Thermal stress in silica-on-silicon disk resonators
Tong Chen, Hansuek Lee, and Kerry J. Vahala
Citation: Appl. Phys. Lett. 102, 031113 (2013); doi: 10.1063/1.4789370
View online: http://dx.doi.org/10.1063/1.4789370
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Thermal stress in silica-on-silicon disk resonators
Tong Chen, Hansuek Lee, and Kerry J. Vahala
a)
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125,
USA
(Received 4 December 2012; accepted 8 January 2013; published online 24 January 2013)
The thermal expansion mismatch of thermal grown silica on a silicon wafer is well known to induce
compressive stress upon cooling from the growth temperature to room temperature. In this Letter, we
investigate how this stress impacts silica disk structures by comparison of measurements with both a
finite element and an analytical model. The disk structures studied are also whispering gallery optical
resonators, and proper control of stress is critical to obtain high-Q resonances. Based on our analysis,
thicker oxide layers and proper control of undercut enable ultra-high-Q optical performance and
mechanical stability.
V
C
2013 American Institute of Physics
.[
http://dx.doi.org/10.1063/1.4789370
]
For the past few decades, numerous experimental and
theoretical studies have considered stress behavior and its
impact on devices.
1
–
3
In the silicon system, the oxide silica is
grown at temperatures near 1000
C. Upon cooling to room
temperature, the difference between the thermal expansion
coefficients of the silica and silicon causes a well-known com-
pressive stress in the oxide. While this stress typically does
not cause yield, reliability, or performance issues in photonic
devices, certain whispering gallery style optical resonators
and waveguides rely upon undercut of the oxide to create opti-
cal confinement.
4
–
6
The undercut silica layer creates an air-
cladding to guide the light (Fig.
1(a)
), and if there is sufficient
thermal-induced stress, then this thin air-cladding silica layer
may buckle and form crown-like patterns. Understanding the
buckling behavior is important in optimizing the device per-
formance. Specifically, oxide undercuts must be deep enough
in these structures to isolate the optical mode from the silicon
support pillar, however, the crowning behavior can occur at a
critical undercut value and thereby interfere with optical per-
formance. Likewise, in MEMS systems with free standing
structures, residual stress can physically warp devices to a
degree that renders them no longer useful.
7
,
8
In this work, we
measure the buckling behavior of silica disks structures and
compare with two models so as to create design guidelines
that eliminate buckling in both resonator and waveguide struc-
tures. Specifically, by proper selection of oxide thickness and
undercut, excellent optical performance is obtained (over 800
million optical quality factor for disk resonator structures of
7.5 mm diameter and 10
l
m oxide thickness
4
).
Earlier work on silica wedge-shaped resonators achieved
Q factors as high as 50 million
9
in devices with diameters
around 100
l
m. Those structures featured a lithographically
defined oxide disk of 2
l
m thickness that had been partially
undercut using the silicon selective etchant xenon diflouride.
The ability to extend both the Q factor and the resonator di-
ameter to larger values (Q greater than 100 million and reso-
nator diameter to the mm-cm size range) is important for
applications such as microcombs and rotation sensing.
10
,
11
However, due to thermal stress, simply scaling the previous
device to larger diameters fails to provide satisfactory
performance. Figure
1(b)
shows a top view (interference
contrast mode) image of a silica wedge resonator having
500
l
m diameter and 2
l
m oxide thickness. With less than
70
l
m undercutting of the silicon, the silica layer starts to
buckle and features a crown-like pattern. Further, the number
of nodes in the buckled pattern decreases with the deepness
of the undercut. For example, as shown in Figs.
1(b)
–
1(e)
,
the number of nodes in the buckled structures are 18, 10, 8,
and 6 for 70
l
m
;
120
l
m
;
155
l
m, and 180
l
m undercuts.
Ultimately, if the undercut is deep enough, the resonator will
return to the unbuckled configuration (Fig.
1(f)
).
