abdolvand trans2003

2A3.1
THERMOELASTIC DAMPING IN TRENCH-REFILLED POLYSILICON
RESONATORS
Reza Abdolvand 1, Gavin K. Ho 1, Ahmet Erbil 2, Farrokh Ayazi 1
Georgia Institute of Technology
1
School of Electrical and Computer Engineering, 2 School of Physics
Atlanta, Georgia 30332-0250 USA
rezaa@ece.gatech.edu; gavinho@ece.gatech.edu; Tel: (404) 385-0962
the thermal path due to the presence of a void in the
middle of the trench refilled (TR) structures. Our analysis
does not include the intracrystalline thermoelastic
damping in polysilicon structures [7] that will come into
effect at very high frequencies (GHz range).
ABSTRACT
This paper presents quality factor measurement results for
trench-refilled (TR) polysilicon beam resonators which
demonstrate an increase in Q as their resonant frequency
increases, a trend opposite to what is expected from solid
beams of the same dimensions. This phenomenon is
believed to be due to a multi-path thermoelastic damping
characteristic. The frequency-dependant behavior of QTED
in trench-refilled resonators with voids is analyzed, for
which the measurement data is agreeable. A multiphysics
FEMLAB simulation was also performed to illustrate the
thermal gradients in a resonating TR beam. We show that
in a particular frequency range, trench-refilled polysilicon
beams can have a higher Q than their silicon counterpart
and the thickness of the resonator can be tuned to produce
a desired quality factor behavior.
TRENCH-REFILLED POLYSILICON RESONATORS
Thick polysilicon structures with high aspect ratio can be
created by refilling narrow trenches (2-6µm wide) with
conformal LPCVD polysilicon layers.
Using the
HARPSS process, thick silicon electrodes can be placed
in the sub-micron vicinity of trench-refilled polysilicon
structures to implement high performance capacitive
MEMS devices, such as resonators and inertial sensors
[1,4]. Figure 1 is a SEM picture of a 35µm thick
clamped-clamped TR polysilicon resonator that is
fabricated in HARPSS. The drive and sense electrodes for
this capacitive device is made of thick single crystal
silicon (SCS) with 0.9µm inter-electrode gap spacing.
Discussion on the fabrication process can be found in [4].
INTRODUCTION
Micromechanical resonators are a promising alternative to
the existing resonator technology mainly due to their
small size, superior performance and large scale
integration capability. Characterization of the quality
factor (Q) of these devices is of great interest to the
MEMS community as many applications exist for such
devices, e.g., resonant sensors [1] and micromechanical
filters [2,3].
sense
SCS
polysilicon
As the physical dimensions of a micromechanical beam
resonator are decreased, the flexural resonant frequency
increases and the resonator Q generally is reduced.
Interestingly, 80µm thick HARPSS polysilicon beam
resonators previously reported in [4] showed an increase
in Q with frequency, contrary to what is normally
expected from solid beams of the same dimensions [5].
Electrical losses, support loss, and air damping have been
ruled out as contributing factors to the observed increase
in Q. Of the remaining factors, thermoelastic damping
(TED) has been identified as the most probable cause,
mainly due to its significant contribution to the quality
factor of high aspect-ratio silicon and polysilicon beams 1
to 10Pm wide in the frequency range of 10kHz-500MHz.
width
drive
Figure 1. TR polysilicon beam resonator fabricated using
the HARPSS process (L=300Pm, w=5Pm, h=35Pm).
In the process of refilling the trenches with LPCVD
layers, a void is usually created in the middle of the
polysilicon structures. Although there have been reports
of void-less TR polysilicon formation through accurate
control of the trench profile [8], structures created by
refilling non-optimized trenches will always yield such a
void. Figure 2 is a cross section of a TR polysilicon
beam, illustrating the existence of such a void in the
In this paper, we present a variation to the original TED
characteristic of beam resonators introduced by Zener [6]
to extend its applicability to trench-refilled (TR)
resonators. This variation is caused by a discontinuity in
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2A3.1
middle of the beam. This void does not change the
resonance frequency of the beam compared to a solid
beam of same dimension, but it affects the heat transfer
across the width of the beam and hence the thermoelastic
damping behavior.
P2
P1
SCS
(a)
(b)
(Void size and width-to height ratio not to scale)
Void
Figure 3. Two thermal paths exist in (a) a hollow beam
and one thermal path in (b) a solid homogenous beam.
Thickness
In the hollow structure, one thermal path is across the
thickness of the beam wall (P1) and the other exists
around the void (P2). P1 and P2 illustrate the flow of heat
from hot to cool regions within the structure. The
individual thermal paths have a unique frequencydependent behavior, and we apply the principle of
superposition to analyze the overall behavior.
