Airy Function-Based Model for 2-DEG Charge and Surface Potential in N-Polar Gallium Nitride Heterostructures

Abstract

A systematic, analytical method to obtain the charge density and surface potential in an N-polar GaN/AlGaN heterostructure is presented. This method also provides the wave functions, sub-band energies, and band diagram. The model accounts for the greater leakage of the wave functions into the spacer and barrier layers in the N-polar structure as compared to the Ga-polar structure. The framework is based on the solution of Schr ödinger-Poisson equations for an approximately triangular well potential with finite confinement, achieved using first-order perturbation theory. No fitting parameters are used. Results obtained using our model are in good agreement with numerical results (obtained using the finite difference method) for a wide range of bias conditions (from off to on states) and AlxGa 1−x N spacer mole fractions.

A Closed-Form Model for 2-DEG Charge and Surface Potential in N-Polar Gallium Nitride Heterostructures

“…We define [28] V of f as the gate voltage at which the Fermi level is aligned with the bottom of the quantum well (i.e ε f = 0),…”

“…Fig. 2 depicts the flow of our earlier analytical model [28]. The blocks shaded yellow involve implicit/non-closed-form expressions.…”

“…Our strategy in [28] was to describe the wave function in the channel, spacer and barrier layers as linear combinations of Airy functions, which are evaluated based on an approximate, effective electric field. The solution of a determinant equation, which enforces continuity conditions on the wave functions, yields very good solutions for the wave functions and an initial estimate of the sub-band energies.…”

“…This article is organized as follows. Section II describes a typical N-polar GaN heterostructure and reviews key aspects of our earlier, analytical model [28]. The modifications required to convert the analytical model into a closed-form model are presented in Section III.…”

A Closed-Form Model for 2-DEG Charge and Surface Potential in N-Polar Gallium Nitride Heterostructures

“…As a simplification, we consider only the lowest sub-band (with energy ε 0 ), since the second sub-band (energy ε 1 ) lies above the Fermi level [28] and hence does not contribute significantly to the charge in the 2-DEG. Further, in structures having a spacer layer, we ignore the leakage of the wave functions into the barrier layer, corresponding to the 2 × 2 method in [28]. The wave functions in the channel and spacer layers can then written in terms of Airy functions [31], [32] as…”