of 10
ARTICLE
Received 22 Jan 2016
|
Accepted 28 Jun 2016
|
Published 3 Aug 2016
Quantum electromechanics on silicon nitride
nanomembranes
J.M. Fink
1,2,
w
, M. Kalaee
1,2
, A. Pitanti
1,2,
w
, R. Norte
1,2,
w
, L. Heinzle
3
, M. Davanc
̧
o
4
, K. Srinivasan
4
& O. Painter
1,2
Radiation pressure has recently been used to effectively couple the quantum motion of
mechanical elements to the fields of optical or microwave light. Integration of all three
degrees of freedom—mechanical, optical and microwave—would enable a quantum
interconnect between microwave and optical quantum systems. We present a platform based
on silicon nitride nanomembranes for integrating superconducting microwave circuits with
planar acoustic and optical devices such as phononic and photonic crystals. Using planar
capacitors with vacuum gaps of 60 nm and spiral inductor coils of micron pitch we realize
microwave resonant circuits with large electromechanical coupling to planar acoustic
structures of nanoscale dimensions and femtoFarad motional capacitance. Using this
enhanced coupling, we demonstrate microwave backaction cooling of the 4.48 MHz
mechanical resonance of a nanobeam to an occupancy as low as 0.32. These results indicate
the viability of silicon nitride nanomembranes as an all-in-one substrate for quantum
electro-opto-mechanical experiments.
DOI: 10.1038/ncomms12396
OPEN
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91
125, USA.
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA.
3
Department of Physics, ETH Zu
̈
rich,
CH-8093 Zu
̈
rich, Switzerland.
4
Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, Maryland
20899, USA.
w
Present addresses: Institute of Science and Technology Austria (IST Austria), 3400 Klosterneuburg, Austria (J.M.F.); NEST, Istituto
Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy (A.P.); Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA
, Delft,
The Netherlands (R.N.). Correspondence and requests for materials should be addressed to J.M.F. (email: jfink@ist.ac.at) or to O.P.
(email: opainter@caltech.edu).
NATURE COMMUNICATIONS
| 7:12396 | DOI: 10.1038/ncomms12396 | www.nature.com/naturecommunications
1
T
hin films of silicon nitride (Si
3
N
4
), when grown
stoichiometrically via low-pressure chemical vapour
deposition (LPCVD) on silicon substrates, can be used to
form membranes with large tensile stress (
E
1 GPa), thickness
down to tens of nanometres and planar dimensions as large as
centimetres
1
. The large tensile stress of these films allows one to
pattern membranes into extreme aspect ratio nanostructures,
which maintain precise planarity and alignment
2,3
. The high
tension also results in a significant reduction in mechanical
damping
4–7
, with
Q
-frequency products as large as 2

10
13
and
3

10
15
Hz having been observed at room temperature
8
and
milliKelvin temperatures
9
, respectively. As an optical material,
Si
3
N
4
thin films have been used to support low loss guided modes
for microphotonic applications, with a measured loss tangent in
the near-infrared of
o
3

10

7
(ref. 10).
Owing to their unique elastic and dielectric properties, Si
3
N
4
nanomembranes have recently been used in a variety of cavity-
optomechanical and cavity-electromechanical experiments
11
involving the interaction of membrane motion and radiation
pressure of either optical or microwave light. These experiments
include optical back-action cooling of a millimetre-scale
membrane close to its quantum ground state of motion
8,12–14
,
measurement of radiation pressure shot noise
15
and optical
squeezing
16
, and parametric conversion between optical
and microwave photons
17
. Thin-film Si
3
N
4
has also been
patterned into various other optomechanical geometries, such
as deformable photonic crystals
18
, nanobeams coupled to
microdisk resonators
19
and optomechanical crystal cavities,
which can be used to co-localize (near-infrared) photons and
(GHz) phonons into wavelength-scale modal volumes
20,21
.
Here we explore Si
3
N
4
nanomembranes as a low-loss substrate
for integrating superconducting microwave circuits and planar
nanomechanical structures. In particular, we exploit the thinness
of the nanomembrane to reduce parasitic capacitance and greatly
increase the attainable impedance of the microwave circuit. We
also use the in-plane stress to engineer the post-release geometry
of a patterned membrane
21,22
, resulting in planar capacitors with
vacuum gaps down to tens of nanometres. Combining the large
capacitance of planar vacuum gap capacitors and the low stray
capacitance of compact spiral inductor coils formed on a Si
3
N
4
nanomembrane, we show theoretically that it is possible to realize
large electromechanical coupling to both in-plane flexural modes
and localized phononic bandgap modes of a patterned beam
structure. Two-tone microwave measurements of an 8 GHz LC
circuit at milliKelvin temperatures in a dilution refrigerator
confirm the predictions of strong electromechanical coupling to
the low-frequency flexural mode of such a beam and microwave
backaction damping is used to cool the mechanical resonance to
an average phonon occupancy of
n
m
¼
0.32. These results,
along with recent theoretical and experimental efforts to realize
Si
3
N
4
optomechanical crystals
21,23
, indicate the viability of
Si
3
N
4
nanomembranes as an all-in-one substrate for quantum
electro-opto-mechanical experiments. Such membrane systems
could be used, for instance, to realize a chip-scale quantum
optical interface to superconducting quantum circuits
17,24–28
.
Results
Device design and fabrication
. The key elements of the
membrane microwave circuits studied in this work are shown
schematically in Fig. 1a. The circuits are created through a series
of patterning steps of an aluminum-coated 300 nm-thick Si
3
N
4
nanomembrane and consist of a mechanical beam resonator,
a planar vacuum gap capacitor, a spiral inductor (
L
) and a 50
O
coplanar waveguide feedline. The vacuum gap capacitor, formed
across the nanoscale cuts in the membrane defining the beam
resonator, is connected in parallel with the coil inductor to create
an LC resonator in the microwave C band. Each LC resonator sits
within a 777
m
m

