Optimum Radiative Properties of Nanoscale Multilayer
Semiconductors
S.A.A.Oloomi1, A.Saboonchi2, and A. Sedaghat3
Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, 84156-83111, I.R. of Iran
Email : AmirOloomi@me.iut.ac.ir
Abstract: A great number of the optical components and semiconductors are being covered with thin films, so study of
the surfaces which covered by thin films are very important. This work uses transfer-matrix method for calculating the
radiative properties. Lightly doped silicon is used and the coherent formulation is applied. The empirical expressions
for the optical constants of lightly doped silicon are employed. Silicon dioxide and silicon nitride are used as thin film
coatings. It is possible to choose the suitable coating for maximum emittance, minimum transmittance and or minimum
reflectance. It depends on industrial usages. This paper considered effects of thin films’ number with various
compositions of coating materials (9 cases). High emittance is needed for suitable thermal balance of the thin-film
solar cells for space applications. Optical coatings that provide high emittance must be formed on the solar cells to
overcome that problem by increasing number of thin film layers. Radiative properties are complex function of
wavelength. Increasing number of thin film layers lead to more complexity and dependency on wavelength regard to
wavelength interferences. Results showed that increasing number of thin film coating is more effective on radiative
properties than increasing thickness. For example in the constant total thickness, it is possible to reach greater
emittance by increasing layers’ numbers. From results, the average emittance for bare silicon from 0.0444 reaches to
0.1117 with coatings in case number 8.
Keywords: Thin Films’ Number, Different Coatings, Reflectance, Emittance, Transmittance, Optimum Properties.
I. Introduction
Optical and thermal radiative properties are fundamental physical properties that describe the interaction between
electromagnetic waves and matter from deep ultraviolet to far-infrared spectral regions [1]. In fact, surface modification
by coatings can significantly affect the radiative properties of a material [1-3]. For lightly doped silicon, silicon dioxide
coating has higher reflectance than silicon nitride coating for visible wavelengths. In visible wavelengths the reflectance
increases as the temperature increases due to decreasing emittance; but in infrared wavelengths, the reflectance and
transmittance decrease as the temperature increases [3]. This work uses incoherent formulation for calculating the
radiative properties of semiconductor materials related to the recent technological advancements that are playing a vital
role in the integrated-circuit manufacturing, optoelectronics, and radiative energy conversion devices. Lightly doped
silicon is used and the empirical expressions for the optical constants of lightly doped silicon are employed. Silicon
dioxide ,silicon nitride and gold are used as thin film coatings. This paper considered effects of thin films’ number with
various compositions of coating materials to gain optimum radiative properties of nanoscale multilayers.(9 cases).
II. Modeling
II.I. Coherent Formulation
When the thickness of each layer is comparable or less than the wavelength of electromagnetic waves, the wave
interference effects inside each layer play an important role in accurate prediction of the radiative properties of the
multilayer structure of thin films. The transfer-matrix method provides a convenient way to calculate the radiative
properties of the multilayer structure of thin films. Assuming that the electromagnetic field in the jth medium is a
summation of forward and backward waves in the z-direction, the electric field in each layer can be expressed by


 A1e iq1 z z  B1e iq1 z z e (iqx xit ) , j  1
Ej  
iq ( z  z )
iq ( z  z ) ( iq x it )
, j  2,3,...N
 A j e jz j 1  B j e jz j 1 e x


(1)
here, A j and B j are the amplitudes of forward and backward waves in the j th layer. Detailed descriptions of how to
solve Eq. (1) for A j and B j is given in [1].
II.II. Incoherent Formulation
1PhD Student, Department of Mechanical Engineering, Isfahan University of Technology, Isfahan; Corresponding Author, Email:
AmirOloomi@me.iut.ac.ir; Tel: +989133593179.
2Associate Professor, Department of Mechanical Engineering, Isfahan University of Technology, Isfahan; Email: AhmadSab@cc.iut.ac.ir; Tel:
+989131130584.
