AC 2010-1626: ON TEACHING THE OPERATING PRINCIPLES OF PIEZORESISTIVE SENSORS

Richard Layton, Rose-Hulman Institute of Technology Richard A. Layton is the Director of the Center for the Practice and Scholarship of Education (CPSE) and an Associate Professor of Mechanical Engineering at Rose-Hulman Institute of Technology. He earned a B.S. in Engineering from California State University, Northridge, and received his M.S. and Ph.D., both in Mechanical Engineering, from the University of Washington, Seattle. His areas of scholarship include student team management, assessment, education, and remediation, undergraduate engineering laboratory reform focused on student learning, data analysis and visualization, and engineering system dynamics. His work has been recognized with multiple best-paper awards. He conducts workshops in student team-building, team-formation and peer evaluation, in laboratory assessment, and in effective teaching. Prior to his academic career, Dr. Layton worked for twelve years in consulting engineering, culminating as a group head and a project manager. He is a guitarist and songwriter and a member of the rock band “Whisper Down”.

Thomas Adams, Rose-Hulman Institute of Technology Thomas M. Adams is an Associate Professor of Mechanical Engineering at Rose-Hulman Institute of Technology. He earned a B.S. in Mechanical Engineering from Rose-Hulman Institute of Technology, and received his M.S. and Ph.D., both in Mechanical Engineering, from the Georgia Institute of Technology. His areas of expertise include heat transfer and energy systems, MEMS, and microfluidics. He has worked extensively to bring the field of MEMS and microscale technology to an undergraduate audience. He is the recipient of best paper awards for both educational and technical papers and has been awarded the Dean’s Outstanding Teaching Award at Rose-Hulman Institute of Technology. He is an avid fingerstyle/jazz guitarist, an amateur body-builder, and a yoga instructor. Page 15.923.1

© American Society for Engineering Education, 2010 On Teaching the Operating Principles of Piezoresistive Sensors

Abstract

We present an approach to teaching the operating principles of piezoresistive sensors that addresses many of the limitations of the treatments encountered in most instrumentation and MEMS textbooks. Namely, we direct the presentation to an undergraduate audience rather than a research-level audience and at the same time we avoid oversimplifying the development of the principles of operation. To this end, we make a discussion of bridge analysis central to the development, use a strain-formulation for gage factor and piezoresistor placement rather than the more common stress-formulation, and keep the associated physics and mathematics at an appropriate level for sophomore engineering undergraduates. In so doing, we maintain accessibility and coherence throughout. We present several sets of learning objectives and strategies for teaching the material that can be tailored to suit the needs of a particular course.

Introduction

Piezoresistive sensors are commonplace—the dominant commercial applications are piezoresistive accelerometers for automotive airbag deployment and piezoresistive pressure sensors for both automotive and medical applications1. Because of this widespread use, particularly in micro-electro-mechanical systems (MEMS) applications, undergraduate engineering programs whose learning outcomes include instrumentation technologies generally include an introduction to the basic operating principles of piezoresistive sensors. In our opinion, however, the exposition of these principles in popular textbooks for instrumentation systems and MEMS are generally inadequate—authors tend to either oversimplify, leaving a student unaware of operational details, or write for a research-oriented audience, making the material inaccessible to undergraduates. In this paper we present an approach to teaching the operating principles of piezoresistive sensors that addresses these issues.

The distinguishing features of our approach are its accessibility and coherence. First, the technical content and mathematics are appropriate for sophomore-level engineering undergraduates. Second, the technical material is presented coherently and completely, that is, each step of the exposition is motivated by the results of the previous step. Third, since mechanical strain is the physical phenomenon relating input to output, a strain-formulation is used for gage factor and for the placement and orientation of the piezoresistors instead of the stress-formulation found in most textbooks. In this paper we share our approach with the instrumentation education community in the hope that its accessibility and coherence will help improve the teaching of the operating principles of this one important type of sensor.

Limitation

We have no student learning data to specifically support our assertion that the approach we present has greater coherence and accessibility for undergraduates than any other. However we do make the case in the following section that our work makes a contribution via a synthesis of Page 15.923.2 the strengths of widely-used texts. Also, in recent years we have seen a steady increase in our accreditation program-outcome measures supported by our measurement systems course, although this material on piezoresistive sensors would contribute at most two hours of content to the course. Based on these broad measures, we are satisfied that a presentation of sensor operating principles like the one developed here contributes to meeting our learning objectives. We plan to develop an approach for measuring success for the next offering of the course.

Background

In doing our literature survey for a chapter on piezoresistive sensors for our recently published book2 on introductory micro-electro-mechanical systems (MEMS), we sought a treatment suitable for a sophomore-level audience. Though several authors give excellent developments on particular aspects of the topic, none were quite what we wanted. In our opinion, the general problems are that authors of textbooks on measurement and instrumentation systems tend to give good coverage to metallic strain gages but only a passing or no reference to semiconductors. (In some indices, the term “piezoresistance” does not even appear.) In the case of MEMS textbooks, however, the authors tend to give good coverage to the piezoresistive effect, but write for an advanced audience.

