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In many thin-film photovoltaic devices, the photoactive layer has a spatially varying refractive index in the substrate-normal direction, but measurement of this variation with high spatial resolution is difficult due to the thinness of these layers (typically 200 nm for organic photovoltaics). We demonstrate a new method for reconstructing the depth-dependent refractive-index profile with high spatial resolution (~10 nm at a wavelength of 500 nm) in thin (200 nm) photoactive layers by depositing a relatively thick index-matched layer (1-10 μm) adjacent to the photoactive layer and applying the Inverse Wentzel-Kramers-Brillouin (IWKB) method. This novel technique, which we refer to as index-matched IWKB (IM-IWKB), is applicable to any thin film, including the photoactive layers of a broad range of thin-film photovoltaics.
© 2014 Optical Society of America
1. Introduction
The performance of many thin-film photovoltaics (PV) is significantly influenced by the variation of material properties throughout the depth (i.e., the vertical, or, substrate-normal, direction) of the photoactive layer. Examples of the spatial variation of the photoactive layer material properties include grading of the Ga concentration (and concomitantly the bandgap energy) in copper indium gallium selenide (Cu(In1-x,Gax)Se2 − CIGS) solar cells [1], as well as vertical phase segregation gradients (donor/acceptor ratios that vary with depth) in bulk heterojunction organic photovoltaics (OPVs) [2, 3]. The accurate measurement of the resulting “depth profiles” for optical and electronic properties such as refractive index [4] and charge collection probability [5–8] is critical to the understanding, modeling, and optimization of these devices. Moreover, in many cases, the optical refractive index profile alone can yield information on depth-dependent material variation from which the depth profile of electronic properties can be inferred [9, 10].
In this paper we describe a novel method, which we refer to as index-matched IWKB (IM-IWKB), for the reconstruction of refractive index profiles (RIPs) with high spatial resolution within the photoactive layers of thin-film photovoltaics for films as thin as 200 nm. While the Inverse Wentzel-Kramers-Brillouin (IWKB) method has been used in the past to successfully reconstruct RIPs of various films [11–14], this approach is only valid for relatively thick (larger than 2 μm) films for visible wavelengths, since the number of points in the reconstructed RIP is equal to the number of modes supported by the film at that wavelength. For the ultra-thin layers relevant to many thin-film photovoltaics (typically 100 nm – 200 nm for OPVs, for example), it is either impossible to reconstruct the profile at all, or the result of the reconstruction has poor spatial resolution due to the low number of guided modes.
The IM-IWKB method described below deposits a relatively thick (1-10 μm) index-matched, non-absorbing layer below the thin photoactive layer of interest, and then applies the IWKB method to this system, resulting in a spatial resolution for the reconstruction of the RIP that is improved by more than a factor of ten compared to the traditional IWKB without the use of an index-matched layer. While RIPs for such thin layers can be obtained using spectroscopic ellipsometry with multiple-layer models [2, 3], our approach requires no free parameters, which is a significant advantage compared to the parameterized, model-based technique used in ellipsometry. This generally allows for much higher spatial resolution using our approach compared to ellipsometry, where the number of free parameters necessarily increases with the desired spatial resolution.
Below we first describe the mathematical basis of our approach, and then apply it to a computational example that assumes an RIP of a “canonical” OPV system to demonstrate the method’s accuracy and limitations for the typical system of interest. Finally, we describe our experimental implementation of this method and results for OPV devices fabricated in our lab.
2. Methodology
2.1 Index-matched IWKB (IM-IWKB) for improved spatial resolution
Consider a monotonically decreasing refractive index profile consisting of glass substrate, index-matched layer, photoactive layer, and air as shown in Fig. 1.According to the IWKB method [12], the characteristic equation of the ith mode is given by
where n(x) is the actual refractive index profile as a function of position x, Ni = βi/k is the effective index of the ith order mode with propagation constant βi, k = 2π/λ is the free space wavenumber, λ is the free-space wavelength of light, xi is the turning point position of the ith order mode as given by Eq. (2), x0 and N0 are the turning point and effective index of the glass substrate, and ϕ0 and ϕi ≡ ϕt = constant (for all i) are the phase contributions from the turning points x0 and xi [12]. The profile is assumed to be monotonically decreasing (true for the experimental case of an OPV bulk heterojunction that we consider later), and therefore the first turning point (x0) is always located at the beginning of the structure (the interface between substrate and index-matched layer in this case). The values of ϕ0 and ϕt are given by where r0 = 1 for transverse-electric (TE) modes, and r0 = (N0/nglass)2 for transverse-magnetic (TM) modes.
