The non-destructive estimation of doping concentrations in semiconductor devices is of paramount importance for many applications ranging from crystal growth, the recent redefinition of the 1kg to defect, and inhomogeneity detection. A number of technologies (such as LBIC, EBIC and LPS) have been developed which allow the detection of doping variations via photovoltaic effects. The idea is to illuminate the sample at several positions and detect the resulting voltage drop or current at the contacts. We model a general class of such photovoltaic technologies by ill-posed global and local inverse problems based on a drift-diffusion system that describes charge transport in a self-consistent electrical field. The doping profile is included as a parametric field. To numerically solve a physically relevant local inverse problem, we present three different data-driven approaches, based on least squares, multilayer perceptrons, and residual neural networks. Our data-driven methods reconstruct the doping profile for a given spatially varying voltage signal induced by a laser scan along the sample's surface. The methods are trained on synthetic data sets (pairs of discrete doping profiles and corresponding photovoltage signals at different illumination positions) which are generated by efficient physics-preserving finite volume solutions of the forward problem. While the linear least square method yields an average absolute $\ell^\infty$ error around $10\%$, the nonlinear networks roughly halve this error to $5\%$, respectively. Finally, we optimize the relevant hyperparameters and test the robustness of our approach with respect to noise.

翻译：半导体器件中掺杂浓度的非破坏性估计对于从晶体生长、最新1千克重新定义到缺陷及不均匀性检测等众多应用至关重要。已有多种技术（如LBIC、EBIC和LPS）能够通过光伏效应检测掺杂变化。其原理是在样品多个位置进行光照，并检测接触端产生的电压降或电流变化。我们基于描述自洽电场中电荷传输的漂移-扩散系统，将此类光伏技术的一般类别建模为病态的全局和局部反问题。掺杂分布作为参数化场包含在内。为数值求解物理相关的局部反问题，我们提出了三种基于最小二乘法、多层感知器和残差神经网络的数据驱动方法。我们的数据驱动方法能够根据激光沿样品表面扫描引起的空间变化电压信号重建掺杂分布。这些方法在合成数据集（由前向问题的高效保物理有限体积解生成的离散掺杂分布与对应照明位置光电压信号对）上进行训练。线性最小二乘法产生的平均绝对$\ell^\infty$误差约为$10\%$，而非线性网络则将该误差大致减半至$5\%$。最后，我们优化了相关超参数并测试了方法对噪声的鲁棒性。