Piezoresistive pressure sensors
SINTEF Electronics and Cybernetics
1 - 29/10/2003
Two sensing principles
y Piezoresistive
y measure mechanical stress in
doped resistor-area
y diaphragm pressure sensor
y bending beam due to
yvolume forces (e.g. acceleration)
yend force (e.g. protein attached)
y capacitive
y measure deflection
(distance to other capacitor plate)
y diaphragm pressure sensor
y bending beam due to
yvolume forces (e.g. acceleration)
yend force (e.g. protein attached)
y Comb-structure (mass and large
capacitor) attached to beam
(flexible mechanical element) is a
common structure
SINTEF Electronics and Cybernetics
2 - 29/10/2003
Relation between stress and strain
in a coordinate system with
axes equivalent to the axes of the unit cell
F
σ x  C11 C12
σ  
 y  C12 C11
σ z  C
C12
12
 =
τ yx  0
0
  
0
τ zx  0
τ xy  0
0
C12 0 0 0
C12 0 0 0
C11 0 0 0
0 C44
0 0
0 0 C44
0
0 0 0 C44
 ε x 
 ε 
 y 
 ε z 
 
 γ yz 
 
 γ zx 
 γ xy 
SINTEF Electronics and Cybernetics
3 - 29/10/2003
Small deflections
Beam equation, plate equation
y Beam equation:
d 4w q
=
4
dx
EI
F
y Plate equation:
∂4w
∂4w
∂4w
D ( 4 + 2 2 2 + 4 ) = P ( x, y )
∂ x∂ y ∂y
∂x
p
y Find displacement and stress
SINTEF Electronics and Cybernetics
4 - 29/10/2003
General: Finite Element Analysis
y Solve Navier’s equation
(partial differential equation)
r
2r
(λ + µ )∇(∇ ⋅ u ) + µ∇ u = 0
y Divide domain into elements
y Approximation of function (solution
to partial differential equation) over
domain
y Simple function over each element
(linear, parabolic)
y Connect elements at nodes
SINTEF Electronics and Cybernetics
5 - 29/10/2003
Example: beam with end load
SINTEF Electronics and Cybernetics
6 - 29/10/2003
Resistor change in metal strain gauges
Mainly due to change of resistor FORM (geometric effect)
L
F
L+∆L
a
a
R0=ρ L/a2
∆a/a =-ν ∆L/L
a+∆a
R=R0+R0 ∆L/L+3R0∆a/a
∆R/R ≈ 2 ∆L/L
∆R/R ≈ 2 ε
ν=0.3
SINTEF Electronics and Cybernetics
7 - 29/10/2003
For silicon: large RESISTIVITY change with stress
(not mainly a geometric resistor-change factor)
Metall:
Silisium:
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+ -
∆R/R ≈ 2 ε
+
-
-
+
∆R/R ≈ 90 ε
SINTEF Electronics and Cybernetics
8 - 29/10/2003
Electronics (Chapter 14.1 - 14.4)
Doped resistors
Define a p-type circuit
in a n-type wafer
Wafer must be at positive
potential relative to the circuit
SINTEF Electronics and Cybernetics
9 - 29/10/2003
The resistivity changes with the mechanical stress
y E - electric field, three components
y j - current density, three components
y ρ0 - resistivity, unstressed silicon
y When mehanical stress is applied, the
resistivity changes depending on the
stress in different directions
d1 d 6 d 5   j1 
 E1 
 j1 
 E  = ρ  j  + ρ d d d   j 
0 2
0 6
2
4  2 
 2
d 5 d 4 d 3   j3 
 E3 
 j3 
SINTEF Electronics and Cybernetics
10 - 29/10/2003
Silicon: Three independent piezoresistive
coefficients
y Example of piezoresistive
coefficients:
y doping: p-type
y sheet resistivity: 7.8 Ωcm
y value of Π11= 6.6 10-11 Pa-1
y value of Π12=-1.1 10-11 Pa-1
y value of Π11= 138 10-11 Pa-1
d1  Π11 Π12 Π12 0 0 0
 d  Π
Π11 Π12 0 0 0
 2   12
d 3  Π12 Π12 Π11 0 0 0
 =
0 0 Π 44 0 0
 d 4  0
 d  0
0 0 0 Π 44 0
 5 
d 6  0
0 0 0 0 Π 44
 σ x 
 σ 
 y 
 σ z 
 
