Electronics and Cybernetics 1

Piezoresistive pressure sensors

Electronics and Cybernetics 2

Two sensing principles

Piezoresistive

measure mechanical stress in doped resistor-area

diaphragm pressure sensor

bending beam due to

volume forces (e.g. acceleration)

end force (e.g. protein attached)

capacitive

measure deflection (distance to other capacitor plate)

diaphragm pressure sensor

bending beam due to

volume forces (e.g. acceleration)

end force (e.g. protein attached)

Electronics and Cybernetics 3

Piezoresistive sensing applications

Veieceller AkselerometerTrykksensorer

Electronics and Cybernetics 4

Relation between stress and strain in a coordinate system with axes equivalent to the axes of the unit cell

xy

zx

yz

z

y

x

xy

zx

yx

z

y

x

CC

C

CCC

CCC

CCC

44

44

44

111212

121112

121211

000000000000000

000

000

000

F

x = E x 1D hooke’s law

Electronics and Cybernetics 5

Small deflections Beam equation, plate equation

Beam equation:

Plate equation:

Find displacement and stress

qdx

wdEI 4

4

F

p

),()2( 4

4

22

4

4

4

yxqyw

yxw

xwD

Electronics and Cybernetics 6

General: Finite Element Analysis

Solve Navier’s equation (partial differential equation)

Divide domain into elements

Approximation of function (solution to partial differential equation) over domain

Simple function over each element (linear, parabolic)

Connect elements at nodes

0)()( 2 uu

Electronics and Cybernetics 7

Example: beam with end load

Electronics and Cybernetics 8

Resistor change in metal strain gauges

Mainly due to change of resistor FORM (geometric effect)

aa

L

R0 =

L/a2

F

a+a

L+L

R=R0 +R0 L/L+3R0 a/a

a/a =- L/L

=0.3

R/R

2 L/L R/R

2

Electronics and Cybernetics 9

For silicon: large RESISTIVITY change with stress (not mainly a geometric resistor-change factor)

+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -

+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -

+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -

+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ - + -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

+ -

R/R

2

Metal:

+ +- -

R/R

90

Silicon:

Electronics and Cybernetics 10

Electronics (Chapter 14.1 - 14.4) Doped resistors

Define a p-type circuitin a n-type wafer

n-type wafer must be at positivepotential relative to the p-type circuit

Reverse biased diode no current between circuit and wafer/substrate

+V-

n p

Electronics and Cybernetics 11

The resistivity changes with the mechanical stress

E - electric field, three components

j - current density, three components

0 – homogeneous resistivity, unstressed silicon

When mechanical stress is applied, the resistivity changes depending on the stress in different directions and the piezo coefficients

3

2

1

345

426

561

0

3

2

1

0

3

2

1

jjj

ddddddddd

jjj

EEE

V1 V2

Electronics and Cybernetics 12

Silicon: Three independent piezoresistive coefficients

Example of piezoresistive coefficients:

doping: p-type

sheet resistivity: 7.8 cm

value of 11 = 6.6 10-11 Pa-1

value of 12 =-1.1 10-11 Pa-1

value of 44 = 138 10-11 Pa-1

Equations 18.3, 18.4, 18.5 in Senturia

xy

zx

yx

z

y

x

dddddd

44

44

44

111212

121112

121211

6

5

4

3

2

1

000000000000000

000

000

000

3

2

1

345

426

561

0

3

2

1

0

3

2

1

jjj

ddddddddd

jjj

EEE

Electronics and Cybernetics 13

Dependence of piezoresistivity on doping

Electronics and Cybernetics 14

Long, narrow resistors

Pre-calculated “piezocoefficients” that enables the designer to calculate the resistivity change for a long, narrow resistor

Often given for a resistor that is placed in the <110> direction

Transverse and longitudinal coefficients

ttllRR

Electronics and Cybernetics 15

Resistors in <110> direction

Much used direction for piezoresistors, bulk micromachining

Pre-calculated longitudinal and transverse piezo-coefficients

positive: tensile stress

negative: compressible stress

positive: increased resistivity with tensile stress

negative: decreased resistivity with tensile stress

p-type silicon: 44 dominates

)(2

)(2/1

)(2/1

44

441211

441211

tl

t

l

RR

Electronics and Cybernetics 16

Beam accelerometer

Long resistors in <110> direction

Piezoresistance coefficients

Electronics and Cybernetics 17

Cantilever with piezoresistors

length 200 m

width 20 m

thickness 5 m

point load at free end

p-type piezoresistor

length 20 m

width 2 m

depth 0.2 m

R/R = l l =0.02

Electronics and Cybernetics 18

Placement of piezoresistors on diaphragm

Over/under konfigurasjoner ikke praktisk i silisium.

Men vi kan snu retningenpå motstandene

Electronics and Cybernetics 19

Stress level at point in structure: Mises yield criterion

213

232

2212/1 mieses

Electronics and Cybernetics 20

x , normal stress on x=const. planes

Electronics and Cybernetics 21

y , normal stress on y=const. planes

Electronics and Cybernetics 22

Piezoresistor placed normal to diaphragm edge

Apply pressure from above

Diaphragm bends down

Piezoresistor is stretched longitudinally

l is positive, tensile stress

Rough argument for mechanical stress in transversal direction: stress must avoid contraction: t = l

Transverse stress is tensile/positive

Change in resistance:

(t is negative)

Resistance increases

ltlRR )(/ 11 ttllR

R

Electronics and Cybernetics 23

Piezoresistor placed parallel to diaphragm edge

Apply pressure from above

Diaphragm bends down

Piezoresistor is stretched transversally

t is positive

Rough argument for mechanical stress in longitudinal direction: stress must avoid contraction: l = t

Tensile, positive stress in longitudinal dir.

Change in resistance:

(t is negative)

Resistance decreases

tltRR )(/ 22

Electronics and Cybernetics 24

Membrane pressure sensor

Electronics and Cybernetics 25

Wheatstone bridge circuit

<100> direction

RRVV

RRRRRRRRRRRR

RRR

RRRVV

s

s

0

44

33

22

11

43

3

12

20

3.668.71

t

l

stress lecompressib negative, stress tensilepositive,

Electronics and Cybernetics 26

Averaging over stress and doping variations

Senturia 18.2.5

dzz

zLWRRR

jz

lloe0 ,

00 )()(

1

3

2

1

345

426

561

0

3

2

1

0

3

2

1

jjj

ddddddddd

jjj

EEE

Electronics and Cybernetics 27

Motorola MAP sensor

Electronics and Cybernetics 28

Potential across piezoresistor

Electric field in transverse direction

3

2

1

345

426

561

0

3

2

1

0

3

2

1

jjj

ddddddddd

jjj

EEE

Electronics and Cybernetics 29

Coventor tutorials-choose one

In files mems_analy.pdf and mems_supp.pdf (2004 version)

In files MEMS Design and Analysis Tutorials vol1 +2 (2005 version)

Mirror design (electrostatic forces)

Beam simulation analysis (pull-in)

Modal and harmonic analysis (resonant frequencies, modal shapes)

Temperature analysis (temperature distribution, different coefficients)

MemHenry (magnetic sensor)

MemPackage (thermal and mechanical effects)

MemPZR (piezoresistors)

MemETherm (Joule heating, thermal expansion)

AutoSpring (Extract spring constants)

MemDamping (Squeeze film, slide damping)

MemTrans (Transient analysis)