# PC3D

## About PC3S

PC3S was initially developed in 2020 to allow researchers to calculate the surface parameters that are required as inputs to PC3D based on the physical cell structure, including 3D surface texture, doping profiles, surface recombination velocity, and optical surface coatings. Ideally, experimental data should be used to obtain the surface parameters for PC3D, but in many cases it is not feasible to conduct the large number of experiments necessary to explore the desired range of physical cell structure. Before PC3S, the primary option for calculating the PC3D surface parameters was to use PC1D. That is still an option, but PC3S provides a superior, fully 3D treatment of pyramidal surface texture, and includes macros that automate the task of extracting the PC3D surface parameters.

Similar to PC3D, PC3S uses Microsoft Excel as the user interface, but all calculations are performed in code. The program code is fully open-source (accessible directly within Excel) and available free of charge. The author asks only that the program be cited in any resulting publications, e.g. P. A. Basore, PC3S 2020 [Online] http:// www.pc3d.info {append the date that the program was downloaded}.

PC3S implements the fully-coupled drift-diffusion equations in three dimensions over a solution region having a faceted surface that is periodic in both lateral dimensions. The facet angle is adjustable over the range 0 to 60 degrees, including the important angle of 54.7 degrees, which is the angle of exposed (111) planes on a (100) surface, as found on anisotropically etched single-crystal silicon. The numerical method is a relaxation technique that is particularly well-suited to solar cells, where the separation of quasi-Fermi potentials typically does not deviate strongly from the flat-band condition.

The most challenging aspect of 3D semiconductor device simulation is the current density. This is relatively easy in 1D because the current density of each carrier can be represented as the gradient of a scalar field, a property that can be exploited to achieve high accuracy in current density despite small errors in the potentials, as typically occurs when using a discrete grid of solution nodes. This property is used to maximum advantage PC1D. In 3D, however, the curl (del-cross) of the vector current density field is not zero, so the current density cannot be represented as the gradient of a scalar field. Even the tiniest errors in the potentials, especially for the majority carrier, lead to large errors in the current density. This hypersensitivity is addressed in PC3S by assuming that although the current density cannot be represented as the gradient of a scalar field, the error between the calculated current density and the true current density can be represented as the gradient of a scalar field. This error-field is called Gamma, and Gamma can be solved by requiring the current density to satisfy boundary conditions that apply to the solution volume as a whole, in particular the requirement that the carrier flux leaving the volume must equal the total generation minus the total recombination within the volume.

For studying the physics within the textured region, PC3S offers several solution modes. Bipolar mode applies voltage between the top and bottom of the solution region, which is representative of a doped emitter. Floating mode applies voltage (quasi-Fermi splitting) entirely at the bottom of the device, which is representative of a non-contacted surface. Open mode allows the quasi-Fermi levels to float nonuniformly so that the total recombination in the solution region equals the total photogeneration in the solution region. FlatBand mode is a fast way to assess recombination loss as a function of applied voltage for the high-efficiency limit where the quasi-Fermi separation is uniform throughout the volume. Lastly, Lateral mode applies a voltage left-to-right across the solution region, which induces lateral current flow that can be used to determine the effective sheet resistance (sheet rho) of the textured surface. Interestingly, the sheet resistance of a pyramidally textured surface is similar to that obtained when the same emitter diffusion process is used on a planar surface. PC3S reveals how the current follows a longer path on the texture surface, but is spread across a larger cross-sectional area, to achieve nearly the same result as for planar.

The optical model allows up to three layers between the silicon substrate and the encapsulant and accounts for oblique incidence and reflectance due to the surface texture angle. Light is assumed to be incident from directly above the solution region, with equal polarization components aligned with the width and length axes. The wave-based optical solution within each layer allows for an index of refraction and an extinction coefficient, both which can be a function of photon energy (wavelength). The two polarization modes are treated separately, both for the initial incidence and for reflection thereof that is then incident on an adjacent facet. Reflections beyond second-facet incidence are not considered, as they are typically small and real textured surfaces do not have a perfectly periodic shape anyway. Secondary incidence only occurs for texture angles exceeding 30 degrees, and only illuminates a portion of the adjacent facet when the texture angle exceeds 45 degrees.

PC3S calculates the spectral transmission that is needed as input to PC3D by accounting for losses in the near-surface region due to reflectance, absorption in the coating layers, and incomplete collection of carriers photogenerated in the near-surface region. The latter is calculated using the reciprocity theorem: A dark bias is applied and the electron and hole densities are calculated throughout the solution volume. The carrier-collection efficiency at each location is the excess pn product relative to flat-band conditions. This spatially-resolved carrier-collection efficiency is convolved with the photogeneration profile to get the carrier collection efficiency for that wavelength of light.

You can gain access to the program code for PC3S in the same manner as for PC3D, as explained at the end of the About PC3D page of this website.