The Thermal Domain II
Joel Voldman*
Massachusetts Institute of Technology
*(with thanks to SDS)
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 1
Outline
> Review
> Lumped-element modeling: self-heating of resistor
> Analyzing problems in space and 1/space
• The DC Steady State – the Poisson equation
» Finite-difference methods
» Eigenfunction methods
• Transient Response
» Finite-difference methods
» Eigenfunction methods
> Thermoelectricity
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 2
The generalized heat-flow equation
> Last time we generated
V
a general conservation
equation
b
g
> Include a flux that
depends on a “force”
gradient
> And a “capacity”
∂b
= −∇ ⋅ F + G
∂t
~
dQ
~
+ ∇ ⋅ JQ = P
sources
dt
n
F
S
Image by MIT OpenCourseWare.
relation
J Q = −κ∇T
∂Q
=C
∂T
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 3
The generalized heat-flow equation
> We get a generalized
conduction equation
• Assume homogeneous
region
> Applies to
• Heat flow
• Mass transport (diffusion)
• Squeezed-film damping
> Provides a rich set of
solution methods
∂T
− κ∇ 2T = P
sources
∂t
1
∂T
− D∇ 2T = P
∂t
C sources
C
[m2/s]
κ
D= ~
C
[W/m-K]
[J/K-m3]
Thermal diffusivity
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 4
Thermal domain lumped elements
> Thermal resistor
• Resistance to heat flow
• Three types
» Conduction
» Convection
» Radiation
IQ
RT
+
ΔT
IQ
CT
-
CT = Cm ρ mV
> Thermal capacitor
• Store thermal energy
• Specific heat × volume ×
ΔT
+
> Electrothermal transducer
• Converts electrical dissipation
IQ
I
density
into heat current
1 L
RT =
κ A
+
V
2
IR
R
-
+
ΔT
-
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 5
Outline
> Review
> Lumped-element modeling: self-heating of resistor
> Analyzing problems in space and 1/space
• The DC Steady State – the Poisson equation
» Finite-difference methods
» Eigenfunction methods
• Transient Response
» Finite-difference methods
» Eigenfunction methods
> Thermoelectricity
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 6
Measuring temperature with the bolometer
> So far, we know how to
convert an input heat flux
into a temp change
Silicon nitride and
vanadium oxide
50 µm
0.5 µm
IR
Radiation
Y-metal
2.5 µm
> How do we convert that
temp change back into
the electrical domain?
CT
ΔT
-
E
X-metal
Monolithic bipolar
transistor
Image by MIT OpenCourseWare.
RT I Q
⎛
1 ⎞
⎟⎟ =
ΔT = I Q ⎜⎜ RT //
CT s ⎠ 1 + RT CT s
⎝
+
IQ
B
RT
ΔTss = RT I Q
τ = RT CT
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 7
TCR
> Resistance changes with
temperature (TCR)
• Beware, TCR is not constant!
> We can use resistor as a
R (T ) = R0 [1 + α R (T − T0 ) ]
R − R0
ΔR =
= α R ΔT
R0
hotplate or a temperature
sensor
V2O5
-200
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 8
Coupling back into the electrical domain
> We can define a
transducer that uses
TCR to convert back
into electrical domain IQ
> In order to measure
electrical R, we need
to introduce a voltage
& current
I
+
+
ΔT CT
RT
R(Δ T) V
-
-
R(ΔT ) = R0 (1 + α R ΔT )
> This current will
couple back and
induce its own ΔT
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 9
Thermo – electrical coupling
> This is our prior
electrothermal transducer
IQ
I
+
V
2
IR
R
-
+
+
IQ
+
ΔT CT
RT
R(Δ T) V
-
-
+
+
ΔT
I
-
2
I
> We can add in the current R
source due to Joule heating
> The current source is
IQ
ΔT CT
-
RT
R(Δ T) V
-
dependent on R, which is
dependent on ΔT, and so on
> What we will want is for IQ»I2R
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 10
I
Example: Self-heating of a resistor
> First, assume input IQ=0
+
> Two ways to drive the resistor,
V R(Δ T)
-
I
current source or voltage
source – it sometimes matters
RT
ΔT
=
I Q 1 + RT CT s
ΔT (1 + RT CT s ) = I Q RT
dΔT
ΔT +
RT CT = I Q RT
dt
I Q = I 2 R = I 2 R0 (1 + α R ΔT )
ΔT
RT
CT
-
temp change is output
IQ=I2R
+
> Now, electrical port is input,
Current-source drive
Expand out
into D.E.
