optical characterization of dielectric and semiconductor thin films by use of transmission...

Optical characterization of dielectric andsemiconductor thin films by use of transmission data

Jorge I. Cisneros

A method to calculate the optical functions n~l! and k~l! by use of the transmission spectrum of a dielectricor semiconducting thin film measured at normal incidence is described. The spectrum should include thelow-absorption region and the absorption edge to yield the relevant optical characteristics of the material.The formulas are derived from electromagnetic theory with no simplifying assumptions. Transparentfilms are considered as a particular case for which a simple method of calculation is proposed. In thegeneral case of absorbing films the method takes advantage of some properties of the transmittance T~l! topermit the parameters in the two regions mentioned above to be calculated separately. The interferencefringes and the optical path at the extrema of T~l! are exploited for determining with precision the refractiveindex and the film thickness. The absorption coefficient is computed at the absorption edge by an efficientiterative method. At the transition zone between the interference region and the absorption edge artifactsin the absorption curve are avoided. A small amount of absorption of the substrate is allowed for in thetheory by means of a factor determined from an independent measurement, thus improving the quality ofthe results. Application of the method to a transmission spectrum of an a:SixN12x:H film is illustrated indetail. Refractive index, dispersion parameters, film thickness, absorption coefficient, and optical gap aregiven with the help of tables and graphs. © 1998 Optical Society of America

OCIS codes: 310.6860, 160.4760, 160.6000, 160.2750.

1. Introduction

The optical characterization of a thin film by trans-mission and reflection measurements is a well-knownproblem. Rigorous expressions for the transmit-tance T and the reflectance R of a thin uniform filmwith smooth interfaces and of constant thickness de-posited onto a transparent substrate were estab-lished by Heavens.1 Once the experimental valuesof R and T are known, the mathematical problemconsists in the calculation of the real and imaginaryparts, n and k, respectively, of the complex refractiveindex of the film. A difficulty is that the expressionscannot be inverted to permit the direct calculation ofn and k from the experimental values of R and T. Asa consequence many numerical and graphic methodswere developed to facilitate the resolution of theequations.

Another mathematical inconvenience in the reso-

The author is with the Instituto de Fisica “Gleb Wataghin,”Universidade Estadual de Campinas, Caixa Postal, 6165 Codigo deEnderacamento Postal, 13083-970, Campinas, Sao Paulo, Brazil.

Received 7 August 1997; revised manuscript received 25 March1998.

0003-6935y98y225262-09$15.00y0© 1998 Optical Society of America

5262 APPLIED OPTICS y Vol. 37, No. 22 y 1 August 1998

lution of the coupled equations that contain R and Tis the appearance of a multiplicity of solutions n, k fora single pair of experimental values R, T.2–7 Severalcriteria are used for selection of the best solution, butwhen the solutions are too similar the choice is diffi-cult. The use of the optical path instead of the phys-ical thickness, as proposed in Ref. 7, drasticallyreduces the number of multiple solutions, facilitatingthe choice of the best one.

To perform the calculations, one needs to know thevalue of the film thickness one needs to know but anaccurate value of this parameter, compatible with theintended accuracy of the results, is frequently diffi-cult to obtain. Furthermore, to avoid errors owing toinhomogeneity in film thickness, one needs to makean independent measurement of the thickness ex-actly in the same place where the light beam passed.

The simplicity of measuring the transmittance andthe better accuracy of this measurement with respectto the reflectance suggest the convenience of the useof a single transmission spectrum for calculation of nand k. Swanepoel8 followed this idea to develop hismethod.

In this paper I describe a new method of calculationof n, k, and the film thickness of dielectric and semi-conducting thin films deposited onto transparent orsemitransparent substrates by using transmission

tnTstmrf

trsn

fcaniSa

spectra measured at normal incidence. This choice ofincidence direction is justified because no polarizationeffects appear; moreover, most commercial equipmentuses this incidence direction. This method was ap-plied in the laboratories of the Institute of Physics ofthe University of Campinas, Brazil to several kinds ofmaterial such as amorphous silicon and germaniumand related compounds, III–V amorphous semiconduc-tors, polymer films of different compositions, electro-chromic oxides, and a few crystalline semiconductingfilms.9–17

The mathematical procedures used in this studymake use of some properties of the transmittance thatpermit the separate calculation of the optical parame-ters in two regions: the low- and the high-absorptionregions, respectively. The interference fringes of thelow-absorption region are used to calculate the refrac-tive index ~real and imaginary parts! and the filmhickness. In this region the small absorption is noteglected, so the accuracy of these results is improved.he film thickness thus obtained improves the preci-ion of the results compared with those from methodshat use thickness measured by an independentethod. The dispersion of n~l! is determined in this

egion as well, permitting the parameterization of thisunction by means of any of the available models.

