Semiconductor Sensors: II—Piezoresistive Devices

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-17, NO. 6, NOVEMBER 1970

415

Semiconductor Sensors: Il-Piezoresistive Devices ANDREA TARONI, MARIA PRUDENZIATI, GIANNI ZANARINI, MEMBER, IEEE

Abstract-This paper presents a survey of the recent developments of piezoresistive sensors. First, the physical concepts which are essential for the understanding of these devices are recalled, and the characteristics and limitations (particularly temperature limitations) of semiconductor strain gauges are discussed. A further section is devoted to strain-gauge pressure transducers (the most important application of strain gauges), and, finally, possible future developments are outlined.

A as E G J m

LIST OF SYMBOLS cross section (cm2) thermal expansion coefficient (0K-1) electric field (V/cm) gauge factor current density (A/cm2) temperature coefficient of the gauge factor

(0K-1)

R resistance (ohms) Ro resistance at reference pressure (ohms) RT resistance at temperature T (ohms) S relative elongation T temperature (OK) To reference temperature (OK) Tij mechanical stress (dyn/cm2) Y Young's modulus (dyn/cm2) I length (cm) aT temperature coefficient of semiconductors (0K-1)

wrj

piezoresistivity coefficients (cm2/dyn)

p resistivity (Q2.cm) po resistiVity of the material not under pressure

(Q-cm).

AND

involved, and, as a consequence, by providing a deeper understanding of the present performance and limitations of these devices and of possible future developments.

II. PIEZORESISTIVE EFFECT The application of pressure to the crystal structure of a semiconductor material temporarily alters its symmetry, and consequently modifies the conduction mechanism which is highly sensitive to these variations; the observed macroscopic effect is a resistivity variation [1]- [6]. This variation can be positive or negative; i.e., the pressure may cause either an increase or a decrease of resistivity, according to the material and the crystallographic direction along which it is applied. For a qualitative explanation of the piezoresistivity effect let us consider Fig. 1, which shows the band structure of n-type silicon, and suppose that a mechanical stress is applied along the [100] direction. The consequent crystal elongation in this direction destroys its cubic symmetry so that the six energy minima in the band structure are no longer at the same level; in particular, the two minima along the considered direction are raised with respect to the four nearest neighbors, with a consequent migration of electrons toward the lower energy valleys [1]. If an electric field is applied along the direction of the mechanical stress, a resistivity decrease can be observed when the stress is increased, because of the fact that the carriers moved to the four valleys normal to those of the [100] direction are characterized by a smaller effective mass and consequently by a higher mobility. When the mechanical stress and the electric field are applied along the [111] direction, the equienergetic ellipsoid axes all form the same angle with this direction. There is no relative displacement of the energy minima, and therefore repopulation effects of the valleys do not take place. The resistivity variations are in this case due to mobility variations with the applied mechanical stress, but this effect is negligible with respect to the

I. INTRODUCTION 7HILE the importance of semiconductor sensors is being more and more acknowledged, piezoresistive devices (and their most important application, strain-gauge pressure transducers) are still far less popular than, for instance, thermoresistive devices. This results from a number of reasons such as overestimation of the limitations (particularly temperature limitations) and some difficulty in understanding their operating principles (for which more than a min- repopulation effect. For a quantitative explanation of the piezoresistivity imum amount of physical knowledge is required). This paper is intended to overcome these two mis- effect let us consider the three-dimensional model understandings by explaining the physical concepts shown in Fig. 2, representing the basic elements of a cubic symmetrical crystal [2]. (The axes 1, 2, and 3 are the semiconductor crystallographic axes.) Manuscript received January 13, 1970; revised June 8, 1970. Suppose we first apply a mechanical stress Til to the of Institute are with the Physics, A. Taroni and M. Prudenziati University of Modena, Modena, Italy. cubic faces normal to the 1 axis and then measure the G. Zanarini is with the Institute of Physics, University of Bologna, resistivity along this direction. The relation besample Bologna, Italy.

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, NOVEMBER 1970

416

similar to (1) may be written; i.e., po(l + lrhTh) where the piezoresistivity coefficient the 7rwj through the relation p

'kll [II-h

I'-'

(3)

Wrh depends upon

1r :=1- + 2(Q44 + 1i2 -7rl) [12m2 + 12n2 + m2n2] (4) in which 1, m, n are the direction cosines of the direction h with respect to the crystallographic axes. I~~~~Elo As can be seen, the quantity inside the square brackets on the right side has a maximum for l= m = n = 1/V3 (values corresponding to the [111] direction), while it has a minimum equal to zero for the crystalFig. 1. Schematic diagram of constant energy surfaces with respect to cristallographic axes for n-type Si. The effect of stress on the lographic axes directions, so that Wrh shows a maximum valley energies shown is indicated by dotted ellipsoids. either for the [111] direction (if W44+WT12 -711> 0) or for the crystallographic axes directions (if W44+W12-W11