ΔT +
d ΔT
RT CT = I 2 R0 (1 + α R ΔT ) RT
dt
Plug in for
IQ
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 11
Example: Self-heating of a resistor
> First-order system with feedback results
d ΔT
RT CT = −ΔT (1 − I 2 R0α R RT ) + I 2 R0 RT
dt
2
−
I
R0α R RT ) I 2 R0
1
(
d ΔT
+ ΔT
=
dt
RT CT
CT
dy
+ ay = b
dt
τI =
RT CT
1 − I 2 R0α R RT
(
ΔTSS , I
)
)
Recognize
D.E. form
Pick out quantities
of interest
I 2 R0
R0 RT I 2
CT
=
=
1 − I 2 R0α R RT
1 − α R R0 RT I 2
RT CT
(
Collect terms
and rearrange
This blows up when
I2 =
1
α R R0 RT
For αR>0
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 12
Example: Self-heating of a resistor
> What changes for voltage-drive?
V R(Δ T)
ΔT
RT
-
-
dΔT
ΔT +
RT CT = I Q RT
dt
+
-
+
ΔT (1 + RT CT s ) = I Q RT
Same
T.F.
V
IQ=I2R
+
RT
ΔT
=
I Q 1 + RT CT s
IQ is now
different
V2
V2
IQ =
=
R R0 (1 + α R ΔT )
IQ ≈
Voltage-source drive
2
V
(1 − α R ΔT )
R0
This leads to
negative feedback
for αR>0
ΔTSS ,V
RT V 2 / R0
=
1 + α R RT V 2 / R0
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 13
CT
Results of modeling
> A positive TCR resistor driven from a current source can go
unstable – fuse effect
> When dealing with the electrostatic actuator, we observed that
very different behavior was found depending on whether the
system was voltage-driven or current-driven
> Here we see that, depending on the way the electrical domain
couples to the thermal energy domain, it is also important to
look at the drive conditions of a system.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 14
Back to the bolometer
> Assume we want to measure
IQ=1 nW with 1% accuracy
> This limits current one can use
for measurement
ΔTSS , I
R0 RT I 2
2
=
≈
R
R
I
0 T
1 − α R R0 RT I 2
ΔTSS ,V
RT V 2 / R0
2
=
≈
R
V
/ R0
T
2
1 + α R RT V / R0
> For Honeywell bolometer,
ΔRmeas
R0~50 kΩ, αR~-2%/K, RT~107
K/W
ΔRsignal = α R RT I Q
> Input signal will create ΔT=10
mK
> This produces ΔRsignal=2x10-4,
ΔRmeas
V2
= α R RT
≤ 0.01α R RT I Q
R0
V≤
I Q R0
or a 10 Ω resistance change
> Voltage must be < 0.7 mV
V2
= α R RT
R0
100
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 15
Outline
> Review
> Lumped-element modeling: self-heating of resistor
> Analyzing problems in space and 1/space
• The DC Steady State – the Poisson equation
» Finite-difference methods
» Eigenfunction methods
• Transient Response
» Finite-difference methods
» Eigenfunction methods
> Thermoelectricity
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 16
Space and reciprocal space
> We have thus far focused on “big” lumped-element
modeling to design and analyze systems
> This isn’t the only way to proceed
> We can chop up the model into many small “lumped
elements Æ discretize in space
> Or we can approximate the answer using series
methods Æ discretize in reciprocal space
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 17
DC Steady State
> The Poisson Equation
> Boundary conditions
∂T
1
2
− D∇ T = P
∂t
C sources
• Dirichlet – sets value on
At steady state
boundary
Fixes T (r ) boundary
• Neumann – sets slope on
boundary Î Flux
Fixes
dT
dn
1 ~
D∇ T = − ~ P
C sources
2
boundary
• Mixed – sets some function of
value and slope
> The Poisson Equation is linear
• Can use superposition methods
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 18
Finite-Difference Solution
1 ~
D∇ 2T = − ~ P
C sources
d 2T
1 ~
D 2 =− ~P
dx
C sources
> We can generate an equivalent
circuit by discretizing the
equation in space
> A numerical algorithm with a
circuit equivalent
> In 1-D, divide bar into N
segments and N+1 nodes
Ie
0
A
T=0
L
x
d 2T
dx 2
≅
xn
T ( xn + h) + T ( xn − h) − 2T ( xn )
h2
x1
x2
xn-1
xn
xn+1
h
xN+1
x
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 19
Equivalent Circuit
> Can create an equivalent
circuit for this equation
T ( xn + h) + T ( xn − h) − 2T ( xn )
P ( xn )
=−
2
κ
h
2 P ( xn )
(Tn+1 − Tn ) + (Tn−1 − Tn ) = −h
κ
I2
n-1
~
I S ,n = (hA)P ( xn )
I1
Rn - 1
IS,n
n
Rn
IS,n + 1
n+1
Rn + 1
Define local
current source
(Tn+1 − Tn ) + (Tn−1 − Tn ) = −h
IS,n + 2
(Tn+1 − Tn ) + (Tn−1 − Tn ) + I
Image by MIT OpenCourseWare.