The absorption coefficient is computed at the absorp-ion edge by use of the results of the low-absorptionegion. Spurious modulations or peaks usually ob-erved in the absorption coefficient curve are elimi-ated by judicious correction of the refractive index.A theoretical background is included in Section 2 to

acilitate the explanations of the mathematical pro-edures. Transparent films are treated in Section 3s a special case. A simple method for the determi-ation of the refractive index and the film thickness

s developed for this case. In this section, and inection 4 as well, the experimental data of an-SixN12x:H film are used to show the different cal-

culations. Section 4 describes the calculations of theabsorbing film, giving the results of the absorptioncoefficient and the optical gap of our particular ex-ample. Discussions of the accuracy and the applica-bility of the method are contained in Section 5.Some recommendations on the properties of samplesto be characterized are given as well. The appen-dixes contain the formulas for the transmittance andreflectance of both transparent and semi-transparentsubstrates, which are useful for completion of thedata necessary to optimize the results.

2. Theoretical Background

The formulas used in the calculation of the opticalparameters of a thin film deposited onto a thick sub-strate that use experimental transmission spectrashould include the reflections from the second inter-face of the substrate; otherwise the results will beaffected by systematic and significant errors.7,8

The derivation of the expressions for the transmit-tance and reflectance is made in two steps. The firstone gives the well-known transmittance and reflec-tance of the film with smooth and parallel faces

bounded by two semi-infinite media.18 In the secondstep the multiple incoherent reflections from thesubstrate–air interface are included to yield the finalexpressions.

To permit the characterization of both transparentand absorbing films, a complex refractive index of thefilm is used throughout this study. The analysis ofthe optical spectrum of the sample, which usuallyincludes the interference fringe region and the ab-sorption edge, yields the refractive index and the ab-sorption coefficient of the film. The thickness of thethin film is also obtained, frequently with a betterprecision than for the result obtained from an inde-pendent measurement.

For the first step of the calculation ~semi-infinitesubstrate! we consider a film of complex refractiveindex N2 5 n2 1 ik2 and thickness d, bounded by twosemi-infinite transparent media of real refractive in-dices N1 5 n1 and N3 5 n3; see Fig. 1~a!. Usually theincident medium is air with n1 5 1.

The amplitudes of the reflected and the transmit-ted electric fields with respect to the incident field ateach interface ij are given by the correspondingFresnel coefficients:

rij 5Ni 2 Nj

Ni 1 Nj, tij 5

2Ni

Ni 1 Nj. (1)

The refractive indices Nj 5 nj 1 ikj with j 5 1, 2, 3correspond to the incident medium, film, and sub-strate, respectively, according to Fig. 1.

The amplitudes of the electric field of the wavestransmitted and reflected by the film are given by thefollowing coefficients, where both incidence direc-tions, ~123! and ~321!, are considered:

t123 5t12t23 exp~iCy2!

1 1 r12r23 exp~iC!, (2)

t321 5t32t21 exp~iCy2!

1 1 r32r21 exp~iC!, (3)

r123 5r12 1 r23 exp~iC!

1 1 r12r23 exp~iC!, (4)

r321 5r32 1 r21 exp~iC!

1 1 r32r21 exp~iC!, (5)

Fig. 1. Optical parameters and directions of the transmittanceand reflectance adopted for the film–substrate assembly: ~a!semi-infinite substrate, ~b! finite substrate.

1 August 1998 y Vol. 37, No. 22 y APPLIED OPTICS 5263

i

tcscd

r

d

wtfst

A~o

Table 1. Determination of Refractive Index n and Film Thickness d of

5

where the phase difference of the wave between thetwo interfaces Cy2 of the film is defined by

c 5 4pN2 dyl (6)

and l is the vacuum wavelength of the light.The complex angle C is separated into its real and

imaginary parts:

Re~C! 5 4pn2 dyl 5 w, (7a)

Im~C! 5 4pk2 dyl 5 ad. (7b)

a 5 4pk2yl is usually called the absorption coefficientof the film; w is referred to as the phase angle.

