NANO LETTERS
Analytic Model for Diffuse Reflectivity of Silicon Nanowire Mats
2009 Vol. 9, No. 10 3494-3497
Robert A. Street,* William S. Wong, and Christopher Paulson Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, California 94304 Received May 27, 2009
ABSTRACT Disordered silicon nanowires are diffuse optical reflectors, with reflectivity modified by the nanowire absorption. We present an analytical model which describes the reflectivity, absorption, and transmission of a nanowire mat, across a wide spectral range, and including substrate effects. The model provides the ability to predict the optical properties of other nanowire mat structures, including core/shell heterostructures.
The optical properties of nanowires and nanotubes are strikingly different from those of thin films and are the subject of active research because of their application to solar cells and other optical devices.1-3 Ordered vertical nanowires have low reflectivity, and their optical interactions can be calculated by solving Maxwell’s equations for the ideal structures.4-6 Disordered nanowire mats have different properties with significant diffuse reflectivity arising from multiple scattering.5,6 Silicon mats have a yellow or brown appearance due to the interaction between scattering and absoption in the nanowires. It is not practical to calculate the optical properties of such mats from first principles. Instead, we describe an analytical model for the reflectivity, absorption, and transmission of disordered mats and show its application to silicon nanowires. The model readily predicts the optical properties of other strongly scattering disordered nanowire systems. The model assumes a uniform scattering mat in front of a partially reflecting surface (the silicon wafer), as illustrated in Figure 1. The various reflectance, absorption, and transmission parameters associated with the mat and the substrate are indicated in the figure. The optical scattering cross section and the optical absorption in the silicon nanowires both increase at shorter wavelength. The spectral range of interest is from ∼350 nm, where the optical absorption coefficient of silicon is ∼106 cm-1, and there is complete absorption within a single nanowire, to long enough wavelengths when the nanowire mat is essentially transparent. To determine the reflectivity and absorption of the mat, initially without considering the substrate, we define a distribution function P(N), which is the probability that the photon will be reflected out of the mat after N scattering events, in the absence of absorption. At each scattering event, there is a probability of absorption in the nanowire of * Corresponding author,
[email protected]. 10.1021/nl901683y CCC: $40.75 Published on Web 07/09/2009
2009 American Chemical Society
Figure 1. Schematic diagram of a nanowire mat on a silicon substrate, showing the various reflectivity, transmission, and absorption parameters.
exp(-RdNW), where R is the silicon absorption coefficient and dNW is an absorption length, expected to be roughly the nanowire diameter. Hence RN )
∫
∞
0
P(N) exp(-NRdNW) dN
(1)
where N is assumed to be large enough to be represented as a continuous variable. Since light is scattered more or less at random in the mat, the probability that the photon is reflected out of the mat is independent of the number of scattering events, from which it follows that P(N) )
1 exp(-N/N0) N0
(2)
Assuming negligible transmission of light through the mat RN )
1 N0
∫
∞
0
exp(-N/N0) exp(-NRdNW) dN )
1 1 + N0RdNW (3)
Equation 3 has a particularly simple form; RN is ∼1/R when R is large and RN is ∼1 when R is small.
Transmission through the mat needs a light-scattering model. The scattering is determined by the Rayleigh-Mie theory7 and is a complex function of the wavelength, the nanowire size, and the density of the mat. Fortunately, provided the scattering is strong enough that the mat is minimally transmitting, the details of the scattering are not important. On the other hand, at long enough wavelength where the mat is transmitting, a simple Rayleigh-type model is accurate enough. Scattering depends on wavelength as λ-4 for particles and λ-3 for rods,8 so we apply the rod model. The transmitted light through the mat is therefore modeled by the relation9 T ) exp(-σwλ-3)[1 - A(w)]
(4)
where w is the total width of the mat and σ is the scattering cross section for the particular mat. The exponential term represents the transmission in the absence of nanowire absorption, and the A(w) term is the absorption of the mat. We make the further approximation that the absorption is given by 1 - RN from eq 3, to obtain T)
1 exp(-σwλ-3) 1 + N0RdNW
(5)
The approximation for A(w) is valid provided the mat is thick enough that there is substantial transmission only when N0RdNW < 1. The optical properties of the substrate also need to be considered when there is significant transmission through the mat. A fraction T of the incident light is transmitted through the mat. Of this transmitted light, a fraction RS is reflected from the substrate and the remainder is assumed to be absorbed by the substrate (see Figure 1). The light reflected by the substrate is again either transmitted through the mat with probability T or reflected back to the substrate with RN. Including multiple reflections between the nanowires and the substrate, the total reflection, R, is given by R ) RN + TRST + TRSRNRST + T(RSRN)2RST + ... ) RST2 RN + 1 - RSRN
(6)
Equations 3, 5, and 6 give the total reflectivity and absorption. The model was compared to measured reflectivity data. Silicon nanowires were grown at 450 °C on (100) and (111) silicon wafers using the VLS method and a gold catalyst which is either nanoparticles in solution or an evaporated and annealed gold film. The side view scanning electron micrograph of a typical nanowire mat, in Figure 2, shows that the nanowires are disordered and have diameters ranging from 50 to 200 nm and the mats are about 10 µm thick. Data are reported only for samples with a uniform diffuse appearance. Measurements of diffuse reflectance are made with an integrating sphere attachment to a spectrophometer, over a Nano Lett., Vol. 9, No. 10, 2009
Figure 2. Photograph of a disordered silicon nanowire mat.
