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.

翻译：针对半导体器件的稳态线性化单极漂移-扩散模型，给出了无限维空间中反演掺杂剖面问题的严格贝叶斯数学表述。其目标是估计掺杂剖面的后验概率分布并计算其后验均值，从而能够通过电压-电流测量值重建掺杂剖面。通过证明半导体模型关于未知参数的有界性和连续性性质，表明了该贝叶斯反演问题的适定性。提出了一种基于物理信息先验模型的预处理Crank-Nicolson马尔可夫链蒙特卡洛方法，用于掺杂剖面的贝叶斯估计。二维二极管的数值结果验证了所提方法的有效性。