Adapted from Figure 12.1 in Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 302. ISBN: 9780792372462.
Rn
Rn −1
I S ,n
Aκ
S ,n
=0
This is KCL at a node
Define local
resistance
Rn = Rn −1 =
1 h
κ A
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 20
Equivalent Circuit
> Let’s apply this to 1-D
self-heated resistor
x1
x2
xn-1
xn
Gn=1/Rn
xn+1
h
xN+1
x
Gn=Gn+1
At node N:
(Tn+1 − Tn ) + (Tn−1 − Tn ) = − I
Rn
Rn −1
S ,n
Gn (Tn +1 − Tn ) + Gn −1 (Tn −1 − Tn ) = − I S ,n
− GTn +1 + 2GTn − GTn −1 = I S ,n
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 21
Example: Self-heated resistor
> Set up conductance
matrix
> Solve
> Very appropriate for
MATLAB
> Can even generate
the conductance
matrix with MATLAB
scripts
> At edges, impose
B.C’s
0 ...
⎡1 0
⎢- G 2G - G ...
⎢
⎢0 - G 2G
⎢
⎢
⎢0
2G - G 0
⎢
- G 2G - G
⎢0
⎢0
0 - G 2G
⎢
⎢
⎢0
⎢
⎢0
⎢
⎣0
0 ⎤⎛ T1 ⎞
⎛ 0⎞
⎟
⎜
⎜ ⎟
0 ⎥⎜ T2 ⎟
⎜1 ⎟
⎥
⎜1 ⎟
0 ⎥⎜ T3 ⎟
⎟
⎜
⎜ ⎟
⎥
⎟
⎜ ⎟
⎥⎜
⎟
⎜
⎜1 ⎟
0 ⎥⎜ Tn −1 ⎟
⎜ ⎟
⎥
~
0 ⎥⎜ Tn ⎟ = hAPo ⎜1 ⎟
⎟
⎜
⎜ ⎟
0 ⎥⎜ Tn +1 ⎟
⎜1 ⎟
⎥
⎟
⎜ ⎟
⎥⎜
⎟
⎜
⎜ ⎟
2G - G 0 ⎥⎜ TN −1 ⎟
⎜1 ⎟
⎥
⎜1 ⎟
- G 2G - G ⎥⎜ TN ⎟
⎟
⎜⎜ ⎟⎟
⎥⎜
… 0 0 1 ⎦⎜⎝ TN +1 ⎟⎠
⎝ 0⎠
GΤ = P
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 22
Eigenfunction Solution
> This is a standard method for solving linear partial differential
equations
> It leads to what amount to series expansion solutions, discretized in
reciprocal space
> Typically problems converge with only a few terms – THIS IS WHY IT IS
USEFUL
∇ 2T = −
P ( x)
κ
d 2ψ i
Eigenfunctions of ∇ :
= λψ
i i
2
dx
Can use any linear combination of e ± jkx , including s in( kx) and cos( kx)
Values of k are determined by the boundary conditions
2
Eigenfunctions can be made orthonormal
*
ψ
∫ jψ i dx = δ ij
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 23
Eigenfunction Expansion
Ie
∞
ψ n ( x) = cn sin ( kn x )
sin( k n L) = 0 ⇒ k n =
Apply BC at x=0, L:
L
T=0
nπ
for n = 1,2,3,...
