Determination of thickness, refractive index, and thickness irregularity for semiconductor thin films from transmission spectra Akram K. S. Aqili and Asghari Maqsood
A simplified theoretical model has been proposed to predict optical parameters such as thickness, thickness irregularity, refractive index, and extinction coefficient from transmission spectra. The proposed formula has been solved for thickness and thickness irregularity in the transparent region, and then the refractive index is calculated for the entire spectral region by use of the interference fringes order. The extinction coefficient is then calculated with the exact formula in the transparent region, and an appropriate model for the refractive index is used to solve for the extinction coefficient in the absorption region 共where the interference fringes disappear兲. The proposed model is tested with the theoretical predicted data as well as experimental data. The calculation shows that the approximations used for solving a multiparameter nonlinear equation result in no significant errors. © 2002 Optical Society of America OCIS codes: 310.0310, 310.6860, 130.5990.
1. Introduction
2. Theoretical Background
Many methods have been proposed in the past 25 years to determine the optical parameters of thin films from transmission data,1–12 rather than from transmission and reflection data. This is due to the simplicity of calibration of the spectrophotometer, which yields accurate experimental results. One of the simplest methods, introduced by Swanepoel,1 used the transmission envelope to solve the equation, analytically, for the thickness and the refractive index. However, including the thickness variation in his formula2 required a numerical solution for every data point. Another interesting method by Cisneros3,4 for a uniform film includes the effect of substrate and solves an equation numerically for the optical parameters. In this paper we present a simple approach for determining thickness, thickness irregularity, refractive index, and extinction coefficient for semiconductor thin films.
The expression for transmittance, including the reflection from the second interface of the substrate and the effect of a finite substrate, which is valid for transparent as well as weakly absorbing substrates,3 is
A. K. S. Aqili and A. Maqsood 共
[email protected]兲 are with the Thermal Physics Laboratory, Department of Physics, Quaid-iAzam University, Islamabad, Pakistan. Received 17 October 2000; revised manuscript received 9 March 2001. 0003-6935兾02兾010218-07$15.00兾0 © 2002 Optical Society of America 218
APPLIED OPTICS 兾 Vol. 41, No. 1 兾 1 January 2002
T⫽
共1 ⫺ 兲T 123U , 1 ⫺ R 321U 2
(1)
R 321 ⫽ r 321r *321 ,
(2)
T 123 ⫽ 共n 3兾n 1兲t 123t *123 ,
(3)
where r321 and t123 are the amplitude of the electric field of the wave reflected and transmitted in 321 and 123 directions, respectively 共illustrated in Fig. 1兲. These parameters are given by
t 123 ⫽
t 12t 23 exp共i兾2兲 , 1 ⫹ r 12r 23 exp共i兲
(4)
r 321 ⫽
r 32 ⫹ r 21 exp共i兲 . 1 ⫹ r 32r 21 exp共i兲
(5)
where the parameters A1, B1, B2, C1, C2, C3, D1, D2, D3, U, and are given in Appendix A. 3. Solving for d, , n, and ␣
In the transparent region of the transmission spectra, maximum and minimum transmission occur at1–5 ⫽ m, where m ⫽ 1, 2, 3, . . . , 4nd兾 m ⫽ m.
Fig. 1. Optical parameters and directions of the transmittance and reflectance, adopted in Eqs. 共1兲–共6兲.
Here rij and tij are the Fresnel coefficients of the reflected and the transmitted wave, respectively, in different regions3,13 r ij ⫽
Ni ⫺ Nj , Ni ⫹ Nj
t ij ⫽
2N i . Ni ⫹ Nj
(6)
The complex refractive index is N i ⫽ n i ⫹ ik i , where ni is the real part and ki is the imaginary part 共extinction coefficient兲 of the complex refractive index of air 共n1, k1兲, film 共n2, k2兲, and substrate 共n3, k3兲. is the phase difference of the wave between two interfaces, ⫽ 4N 2 d兾 ⫽ 4n 2 d兾 ⫹ 4k 2 di兾 ⫽ ⫹ i␣d, where d is the film thickness, is the wavelength, ␣ is the absorption coefficient, and is the phase angle. The modified Fresnel coefficients of reflected and transmitted waves at a rough film surface,14 共1–2兲
m⬵
(11)
For n2 ⬎ n3, m is even at maximum transmission and odd at minima. Equation 共11兲 could be rewritten as n 2共m, 兲 ⫽ m兾4d.