The values of the reflectance and the transmittanceof the film, according to electromagnetic theory, aredefined by

R123 5 r123r123*, (8)

R321 5 r321r321*, (9)

T123 5 ~n3yn1!t123t123*. (10)

The transmittances in both directions are equal whenthe substrate is transparent.

The effect of the finite substrate @see Fig. ~1b!#, isntroduced, according to Knittl,19 by means of the

following expressions, which are valid for transpar-ent and weakly absorbing substrates:

R 5R123 1 ~T123

2 2 R123R321!rU2

1 2 rR321U2 , (11)

R9 5~1 2 2r!R321U

2 1 r

1 2 rR321U2 , (12)

T 5~1 2 r!T123U1 2 rR321U

2 . (13)

R and R9 are the reflectances in the ~123! and ~321!directions, respectively, and T is the value of theransmittance of the assembly ~film 1 substrate!, ac-ording to Fig. ~1b!. Any weak absorption in theubstrate is taken into account by the factor U, whichan be determined separately as indicated in Appen-ix B. The factor

r 5 @~n1 2 n3!2 1 k3

2#y@~n1 1 n3!2 1 k3

2#

is the reflectivity of the 1–3 interface. Usually, inthe substrates used for transmission experiments,the absorption is zero or small; in these cases theterm k3

2 can be dropped.Substituting R321 and T123 from Eqs. ~9! and ~10!,

espectively, into Eq. ~13! and after a straightforwardbut tedious calculation, one gets for the transmit-tance

T 5A exp~ad!

B exp~2ad! 1 C exp~ad! 1 D, (14)

where A, B, C, and D ~given in Appendix A! arealgebraic functions of the optical constants and thick-


ness of the film and adjacent media. In the low-absorption region of the spectrum the factor C in Eq.~14! is responsible for the modulation of the trans-mittance @see Eq. ~A3! below#. Equation ~14!, whichdoes not involve any approximation, is used through-out this study and applied to the different regions ofthe spectrum, independently of the intensity of theabsorption.

3. Treatment of Transparent Films

The simplest case of thin-film calculation with atransmission spectrum occurs when interest is lim-ited to a region where absorption can be neglected.The sequence of steps indicated below can be followedin Table 1, which contains the results for a siliconnitride film.

When k2 5 0 the film is said to be transparent, andthe fully modulated interference fringes appear.The extrema ~maxima and minima! of the transmit-tance occur at wavelengths lm, which satisfy the con-

ition18

4pn2 dylm 5 mp, (15)

here m 5 1, 2, 3 . . . are the interference orders ofhe fringes. Although Eq. ~15! is strictly valid onlyor a dispersion-free film, in practice small disper-ions do not invalidate the expression. At the ex-rema the phase angle reduces to

w 5 mp. (16)

pplying the above condition [Eq. (15)] to Eq. ~13! or14!, one gets for the even and the odd interferencerders

Teven 52n1 n3

n12 1 n3

2 , (17)

Todd 54n1 n2

2n3

~n12 1 n2

2!~n22 1 n3

2!. (18)

By comparison of Eqs. ~17! and ~B1! below, one seesthat the transmittance of the film–substrate assem-bly equals the transmittance of the naked substrateat the wavelengths that correspond to the even or-ders. In other words, the modulated transmittancecurve of the film is tangential to the flat transmissioncurve of the substrate at these points. Furthermore,it can be shown that Todd , Teven when n2 . n3, which

2

an a-SixN12x Film by the Method of Section 3 ~Transparent Film!

l ~nm! m n3 Todd n2 d ~nm!

1887 3 1.462 0.8034 1.881 7531149 5 1.464 0.7976 1.898 757823.0 7 1.468 0.7912 1.919 756645.2 9 1.473 0.7843 1.941 748531.9 11 1.479 0.7761 1.967 744435.5 13 1.486 0.7669 1.997 738392.6 15 1.494 0.7550 2.036 732355.2 17 1.503 0.7357 2.098 719

ffiSmw

n

Wtsr

aao

wt

means that the transmission curve of the assemblylies beneath the transmission of the substrate. Theopposite is true for n2 , n3. These two oppositesituations are demonstrated in Fig. 2. The firstcase, for which Teven corresponds to the maxima, isrequently found when a semiconducting or dielectriclm is deposited onto a glass or a quartz substrate.amples deposited upon silicon substrates for IReasurements are examples of the second case,here Teven are minima.The interference orders m can be determined from

Eq. ~15!. The dispersion of the refractive index ofany dielectric or semiconductor is small in its regionof transparency; therefore for any three consecutiveextrema we have