L
L
1 = ∫ ψ ( x)dx = ∫ cn2 sin 2 (k n x)dx ⇒
0
Plug into DE:
A
n =1
Eigenfunctions
for this problem:
Normalize:
0
T ( x) = ∑ Anψ n ( x)
Assume:
2
n
0
L
x
2
sin(k n x)
L
d 2T
P( x)
=−
2
dx
κ
∞
P( x)
2
∑ kn Anψ n ( x) =
1
κ
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 24
Eigenfunction Expansion
∞
2
k
∑ n Anψ n ( x) =
1
Multiply by
orthogonal
eigenfunction and
integrate:
~
P ( x)
κ
∞ L
L
1
0
2
k
∑ ∫ n Anψ n ( x)ψ m ( x)dx = ∫
0
k Am =
2
m
Am =
1
k m2κ
1
κ
L
∫
0
~
P ( x)
κ
ψ m ( x)dx
2
⎛ mπx ⎞ ~
sin ⎜
⎟P ( x)dx
L ⎝ L ⎠
2
⎛ mπx ⎞ ~
sin
⎜
⎟P ( x)dx
∫
L0 ⎝ L ⎠
L
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 25
Eigenfunction Expansion
For uniform
power density:
~
L
P0 2
⎛ mπx ⎞
Am = 2
sin ⎜
⎟dx
∫
k mκ L 0 ⎝ L ⎠
~
P0 2
n
An = 3
1 − (− 1)
k nκ L
(
)
∞
T ( x) = ∑ Anψ n ( x)
n =1
~
P
2
2
⎛ nπx ⎞
n
T ( x) = ∑ 30
1 − (− 1)
sin ⎜
⎟
L
L ⎝ L ⎠
n =1 k n κ
~
4 L2 P0
1
⎛ nπx ⎞
T ( x) = 3 ∑ 3 sin ⎜
⎟
π κ n odd n
⎝ L ⎠
∞
(
)
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 26
The Details
Final answer: T ( x) =
Power density:
At x=L/2:
Tmax
4 Po L2
κπ 3
∑
n odd
sin ( nπ x / L )
n3
I e2 R
I e2
=
Po =
volume σ e A2
2 2
⎛ 4 ⎞ Ie L
=⎜ 3 ⎟
2
⎝ π ⎠ σ eκ A
⎡ 1 1
⎢⎣1 − 33 + 53 +
⎤
⎦⎥
Even if we consider only the first term in the expansion, we find
Tmax
2 2
⎛ 1 ⎞ Ie L
=⎜
⎟
2
⎝ 7.75 ⎠ σ eκ A
2 2
⎛ 1 ⎞ Ie L
compared to the exact solution of ⎜ ⎟
2
⎝ 8 ⎠ σ eκ A
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 27
Outline
> Review
> Lumped-element modeling: self-heating of resistor
> Analyzing problems in space and 1/space
• The DC Steady State – the Poisson equation
» Finite-difference methods
» Eigenfunction methods
• Transient Response
» Finite-difference methods
» Eigenfunction methods
> Thermoelectricity
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 28
Transient Modeling
> Finite-difference method
• Simply add a thermal capacitance to ground at each node of
the finite-difference network. These circuits can be analyzed
with SPICE or other circuit simulators.
n-2
IS,n - 2
Rn - 2
IS,n - 1
n-1
n
Rn - 1
Rn
IS,n
IS,n + 1
n+1
Rn + 1
n+2
IS,n + 2
Image by MIT OpenCourseWare.
C = hAρ mCm
Adapted from Figure 12.1 in Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 302.
ISBN: 9780792372462.
Node volume
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Transient Modeling
> Finite-difference method
• What does the matrix representation look like now?