(12)
We can calculate the interference fringes order 共m兲 by assuming that the refractive index varies slowly with the wavelength3,4 in this region so that 共m ⫺ 1兲 m⫺1 ⬇ m m ⬇ 共m ⫹ 1兲 m⫹1 3 m ⬵
m⫹1 m⫺1 ⬵ . m⫺1 ⫺ m m ⫺ m⫹1
(13)
Using the conditions that m is even for maxima and odd for minima makes it easy to find the value of m. In some cases, in which it is not easy to decide the value of m, for example, in thicker films, one could use a model for variation of the refractive index with the wavelength1,2,6 as n ⫽ n 0 ⫹ g兾 2,
(14)
where n0 and g are constants, and Eq. 共12兲 may be written as 4dn m ⫽ m m ,
4dn m⫹1 ⫽ 共m ⫹ 1兲 m⫹1 ,
4dn m⫺1 ⫽ 共m ⫺ 1兲 m⫺1 .
(15)
Substituting Eq. 共14兲 into Eq. 共15兲 and solving for m gives
3 2 2 3 3 2 3 2 m⫺1 m⫹1 ⫹ m⫺1 m⫹1 ⫺ m⫺1 m ⫺ m⫹1 m 3 2 2 3 2 2 2 2 2 . 共 m⫹1 ⫺ m兲共 m⫺1 m⫹1 ⫺ m⫺1 m⫹1 ⫹ m⫺1 m ⫺ m⫺1 m⫹1 m ⫺ m⫺1 m ⫹ m⫹1 m 兲
(16)
r⬘12 ⫽ r 12 exp关⫺2共2兾兲 2n 12兴 ⫽ r 12,
(7)
Then m could be listed for all the spectra where the interference fringes appear. Knowing the value of m, we replace the refractive index of the film 共n2兲 at the extremes in Eq. 共10兲 with
r⬘21 ⫽ r 21 exp关⫺2共2兾兲 2n 22兴 ⫽ r 21,
(8)
n 2共m, 兲 ⫽ m m兾4d.
t⬘12 ⫽ t 12 exp关⫺ ⁄ 共2兾兲 共n 1 ⫺ n 2兲 兴 ⫽ ␥t 12.
(9)
Substituting Eq. 共17兲 into Eq. 共10兲 and setting k2 ⫽ 0 gives two equations for TM 关for m even 3 cos共兲 ⫽ 1兴 and Tm 关for m odd 3 cos共兲 ⫽ ⫺1兴. We solve these two equations by minimizing ⌬, where
interface, where the rms height of surface irregularity ⬍⬍ , are
12
2
2
Substitution of Eqs. 共2兲–共9兲 into Eq. 共1兲 and proceeding with careful and lengthy calculation will result in an expression for transmittance in the following simplified form, T⫽
A 1 exp共␣d兲 B 1 exp共2␣d兲 ⫹ C 1 exp共␣d兲 ⫹ D 1 ⫻
B 2 exp共2␣d兲 ⫹ C 2 exp共␣d兲 ⫹ D 2 , B 2 exp共2␣d兲 ⫹ C 3 exp共␣d兲 ⫹ D 3
(10)
⌬ ⫽ 共T m ⫺ T me兲 2 ⫹ 共T M ⫺ T Me兲 2.
(17)
(18)
TMe and Tme are the experimental transmittance data at maxima and minima, respectively. Solving Eq. 共18兲 for one consecutive maxima and minimagives the value of d and . Then the refractive index can be calculated from Eq. 共17兲 for all maxima and 1 January 2002 兾 Vol. 41, No. 1 兾 APPLIED OPTICS
219
Fig. 2. Calculated transmittance curve of ␣-Si:H versus wavelength 共兲, along with interference fringes order 共m兲.
minima, since this equation is valid for absorption films for which the value of m could neatly be defined.3 Fitting the values of the refractive index to some known model gives the values of the refractive index for the entire spectra. Knowing the values of d, , and n2, we can solve Eq. 共10兲 for k2 by minimizing ⌬1, ⌬ 1 ⫽ 关T共k兲 ⫺ T e兴 2.
