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We show that nearly perfect absorption can be achieved in a simple structure with highly doped silicon on a sapphire (SOS) substrate. An SOS structure with the n-Si film being 600 nm thick and having doping concentration of 2e19 cm−3 has an absorption peak of 96% in the film at a wavelength of 12.1 μm. More generally, 95% absorption in the n-Si can be achieved and tailored to specific wavelengths in the range of 11.6-15.1 μm utilizing dopings of 1-2.4e19 cm−3 and film thicknesses of 600-1000 nm. Regions of 90% absorption can be achievable down to 11 μm and up to as much as 22 μm with tailoring of doping and film thickness. It is also shown that choice of substrate with large k/n (imaginary over real part of refractive index) is imperative for high absorption in the thin-film and will play a role in tailoring possibilities. Shown here are results for n-Si, but in general these results also apply to p-Si and the methods may be used to investigate structures with alternative films or substrates. This investigated SOS structure has high potential since desired film thickness and doping investigated here for perfect absorption can be purchased commercially and easily tuned by etching the silicon film.
©2013 Optical Society of America
1. Introduction
The theory of ultra-thin-film perfect absorbers of VO2 for the ~11-12 μm infrared wavelength range has been developed recently by M. A. Kats and associates [1] who found experimentally that a λ/65 film of VO2 on an opaque sapphire substrate could be thermally tuned into a 99.75% absorption state at normal incidence, although only ~90% of the absorption occurs in the thin film. This type of perfect absorber is very worthwhile since it requires no patterning, easing fabrication constraints. Although VO2 is interesting and the thermal tuning is valid, we believe that there is a much wider range of ”perfect absorber” applications that calls upon a commercially available thin film of crystalline silicon on a realistic substrate of sapphire (SOS). We have extended the theory to SOS materials over a wider long-wavelength infrared range of use since (1) the silicon film is not required to be as thin as that for VO2, (2) the desired film thickness for perfect absorption or minimum reflectivity can be selected readily from the vendor or by etching a thick film, and (3) the substrate thickness of sapphire (Al2O3) can be that of a standard wafer. In general p-type or n-type doping of Si can be employed to tune the film for optimum absorption, although n-type is focused on here for simplicity. Reference [1] points out that “modulation” of the IR absorption in VO2 can be accomplished by a thermally induced phase change. Using the inherent plasma frequency tunability of highly doped silicon, which is also CMOS compatible, has been of interest recently for long-wave infrared plasmonics [2–4]. Strong-to-perfect absorption has also been investigated for thin film doped-Si on SiO2 with a focus on implementation of a high index germanium layer to increase absorption [5] which has merit. Structured perfect absorbers [6] may also be designed for long wavelengths but with increased fabrication difficulty. The investigations presented here focus on the tunability of the simpler SOS structure via doping along with thickness design, to enable near perfect absorption strictly in the thin-film at selective wavelengths. Silicon does not possess the aforementioned phase transition, but is nevertheless amply capable of electro-optical as well as thermo-optical modulation of perfect absorption–for example by electrical injection of electrons and holes into Si, which is an alternative approach to that of the “lattice change” of VO2.
2. Absorption theory of the two-layer SOS device
The perfect absorber in its simplest form is a thin layer of heavily doped Si on a finite-thickness substrate “layer” as illustrated in Fig. 1 (left) where unpolarized infrared light is assumed to be normally incident upon the Si film with thickness d. The sapphire has a complex index of refraction, η = n + ik, which is a function of free-space wavelength λ. Here the extinction coefficient k is a measure of infrared absorption and, because the optical phonon spectrum of sapphire has strong resonance features, we find from the literature [7] that the k exhibits peaks in the region of interest (Fig. 2 left), and more noteworthy for this work that k/n exhibits four peaks as shown in Fig. 2 (right). Here we note that the k/n ratio exceeds 10 in three peak spectral zones that are shown, which will be investigated. While the k/n of the substrate is important, to attain near perfect absorption in the film it is necessary to employ the “plasmonic” properties of the film. This is done by introducing n-type or p-type doping of sufficient strength to move the plasma frequency into the long-wave range [2–4], or to wavelengths just longer than that of the desired near perfect absorption.
Fig. 2 Wavelength dependent (left) n and k, and (right) k/n for sapphire [7]. The dashed line indicates k/n = 10 and the wavelength regions with k/n > 10 are indicated.