For example, measurement and instrumentation systems texts by Beckwith et al.3, Holman4, Northrop5, and Wheeler and Ganji6 all give good developments of metal strain gages and bridge analysis, but only passing reference to piezoresistance and semiconductors. Texts by Figliola and Beasley7 and Doebelin8 are similar, and though they include brief discussions of piezoresistive coefficients, they do so without a coherent connection to their strain-gage material. Among these texts, only Doebelin and Northrop use the Taylor series—the proper mathematical tool, in our opinion—for exploring small changes in variables such as sensor output voltage, electrical resistance, and area change. And while the texts by Holman and by Wheeler and Ganji pay some attention to the dimensional geometry associated with strain, none of these texts develops the geometry in detail.

The text on sensors by Busch-Vishniac9 is an exception. The author provides a fairly complete development of the piezoresistive effect and of metal and semiconductor piezoresistors, though written for an advanced audience.

The best-known MEMS texts have the same “deficiency” (for our purposes) of tending to be written for an advanced audience. All cover the piezoresistance principles of adequately, sometimes going deeper into molecular behavior than is needed by our audience. Standouts in this area include authors Maluf10 and Senturia11. Also because of their advanced audience, like Busch-Vishniac9, these books tend to omit or superficially treat bridge analysis—an important topic in our approach. We also find that MEMS texts tend to cover deformation mechanics in more detail than that needed by our audience, e.g., texts by Senturia and by Madou12.

Nevertheless, some of these MEMS texts provide important material for our approach that is generally missing from conventional treatments on measurement and instrumentation systems3–8. Such attributes include deriving general mathematical models of piezoresistance to include both 9 metal and semiconductor piezoresistors , developing a strategy for placement of piezoresistors Page 15.923.3 on the mechanical system subjected to strain11,12, development of the geometry of piezoresistors under strain10,13, some details of the configuration of piezoresistive sensors (how they are put together)9,10,13, and numerical values for coefficients of piezoresistance and elastoresistance9,10,12. These developments tend to use a stress formulation (using the π piezoresistance coefficients) rather than the strain formulation we recommend here (using the γ elastoresistance coefficients). One last small but notable shortcoming of these texts, in common with the measurement and instrumentation systems previously described, is their tendency to neglect the application of the Taylor series for exploring small changes in variables.

We draw on the individual strengths of these references to synthesize a complete, coherent, and balanced approach to teach the operating principles of piezoresistive sensors. We use a “system- level” perspective and attempt to develop each topic to roughly the same depth of detail using mathematics, physical principles, and engineering analyses suitable for our sophomore-level engineering audience.

Basic principles of operation

We treat our sensor as an input-output system, illustrated in Figure 1. A mechanical input (pressure, force, or acceleration for example) is applied to a mechanical structure of some kind (a beam, a plate, or a diaphragm) causing the structure to experience mechanical strain. Small piezoresistors secured to the structure undergo the same mechanical strain, changing their electrical resistances. These resistors are wired together in a Wheatstone bridge—an electrical circuit designed for detecting small changes in resistance. The bridge requires a constant DC voltage input and produces a measurable DC voltage output whose magnitude is proportional to the magnitude of the mechanical measurand.

constant DC voltage input to bridge

mechanical input piezoresistive mV output transducer from bridge

Figure 1: System inputs and outputs for a piezoresistive transducer.2

The primary physical phenomenon that makes this possible is piezoresistance: the material property that the electrical resistance of a material changes when the material is subjected to mechanical deformation, illustrated in Figure 2. An electrical resistor is fabricated from a piezoresistive material and wired into one leg of a bridge circuit. A force f deforms the material, often in bending, causing strain in the material, changing its electrical resistance. Recalling Ohm’s law, e = iR, where e is voltage, i is current, and R is resistance, the change in resistance ΔR due to mechanical deformation is detected via a change in bridge output voltage Δe.

Page 15.923.4

Figure 2: Conceptual schematic of piezoresistance: electrical resistance varies with mechanical strain.2

Teaching this material

Our presentation is adaptable to a number of teaching strategies depending on the learning objectives of a particular course. Instructors can select the level of detail appropriate to their course and students. At any level of detail however, we suggest that the lesson cover the major aspects of the transducer operation shown in Figure 3, illustrating our concept of a coherent development, that is, each step of the exposition is motivated by the results of the previous step.

Piezoresistance Bridge to dR related Input-output property of detect dR, to mechanical system, Fig. 1 materials, Fig. 2 Fig. 4 strain, Eq. 14

Placement of Model relating Gage factor F as a Signal piezoresistors, bridge output to measure of sensitivity conditioning e.g., Fig. 7 strain, Eq. 29 of dR to strain, Eq. 22

Figure 3: Topic flow chart for presenting the operating principles of piezoresistive sensors.

