Nonequilibrium Green’s Function Modeling of Trap-Assisted Tunneling in ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$/GaN Light-Emitting Diodes

This work presents an investigation of carrier transport in $\mathrm{Ga}\mathrm{N}$-based light-emitting diodes in the subthreshold forward-bias regime where tunneling processes are relevant. A quantum kinetic theory of trap-assisted tunneling is developed within the framework of the nonequilibrium Green’s function formalism. Based on fully nonlocal scattering self-energies computed in the self-consistent Born approximation and a multiband description of the electronic structure, the model provides access to spectral quantities, such as the local density of states and the current density, which are essential to understand the nature of the tunneling process. The quantum nonradiative recombination rates can be reproduced by the conventional Shockley-Read-Hall theory, provided that the classical charge is replaced with the correct quantum charge, which means that trap-assisted tunneling can be described with drift-diffusion solvers complemented with appropriate quantum corrections for the calculation of the local density of states. The subthreshold $I$-$V$ characteristics and ideality factors predicted by the quantum kinetic model are in agreement with measurements.

Figure 1

Local density of states (left panel, color maps) of a single-quantum-well LED in weak forward bias, showing a complex structure of bound and quasibound states. Contour lines (black curves) show the extension of these resonant states in the energy gap. The green line marks the energy level of the defect, which is assumed to be at midgap (conduction and valence edges are represented by blue and red lines, respectively) and uniformly distributed in the structure. Sharp phonon resonances related to multiphonon relaxation can be observed in the logarithmic plot of the LDOS evaluated at $z=23\mathrm{nm}$ (right panel), where $U_{\mathrm{SRH}}$ is maximum.

Figure 2

SRH scattering current computed with NEGF at a forward bias of 2 V (left panel) and 2.4 V (right panel). The continuous stripes, corresponding to current-carrying channels, switch bands where $U_{\mathrm{SRH}}$ is maximum. Black lines represent the corresponding energy-integrated recombination rates (arbitrary units). All simulations are performed at room temperature. The total energy-integrated currents are perfectly conserved.

Figure 3

SRH recombination rates computed with different models at a forward bias of $2\mathrm{V}$ (blue lines, left axis), and the electric field profile (red line, right axis). The SRH rate computed with NEGF (solid blue line) is well reproduced by the quantum-corrected SRH rate (dashed blue line) computed from Eq. (20) with the NEGF carrier densities. Neglecting tail states, which corresponds to the classical limit (i.e., SRH without the inclusion of tunneling), significantly underestimates the recombination rate (dotted blue line).

Figure 4

Electron and hole densities at different forward bias voltages.

Figure 5

The $I$-$V$ characteristics (left axis) and corresponding ideality factor (right axis), computed with the NEGF approach (solid lines) and from measurements [7] (circles). A series resistance $R_s=1\mathrm{\Omega }$, accounting for nonideal contacts and buffer layers, is selected to match the slope of the experimental $I$-$V$ curve at high voltages.