Localization landscape theory of disorder in semiconductors. III. Application to carrier transport and recombination in light emitting diodes

This paper introduces a novel method to account for quantum disorder effects into the classical drift-diffusion model of semiconductor transport through the localization landscape theory. Quantum confinement and quantum tunneling in the disordered system change dramatically the energy barriers acting on the perpendicular transport of heterostructures. In addition, they lead to percolative transport through paths of minimal energy in the two-dimensional (2D) landscape of disordered energies of multiple 2D quantum wells. This model solves the carrier dynamics with quantum effects self-consistently and provides a computationally much faster solver when compared with the Schrödinger equation resolution. The theory also provides a good approximation to the density of states for the disordered system over the full range of energies required to account for transport at room temperature. The current-voltage characteristics modeled by three-dimensional simulation of a full nitride-based light emitting diode (LED) structure with compositional material fluctuations closely match the experimental behavior of high-quality blue LEDs. The model allows also a fine analysis of the quantum effects involved in carrier transport through such complex heterostructures. Finally, details of carrier population and recombination in the different quantum wells are given.

Figure 1

Schematic of the local density of states (LDOS) arising from the landscape potential $1/u_e$ for electrons. For simplicity, the effective potential is shown in 1D.

Figure 2

Flowchart of Poisson and drift-diffusion equations by applying the LL theory.

Figure 3

(a) Schematic full LED structure. (b) In-plane aluminum distribution in the ${\mathrm{Al}}_{0.15}{\mathrm{GaN}}_{0.85}\mathrm{N}$ EBL layer. (c) In-plane indium distribution in the third ${\mathrm{In}}_{0.14}{\mathrm{GaN}}_{0.86}\mathrm{N}$ QW layers. The Al and In distributions are generated by random numbers.

Figure 4

The In, Al, and Ga atoms at each cation lattice site are assigned randomly by a random number generator. This possibility to obtain each atom is decided by the average alloy composition. The local composition at each atom site is determined by the Gaussian averaging method. If the mesh node does not coincide with the atom grid position, the linear interpolation of grid map will be used to determine the composition.

Figure 5

In-plane potential energy maps computed in the midplane of the third QW of the LED structure at 2.8-V bias. (a), (c) The conduction and valence band potentials, respectively, solved by classical Poisson and DD models. (b), (d) The effective confining potentials solved by the LL theory corresponding to the conduction band and valence band potentials, respectively. The location of the plane is displayed in Fig. 6.

Figure 6

(a) Band diagrams of the LED structure at 2.8-V bias along the $z$ direction corresponding to $E_c$ and $E_v$ (black and blue lines, respectively), $1/u_e$ and $1/u_h$ (red and green lines, respectively) computed self-consistently. (b) Details of the zoomed band diagram, where the variations of the effective potentials $1/u$ with respect to the original band edge potentials can be observed. The electron and hole quasi-Fermi levels are shown by dashed lines.

Figure 7

Electron and hole carrier densities computed in the midplane of the third QW with compositional disorder at $20\mathrm{A}{\mathrm{cm}}^{-2}$ current density using: (a), (c) the classical Poisson and drift-diffusion models; (b), (d) the landscape theory implemented in Poisson-DD.

Figure 8

A comparison of Poisson-DD equations solutions for the I-V characteristics assuming homogeneous QWs (black curves), disordered QWs described by the real potentials (red curve), or by the $1/u$ effective confining potentials (blue curve).

Figure 9

Carrier distribution in the 6 QWs of the LED for electrical injection at $20\mathrm{A}{\mathrm{cm}}^{-2}$ obtained using a classical Poisson-DD model for uniform layers and the landscape theory implement in Poisson-DD for a structure with random alloy fluctuations. The $x$ and $y$ axes are shifted for illustration.

Figure 10

Variation of the effective radiative coefficient with carrier injection in the last QW $(p$ side).

Figure 11

(a), (b) The perspective views along the $z$ direction of vertical transport of $1/u_e$ and $z$ component of current $\left(J_z\right)$, respectively. (c), (d) The $1/u_e$ and $J_z$ value in the midplane $(x\text{−}y$ plane) of the third QW. All figures are solved by $1/u$-Poisson-DD model, where the LED current density is $20\mathrm{A}{\mathrm{cm}}^{-2}$.

Figure 12

Ideality factors corresponding to the I-V characteristics shown in Fig. 8 computed from Eq. (11).

Figure 13

Ratio of Auger and leakage currents to total injected current.

Figure 14

IQE curve for the full structure LED.

Figure 15

The in-plane local radiative recombination rates in the third QW computed from (a) the classical Poisson-DD model; (b) $1/u$-Poisson-DD model. Current density is fixed at $20\mathrm{A}{\mathrm{cm}}^{-2}$.

Figure 16

Distribution among the different QWs of (a) IQE. (b) Recombination current density. LED current density is $20\mathrm{A}{\mathrm{cm}}^{-2}$.