 τ yx 
 
 τ zx 
 τ xy 
SINTEF Electronics and Cybernetics
11 - 29/10/2003
Dependence of piezoresistivity on doping
SINTEF Electronics and Cybernetics
12 - 29/10/2003
Long, narrow resistors
y Pre-calculated “piezocoefficients”
that enables the designer to calculate
the resisivity change for a long,
narrow resistor
y Often given for a resistor that is
placed in the <110> direction
y Transverse and longitudinal
coefficients
∆R
= π lσ l + π tσ t
R
SINTEF Electronics and Cybernetics
13 - 29/10/2003
Resistors in <110> direction
y Much used direction for
piezoresistors
y Pre-calculated longitudinal and
transverse piezo-coefficients
y σ positive: tensile stress
y σ negative: compressible stress
y π positive: increased resistivity with
tensile stress
y π negative: decreased resistivity with
tensile stress
y p-type silicon: π44 dominerer
π l = 1 / 2(π 11 + π 12 + π 44 )
π t = 1 / 2(π 11 + π 12 − π 44 )
∆R π 44
≈
(σ l − σ t )
R
2
SINTEF Electronics and Cybernetics
14 - 29/10/2003
Piezoresistive sensing applications
Trykksensorer
Veieceller
Akselerometer
SINTEF Electronics and Cybernetics
15 - 29/10/2003
Beam accelerometer
y Long resistors in <110> direction
y Piezoresistance coefficients
SINTEF Electronics and Cybernetics
16 - 29/10/2003
Placement of piezoresistors on diaphragm
Over/under konfigurasjon
er ikke praktisk i silisium.
Men vi kan snu retningen
på motstandene
SINTEF Electronics and Cybernetics
17 - 29/10/2003
Piezoresistor placed normal to diaphragm edge
y
y
y
y
y
Apply pressure from above
Diaphragm bends down
Piezoresistor is stretched longitudinally
σl is positive
Rough argument for mechanical stress in
transversal direction: stress must avoid
contraction: σt= σlν
y Reasonable arguments indicates that
transverse stress is compressible
y σt is then negative
y Change in resistance:
∆R1 / R1 = (π l − νπ t )σ l
y (πt is negative)
y Resistance increases
SINTEF Electronics and Cybernetics
18 - 29/10/2003
Piezoresistor placed parallel to diaphragm edge
y
y
y
y
y
y
y
Apply pressure from above
Diaphragm bends down
Piezoresistor is stretched transversally
σt is positive
Rough argument for mechanical stress
in longitudinal direction: stress must
avoid contraction: σl= σtν
Reasonable arguments suggest
compressible stress in longitudinal dir.
Change in resistance:
∆R2 / R2 = (π t −νπ l )σ t
y (πt is negative)
y Resistance decreases
SINTEF Electronics and Cybernetics
19 - 29/10/2003
Membrane pressure sensor
SINTEF Electronics and Cybernetics
20 - 29/10/2003
Wheatstone bridge circuit
y <100> direction
π l = 71.8
π t = −66.3
σ positive, tensile stress
σ negative, compressible stress
 R
R 
V0 =Vs  2 − 3 
 R2 + R1 R3 + R4 
R1 = R1 − ∆R
R2 = R2 + ∆R
R3 = R3 − ∆R
R4 = R4 + ∆R
 ∆R 
V0 =Vs  
 R
SINTEF Electronics and Cybernetics
21 - 29/10/2003
Cantilever with piezoresistors
y length 200 µm
y width 20 µm
y thickness 5 µm
y point load at free end
y
y
y
y
p-type piezoresistor
length 20 µm
width 2 µm
depth 0.2 µm
y ∆R/R = πl σl =0.02
SINTEF Electronics and Cybernetics
22 - 29/10/2003
Motorola MAP sensor
SINTEF Electronics and Cybernetics
23 - 29/10/2003
Potential across piezoresistor
Electric field in transverse direction
SINTEF Electronics and Cybernetics
24 - 29/10/2003
Coventor tutorials-choose one
y In files mems_analy.pdf and mems_supp.pdf
y
y
y
y
Mirror design (electrostatic forces)
Beam simulation analysis (pull-in)
Modal and harmonic analysis (resonant frequencies, modal shapes)
Temperature analysis (temperature distribution, different coefficients)
y
y
y
y
y
y
y
MemHernry (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)
SINTEF Electronics and Cybernetics
25 - 29/10/2003
SINTEF Electronics and Cybernetics
26 - 29/10/2003
SINTEF Electronics and Cybernetics
27 - 29/10/2003
SINTEF Electronics and Cybernetics
28 - 29/10/2003
SINTEF Electronics and Cybernetics
29 - 29/10/2003
SINTEF Electronics and Cybernetics
30 - 29/10/2003