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Modeling Piezoresistivity in Silicon and Polysilicon November 19, 2005 10:41 Journal of Applied Engineering Mathematics Volume 2 MODELING PIEZORESISTIVITY IN SILICON AND POLYSILICON Gary K. Johns Mechanical Engineering Department Brigham Young University Provo,Utah 84602 ABSTRACT INTRODUCTION The piezoresistive effect of silicon is often utilized in sen- Piezoresistivity is a material property that couples bulk elec- sors. Due to the crystalline nature of silicon, the sensitivity of trical resistivity to mechanical strain [1]. In other words, as a ma- the piezoresistive effect depends on many things including the terial is stretched, compressed, or distorted in any way, the elec- direction and magnitude of the applied stress and the orientation trical resistivity changes. Almost all materials have this property, of the crystallographic plane. A method of modeling the piezore- even though in some it is so small that it is virtually undetectable. sistive effect in silicon and poly crystalline silicon is presented. However, other materials show a large change in resistivity for a This mathematical model includes partial derivatives and linear relatively small strain. One of these materials is silicon. The algebra to combine basic electrical and mechanical equations to piezoresistive effect in silicon makes it a good candidate to use describe the change in resistivity as a function of stress. An ex- as a sensor. A sensor element is a device that converts one form periment was used to verify the model. The analytical solution of energy into another [2]. An element made from silicon can and experimental results agree. The model presented in this pa- be designed to be a sensor since the piezoresistive property links per is adequate to predict the behavior of a piezoresistive sensor strain energy to change in resistance (electrical energy). This under uniform stress. change in resistance can be related to a physical phenomenon such as displacement, pressure, or force. A mathematical model of the piezoresistive effect will be presented, which may be used NOMENCLATURE to design piezoresistive sensors in silicon or polysilicon. Ei = Electric Field BACKGROUND J = Current i In 1856, William Thomson, more commonly known as Lord ρ = Resistivity Kelvin, first discovered the piezoresistive effect when he noticed σi = Stress that the electrical resistance of copper and iron wires changed when subjected to a mechanical strain. This phenomenon be- τi = Shear Stress came known as piezoresistivity: “piezo” meaning “to push” in R = Electrical Resistance Greek, and “resistivity” from the change in electrical resistance. L = Length In 1954, Charles Smith, working for Bell Laboratories, published A = Cross Sectional Area a paper [3] exposing the piezoresistive properties of silicon and germanium. He set up experiments in order to derive a method of li,mi,ni = Directional Cosines modeling this effect; from which he derived piezoresistive coeffi- ε = Strain cients that relate change in resistance to stress. His experimental πi j = Piezoresistive Coe f ficients setup is illustrated in Figure 1. He also found that silicon and Journal of Applied Engineering Mathematics April 2006, Vol.2 1 Copyright © 2005 by EngT503 BYU germanium are much more sensitive to the piezoresistive effect MODELING PIEZORESISTIVITY ALONG CRYSTAL than most metals. In the 1960’s, silicon began to be used as a AXES sensor on thin membranes by diffusing it on areas of the mem- Modeling the piezoresistive effect of silicon requires linking brane that would experience the most stress in order to measure equations related to the electrical properties of a material with pressure. It was also doped on cantilever beams to enable mea- equations describing the stress or strain and volumetric changes surement of other phenomena (see Figure 2). Since then, many of a material. Starting with Ohm’s law (E = IR), the electric field sensors have been developed that are based on the piezoresistive vector of an anisotropic cubic crystal can be derived. It is related effect of silicon. to the current vector by a resistivity tensor [4]. E1 ρ1 ρ6 ρ5 J1 E2 = ρ6 ρ2 ρ4 J2 (1) E3 ρ5 ρ4 ρ3 J3 The resistivity tensor is symmetric and is therefore made up of only six components. When the crystal is in an unstressed state, the resistivity components along the diagonal of the matrix, which represent the resistivity along the <100> axes of the cubic, are equal to each other. The other three components are also equal to each other and are 0. ρ1 = ρ2 = ρ3 = ρ, ρ4 = ρ5 = ρ6 = 0 (2) When the cubic is stressed, each component of