Langmuir 2002, 18, 4165-4170
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Structure Characterization of Porous Silicon Layers Based on a Theoretical Analysis Caide Xiao,† Rabah Boukherroub,‡ James T. C. Wojtyk,‡ Danial D. M. Wayner,‡ and John H. T. Luong*,† National Research Council Canada, Biotechnology Research Institute, Montreal, Quebec, H4P 2R2 Canada, and Steacie Institute of Molecular Sciences, Ottawa, Ontario, K1A 0R6 Canada Received January 4, 2002. In Final Form: March 11, 2002 A theoretical framework is presented to allow for the determination of the basic structural parameters of a porous silicon thin layer using constructive-destructive positions of Fabry-Perot fringes in air. The structural parameters include film thickness, porosity, and refractive index between 350 and 2000 nm. The model is general and can be applied to determine the optical properties of any thin layer that exhibits a Fabry-Perot fringe pattern. Such information is of importance to fabricate practical sensing devices with low costs for various applications in biomedicine and analytical and environmental chemistry.
1. Introduction Since the first report of visible photoluminescence of porous silicon (PSi),1 there has been considerable effort to determine structural, optical, and electronic properties of this material. After an initial focus on optoelectronic properties,2 recent advances in organic modification of silicon surfaces, including PSi,3-13 have led to possible applications of such materials for chemical/biochemical sensing.14-21 One of the ongoing challenges is the structural characterization of the porous silicon. Various methods * To whom correspondence should be addressed. † Biotechnology Research Institute. ‡ Steacie Institute of Molecular Sciences. (1) Canham, L. T. Appl. Phys. Lett. 1990, 57, 1046. (2) Cullis, A. G.; Canham, L. T.; Calcott, P. D. J. J. Appl. Phys. 1997, 82, 909. (3) Buriak, J. M.; Allen, M. J. J. Am. Chem. Soc. 1998, 120, 1339. (4) Bateman, J. E.; Eagling, R. D.; Worall, D. R.; Horrocks, B. R.; Houlton, A. Angew. Chem., Int. Ed. Engl. 1998, 37, 2683. (5) Linford, M. R.; Chidsey, C. E. D. J. Am. Chem. Soc. 1993, 115, 12631. (6) Effenberger, F.; Gotz, G.; Bidlingmaier, B.; Wezstein, M. Angew. Chem., Int. Ed. Engl. 1998, 37, 2462. (7) Boukherroub, R.; Bensebaa, F.; Morin, S.; Wayner, D. D. M. Langmuir 1999, 15, 3831. (8) Kim, N. Y.; Laibinis, P. E. J. Am. Chem. Soc. 1997, 119, 2297. (9) Henry de Villeneuve, C.; Pinson, J.; Bernard, M. C.; Allongue, P. J. Phys. Chem. B 1997, 101, 2145. (10) Bansal, A.; Li, X.; Lauerman, I.; Lewis, N. S. J. Am. Chem. Soc. 1996, 118, 7225. (11) Zazzera, L. A.; Evans, J. F.; Deruelle, M.; Tirrell, M.; Kessel, C. R.; McKeown, P. J. Electrochem. Soc. 1997, 144, 2184. (12) Warntjes, M.; Vieillard, C.; Ozanam, F.; Chazalviel, J.-N. J. Electrochem. Soc. 1995, 142, 4138. (13) Lee, E. J.; Bitner, T. W.; Ha, J. S.; Shane, M. J.; Sailor, M. J. J. Am. Chem. Soc. 1996, 118, 5375. (14) Motohashi, A.; Kawakami, M.; Aoyagi, H.; Kinoshita, A.; Satou, A. Jpn. J. Appl. Phys., Part 1 1995, 34, 5840. (15) Ben-Chorin, M.; Kux, A.; Schecter, I. Appl. Phys. Lett. 1994, 64, 481. (16) Watanabe, K.; Okada, T.; Choe, I.; Sato, Y. Sens. Actuators, B 1996, 33, 194. (17) Doan, V. V.; Sailor, M. J. Science 1992, 1791. (18) Lin, V. S.-Y.; Motesharei, K.; Dancil, K.-P. S.; Sailor, M. J.; Ghadiri, M. R. Science 1997, 278, 840. (19) Janshoff, A.; Dancil, K.-P. S.; Steinem, C.; Greiner, D. P.; Lin, V. S.-Y.; Gurtner, C.; Motesharei, K.; Sailor, M. J.; Ghadiri, M. R. J. Am. Chem. Soc. 1998, 120, 12108. (20) Dancil, K.-P. S.; Greiner, D. P.; Sailor, M. J. J. Am. Chem. Soc. 1999, 121, 7925. (21) Snow, P. A.; Squire, E. K.; Russell, P. St. J.; Canham, L. T. J. Appl. Phys. 1999, 86, 1781.
