Extension of the Shockley-Queisser Limit for Nanostructured Solar Cells

Abstract

This article extends the Shockley-Queisser limit to nanostructured solar cells using the quantum phase space formalism. The parameter B_ll represents the momentum variance in each confinement direction and acts as a variance-covariance matrix linking the nanostructure geometry to thermodynamic properties. Electron-electron interactions are included via an exchange-correlation energy with an adjustable coefficient theta. The authors derive an analytical expression for the maximum efficiency as a function of size, shape, temperature, and doping. For the cylindrical geometry, the exact confinement energy uses the first zero of the Bessel function j_0,1. Numerical simulations are performed with Python 3.8.1, NumPy, and Matplotlib for PbS quantum dots in four geometries: cube, square parallelepiped, cylinder, and sphere. The integral is evaluated using an exact convergent series expansion. Results show that the maximum efficiency reaches 48.7 percent for a 5 nanometre cube, 49.0 percent for flattened parallelepiped and cylinder shapes, and 49.1 percent for a 3 nanometre sphere. These values greatly exceed the bulk PbS efficiency of 15.8 percent and surpass classical Shockley-Queisser limits. For constant-volume shapes, two efficiency peaks appear corresponding to different aspect ratios. The model correctly returns to classical values for large sizes. This approach provides a theoretical framework for optimising nanostructured solar cells and demonstrates that quantum confinement offers a promising route to surpass traditional photovoltaic limits.