3Assistant Professor, Department of Mechanical Engineering, Isfahan University of Technology, Isfahan; Email: Sedaghat@cc.iut.ac.ir; Tel:
+989131676507.
When the thickness of silicon substrate is much greater than the coherent length, and the considered wavelength falls in
the semitransparent region of silicon, interferences in the substrate are generally not observable from the measurements.
In this case, the incoherent formulation or geometric optics should be used to predict the radiative properties of the
silicon substrate. Two ways to get around this problem are to use the fringe-averaged radiative properties and to treat
thin-film coatings as coherent but the substrate as incoherent [1]. Consequently, the radiative properties of the silicon
wafer with thin-film coatings in the semitransparent region can be expressed as [4]
 i2 t2  bs
   ta 
1  i2  ts  bs
i t b

1  i2  ts  bs
(2)
(3)
  1   
(4)
II.III. Optical Constants
The Jellison and Modine (J-M) expression of the refractive index for a wavelength between 0.4 µm and 0.84 µm is
given in [5]. Li [6] developed a functional relation, for the refractive index of silicon that covers the wavelength region
between 1.2 µm and 14 µm. The J-M expression is used in this study to calculate the refractive index of silicon for the
wavelength region from 0.5 µm to 0.84 µm but Li’s expression is employed for wavelengths above 1.2 µm. For a
wavelength range of 0.84 µm to 1.2 µm, we use a weighted average based on the extrapolation of the two expressions.
The optical constants of silicon dioxide and silicon nitride are mainly based on the data collected in Palik [7].
III. Results
Figure 1 compares the reflectance and transmittance of thick silicon substrate with 700m thickness and coated by
silicon dioxide thin film with
results in [8].
300nm thickness in two different coating cases and tow different temperatures with the
0.5
0.4
Reflectance
25ºC
0.3
500ºC
0.2
0.1
Bare Silicon
Top Side
Both Sides
0
1
2
3
4
5
6
7
Wavelength (µm)
1
Bare Silicon
Top Side
Both Sides
0.9
Transmittance
0.8
0.7
0.6
0.5
0.4
25ºC
0.3
0.2
500ºC
0.1
0
1
2
3
4
Wavelength (µm)
5
6
7
Figure 1. A comparison of the calculated results (Left side) with results of [8] (Right side)
The Electromagnetic waves are incident at   0 . The calculated results are in good agreement with results in [8].
Because the refractive index of silicon dioxide (around 1.45) is smaller than that of silicon, the reflectance with a
coating is always lower than that of bare silicon (Figure 1). The oscillation in the reflectance is due to interference in the

silicon dioxide coating. The free spectral range is determined by
 / 2  (2n f d f )1 , where  is the separation
between adjacent interference maxima and n f and d f are the refractive index and thickness of the thin film. The
spectral separation  increases toward longer wavelengths. As the film thickness increases, the free spectral range
decreases, resulting in more oscillations with the thicker silicon dioxide film. Therefore oscillations increased toward
longer wavelengths. Table 1 shows the used cases in order to comparing the radiative properties of nano scale
multilayer.
