Coulomb Effects at Saddle-Type Critical Points

The effect of the Coulomb interaction between electron and hole at an $M_1$ (saddle-type) critical point is studied in the effective-mass approximation, using the adiabatic method first treated by Velický and Sak. The complete adiabatic potential for the heavy-mass degree of freedom is computed. The method is tested by calculating the binding energy for positive heavy mass and comparing it with Kohn and Luttinger's variational result. Reasonable accuracy for mass ratios greater than 5 is obtained. For negative heavy mass, the two-dimensional light-mass coordinates give a bound state which results in an effective repulsive potential. The contribution to ${\epsilon }_2$, the imaginary part of the dielectric function, is computed, and a peak is found at the energy of the two-dimensional bound state. A sharp drop in ${\epsilon }_2$ above the peak is found, which contrasts with the sharp drop below the peak for $M_0$ singularities. Quantitative agreement is found with the structure in ${\epsilon }_2$ for CdTe at 3.5 eV measured by Marple and Ehrenreich, and it is expected that other $L$-point transitions in III-V semiconductors can be similarly interpreted.