IN – Track B



Understanding Doping Variations in Silicon Crystals

Project Heads

Nella Rotundo, Patricio Farrell, Natascha Dropka

Project Members

Stefan Kayser (WIAS)

Project Duration

01.01.2020 – 31.12.2020

Located at




It is impossible to measure the temperature distribution within a crystal during growth. Yet, to improve crystal growth this is paramount. Due to temperature fluctuations, microscopic variations appear in the doping concentration. These striations follow the solid-liquid interface and can be measured even in the cooled-down crystal. Traditional techniques to measure striations have a poor spatial resolution, take long time or are inherently destructive. To overcome these limitations, the lateral photovoltage scanning method (LPS) has been proposed. This opto-electrical measurement procedure detects doping inhomogeneities at wafer-scale and room temperature in a non-destructive fashion. The LPS method excites the semiconductor crystal with a laser, creating a voltage difference at the sample edges which is proportional to the local doping variation.



The goal of this project is to efficiently model and simulate the LPS technique.


Simulation strategy

We solve the forward LPS model for a given doping profile via a finite volume discretization and Newton solver embeddings with the open-source ddfermi software tool ddfermi. After solving for the equilibrium solution, we first apply a bias and then turn on the laser. Once we reach the prescribed laser power, we sweep the sample for different laser spot positions. For every laser spot position we need to ensure that potentials at the boundaries satisfy a nonlinear boundary condition derived from modified nodal analysis (MNA). To achieve this, we employ the secant method.



Tauc made three main theoretical predictions: First, the LPS voltage depends on local doping variations. Second, LPS voltage depends logarithmically on moderate laser intensities. And third, eventually LPS voltage saturates for higher laser intensities due to the screening effect. We proved all three predictions mathematically. Since several approximations, we also investigated the limits of the approach. Furthermore, we were able to qualitatively reproduce all three of Tauc’ predictions with our computational setup.


Our code runs about two orders of magnitudes faster than earlier implementations based on commercial software. It also performs well for small doping concentrations which previously could not be simulated at all due to numerical instabilities. We present a convergence study which shows that the LPS voltage converges quadratically. Finally, our simulations provide experimentalists with reference laser powers for which meaningful voltages can still be measured. For higher laser power the screening effect does not allow this anymore.


Selected Publications

    1. A. Lüdge and H. Riemann, “Doping Inhomogeneities in Silicon Crystals Detected by the Lateral Photo-voltage Scanning (LPS) Method”, Inst. Phys. Conf. Ser., vol. 160, pp. 145–148, 1997
    2. J. Tauc, “The Theory of a Bulk Photovoltaic Phenomenon in Semiconductors,” Czech J Phy, vol. 5, pp. 178– 191, 1955.
    3. S. Kayser, A. Lüdge, and K. Böttcher, “Computational simulation of the lateral photovoltage scanning method,” In Proceedings of the 8th International Scientific Colloquium, pp. 149–154, 2018.

Our publications can be found here: Project related publications



Simulated charge carrier distribution due to impurities.

Alignment between gradient of doping concentration and LPS voltage for a laser beam sampling along the x axis.

Difference of generation and recombination rates as laser hits the crystal from above.

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