1
–
Supplementary Information
–
Nanomechanical Torsional Resonators
for Frequency-Shift Infrared Thermal Sensing
X. C. Zhang
1
, E. B. Myers
1
, J. E. Sader
1,2
, M. L. Roukes
1*
1
Kavli Nanoscience Institute and Condensed Matter Ph
ysics, California Institute of Technology,
Mail Code 14933, Pasadena, CA 91125, USA
2
Department of Mathematics and Statistics, The Unive
rsity of Melbourne, Victoria 3010,
Australia
1. Temperature Coefficient of Frequency (
TCF
) for the torsional and flexural
modes of the paddle resonator
The
TCF
of the flexural mode possesses an additional therm
al stress term which is absent for
the torsional mode. The mode(dependent thermal sens
itivities originate from the distinct roles
that stress plays in the motion. Figure S1 illustra
tes how the tensile force
T
enters into the
equation of motion of the flexural mode, but not in
to the torsional motion
Fig. S1. (a), in the flexural mode, the tensile fo
rce
T
has a component along the direction of
beam motion
z
, and thus contibutes to its resonant frequency and
TCF
, see text. In the equations,
is the displacement in the
direction
.
For the torsional mode (b), the tensile force
T
is along
the direction of torsional axis, and generates no t
orque. Therefore, the tensile force dose not
affect the torsional resonant frquency. This leads
to an exceptionally linear torsional mode with a
reduced
TCF
.
*
E(mail: roukes@caltech.edu
(b). Torsional mode
2
1.1
The flexural mode resonant frequency and
TCF
of a stressed torsional resonator
Fig. S2. A plane view diagram of the torsional devi
ce with the geometry labels. Origin of the
coordinate system is at the center of the left hand
clamped end.
We derive the resonant frequency of a doubly clampe
d torsional resonator with a built(in
stress
T
. A schematic of the torsional device is given in F
ig. S2. The coordinate system employed
in this manuscript is also defined, with the origin
set at the left end of the beam. The governing
equation for the beam deflection function
w
is
=
0 ,
S
1
where
/
is the reduced distance along the beam length
L
,
I
is the areal moment of inertia,
E
is the Young’s modulus,
t
is time, and
is the mass per unit length.
Since the width of the paddle
L
p
greatly exceeds that in the arm
w
r
, we consider the paddle to
be rigid relative to the narrow arms, and assume th
e inertia of the rods is negligible. We therefore
ignore the inertial term in Eq. S1, and include the
inertia only through the shearing force at the
contact point between the rods and the paddle. This
gives the governing equations and boundary
conditions:
L
r
= Torsion rod length
w
r
= Torsion rod width
h
= Device thickness
L
p
= Paddle length
w
p
= Paddle width
L
= Device whole length
Clamp
Clamp
"
#
$
#
"
%
$
%
$
&
'
3
=
0 ,
S2
with the usual clamped boundary conditions:
)
*
+,-
=
.
+,/
0
1
1
1
0 .
S3a
Applying Newton’s 2
nd
law at the paddle
5
5
5
6
+,/
0
1
1
1
1
2
7
8
∂
∂
6
+,/
0
1
1
1
,
S3b
where
;
1
1
1
;
/
,
M
p
is the mass of the paddle, and the factor of ½ in
Eq. S3b accounts for the
symmetry of the resonator , i.e. that there are two
identical narrow torsion rods.
The displacement is expressed in terms of an explic
it time dependence
exp >?
:
,
W
ξ
BCD
-i
?
t
.
S4
Substituting Eq. S4 into Eq. 2 and 3 gives
I
J
I
υ
I
J
I
0 ,
S5
with boundary conditions
)J
IJ
I
*
+,-
=
IJ
.
+,/
0
1
1
1
0 ,
S6a
5
I
5
J
I
5
6
+,/
0
1
1
1
1
2
7
8
?
J.
+,/
0
1
1
1
,
S6b
where the normalized tension parameter
υ
is
N
S7
Note that the required resonant frequency
?
only appears in the boundary condition S6b. This
property allows for an explicit determination of
?
by solving the system in Eqs. S5(7:
4
?