Now we derive analytical theories to be utilized in terms of the complex indices and thicknesses of film and substrate. Optical constants for heavily doped n-type silicon are determined via the Drude model. In the IR range of interest k of undoped silicon is at most ~10−3 [8] and k of highly doped silicon is on the order of unity and, therefore, the material is dominated by free carrier effects which validates the model. The complex refractive indices are determined by
where ω = 2πc/λ is the infrared frequency and ε∞ is the high frequency dielectric constant which is taken to be 11.7. The plasma frequency is determined bywhere λp is the plasma wavelength, N is the doping density (free electron concentration for n-type), e is the electron charge, εo is the permittivity of free space, and m* is the effective mass which for n-Si is taken to be 0.272 [9] times the electron mass. The relaxation time is determined bywhere μ is the carrier mobility and is determined as function of N according to Reference [10]. Figure 3 presents n and k for n-Si with N = 1e19 and 5e19 cm−3 as calculated using Eqs. (1)-(3). These materials become metallic at wavelengths greater than the crossing of n and k,with these dopings enabling this crossing in the 10-22 μm range. These dopings are specifically pictured since these structures operate near this metallic regime as will be discussed later.The absorption in the thin-film is found analytically via standard Fresnel equations for conventional thin-film wave propagation in absorbing media [11]. For this work, the light is unpolarized and is incident normal to the surface. The reflection and transmission coefficients can be found first for each single interface (01, 12, and 23 in Fig. 1) in the structure via
For the above general interface, the incident and transmitted mediums are referred to with the index of “i” and “j”, respectively. The “ith” layer always indicates the medium directly above the “jth” layer. A phase constant is calculated for the layers with finite thickness (1 and 2 from Fig. 1) viaThe reflection and transmission coefficients for multiple interfaces,are calculated backwards starting from the “jth” layer being the final medium in the structure, indicated by the index “f”, and which will either be the sapphire substrate or air in this work. For the simplified case with sapphire being the substrate, or the sapphire being treated as infinitely thick, Eq. (6) is used to determine r0,2 and t0,2 which are the final reflection and transmission coefficients. In the case of the structure with a finitely thick sapphire substrate, or where the final medium is air, Eq. (6) is first used to calculate r1,3 and t1,3. These are then used in conjunction with r0,1 and t0,1, again in Eq. (6), to determine r0,3 and t0,3. The overall reflection and transmission for either case is then found bywith the absorption then being found byUtilizing Eqs. (1)-(8) along with the frequency−dependent n and k of the substrate, we are now in position to calculate the absorption of the two-layer device.3. Arbitrary film on sapphire
Moving away for a moment from silicon, Fig. 4 presents contour absorption plots calculated for films with arbitrary optical constants n and k on sapphire at a specific film thickness and wavelength. For this case, the final medium is an infinitely thick sapphire substrate in order to first understand guidelines of behavior. Black scatter points are also plotted, corresponding to Drude optical constants for n-Si, to illustrate where this material may be useful in this type of absorption scheme. The n = k curve is plotted as a dashed line, above which the film becomes metallic. Figure 4 (top) presents the absorption contour as a function of n and k for a film that is 600 nm thick and at a wavelength of 11.75 μm. In this case the absorption maxima, approximately ~95%, lies on the n-Si curve in the N range of 2-3e19 cm−3. It is noted that the absorption of ~95% using n-Si on sapphire at 11.75 μm is a 5% improvement on those reported by [1] for the VO2 film that also is on sapphire at the same wavelength. Figure 4 (bottom-left) presents a similar absorption contour for a 600 nm thick film with the wavelength now shifted to 13.9 μm. In this case the absorption maxima lays offset from the n-Si optical constant curve. In order to optimize for a wavelength of 13.9 μm, the thickness must be modified to match optical constants available to n-Si. Figure 4 (bottom-right) demonstrates this with an adjustment of the film thickness to 850 nm. At this wavelength the absorption maxima occurs within the n-Si doping range of 1-2e19 cm−3. This is proof of principal that n-Si has strong tailoring possibilities for such strongly absorbing thin film structures.
Fig. 4 Absorption contour of the thin film with arbitrary optical constants on infinitely thick sapphire. (top) λ = 11.75 microns, 600 nm thick, (bottom-left) λ = 13.9 microns, 600 nm thick (bottom-right) λ = 13.9 microns, 850 nm thick film. The black curve indicates the Drude optical constants for n-Si with carrier concentrations labeled in units of cm−3. The dashed line illustrates n = k.