have been used to determine the film thickness, the porosity, and the average pore diameter. Although scanning electron microscopy (SEM) and atomic force microscopy (AFM) are suitable for the determination of these first two parameters, they are expensive and destructive techniques since cross-sectional views are required.22 Porosity has been determined by gravimetry, gas adsorption (Brunauer-Emmett-Teller (BET)) measurements, and X-ray reflectivity.17-22 Gravimetry is not desirable, as this is also a destructive technique with low accuracy. BET measurements are limited to certain size regimes and cannot be universally applied. X-ray reflectivity is more useful and nondestructive. In principle, optical methods may be used to determine both the porous layer thickness and the porosity. Janshoff et al.19 used spectroscopic interferometry in a variety of solvents and deduced the porosity using effective medium theory. We describe, herein, a theoretical framework that will allow these parameters to be estimated by a single spectroscopic interferometric measurement in air. Flat thin films of PSi with reasonably well-defined pore morphologies can be prepared from a single-crystal silicon wafer by chemical and/or electrochemical etching in HFbased solutions. The resulting porosity, pore size, and PSi layer thickness depend on the etching conditions such as the current density, the composition of the etching solution, and the etching time as well as the type, doping level, and orientation of the substrate.22 PSi samples with layer thicknesses from 1 to 5 µm, with pore diameters of hundreds of nanometers and porosities of up to 80%, can be prepared easily and routinely. The thin PSi layer is uniform and sufficiently transparent to display FabryPerot fringes in its optical reflection spectrum.17 Such a feature may provide new opportunities in both fundamental research and practical applications of PSi. The technology using a biological material with PSi can be “core” to a host of analyzers and methods that cover the spectrum of biomedicine and analytical chemistry. Already, PSi-based biosensors have been developed for detection of DNA hybridization,18 biotin-avidin interactions,19 and antigen-antibody interactions.20 With a (22) Pits and Pores: Formation, Properties and Significance for Advanced Luminescent Materials; Schmuki, P., Lockwood, D. J., Isaacs, H., Bsiesy, A., Eds.; Electrochemical Society: Pennington, NJ, 1997.
10.1021/la025508p CCC: $22.00 © 2002 American Chemical Society Published on Web 04/17/2002
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and chemically stable for multiple uses and for providing reproducible results. This paper presents a theoretical analysis of PSi optical interferometric biosensors which allows estimation of the PSi layer thickness, refractive index, and porosity. The theoretical approach devised is general and can be applied to any thin layer whose optical properties are known or can be estimated. 2. Theoretical Analysis Structure Determination for a PSi Layer. Considering the incident light beam that is normal to the PSi layer, that is, the incident angle (θ) is zero (Figure 1), from Fresnel’s formulas, the reflectivity of light on the ambient interface and on the internal surface is
|r01|2 )
|
|r12|2 )
|n + n |
|
n1 - n0 n1 + n0
2
and Figure 1. Schematic of a porous silicon based optical interferometric biosensor and dispersion function of silicon (θ ) 0°, L < 5000 nm).
n2 - n1 2
2
1
, respectively. Light beams reflected from the two interfaces have a phase difference δ ) 4πn1L/λ. The reflectivity23 of light of the optical system with a thin film can be expressed as
R(λ) )
Figure 2. An optical interferogram of a porous layer of silicon in air (solid line) and a reconstructed interference pattern for a PSi layer with thickness 3000 nm (dotted line). The curve with diamond symbols is the calculated effective optical thickness. The last two curves were obtained from the theoretical analysis.