24
7
8
;
5
PN
S8
where
P
N
;
1
1
1
υ
12
R1
2
;
1
1
1
√
N
tanh
;
1
1
1
√
N
2
U
VW
S9
We can remove the scaling with respect to the total
beam length
L
. Eqs. S8(9 then become:
?
24
7
8
;
5
Y
Z
,
S10
where
Y
Z
τ
12
R1
2
√
Z
tanh
√
Z
2
U
VW
,
S1
1a
Z
;
.
S11b
For a rectangular cross section resonator,
;
\
5
/12
, Eq. S10 then simplifies to
?
2
;
\
5
7
8
;
5
Y
Z
,
S12
Using
tanh
C
C
]
^
5
W_
C
_
W`
5W_
C
`
⋯
, in the limit of small tension, Eq. S11a becomes
Y
Z
1
Z
10
b
Z
.
S13
Equation S12 and S13 show that the resonant frequen
cy reduces to the result of Evoy et al.
1
at
zero tension case, i.e.,
?
-
c
de
0
f
^
g
h
/
0
^
. Substituting Eq. S12 into S13 gives the resonant
frequency for a low stressed device
?
≈
i
2
;
\
5
7
8
;
5
j1
Z
10
k
l ?
-
j1
Z
20
k .
S14
We can rewrite Eq. S11b as
5
Z
12m
;
\
,
S15
where
σ=T/A=
(
σ
i
αE3T
) is the tensile stress of the beam,
σ
i
is the intrinsic stress resulting from
the wafer growth process,
αE3T
is the thermal stress term that accounts for the I
R (or any other
sources) heating induced beam softening effect, wh
ere
n
W
o
po
pq
is the linear thermal expansion
coefficient. The 1
st
derivative of Eq. S15 with the temperature yields:
rs|
flexural
1
P
-
IP
I
n x
2
3
5
y
;
\
z
. n,
S16
where
x
W
d
pd
pq
is the thermal coefficient of Young’s modulus.
1.2
TCF
of the torsional mode
The intrinsic resonant frequency of the torsional m
otion is given by
P
-
1
2{
c
|
S17
where
I
is the moment of inertia of the resonator,
κ
is the torsional spring constant.
Moment of Inertia
The moment of inertia for the two torsion rods is
}
1
6
~
}
;
\
,
S18a
where
~
}
is the mass,
w
r
is the width, and
h
is the thickness of the torsional rod.
The paddle’s moment of inertia is:
8
1
12
7
8
8
,
S18b
here
~
8
and
8
are the mass and length of the paddle, respectivel
y. The total moment of inertia
is:
6
}
8
l
8
1
12
7
8
8
.
S19
The last approximation holds true if
8
≫
;
, \
, which is the case for our device.
Torsional spring constant
For macroscopic beams of rectangular cross sections
, the torsional spring constant can be
calculated by
2
| 2
/
;
,
(S20)
where
G
s
is the shear modulus of the rod, which is related
to Young’s modulus by the relation
d
h
W
, N
8
is Poisson
s ratio.
;
is the length of each rod,
K
is the torsional moment of
inertia. For a rectangular cross section with sides
of length
h
and
w
r
, where
h>w
r
,
the torsional
moment is given by
2
\
;
5
1
3
0.21
;
\
1
\
12w
;
for \ w
;
.
S21
Combing all above, the torsional resonant frequency
is given by
P
1
2{
i
}
l
1
2{
i
1
2{
i
24
7
8
\
;
5
8
;
.
S22
The first derivative with temperature of Eq. S22 yi
elds
TCF
|
torsion
n x
2
S23
2. NETD limited by temperature fluctuation noise p
rocess
The spectral density of the thermodynamically drive
n temperature fluctuations are
3
q
?
2
{
4
J
1 ?
Z
,
(S24)
where
BW
is the measurement bandwidth. The temperature flu
ctuations can be converted into
noise equivalent power by
q
.
q
W
?.