(19)
Here T共k兲 is the transmittance formula 共10兲, and Te is the experimental transmittance value. 4. Simulation for ␣-Si:H
For checking the accuracy of the presented method, typical values2,6 for n共兲, ␣共兲, d, and , were used to reproduce the transmission curve with Eq. 共10兲. The values were n2 ⫽ 2.6 ⫹ 3 ⫻ 105兾2, Log共␣兲 ⫽ ⫺8 ⫹ 1.5 ⫻ 106兾2, d ⫽ 1000 nm, ⫽ 15 nm, n3 ⫽ 1.53, and n1 ⫽ 1. Interference fringes 共m兲 calculated with relation 共13兲 and listed for all extremes are shown in Fig. 2. Solving Eq. 共18兲 in the transparent region as shown in Fig. 3 gave d ⫽ 999.63 nm and ⫽ 15.014 nm. The value of d was used to determine the refractive index by Eq. 共17兲. A plot of n2共兲 versus 1兾2 is shown in Fig. 4. A linear fitting gives n2 ⫽ 2.60138 ⫹ 3.00859 ⫻ 105兾2. From Table 1, which shows the calculated values of the thickness 共d兲 and thickness irregularity 共兲 for different values of the interference order 共m兲 in the transparent and in the low-absorption regions, one can note that the calculated values of d are within good accuracy, better than 0.1%, whereas the accuracy of is better in the transparent region. The calculated values of d, , and n2 are used to determine the absorption coefficient 共␣兲 from Eq. 共19兲 in the transparent and in the low-absorption regions, whereas in the high-absorption region, where the interference fringes disappear, the fitting parameters of the refractive index have been used. Table 2, which gives the comparison of the calculated and the exact values of the absorption coeffi220
APPLIED OPTICS 兾 Vol. 41, No. 1 兾 1 January 2002
Fig. 3. Solution of Eq. 共18兲 for d and , 共a兲 large range and 共b兲 small range.
cient for different wavelengths, shows that the calculated values of ␣ have good accuracy in highand low-absorption regions, whereas in the highly transparent region the accuracy was less because of a small value of ␣ 共⬃10⫺6–10⫺8 nm⫺1兲. However, fitting the calculated values 共Fig. 5兲 gave good accuracy, since the values of ␣ in the transparent region have less weight than in the other regions.
Fig. 4. Linear fitting of the refractive index 共n兲 versus 1兾2 for ␣-Si:H.
Table 1. Calculated Values of Thickness 共d兲 and Thickness Irregularity 共兲 Obtained by Solution of Eq. 共18兲 for Different Values of Interference Fringes Orders, for ␣-Si:H
Interference Orders 共m兲
M 共nm兲
TM
m 共nm兲
Tm
6,7 8,9 10,11 12,13 14,15 16,17 18,19 20,21 22,23
1796 1379 1134 973 859 775 710 659 617
0.90690 0.89955 0.88973 0.87727 0.86191 0.84243 0.81564 0.77156 0.68630
1556 1241 1044 911 814 741 683 636 597
0.54687 0.52515 0.50099 0.47532 0.44872 0.42120 0.39211 0.35808 0.31002
d 共nm兲
共nm兲
999.63 999.91 999.631 999.764 999.90 1000.58 1000.60 1000.7 1000.83
15.014 15.029 15.048 15.098 15.190 15.432 15.996 17.3682 20.571
Table 2. Comparison between Calculated and Exact Values of the Absorption Coefficient in Different Regions, for ␣-Si:H
共nm兲 500 505 510 515 520 525 530 535 540 545 550 555 560 565 570
␣exact 共nm兲⫺1
␣calculated 共nm兲⫺1
0.01000 0.00762 0.00585 0.00452 0.00353 0.00277 0.00219 0.00174 0.00139 0.00112 9.088 ⫻ 10⫺4 7.400 ⫻ 10⫺4 6.069 ⫻ 10⫺4 4.999 ⫻ 10⫺4 4.138 ⫻ 10⫺4
0.00999 0.00752 0.00584 0.00452 0.00352 0.00276 0.00220 0.00174 0.00139 0.00110 8.996 ⫻ 10⫺4 7.558 ⫻ 10⫺4 6.301 ⫻ 10⫺4 5.024 ⫻ 10⫺4 3.875 ⫻ 10⫺4
5. ZnTe Thin Film
The transmission spectra15 of ZnTe thin film, prepared by two-sourced thermal evaporation on Corning 7059 glass substrate, recorded with a Perkin-Elmer, Lambda19, UV-VIS-NIR 共UV–visible–near-IR兲 spectrophotometer with UV-WinLab software, for range 400 –2000 nm is shown in Fig. 6. The interference order 共m兲 calculated from relation 共13兲 is also listed for all extremes in Fig. 6. The solution of Eq. 共18兲 gives d ⫽ 745.8 nm and ⫽ 6.17 nm, where the refractive index of the substrate 共n3兲 was 1.53. The calculated refractive index of the ZnTe film, using Eq. 共17兲, is shown in Fig. 7. Many models3,4,8,10,12,16 –21 could be used for fitting of such film, but the single-oscillator model3,4,12,16 –19 is found to have the best fitting for the refractiveindex values.