Table 1. The Cases for comparing radiative properties
Case
Case Definition
Number
Case 1:Bare Silicon with 500m thickness
1
Case 2: Three Layers include Silicon substrate with 500m thickness and coated with silicon dioxide with
400nm thickness from both sides
Case 3: Three Layers include Silicon substrate with 500m thickness and coated with silicon nitride with
400nm thickness from both sides
Case 4: Five Layers include Silicon substrate with 500m thickness and coated with silicon nitride on silicon
dioxide with 200nm thickness from both sides
Case 5: Five Layers include Silicon substrate with 500m thickness and coated with silicon dioxide on silicon
nitride with 200nm thickness from both sides
Case 6: Seven Layers include Silicon substrate with 500m thickness and coated with silicon dioxide on
silicon nitride on silicon dioxide with 133nm thickness from both sides
Case 7: Seven Layers include Silicon substrate with 500m thickness and coated with silicon nitride on
silicon dioxide on silicon nitride with 133nm thickness from both sides
Case 8: Nine Layers include Silicon substrate with 500m thickness and coated with silicon nitride on silicon
dioxide on silicon nitride on silicon dioxide with 100nm thickness from both sides
Case 9: Nine Layers include Silicon substrate with 500m thickness and coated with silicon dioxide on
silicon nitride on silicon dioxide on silicon nitride with 100nm thickness from both sides
2
3
4
5
6
7
8
9
Reflectance
1
0.9
0.8
0.7
Case 1
Case 4
Case 7
Case 2
Case 5
Case 8
Case 3
Case 6
Case 9
0.6
0.5
0.4
0.3
0.2
0.1
Wavelength(µm)
0
1
2
3
4
5
6
7
Figure 2. A comparison of the reflectance of different cases
8
9
10
This paper considered the radiative properties of silicon coated with silicon dioxide and silicon nitride at room
temperature for 9 layers with different coating procedures and coherent formulation is used. The results for reflectance
are shown in figure 2, transmittance in figure 3 and emittance in figure 4 for wavelengths between 1m to 10 m . The
fluctuations in the results are observed because of the wave’s interferences, these fluctuations are in the shape of sinus
curves and with increasing wavelength, the distance between peaks grows (figures 2 to 4). Interferences in the substrate
are generally not observable in the incoherent formulation. This is the major difference between coherent and incoherent
formulations [9]. The average radiative properties consist of reflectance; emittance and transmittance are shown in table
2.
Table 2. Comparison of average radiative properties for all cases
Radiative Properties/ Cases Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9
0.4379
0.3313
0.2487
0.2622
0.2619
0.2886
0.2501
0.2485
0.2802
Average Reflectance
0.5178
0.5473
0.6938
0.6372
0.6239
0.6023
0.6539
0.6398
0.6158
Average Transmittance
0.0443
0.1214
0.0575
0.1007
0.1142
0.1091
0.0959
0.1117
0.1040
Average Emittance
Transmittance
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Case 1
Case 4
Case 7
0.2
Case 2
Case 5
Case 8
Case 3
Case 6
Case 9
0.1
Wavelength (µm)
0
1
2
3
4
5
6
7
8
9
10
Figure 3. A comparison of the transmittance of different cases
Case 1
Case 4
Case 7
Emittance
1
0.9
0.8
Case 2
Case 5
Case 8
Case 3
Case 6
Case 9
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Wavelength (µm)
0
1
2
3
4
5
6
7
Figure 4. A comparison of the emittance of different cases
8
9
10
IV. Conclusion
From the results:
1. It is possible to choose the appropriate coating for maximum emittance, minimum reflectance and minimum
transmittance.
2. High emittance is needed for suitable thermal balance of the thin-film solar cells for space applications. It is possible
to reach to this issue by increasing number of thin film layers.
3. Increasing number of thin film coating leads to increasing the emittance that is due to waves’ interferences in the
layers.
4. Radiative properties are complex function of wavelength. Increasing number of thin film layers lead to more
complexity and dependency on wavelength based on wavelength interferences phenomena.
5. Effect of increasing layers in order to controlling radiative properties is more than the effect of changing the thickness
of layer. For example, it is possible to reach greater emittance by increasing number of coating layers in constant total
thickness.
6. The emittance of nine coating layers is more than seven coating layers, more than five coating layers, more than three
coating layers and more than one coating layer.
7. The average emittance for bare silicon is 0.0443. It reaches to 0.1117 by using thin film coating in case number 8.
8. The reflectance in the wavelength   9.5m decreases from 0.6887 to 0.3598 with selecting suitable coating. (By
choosing coating of case number 8)
9. The transmittance in the wavelength   9.5m decreases from 0.2976 to 0.1852 with selecting suitable coating.
(By choosing coating of case number 9)
10. The emittance in the wavelength   9.5m increases from 0.0137 to 0.4406 with selecting suitable coating. (By
choosing coating of case number 9)
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