(S25)
7
The amount of power received by an IR detector
δP
t
can be related to the temperature
difference
δT
of a target relative to its surroundings (assuming
blackbody radiation) by the
following formula
3
p
4s
y
I
I
z
,
,
(S26)
where
A
d
is the detector area,
F
is the focal ratio of the optics, and
j
p
pq
k
,
is the slope of the
function
P
=
f
(
T
t
), where
P
is the power radiated by a blackbody target within
the spectral band
from
λ
1
to
λ
2
. For the long wavelength infrared band from 8(14
Fm,
dP/dT
= 2.62 W/m
2
⋅
K. The
NETD limited by the temperature fluctuation can be
determined by combing Eq. S25 and S26.
The calculated NETD limited by the temperature fluc
tuations versus thermal conductance is
depicted by the blue curve in Fig. 1d. The adopted
parameters in the calculation are:
?
=10 Hz,
BW
=1 Hz,
0.3
,
F
=0.5. The device has the same geometry as shown in
Fig. 1b.
3. Device fabrication process
The pattern of the torsional device is defined by t
he electron beam lithography, followed by
the gold film, Strontium Fluoride (SrF
2
) film deposition, and lift(off process. The device
is
released in a Sulfur Hexafluoride (SF6) based induc
tively coupled plasma (ICP) dry etching step
with SrF
2
as the etching mask. The obtained narrowest suppor
ting rod is 50 nm. The ICP etching
offers a high lateral etching ratio between Si and
SiN over 100:1, a Si lateral etching rate ~ 1
Km/min, a uniform and smooth etched surface over a
whole 4 inch wafer.
4. Modeling of the optical interference technique
We develop a simple interference model to account f
or the optical transduction nonlinear
effect. Assuming the interference cavity, commonly
formed by the device top surface and the
8
substrate with a separation of
d
, the laser electric fields reflected from the cavi
ty’s top and
bottom surfaces are:
W
%,
W
%
B
¡
∅
£
%
V¤
¥
¦
,
27
%,
%
B
¡
∅
£
%
§pV¤
¥
¦
,
28
where
̈
-
%
is the initial phase,
k
is the wave vector, and
?
is the angular frequency of the laser
light. The summed field is
©
ª,
W
%,
%,
, and the interference intensity
%
is
given by
%, I
«©%,©
∗
%,
I
W
%
%
2
W
%
%
cos
2I
29
This sets the background DC intensity that the phot
odetector measures. When the device vibrates
at an amplitude of
®C
, the interference intensity,
%
,
becomes modulated. The modulation
depth in
%
,
∆
,
is recorded by the photodetector as the measure of
the device resonant
amplitude, i.e. the optical displacement signal:
∆
|
%,I C
%, I C
|
4
W
%
%
sin2I sin2C sin2C
30
5. Optical nonlinearity caused resonance peak spli
tting versus actuation level
We quantify the resonance splitting caused by the o
ptical nonlinearity as a function of the
actuation level based on a driven damped resonator
model. The amplitude of a driven damped
resonator is given by
°?
s
?
7
±²²
³
?
?
-
?
?
-
/ ́
μ
31
At the critical drive, the device reaches its maxim
um optical response at an amplitude
C
¶
· 8
⁄
according to Eq. S30. The two frequencies correspon
ding to this amplitude according to above
equation are
9
?
®
?
-
¹
y1
1
2 ́
z ®
1
́
i
y
C
C
¶
z
1
l ?
-
®
?
-
2 ́
i
y
C
C
¶
z
1
32
Finally the splitting is
∆?
?
-
́
i
y
C
C
¶
z
1
?
-
́
i
y
º
º
¶
z
1
,
33
V
is the applied RF voltage on the piezo(ceramic disk
,
V
c
is the voltage corresponding to
°
¶
· 8
⁄
.
The piezo(ceramic disk acts as a linear actuator i
n the whole power range of our
experiments.
6. Dynamic range (DR) of the torsional mode
Since the torsional mode is inherently linear, its
DR range should be limited by its elastic limit,
i.e.
the yield strength of the torsional rod
Z
»
. The experimental elastic limit of SiN is about 12
GPa.