n2 ⫽ 1 ⫹
2 Em Ed 共n 02 ⫺ 1兲 E m ⫽ 1 ⫹ , 2 2 Em ⫺ 共h兲 2 Em ⫺ 共h兲 2
共nm兲
␣exact 共nm兲⫺1
␣calculated 共nm兲⫺1
582 595 597 600 617 636 659 683 710 741 776 814 859 911 972 1044 1134
2.682 ⫻ 10⫺4 1.783 ⫻ 10⫺4 1.670 ⫻ 10⫺4 1.468 ⫻ 10⫺4 8.714 ⫻ 10⫺5 5.109 ⫻ 10⫺5 2.844 ⫻ 10⫺5 1.643 ⫻ 10⫺5 9.454 ⫻ 10⫺6 5.390 ⫻ 10⫺6 3.090 ⫻ 10⫺6 1.835 ⫻ 10⫺6 1.078 ⫻ 10⫺6 6.410 ⫻ 10⫺7 3.890 ⫻ 10⫺7 3.516 ⫻ 10⫺6 1.467 ⫻ 10⫺7
2.695 ⫻ 10⫺4 1.878 ⫻ 10⫺4 1.592 ⫻ 10⫺4 1.319 ⫻ 10⫺4 8.841 ⫻ 10⫺5 5.138 ⫻ 10⫺5 2.901 ⫻ 10⫺5 1.473 ⫻ 10⫺5 9.262 ⫻ 10⫺6 3.297 ⫻ 10⫺6 2.902 ⫻ 10⫺6 1.105 ⫻ 10⫺6 9.538 ⫻ 10⫺7 2.759 ⫻ 10⫺7 3.659 ⫻ 10⫺7 1.878 ⫻ 10⫺7 9.973 ⫻ 10⫺8
quency, and n0 is the refractive index of an empty lattice at infinite wave length. The calculated values of the refractive index along with fitting to Eq. 共20兲 is shown in Fig. 7.
(20)
where Em, Ed are the oscillator and dispersive energies, h is the Planck constant, is the photon fre-
Fig. 5. Fitting of calculated absorption coefficient 共␣兲 values for ␣-Si:H. 1 January 2002 兾 Vol. 41, No. 1 兾 APPLIED OPTICS
221
Fig. 6. Experimental transmission spectra of ZnTe film with interference fringes order 共m兲.
Fig. 8. Calculated values of the absorption coefficient, with fitting to Urbach relation.
The values of ␣, calculated by Eq. 共19兲, are fitted to the Urbach relation,16,22
Eg obtained was 2.234 eV. Figure 10 shows the transmittance curve reproduced by Eq. 共10兲 by use of calculated values of d and and the fitting parameters of n2 and ␣, in the Urbach region, along with experimental transmission data, which have a good matching.
␣ ⫽ ␣ 0 exp共h兾E e兲 ⫽ exp共a ⫹ b兾兲,
(21)
where ␣0, Ee, a, and b are constants related to the characteristic slope of ␣. It is obvious, from Fig. 8 that the Urbach relation has good fitting for ␣ ⬍ 0.003 nm⫺1 共30,000 cm⫺1兲. Near the absorption edge the optical energy gap 共Eg兲 for allowed direct transition could be calculated, with the well-known dependence22 ␣2 ⬃ 共h ⫺ Eg兲. The energy gap is obtained by means of extrapolating the square of the absorption coefficient 共␣2兲 versus incident photon energy 共h兲. Figure 9 shows the energy gap with the values of ␣ calculated by Eq. 共19兲, Eg ⫽ 2.47 eV, and ␣ is calculated with the following approximation23–25 near the absorption edge, T ⬃ exp共⫺␣d兲.
