Experimental Characterization of the Temperature Dependence of the Piezoresistive Coefficients of Silicon

Abstract

In this work, the dependence of the silicon piezoresistive coefficients p11, p12, and p44 on temperature is investigated. Experimental calibration results for the piezoresistive coefficients of silicon as a function of temperature are presented and compared and contrasted with existing values from the literature. Stress-sensing test chips are widely used to investigate die stresses occurring to assembly and packaging operations. They incorporate resistor- or transistor-sensing elements that are able to measure stresses through the observation of the changes in their resistivity/mobility. The piezoresistive behavior of such sensors can be completely characterized through the use of three piezoresistive (pi) coefficients, which are electro-mechanical material constants. In most prior investigations, calibration of the piezoresistive coefficients has been performed at room temperature. Such restriction limits the accuracy of test chip stress measurements made at other temperatures. In this work, we have performed an extensive experimental study on temperature dependence of the piezoresistive behavior of silicon. Calibration has been performed using four-point bending of chip-on-beam specimens. A special four-point bending apparatus has been constructed and integrated into an environmental chamber capable of temperatures from -185 to 300C. Finite element analysis has been used to calculate the stress states applied to the calibration samples. Our test results show that the piezoresistive coefficients for p- and n-type silicon decrease monotonically when temperature is increased from -150 to 125C. Our goals in this work are to enable packaging stress measurements over a wide range of temperature, to obtain a comprehensive set of piezoresistive coefficients over a broad range of temperature, to resolve sign issues and demonstrate proper methods for relating coefficients at different temperatures, and to obtain consistent formulation of Resistance valid over wide temperature range.