4
The maximum shear stress occurs at the outer surfa
ce of the torsional rod can be expressed
as
2
Z
¼]
¼]
{ª
5
,
for circular rod with a radius r
34°
Z
¼]
3
¼]
2\
x j
\
k for rectangular rod with cross section length
\ ¿ , S34b
Where
x
j
e
0
f
k 1 0.6095
e
f
0.8865
e
f
1.8023 j
e
f
k
5
0.91
e
f
,
¼]
is the
maximum torque applied to the torsional rod under t
he elastic limit, i.e.,
Z
¼]
À Z
»
. The
maximum displacement angle
Á
¼]
¼]
|
⁄
, where
|
is the torsional constant expressed in Eq.
S20, can be formulated as:
Á
¼]
ª
Z
»
,
for circular rod
35°
10
Á
¼]
1
3x j
\
k . x
j
\
k
.
.
Z
»
, for rectangular rod ,
S35b
We simply consider the thermomechanical noise as th
e lower limit of the maximally(attainable
DR. The spectral density of the angular displacemen
t noise on resonance
Â
W/
Ã
4
. ́ ?
-
|
⁄
is
Â
W/
64
́
3{
5
5
.
8
;
5
8
5
ª
WW
W/
,
for circular rod
36°
Â
W/
́
x
5
5
.
8
;
5
8
5
\
;
Ä
W/
, for rectangular rod
S36b
where
8
,
;
,
8
,
;
are the length and width of the torsional rod and
paddle, respectively.
The DR of the torsional mode is defined as
DR dB 20 log y
Â
ÈÉÊ
Ã
Ë
Ì
Í
z
,
BW
=1Hz is the
measurement bandwidth. It can be formulated as:
ÎÏ 20 log Ð
1
2
√
2
. Z
»
.
3{
5
4
́
.
%
Ñ
$
%
$
#
"
#
Ò
W/
Ó,
for circular rod
37°
ÎÏ 20 log Ð
1
3
√
2
x
. Z
»
.
3{
5
́
x
.
Ô
Õ
$
%
"
%
Ö
$
#
"
#
Ò
W/
Ó, for rectangular rod S37b
7. Allan deviation: its definition and measurement
method
The Allan deviation (AD) is a time variance of the
measured frequency of a source, each
measurement averaged over a time interval
τ
A
. It is defined as
m
×
W
²
Ø
W
ÙVW
∑
P
1
1
1
P
VW
1
1
1
1
1
1
Ù
,
,
5
where
P
1
1
1
is the average frequency measured over the
m
th
time interval of length
3t
=
τ
A
, and
f
c
is
the nominal carrier frequency.
11
To measure AD, the device was driven at its resonan
t frequency
f
0
by a CW signal, and the
real time phase fluctuations were recorded in an op
en(loop configuration by a network analyzer.
The measured phase can be converted to frequency by
dividing the predetermined slope 2
Q
/
f
0
in
the linear region of the phase(frequency resonance
curve. The fastest sampling period in our
measurements is limited to 100 ms by the data bus t
ransfer speed.
REFERENCES
1
Evoy S.; Carr D. W.; Sekaric L.; Olkhovets A.; Par
pia J. M.; Craighead H. G.
J. Appl. Phys
1999, 86, 6072(6077.
2
Young W. C.; Budynas R. G.; Sadegh A.
Roark's Formulas for Stress and Strain
, 8
th
Ed., New
York: McGraw(Hill, C2012.
3
Krause P. W.; Skatrud D. D. “Uncooled Infrared Im
aging Arrays and Systems”,
Semicond. &
Semimetals
47, Academic press, 1997.
4
Nathenson, D. I., “Experimental Investigation of H
igh Velocity Impacts on Brittle Materials”,
PhD thesis, Case Western Reserve University, May 20
06.
5
Cleland A. N.; Roukes M. L. J. Appl. Phys. 2002, 9
2, 2758(2769.