(22)
Fig. 7. Calculated values of the refractive index of ZnTe film, with fitting to Eq. 共20兲. 222
APPLIED OPTICS 兾 Vol. 41, No. 1 兾 1 January 2002
6. Discussion and Comments
The following points summarize our findings: 1. The formulas include most of the film and substrate parameters, which affect the transmission spectra, and are given in a simplified form. 2. The values of the interference fringes order 共m兲, for films of thickness below ⬃2000 nm, could simply be determined with relation 共13兲 by approximation of the resulting value to the closest even integer for maxima and odd integer for minima. For thicker films where the interference fringes are so close, relation 共16兲 could be used. 3. The thickness, the thickness irregularity, and the refractive index of the film 共with good accuracy兲
Fig. 9. Plot of ␣2 versus photon energy 共h兲 for ZnTe film.
t ⫽ 共n 2 ⫹ n 3兲 2 ⫹ k 22, Y ⫽ n 22 ⫺ n 12 ⫹ k 22,
Z ⫽ n 22 ⫺ n 32 ⫹ k 22,
⫽ 关共n 1 ⫺ n 3兲 2 ⫹ k 32兴兾关共n 1 ⫹ n 3兲 2 ⫹ k 32兴, n 3 ⫽ n 1关1兾T s ⫹ 共1兾T s2 ⫺ 1兲 1兾2兴, U ⫺1 ⫽
冋
共1 ⫺ 兲 2 共1 ⫺ 兲 4 ⫹ ⫹ 2 2T s 4T s2
册
1兾2
,
where Ts is the transmittance of the substrate and, for transparent substrate U ⫽ 1, and k3 ⫽ 0.
Fig. 10. Calculated transmittance curve, with experimental transmission data, versus wavelength for ZnTe film.
This research was supported in part by the Abdus Salam International Center for Theoretical Physics 共ICTP兲, the Pakistan Atomic Energy Commission 共PAEC兲, and Dr. A. Q. Khan Research Laboratories 共KRL兲. References
can be calculated by simple solution of one numerical equation. 4. Calculated values of the absorption coefficient or the extinction coefficient of the film, in the absorption region, where the interference fringes no longer exist, depend on the selection of model for the refractive index. With the transparent region it is calculated by use of the values of the refractive index calculated in this region. 5. Simulation of a tested theoretical model proved that there is no significant error due to approximation used, consider k ⫽ 0 in transparent region and m is exactly integer, even at wavelengths below 1000 nm 共but not at lower wavelengths where the absorption is too high兲. 6. Comparing the rms of thickness irregularity 共兲 with the value of thickness variation 共⌬d兲 calculated with the Swanepoel formula, with similar a approach,15 indicates that ⌬d ⬃ 2, which is due to the definition of the parameter in both formulas, whereas the values of the thickness and the refractive index were approximately same. Appendix A
The values of the parameters in Eq. 共10兲 are as follows: A 1 ⫽ ␥ 2关16n 1 n 3共1 ⫺ 兲共n 22 ⫹ k 22兲U兴, B 1 ⫽ st ⫺ sU 2,
B 2 ⫽ st,
C 1 ⫽ 兵关2共4n 3 k 22 ⫺ ZY兲cos ⫹ 4k 2共n 3 Y ⫹ Z兲sin 兴 ⫺ U 2关4k 2共Z ⫺ n 3 Y兲sin ⫺ 2共ZY ⫹ 4n 3 k 22兲cos 兴其, C 2 ⫽ 关2共4n 3 k 22 ⫺ ZY兲cos ⫹ 4k 2共n 3 Y ⫹ Z兲sin 兴, C 3 ⫽ 关2共4n 3 k 22 ⫺ ZY兲cos ⫹ 4k 2共n 3 Y ⫹ Z兲sin 兴, D 1 ⫽  2关u ⫺ tuU 2兴, D 3 ⫽ 2关u兴,
D 2 ⫽  2关u兴,
u ⫽ 共n 1 ⫺ n 2兲 2 ⫹ k 22,
⫽ 共n 2 ⫺ n 3兲 2 ⫹ k 22,
s ⫽ 共n 1 ⫹ n 2兲 2 ⫹ k 22,
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