Fig. 1 Refractive index profile n(x) of an index-matched layer and an unknown photoactive layer on a glass substrate, as a function of position x. (Ni, xi) represents the (effective index, turning point) for guided mode i.
The recursive equation
where Navg,i = , relies on backward averaging of the effective indices to obtain values between the measured ones. To verify the reliability of our IM-IWKB method in the Numerical Results section below, we use Finite-Difference Time-Domain (FDTD) calculations [15] to construct a sample structure with a known RIP and solve for the effective indices of the guided modes. These effective index values are then used to reconstruct the RIP using the IM-IWKB method outlined above. In the Experimental Results section below, the effective index values of a test structure are measured using prism coupling [16–18], and these values are then used to reconstruct the RIP using the IM-IWKB method outlined above. It is important to note that if the index gradient of the photoactive layer was in the opposite direction (highest index at the air-layer interface and lowest at the substrate-layer interface), then we would need to put the index-matched layer between the photoactive layer and the air.2.2 Measurement of the effective indices for modes within the film
Experimentally, we use a prism coupler (Metricon Model 2010) to measure the effective indices of the guided modes in a sample, as shown in Fig. 2 [16, 17]. This setup consists of a prism that is contacted to the thin film sample, with the air gap (typically measuring around 100 nm) between the prism and sample controlled via the pressure in a pneumatic coupling head that pushes the sample into the face of the prism. A pressure of 45 psi is typically used, yielding an air gap of about 100 nm. In order to make sure this pressure does not deform the film under study, we tune the incident angle of the laser on the prism to coincide with a guided mode (indicated by a dip in the intensity of the light reflected by the prism as measured by the photodetector) and verify that as the pressure is swept over a range (30 psi – 55 psi) centered on the pressure used in the experiment (45 psi), the incident laser angle at which the dip occurs remains unchanged. The fact that the angle remains unchanged as the pressure is varied indicates that over the measured range of angles, the thickness of the film does not vary significantly. As the laser angle is swept, the light undergoes total internal reflection within the prism and the photodetector measures the intensity of the reflected light. When the phase velocity of the incident light is equivalent to the phase velocity of a guided mode, resonant energy transfer occurs from the incident light into the structure via “optical tunneling” [19]. For the setup shown in Fig. 2, we detect the reflected optical intensity as the laser angle (and therefore the incident phase velocity) is scanned, resulting in intensity minima measured by the detector at angles that allow resonant coupling to the guided modes of the sample. The position of these minima determine the effective indices of the guided modes, which are the propagation constants (divided by the vacuum wavevector) of the electromagnetic wave within the plane of the thin film.
2.3 Sample preparation and characterization
The test films used for the experiments shown in Fig. 4 were deposited on the AlN side of AlN (5 μm ± 5% thicknesses were used)/sapphire(430 μm) substrates (Kyma Technologies, wafer ID:B6523-1). The thickness of the AlN layer was determined by spectroscopic ellipsometry. The RMS surface roughness of the AlN is 50 nm. The test photoactive layer films were bulk heterojunction (BHJ) layers that are typically used as the photoactive layer in OPVs, consisting of solution-processed 1:1 ratio poly(3-hexythiophene-2,5-diyl) (P3HT) and phenyl-C61-butyric acid methyl ester (PCBM) (20 mg/ml) in dichlorobenzene solvent [8]. The P3HT:PCBM was spin-coated (700 rpm for 40 s) on the AlN. The devices were thermally annealed at 150 °C for 10 min in a glove box for post processing prior to measurement. Although we use a sapphire substrate in these experiments, any substrate can be used as long as it has minimal absorption at the wavelength of interest, and as long as its refractive index is lower than the lowest refractive index of the photoactive layer. We choose AlN as the index-matched layer for the following reasons: (1) The refractive index (n = 2.0039) of AlN is higher than and very close to the highest refractive index of the P3HT:PCBM photoactive layer. In order to use the IM-IWKB method, the refractive index of the whole structure (including the index-matched layer) should decrease monotonically. Therefore the refractive index of the index-matched layer should be higher than the highest refractive index of the P3HT:PCBM photoactive layer. (2) The thickness of index-matched layer (AlN, 5 μm) should be large enough to obtain a sufficiently large number of data points.