To assess the impact of buckling on the optical perform-
ance of the resonators, the intrinsic optical Q factor
12
of normal
and buckled samples was measured. Devices were coupled to
an optical fibre using a fibre taper,
13
,
14
and spectral lineshape
data were obtained by tuning an external cavity semiconductor
laser (1550 nm) across an optical resonance while monitoring
transmission on an oscilloscope (see sample scan in Fig.
1(g)
).
To accurately calibrate the laser scan in this measurement, a
portion of the laser output was also monitored after transmis-
sion through a calibrated Mach-Zehnder interferometer having
a free-spectral-range of 6.72 MHz. The buckled resonators
(Figs.
1(b)
–
1(e)
) show low Q factors, typically below 1 million.
Moreover, some of these samples were observed to crack over
a period of a few days. In contrast, an unbuckled 500
l
mdiam-
eter resonator with 2
l
m thickness (corresponding to an under-
cut of 55
l
minFig.
1(a)
) featured a 38 million Q factor, and
this Q-factor performance was preserved for more than 1 year.
The spectral scan for this resonator is shown in Fig.
1(g)
.
To model thermal stress, we take the difference between
oxide growth temperature (1000
C) and the simulation tem-
perature as a control parameter (
D
T
) and simulate the equi-
librium state of the resonator via a finite element model in
COMSOL MULTIPHYSICS
(
http://www.comsol.com/
). The follow-
ing is defined as an order parameter
X
¼
1
V
ððð
silica
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
@
w
@
x
2
þ
@
w
@
y
2
s
dV
;
(1)
where
w
is the vertical component of deformation due to ther-
mal stress,
V
is the volume of silica, and integration is carried
over the whole silica layer. The order parameter is a simple
a)
Author to whom correspondence should be addressed. Electronic mail:
vahala@caltech.edu.
0003-6951/2013/102(3)/031113/4/$30.00
V
C
2013 American Institute of Physics
102
, 031113-1
APPLIED PHYSICS LETTERS
102
, 031113 (2013)
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measure of the irregularity of the silica induced by the stress.
Figure
2
is a plot of the numerically simulated order parameter
plotted versus
D
T
. The plot shows a second-order phase tran-
sition behavior as
D
T
increases in a resonator of diameter
500
l
m, silica layer thickness 2
l
m, and undercut of 70
l
m.
Above a critical temperature difference (
D
T
c
), the silica layer
buckles and features a crown-like pattern with 16 nodes. In
contrast, the silica layer bends uniformly downward when
D
T
is smaller than the critical value. Since the silica is grown at
1000
C, the equilibrium state of this resonator at room tem-
perature will have a buckled pattern.
The equilibria of resonators with differing amounts of
undercut were also calculated for comparison with the meas-
urements in Fig.
1
(see, Figs.
1(h)
and
1(i)
). Ultimately, if
the undercut is deep enough, the silica layer will uniformly
bend downward. The simulated configurations have 16, 10,
6, and 6 nodes for 70
l
m
;
120
l
m
;
155
l
m, and 180
l
m
undercuts and are more or less consistent with measure-
ments. The discrepancy might result from the slight irregu-
larity in the etched silicon pillar.
In addition to the finite element analysis, an analytic
model was studied to provide guidance on device design. The
approach is based on a two-dimensional buckling model of
the annular disk and the energy method. The energy model
employed is especially useful when a rigorous solution of the
Kirchkoff equation is unknown or it is required to find only an
approximate value of the critical temperature difference for
buckling.
15
–
17
Basically, if the work done by thermal stress is
smaller than the strain energy of bending for every possible
shape of buckling, then the unbuckled equilibrium is stable. If
the same work becomes larger than the energy of bending for
any shape of deformation, then the structure tends to be unsta-
ble and starts buckling. In a silica disk resonator with radius
b
,
thickness
t,
and for a silicon pillar of radius
a
,thebuckling
shape can be approximated by a sine curve along the circum-
ference of a plate.