Tn-1
Tn
Rn-1
IS,n
Rn
Tn+1
Cn
I S ,n + Gn −1 (Tn −1 − Tn ) + Gn (Tn +1 − Tn ) − CnTn = 0
−Gn −1 (Tn −1 − Tn ) − Gn (Tn +1 − Tn ) + CnTn = I S ,n
GΤ + CT = P
T = AT + BP
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 30
Transient Modeling (continued)
> Eigenfunction method:
T ( x, t ) = Tˆ ( x)Y (t )
• Spatial response same
as before
dT
1 ~
2
− D∇ T = ~ P
dt
C sources
• Use impulse response in
time to eventually get
Laplace transfer function
• Use separation of
variables to separate
space and time
~
~
P ( x, t ) = Q0 ( x)δ (t )
Separate
variables
dT
− D∇ 2T = 0
dt
[J/m3]
t>0+
2 ˆ
dY
(
t
)
d
T ( x)
Tˆ ( x)
− DY (t )
=0
2
dt
dx
1 d 2Tˆ ( x)
1 dY (t )
=
−
α
=
D
2
Y (t ) dt
Tˆ ( x) dx
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 31
Transient Modeling (continued)
> Eigenfunction method:
• Time response is a sum
of decaying
exponentials
• Time and space are
linked via eigenvalues
1 d 2Tˆ ( x)
1 dY (t )
= −α =
D
2
ˆ
Y (t ) dt
T ( x) dx
Y (t ) = e −αt
T ( x, t ) = Tˆ ( x)e −αt
d 2Tˆ
D 2 = −αTˆ
dx
2
Tˆ ( x) = ∑ An
sin (k n x )
L
n
α
⎛ nπ ⎞
2
2
kn D = α ⇒ kn = = ⎜
⎟
D ⎝ L ⎠
2
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 32
Transient Modeling (continued)
> Eigenfunction method:
• Match I.C. at t=0 to get
T ( x, t ) = ∑ An
n
series coefficients
• T(x,0) is related to
instantaneous heat input
and heat capacity
T ( x,0) = ∑ An
n
~
Q0
An = ~
C
An ,odd
2
sin (k n x )e −α nt
L
~
Q0
2
sin (k n x ) = ~
L
C
L
2
sin (k n x )dx
∫
L0
~
~
Q0 2 2 Q0 2 2 L
= ~
= ~
C L k n C L nπ
~
4 Q0
−α n t
(
)
sin
k
x
e
Tˆ ( x, t ) = ∑
~
n
π
n
C
n , odd
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 33
Example: Impulse Response
> Uniformly heated bar, an impulse in time
> Result is a series of decaying exponentials in time
z
Uniform internal heat generation
x
y
L
Heat flow from ends
W
Image by MIT OpenCourseWare.
Adapted from Figure 12.3 in Senturia, Stephen D. Microsystem Design. Boston,
MA: Kluwer Academic Publishers, 2001, p. 308. ISBN: 9780792372462.
~
Q
T ( x, t ) = ~0
C
where
⎛ 4
⎜
∑
n odd⎝ nπ
⎞ ⎛ nπx ⎞ −α nt
⎟ sin ⎜
⎟e
⎠ ⎝ L ⎠
n 2π 2 D
αn =
L2
lower spatial frequencies
decay slower
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 34
Using the Eigenfunction Solution
> We can go from solution to
equivalent circuit
> First, we will lump
• Heat current conducted out
as output
• Choose heat current
source as input P = Q0δ (t )
> Then take Laplace
⎛ W H ∂T
I Q = −κ ⎜⎜ ∫ ∫
⎝ 0 0 ∂x
∂T
dzdy − ∫ ∫
∂x
x =0
0 0
~
Q
8
⎛
⎞
I Q (t ) = ⎜ κWH ∑ e −α nt ⎟ ~0
L n odd
⎝
⎠C
WH
8κWH
1
I Q ,n ( s) =
αnL 1+ s
Y (s) =
> Then identify equivalent
1
1+ s
• This is NOT unique
~
Q0
~
C
X (s)
αn
circuit for 1st order system
X(s)
αn
⎞
dzdy ⎟⎟
x=L
⎠
C
R
Y(s)
Image by MIT OpenCourseWare.
Adapted from Figure 12.4 in Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 310. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 35
Using the Eigenfunction Solution
> Each term in the
eigenfunction solution has a
simple circuit representation
> This means that if the
eigenfunction solution
converges with a few terms,
the lumped circuit is very
simple
+
Q0,n(s)
Tn
Cn
Rn
IQ,n(s)
Image by MIT OpenCourseWare.
Adapted from Figure 12.4 in Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 310. ISBN: 9780792372462.
~
Cn = (mode shape )(volume)C
2
~
~
⎛ nπx ⎞
Cn = C ∫ ∫ ∫ sin ⎜
LHWC
⎟dxdydz =
nπ
⎝ L ⎠
0 0 0
HW L
n 2π 2 D
1
= αn =
RnCn
L2
⇓
⎛ 1
Rn = ⎜
⎝ nπ
Q0,n ( s ) =
⎞ L/2
⎟
⎠ κWH
8κWH Q0 8κWHL Q0
= 2 2
αn L C
nπ D C
⎛ 8 ⎞
Q0,n ( s ) = ⎜ 2 2 ⎟ (WHL ) Q0
⎝n π ⎠
⎛ 8 ⎞
Q0,n (t ) = ⎜ 2 2 ⎟ (WHL ) Q0δ (t )
⎝n π ⎠
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 36
A Three-Mode Equivalent Circuit
> For the first three terms in the eigenfunction
expansion, we combine the three single-term circuits
appropriately
100
C3
IS,2(t)
R3
C2
IS,1(t)
10-1
R2
R1
IQ
C1
Magnitude
IS,3(t)
10-2
10-3
10-4
10-3
Image by MIT OpenCourseWare.
