The solar absorptivity of silicon is 0.46 for doubly polished wafers, and 0.48 for wafers with a matte finish on one side. The IR properties are summarized below, giving the fraction of an incident IR beam that is transmitted, reflected and absorbed by a wafer of thickness 0.048 cm:

	Doubly polished	Matte finish
T=transmission	0.48	0.43
R=reflection	0.46	0.39
Absorption	0.06	0.18

Thus the all important ratio of solar absorptance to IR emittance is 7.7 for doubly polished wafers, and 2.67 for wafers with a matte finish on one side.

If silicon is doped with a dopant level of ~ 1 x 10¹⁸ cm^-3 the iR absorptance increases to 28% absorption of the incident beam over the spectrum of a body at 100 °C. This corresponds to a resistivity of the rough order of 0.02 ohm-cm.

An SiO2 film of roughly 2 microns thickness would also provide an IR absorptance of ~ 28% of an incident beam over the spectrum of a body at 100 °C.

1. Introduction

Silicon is a unique material with rather odd optical-thermal properties, and since silicon is employed as solar wind collectors in the Suess-Urey mission and must face the sun, the estimate of its temperature requires careful analysis. Since the surface finish affects the thermo-optical properties, it is important to describe the surface finish in any report on properties of silicon. Silicon wafers, as received from the factory, generally have a highly polished front surface and a matte finish rear surface. It is not clear how the matte finish varies from lot to lot for one manufacturer, or from manufacturer to manufacturer. However, wafers from a single lot seemed to have very repetitive matte finished surfaces. These were examined under an electron microscope and it was found that they had structure at the level of several microns. A typical photomicrograph is shown in Figure 1.

It is also important to specify whether the silicon contains dopants. The IR absorptivity of silicon increases linearly with dopant conentration and is pretty much independent of the specific dopant used. For low dopant concentrations, the IR absorptivity is essentially the same as that of pure silicon.

Figure 1. Typical structure in matte finish surface of silicon wafer

2. Reflectivity Measurements

Silicon is quite reflective in the UV and short end of the visible (as high as ~ 70%) and the reflectivity falls off with increasing wavelength through the visible and into the near IR. Longward of about 1.1 microns, silicon is roughly 30% reflective at all wavelengths from the first surface. The reflectivity from the first surface of silicon in air is shown in Figure 1.

The apparent reflectivity of a thin wafer of silicon will be the sum of reflectance from the first and second surfaces, as shown in Figure 2. Thus the total observed reflectivity is the sum of contributions from both surfaces. One may hope to reduce contributions from the backing material by using a black felt, but the discontinuity where the silicon meets the backing will still produce reflections.

In general, experiments carried out by JPL or under contract to JPL measure the apparent IR reflectivity of the material used in the experiment.

Figure 1. Literature data on reflectivity from the first surface of silicon as a function of wavelength.

Figure 2. Total reflectivity is a sum of reflectance from front and back surfaces.

For an opaque material, all reflection comes from the first air/material interface. The interpretation of experiments is then simple. Typical data are given in Table 1.

Table 1. Measured reflectances of various backing materials

Material or system	JPL Solar reflectance	JPL IR reflectance
Gold Backing	.95	.83
Al-Kapton(i) Backing	.33	.70
Al-Kapton(ii) Backing	.79	.48
Black Felt Backing	.09	.04

However, when a transparent material like silicon is used, we again measure the apparent IR reflectivity of the material used in the experiment but we must provide a backing material (which of course could be air).

The measured reflectivities are given in Table 2. The JPL measurements were made with light sources that approximated either the sun, or an IR source characteristic of a body near room temperature (5 - 25 microns). The SOC measurements were made with a spectrometer and were made as a function of wavelength. The IR data reported in the table average these spectral data over the spectrum of a body at 100 °C.