surface area of 100-1000 times higher than that of an unetched silicon surface,17 significantly more biomolecules can be immobilized on a PSi layer to enhance the detection sensitivity of such biosensors. A porous silicon-based interferometric biosensor consists of three optical media: silicon bulk, PSi layer, and ambience with the refractive index of n2(λ), n1(λ), and n0(λ), respectively (Figure 1). As the PSi film is transparent to light, white light reflected from the two interfaces (ambience and internal) would interfere with each other to produce well-resolved Fabry-Perot fringes or interference patterns (Figure 2). The binding of target ligands to corresponding recognition receptors immobilized in the PSi porous layer would increase the refractive index of the porous layer to induce peak shifts of the Fabry-Perot fringes to longer wavelengths. As a prerequisite in sensing applications, the pore size must be sufficiently large to allow biomolecules to enter the pores freely but small enough to retain optical reflectivity of the porous layer. It is of also importance that the material is mechanically
|
r01 + r12eiδ
|
1 + r01r12eiδ
2
(1)
If the ambience of the PSi layer is the air, n0 is equal to 1. However, the refractive index24,27 of the bulk silicon is a complex number (n2 ) η2 + ik2) and dependent on wavelength. The dispersion functions of η2 and k2 are shown in Figure 1 where η2 is about 3.5 and k2 is less than 10-5 for λ > 1200 nm. Since a PSi layer is filled with air or an aqueous buffer, the refractive index of the PSi layer is also a complex number (n1 ) η1 + ik1) and it is sensitive to the porosity (p) and wavelength (λ). As p varies from 0 to 1, η1 has the range of n0 to η2 and k1 lies between 0 and k2. For simplicity, a weighted average has been used. In principle, the indices of refraction should be evaluated in the framework of effective medium theory. However, this has only a negligible effect on the estimated parameters. For ease of use, we prefer the more simple approach. For a PSi optical system, the PSi layer thickness (L) is a constant, and the wavelength of white light continually distributes from visible to IR bands. The phase difference (δ) of reflected light beams would change with λ, and δ might be equal to 2mπ at some wavelengths. The variable m is the interference order of a peak (constructive interference) or a valley (destructive interference) in an interference pattern. The effective optical thickness (EOT ) η1L) of the PSi layer can be obtained at peaks and valleys of the interference pattern:
n1L ) mλ/2
(2)
(23) Born, M.; Wolf, E. In Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed.; Permagon Press: Oxford, 1980. (24) Handbook of Chemistry and Physics, 78th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1997. (25) Theiss, W. Surf. Sci. Rep. 1997, 29, 91. (26) Aspnes, D. E. Thin Solid Films 1982, 89, 249.
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For a peak, m is an integer and dependent on λ. According to eq 1, the reflectivities at all peaks of an interference pattern display a constructive profile.23
Rc(λ) )
|n + n | n2 - n0 2
2
|
i
|
n12 - n0n2
2
(8)
If noise is not taken into consideration, each value of Li at the respective peaks and valleys should be identical. In practice, noise exists and leads to a dispersion of Li values, and the average thickness should be j
1
∑ Li
j + 1 i)0
The spectral order m0 could be obtained by minimization of
S(m0) )
1
j
∑ (Li - Lh )2
j + 1 i)0
using regression analysis as follows:
x
x x
Rd(λ)
n2 + n0 n2 - n0
Rc(λ)
Rd(λ)
m0 ) n0n2
(5)
Rc(λ)
I(λ) - Ib(λ) f(λ)I0(λ)
n2 + n0 + n2 - n0 n2 + n0 n2 - n0
x x
(6)
n0n2 Id(λ) - Ib(λ)
λi
j
∑
)(
j
∑
1
Ic(λ) - Ib(λ)
with Id(λ) and Ic(λ) defined as the intensity of the destructive and constructive peak, respectively. Therefore, knowing n0 and n2, the refractive index of the PSi layer can be determined from the interference pattern in the region where λ > 1200 nm. The peaks and valleys on the right of an interference pattern should be chosen for calculation of n1(λ), and the total number of chosen peaks and valleys is j + 1. From right to left, the spectral orders of these chosen peaks and valleys are set to m0, m0 + 1/2, ..., m0 + j/2 and the corresponding wavelength positions are λ0, λ1, ..., λj. If the spectral order series starts with a peak, m0 is an integer, whereas m0 is an integer plus a half when the series begins with a valley. For example, the last two valleys and the last peak were chosen in the right of Figure 2, and j is equal to 2. (27) Gray, D. E. in American Institute of Physics Handbook, 2nd ed.; McGraw-Hill: Toronto, 1963.
