Index-matched IWKB method for the measurement of spatially varying refractive index profiles within thin-film photovoltaics

Y. T. Pang; M. Bossart; M. D. Eisaman

doi:10.1364/OE.22.00A188

Optics Express
Vol. 22,
Issue S1,
pp. A188-A197
(2014)
•https://doi.org/10.1364/OE.22.00A188

Index-matched IWKB method for the measurement of spatially varying refractive index profiles within thin-film photovoltaics

Y. T. Pang, M. Bossart, and M. D. Eisaman

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Y. T. Pang, M. Bossart, and M. D. Eisaman, "Index-matched IWKB method for the measurement of spatially varying refractive index profiles within thin-film photovoltaics," Opt. Express 22, A188-A197 (2014)

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Abstract

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.

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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. (N_i, x_i) represents the (effective index, turning point) for guided mode i.

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Fig. 2 Prism coupler setup used to measure the effective indices of the sample.

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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 t_IM, n = 1.93)/photoactive layer(thickness t_PA, “Actual Profile” (solid black line): n_PA(x) = 1.93−0.13(x/t_PA)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) t_IM = (1 μm, 5 μm, 10 µm); t_PA = 1μm. (b) t_IM = 10 μm; t_PA = 1μm, n_PA(x) = (parabolic, exponential, Gaussian). (c) t_IM = 5 μm; t_PA = (200 nm, 500 nm, 1 µm) normalized to 1. (d) t_IM = 5 μm; t_PA = 200 nm; λ = (500 nm, 650 nm, 829 nm). (e) t_IM = 5μm, n_IM = (1.95, 2.0, 2.03), Δn = (0.02, 0.07, 0.1); t_PA = 1 μm. (f) Spatial resolution (defined as the average spacing between successive points in the reconstruction) for the IM-IWKB reconstruction t_IM = 1-10 μm; t_PA = (1 μm, 200 nm).

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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, t_IM = 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.

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Equations (8)

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k \int_{0}^{x_{i}} {[v^{2} (x) - N_{i}^{2}]}^{\frac{1}{2}} d x = (i - 1) π + ϕ_{0} + ϕ_{t}, \begin{matrix}  \end{matrix} i = 1, 2, 3...

n (x_{i}) = N_{i}

v (x) = {\begin{matrix} n (x) \begin{matrix} \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} (T E) \end{matrix} \\ n (x) [1 + \frac{n (x) \dot{n} (x) - 2 {\ddot{n}}^{2} (x)}{k^{2} n^{4} (x)}] \begin{matrix}  \end{matrix} (T M) \end{matrix}

ϕ_{0} = \tan^{- 1} {r_{0} {[\frac{N_{i}^{2} - n_{g l a s s}^{2}}{N_{0}^{2} - N_{i}^{2}}]}^{\frac{1}{2}}}

ϕ_{t} = \frac{π}{4}

x_{i} = \frac{(i - 1) π + ϕ_{0} (N_{i}) + ϕ_{t} - \sum_{j = 1}^{i - 1} k {x_{j} [{(N_{a v g, j}^{2} - N_{i}^{2})}^{\frac{1}{2}} - {(N_{a v g, j + 1}^{2} - N_{i}^{2})}^{\frac{1}{2}}]}}{k {(N_{a v g, i}^{2} - N_{i}^{2})}^{\frac{1}{2}}}, \begin{matrix}  \end{matrix} i = 1, 2, 3

v_{P 3 H T} \frac{n_{P 3 H T} - N_{i}}{n_{P 3 H T} + 2 N_{i}} + v_{P C B M} \frac{n_{P C B M} - N_{i}}{n_{P C B M} + 2 N_{i}} = 0, \begin{matrix}  \end{matrix} i = 1, 2, 3...

v_{P 3 H T} + v_{P C B M} = 1

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Equations (8)

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