Table 2. Measured Reflectivities by JPL and JPL Contractors

Material or system	JPL Solar reflectance	JPL IR reflectance	SOC IR reflectance	SOC IR transmittance
Gold Backing	.95	.83
Al-Kapton(i) Backing	.33	.70
Al-Kapton(ii) Backing	.79	.48
Black Felt Backing	.09	.04
Si as received - matte finish on one side
Si on Gold	.39	.72
Si on Al-Kapton(i)	.39	.50
Si on Black Felt	.36	.43	0.39
Silicon (no backing)	.36	.43		0.43
Si polished on both sides (very lightly doped)
Si on Gold	.41	.87
Si on Al-Kapton(ii)	.42	.62
Si on Black Felt	.38	.47	0.46
Silicon (no backing)	.38	.46		0.48

3. Absorptivity Measurements

Silicon is highly absorptive across the UV and visible, and for all intents and purposes may be considered totally opaque in the visible. However, the absorption coefficient falls very sharply over many orders of magnitude from about 0.9 micron to 1.1 microns. For wavelengths in the range 1.1 to 2.5 microns, silicon is nearly perfectly transparent. This implies that silicon is opaque to the bulk of the solar spectrum from the UV through the visible, but is highly transparent for the near IR-tail of the solar spectrum. Longward of 2.5 microns, the absorption coefficient increases significantly, but remains moderately small, varying in the range 0.5 to 3 cm-1 longward of 10 microns. This implies that thin disks of silicon are moderately transparent to IR radiation emitted by black bodies at room temperature and above.

Measured values in the literature are shown in Figures 3 and 4. When these values are averaged across either the solar spectrum or the black body spectrum of a body at 100°C, the absorptivity of thin wafers can be estimated, as is given in §4.

Figure 3. Literature values of absorptivity of silicon vs. wavelength.

Figure 4. Literature values of absorptivity of silicon vs. wavelength with expanded scale to show high absorption in the visible.

4. Absorptivity and Transmissivity of Thin Silicon Wafers

The transmission of silicon for the solar spectrum divides between essentially no transmission shortward of 1.1 microns, and perfect transmission from 1.1 microns out to 2.5 microns, for an overall average transmission coefficient of of 25%. However, this 25% figure is for the remainder, after subtracting off 36% reflection. Thus 0.25 x 0.64 = 16% of the incident solar intensity is transmitted, independent of silicon thickness, and essentially all of this transmission lies between 1.1 and 2.5 microns. The solar spectrum is thus roughly 36% reflected, 16% transmitted, and and 48% absorbed by doubly polished silicon wafers. For silicon wafers with a matte finish on one side, the reflectivity increases to ~ 38% and assuming that the transmission is unchanged at 0.16, the absorptance is 0.46.

The absorption by doubly polished silicon of IR radiation from a body at 100 °C is obtained by averaging the absorption over the spectrum emitted by such a body, and is thickness-dependent. For three thicknesses, the absorption is:

Thickness cm	Absorption of beam entering Si
0.048	0.065
0.096	0.133
0.144	0.175

These absorption factors are not based on the incident beam, but rather upon that part of the incident beam which is not reflected at the first air/silicon interface. Since the reflection coefficient at this coefficient is flat across the IR at ~ 0.30, the above transmission factors should be multiplied by 0.7 to get the fractional absorption of the incident beam.

We may now draw the following picture of what happens when IR radiation of intensity [1.0] from a body at 100 °C impinges on a 0.048 cm thick doubly polished silicon wafer.

(i) At the air/silicon interface, a flux of intensity 0.3 is reflected, and a flux of intensity 0.7 enters the silicon

(ii) An amount of flux ~ 0.046 is absorbed in the silicon

(iii) Of the residual flux of 0.654 which reaches the back surface of the silicon, 30% is reflected back into the silicon, and 70% is transmitted. Thus the transmitted flux is 0.7 x 0.654 = 0.46. The reflected flux is 0.20.

(iv) Of the flux reflected back into the silicon at the second surface, a fraction 0.065 is absorbed, so that the additional flux absorbed is 0.065 x 0.20 = 0.013. The flux reaching the front surface at the inside is 0.187.

(v) Of the flux impinging on the inside of the front surface, 70% is transmitted out to the air. Thus, an additional 0.70 x 0.187 = 0.13 contributes to the apparent reflection from a silicon wafer. The remaining 0.30 x 0.187 = 0.056 makes another pass through the silicon.

Ultimately, when a flux of 1.0 impinges on a silicon wafer, the fluxes transmitted, reflected and absorbed are:

T = 0.48

R = 0.46

A = 0.06

We may take these to be the transmission, reflection and absorption coefficients for IR by a 0.048 cm thick doubly polished wafer of silicon.

Note that the transmission coefficient calculated by this procedure is in perfect agreement with that measured by SOC as given in Table 2. The observed reflection coefficient of 0.46 is also in good agreement.