3. Numerical results and discussion
In order to demonstrate the benefits and reliability of our methods, we use FDTD calculations [15] to simulate a sample structure with a known RIP and solve for the effective indices of the guided modes. We first consider the structure: substrate(semi-infinite thickness, refractive index n = 1.76)/index-matched layer(1 μm, 5 μm, 20 µm, n = 1.93)/photoactive layer(1 μm)/air shown in Fig. 3(a) at a wavelength of 500 nm. For this example, all layers are considered to be non-absorbing (purely real refractive index), including the RIP of the 1 μm photoactive layer, which is defined as nPA(x) = 1.93 − 0.13(x/tPA)2 (shown as the black line labeled “Actual Profile” in Fig. 3(a)), where tPA is the thickness of the photoactive layer, and x is the position within the photoactive layer with x = 0 representing the photoactive-layer-index-matched-layer interface. From Fig. 3(a), we see that by varying the thickness tIM of the index-matched layer, we can increase the spatial resolution and accuracy of the reconstruction − (0.0095, 0.0072, 0.0068) are the root mean squared differences, ΔRMS, between the reconstructed profile and the actual profile, and (200nm, 63nm, 33nm) are the spatial resolutions (spacing between successive points in the reconstruction) for (1 μm, 5 um,10 μm) thick index-matched layers, respectively. In this reconstruction, the deviations of the higher-order modes near the air interface from the actual profile become larger because the phase contributions from the turning points near air interface are not exactly π/4 due to the weaker confinement of these modes.
Fig. 3 Reconstructed RIPs for TE-polarized light performed by the IM-IWKB method using guided mode effective indices obtained via FDTD simulation for the structure: substrate(semi-infinite, n = 1.76)/index-matched layer(thickness tIM, n = 1.93)/photoactive layer(thickness tPA, “Actual Profile” (solid black line): nPA(x) = 1.93−0.13(x/tPA)2 unless otherwise stated)/air. Wavelength λ assumed to be 500 nm unless otherwise stated. The number in parentheses in the legend is the root mean squared difference between the reconstruction and the actual profile. (a) tIM = (1 μm, 5 μm, 10 µm); tPA = 1μm. (b) tIM = 10 μm; tPA = 1μm, nPA(x) = (parabolic, exponential, Gaussian). (c) tIM = 5 μm; tPA = (200 nm, 500 nm, 1 µm) normalized to 1. (d) tIM = 5 μm; tPA = 200 nm; λ = (500 nm, 650 nm, 829 nm). (e) tIM = 5μm, nIM = (1.95, 2.0, 2.03), Δn = (0.02, 0.07, 0.1); tPA = 1 μm. (f) Spatial resolution (defined as the average spacing between successive points in the reconstruction) for the IM-IWKB reconstruction tIM = 1-10 μm; tPA = (1 μm, 200 nm).
Next we consider the structure: substrate(semi-infinite, n = 1.76)/index-matched layer(tIM = 10 µm, n = 1.93)/photoactive layer(tPA = 1 µm, parabola (ΔRMS = 0.0068), exponential (ΔRMS = 0.0147), Gaussian((ΔRMS = 0.0090) profile)/air in Fig. 3(b) in order to test the performance of the IM-IWKB method works for different profiles. As shown in Fig. 3(b), the IM-IWKB method performs well for any functional form, as long as that form satisfies the slowly varying constraint of the WKB method.
Next we consider the structure: substrate(semi-infinite, n = 1.76)/index-matched layer(tIM = 5 μm, n = 1.93)/photoactive layer(tPA = 200 nm, 500 nm, 1 µm)/air in Fig. 3(c) at a wavelength of 500 nm. From the figure, we see that ΔRMS gets larger as the photoactive layer becomes thinner. However, the reconstruction is still very accurate for a 200 nm photoactive layer (ΔRMS = 0.0133). Decreasing the laser wavelength can also increase the spatial resolution and accuracy of the reconstruction. Figure 3(d) demonstrates this effect for the structure: substrate(semi-infinite, n = 1.76)/index-matched layer(5 μm, n = 1.93)/photoactive layer(200 nm)/air at laser wavelengths of 829 nm (ΔRMS = 0.0249), 650 nm (ΔRMS = 0.0182) and 500 nm (ΔRMS = 0.0133). The absorption spectrum of the material used in the structure is the key limitation to determining which wavelengths are experimentally feasible. Wavelengths that are strongly absorbed cannot be used to measure the effective indices since the minimum in the reflected signal that indicates the effective index becomes increasingly difficult to identify with increasing absorption.
