with surface wrinkles and folds

*With a simple model for in-plane propagation in a multilayer stack (an organic solar cell), we derive the admissibility criteria for wave-guided modes to propagate in the plane, and explain how the incorporation of surface folds enables sunlight to propagate into in-plane modes and enhance the photoconversion efficiency.*

In order to quantify the influence of wrinkles and folds on the light harvesting characteristics of polymer solar cells, we simulated the propagation of light in these devices using a finite-element solver, COMSOL, in two-dimensions describing the optical stack (Fig. S5). We first defined a window of interest in which layers representing the material stack within the device structure are incorporated, including the NOA substrate, semi-transparent gold as bottom electrode (15nm), PEDOT:PSS as hole transport layer, P3HT:PCBM photoactive layer, and aluminum top electrode. In our simulations, we specified the surface topographies representative of those obtained from line traces (e.g., Fig. 1) during atomic force microscopy experiments after the sequential deposition of individual layers.

*Figure 1 - Three-dimensional renditions of atomic force microscope micrographs and one-dimensional scans of representative wrinkled and composite surfaces.*

*Figure 2 - Two-dimensional geometry used for FEM digital computation of light propagation in wrinkled and composite structures. A window of interest (white) is surrounded by non-reflective perfectly matched layers (6, in green). An incoming plane wave (red line) propagates through the patterned structure [NOA(1), 15nm of gold(2), 140nm PEDOT:PSS(3), 180nm P3HT-PCBM(4) and a reflective metallic back-cover(5).*

On one side of the window (bottom of the rectangular windows in Fig.2, the boundary condition is given by an incoming plane wave propagating along the vertical (z) axis where:

(1)

Imposing the complex refractive indices, associated with each layer, the topography, and the boundary conditions prescribed above, the electromagnetic wave equation:

(2)

has a unique solution in the window of interest. Low error levels are guaranteed by a Shannon-type criterion when the mesh size is much smaller than the wavelength. For the simulations, we chose a mesh size of 60nm, which represented a good compromise between resolution and computation time. Our simulations return the general expression for the electric field, **E**, and the magnetic field, **B**. The net photon flux diagram shown in Fig. 3 are deduced from **E** and **B** by:

(3)

We further take into account the effects of black-body-type broad-band radiation of the solar spectrum by running the simulations over many discrete wavelengths. Herein, we solve Maxwell's Equation in the case of a coherent plane wave, i.e., we assume infinite spatial and temporal coherence. Because the optical stack is thin, temporal coherence of sunlight (typically a few wavelengths) is a valid assumption. By considering infinite spatial coherence, however, the incident radiation is assumed to be a plane wave with normal incidence on the optical stack of interest. This point-source approximation is accurate for direct exposure because of the small angular diameter (32') of the sun seen from earth. Under the scenario in which the polymer solar cells are exposed to diffused sunlight, the model can be modified easily to take into account the different angles of the incoming radiation.

For simplicity, we consider a scalar model in which light propagation in the optical stack satisfies the Maxwell equation:

(4)

Stationary solutions satisfy, for waves propagating in the plane of the optical stack:

(5)

Within our optical stack, the active layer comprising P3HT and PCBM has the highest refractive index. We can thus consider the optical stack to exhibit the characteristics of a planar waveguide, with P3HT:PCBM as the confinement layer.

*Table 1 - The refractive indices and thicknesses of individual layers within the optical stack.*

In a planar waveguide, a propagating mode in the P3HT:PCBM layer obeys the condition

(6)

In this stack, a guided solution satisfies the following boundary conditions:

At x = 0, the reflective coating makes a boundary condition **E**= **0**, hence continuity of **E** and **B** at the interface implies:

We thus deduce from the continuity equations that the guided modes satisfy:

(6)

This equation admits a solution if:

(7)

Since the thickness of the P3HT:PCBM active layer is d_M = 180nm, there exists a trapped mode for each wavelength < 890nm.

*Table 2 - Maximum cut-off wavelength for in-plane propagation modes of various bulk-heterojunction materials pairs commonly used in polymer solar cells extracted from typical active layer thicknesses and refractive indices reported in the literature. The absorption edges of these bulk heterojunction materials pairs are also provided.*

The principles of in-plane light trapping are not limited to the materials used in this experiment. We have carried out this analysis with other common bulk-heterojunction materials pair. Given typical active layer thicknesses and refractive index difference between poly[2-methoxy-5-(3,7-dimethyloctyloxy)-1,4-phenylenevinylene], MDMO-PPV, with PCBM and PEDOT:PSS, as well as that between poly[2,6-(4,4-bis- (2-ethylhexyl)-4H -cyclopenta[2,1-b;3,4-b]dithiophene) -alt- 4,7(2,1,3-benzothiadiazole)], PCPDTBT, with PCBM and PEDOT:PSS, waveguiding modes are supported at wavelengths up to 300nm and 50nm beyond the absorption edges of the active layers, respectively (see Table 2). The introduction of deep folds should thus also increase light coupling into and waveguiding within active layers, effectively enhancing light absorption in these polymer solar cells.

The experiments have shown that for the range of wavelengths predicted by equation (7), light trapping behavior has been observed in solar cells with wrinkles and surface folds. In-plane propagation of light, with folds as point of entry, significantly increase the propagation length of trapped photons, and enhance photoconversion despite low bulk absorption levels in the active layer.

Wrinkles and deep folds as photonic structures in photovoltaics

J. B. Kim*, P. Kim*, N. C. Pegard*, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, and Y-L. Loo

Nature Photonics **6** 327-332 (2012)

*These authors contributed equally to this work

Incorporation of Wrinkles and Folds to Enhance Efficiency and Bendability of Solar Cells

J.B. Kim, P. Kim, H.A. Stone, N. C. Pegard, J.W. Fleischer, Y.-L. Loo

Provisional Patent Application 61-635,540 - 2012