18
,
19
Assume that the deflection of the annu-
lar disk in the vertical direction is
w
¼
C
ð
r
a
Þ
2
cos
ð
n
h
Þ
;
(2)
where
C
is the amplitude of buckling, 2
n
is the number of nodes
in the crown-like pattern, and the deflection obeys the clamped
boundary condition at
r
¼
a
. With respect to a polar coordinate
system
ð
r
;
h
Þ
with origin at the center, the components of stress
induced by the thermal expansion mismatch are
20
r
r
ð
r
Þ¼
b
2
a
2
a
2
b
2
r
2
1
r
;
(3)
r
h
ð
r
Þ¼
b
2
a
2
a
2
b
2
r
2
þ
1
r
;
(4)
where
r
¼
D
a
D
T
1
Si
E
Si
þ
1
E
SiO
2
1
þ
q
2
1
q
2
þ
SiO
2
E
SiO
2
(5)
¼
D
a
D
T
h
ð
q
Þ
(6)
and where
q
¼
a
=
b
;
Si
and
SiO
2
are the Young’s moduli of
silicon and silica;
E
Si
and
E
SiO
2
are the Poisson’s ratio of sili-
con and silica;
D
a
is the difference of the thermal expansion
0
200
400
600
800
1000
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
Δ
T (K)
Ω
(A.U.)
Δ
T=
Δ
T
c
Before Buckling
After Buckling
FIG. 2. Plot of the order parameter (Eq.
(1)
) versus the change in tempera-
ture relative to the growth temperature. A second-order phase transition is
apparent at a critical temperature difference (
D
T
c
). Above this value, the
disk buckles and features a crown pattern with several nodes. On the other
hand, the silica layer bends uniformly down when
D
T
is smaller than the
critical value. In this FEM simulation, the resonator has diameter 500
l
m,
thickness 2
l
m, and undercut 70
l
m.
FIG. 1. (a) SEM image of an unbuckled silica disk (500
l
m diameter and 2
l
m oxide thickness) resonator on a silicon pillar. The undercut is about 55
l
m.
(
b)-
(f) Top-view microscope images of resonators (500
l
m diameter and 2
l
m oxide thickness) with different buckled oxide configurations. The undercuts of these
devices are 70
l
m, 120
l
m, 155
l
m, 185
l
m, and 225
l
m respectively. (g) Spectral scan for the resonator in panel (a). The measured linewidth corresponds to
an optical Q factor of 37 million. The red curve gives a Lorentzian fitting of the experimental transmission and the cyan line shows a sinusoidal fitting o
f the in-
terferometer output. (h)-(i) Finite element simulation results of resonators (500
l
m diameter and 2
l
m oxide thickness) showing different buckling configura-
tions. The undercut of these devices match those in the panels (b)-(f).
031113-2 Chen, Lee, and Vahala
Appl. Phys. Lett.
