An Empirical Model of the Silicon Bandgap Dependence on the External Pressure
Keywords:deformation, pressure, band gap, delamination, model, approximation, conductance, current density, semiconductor
One of the factors that affect the electrical characteristics and parameters of integrated circuits is the external pressure, which most often results from delamination processes in the thickness of the chip body. Under the influence of pressure, the band gap of the semiconductor changes, which leads to changes in the electrical parameters of the active and passive components of the IC. Current models of bandgap dependence on pressure are very simplified, do not provide modeling accuracy in a wide range of pressure values, which does not allow to development of adequate mathematical models of semiconductor components for further study of the effect of pressure on the electrical characteristics and parameters of integrated circuits. To build a more accurate mathematical model of the dependence of the bandgap of silicon on the external pressure, the effect of pressure on the electrical parameters of the semiconductor resistor and diode in the integrated circuit was experimentally investigated. The choice of passive and active components for experimental research made it possible to eliminate the influence of the features of technological manufacturing processes and to obtain more reliable data for further construction of the approximation model. The studies were performed in the pressure range from 0 to 25 GPa. Measurements were made using a specially designed measuring stand. The stand makes it possible to perform high-precision measurements of the resistance of integral resistors, VAC diodes, and transistors under the influence of controlled pressure applied to the surface of the passive or active component implemented in the IC crystal. The measurement error of this stand is determined by the error of the multimeter and is +/- 0.001 V for voltage and +/- 0.0001 A for current. An error of +/- 0,025 N is introduced by the force sensor. The expressions were obtained to determine the bandwidth value due to the experimental values of the resistor and current resistance through the diode at zero pressure and a certain amount of pressure, which allowed us to construct the bandgap of silicon on the pressure. It is shown that the classical linear model does not reflect the real nonlinear nature of such a dependence. A clarifying nonlinear coefficient is proposed and a series of computational experiments are conducted to select the most optimal method of approximation of the experimental data. In the computational experiment, such methods of approximation as power, logarithmic, hyperbolic, and exponential were investigated. All calculations were performed using MATLAB R2016a. To increase the approximation accuracy, 25 experimental data points were used in the calculations. The smallest relative error of approximation was obtained for hyperbolic approximation. Using the hyperbolic approximation, an empirical model of the dependence of the bandgap of silicon on the external pressure is constructed, the error of which does not exceed 2 % in the pressure range from 0 to 25 GPa. Using the proposed empirical model of the dependence of the width of the silicon band gap on the pressure, empirical models of the corresponding dependences of the resistance of the integrated resistor and the current through the integrated diode are constructed. Adjustment coefficients are proposed which allowed reducing the relative error of the model for resistance up to 11%, for current up to 25%. The obtained accuracy of the models allows us to use them for further studies of the influence of pressure on the electrical characteristics of integrated circuits using circuit analysis and optimization.
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