Derivation of built-in polarization potentials in nitride-based semiconductor quantum dots

We present a simple analytical approach for the calculation of the built-in strain-induced and spontaneous potentials in nitride-based semiconductor quantum dots. We derive the built-in potentials and electric fields in terms of volume or surface integrals. We describe using a number of simplifying assumptions the general properties of piezoelectric and spontaneous fields in $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ and $\mathrm{In}\mathrm{N}∕\mathrm{Ga}\mathrm{N}$ quantum dots and obtain analytic solutions to the potential along and close to the axis of symmetry in spherical, cylindrical, cuboidal, truncated-cone, and ellipsoidal dots. We show that the potential distribution in a hexagonal quantum dot is well represented by that of an equivalent dot with circular symmetry. We demonstrate that the built-in electric fields in nitride dots can provide a strong additional lateral confinement for carriers localized in the dot. This additional lateral confinement strongly modifies the electronic structure and optical properties of nitride-based quantum dot structures.

Figure 1

Contour plot showing the built-in potential in the $x\text{−}y$ plane at $z=0$ due to a square and a circular $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ planar interface.

Figure 2

(a) Built-in potential along the central $z$ axis due to a square $(z<0)$ and a circular $(z>0)$ $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ planar interface and (b) the difference between the two, $\Delta \phi ={\phi }_{\mathit{Circ}}-{\phi }_{\mathit{Sqr}}$ (for all $z$).

Figure 3

Contour plot in the $x\text{−}y$ plane for a $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ truncated cone dot and a truncated hexagonal pyramid dot.

Figure 4

Contour plot in the $r\text{−}z$ plane comparing the built-in potential of a $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ truncated cone dot (left) and a truncated hexagonal pyramid dot (right) of equal volumes, taken along the line of equal radii.

Figure 5

(a) Built-in potential along the central $z$ axis for a $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ truncated cone and a truncated hexagonal pyramid (indistinguishable in this plot) and (b) the difference between the two, $\Delta \phi ={\phi }_{\mathit{Pyr}}-{\phi }_{\mathit{\text{Cone}}}$.

Figure 6

Spontaneous and strain-induced contributions to the total built-in potential for a $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ QD (left) and an $\mathrm{In}\mathrm{N}∕\mathrm{Ga}\mathrm{N}$ QD (right).

Figure 7

Surface over which the integration is performed.