Suggested learning objectives

Objectives set 1: To teach this material in a brief exposition, with the least amount of detail, one can present just the outline of the operating principles shown in Figure 3. Using the suggested figures and equations, an instructor can cover the basic principles of operation in about 15–20 minutes, providing students an effective overview without detailed modeling or analysis. Appropriate learning objectives for this sort of lesson might include: • List the inputs and outputs of the sensor. • Define piezoresistance. • Explain the purpose of the bridge. • Explain how a piezoresistive sensor works. • Describe the purpose of signal conditioning.

This 15–20 minute exposition is the approach we use in teaching the mini-labs of our measurement systems course. In a mini-lab, a conventional lecture is replaced with a short 17 lecture commingled with a guided hands-on experience in a 2-hour studio format . While we Page 15.923.5 have not measured the efficacy of the mini-lab specifically compared to a traditional lecture- homework format, our measure of program outcomes supported by this course has steadily improved over the years. We assert, therefore, that even a brief exposition of a sensor’s operating principles can meet course learning objectives if the presentation is coherent and accessible.

Objectives set 2: To teach this material in greater detail, one might teach the bridge analysis and parts of the case study in detail in class with other aspects assigned as homework. Appropriate learning objectives for this sort of lesson might include all of those listed above plus: • Given a manufacturer’s specification sheet, determine the sensitivity of the sensor. • Derive a model for a half-bridge configuration. • Given representative values of gage factor and representative stress and strain values in the regions of maximum stress for a pressure diaphragm, determine the sensitivity of the pressure transducer in mV/V/Mpa. • Explain the possible purposes of signal conditioning in a piezoresistive sensor.

Objectives set 3: In a lesson at the highest level of detail, one might teach most of the material presented in this paper, leaving some content as exercises for the student. For example, one could teach the change-in-geometry material for a rectangular cross section, but leave the circular-cross section material as an exercise for students. Appropriate learning objectives for this sort of lesson might include all of those listed above plus: • Derive the change-in-geometry ΔA/A expression for a conductor of circular cross-section. • Given a circuit of a Wheatstone bridge with a null-offset or temperature compensation, derive the bridge model. • Given a diaphragm geometry and locations and magnitudes of maximum stress and strain, determine the position and orientation the piezoresistors and their wiring configuration in the bridge to produce maximum sensitivity. • Given appropriate material properties, estimate the gage factor of a piezoresistor. • Given a configuration of p-type Si resistors on a square diaphragm where two resistors are located side by side to sense the maximum stress σC, and given design values and stress-strain values (like in the case study that concludes the paper), determine the new bridge equation, the value of ΔR/R for each resistor, the value of Δeo/ei, and the sensor sensitivity.

The mini-lab

In our junior-level measurements course, we present this material in a 2-hour mini-lab on pressure sensors. We spend the first 15-20 minutes developing the operating principles to meet the first set of learning objectives described above. The remainder of the lab period is a hands- on, collaborative activity in which the students explore the operation of the transducer by informal experimentation, determining the sensor’s sensitivity and resolution followed by an applications question and an elementary uncertainty analysis.

The mini-lab concept is simple one: instead of lecturing about a transducer or class of transducers, replace the lecture with a hands-on activity with one particular transducer from the class, e.g., a load cell, a pressure transducer, a potentiometer, etc.18 We ask the students,

working in pairs, to connect the transducer to appropriate input and output devices, but we do not Page 15.923.6 provide detailed procedures. Our goal is to have students interact with the sensor in a mode we might call “guided discovery”—students are given general guidelines about what they are to find and the professor circulates and answers questions as they work. Students determine characteristics such as sensitivity, range, and resolution, compare their informal experimental findings to the manufacturer’s specifications (giving them practice at reading and interpreting specifications), and answer questions regarding an application, an uncertainty analysis, and at least one question requiring critical thinking.

The pressure mini-lab apparatus is illustrated in Figure 4. Students assemble the apparatus, connect the transducer to a power supply and a digital multimeter, and answer these questions: 1. Obtain readings for an informal calibration curve and estimate sensitivity. Quantitatively compare the experimental sensitivity to the expected sensitivity you found in the prelab. 2. Remove the plastic tube from port 2 and connect it to port 1, leaving port 2 open to the atmosphere. Discover if the transducer can be used this way. Explain. 3. Identify a way to increase the sensitivity of the transducer (not the manometer). Be specific and quantitative. 4. Determine the resolution of the measurement system. Explain your approach. 5. Suppose this pressure transducer were connected to a Pitot-static tube to measure airspeed (see the text, p. 341-2, assume C = 1). Excitation voltage is 10 V and air density of 0.002377 slug/ft3. What is the maximum airspeed this transducer could measure? 6. For an airspeed of 88 ft/sec (60 mph), determine the uncertainty in the airspeed. Neglect any uncertainty in density. Show all your work.

This concludes our discussion of the context in which we teach transducer operating principles generally. We turn now to the detailed modeling of the piezoresistive sensor in particular.

port 2 port 1, open to tee atmosphere tubing differential open to pressure transducer atmosphere clip

bellows for applying manometer: for the pressure “reference” pressure measurement

Figure 4: Apparatus for the pressure mini-lab. The differential pressure transducer is a piezoresistive type.

