On the example of semiconductor crystals Ge, Si, PbTe, PbS, InSb with different levels of doping and different types of conductivity, the geometry of the piezoresistive effect was optimized, namely, such directions of voltage measuring and uniaxial pressure applying were determined, which ensure the maximum achievable value of the effect. The optimization is based on an approach using the construction and analysis of extreme surfaces that represent all possible maxima of the objective function (the magnitude of the effect) under different spatial orientations of interacting factors. The optimization parameters were the angles that determine the directions of the unit vectors of the directions of current and uniaxial pressure applying. The directions of the radius vectors of the points on the extreme surface coincide with the ones in which the electric voltage is measured, and the length of this radius vector for each point was determined by setting such optimization parameters for which the magnitude of the effect for a given direction of voltage measuring would be maximal. It is shown that the optimal interaction geometry in most of the studied cases is longitudinal, and only for n-PbTe, p-InSb crystals it is transverse (although not identical), and the optimal directions for the studied crystals are <100>, <110> or <111> depending on the composition of the crystal and the type of doping. Despite the fact that all investigated crystals belong to the same point symmetry group (m3m), the shapes of the extreme surfaces for them are significantly different, which is caused by different ratios between the piezoresistive coefficients. Typical forms of extreme surfaces have been identified, and in order to explain the obtained results, an analysis of limiting cases that differ in the ratio of piezoresistive coefficients has been carried out. Based on this analysis, four main types of extreme surfaces were established. A scheme has been built that allows, in the case of cubic crystals, to estimate the type of extreme surface and the corresponding optimal directions of voltage measuring, current density (for cubic crystals, these directions coincide) and uniaxial pressure applying. On the basis of this scheme, the forms of extreme surfaces obtained for the investigated crystals are explained.

[1] Barlian, A., Park, W.-T., Mallon, J., Rastegar, A. and Pruitt, B. (2009) ”Review: Semiconductor piezoresistance for microsystems”, Proc. IEEE, vol. 97, no. 3, pp. 513-552. doi: 10.1109/JPROC.2009.2013612

[2] Doll, J. and Pruitt, B. (2013) Piezoresistor Design and Applications. Springer Science+Business Media, New York. doi: 10.1007/978-1-4614-8517-9

[3] Fiorillo, A., Critello, C. and Pullano S. (2018) ”Theory, technology and applications of piezoresistive sensors: A review”, Sensors and Actuators A, vol. 281, pp. 156-175. doi: 10.1016/j.sna.2018.07.006

[4] Li, J., Fang, L., Sun, B., Li, X. and Kang S. (2020) ”Review – Recent progress in flexible and stretchable piezoresistive sensors and their applications”, Journal of The Electrochemical Society, vol. 167, no. 3, 037561. doi: 10.1149/1945-7111/ab6828

[5] Buryy, O., Andrushchak, A., Kushnir, O., Ubizskii, S., Vynnyk, D., Yurkevych, O., Larchenko, A., Chaban, K., Gotra, O. and Kityk, A. (2013) ”Method of extreme surfaces for optimizing the geometry of acousto-optic interactions in crystalline materials: Example of LiNbO3 crystals”, J. Appl. Phys., vol. 113, no. 8, 083103. doi: 10.1063/1.4792304

[6] Buryy, O., Andrushchak, A., Demyanyshyn, N. and Mytsyk B. (2016) ”Optimizing of piezo-optic interaction geometry in SrB4O7 crystals”, Optica Applicata, vol. 46, no. 3, pp. 447-459. doi: 10.5277/oa160311

[7] Andrushchak, A., Buryy, O., Andrushchak, N., Hotra, Z., Sushynskyi, O., Singh, G., Janyani, V. and Kityk, I. (2017) ”General method of extreme surfaces for geometry optimization of the linear electro-optic effect on an example of LiNbO3:MgO crystals”,. Appl. Opt., vol. 56, no. 22, pp. 6255-6262. doi: 10.1364/AO.56.006255

[8] Andrushchak, N., Buryy, O., Danylov, A., Andrushchak, A. and Sahraoui, B. (2021) ”The optimal vector phase matching conditions in crystalline materials determined by extreme surfaces method: Example of uniaxial nonlinear crystals”, Opt. Mat., vol. 120, 111420. doi: 10.1016/j.optmat.2021.111420

[9] Sirotin, Yu. and Shaskolskaja, M. (1983) Fundamentals of crystal physics. Imported Pubn., Moscow.

[10] Microelectronic sensors of physical values (2003). Ed. by Z. Hotra. Vol. 2. Liga-press, Lviv (in Ukrainian).

[11] Press, W., Flannery, B., Teukolsky, S. and Vetterling, W. (1989) Numerical Recipes in Pascal. The art of Scientific Computing. Cambridge University Press, Cambridge.