~m 2 1!lm21 > mlm > ~m 1 1!lm11. (19)

From Eq. ~19!, the following relations for the calcu-lation of m, which can be used interchangeably, arederived:

m >lm21 1 lm11

lm21 2 lm11,

m >lm21

lm21 2 lm,

m >lm11

lm 2 lm11. (20)

A few consecutive values of m are determined, pref-erably at the long-wavelength side of the spectrumwhere the precision of relations ~20! is better; then therest of the sequence, for the remaining extrema, isreliably assigned. Some remarks about the determi-nation of the values of m are useful at this point: ~a!In the assignment of the interference orders the parityof m should always be observed; i.e., the order of theextrema where the transmission of the film touchesthe transmission of the substrate should be even num-bers. ~b! Small values of m are found in thinner films;if this is the case ~m , 15! the uncertainty of m asdetermined by relations ~20! is small. ~c! If m . 15,greater uncertainties may arise, which could induce

Fig. 2. Calculated transmittance of a thin transparent film whenn2 , n3 ~upper curve! and n2 . n3 ~lower curve!, showing the points

here the transmittance of the film is tangential to the transmit-ance of the substrate ~even orders of interference!. The interfer-

ence order m is indicated at each extremum. The refractiveindices are n1 5 1, n2 5 2, n2 5 3.8, and n3 5 3.

the operator to accept two m sequences instead of asingle one, for instance, m 5 20, 22, 24, . . . andm 5 22, 24, 26, . . . for the same set of even-orderextrema. If this is the case, the uncertainty in m willbe reflected in the error of the refractive index and ofthe film thickness calculated thereafter.

The optical path in the film at each lm as deter-mined from Eq. ~15!,

s~lm! 5 dn2~lm! 5 mlmy4, (21)

is known from the experimental spectrum. It willhelp in the complete determination of the refractiveindex.

We calculate the refractive index of the film at theodd-order extrema, using the experimental valuesTodd. From Eq. ~18! we get a biquadratic equation in

2 whose positive roots are

n2 5 @b 6 ~b2 2 n12n3

2!1y2#1y2, (22)

where

b 52n1 n3

Todd2

n12 1 n3

2

2. (23)

hen the transmission spectrum imposes the condi-ion that n2 . n3, only one of the roots @Eq. ~22!#atisfies this condition. In the opposite case bothoots satisfy the condition n2 , n3; then, we need

additional information to find the best solution: anadditional measurement changing n1 ~a liquid in-stead of air, for instance! or an additional sampledeposited onto a substrate with different n3.

Knowledge of n2 at the odd orders and the opticalpath @Eq. ~21!# at these points allows one to calculate

set of values of d, the mean value of which isdopted as the film thickness. Finally, at the even-rder extrema, n2 is determined by means of the op-

tical path and the film thickness already determined.With this step the calculation of the refractive indexof the transparent film at the initially selected ex-trema is completed.

Frequently a model function for the refractive in-dex is needed, as is shown in Section 4. We choosethe single-oscillator model that is due to Wemple andDidomenico20 ~W-D! in which the dispersion of therefractive index is related to the photon energy,E 5 hcyl, according to the equation

n2~E! 5 1 1Em Ed

Em2 2 E2 , (24)

where the parameters Em and Ed are the oscillatorand the dispersive energies, respectively. We caneasily determine these parameters by plotting ~n2 21!21 versus E2. This procedure gives statisticallysignificant results whenever the number of extremais not small.

To illustrate this procedure we use transmissiondata of a hydrogenated amorphous silicon–nitrogencompound deposited onto a quartz substrate. Fig-ure 3 shows the complete spectrum as measured witha Perkin-Elmer Lambda 9 spectrophotometer. Thenear-infrared, visible, and ultraviolet ranges were


a

wcms

tpfif

o

fisw

5

covered in the measurement. The film will be con-sidered absorptionless for the purpose of this sectionin the wavelength interval from 400 to 2000 nm.Table 1 shows the results of the present calculations:the interference orders, the refractive index n2, andthe thickness d at the minima ~odd-order extrema!re determined by means of relations ~20! and Eqs.

~22! and ~15!, respectively. The refractive index ofthe substrate was computed from its transmittancewith the assumption that it has zero absorption @Eq.~B3!# below. No problems arise in determination ofthe values of m in this film because the uncertaintiesof the results of relations ~20! are so small.