Detailed modeling

Consider the piezoresistive sensor as the input-output system shown in Figure 1. We begin by modeling the bridge to obtain Δe as a function of ΔR. This result motivates the next step relating

ΔR to mechanical strain, producing a model of resistance as a function of geometry and Page 15.923.7 resistivity. The geometry term motivates a development of the gage factor for metal resistors (strain-gage type sensors) and the resistivity term motivates a discussion of gage factor for semiconductor resistors (MEMS sensors). We use gage factor to obtain a model relating output voltage to input strain. The strain-formulation of the sensor model is used to motivate a discussion about the physical placement and orientation of piezoresistors on the mechanical structure. The strain formulation is, to our knowledge, unique.

Bridge analysis

A Wheatstone bridge is an electrical circuit that enables the detection of small changes in resistance. Beckwith3 and Holman4 develop models of several different types of Wheatstone bridges. Figure 5 illustrates the bridge we examine: a voltage–sensitive, deflection–type circuit with a constant– voltage DC input, and ideal resistances (i.e., no impedance elements) in the arms of the bridge.

constant DC voltage input to bridge + ei − piezoresistive transducer

A i R2 R1 m i i2 1 + mechanical input causes mV output a change in one or more eo of the resistances from bridge B i D − 3 i4

R3 R4

C

Figure 5: Wheatstone bridge circuit inside a piezoresistive sensor. One or more of 2 the resistors R1 through R4 may be piezoresistors affected by the mechanical input.

The input voltage ei is supplied by a constant DC source. The four arms of the bridge each contain a resistor, R1 through R4 (at least one of which is made of a piezoresistive material). The output voltage eo is the difference between the voltages at nodes A and C, that is,

eo = eA − eC . (1) To obtain an expression for the output, therefore, we need expressions for the voltages at A and C. Using nodal analysis (applying Kirchhoff’s current rule) at those two nodes,

at A : i1 + im = i2 , at C : i3 + im = i4 . (2) The output voltage is usually measured by a voltmeter with a high resistance, making the current im small enough to be negligible. Thus,

i1 = i2 , i3 = i4 . (3) Page 15.923.8

The behavior of resistors is described using Ohm’s law, Δe = iR, where Δe is the difference in voltage across the two ends of the resistor. Substituting for each current in (3) yields, e − e e − e e − e e − e A D = B A , B C = C D . (4) R1 R2 R3 R4 The voltage at B is the input voltage and the voltage at D is the reference zero voltage. Making these substitutions yields e e − e e − e e A = i A , i C = C . (5) R1 R2 R3 R4 Rearranging these two equations to solve for the voltages at A and at C and substituting into (1) yields the input-output relationship ⎛ R R ⎞ ⎜ 1 4 ⎟ eo = ⎜ − ⎟ei . (6) ⎝ R1 + R2 R3 + R4 ⎠ The first conclusion we can draw from this analysis so far is the condition for bridge balance. If there is no mechanical input to the sensor, we would like to have zero voltage output. Zero output occurs if the parenthetical term in (6) is zero, that is, R R 1 − 4 = 0 . (7) R1 + R2 R3 + R4 We can rearrange this relationship to obtain

R1R3 = R2R4 . (8) This condition can be met by the designer in more than one way. For instance, in what is called a full-bridge, all four resistors are identical piezoresistors with identical values of R. Or in a half- bridge, R1 and R2 could be identical piezoresistors with R3 and R4 being identical fixed resistors. In either case, selecting R1, R2, R3, and R4 such that (8) is true, we have eo = 0 when there is no mechanical input to the transducer.

Continuing our bridge analysis, we know that the sensor is designed such that the resistors undergo a small change in resistance ΔR that produces a small change in output voltage Δeo. Small changes like these are readily (and rigorously) expressed mathematically using a Taylor- series expansion about the balanced condition. If we assume that all the resistances are subject to change due to applied strain, then the Taylor series has the form,

∂eo ∂eo ∂eo ∂eo Δeo = ΔR1 + ΔR2 + ΔR3 + ΔR4 + higher order terms . (9) ∂R1 ∂R2 ∂R3 ∂R4 We assume that the higher-order terms are negligible because they involve products and integer powers of ΔR. The magnitude of ΔR is small and so the products and powers of ΔR are smaller still—hence negligible.

Page 15.923.9 We obtain the partial derivative terms from (6), substitute them into (9), neglect the higher-order terms, and divide by ei to obtain

Δeo R2 R1 R4 R3 = 2 ΔR1 − 2 ΔR2 + 2 ΔR3 − 2 ΔR4 . (10) ei ()R1 + R2 ()R1 + R2 ()R3 + R4 ()R3 + R4 This general form of the bridge model accommodates any combination of values of the resistances R1 through R4. To develop insight into the design of the sensor, however, it helps at this point to study a particular design—the full-bridge with four identical piezoresistors, that is, R1 = R2 = R3 = R4 = R. With these substitutions, (10) becomes Δe 1 ⎛ ΔR ΔR ΔR ΔR ⎞ o = ⎜ 1 − 2 + 3 − 4 ⎟ . (11) ei 4 ⎝ R R R R ⎠

As a point of pedagogy: the bridge analysis relates the electrical input ei, the electrical output Δeo, and the relative change in resistance ΔR/R of the piezoresistive material. Our next logical step, therefore, is to examine how the resistance term ΔR/R relates to the mechanical properties of the piezoresistive material.