∑
( ) ( 2
j + 1 i)0 n1(λi)
)
-
1
∑
( )
j
∑
λi
)
2
2
j + 1 i)0 n1(λi)
(9)
From m0, spectral orders for all peaks and valleys in the interference pattern could be expressed as m0, m0 + 1/ , ..., m + K/2, where K + 1 was the total peaks and 2 0 valleys considered over the entire interference pattern. The effective optical thickness of the PSi layer at these peaks and valleys could be obtained: EOTi ) (m0 + (i/2))λi/2, (i ) 0, 1, ..., K). Finally, the real part of n1(λ) can be expressed as 0
(7)
j
λi
(m η (λ ) )
Id(λ) - Ib(λ) Ic(λ) - Ib(λ)
(
j λi λi 1 i i j + 1 i)0 n1(λi) j + 1 i)0 n1(λi) j + 1 i)0 n1(λi)
1
1
in which I0(λ) is the emitting spectrum of a light source, f(λ) is the optical detector’s sensitivity, R(λ) is the reflectivity of the PSi layer, and Ib(λ) is the dark background of the optical system. From eqs 5 and 6, it follows that
x
(i ) 0, 1, ..., j)
2n1(λi)
L h)
The relative intensity of reflected light can be generally expressed as I(λ) - Ib(λ) ) f(λ) R(λ) I0(λ) or
n1(λ) )
i
(4)
n12 + n0n2
n2 + n0 + n2 - n0
R(λ) )
0
0
In biosensing applications, the thickness of a thin PSi layer is about 1-5 µm. For silicon in a condition of λ > 1200 nm, 4πk2L/λ is less than 5.5 × 10-4 or 0.999 e e-4πk2L/λ e 1. Consequently, both k1 and k2 can be neglected, so the refractive indices of silicon bulk and the PSi layer can be treated as real numbers (n2 ) η2 and n1 ) η1). From the constructive and destructive profiles determined by eqs 3 and 4, the refractive index of the PSi layer can be derived:
n1(λ) )
i m + ( ))λ ( 2 L )
(3)
For a valley, m is an integer plus 1/2. As shown in eq 1, the reflectivities at all valleys of an interference pattern display a destructive profile.23
Rd(λ) )
Upon the estimation of the refractive indices of the PSi layer at λ0, λ1, ..., λ from eq 7, the thickness of the PSi layer could be determined from eq 2:
1
i
+ 2L h
i λ 2 i
)
(i ) 0, 1, ..., K)
(10)
From the η1(λ) series obtained, η1(λ) can be expressed as a simple weighted average of n0(λ) and η2(λ),
η1(λ) ) pn0(λ) + (1 - p)η2(λ)
(11)
The porosity p obtained by minimization of K
S(p) )
[p n0(λi) + (1 - p) η2(λi) - η1(λi)]2 ∑ i)0
can be expressed as K
p)
[η2(λi) - n0(λi)][η2(λi) - η1(λi)] ∑ i)0 K
(12)
[η2(λi) - n0(λi)] ∑ i)0
2
Consequently, the refractive index of the PSi layer can be
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estimated as a simple weighted average of n0(λ) and n2(λ), that is, n1(λ) ) pn0(λ) + (1 - p)n2(λ). 3. Experimental Section 3.1. Etching Procedure. Heavily doped p-type silicon wafers (〈100〉, boron-doped, 0.0018-0.0022 Ω.cm, International Wafer Service) were used to prepare porous silicon samples by an anodic etch in ethanolic HF solution (HF/EtOH ) 3:1 v/v) made from 48% HF in water. Prior to the etching procedure, the wafers were cleaned with a 3:1 (v/v) of H2SO4/H2O2 for 30 min, rinsed with copious amounts of Milli-Q water, and then immersed in HF solution for 60 s. These wafers were dried under nitrogen gas, contacted on the backside with an In/Ga eutectic, and mounted in a Teflon etching cell. Porous silicon layers were fabricated using a two-electrode setup at a constant current of 250 mA/cm2 for 10 s in the absence of light using a Pt mesh counter electrode to ensure a homogeneous electric field. The etched wafer was removed from the cell, rinsed with EtOH, and dried under nitrogen. 3.2. Chemical Modification of the PSi. The freshly prepared PSi surface was placed under argon in a Schlenk tube containing a deoxygenated neat 1-decene or ethyl-undecylenate and allowed to react at 115 °C for 16 h. The excess of unreacted and physisorbed reagent was removed by rinsing, at room temperature, with tetrahydrofuran and 1,1,1-trichloroethane, and then the sample was dried under a stream of nitrogen. 