The material of the film characterized in this sec-tion is not completely transparent in the chosen spec-tral range. The effects of this small absorption are asystematic error in the refractive index and a devia-tion of the film thickness as a function of the wave-length, as is illustrated in Table 1. The best valuesof n2 and d correspond to the smaller orders wherethe absorption is quite small.

4. Treatment of Absorbing Films

The calculation of the real or the imaginary parts ofthe refractive index of the film with a given experi-mental value of the transmittance is done numeri-cally by means of the equation

T~n2, k2, l, d, n1, n3! 2 Texp 5 0, (25)

here either n2 or k2 is chosen as an unknown. Onean use any efficient root-finding procedure to deter-ine the desired parameter. The analytical expres-

ion for the transmission to be used in Eq. ~25! isgiven by Eq. ~14! with Eqs. ~A1!–~A4! in Appendix A.An alternative, and computationally faster, methodto determine k2 ~or a! is as follows: Eq. ~14! is rear-ranged to give

exp~ad! 51

2B HS ATexp

2 CD 1 FS ATexp

2 CD2

2 4BDG1y2J,

(26)

where the exponential dependence on a is now iso-lated. We determine an approximate value ofexp~ad! by calculating the second member of Eq. ~26!,


using an arbitrary value of k2 ~or a!. This result isused to repeat the calculation with the new value ofk2 @Eq. ~7b!#. The repetition of this procedure re-sults in an efficient iterative method that convergesrapidly. Three or four iterations are sufficient toyield a value of the absorption coefficient with satis-factory precision.

Here, for the purpose of the calculation, we dividethe spectrum into two parts: the low-absorption re-gion, where the interference dominates with the typ-ical fringes produced by the coherent reflections atthe interfaces of the film, and the absorption edge,where the abrupt increase in the light absorptioncauses the simultaneous diminution of the modula-tion of the fringes and of the absolute intensity of thetransmitted radiation until it becomes negligible.These two regions are treated separately.

A. Low-Absorption Region

Here we calculate the refractive index and the ab-sorption coefficient at the maxima and minima of thetransmission in the interference region. This part ofthe calculation parallels the procedure followed inSection 3, with the difference that the absorption isnot ignored.

This calculation is simplified by consideration oftwo properties of the transmission:

~1! Equation ~15!, which states the condition forhe occurrence of maxima and minima in the trans-arent films, is approximately true in absorbinglms too, with the condition that the interferenceringes be neatly defined.7 Then we can define and

calculate the interference orders m and set the val-ues of the phase angle w 5 mp at each lm, as wasdone in Section 3, even if the film is not strictlytransparent.

~2! The value of the transmittance at the even-rder extrema depends strongly on k2 but only

weakly on n2. Consequently, by use of an approxi-mate value of the refractive index n2 it is possible todetermine reasonably accurate values of the extinc-tion coefficient k2 at these extrema with the help ofEq. ~26!. The approximate value of n2 that is neces-sary for this purpose can be given by the transparentfilm approximation, Eq. ~22!. This property, in ad-dition to the possibility of the determination of a inthe low-absorption region, permits an improvementin the calculation of n2.

In Table 2 we present the absorption coefficient atthe even-order extrema ~maxima of the transmis-sion!; the data correspond to those used in Section 3.The approximate values of n2 are taken from Table 1.The calculation of a was done with Eq. ~26! and con-dition ~16!; in this case we chose as the initial value a05 0, and no iteration was necessary because of thesmallness of the absorption coefficient. It should benoted that in this part of the calculation knowledge ofthe film thickness is not needed because of the use ofcondition ~16!.

The absorption data of Table 2 permit the calcula-

Fig. 3. Experimental transmission spectrum of an a-SixN12x:Hlm, showing the interference ~low-absorption! region and the ab-orption edge. The transmission is plotted as a function of theave number to show the details of the absorption edge.

mi

wo

est

riet

Table 2. Calculation of the Absorption Coefficient of the a-Si N Film Table 4. Calculation of the Absorption Coefficient at the Absorption

tion of the refractive index at the odd-order extrema~minima of transmission! by numerical solution ofEq. ~25!. It is expected that these n2 will be betterthat those of Section 3, where the absorption was nottaken into account. These results are shown in Ta-ble 3, where the rightmost column contains the valueof the thickness obtained from Eq. ~15! at each min-imum. As expected, the spread of the values of d is

uch smaller than that in Table 1. The mean values d 5 760 6 2 nm.

So far we have obtained the values of n2 only at oddorders, but knowledge of the thickness permits deter-mination of n2 at the even orders as well. Thesevalues are not included in Table 3.