Relating electrical resistance to mechanical strain.

Given a physical material of length L and constant cross-sectional area A, its electrical resistance R is given by ρL R = , (12) A where ρ is the material’s resistivity. The geometry of the resistor is illustrated in Figure 6 for both a rectangular cross-section (as in a thin plate) and a circular cross-section (as in a thin wire).

L uniform cross-sectional area A

L rectangular circular conductor conductor

Figure 6: Generalized resistor geometry.2

A change in resistance ΔR is produced by changes in any of the three quantities—resistivity, length, or area. Small changes are modeled using a Taylor-series expansion. From (12) we obtain ∂R ∂R ∂R ΔR = Δρ + ΔL + ΔA + higher order terms . (13) ∂ρ ∂L ∂A We obtain the partial derivative terms from (12), neglect the higher-order terms of the series, and divide by R to obtain ΔR Δρ ΔL ΔA Page 15.923.10 = + − . (14) R ρ L A The first term on the right-hand side of (14) represents the piezoresistive property—a change in resistance due to a change in resistivity Δρ due to the application of mechanical stress. The second two terms represent changes in resistance due to changes in the geometry (length and area) of the resistor. In semiconductors, the first term dominates. In metals, the geometric terms dominate8,9. In the next two sections we develop the model for both metals and semiconductors.

Gage factor for metal resistors.

In metals, the relative change in resistance ΔR/R is due primarily to the changing geometry. In this case we can neglect the Δρ/ρ term in (14) and model the relative change in resistance using ΔR ΔL ΔA = − . (15) R L A Piezoresistors made of metals are most commonly encountered in the form of strain gages. Strain gages are fabricated to be most responsive to uniaxial strain. Thus our analysis of the geometric effect in (15) is based on uniaxial strain applied to an isotropic material.

When a material is subjected to stress in one direction (we’ll use a rectangular cross-section to illustrate), the length L of the material increases by a small amount ΔL and the height h and width w decrease by the amounts Δh and Δw, as illustrated in Figure . Strain εL in the direction of the applied stress is defined as the ratio of the change in length to the original length, that is, εL =−ΔL/L. original area A, height h and width w new area, height h−Δh and width w−Δw L + ΔL L

applied stress σ

Figure 7: Dimensional changes due to applied stress in one direction.2

The cross-sectional area of the unstressed specimen is A = hw. A small change in area ΔA is modeled using (yet again) a Taylor-series expansion, ∂A ∂A ΔA = Δh + Δw + higher order terms . (16) ∂h ∂w The higher-order terms involve products and integer powers of Δh and Δw that are negligible in magnitude compared to the first-order terms. Neglecting them and dividing by A yields ΔA Δh Δw = + . (17) A h w Page 15.923.11

The transverse strain terms Δh/h and Δw/w for metals and cubic crystals can be expressed in terms of Poisson’s ratio ν and the axial strain εL by Δh Δw = −νε and = −νε , (18) h L w L where the negative signs indicate that both h and w decrease with positive axial strain εL.

Substituting (18) in (17) and simplifying yields ΔA = −2νε . (19) A L To show that this relationship is not unique to the rectangular conductor, we outline the analysis for a circular conductor. Cross-sectional area A for a circular conductor is given by A = πr2, where r is the conductor radius. Applying a Taylor series as before yields ΔA Δr = 2 . (20) A r

Radial strain is a function of axial strain, Δr/r = −νεL. By substitution in (20) we obtain ΔA = −2νε . (21) A L This result (21) for a circular conductor is identical to result (19) for a rectangular conductor.

Returning to our change-of-resistance relationship (15), substituting (21) and εL = ΔL/L yields ΔR = ()1+ 2ν ε . (21) R L This relationship models the change in resistance in metals used in strain gages as a function of a material property, Poisson’s ratio, and the applied mechanical input, uniaxial strain.

Sensitivity, in a sensor, is the ratio of change in output to change in input. For the resistive sensing element the output is the relative change in resistance ΔR/R due to the input strain εL. Thus the sensitivity of the element depends on the ratio of ΔR/R to εL. In piezoresistive applications, this ratio is called the gage factor, F, a dimensionless number defined as ΔR R F = . (22) ε L Applying this definition to (21), we obtain the expression for gage factor for metal strain gages F = 1+ 2ν . (23) Gage factor is used to compare the predicted performance of candidate materials as piezoresistors (higher F means greater sensitivity) and to guide us in positioning the piezoresistors on the mechanical structure that is subjected to the input strain—a topic to which Page 15.923.12 we return after developing an expression gage factor for semiconductors.

Gage factor for semiconductor resistors.

