A rigorous Bayesian formulation of the inverse doping profile problem in infinite dimensions for a stationary linearized unipolar drift-diffusion model for semiconductor devices is given. The goal is to estimate the posterior probability distribution of the doping profile and to compute its posterior mean. This allows for the reconstruction of the doping profile from voltage-current measurements. The well-posedness of the Bayesian inverse problem is shown by proving boundedness and continuity properties of the semiconductor model with respect to the unknown parameter. A preconditioned Crank-Nicolson Markov chain Monte-Carlo method for the Bayesian estimation of the doping profile, using a physics-informed prior model, is proposed. The numerical results for a two-dimensional diode illustrate the efficiency of the proposed approach.

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