W. Shockley

In order to find an upper theoretical limit for the efficiency of p-n junction solar energy converters, a limiting efficiency, called the detailed balance limit of efficiency, has been calculated for an ideal case in which the only recombination mechanism of hole-electron pairs is radiative as required by the principle of detailed balance. The efficiency is also calculated for the case in which radiative recombination is only a fixed fraction fc of the total recombination, the rest being nonradiative. Efficiencies at the matched loads have been calculated with band gap and fc as parameters, the sun and cell being assumed to be blackbodies with temperatures of 6000°K and 300°K, respectively. The maximum efficiency is found to be 30% for an energy gap of 1.1 ev and fc = 1. Actual junctions do not obey the predicted current-voltage relationship, and reasons for the difference and its relevance to efficiency are discussed.

The statistics of the recombination of holes and electrons in semiconductors is analyzed on the basis of a model in which the recombination occurs through the mechanism of trapping. A trap is assumed to have an energy level in the energy gap so that its charge may have either of two values differing by one electronic charge. The dependence of lifetime of injected carriers upon initial conductivity and upon injected carrier density is discussed.

The method of effective mass, extended to apply to gradual shifts in energy bands resulting from deformations of the crystal lattice, is used to estimate the interaction between electrons of thermal energy and the acoustical modes of vibration. The mobilities of electrons and holes are thus related to the shifts of the conduction and valence-bond (filled) bands, respectively, associated with dilations of longitudinal waves. The theory is checked by comparison of the sum of the shifts of the conduction and valence-bond bands, as derived from the mobilities, with the shift of the energy gap with dilation. The latter is obtained independently for silicon, germanium and tellurium from one or more of the following: (1) the change in intrinsic conductivity with pressure, (2) the change in resistance of an $n\ensuremath{-}p$ junction with pressure, and (3) the variation of intrinsic concentration with temperature and the thermal expansion coefficient. Higher mobilities of electrons and holes in germanium as compared with silicon are correlated with a smaller shift of energy gap with dilation.

The energies and motions of grain boundaries between two crystallites are investigated theoretically using the dislocation model of grain boundaries. Quantitative predictions made for simple boundaries for cases in which the plane of the boundary contains the axis of relative rotation of the grains appear to agree with available experimental data. The quantitative expression for energy per unit area for small angles is approximately $[\frac{\mathrm{Ga}}{4\ensuremath{\pi}(1\ensuremath{-}\ensuremath{\sigma})}]\ensuremath{\theta}[A\ensuremath{-}\mathrm{ln}\ensuremath{\theta}]$ where $G$ is the rigidity modulus, $a$ the lattice constant, $\ensuremath{\sigma}$ Poisson's ratio, $\ensuremath{\theta}$ the relative rotation and $A$ approximately 0.23. Grain boundaries of the form considered may permit intercrystalline slip and may act as stress raisers for the generation of dislocations.

For certain p-n junctions, it has been observed that the measured current-voltage characteristics deviate from the ideal case of the diffusion model. It is the purpose of this paper to show that the current due to generation and recombination of carriers from generation-recombination centers in the space charge region of a p-n junction accounts for the observed characteristics. This phenomenon dominates in semiconductors with large energy gap, low lifetimes, and low resistivity. This model not only accounts for the nonsaturable reverse current, but also predicts an apparent exp (qV/nkT) dependence of the forward current in a p-n junction. The relative importance of the diffusion current outside the space charge layer and the recombination current inside the space charge layer also explains the increase of the emitter efficiency of silicon transistors with emitter current. A correlation of the theory with experiment indicates that the energy level of the centers is a few kT from the intrinsic Fermi level.