MATHEMATICAL MODEL FOR CALCULATING BORON DIFFUSION COEFFICIENT IN SILICON

Abstract

The article presents the development and substantiation of a mathematical model for calculating the boron diffusion coefficient in silicon, taking into account the influence of the main technological parameters involved in p–n junction formation. The relevance of the study is determined by the increasing requirements for accuracy in reproducing the parameters of semiconductor structures in the production of modern electronic devices and integrated circuits. Traditional approaches to determining the diffusion coefficient are mainly based on temperature dependence and do not consider the effect of the implanted dopant dose, which leads to errors in predicting the junction depth and concentration profile. The paper analyzes existing literature data concerning the proportionality factor and activation energy of boron diffusion, identifying their variability and dependence on technological process conditions. Calculations were performed for n-type silicon wafers with different resistivity values in the drive-in temperature range of 1050–1150 °C, implantation doses of 4–100 μC/cm², and thermal treatment durations of 75–225 minutes. Based on analytical transformations of diffusion equations and the determination of concentrations through resistivity values, numerical values of diffusion coefficients were obtained for various process regimes. It has been established that the proportionality factor D₀ depends on the dopant dose and increases according to a power law with an exponent of 0.331, while the activation energy E also increases with increasing dose according to an exponent of 0.0091 and is practically independent of temperature and process duration within the studied range. Generalization of the obtained results made it possible to formulate an analytical expression for the boron diffusion coefficient that simultaneously accounts for both temperature and implantation dose. The proposed mathematical model improves the accuracy of engineering calculations of diffusion layer parameters and can be applied to optimize technological regimes for manufacturing semiconductor structures with specified electrophysical characteristics. The obtained results are of practical importance for the design and improvement of microelectronic device fabrication technologies