3.3. SEM Characterization. SEM images were obtained with a Hitachi S4700 field emission microscope (Hitachi, Tokyo, Japan) using an accelerating voltage of 5.0 keV. Top and cross-sectional views were obtained at varying magnifications to examine the pore morphology. 3.4. Interferometric Reflectance Spectra. Interferometric reflectance spectra of a porous silicon layer were obtained using an IR spectrometer (Control Development, South Bend, IN) in which the optical fiber bundle is trifurcated and serves to direct incandescent light to the sample as well as lead the reflected light to the detectors (a Si linear array and an InGaAs array, made by Control Development). The use of these detectors allows two stages of normalization before the reflectance spectrum is taken of the porous silicon sample. The first is subtraction of a dark spectrum (signal obtained with no light present), and the second is normalization to a reference spectrum (Spectralon, Labsphere). To calculate sample reflectance, the program calculates minus log[I(λ)/Iref(λ)]. The illumination and detection of the reflected light was performed using an incident angle normal to the PSi surface in the range of 380-2200 nm.
4. Results and Discussion 4.1. Optical Interferogram of Porous Silicon. PSi samples were prepared by electrochemical etching and passivated by thermal reaction with decene or ethyl undecylenate.25 The thickness of the PSi layer is 3000 nm obtained from a cross-sectional SEM image showing the morphology of characteristic pores (Figure 3). From the original interference patterns obtained (-log I/Iref vs λ with Iref ) 2000), this spectrum was transformed into a linear scale as shown in Figure 2. The first two valleys and the first peak from the right of the spectrum were then chosen to calculate the spectral order (m0) of the first valley on the right of Figure 2. The positions of these valleys and peaks are (1808.2, 299.595), (1953.38, 2070.6), and (2129.25, 314.446), and the value of j in eq 8 should be equal to 2. With n0(λ) and n2(λ) known from physical data sources,21,26 n1(λ) can be obtained from eq 7 if [Imin(λ) Ib(λ)]/[Imax(λ) - Ib(λ)] is known. Since |dη2/dλ| < 10-4 for λ around 1800 nm, the reflectivity was almost constant for peaks and valleys, respectively. The average light intensity (2049) of the first two peaks was set for the peak profile Ic(λ), and the average light intensity (307) of the first two valleys was set for the valley profile Id(λ). Only Ib(λ), the dark background of the optical system, was not
Figure 3. Scanning electron micrograph of a porous silicon sample (cross-sectional view). Table 1. Calculation Results of Different Backgrounds Selected Ib(λ)
m0
L h (nm)
p
0 307 293
5.529 5.551 5.546
2541 ( 2.0 3155 ( 3.1 3004 ( 2.8
47.5% 64.5% 61.0%
known. If Ib(λ) was set to zero, the calculated PSi layer thickness was 2541 nm, whereas if Ib(λ) was set to Id(λ), the calculated thickness was 3155 nm. When Ib(λ) was set to 293, the calculated thickness matched the value measured by cross-sectional scanning electron microscopy (Figure 3). As shown in Table 1, both the thickness and the porosity were very sensitive to the dark background Ib(λ), but m0 was not sensitive to Ib(λ). For the three Ib(λ) values selected in the table, the estimated m0 values were 5.529, 5.551, and 5.546, respectively. Since the m0 was the spectral order of the first valley on the right of Figure 2, it should be about 5.5. Only at the selected m0,
S(m0) )
1
j
(Li - L h )2 ∑ j + 1 i)0