Figure 2. SEM picture of a broken resonator, showing
the void in the polysilicon trench-refilled beam.
THERMOELASTIC DAMPING
The quality factor of a micromechanical resonator is
determined from various loss mechanisms and can be
expressed by the following:
For the high frequency characteristic along P1, we assume
the effective thermal path d1 to be half the width of the
beam. In the low frequency TED mechanism along P2,
we predict the effective thermal path d2 to be equal to the
height h of the beam. Note that this is an average value,
and actual conditions will result in broadening of the
frequency dependent loss. The corresponding relaxation
time constants are:
UC p w / 22
UC p h 2
(6), (7)
, W TH 2
W TH 1
2
NS
NS 2
1
§ 1
1
1
1
1 ·
¸¸ (1)
Q ¨¨
© QTED QSUPPORT Q AIR QSURFACE QOTHER ¹
For operation in vacuum, 1/Qair is very small and hence is
negligible. Support loss for the high aspect-ratio beams
discussed in this paper is less significant than
thermoelastic damping.
According to Zener [6],
thermoelastic loss of a solid homogeneous beam in the
fundamental flexural mode is approximated by:
ZW TH
Q 1 '
(2)
2
1 ZW TH where,
2
UC p d 2
EDTH To
'
, W TH
(3), (4)
UC p
NS 2
The relaxation strength ' of the two mechanisms is
assumed to be equal to the relaxation strength of the
homogeneous beam (3). Applying the principle of
superposition, the overall frequency-dependant quality
factor of a trench refilled polysilicon beam is:
QTED
' is the relaxation strength and W7+ is the thermal
relaxation time constant with d as the effective thermal
path length. In the case of a flexural solid beam, d is
equal to the width of the beam. The thermoelastic loss of
a beam has a Lorentzian behavior (2), and the maxima
occurs at ZWTH=1. Thus, the frequency of minimal Q is
1
NS
.
f Q min
(5)
2SW TH 2 UC p d 2
QTED
1
'
1
1
QTED1 QTED 2
ZW TH 1
ZW TH 2
'
.
2
2
1 ZW TH 1 1 ZW TH 2 (8)
(9)
RESULTS
Two sets of TR polysilicon beam resonators with
thicknesses of 35µm and 90µm were fabricated using the
HARPSS process. Each set included beams with various
width and length to study the resonator Q. According to
(9), the quality factor of a TR resonator has two distinct
minima that correspond to the two thermal relaxation
mechanisms. Therefore, an increase in Q is expected
beyond the low frequency fQmin (for d=h). Table 1
provides a summary of the measured Q of the fabricated
resonators, which show an increase in the Q of 35Pm
thick devices with increasing frequency.
Effect of the Void
The void created in TR polysilicon beams causes a
discontinuity in thermal transport across the width,
altering the thermoelastic behavior of the resonator. We
predict two thermal paths will be present in a vibrating
TR beam, as opposed to only one thermal path in a solid
resonator (Figure 3).
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2A3.1
Table 1. Quality factor measurements for TR polysilicon
beam resonators in the fundamental mode (in vacuum).
Length
[Pm]
900
700
500
300
900
700
500
300
150
900
700
500
900
700
500
300
Width
[Pm]
Height
[Pm]
4.5
35
3.5
35
7.0
90
5.5
90
Frequency
(kHz)
43
77
157
464
31
59
119
364
1150
73
119
231
56
99
190
532
resonators in their fundamental and high order modes
were taken and listed in Table 2.
Q
Table 2. Quality factor measurements of two trenchrefilled polysilicon beam resonators in flexural modes.
3100
3700
6200
10100
3000
3300
4000
6200
5600
8500
7000
5800
6000
7800
6700
4800
Length
[Pm]
Width
[Pm]
Height
[Pm]
900
5.5
35
1100
7.0
35
Quality Factor
Quality Factor
4.5PmX35Pm Test Data
4.5PmX35Pm TR Poly QTED
§ w·
Q
12200
14700
12300
9000
6500
13000
26000
5600
900PmX5.5PmX35Pm Test Data
5.5PmX35Pm TR Poly QTED
3.5PmX35Pm Test Data
3.5PmX35Pm TR Poly QTED
10
1
3
5
7
1
3
4 (?)
5
Frequency
(kHz)
59.4
313.6
760.7
1400
48.5
259.8
406
631.8
The data of Table 2 is plotted with the predicted TED in
Figure 6, which shows very good agreement considering
that each set of data has come from one single resonator.
The support loss for all these measurements should be
negligible due to very high aspect ratio of the beam.
The measurement data is plotted with the predicted TED
behavior in Figure 4 (35Pm) and Figure 5 (90Pm). Both
plots show agreement between the measurement data and
the trend predicted by QTED for D=2.3×10-6K-1, E=150GPa
[9], N=40W/mK [10], Cp=713J/kgK, and U=2328kg/m3.