The complete set of refractive indices ~at even andodd orders! is plotted in Fig. 4 as a function of thephoton energy. The inset shows the W-D plot usedin determining the parameters of the model:Ed 5 23.7 6 0.3 eV and Em 5 9.28 6 0.01 eV. Thesmall amount of spread that we can observe in Fig. 4

Fig. 4. Refractive index of the film of Fig. 3, including the exper-imental points and the W-D fitting. Inset, W-D plot used in de-termining the parameters Em and Ed of the model.

x 12x

for Even Orders ~Subsection 4.A, Low Absorption Region!a

l ~nm! m n3 n2 Texp a ~cm21!

1439.0 4 1.463 1.89 0.9273 56.8961.5 6 1.466 1.91 0.9218 120.5724.6 8 1.470 1.93 0.9163 181.1584.8 10 1.475 1.95 0.9111 234.0490.2 12 1.482 1.98 0.9043 306.4424.6 14 1.490 2.02 0.8968 385.7375.9 16 1.498 2.06 0.8828 552.0339.0 18 1.507 2.13 0.8524 1077.0

aThe values of n2 were taken from Table 1.

Table 3. Calculation of Refractive Index n2 and Thickness d for theOdd Orders ~Subsection 4.A!a

l ~nm! m Todd n2 d ~nm!

1149 5 0.7976 1.886 762823.0 7 0.7912 1.901 758645.2 9 0.7843 1.916 758531.9 11 0.7761 1.928 759435.5 13 0.7669 1.939 760397.6 15 0.7555 1.953 763355.2 17 0.7357 1.991 758

aThe values of a were obtained from Table 2 by interpolation.

is reflected in the smallness of the errors of Em andEd, demonstrating the applicability of the model.The results of this section are used in Subsection 4.Bto characterize the absorption edge.

B. Optical Absorption Edge

When the absorption increases at the absorption edge,the number of multiply reflected beams that effectivelycontribute to the transmitted light diminishes gradu-ally until the interference becomes irrelevant. Simul-taneously the intensity of the transmitted lightdecreases rapidly, as can be seen from Fig. 3.

The iterative procedure associated with Eq. ~26! isadopted to compute the absorption coefficient. Forthis purpose it is necessary to know at each wave-length the transmittance, the refractive index of thesubstrate, and the refractive index of the film and itsthickness as determined in the low-absorption region.Absorption in the substrate is taken into account bymeans of the factor U introduced in Eqs. ~11!–~13!,

hich can be evaluated by means of the transmissionf the substrate, Eq. ~B6! below.Selected values of a as a function of the photon

nergy are contained in Table 4. The quartz sub-trate of this sample is partially absorbing in part ofhe ultraviolet region. The values of U that we used

to correct this effect are included in that table. Fig-ure 5 shows the absorption coefficient as a function ofenergy. The graph includes the results at the ab-sorption edge and in the low-absorption region.

At the higher-absorption end of the spectrum,where no traces of interference are found, the trans-mittance is a monotonically increasing curve. Inthis range a smooth curve for a is always obtained.On the other hand, the transmittance in the tran-sition region, where the interference effects are stillpresent, is slightly modulated. In this part of thespectrum the a-versus-E curve may show some spu-ious peaks or modulations because of the use of anncorrect refractive index in the calculation. Thisffect appears because the refractive index given byhe W-D model may be different from the true val-

Edge ~Subsection 4.B!

l ~nm!E

~eV! U Texp

a~cm21!

339.0 3.658 1 0.8524 926309.6 4.005 1 0.8004 1 651287.4 4.315 1 0.6981 3 227269.5 4.600 0.9930 0.5546 5 550246.9 5.022 0.9545 0.2661 13 930241.0 5.146 0.9404 0.2004 17 300235.3 5.270 0.9437 0.1380 22 520229.9 5.394 0.9621 0.0906 27 900224.7 5.518 0.9642 0.0528 35 230219.8 5.642 0.9458 0.02679 43 660215.1 5.766 0.9209 0.01122 54 760210.5 5.890 0.8895 0.00372 68 710206.2 6.014 0.8569 0.00100 85 420202.0 6.138 0.8341 0.00026 102 800


a

p1

sti

talti

ctbaAo

vbTt

otu

5

ues at the absorption edge. One can easily correctthis artifact by changing the parameters of themodel slightly, as was done in the treatment of ourdata, thus yielding a rather smooth curve in theinterval from 4 to 5 eV where the spurious peaks areexpected.