In the experiment setup, the refractive index of the photoactive layer at the index-matched-layer interface (nPA(x = 0)) and the refractive index of the index-matched layer (nIM) are not matched perfectly. If the IM-IWKB method is used on such a profile, there will be an error introduced because the slowly varying assumption of the IWKB method is violated at this interface. However, if we can make the refractive index step (Δn = nIM − nPA(x = 0)) at the interface small enough, the introduced error can be neglected as shown in Fig. 3(e). The structure assumed in Fig. 3(e) is: substrate(semi-infinite, n = 1.76)/index-matched layer(5 μm with Δn = 0.02, 0.07 and 0.1 refractive index steps)/photoactive layer(1 μm)/air. From Fig. 3(e), we see that while step sizes of Δn = 0.1 and Δn = 0.07 yield relatively large ΔRMS values (0.0132 and 0.0144, respectively), reducing the step to Δn = 0.02 improves the accuracy of the reconstruction (ΔRMS = 0.0074).
Finally, in Fig. 3(f) we show that the spatial resolution of the IM-IWKB method increases with increasing thickness of the index-matched layer. The spatial resolution is defined as the average spacing between successive points in the reconstruction, and is calculated by dividing the thickness of the photoactive layer by the number of modes whose turning points lie within the photoactive layer. Relative to the case of a 1µm thick index-matched layer, a 10µm thick index-matched layer improves the spatial resolution of the reconstruction for the 1 μm and 200 nm thick photoactive layers by a factor of 6 and 9.4, respectively.
4. Experimental results and discussion
The bulk-heterojunction active layers that are typically employed in OPV devices are comprised of a mixture of electron donor and electron acceptor materials [20]. A canonical material choice for a OPV photoactive layer employs poly(3-hexythiophene-2,5-diyl) (P3HT) as the electron donor and phenyl-C61-butyric acid methyl ester (PCBM) as the electron acceptor. In this section, we describe experiments that use the IM-IWKB method to reconstruct the RIP of P3HT:PCBM bulk heterojunction photoactive layers. These experiments use a prism coupler to measure the guided-mode effective indices (as shown in Fig. 2) of P3HT:PCBM thin-film samples on top of an indexed-matched layer of AlN, and then use these effective indices as input to the IM-IWKB reconstruction technique. It is well known that these layers exhibit vertical phase segregation gradients (donor/acceptor ratios that vary with depth) [2, 3]. From knowledge of the refractive indices of pure P3HT (1.780 for TE polarization at a wavelength of 829 nm) and pure PCBM (1.95425 for TE at 829 nm) obtained via spectroscopic ellipsometry, the RIP measured using IM-IWKB, and Bruggeman effective medium theory [21], we obtain the concentration depth profiles for PCBM and P3HT throughout the photoactive layer. In order to take into account the influence of annealing on P3HT due to the change in crystallinity, the pure P3HT is also thermally annealed at 150°C for post processing prior to measurement. Figure 4(a) presents measurements of the reflected light intensity at 829 nm for TE polarization as measured by the prism coupler setup shown in Fig. 2. For the structure: sapphire(430 μm)/index-matched layer(AlN, tIM = 5 μm ± 5%)/photoactive layer (P3HT:PCBM, 200 nm)/air, the reflected intensity shows 13 minima, each of which corresponds to a guided mode effective index. From these 13 effective index values, we use the IM-IWKB method described above to reconstruct the RIP as shown in Fig. 4(b). The thickness of the index-matched layer is around 5.3 µm from ellipsometry measurements and by subtracting this thickness from the thickness of the entire structure, we can obtain the RIP in the photoactive layer region as shown in Fig. 4(c). From the reconstructed RIP, we measure that the refractive index of the P3HT:PCBM photoactive layer changes from 1.98 to 1.8, in going from the index-matched layer interface to the air, and that the thickness is around tPA » 170 nm (5.3 µm to 5.47 µm), While ellipsometry measurements yield tPA = 200 nm, from Fig. 3(d), we see that the thickness of the photoactive layer (200 nm) suggested by the reconstructed RIP underestimates the actual thickness by about 30nm. The refractive index of index-matched layer (AlN) is 2.00390.0001 as measured by single-film prism coupler measurements. There is a refractive index step Δn ≥ (2.0039 − 1.977) = 0.0269 between the index-matched layer and photoactive layer, since we know from Fig. 3(e), that the highest refractive index (1.977) at the index-matched layer-photoactive-layer interface is overestimated. As mentioned earlier, the P3HT:PCBM films are typically thermally annealed at 150 °C for 10 min in a glove box for post processing prior to measurement. Figure 4(d) shows the reconstructed RIP of the structure: sapphire(semi-infinite)/index-matched layer(AlN, tIM = 5.3 μm)/photoactive layer (P3HT:PCBM, 200 nm)/air with and without thermal annealing, demonstrating that the slope of the RIP with annealing is larger than that without annealing. From these RIPs, and knowledge of the refractive indices of pure PCBM (1.954 for TM and TE at 829 nm) and P3HT (1.596 for TM and 1.780 for TE at 829 nm), we use Bruggeman effective medium theory as shown in Eqs. (7) and (8) (where vP3HT and vPCBM are the volume fractions of P3HT and PCBM, and nP3HT and nPCBM are the refractive indices of pure P3HT and pure PCBM) [21] to determine the depth profile of the PCBM volume fraction in the photoactive layer, as shown on the right-hand vertical axis in Fig. 4(d).