Adapted from Figure 12.5 in Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 312. ISBN: 9780792372462.
1 Term
3 Terms
100 Terms
10-2
10-1
100
101
102
103
ωR1C1
Image by MIT OpenCourseWare.
Adapted from Figure 12.6 in Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 313. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 37
Outline
> Review
> Lumped-element modeling: self-heating of resistor
> The DC Steady State – the Poisson equation
• Finite-difference methods
• Eigenfunction methods
> Transient Response
• Finite-difference methods
• Eigenfunction methods
> Thermoelectricity
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 38
Microscale temperature measurement/control
> We have seen that a resistor can be used as
a temperature sensor and hotplate
> There are other techniques to measure or
control temperature at microscale
• Couple temperature to material properties
> Sensors
• TCR: temperature Æ resistance change
• Thermal bimorph: temperature Æ deflection
• Thermoelectrics: temperature Æ induced voltage
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 39
Coupled Flows
> In an ideal world, one driving
force creates one flux
> In our world, multiple forces
create multiple fluxes
J Q = −κ∇T
J e = −σ e∇ϕ
• Drift-diffusion in
semiconductors or
electrolytes
> In general, all the different
fluxes are coupled
> If you set it up right, the Lij
J e = − z n qe Dn∇n − qe nμ n∇ϕ
n
J i = ∑ Lij F j
j =1
matrix is reciprocal
• The Onsager Relations
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 40
Quantities in the Onsager Relations
> To explain thermoelectrics,
J e = − L11∇ϕ − L12∇T
we must look at coupling
between heat flow and
electric field
JQ
T
= − L21∇ϕ − L22∇T
> This is written in a standard
form
Resistivity
Seebeck coefficient
− ∇ϕ = ρ e J e + α S ∇ T
J Q = Π J e − κ∇T
Peltier coefficient
Thermal conductivity
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 41
Thermocouples
> Analyze the potential gradient around
a closed loop under the assumption
of zero current (Je=0)
> Thermocouple voltage depends on
the difference in Seebeck Coefficient
between the two materials, integrated
from one temperature to the other
> It is a BULK EFFECT, not a junction
effect
Isothermal voltmeter
b
V
Hot
αS,1
αS,2
a
Cold
Image by MIT OpenCourseWare.
Adapted from Figure 11.10 in Senturia, Stephen D.
Microsystem Design. Boston, MA: Kluwer Academic
Publishers, 2001, p. 294. ISBN: 9780792372462.
−∇ϕ = α S ∇T
⇓
Tb
Vab = ∫ α S (T )dT
Ta
> It is possible to make thermocouples
by accident when using different
materials in MEMS devices in regions
that might have temperature
gradients!
Go around the loop
VTC =
TH
∫ (α
S ,2
− α S ,1 ) dT
TC
VTC = (α S ,2 − α S ,1 ) ΔT
For small temp rises
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 42
MEMS Thermocouples
> Many thermocouples in series
create higher sensitivity (V/K)
> These are known as thermopiles
> In MEMS thermopiles, often use
Al/Si or Al/polySi
> Able to get good thermal
isolation of sensing element
Courtesy of Thermometrics Corporation. Used with permission.
Thermometrics commercial
Si thermopile
> Number of thermocouples is
limited by leg width
• Increasing leg width decreases
thermal resistance and thus
temperature response
Image removed due to copyright restrictions.
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 43
Conclusions
> The thermal domain is a great way to transfer energy
around
• Except that you have to pay the tax
> We can model thermal problems using
• Equivalent circuits via lumped element models in space
» “Big” and “small”
• Equivalent circuits via lumped element models in reciprocal
space
For Further Information
> Introduction to Heat Transfer, Incropera and DeWitt
> Analysis of Transport Phenomena, William Deen
> Solid-State Physics, Ashcroft and Mermim
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J. Voldman: 2.372J/6.777J Spring 2007, Lecture 13 - 44