lecture 9: piezoresistivity - university of victoriamech466/mech466-lecture-9.pdf · lecture 9:...

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MECH 466Microelectromechanical Systems

University of VictoriaDept. of Mechanical Engineering

Lecture 9:Piezoresistivity

© N. Dechev, University of Victoria

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Examples of Piezoresistive Sensors

Resistance Change due to Geometric Change

Resistance Change due to Resistivity Change (Piezoresistive effect)

Metal Strain Guages

Doped Silicon Piezoresistors

Stress Distribution in Beams

Piezoresistivity Overview

© N. Dechev, University of Victoria

3© N. Dechev, University of Victoria

Piezoresistive Sensors

Operate on a sensor principle whereby an electrical resistor will change its resistance when it is subjected to a strain (deformation).

Piezoresistive sensors are used as part of many MEMS devices, including:

-Pressure sensors

-Accelerometers

-Tactile sensors

-Flow Sensors

In other words, these “Piezoresistive-based” applications are sensitive to phenomena that cause beams or thin plates to deform, and this deformation can be measured by resistance change.


Pressure Sensors

MEMS pressure sensor consist of a ‘thin plate’ of silicon with a pressure differential across the plate. The resulting deformation causes strain along the edges of the plate.

With careful design, regions along the edge of the plate can be ‘doped’ to create resistors, which will subsquently exhibit resistance change in proportion to the applied strain.

thin diaphragm

thin diaphragm

thindiaphragm

strainsensors

Diagram of Piezoresistive-based Pressure Sensor [Chang Liu]


Pressure Sensors

The deformation of the ‘thin plate’ on the pressure sensor, can be visualized in the following image:

Total deformation of thin plate. Red = high deformation. Dark Blue = none.


Pressure Sensors

The state of strain of the same ‘thin plate’ is shown here.

Note the locations of the maximum strain along the edges.

Total Strain of a thin plate. Red = high strain. Dark Blue = no Strain.


Example: Piezoresistive Accelerometer

This consists of a relatively large moving mass, connected to a flexible beam that will deform (strain) as a result of acceleration of the mass:

Review Case Study 6.2 in textbook for fabrication details.

Case Study: 6.2: Diagram of Piezoresistive-based Accelerometer [Chang Liu]

piezoresistorproof mass

acceleration


Piezoresistive Sensors

Recall the relationship:

-Which defines the electrical resistance of any conductor.

There are two modes of resistance change when we deform/strain a device:

(1) Mode 1: Physical change in dimensions.

(2) Mode 2: Resistivity ρ is a function of strain ε.

L

w

h


Mode 1: Resistance Change

Mode 1: Physical change in dimensions

Recall Poisson’s ratio, which states that as we elongate a ‘member’, is dimensions change in all three axes, x, y and z.

This relationship is defined as:

F

deformed

F

ΔL

x

z

undeformed-y

L

Undergo resistance change due to strain, primarily due to dimensional change, (i.e. shape deformation), (Mode 1).

Design Principle:


Metal Strain Guages

not very sensitive

good, sensitive design

Mode 2: Resistivity is a function of strain:

Note that this function is non-linear.

For semiconductor piezoresistors, mode ② >> mode ①.

Also, for the purposes of this course, we will assume a linear relationship between resistivity and strain, i.e.:


Mode 2: Resistivity, ρ, is a Function of Strain

Strain, ε

Resistivity,ρ

Strain, ε

Resistivity,ρ

Peizoresistor resistance is defined as:

This equation can be derived as follows:

Due to Poisson’s effect, when a solid body is deformed, it will experience strain in all three axes.

Recall:


Piezoresistor Resistance

Transverse

Longitudinal

€

x

z

-y

Therefore, we can state that the resistance change is:

Note, for Mode 1 materials (i.e. metals), the third term is negligible)

Due to Poisson’s effect:

Hence:

Note that change in resistivity caused by stress is expressed as:



Where:First term is the increased lengthSecond term is the decreased areaThird term is the change in resistivity

Where:pi = is called the piezoresistive coefficient of the material[sigma] = stress tensor

Hence:

We now define: G as the relative change of resistance per unit strain, and refer to it as the “gauge factor”

Hence we can write:

For both directions, the total resistance change is the summation of changes due to longitudinal and transverse strain components.



Note: valid for 1-D resistance change

Note: valid for 1-D resistance change

Definition of longitudinal piezoresistor:

Definition of transverse piezoresistor:



Note: resistor orientation w.r.t. applied loadF

F

F

For

F

F

Note: resistor orientation w.r.t. applied load

Undergo resistance change due to strain, primarily due to change in the physical resistivity, ρ, of the material, (Mode 2).

The value of is highly dependent on the doping concentration of the doped silicon.


Single Crystal Doped Silicon Piezoresistors

For a 3-D solid mass, more general relationships exist.

The resistivity ρij can be expressed as:

Note: the reference for this material is available from the book: [M-H Bao, 2000, Micro mechanical transducers Handbook of Sensors and Actuators vol 8, Editor: S Middelhoek (Amsterdam: Elsevier) chapter 5]




The general relationship between ‘resistivity’ and ‘stress’ is:



For the case of single crystal silicon, where the x, y and z axes line up with the <100> family of directions, we have the relation:

where values for π can be found in Table 6.1 of the textbook.

Note: for other directions, such as <110>, or <111>, transformation matrices are required as described in the work: [M-H Bao, 2000, Micro mechanical transducers Handbook of Sensors and Actuators vol 8, Editor: S Middelhoek (Amsterdam: Elsevier) chapter 5]



See Class Notes for Solution

Example: Longitudinal Piezoresistor

A longitudinal resistor has been fabricated in silicon crystal by doping, and lies in the <111> direction. The piezoresistor is n-type, with a doping concentration of 11.7 Ωcm.

Question: Find the longitudinal guage factor of the resistor, using Table 6.1 and Table 6.2 in the course textbook.


Review of Beam Stress

Recall the distribution of stress within a beam, under a condition of ‘pure bending’:

w

F

x

F

F

M=Fx


Review of Beam Stress

The stress within the beam increases linearly, as we move from the beam ‘neutral axis’ to the top and bottom faces.

F

F

F

Also, the stress within the beam is proportional to the moment M, at that position on the beam.


Design of Piezoresistive Sensors with Micro-Beams

Any well designed sensor should not ‘interfere with’ or ‘alter’, the system it is measuring.

The sensor should be located in such a way as to maximize its sensitivity to the phenomena to be measured.

And, it should minimize the measurement noise.



To create a strain sensor on a micro-beam, we can dope the beam in an ‘appropriate location’ to create a piezoresistor.

Consider the following scenarios for various doping locations on a beam:

(A)

Doped Region (Piezoresistor)

Deposited Material (Piezoresistor)

Good Designs:- Sensor element is minimally disruptive to the structure.-Sensor is placed at best location to maximise the sensitivity to strain.(B)



The following designs can be considered as Poor Designs:

(C)

- Sensor Design (in this case size) changes the physical properties of the beam

(D)

Very ThickDeposited Material

Very Deep Doping -Sensor Location (in this case sensitive region is located on both sides of the neutral axis) makes for poor sensitivity, since the tensile and compressive strain-resistances cancel each other out.

lecture 9: piezoresistivity - university of victoriamech466/mech466-lecture-9.pdf · lecture 9:...

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