In Fig. 4(d), vPCBM is slightly larger than one in the first 75 nm of the photoactive layer because the reconstruction overestimates the actual profile in this region in the presence of a refractive-index step Δn ≥ 0.0269, as shown in Fig. 3(e). From Fig. 4(d), it appears that higher-resolution data near the index-matched layer in the 0 − 0.05 μm range may be desired. Additional data points in this range could be obtained by increasing the thickness of the index-matched layer. The thickness of the index-matched layer can be chosen to achieve the desired minimum resolution in a given region of the photoactive layer.Fig. 4 Experimental reconstruction of the RIP by the IM-IWKB method, using guided mode effective indices measured the prism coupler setup shown in Fig. 2. (a) Reflection spectrum for structure: sapphire(430 μm)/index-matched layer(AlN, tIM = 5.3 μm)/photoactive layer (P3HT:PCBM, 200 nm)/air. (b) RIP reconstruction for the structure in (a). (c) RIP reconstruction of the photoactive layer region for the structure in (a). (d) RIP reconstruction for the structure in (a) with (red) and without (green) thermal annealing. In panels (d), the right-hand axis shows the PCBM volume fraction calculated from the RIP by applying Bruggeman effective medium theory as described in the text.
5. Conclusion
We have demonstrated a new method (IM-IWKB) for reconstructing the depth-dependent refractive-index profile in thin films (200 nm) with high spatial resolution (10 nm possible at a wavelength of 500 nm) by depositing a relatively thick index-matched layer (1-10 μm) adjacent to the thin film of interest and applying the Inverse Wentzel-Kramers-Brillouin (IWKB) method. We have applied this method to 200 nm thick bulk heterojunction layers of P3HT:PCBM, a common absorber layer used in OPVs and shown that by combining the RIP reconstruction with effective medium theory, the depth dependent profiles of donor (P3HT) and acceptor (PCBM) volume fractions can also be determined. By reconstructing the volume fraction profile both before and after the annealing of the P3HT:PCBM layer, we were able to directly measure, with high spatial resolution, the evolution of phase segregation of the P3HT and PCBM phases during the annealing process. In comparison with alternative approaches such as ellipsometry, the IM-IWKB method achieves high spatial resolution with no free parameters as it reconstructs the refractive index profile by applying the IWKB method to direct measurements. The IM-IWKB technique is applicable to any thin film, including the photoactive layers of a many thin-film photovoltaics, and we believe it has the potential to be a broadly useful and non-destructive technique for measuring optical and material properties with high spatial resolution.
Acknowledgments
This work was partially supported by the U. S. Department of Energy, Sustainable Energy Technologies Department under contract DE-AC02–98CH10886. Research was carried out in part at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-AC02–98CH10886. M. B. was partially supported by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists (WDTS) under the Science Undergraduate Laboratory Internships Program (SULI). The authors would like to thank N. Dissanayake for helpful discussions and A. Ashraf for spectroscopic ellipsometry measurements.
References and links
1. O. Lundberg, M. Edoff, and L. Stolt, “The effect of Ga-grading in CIGS thin film solar cells,” Thin Solid Films 480–481, 520–525 (2005). [CrossRef]
2. D. S. Germack, C. K. Chan, R. J. Kline, D. A. Fischer, D. J. Gundlach, M. F. Toney, L. J. Richter, and D. M. DeLongchamp, “Interfacial segregation in polymer/fullerene blend films for photovoltaic devices,” Macromolecules 43(8), 3828–3836 (2010). [CrossRef]
3. A. Ashraf, D. M. N. M. Dissanayake, D. S. Germack, C. R. Weiland, and M. D. Eisaman, “Confinement-induced reduction in phase segregation and interchain disorder in bulk heterojunction films,” ACS Nano (2013).
