Hamiltonian plasma-harmonic oscillator theory: Generalized depth profilometry of electronically continuously inhomogeneous semiconductors and the inverse problem

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Abstract
The Hamilton–Jacobi formalism of the propagation of an electron‐hole photoexcited plasma in continuously inhomogeneous semiconductors with arbitrary depth profiles in carrier diffusivity and/or minority‐carrier lifetime is presented. The theoretical model is based on the variational formulation of the canonical Hamiltonian for the evolution of carrier plasmas and shows that propagating plasma waves can be formally described by a plasma‐harmonic oscillator, thus generalizing existing theoretical treatments of photoexcited carrier diffusion in electronic solids. Simple analytical expressions for the free‐carrier diffusion magnitude and phase frequency dependencies in the case of exponential carrier diffusivity and minority carrier lifetime profiles are obtained. The effect of continuously varying electronic properties on the surface plasma density magnitude and phase frequency behavior is demonstrated through computer simulations and very good quantitative agreement is obtained with photothermal radiometric data from an ion‐implanted Si wafer allowing the reconstruction of the lifetime depth profile.