was less than 10. If |m0 - 5.5| > 1, the value of S(m0) would be thousands. As an example, the function S(m0) was 1280(m0 - 5.551)2 + 3.019 when the Ib(λ) was 307. From the calculated value of m0, the spectral orders of all peaks and valleys could be obtained. From right to left of the spectrum shown in Figure 2, the spectral orders of peaks were 6, 7-30, respectively. As the wavelengths of these peaks and valleys were already known from the spectral data, the EOT of the PSi layer could be obtained from (1/2)(m0 + (i/2))λi, (i ) 0, 1, ..., 49), and EOT was also not sensitive to Ib(λ). 4.2. Estimation of Porosity. From a PSi layer thickness shown in Table 1 and EOT at peaks and valleys, the real part of n1(λ) could be obtained from η1(λi) ) (1/2L h )(m0 + (i/2)), (i ) 0, 1, ..., 49). Since η1(λ) could be regressed to a weighted average of n0(λ) and η2(λ) as shown in Figure 4, the imaginary part of n1(λ) was also obtained from the same weighted average: k1(λ) ) 0 × p + (1 p)k2. The calculated porosity of 60-65% agrees with the value of 59 ( 5% determined by X-ray reflectivity. Note that the resulting porosity is dependent on several operating parameters including the current density used during the etching step. At a current density of 250 mA/ cm2, a porous silicon layer with a porosity of 70% was
Characterization of Porous Silicon Layers
Figure 4. Calculated dispersion function curves of the porous silicon layer using, from top to bottom, Ib(λ) ) 0, 293, and 307. The refractive index of the porous silicon was fit to the general form n1 ) pn0 + (1 - p)n2 (see the text).
al.19
reported by Janshoff et From the calculated values for the thickness (3004 nm) and n1(λ), eq 1 and eq 6 were then used to simulate the interference pattern. With f(λ) I0(λ) ) 6100 and Ib(λ) ) 293, the simulated interference pattern mimicked well the measured interference pattern between 400 and 1000 nm. The main difference between the experimental and reconstructed spectra is the constructive peak profile from 1000 to 1700 nm. As shown in Figure 2, the IR spectrometer changed the light detector between a Si linear array and an InGaAs array at λ ) 1100 nm. Both of these two detectors were not sensitive around 1100 nm. This shortcoming of the spectrometer resulted in a deviation between the experimental and reconstructed spectra. Notice that the reflectivity at a constructive peak is theoretically independent of the porous silicon layer (eq 3). 4.3. Estimation of the Refractive Index. The morphology of pores in a PSi layer is also very important to PSi applications. The pores may have spherical, cylindrical, or other irregular shapes, depending upon the etching conditions. On the basis of the spherical model and cylindrical model, Lazarouk et al.28 developed two key formulas to calculate the refractive index of PSi layers from known p, n0, and n2 for pores with spherical and cylindrical morphologies. With p ) 10%, 61%, and 85%, the refractive index of the PSi layer was calculated from the two models28 and compared with the value obtained by our weighted-average model (Figure 5). Both η1 and k1 curves from the spherical model are above the curves from the cylindrical model, and the k1 curves from our model lie between the curves of the two models. In the extreme low and high porosity conditions, the k1 curves from our model fit well with those of the spherical and cylindrical model. However, the η1 curves from our model are always above η1 curves from the two models. At p ∼ 50%, the differences of η1 among the three are more pronounced. The η1 curves from our model are closer to the curves derived from the spherical model. Among several other methods developed for determination of optical properties of thin films, of particular interest is a procedure based on a constructive or destruc(28) Lazarouk, S.; Jaguiro, P.; Katsouba, S.; Maiello, G.; La Monica, S.; Masini, G.; Proverbio, E.; Ferrari, A. Thin Solid Films 1997, 297, 97.
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Figure 5. Dispersion curves of the PSi layer calculated from our weighted-average model, spherical model, and cylindrical model with p ) 10%, 61%, and 85%. XXXXX for the spherical model, XXXXX for the cylindrical model, and XXXXX for the weighted-average model.
tive interference pattern of thin films.29 In brief, this procedure is only effective in solving for the optical constants of thin films (