The study of the absorption edge is important foran understanding of the mechanisms of optical ab-sorption in crystalline and amorphous materials. Itis of great practical interest because knowledge of theabsorption coefficient in this region permits the de-termination of the optical gap, one of the most impor-tant optical properties of the material. The shape ofthe absorption edge and the determination of theoptical gap depend strongly on the electronic struc-ture of the material.21–23 In our case the sample is awide-gap amorphous semiconductor, which is ex-pected to show a subbandgap absorption region, anUrbach tail with an exponential increase in absorp-tion, and a Tauc region that one can use to determinethe optical gap in this material. These three regionsare seen in Fig. 5 and occupy roughly the energyintervals ~1, 3.5!, ~3.5, 5.5!, and ~5.5, 6.2! eV, respec-tively.

We can express the absorption in the Urbach tail bythe equation

a 5 a0 exp~EyE0!. (27)

The values of the parameters of Eq. ~27! that corre-spond to the results of Fig. 5 are E0 5 485 6 3 meVnd a0 5 0.43 6 0.03 cm21.According to Tauc,23 in amorphous materials the

absorption that is due to the band-to-band transitionsthat determine the optical gap Eg is given by

EÎε2 5 g~E 2 Eg!, (28)

Fig. 5. Absorption coefficient of the film of Fig. 3. The resultscover the low-absorption region and the absorption edge. TheTauc plot for the determination of the optical gap is shown in theinset.


where ε2 is the imaginary part of the dielectric func-tion. Introducing the absorption coefficient trans-forms Eq. ~28! into

ÎanE 5 b~E 2 Eg!, (289)

where b is a constant related to the structure of theamorphous material. This theory predicts an en-ergy interval where the experimental values havethe linearity established by Eq. ~28! and thus per-mits the determination of the optical gap. The in-set of Fig. 5 is a Tauc plot from Eq. ~289! with the

arameters Eg 5 4.83 6 0.13 eV and b 5 971 66~eV cm!21y2.The determination of Eg by the Tauc method re-

quires the measurement of quite small transmit-tances to yield the linear part of the plot, as can beseen from the bottom few rows of Table 4.

5. Discussion and Comments

The method of optical characterization of thin filmsdiscussed in this paper has the following features:

~1! From the theoretical point of view, all the for-mulas were rigorously derived from electromagnetictheory. No approximations were introduced to facil-itate the calculations. As a consequence, the sameformulas are used in the different regions of the spec-trum, avoiding errors and simplifying the programsused for the calculations.

~2! The interference fringes are fully used in thetransparent region and in the spectral region withweak absorption. Accurate results of the refractiveindex and its dispersion ~from the W-D model, forinstance! are obtained in these two spectral regions.

~3! The absorption coefficient in the low-absorptionregion ~subgap absorption! can be obtained with rea-onable precision if appropriate values of the refrac-ive index and absorption of the substrate arentroduced into the formulas.

~4! An accurate value of the film thickness is ob-ained from the interference fringes of the spectrums an important by-product of the calculations. Al-owance for the absorption of the substrate improveshe accuracy of both film-thickness and refractive-ndex results.

~5! Undesirable spurious peaks of the absorptionoefficient curve in the transition zone lying betweenhe absorption edge and the fringe region are avoidedy the adjustment of the W-D parameters to producesmall variation of the refractive index in this region.monotonically increasing a-versus-E curve is thus

btained, as shown in Fig. 5.~6! The quality of the results was systematically

erified by calculation of the transmittance spectrumy use of the calculated values of n2~E!, a~E!, and d.he coincidence of experimental and calculated spec-ra was usually remarkable.

~7! The mathematics used in this method, in spitef the fact that the full theory is used in the deriva-ion of the formulas, is simple enough to permit these of a good programmable pocket calculator to per-

tacars

rl

2

~s

form all the calculations required for a complete char-acterization.

~8! The paper is self-contained in the sense that allnecessary theoretical formulas, corresponding to bothfilm and substrate, have been included. Detailedexplanations related to the calculations of the indexof refraction, absorption coefficient, film thickness,and optical gap have been given as well.

One should observe some aspects related to thesamples and the substrates to optimize the result ofthe calculations:

~1! The films should be deposited onto thick enoughsubstrates ~ds . 0.4 mm! to prevent the appearanceof interference in the light owing to reflections at theinterfaces of the substrate. These fringes obscurethe spectrum of the film and make analysis of thedata difficult.

