Energy levels of finite-depth quantum wells in an electric field

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Abstract
Numerical calculations of energy levels and wavefunctions for a particle in a finite quantum well subject to an electric field are described. The calculations are restricted to the regime where the tunneling rate out of the well is small. In this regime the results are in good agreement with results of an approximate calculation wherein the finite well is replaced by an infinitely deep well whose width has been adjusted (separately for each level) to obtain the correct zero-field eigenvalue, as recently proposed for the ground state by Miller et al. [D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, and C. A. Burrus, Phys. Rev. B 32, 1043 (1985)]. Over a significant range of well depths and fields (which are the only variables, provided that appropriately normalized units are used), it is found that the difference between the approximate and exact eigenvalues can be accurately estimated from a simple empirical formula. These results should be useful in studies of electro-optic effects in semiconductor quantum-well structures.