4. B. H. Hamadani, S. Jung, P. M. Haney, L. J. Richter, and N. B. Zhitenev, “Origin of nanoscale variations in photoresponse of an organic solar cell,” Nano Lett. 10(5), 1611–1617 (2010). [CrossRef] [PubMed]
5. R. Scheer, C. Knieper, and L. Stolt, “Depth dependent collection functions in thin film chalcopyrite solar cells,” Appl. Phys. Lett. 67(20), 3007–3009 (1995). [CrossRef]
6. R. Scheer, M. Wilhelm, H. J. Lewerenz, H. W. Schock, and L. Stolt, “Determination of charge carrier collecting regions in chalcopyrite heterojunction solar cells by electron-beam-induced current measurements,” Sol. Energy Mater. Sol. Cells 49(1–4), 299–309 (1997). [CrossRef]
7. R. Kniese, M. Powalla, and U. Rau, “Evaluation of electron beam induced current profiles of Cu(In,Ga)Se2 solar cells with different Ga-cotents,” Thin Solid Films 517(7), 2357–2359 (2009). [CrossRef]
8. D. M. N. M. Dissanayake, A. Ashraf, Y. Pang, and M. D. Eisaman, “Mapping spatially resolved charge collection probability within P3HT:PCBM bulk heterojunction photovoltaics,” Adv. Energy Mater. (2013).
9. R. A. Marsh, C. Groves, and N. C. Greenham, “A microscopic model for the behavior of nanostructured organic photovoltaic devices,” J. Appl. Phys. 101(8), 083509 (2007). [CrossRef]
10. W. S. Koh, M. Pant, Y. A. Akimov, W. P. Goh, and Y. Li, “Three-dimensional optoelectronic model for organic bulk heterojunction solar cells,” IEEE J. Photovoltaics 1(1), 84–92 (2011). [CrossRef]
11. Z. Cao, Y. Jiang, Q. Shen, X. Dou, and Y. Chen, “Exact analytical method for planar optical waveguides with arbitrary index profile,” J. Opt. Soc. Am. A 16(9), 2209–2212 (1999). [CrossRef]
12. K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indexes,” J. Lightwave Technol. 3(2), 385–391 (1985). [CrossRef]
13. L. P. Shi, E. Y. B. Pun, and P. S. Chung, “Extended IWKB method for determination of the refractive=index profile in optical waveguides,” Opt. Lett. 20(15), 1622–1624 (1995). [CrossRef] [PubMed]
14. H. Zhu, Z. Cao, and Q. Shen, “Construction of refractive-index profiles of planar waveguides with additional information obtained from surface plasmon resonance,” Appl. Opt. 44(16), 3174–3178 (2005). [CrossRef] [PubMed]
15. I. Lumerical Solution, http://www.lumerical.com/tcad-products/fdtd/.
16. D. M. N. M. Dissanayake, A. Ashraf, Y. Pang, and M. D. Eisaman, “Guided-mode quantum efficiency: A novel optoelectronic characterization technique,” Rev. Sci. Instrum. 83(11), 114704 (2012). [CrossRef] [PubMed]
17. R. Ulrich and R. Torge, “Measurement of thin film parameters with a prism coupler,” Appl. Opt. 12(12), 2901–2908 (1973). [CrossRef] [PubMed]
18. P. K. Tien, R. Ulrich, and R. J. Martin, “Modes of propagating light waves in thin deposited semiconductor films,” Appl. Phys. Lett. 14(9), 291–294 (1969). [CrossRef]
19. Y. Pochi, Optical Waves in Layered Media (John Wiley, 1988).
20. G. Yu, J. Gao, J. C. Hummelen, F. Wudl, and A. J. Heeger, “Polymer photovoltaic cells: enhanced efficiencies via a network of internal donor-acceptor heterojunctions,” Science 270(5243), 1789–1791 (1995). [CrossRef]
21. M. Khardani, M. Bouaïcha, and B. Bessaïs, “Bruggeman effective medium approach for modelling optical properties of porous silicon: comparison with experiment,” Phys. Status Solidi 4(c), 1986–1990 (2007).