102
, 031113 (2013)
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coefficients; and
D
T
is the temperature difference between
oxidation growth temperature and room temperature. For
clarity, we defined that
h
ð
q
Þ¼
1
1
Si
E
Si
þ
1
E
SiO
2
ð
1
þ
q
2
1
q
2
Þþ
SiO
2
E
SiO
2
. The bend-
ing energy is
15
ð
¼
SiO
2
;
D
¼
E
SiO
2
t
3
12
ð
1
2
SiO
2
Þ
Þ
;
U
¼
ðð
rdrd
h
D
2
@
2
w
@
r
2
þ
1
r
@
w
@
r
þ
1
r
2
@
2
w
@
h
2
2
D
ð
1
Þ
@
2
w
@
r
2
1
r
@
w
@
r
þ
1
r
2
@
2
w
@
h
2
þ
D
ð
1
Þ
1
r
@
2
w
@
r
@
h
1
r
2
@
w
@
h
2
:
(7)
The work done by the silica layer forces during the buckling
is found to be
15
T
¼
1
2
ðð
rdrd
hr
r
t
@
w
@
r
2
þ
r
h
t
1
r
@
w
@
h
2
"#
;
(8)
where
t
is the thickness of the silica layer. By equating
U
and
T
, the critical condition of buckling can be written as
D
T
¼
2
t
2
E
SiO
2
b
2
ð
1
2
SiO
2
Þ
D
a
h
ð
q
Þ
F
ð
n
;
q
Þ
G
ð
n
;
q
Þ
;
(9)
where
F
ð
n
;
q
Þ¼ð
1
þ
q
Þ
8
ð
1
þ
Þð
1
þ
q
Þ
þ
2
n
4
ð
1
þ
Þð
1
þ
q
Þð
1
þð
8
þ
q
Þ
q
Þ
n
2
ð
13
þ
12
ð
1
þ
3
q
Þþ
q
ð
43
þð
7
þ
q
Þ
q
ÞÞ
þ
4
ð
2
þ
6
n
4
ð
1
þ
Þþ
n
2
ð
9
þ
6
ÞÞ
q
2
log
ð
q
Þ
;
(10)
G
ð
n
;
q
Þ¼
q
2
1
þ
q
2
4
3
þ
q
ð
16
þ
12
q
þ
q
3
Þ
n
2
9
þ
q
ð
64
þ
36
q
þ
19
q
3
Þ
þ
12
q
2
4
þ
n
2
ð
6
þ
q
2
Þ
log
ð
q
Þ
:
(11)
Figure
3
shows the relation between the normalized critical
temperature difference (
D
a
D
Tb
2
=
t
2
) and parameter
q
¼
a
=
b
for configurations with different numbers of nodes in the
silica/silicon disk system. The normalization used here pro-
vides a dimensionless parameter that characterizes the thresh-
old of buckling. It also reflects the fact that critical
temperature difference depends quadratically on the resona-
tors’ oxide thickness and inversely quadratically on their ra-
dius. At each
q
, the configuration with minimal critical
temperature difference gives the equilibrium state after buck-
ling. As the undercut goes deeper, the equilibrium buckled
configuration will have fewer nodes. This result is consistent
with both experimental observation and the finite element sim-
ulation. Further, the combination of these curves defines an en-
velope function that outlines the boundary between the
unbuckled and buckled state (cf., Fig.
2
). It also explains the
fact that the resonator has an unbuckled configuration for
small silicon pillar (Fig.
1(f)
).
The y-axis parameter in Fig.
3
shows that the critical
temperature difference depends quadratically on the thick-
ness of the silica layer. It implies that thicker oxides will
avoid the buckling and maintain mechanical stability. Inter-
estingly, optical performance, in particular the Q factor, also
improves for thicker oxides. This can be understood to result
because the optical field at the oxide-air interface is gener-
ally weaker for thicker oxide.
4
This reduces both surface
absorption and scattering, thereby increasing the optical Q
factor. From a design perspective, the undercut needs to be
carefully controlled. On one hand, it must be deep enough to
reduce optical loss due to the mode leakage into the silicon
pillar. On the other hand, shallow undercut is more desirable
to avoid buckling. Indeed, we have demonstrated elsewhere
4
that by increasing the oxide thickness to 10
l
m and control-
ling the undercut to be approximately 150
l
m, it is possible
to obtain a record Q factor on a chip of 875 million Q in a
device that is mechanically stable.
FIG. 3. The relation between normalized critical temperature difference
(
D
a
D
Tb
2
=
t
2
) and parameter
q
¼
a
=
b
for configurations with different num-
bers of nodes (see legend) in the silica/silicon system. Insets show examples
from experiments of the buckled configurations with
q
¼
0
:
52, 10 nodes and
q
¼
0
:
26, 6 nodes (see, Fig.
1
).
FIG. 4. The threshold of undercut parameter
ð
n
¼
1
a
=
b
Þ
for buckling.
