

11. D. E. Aspnes and A. A. Studna, "Dielectric Functions and Opti-cal Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSbfrom 1.5 to 6.0 eV," Phys. Rev. B 27, 985-1009 (1983).

12. The value reported in Ref. 1 for ,, of a-GaSb at Xo = 780 nmequals 4.6 - 1.2i instead of the printed value 4.6 - 0.2i: aprinting error.

13. Private communications to C. J. van de Poel, see Ref. 5.14. G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen,

and K. Schouhamer Immink, Principles of Optical Disk Sys-tems (Hilger, Bristol, 1986).

We would like to thank P. van de Werf for preparingthe samples by magnetron sputtering.

5104 APPLIED OPTICS / Vol. 28, No. 23 / 1 December 1989