Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes

We investigate the anisotropic optical gain in non-$c$-plane InGaN quantum wells with 20% indium content including band-gap renormalization and the screening of the quantum confined Stark effect. Waveguide modes and their polarizations are determined as TE and TM modes or extraordinary and ordinary modes, depending on the birefringence and the orientation of the laser diode’s ridge waveguide relative to the $c$ axis. The band structures and optical matrix elements along the polarization directions are calculated using a $6\times 6$ $k\cdot p$ Hamiltonian and a self-consistent Schrödinger-Poisson solver. From these calculations the reduced density of states and the optical gain for the different polarizations are determined in the free-carrier picture with an ad hoc inclusion of the band-gap renormalization and compared to a $c$-plane quantum well. It is found that for high indium concentrations the gain can be significantly increased by going from the $c$ plane to a semipolar or a nonpolar crystal orientation. However, due to birefringence and composition of the topmost valence-band wave function, the ridge has to be oriented along the $\left[\overline{1}\overline{1}23\right]$ direction for semipolar and along the [0001] direction for nonpolar laser diodes.

Figure 1

(Color online) Ridge geometries and crystal orientations of two possible realizations of a laser structure on the $\left(11\overline{2}2\right)$ plane (indicated by the transparent plane on the crystal images). (a) ridge along $\left[\overline{1}\overline{1}23\right]$ and (b) ridge along $\left[1\overline{1}00\right]$.

Figure 2

Waveguide structure with coordinate system.

Figure 3

(Color online) Electric field components of the extraordinary mode of the semipolar waveguide and angle ${\alpha }^{\prime }=\text{arctan}\left(E_{x^{\prime }}/E_{z^{\prime }}\right)$ as a function of $z^{\prime }$ near the waveguide layer ($y^{\prime }$ component divided by the complex unit). The dashed line indicates the extraordinary direction.

Figure 4

(Color online) Electric field components of the ordinary mode of the semipolar waveguide and angle ${\alpha }^{\prime }$ as a function of $z^{\prime }$ near the waveguide layer ($y^{\prime }$ component divided by the complex unit). The dashed line indicates the ordinary direction.

Figure 5

Average angle ${\alpha }_{avg}^{\prime }$ of both modes as a function of the anisotropy factor for a waveguide thickness of 200 nm (black lines) and 100 nm (gray lines)

Figure 6

Definition of crystal angle and coordinates (Ref. 12).

Figure 7

Topmost valence bands in a 3 nm ${\text{In}}_{0.2}{\text{Ga}}_{0.8}\text{N}$ quantum well for different crystal orientations at a sheet-carrier density of $7\times {10}^{12}{\text{cm}}^{-2}$. Subbands that correspond to different bulk valence bands have been labeled A and B.

Figure 8

(Color online) In-plane energy dispersion relation for valence subbands A1 (top) and B1 (bottom) in a semipolar 3 nm ${\text{In}}_{0.2}{\text{Ga}}_{0.8}\text{N}$ quantum well at a sheet-carrier density of $7\times {10}^{12}{\text{cm}}^{-2}$. The energy spacing is 10 meV per solid line.

Figure 9

(a) Piezoelectric polarization $P_{PZ}^{\left(QW\right)}$ and spontaneous polarizations $P_{SP}^{\left(b\right)}$, $P_{SP}^{\left(QW\right)}$ in the barrier and quantum well and (b) unscreened internal field $E_{z^{\prime }}$ in the quantum well depending on the crystal angle $\theta$ and potential [(c), solid lines] with electron and hole probability distributions (dashed lines) for a 3 nm ${\text{In}}_{0.2}{\text{Ga}}_{0.8}\text{N}$ quantum well grown on the $\left(11\overline{2}2\right)$ plane at a sheet-carrier density of $2\times {10}^{12}{\text{cm}}^{-2}$.

Figure 10

Material gain for TE mode (black curves) and TM mode (gray curves) of $c$ plane and semipolar and nonpolar quantum wells with ridge orientation parallel to the projection of the $c$ axis on the QW plane. Estimated threshold gain is indicated as a dashed line. Sheet-carrier densities are (from bottom to top) $5\times {10}^{12}–8.5\times {10}^{12}{\text{cm}}^{-2}$ ($c$ plane), $4\times {10}^{12}–7.5\times {10}^{12}{\text{cm}}^{-2}$ (semipolar), and $3\times {10}^{12}–6\times {10}^{12}{\text{cm}}^{-2}$ (nonpolar) in steps of $0.5\times {10}^{12}{\text{cm}}^{-2}$.

Figure 11

Material gain for extraordinary mode (black curves) and ordinary mode (gray curves) of semipolar and nonpolar quantum wells with ridge orientation perpendicular to the $c$ axis. Estimated threshold gain is indicated as a dashed line. Sheet-carrier densities are (from bottom to top) $4\times {10}^{12}–7.5\times {10}^{12}{\text{cm}}^{-2}$ (semipolar and nonpolar) in steps of $0.5\times {10}^{12}{\text{cm}}^{-2}$.

Figure 12

Absolute squared electron $e1$ and hole $A1$ wave-function overlap at the $\Gamma$ point as a function of charge-carrier density for $c$ plane, semipolar $\left(11\overline{2}2\right)$, and nonpolar $\left(11\overline{2}0\right)$ quantum wells.

Figure 13

In-plane averaged absolute squared momentum matrix elements of the two lowest-energy transitions for TE mode (solid curves) and TM mode (dashed curves) as a function of the wave number for different quantum well orientations (matrix elements not drawn are essentially zero). Note the different scales for $c$ plane and semipolar and nonpolar. The sheet-carrier density is $7\times {10}^{12}{\text{cm}}^{-2}$.

Figure 14

In-plane averaged absolute squared momentum matrix elements of the two lowest-energy transitions as a function of the wave number for ordinary (dashed curves) and extraordinary (solid curves) modes for different quantum well orientations (matrix elements not drawn are essentially zero). The sheet-carrier density is $7\times {10}^{12}{\text{cm}}^{-2}$.

Figure 15

Reduced density of states for different crystal orientations at a sheet-carrier density of $7\times {10}^{12}{\text{cm}}^{-2}$.

Figure 16

Gain spectra for different crystal angles $\theta$ and ridge orientation perpendicular (top and middle) and parallel (bottom) to the projection of the $c$ axis in the QW plane at a charge-carrier density of $7\times {10}^{12}{\text{cm}}^{-2}$. The angle is varied from $0°$ ($c$ plane) to $90°$ ($a$ plane) in steps of $10°$, from left to right (note the different scales).