5
Mode
2
10
5
10
4
1100PmX7.0PmX35Pm Test Data
7.0PmX35Pm TR Poly QTED
W2 v ¨ ¸
©2¹
W1 v h 2
3
10
10 3
10
4
10
4
10
5
10
6
10
7
10
8
Frequency [Hz]
Figure 6. Quality factor of two beams in their
fundamental and high order flexural modes.
3
10
3
10
10
4
10
5
10
6
10
7
10
The smaller Q values measured compared to the predicted
QTED values, as well as part-to-part variation in Q, are
attributed to other sources of loss such as surface
roughness and defects. Figure 7 shows the difference in
the surface roughness of two TR beams from two
different lots. The resonators with the smoother sidewall
demonstrated higher Q values.
8
Frequency [Hz]
Figure 4. Test data and QTED for 35Pm thick beams.
Quality Factor
5.5PmX90Pm Test Data
5.5PmX90Pm TR Poly QTED
10
5
10
4
7.0PmX90Pm Test Data
7.0PmX90Pm TR Poly QTED
3
10
3
10
10
4
10
5
10
6
10
7
10
8
Frequency [Hz]
Figure 5. Test data and QTED for 90Pm thick beams.
To ensure that the quality factor increase is only due to
QTED behavior and irrespective of other phenomena,
quality factor measurements of two individual beam
Figure 7. Lot to lot variations in the surface roughness of
the fabricated TR polysilicon beams.
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2A3.1
FINITE ELEMENT ANALYSIS
10
A finite element analysis was performed to illustrate the
temperature profiles in the vibrating hollow and solid
beams. Figure 8-a shows the temperature profile obtained
for a vibrating clamped-clamped TR beam using 3D
multiphysics in FEMLAB. A 2D temperature profile
exists in a solid beam, invariant along the thickness as
expected (Figure 8-b). The existence of an additional
thermal relaxation mechanism in the hollow beam is
illustrated by the thermal gradient along the thickness.
Temp. (K)
6
5.5PmX90Pm Test Data
5.5PmX90Pm TR Poly QTED
Quality Factor
4.0PmX80Pm Test Data
4.0PmX80Pm TR Poly QTED
Temp. (K)
10
4.5PmX35Pm Test Data
4.5PmX35Pm TR Poly QTED
5
10
4
10
3
10
4
10
5
10
6
10
7
10
8
Frequency [Hz]
Figure 10. Comparison of beams with various thicknesses
CONCLUSIONS
In this paper we applied thermoelastic damping theory to
explain the increase in quality factor with frequency for
polysilicon trench-refilled beams in flexural modes. The
predicted characteristic for hollow TR beams have two Q
minima (compared to only one minimum for solid beams)
corresponding to two individual thermal paths. The
predicted TED characteristic showed good agreement
with the measured data.
(a)
(b) thickness
Figure 8. Steady-state vibration-induced temperature
distribution in (a) hollow and (b) solid beams obtained
with finite element analysis using FEMLAB.
DISCUSSION
From the observed frequency characteristic of a high
aspect-ratio TR beam, we infer that there is a particular
frequency range in which a TED-limited trench-refilled
beam has a larger Q than a solid homogenous beam with
the same dimensions (Figure 9). For devices requiring a
specific operational frequency within this range that have
stringent constraints on geometry, a higher Q can be
obtained from TR polysilicon than single crystal silicon.
ACKNOWLEDGEMENTS
The authors thank Siavash Pourkamali, Seong Yoel No,
Akinori Hashimura and the staff of the Georgia Tech
Microelectronics Research Center for their help.
REFERENCES
1
4.0PmX80Pm Test Data
4.0PmX80Pm TR Poly QTED
4.0Pm SCS QTED
Quality Factor
10
2
5
3
10
4
4
5
Data taken from [4]
3
10 3
10
10
4
10
5
10
6
10
7
10
6
8
Frequency [Hz]
Figure 9. TED-limited quality factor of a trench-refilled
hollow resonator compared to a solid silicon resonator
7
Secondly, since the low frequency TED characteristic is
dependent on the thickness of the resonator, the thickness
can be tuned to produce a greater Q. Comparing beams
80Pm and 90Pm thick to similar beams that have a
thickness of 35Pm, a larger Q is observed in the thicker
resonators (Table 1). The increase in thickness results in
a shift in the low frequency TED mechanism (Figure 10).
8
9
10
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TRANSDUCERS ‘03
The 12th International Conference on Solid State Sensors, Actuators and Microsystems, Boston, June 8-12, 2003
0-7803-7731-1/03/$17.00 ©2003 IEEE
327
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