In semiconductors, the relative change in resistance ΔR/R is due primarily to changes in resistivity, not geometry. Neglecting the geometric terms ΔL/L and ΔA/A in (14) yields ΔR Δρ = . (24) R ρ For a piezoresistor subjected to longitudinal and transverse stresses, the resistivity change is Δρ = π σ + π σ . (25) ρ L L T T where πL and πT are the longitudinal and transverse piezoresistance coefficients of the material. In practice, longitudinal means “in the direction of current” and transverse means “perpendicular to the direction of current”.

Alternatively—and slightly more useful in our discussion of gage factor—resistivity can be expressed in terms of strain, Δρ = γ ε + γ ε . (26) ρ L L T T where γL and γT are the longitudinal and transverse elastoresistance coefficients of the material. The two models, (25) and (26), are related by the relationships between stress and strain in the longitudinal and transverse directions. However, because piezoresistive semiconductor materials are generally anisotropic, the linear Young’s modulus relationship, σ = Eε, does not apply here.

Substituting (26) in (24) we obtain ΔR = γ ε + γ ε . (26) R L L T T This relationship models the change in resistance in semiconductors used in piezoresistive applications as a function of material property, the elastoresistance coefficients, and the applied mechanical inputs, longitudinal and transverse strains.

Dividing by εL, we obtain an expression for the gage factor,

εT F = γ L + γ T . (28) ε L Both the elastoresistance coefficients and the strains depend on the orientation of the resistor. Gage factors for different materials lie on the approximate ranges given in Table 1 (see, for example 9,13). The table shows that semiconductors are more sensitive (higher F) that metal strain gages. The trade-off is that semiconductors are more brittle and have lower values of fracture stress than metals. Page 15.923.13 Table 1: Gage factors for metals and semiconductors 2 ≤ F ≤ 5 for metals, 5 ≤ F ≤ 50 for cermets (ceramic-metal mixtures) 70 ≤ F ≤ 135 for silicon and germanium

Physical placement and orientation of piezoresistors.

Using the definition of bridge factor (22), we rearrange terms to obtain an expression for the relative resistance change: ΔR/R = FεL. This expression holds for both metal and semiconductor piezoresistors. Substituting FεL for each ΔR/R term in the bridge model (11) yields an expression for bridge output in terms of strain only, assuming F is known and constant,

Δeo F = ()ε1 − ε 2 + ε 3 − ε 4 . (29) ei 4 By introducing the concept of gage factor, this model unambiguously relates the sensor’s basic mechanical input (strain) to the sensor’s basic electrical output (Δeo) for both metal and semiconductor piezoresistors. The model shows too that for a given level of strain, a higher gage factor (or sensitivity) produces higher output voltage from the bridge.

The model also leads us to insights regarding placement of piezoresistor on the mechanical structure of the sensor element. Equation (29) indicates that if the strains are equal in magnitude and have the same sign, then the bridge will produce zero output—not a useful outcome. To obtain a useful voltage output, we therefore position resistors 2 and 4 to undergo strain in the opposite sense to the strain of resistors 1 and 3.

For example, consider the piezoresistive accelerometer shown in Figure . The accelerometer housing is secured to a body undergoing the acceleration we want to measure—the vibration of a machine, for example. In response to the acceleration of the body, the seismic mass accelerates causing the cantilever beam to bend. The bending applies strain to the piezoresistors. Physical placement and orientation of the resistors is influenced by three considerations: the bridge configuration (which resistor is wired into which leg of the bridge), the region of maximum stress, and the direction of the strain to which we want the resistor to respond.

resistors oriented to respond to longitudinal strain εL

+ ei − 1 3 cantilever sensor housing beam 2 1 + eo − seismic 34 mass 2 4 region of (under) maximum bridge acceleration stress configuration sensitive axis

Page 15.923.14 Figure 8: Placement of resistors on a beam-type piezoresistive accelerometer sensitive to acceleration in one direction.2 The bridge configuration in this example is determined by the sign of the strain in each resistor location. If the beam bends downwards, the two resistors on top of the beam are in tension (a positive ε for p-type semiconductors) at the same time the resistors underneath the beam are in compression (a negative ε for p-type semiconductors). If the acceleration changes direction, the regions of tension and compression swap. For the strains to add up and not cancel each other out in (29) we place resistors 1 and 3 side by side on one side of the beam (here both are shown on top of the beam) and resistors 2 and 4 on the other side of the beam. Then ε1 and ε3 have the same sign, opposite that of ε2 and ε4, and the bridge produces a total positive or a total negative output voltage Δeo.

The four resistors are placed at the base of the beam because this is the region of maximum stress. Any other placement reduces the sensitivity of the sensor.

The resistors are oriented to be responsive to longitudinal strain, that is, the direction along the longitudinal axis of the cantilever beam. Transverse effects are negligible in this configuration. These three factors—bridge configuration, location and orientation of maximum stress and strain, and the orientation of the resistors that respond to the strain—are used to determine the optimum placement and orientation of piezoresistors in a transducer.