~2! When the main purpose of the exercise is thedetermination of the optical gap, it is convenient toreach the highest values of the absorption coefficientpermitted by the data; thinner films are better in thiscase. Films that are too thin, however, may producefew interference fringes, thus diminishing the accu-racy of the calculated film thickness that enters intothe calculation of the absorption coefficient. So acompromise between the two opposite situations hasto be made.

~3! Thicker films are to be preferred in those casesin which the main interest is focused on the film’sthickness, refractive index, and absorption coefficientin the low-absorption region.

~4! The uniformity of the film’s thickness should beoptimized in the preparation of the samples becausedifferences in thickness in the region of the samplecovered by the light beam of the spectrophotometermay modify the transmission spectrum. One canminimize the effect of small inhomogeneities by dimin-ishing the size of the light beam used in the measure-ment by means of a diaphragm or a beam condenser.The latter alternative should be preferred, if available,because smaller diaphragms diminish the beam en-ergy, impairing the signal-to-noise ratio.

~5! Precise values of the refractive index of thesubstrate @Eq. ~B3! below# and of the absorption fac-or U @Eq. ~B6!, which is frequently not considered#re necessary for improvement of the results, espe-ially in the low-absorption region. The well-knownbsorption bands in the near infrared and ultravioletegions of glass, quartz, and other common sub-trates should not be forgotten.

Appendix A. Transmission of a Thin Absorbing FilmDeposited onto a Thick Semitransparent Substrate

The substitution of Eqs. ~9! and ~10! into Eq. ~13!esults in a lengthy expression. We adopt the fol-owing simplifying definitions to write Eq. ~14!:

A 5 16n3~1 2 r!~n22 1 k2

2!U, (A1)

B 5 st 2 Usvr, (A2)

C 5 @2~4n3 k2 2 ZY!cos w 1 4k2~n3 Y 1 Z!sin w#

2 rU2@4k2~Z 2 n3 Y!sin w 2 2~ZY 1 4n3 k22!cos w#,

(A3)

D 5 uv 2 U2tur, (A4)

u 5 ~n1 2 n2!2 1 k2

2, (A5)

v 5 ~n2 2 n3!2 1 k2

2, (A6)

s 5 ~n1 1 n2!2 1 k2

2, (A7)

t 5 ~n2 1 n3!2 1 k2

2, (A8)

Y 5 n22 2 n1

2 1 k22, (A9)

Z 5 n22 2 n3

2 1 k22. (A10)

Appendix B. Transmittance and Reflectance ofTransparent and Semitransparent Substrates

In the thick plates ~ds . 0.4 mm! commonly used assubstrates for thin films, the light multiply reflectedat the interfaces loses its coherence because of theirregularities of the surfaces. In this case we cancalculate the total transmitted and reflected intensi-ties by summing the partial intensities of the contrib-uting beams.

It can be shown that in the case of transparentsubstrates the transmittance and the reflectance ofthe substrate are given by

TS 51 2 r

1 1 r, (B1)

RS 52r

1 1 r, (B2)

respectively, where r has the same definition as givenin Section 3.

If the transmittance of the substrate and the re-fractive index of the incident medium are known, therefractive index of the substrate is determined by

n3 5 n1F 1Ts

1 S 1Ts

2 2 1D1y2G . (B3)

The equivalent expressions for semitransparentsubstrates are

TS 5~1 2 r!2 exp~2as ds!

1 2 r2 exp~22as ds!(B4)

RS 5 rF1 1~1 2 r!2 exp~22as ds!

1 2 r2 exp~22as ds!G , (B5)

where as and ds are the absorption coefficient and thethickness, respectively, of the substrate.

The factor U 5 exp~2asds! introduced in Eqs. ~9!–11!, which takes into account the absorption of theubstrate in these equations, can be determined by


Tt

10. M. M. Guraya, H. Ascolani, G. Zampieri, J. I. Cisneros, J. H.

1

1

1

1

5

means of the following equation derived from Eq.~B4!:

U21 5~1 2 r!2

2Ts1 F~1 2 r!4

4Ts2 1 r2G1y2

. (B6)

o calculate U it is necessary to know n1 and n3 ando measure the transmittance Ts of the naked sub-

strate.

We acknowledge financial support for this researchby the Fundacao de Amparo a Pesquisa do Estado deSao Paulo and the Conselho Nacional de Desenvolvi-mento Cientıfico e Tecnologico. We thank J. H. Diasda Silva for fruitful discussions and S. Durrant forcarefully correcting the manuscript.

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