4. N-Si on 750 um of sapphire
Motivated by the tailoring possibilities of n-Si, wavelength-dependent absorption spectra are calculated for thin Si films with d = 600 and 850 nm, and N = 2e19 cm−3. Figure 5 presents these results for wavelengths up to 35 μm. Here, infinite sapphire thickness was assumed to analytically observe the absorption in the thin film only (similar to Fig. 4). For the 600 and 850 nm films, the absorption peaks are 95.7 and 95.2% and occur at 12.1 and 12.4 μm, respectively. Increasing the film thickness from 600 to 850 nm also resulted in an increase of full-width half-maximum of the peak by 6%. Figure 6 presents the absorption in the entire structure for the same thin films on 750 μm of sapphire for comparison. This absorption calculation predicts what may be empirically measured from optical transmission and reflection measurements. For the 600 and 850 nm films, the absorption is > 99.9% and 97.6% respectively. Oscillations observed at the low and high wavelength ends of the spectra are Fabry-Perot fringes related to the thickness of the thin-film and substrate, respectively. Further analytical calculations reveal that sapphire substrates as thin as 5 μm give the same peak absorption observed in Fig. 6 for each respective film. Increasing of the substrate thickness from ~5 μm appears to only effect the fringes and baseline, leaving the peak absorption relatively unchanged.
Fig. 5 Absorption of the n-Si thin-film with N = 2e19 cm−3, d = 600 nm (left) and 850 nm (right). The substrate is infinitely thick sapphire.
Fig. 6 Absorption of the entire structure with N = 2e19 cm−3, d = 600 nm (left) and 850 nm (right), and on 750 μm thick sapphire.
For practical fabrication of SOS structure with desired absorption performance, we now investigate the effects of a plus or minus 10% change in both doping and film thickness for the case with N = 2e19 cm−3 d = 600 nm (Fig. 5 left). Simulations indicate that the peak wavelength of 12.1 μm may shift by as much as 3% with the full-width half-maximum and absorption of the peak varying by up to 4% and 2% respectively. This serves to give a guide of effects of non-ideal fabrication while also pointing to the tunability of this particular SOS structure which will be elaborated on later.
5. Absorption depth profile
To supplement the above developed analytical approach, Lumerical FDTD (finite-different time-domain) software was used to model the electromagnetic absorption in the n-Si films with film specifications used in Figs. 5-6 but with the entire structure thickness including sapphire being 11 μm for computational purposes. This decrease in substrate thickness is valid since at the wavelength of peak absorption, this absorption occurs within 5 μm into the substrate from the film interface. Plotted in Figs. 7 and 8 (left) are 1D plots showing the power absorbed as function of the depth, x, of the structures at the peak absorption wavelength. The film boundaries are indicated by vertical red lines. Summing over the power absorbed in the film and dividing by the power absorbed overall in the structure confirms that the absorption occurs in the film for each respective thickness on a realistic substrate to within 1% of that determined analytically according to Figs. 5 (left) and (right) if we assume the overall structure absorbs ~100% and ~98% as seen in Fig. 6 (left) and (right) respectively. We take the results of Fig. 5 to be more accurate though due to inaccuracies of meshing in the FDTD software and truncation of the substrate. Figures 7 and 8 (right) present spatial contours of the power absorbed in the film and partially into the sapphire substrate. The dark red clearly illustrates the thin film with the contour being intentionally scaled to accentuate the order of magnitude lower absorption found in the initial portion of the substrate relative to Figs. 7 and 8 (left). The power absorbed of the light penetrating the substrate decreases by an order-of -magnitude approximately 1 micron into the substrate further indicating that ~1 um of sapphire is generally sufficient. These FDTD calculations demonstrate the validity of analytically calculating the absorption in the thin-film as long as the substrate is significantly thick.
Fig. 7 FDTD calculation of power absorbed for a structure with 600 nm of 2e19 cm−3doped n-Si on 10.4 μm thick sapphire at the absorption peak wavelength of 12.1 μm. (left) is a 1D slice of the contour with the x-axis representing the axis perpendicular to the film and (right) is a contour of the spatial power absorbed in the structure.
Fig. 8 FDTD calculation of power absorbed for a structure with 850 nm of 2e19 cm−3doped n-Si on 10.15 μm thick sapphire at the absorption peak wavelength of 12.5 μm. (left) is a 1D slice of the contour with the x-axis representing the axis perpendicular to the film and (right) is a contour of the spatial power absorbed in the structure.