wavelength range from 350 to 2500 nm. The reflectance measurements are calibrated with a BaSO4 white powder reflector. Reflectivity measurements involve a sample pressed against a sample holder, placed against an aperture of the integrating sphere. There is a visible change in appearance where the sample has touched a surface, and scanning electron microscopy shows that the nanowires are slightly flattened. Multiple measurements of the same sample give qualitatively the same spectrum, but the magnitude of the reflectivity often changes by about 10%, which we attribute to flattening or other distortion of the nanowires. After the gold catalyst is etched off the nanowire mat, the reflectivity spectrum retains the same shape although the magnitude of the reflectivity sometimes changes slightly. We attribute this to small changes in the packing of the nanowires after exposure to the etch, similar to the changes caused by samples that have been pressed against a surface. Hence we do not believe that the gold significantly affects the spectrum. Measurements of the diffuse reflectivity from two different nanowire samples are shown in Figure 3 and are broadly typical of other samples. The data tend to a constant reflectivity of 10-30% beyond 1500 nm, increasing to a peak at 600-900 nm, and then decreasing rapidly at shorter wavelength. There is a small residual reflectivity below 400 nm. The peak reflectivity is 60% in one sample and 35% in the other.10 Figure 3 shows that the analytical model provides an excellent fit to the data, accounting for the peak reflectivity at 600 and 900 nm in the two samples and the decrease at both shorter and longer wavelengths. The fit provides values of N0dNW of 3.3 × 10-4 and 1.8 × 10-4 cm for the two samples. Published values for the silicon absorption coefficients were used.11 Figure 3 also shows how the model applies in two limiting cases. The zero transmission case corresponds to an infinitely thick mat and is given by eq 3. The fit is adjusted to account for the small residual reflection, RNS ∼ 3-4%, of the surface of the mat, by R ) RNS + RN(1 - RNS)
(7)
In the region of high absorption, R is ∼1/R, as predicted by the model. The role of the substrate is highlighted in Figure 3 by assuming zero absorption in the mat, so that RN + T ) 1. 3495
Figure 4. Plot of the total absorption depth as a function of photon energy, extracted from the model fit for sample A.
Figure 3. (a) Diffuse reflectivity spectrum of a silicon nanowire sample (A) and the fit to the model (circles) as described in the text. The fitting parameters are shown. (b) Diffuse reflectivity spectrum of a second silicon nanowire sample (B) and the fit to the model with the fitting parameters shown.
The reflectivity approaches 100% at ∼500 nm and drops to the substrate reflectivity above 1600 nm. The model evidently accounts well for the increasing diffuse reflectivity up to 800-1000 nm in both samples. The RS parameter is close to the known reflectivity of a silicon crystal in the data of Figure 3a but substantially smaller in Figure 3b. Even though the nanowire mat is evidently transparent at wavelength >1500 nm, nanowire scattering remains significant, as measurements show that the reflectivity is diffuse. Significant specular reflectivity only begins near 2500 nm, indicating the start of the region where the nanowires have negligible scattering. The analysis allow various domains of reflectivity and absorption across the spectrum to be identified: Strong Absorption (r > 105 cm-1). Light is highly absorbed by a nanowire, so there is minimal light scattering and the reflectivity is dominated by backscattering from only the top layer of nanowires. Photons scattered from nanowires deeper into the mat have a high probability of interacting with other nanowires and a high probability of absorption. Medium Absorption and Strong Scattering. Individual nanowires are mostly transparent, but light scatters from a large number of wires so the overall absorption is large. The reflectivity increases steadily at increasing wavelength as the absorption coefficient decreases. 3496
Figure 5. Calculated reflectivity for silicon nanowires, and nanowires coated with an increasing fraction of amorphous silicon, based on eq 9, with d2/d1 ) 0.1-2, as indicated. The amorphous silicon suppresses the reflectivity because of its higher absorption coefficient.
