A Systematic Approach for Multidimensional, Closed-Form Analytic Modeling: Effective Intrinsic Carrier Concentrations in Ga(1−x)Al(x)As Heterostructures

A critical issue identified in both the technology roadmap from the Optoelectronics Industry Development Association and the roadmaps from the National Electronics Manufacturing Initiative, Inc. is the need for predictive computer simulations of processes, devices, and circuits. The goal of this paper is to respond to this need by representing the extensive amounts of theoretical data for transport properties in the multi-dimensional space of mole fractions of AlAs in Ga(1−)(x)Al(x)As, dopant densities, and carrier densities in terms of closed form analytic expressions. Representing such data in terms of closed-form analytic expressions is a significant challenge that arises in developing computationally efficient simulations of microelectronic and optoelectronic devices. In this paper, we present a methodology to achieve the above goal for a class of numerical data in the bounded two-dimensional space of mole fraction of AlAs and dopant density. We then apply this methodology to obtain closed-form analytic expressions for the effective intrinsic carrier concentrations at 300 K in n-type and p-type Ga(1−)(x)Al(x)As as functions of the mole fraction x of AlAs between 0.0 and 0.3. In these calculations, the donor density N(D) for n-type material varies between 10(16) cm(−3) and 10(19) cm(−3) and the acceptor density N(A) for p-type materials varies between 10(16) cm(−3) and 10(20) cm(−3). We find that p-type Ga(1−)(x)Al(x)As presents much greater challenges for obtaining acceptable analytic fits whenever acceptor densities are sufficiently near the Mott transition because of increased scatter in the numerical computer results for solutions to the theoretical equations. The Mott transition region in p-type Ga(1−)(x)Al(x)As is of technological significance for mobile wireless communications systems. This methodology and its associated principles, strategies, regression analyses, and graphics are expected to be applicable to other problems beyond the specific case of effective intrinsic carrier concentrations such as interpreting scanning capacitance microscopy data to obtain two-dimensional doping profiles.