These data pertain to silicon wafers that are polished on both sides. For silicon wafers with a matte finish, no literature values are available; however JPL and SOC data can be used to infer T, R and A. It is observed that the SOC measurements of reflectivity and transmissivity are:

	Doubly polished	Matte on one side
R	0.46	0.39
T	0.48	0.43

By inference, the IR absorptance is 0.06 for doubly polished, and 0.18 for matte on one side.

5. Emittance of Silicon Wafers

We have the data for an incident beam of IR radiation:

	Doubly polished	Matte finish
T=transmission	0.48	0.43
R=reflection	0.46	0.39
Absorption	0.06	0.18

By definition, the emittance is equal to the absorptance, so the emittances are:

doubly polished: 0.06

matte finish: 0.18

6. Dependence of IR absorptivity of silicon on dopant level

Spitzer and Fan, Phys Rev 108, 268 (1957) measured the IR absorption coefficient vs. wavelength out to 50 microns of six samples of silicon doped with various dopants (arsenic, antimony, phosphorous) to impurity levels ranging from around 2 x 10¹⁶ cm^-3 to 7 x 10¹⁹ cm^-3.

For each sample, the carrier concentrations were found from experimental measurements of the Hall effect. In general, the ratio of carriers (holes or electrons) to impurity atoms approaches 1 at very small impurity concentrations, and decreases as the impurity concentration increases. At an impurity concentration of of 1 x 10¹⁷ this ratio is about 0.85. At an impurity concentration of 7 x 10¹⁹ it is about 0.15. It is widely believed that the carriers are responsible for the increase in IR absorptivity in silicon.

When the the IR absorptivity data of Spitzer and Fan are plotted vs. carrier concentration we find an almost perfect straight line relationship at any wavelength.

When IR waves impinge on a silicon wafer, 30% is reflected at the first interface. and 70% enter the silicon wafer. Suppose we want to get say 28% absorption of the incident beam in the silicon. This implies that 40% of the 70% must be absorbed. To get say 40% absorption in a 0.048 cm thick wafer, we require an absorption coefficient a(cm^-1) such that exp(-a*0.048) = 0.6. A ~ 10 cm^-1 absorption coefficient leads to this level of absorption. These data indicate that in the IR wavelength range around 10 microns, a dopant level of 1 x 10¹⁸ cm^-3 is required.

Quite a few years later, in 1979, L. Jastrzebski, J. Lagowski and H. Gatos, J. Electrochem. Soc. 126, 260 (1979) again studied the effect of dopants on IR absorption by silicon. They studied absorption from 2 to 11 microns for ten different p-type dopant levels and 8 different n-type dopants. They found that the IR absorptance is roughly independent of the type of dopant but only depends on the level of carriers (holes or electrons). Their results for 10 micron wavelength are summarized in a graph shown below. These data are in pretty good agreement with those of Spitzer and Fan.

Thus it is concluded that a dopant level of ~ 1 x 10¹⁸ cm^-3 is required to get 28% absorption of the incident beam. This corresponds to a resistivity of the rough order of 0.02 ohm-cm.

Figure 5. Dependence of IR absorptance on carrier concentration

Figure 6. Dependence of IR absorptance on impurity concentration

Figure 7. Dependence of IR absorptance on carrier concentration

7. Optical properties of SiO₂ films on thin Si wafers

Refer to the paper:

"The IR optical properties of SiO2 and SiO2 layers on Si," by H. R. Phillip, J. App. Phys. 50, 1053 (1979)

The author points out that a convenient way to characterize IR properties of films is to grow the films on thin wafers of silicon "since the thin silicon substrate is essentially non absorbing at these energies for modest doping levels." [Note: "these energies" refers to the range 7 to 15 microns, which is just the range we are interested in for bodies at ~ 100 °C].

It turns out that SiO2 has a region of strong absorptance between 8 and 30 microns.

Figure 8. Absorption of IR by silicon dioxide films

When measurements are taken for films of SiO2on thin silicon substrates (~ 0.25 mm) the results are as shown below.

To get an IR absorptance of ~ 28% of an incident beam over the spectrum of a body at 100 °C, we would probably need an SiO2 film of roughly 2 microns thickness.

Figure 9. Absorption of IR by thin silicon wafers with various thicknesses of silicon dioxide