resistivity changes. The new resistivity of each component in the resistivity tensor is ρ1 ρ ∆ρ1 Figure 1. Charles Smith’s experimental setup ρ2 ρ ∆ρ2 ρ3 ρ ∆ρ3 = + (3) ρ4 0 ∆ρ4 ρ5 0 ∆ρ5 ρ6 0 ∆ρ6 The link relating change in resistance to stress is the Π matrix. This is the matrix which Charles Smith worked to define. The ex- perimental setup in Figure 1 shows how he derived the piezore- Proof Mass R1 Side Flexure View sistive coefficients for silicon and germanium. The change in resistance is directly related to the stress on the object through Longitudinal Top these piezoresistive coefficients by R4 R2 View Flexible Diaphragm Transverse Top Top View 1 R3 View ∆ρi = [πi j][σ j] (4) ρ a) b) Figure 2. Piezoresistive sensor elements doped onto diaphragms and where the stresses relate to the stresses shown in Figure 3. σ1,σ2, beams and σ3 are along the axes of the cube and σ4,σ5, and σ6 map to the shear stresses τ1,τ2, and τ3. With six stress components and six resistivity components, the π matrix must be a six-by-six matrix. Therefore, 36 π coefficients are required to populate the Journal of Applied Engineering Mathematics April 2006, Vol.2 2 Copyright © 2005 by EngT503 BYU MODELING PIEZORESISTIVITY IN ANY DIRECTION The above equations are used to model the piezoresistive ef- fect of silicon in the direction of the crystal axes. However, it is often convenient to be able to model the piezoresistive effect along other directions as well. Two new π coefficients can be defined in terms of the original 3 π coefficients and an arbitrary direction. These two new π coefficients relate the change in re- sistance to the stress in the longitudinal direction (defined to be the same direction as the current flow) and the transverse direc- tion (defined to be perpendicular to the current flow). πl and πt are 2 2 2 2 2 2 πl = π11 + 2(π44 + π12 − π11) l1 m1 + l1 n1 + m1n1 2 2 2 2 2 2 (8) πt = π12 − (π44 + π12 − π11) l1 l2 + m1m2 + n1n2 Figure 3. Stress Cube l1,m1, and n1 are directional cosines between the longitudinal matrix. However, for the cubic crystal structure of silicon, the direction of the sample and the cubic axes and l2,m2, and n2 are matrix simplifies to the directional cosines between the direction of one component of the transverse stress and the cubic axes [6]. π11 π12 π12 0 0 0 π12 π11 π12 0 0 0 LONGITUDINAL EXAMPLE OF PIEZORESISTIVITY π12 π12 π11 0 0 0 (5) A simple example that demonstrates the modeling of the 0 0 0 π44 0 0 piezoresistive effect can be seen in the uniaxial stress case. The 0 0 0 0 π44 0 setup for the example is shown in Figure 4. The resistance of the 0 0 0 0 0 π44 Only three π coefficients (π11,π12, and π44) are needed. When all of the values of the components are filled in, equation 4 becomes ∆ρ1 π11 π12 π12 0 0 0 σ1 h ∆ρ2 π12 π11 π12 0 0 0 σ2 I b 1 ∆ρ3 π12 π12 π11 0 0 0 σ3 = (6) ρ ∆ρ4 0 0 0 π44 0 0 τ1 dV L ∆ρ5 0 0 0 0 π44 0 τ2 V ∆ρ6 0 0 0 0 0 π44 τ3 As shown in [5], [1], and [4], equations 1, 3, and 6 can now be combined to obtain E1 = ρJ1 + ρπ11σ1J1 + ρπ12 (σ2 + σ3)J1 + ρπ44 (J2τ3 + J3τ2) F E2 = ρJ2 + ρπ11σ2J2 + ρπ12 (σ1 + σ3)J2 + ρπ44 (J1τ3 + J3τ1) E3 = ρJ3 + ρπ11σ3J3 + ρπ12 (σ1 + σ2)J3 + ρπ44 (J1τ2 + J2τ1) (7) which describe the electric field potential along the crystal axes. Figure 4. Tension example The first term in each of the equations is the unstressed condition of the cubic. The next term takes in the effect of the stress along the axis of the respecting electric field component. The effect of unstressed beam, R, is given by the other two axial stresses are presented in the third term, and the last term models how the shear stresses effect the electrical ρL R = (9) potential. A Journal of Applied Engineering Mathematics April 2006, Vol.2 3 Copyright © 2005 by EngT503 BYU where ρ is the bulk resistivity, L signifies length, and A is the In a similar manner the fractional change in resistance in the cross sectional area. By implicit differentiation: transverse direction can be found to be L ρ ρL ∆R = ( E − 1) (16) δR = δρ + δL − δA 2 (10) εt πt A A A R t Dividing equation 10 by the original resistance equation yields MODELING PIEZORESISTIVITY IN POLY CRYS- TALLINE SILICON δR δρ δL δA = + − (11) Many MEMS devices are made from polysilicon instead of R ρ L A pure silicon. The π coefficients used in the silicon Π matrix are no longer valid with polysilicon. Polysilicon can be approxi- Equation 11 can be approximated as mated as several small single crystalline grains of silicon sepa- rated by grain boundaries [5]. The larger the grains, the more the ∆R ∆ρ ∆L ∆A crystalline material acts like single crystalline material.
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