For each thickness, the left side of curve defines a region in which the disk
resonators do not buckle.
031113-3 Chen, Lee, and Vahala
Appl. Phys. Lett.
102
, 031113 (2013)
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To provide guidance to disk resonator design, we calcu-
late the buckling threshold of undercut for different disk res-
onators having 6
l
m and 10
l
m oxide thickness. As shown in
Figure
4
, for each resonator radius there are two thresholds
(lower and higher). If the undercut is smaller than the lower
threshold or larger than the higher threshold, the resonator
remains in the unbuckled configuration. In contrast, if the
undercut is between these two thresholds, the resonator will
buckle (cf., Fig.
1
).
In summary, this work demonstrates and models the
impact of the thermal stress in s
ilica-on-silicon disk resonators.
We provide both analytical and finite element modeling to
understand buckling behavior and offer guidance on perform-
ance improvement. In particula
r, by proper design, stress does
not limit the optical performance of these devices. Although we
only discuss the particular case of the disk resonator, our analy-
sis could be extended to guide the design of other structures.
We gratefully acknowledge the Defense Advanced
Research Projects Agency under the iPhoD program, the Insti-
tute for Quantum Information and Matter, an NSF Physics
Frontiers Center with support of the Gordon and Betty Moore
Foundation, and also the Kavli Nanoscience Institute at Cal-
tech. H.L. thanks the Center for the Physics of Information.
1
N. J. Hoff, J. R. Aeronaut. Soc.
61
, 756–774 (1957).
2
E. I. Grigolyuk and L. A. Magerramova, Izv An SSSR. Mekh. Tverd. Tela
16
, 111–138 (1981).
3
T. R. Tauchert,
Appl. Mech. Rev.
44
, 347–360 (1991).
4
H. Lee, T. Chen, J. Li, K. Yang, S. Jeon, O. Painter, and K. J. Vahala,
Na-
ture Photon.
6
, 369–373 (2012).
5
H. Lee, T. Chen, J. Li, O. Painter, and K. J. Vahala,
Nat. Commun.
3
, 867
(2012).
6
D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala,
Nature
421
, 925–929 (2003).
7
L. Starman, J. Busbee, J. Reber, J. Lott, W. Cowan, and N. Vandelli,
Nanotechnology
1
, 398–401 (2001).
8
W. Spengen,
Microelectron. Reliab.
43
, 1049–1060 (2003).
9
T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala,
Phys. Rev. A
74
, 051802 (2006).
10
T. J. Kippenberg, R. Holzwarth, and S. A. Diddams,
Science
332
, 555–559
(2011).
11
C. Ciminelli, F. Dell’Olio, C. Campanella, and M. Armenise,
Adv. Opt.
Photon.
2
, 370–404 (2010).
12
K. J. Vahala,
Nature
424
, 839–846 (2003).
13
M. Cai, O. J. Painter, and K. J. Vahala,
Phys. Rev. Lett.
85
, 74 (2006).
14
S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala,
Phys.
Rev. Lett.
91
, 043902 (2003).
15
S. P. Timoshenko and J. Gere, “Theory of Elastic Stability” (McGraw-Hill
Book Company, 1961).
16
E. H. Mansfield,
J. Mech. Appl. Math.
13
, 16–23 (1960).
17
K. K. Raju and G. V. Rao,
Comput. Struct.
18
, 1179–1182 (1984).
18
T. X. Yu and W. Johnson,
Int. J. Mech. Sci.
24
, 175–188 (1982).
19
C. Y. Wang,
J. Appl. Mech.
72
, 795–796 (2005).
20
T. Tang, C. Y. Hui, H. G. Retsos, and E. J. Kramer,
Eng. Fract. Mech.
72
,
791–805 (2005).
031113-4 Chen, Lee, and Vahala
Appl. Phys. Lett.
102
, 031113 (2013)
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