6. Optimized absorption via N and d
Figure 9 presents analytically calculated contours of absorption in n-Si films as a function of incident wavelength and doping. The colorbar is chosen to accentuate greater than 70% absorption with dashed and solid black lines also plotted to indicate regions of 90 and 95% absorption, respectively. The stacked plots in Fig. 9 give the contours for n-Si thicknesses of 600, 850, 1000, and 1500 nm with all scales being uniform. The observed results for doping of 2e19 cm−3 and thicknesses of 600 and 850 nm are in agreement with Fig. 5. The 600 nm film may achieve 95% and 90% absorption in the wavelength ranges of 11.6-12.6 μm and 11.1-13.5 μm respectively (see also Table 1). The areas of 95 and 90% absorption increase with a film thickness of 850 nm where peak absorption occurs in the 12.3-14.4 μm and 11.4-15.0 μm wavelength ranges, respectively. Beyond 850 nm, this absorption region begins decreasing in area although with red-shifting. Figure 8 and Table 1 clearly demonstrate how the film may be tailored via doping and thickness to achieve 95% absorption in roughly the 11.6-15 μm range with the thinner and thicker films better enabling the lower and higher ends of that absorption window, respectively.
Fig. 9 Absorption contours as a function of wavelength and doping for the n-Si film. Shown is only absorption > 70% for clarity. The dashed line is the 90% line and the solid black line is 95% absorption. The film thickness of 600, 850, 1000 and 1500 nm show the tailoring possibilities for maximum absorption on 750 um of sapphire.
Table 1. Limits of wavelength and carrier concentration giving 90 and 95% absorption for n-Si thin films on sapphire
Absorption calculations similar to those shown in Fig. 9 were completed with film thickness from 300 to 2200 nm at 50 nm increments. At film thicknesses decreasing from 600 nm, we observed that the 90 and 95% absorbing regions decrease in area with the former remaining only near 11 μm at 400 nm thick and the latter not appearing at even 550nm. At 350 nm thick, no 90% absorption remains. Increasing the thickness to 1500 nm also enables a second, higher wavelength region of greater than 90% absorption (Fig. 9 bottom), which appears between 16.1 and 17.1 μm. This corresponds to a region of k/n > 10 for sapphire according to Fig. 2 (right). Our calculations indicate for this wavelength region that 95% cannot be reached for this simple structure although that may be feasible by adding one or more layers such as a Ge layer [5]. A third region of high k/n for sapphire is found in the range of 21.1-22.3 μm according to Fig. 2 (right). 90% absorption in this region begins with films as thin as 1100 nm with 5-6e18 cm−3 doping. In this wavelength range, absorption never reaches 95%, although it reaches 94% for the doping of 5e18 cm−3 with films 1450-1950 nm thick. The results indicate that the k/n > 10 is an important figure of merit to achieve greater than 90% absorption in thin-films such as these. The wavelength of the absorption region then may also be tailored by choice of substrate.
It is worth noting that for the thinnest film shown in Fig. 9 (600 nm), the doping related to plasma wavelengths for the n-Si, as determined by Eq. (2), crosses over the upper-left hand portion of the 90% absorption region. This is the only region that may be “metallic” with negative permittivity although the permittivity is relatively small. For increasing film thicknesses, the 90% absorption region pushes to lower doping and longer wavelengths, with the latter being in the positive permittivity regime for the n-Si. Therefore, the highly absorptive region is predominantly related to a dielectric, albeit highly absorbing, film.
7. Alternative substrates for perfect absorption using n-Si
We expect that similar ~95% absorption results at long-wave can be attained when a different, finite-thickness opaque substrate is substituted for sapphire in our two-layer device. In choosing a perfect-absorber substrate material for the top n-Si film, we need only to examine the wavelength-dependent n and k data of that material to find infrared regions for which the k/n ratio is high, preferably greater than 10. Applying the large k/n criterion, we identify SiC, GaP, InP, and GaAs as practical choices of substrate for perfect infrared absorption in the two-layer with each having k/n that is at least on the order of 10 near wavelengths of 11.5, 26, 30.5, and 35.5 μm respectively [8]. Heavily doped semiconductors, both group IV and III-V, and transparent-conductor material, such as indium tin oxide or aluminum zinc oxide, may be useful substrates for this two- layer absorber. In those cases, the near metallic substrate may result in the overall loss of transparency limiting potential use as tunable, transmissive filters, although such structures may still enable zero reflectors.
Perhaps the most notable alternative opaque substrate material is SiO2 which is the buried-oxide layer that is used in silicon-on-insulator (SOI) and is beneficial because an SOI wafer may be useful in conjunction with relevant ion implantation. We find that for SiO2 k/n peaks at ~4.5 at 8.9 μm [12]. Using the same analytical simulation methods shown earlier, we find that ~85% absorption can be achieved in the thin film for the simple structure of n-Si on SiO2. It is worth mentioning that introducing a second layer on top of the Si, such as high index germanium, this absorption can be greatly increased [5].