Device case study: a piezoresistive pressure sensor

In this section we study a piezoresistive pressure sensor with operational parameters similar to those seen in automotive applications. We use parameters representative of a class of pressure sensors similar in size and use to the Omega PX409 pressure transducer. To give our numerical example some semblance to reality, we use physical parameters from the literature14,15.

electrical connection stainless steel housing

Exterior view pressure inlet 80 mm

silicone oil structure to reservoir support the Si-wafer stainless steel diaphragm (100) Si-wafer

e pressure i inlet eo

signal conditioning compartment Cross-sectional schematic Page 15.923.15 Figure 9: A commercially available piezoresistive pressure sensor.2 (Based on the Omega PX409 pressure transducer16.) The basic configuration of this piezoresistive sensor is shown in Figure 9 (adapted from 16). The sensor housing is stainless steel, about 80 mm long, with a pressure fitting at one end and an electrical connection for input and output voltage at the other end. Sensors of this type can typically measure pressure ranges from 0–6.9 kPa (1 psi) to 0–34.5 MPa (5000 psi). The DC excitation voltage is usually between 5–10 V. When selecting a particular sensor make and model, one usually has a choice of electrical output signals: 0–100 mV, 0–5 V, or 4–20 mA.

The cross-section shows that the working fluid whose pressure we want to measure enters the sensor through the pressure inlet fitting and imposes pressure on the stainless steel diaphragm. A small volume of silicone oil transfers the pressure from the stainless-steel diaphragm to the (100) Si-diaphragm. The induced stress and strain of the wafer is detected by piezoresistors on the wafer in a bridge configuration, producing an electrical output proportional to the inlet pressure.

In this example, we examine a pressure sensor with a 0–1 MPa (145 psi) full scale input, 0–100 mV full scale output, a 10 VDC excitation, and p-Si piezoresistors.

We begin by discussing the mechanical properties of the diaphragm. In this example, the (100) Si-diaphragm is a square, 1.2 mm on each side, 80µm thick, oriented with the <110> directions bisecting the square as shown in Figure . When pressure is applied from below the largest upwards deflection of the diaphragm is at the center of the square. Consequently, the largest stress σC, located midway along each edge, is directed towards this center. Stress σB, at the same location, is directed parallel to the edge, along the boundary of the diaphragm.

Based on published experimental results for a square diaphragm of this type and size, we 14 estimate σC = 45.0 MPa and σB = 22.5 MPa . The strains in the same two directions are εC = 152 −6 µε and εB = −17 µε, where the symbol µε means microstrain (1 microstrain = 10 strain). The maximum stress σC is lower than the fracture stress of the Si-diaphragm (360 MPa) by safety factor of 8.

location of maximum stress and strain (all four edges) location of maximum <110> deflection

<110> σC σC σB σB (100) Si-diaphragm 1.2 mm 80 µm each side thick Figure 10: Orientation of the Si-diaphragm and points of maximum stress and strain.2

The four p-type piezoresistors are placed at the four locations of maximum stress and strain.

From the bridge model (29), we know we want resistors 1 and 3 to undergo a positive strain at Page 15.923.16 the same time resistors 2 and 4 undergo a negative strain. We accomplish this by orienting the resistors as shown in Figure (adapted from 11,12). Resistors 1 and 3 are oriented with their longitudinal axes in the direction of maximum stress (σC); ε1 and ε3 will be positive. Resistors 2 and 4 are oriented with their transverse axes in the direction of maximum stress (σC); ε2 and ε4 will be negative. + ei − piezoresistor 2 1 + conductors current runs to bridge eo 1 longitudinally − 34 4 i bridge 2 i configuration σT σ <110> L σL 3 σT

<110> (100) Si-wafer Figure 11: Location and orientation of four piezoresistors on the square diaphragm.2

Recall, from (26), that the change in resistance ΔR/R is a function of the longitudinal and transverse elastoresistance coefficients and strain. We group the analysis below into two columns: the left column for resistors 1 and 3 and the right column for resistors 2 and 4. We use (24) and (26) for each case: ΔR ΔR 3,1 = γ ε + γ ε , 4,2 = γ ε + γ ε . (30) R L L T T R L L T T

Comparing Figure to Figure , we see that the longitudinal strain of resistors 1 and 3 is εC (towards the center of the diaphragm) while the longitudinal strain of resistors 2 and 4 is εB (along the boundary of the diaphragm). Substituting for εL yields ΔR ΔR 3,1 = γ ()ε + γ ε , 4,2 = γ ()ε + γ ε . (31) R L C T T R L B T T where parentheses have been used to highlight the substitutions.

The converse is true for the transverse strains. The transverse strain of resistors 1 and 3 is εB (along the boundary) while the transverse strain of resistors 2 and 4 is εC (towards the center). Substituting for εT yields ΔR ΔR 3,1 = γ ()ε + γ ()ε , 4,2 = γ ()ε + γ ()ε . (32) R L C T B R L B T C The elastoresistance coefficients of the resistors in the <110> direction are found in published 9 tables to be γL = 120 <110> and γT = −54 <110>.