Low Absorption and Significant Transmission. The silicon optical absorption coefficient is too low for significant absorption in the nanowire mat. At longer wavelengths, an increasing fraction of the photons reaches the silicon substrate after a small number of scattering events. The model makes several assumptions which place limits on its validity. The mat must be thick enough that at wavelengths when it is significantly transparent, the absorption in the nanowires is small and scattering can be described by the Rayleigh approximation. For the measured samples, there is significant transparency above ∼700-800 nm, for nanowire thicknesses of ∼100 nm, which is evidently a large enough, λ/dNM, for the model to apply. The analytical model allows us to extract various parameters and predict the reflectivity and absorption in other systems. The average number of scattering events in the mat before reflection is N0, which we expect to be reasonably independent of the scattering cross section of a particular mat. For sample A, N0 is about 30, assuming that the nanowire diameter is ∼100 nm. The smaller value of N0dNW for sample B may be due to smaller diameter nanowires. The model also shows that the scattering parameter σw is 3 times smaller for sample B than for sample A, presumably also because it is a less dense mat with thinner nanowires. Nano Lett., Vol. 9, No. 10, 2009
The long wavelength reflectivity in sample B is only ∼10%, for reasons that are not clear since it is lower than expected reflectivity from a flat silicon surface. Possibly there is a rougher silicon layer at the surface due to the initial nanowire growth, which couples more light into the substrate. In some samples, we observe an increase in reflectivity between 1000 and 1200 nm, which is the range that the substrate goes from absorbing to transparent, so that there is evidently some contribution from the back side of the substrate. The total effective absorption length, dTOT, defined by RN ) exp(-RdTOT), is much larger than the nanowire diameter because of the multiple scattering, particularly when an individual wire is weakly absorbing. From eq 3 dTOT
1 ) ln(1 + N0RdNW) R
(8)
Figure 5 shows the predicted reflectivity for silicon nanowires with the parameters used for sample A and different fractions, d2/d1 of deposited amorphous silicon, using the known absorption of a-Si.12 The model assumes no transmission through the mat, and neglects the residual surface reflectivity. The results are consistent with the previously published data,6 and suggest that d2/d1 ∼ 0.3 for the sample measured. In summary, we have developed a simple analytical model for the diffuse reflectivity of thick disordered nanowire mats, which takes into account the nanowire absorption and the optical properties of the substrate. The model explains the reflectivity measurements and allows the optical properties of different nanowire systems to be predicted. Acknowledgment. The authors are grateful to R. Lujan for assistance with nanowire sample deposition. References
The total absorption length is N0dNW when R is small and decreases with increasing absorption. Figure 4 plots the total absorption length for the sample of Figure 3a and shows dTOT increasing from the nanowire diameter at 3.5 eV to more than 3 µm at 1.5 eV. The data are shown on an energy scale to illustrate the approximately linear relation over most of the visible spectrum, which is a coincidence of the shape of R(E) for silicon. The increasing absorption depth at low energy is of particular interest for solar cell applications as it extends the region of high absorption to much lower energy than the equivalent thin film. In previous work we showed that when silicon nanowires are coated with amorphous silicon, the reflectivity of the mat decreases, due to the additional absorption of the deposited layer.6 The effect can be modeled by replacing eq 3 with RN )
1 1 + N0(R1d1 + R2d2)
(9)
where the subscripts denote the silicon core and the deposited film and d1 and d2 are the effective absorption depths.
Nano Lett., Vol. 9, No. 10, 2009
(1) Tsakalakos, L.; Balch, J.; Fronheiser, J.; Shih, M.-Y.; LeBoeuf, S. F.; Pietrzykowski, M.; Codella, P. J.; Sulima, O.; Rand, J.; Kumar, A. D.; Korevaar, B. A. J. Nanophotonics 2007, 1, 013552. (2) Peng, K.; Wu, Y.; Fang, H.; Zhong, X.; Xu, Y.; Zhu, J. Angew. Chem., Int. Ed. 2005, 44, 2737. (3) Theocharous, E.; Deshpande, R.; Dillon, A. C. Lehman. J. Appl. Opt. 2006, 45, 1093.1. (4) Zhu, J.; Yu, Z.; Burkhard, G. F.; Hsu, C.-M.; Connor, S. T.; Xu, Y.; Wang, Q.; McGehee, M.; Fan, S.; Cui, Y. Nano Lett. 2009, 9, 279. Hu, L.; Chen, G. Nano Lett. 2007, 7, 3249–3252. (5) Muskens, O.; Rivas, J. G.; Algra, R. E.; Bakkers, E. P. A. M,; Lagendijk, A. Nano Lett. 2008, 8, 2638. (6) Street, R. A; Qi, P.; Lujan, R.; Wong, W. S. Appl. Phys. Lett. 2008, 93, 163109. (7) Silverman, M. P. WaVes and Grains; Princeton University Press: Princeton, NJ, 1998. (8) Van de Hulst, H. C. Astrophys. J. 1950, 112, 1. (9) The form of the transmission in the absence of absorption depends on the scattering properties. An alternative expression, T ) (1 + Sw)-1, (see ref 7) gives an increased transmission at short wavelengths but leads to a similar fit to the data. (10) In the previous data (ref 6) the reflectivity is overestimated by about 20% due to an inaccurate calibration with a white reflector. (11) www.virginiasemi.com. Note that published values for the optical absorption coefficients for crystalline silicon differ significantly. (12) E.g.: Street, R. A. Hydrogenated Amorphous Silicon; Cambridge University Press: Cambridge, 1990; p 90.
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