Another worthwhile note is that particular heavily doped semiconductors such as Ge and InAs, or heavily doped zinc oxides, all of which have been of interest lately for infrared plasmonics [13–17], may also be useful as tunable thin films for similar structures and should be investigated. In this case, these materials can be heavily doped to the point of pushing the plasma wavelength to slightly longer wavelengths than desired perfect absorption similar to the n-Si doping in this investigation. The aforementioned heavily doped materials all may exhibit various advantages relative to each other for thin-film perfect absorbers such as easing of fabrication constraints, i.e. target film thickness and doping having broad acceptable ranges, compatibility to industry standards, and lattice matching to the potential substrate.
8. Potential applications
The present two-layer device may be lightweight, easy to manufacture, large in area if desired and potentially low in cost. The overall thickness needs only to be ~5 μm, as discussed in section 5, which opens up the possibility of having high transmission through the device within certain spectral ranges. Substrate thickness less than 5 μm with high absorption is still possible and of course is material dependent which will require further investigations. However we can consider this device as a lightweight infrared “band blocking filter” having a main band of absorption and several wavelength regions of transmission. This “blocker” can fully absorb light whose wavelength(s) fall within the principal absorption band, a band that can be narrow. A given long-wave IR source can be absorbed by appropriate tuning of the device’s absorption band center wavelength—tuning obtained by optimizing d and N of the Si thin film and choice of substrate. After the light has been absorbed and converted to heat in the Si thin-film, some of that heat is transferred to the substrate. Thus the device offers substrate-assisted dissipation that reduces the temperature rise in the Si below that which would otherwise have occurred in air-clad Si.
Since the infrared emissivity is proportional to absorption, a resulting peak in emissivity is exhibited at the wavelength of peak absorption in the thin-film structure. Consequently, when the device is deliberately heated, the resulting black body emission spectrum will be modified from that of an ideal black-body radiator whose absorption is relatively spectrally flat. With the emission strength depending upon the tuning mechanisms discussed in this work, the two-layer device can be fashioned into an infrared scene generator by making a photo lithographically formed pattern of thickness variation in the Si film. A scene generator based on a thin-film structure has been demonstrated for doped silicon on SiO2 via etching thickness of top high-index Ge layer [8]. Since the near perfect absorption has a carrier concentration dependence, a pattern or “scene” of infrared absorption can also be created in the device by varying concentration on a point-by-point basis over the spatial extent of the device. The carrier density can be spatially “modulated” in several ways. Regarding electron-hole pairs, electrical PIN injection and optical injection are both feasible. For the latter, a short-wavelength laser beam focused on the Si could generate ~1e19 cm-3 e-h pairs in a local spot. If the silicon is intrinsic or lightly-doped prior to carrier injection, the injected spot would have very high absorption within a background area of low absorption. Hence we could have a dynamic, optically controlled scene.
9. Summary
Presented in this work are results demonstrating the nearly perfect absorption that can be achieved via highly doped, sub-wavelength thickness n-Si on sapphire in the long-wave infrared wavelength range. We find that n-Si on sapphire can achieve absorption in the thin film of ~95% at 11.75 μm, which is a 5% improvement on those reported by [1] for absorption in a VO2 thin film at the same wavelength. More generally we find that this SOS structure can achieve 95% and 90% in the thin-film n-Si at least in the wavelength ranges of 11.6-15.1 and 11.1-15.6 respectively. The data presented here in Fig. 9 and Table 1 are indicative of how film thickness and n-Si doping may be utilized to achieve maximum thin film absorption tailored to wavelength.
The highly absorptive regions of 95% discussed here in the long-wave infrared roughly correspond to regions where k/n > 10 for the sapphire substrate. Other absorption regions of greater than 90% are found also in regions of k/n > 10 at longer wavelengths for sapphire. Comparisons with SiO2 as a substrate indicate that only ~85% absorption maybe found where the SiO2 k/n peaks at ~4.5 at a wavelength of 8.9 μm. Other substrates with high k/n may have similar potential with the choice of substrate proving to be another parameter in tailoring the absorption feature in this simple structure.
This SOS investigated structure has high potential since desired film thickness for perfect absorption or can be selected readily from the vendor or fabricated by etching a thick film on standard sapphire wafers and are likely low cost. This SOS structure also has potential due to silicon’s capability of modulation of perfect absorption, potentially by electrical injection of carriers.
Acknowledgments
This work is supported by Air Force Office of Scientific Research. J.W.C acknowledges AFOSR LRIR No. 12RY10COR (Program Officer Dr. Gernot Pomrenke), R. S. acknowledges Grant Number 9550-10-1-0417, and J.R.H would like to acknowledge support under LRIR No. 12RY05COR.
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