Page 15.923.17 We’re ready to obtain a numerical value for the two ΔR/R terms: ΔR 3,1 = ()120 ()152×10−6 + ()− 54 ()−17 ×10−6 R = 19 1. ×10−3 . (33) ΔR 4,2 = ()120 ()−17 ×10−6 + ()− 54 ()152×10−6 R = −10 2. ×10−3 Thus, as we’d planned, resistors 1 and 3 see a positive change in resistance (of about 2%) and resistors 2 and 4 see a negative change in resistance (of about 1%). As a reality check on the analysis to this point, we compute the gage factor F for resistors 1 and 3, oriented longitudinally in the direction of maximum stress (σC). The gage factor is computed below, with a result that lies in the expected range (70 ≤ F ≤ 135) for silicon. ΔR R F = 1 ε C 19 1. ×10−3 (34) = . 152×10−6 = 126

Next, we determine the ratio of electrical output to input at the full pressure load of 1.0 MPa using the full-bridge model and the values of ΔR/R, Δe 1 ⎛ ΔR ΔR ΔR ΔR ⎞ o = ⎜ 1 − 2 + 3 − 4 ⎟ ei 4 ⎝ R R R R ⎠ 1 (35) = []19 1. − ()−10 2. +19 1. − ()−10 2. ×10−3 . 4 = 14 7. mV/V. With a 10 VDC excitation voltage, the bridge full load output is 147 mV. Since the full load is 1.0 MPa, the sensitivity η is given by 147 mV η = 1 MPa . (36) = 147 mV/MPa .1( 01 mV/psi) To meet our goal of a 0–100 mV full scale output, we add an amplifier circuit to attenuate (reduce) the 147 mV bridge output by a factor of 100/147 = 0.682. This factor is called a gain K and is often reported in the units of decibels (dB) as follows, ⎛100 ⎞ K = 20log10 ⎜ ⎟ ⎝147 ⎠ . (37) = − .3 32 dB Page 15.923.18 The simplest op-amp “textbook” circuit that provides positive attenuation comprises two inverting amplifiers in series, as shown in Figure . We have to select the resistors Rf and Ro to obtain the desired gain of 0.0682 (or −3.32 dB). Rf Rf

voltage signal Ro R − o from bridge − + transducer + output

Figure 12: Simple non-inverting attenuation circuit.2

The gain of an inverting amplifier is the ratio of Rf to Ro, that is, R K = − f . (38) Ro and therefore the gain of the two inverting op-amps in series is given by

2 ⎛ R f ⎞⎛ R f ⎞ ⎛ R f ⎞ K = ⎜− ⎟⎜− ⎟ = +⎜ ⎟ . (39) ⎝ Ro ⎠⎝ Ro ⎠ ⎝ Ro ⎠ We write a computer program to sort through the combinations of readily obtainable resistors looking for values of K close to 0.682. We find that selecting Rf = 6.2 Ω and Ro = 7.5 Ω produces a gain K = 0.683, which is within 0.2% of the desired value, close enough to be useful. This amplifier circuit is placed in the signal-conditioning compartment of the sensor. Our electrical output-to-input ratio now includes the amplifier gain, Δe K ⎛ ΔR ΔR ΔR ΔR ⎞ o = ⎜ 1 − 2 + 3 − 4 ⎟ ei 4 ⎝ R R R R ⎠ .0 683 (40) = []19 1. − ()−10 2. +19 1. − ()−10 2. ×10−3 . 4 = 10 0. mV/V. With a 10 VDC excitation voltage, the bridge full load output is 100 mV and the sensitivity η is now, as desired, η = 100 mV/MPa 0( .69 mV/psi) . (41) As a point of pedagogy: we have purposefully selected an oversimplified attenuation circuit to keep the analysis accessible to our audience. In teaching this material to our students we note that real signal conditioning is generally more complex and accomplishes more than just setting a gain. Senturia11, for example, shows a schematic of a signal conditioning circuit for a Motorola pressure sensor, though without mathematical analysis.

In summary, this example illustrates how each of the important factors—bridge configuration, location and orientation of maximum stress and strain, and the orientation of the resistors that respond to the strain—affect the design and sensitivity of a piezoresistive transducer. Page 15.923.19

Conclusion

The goal of this paper is to present an approach to teaching the operating principles of piezoresistive sensors that addresses the issues we’ve encountered in most instrumentation texts of either oversimplifying the presentation or writing for a research-level audience. In our opinion, the level of detail we provide and basing each step of the development on principles familiar to the sophomore-level undergraduate engineering student meets this goal.

First, the technical content and mathematics are appropriate for sophomore-level engineering undergraduates. Second, the technical material is presented coherently, that is, each step of the exposition is motivated by the results of the previous step. Third, since mechanical strain is the physical phenomenon relating input to output, a strain-formulation is used for gage factor and for the placement and orientation of the piezoresistors instead of the stress-formulation found in most textbooks. We share this approach with the instrumentation education community in the hope that its accessibility and coherence will help improve the teaching of the operating principles of this one important type of sensor.

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