Surface optical phonons in a triaxial ellipsoidal quantum dot

The surface optical phonon eigenmodes in ellipsoidal quantum dots were calculated using the dielectric-continuum model. The problem is exactly solvable in terms of ellipsoidal coordinates and the eigenmodes are written in terms of Lamé polynomials. A study of the mode frequency dependence on the shape of the dots is carried out. This represents a generalization over earlier work on spheres and spheroids. Differences in the results obtained using an optically active and an optically passive host is presented. The size independence of the mode frequencies for a quantum dot of arbitrary shape is proven.

Figure 1

A few Lamé functions both inside and outside an ellipsoid: $(m,p)=(1,0),(1,1),(1,2)$.

Figure 2

Frequencies of the $(m,p)=(1,0)(1,1),(1,2),(2,4)$ SO modes (top to bottom) as a function of asymmetry using optically active host (left two panels, left for barrier-like modes, middle for dot-like modes), and a constant host dielectric constant (right panel). Note the different vertical scale for the barrier modes (left panel). The dot is spherical in the limit $\alpha ,\beta \to 1$. The horizontal lines are the exact spherical results.