777
m
m square membrane and is surrounded
on all sides by a ground plane. The coplanar waveguide feedline is
terminated by extending the centre conductor from one side of
the membrane to the other, where it is shorted to the ground
plane. Electrical excitation and readout of the LC resonator is
provided by inductive coupling between the centre conductor and
the spiral inductor. It s noteworthy that although thinner
membranes could have been used, our choice of a 300 nm-thick
membrane allows for compatibility with single-mode near-
infrared photonic devices and is guided by an ultimate goal of
integrating planar optical components with electromechanical
ones as per ref. 23.
The electromechanical coupling between the beam resonator
and the LC circuit in general depends on the particular resonant
mode of the beam and is given in terms of the linear dispersion
(
g
EM
) of the microwave circuit resonance frequency (
o
r
) with
respect to modal amplitude coordinate
u
,
g
EM
¼
@
o
r
@
u
¼
Z
o
r
2
C
m
@
C
m
@
u
:
ð
1
Þ
Here,
C
m
is the vacuum gap capacitance across the beam,
C
tot
is
the total capacitance of the circuit and
Z

C
m
/
C
tot
is the motional
participation ratio. In the case of uniform in-plane beam motion
and assuming
C
m
behaves approximately as a parallel plate
capacitor, the cavity dispersion simplifies to
g
EM
¼
Z
(
o
r
/2
s
0
),
where
s
0
is the nominal capacitor gap size. The vacuum
coupling rate, describing the interaction between light and
mechanics at the quantum level, is given by
g
0

g
EM
x
zpf
, where
x
zpf
¼
(
/2
m
eff
o
m
)
1
/
2
is the zero-point amplitude,
m
eff
is the
motional mass and
o
m
is the mechanical resonance frequency of
a given mechanical mode of the beam.
In this work we consider a patterned beam resonator of
width
W
¼
2.23
m
m and length
l
b
¼
71.4
m
m, which supports two
in-plane resonant modes, which can be coupled efficiently to
microwave or optical cavities
23
. The beam unit cell, shown in
Fig. 1b, has a lattice constant
a
and contains a central hole of
width
W
x
and height
W
y
. A pair of upper and lower aluminum
wires of thickness 65 nm and width 170 nm at the edges of the
beam form one half of the vacuum gap capacitor electrodes.
Simulations of the mechanical modes of the beam are performed
using a finite-element method solver
29
and include the internal
stress of the nitride film (
s
E
1 GPa).
The simulated fundamental in-plane flexural mode of the
patterned and wired beam, a displacement plot of which is
inserted into the microwave circuit of Fig. 1e, occurs at a
frequency of
o
m
/2
p
¼
4.18 MHz. As shown in Fig. 1c,d, a higher
frequency mode also results from Bragg diffraction of acoustic
waves due to the patterning of holes along the beam’s length. In
the structure studied here, the nominal hole parameters are
chosen to be
a
¼
2.23
m
m and
W
x
¼
W
y
¼
1.52
m
m, which results
in a 100 MHz phononic bandgap around a centre frequency of
450 MHz. A defect is formed in the phononic lattice by increasing
the hole width (
W
x
) over the central 12 holes of the beam,
resulting in a localized ‘breathing’ mode of frequency
o
m
/2
p
¼
458 MHz that is trapped on either end by the phononic
bandgap. From the simulated motional mass of both mechanical
resonances, the zero-point amplitude is estimated to be
x
zpf
¼
8.1
and 4.2 fm for the flexural and breathing modes, respectively.
As motional capacitance scales roughly with mechanical
resonator size, realizing large electromechanical coupling to
nanomechanical resonators depends crucially on minimizing
parasitic capacitance of the microwave circuit as per equation (1).
Using a planar spiral inductor coil of multiple turns greatly
increases the coil inductance per unit length through mutual
ARTICLE
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12396
2
NATURE COMMUNICATIONS
| 7:12396 | DOI: 10.1038/ncomms12396 | www.nature.com/naturecommunications
inductance between coil turns and, consequently, reduces coil
capacitance. One can determine the capacitance (
C
l
) and
inductance (
L
) of a given coil geometry by numerically simulating
its self resonance frequency with and without a known small
shunting capacitance. Figure 1f displays a method of moments
numerical simulation
30
of the self resonance frequency (
o
coil
)
of a series of square planar coil designs with constant area
(
A
coil
¼
87
m
m

87
m
m) but varying wire-to-wire pitch
p
,or
equivalently, coil turns
n
. Here we assume a coil wire width and
thickness of 500 and 120 nm, respectively, deposited on top of the
300 nm nitride membrane. Although the coil capacitance is
roughly constant at
C
l
¼
2.1 fF, the coil inductance varies over
three orders of magnitude, in good agreement with an analytical
model for planar inductors
31
. An additional stray capacitance of
C
s
¼
2.2 fF is estimated for the full integrated microwave circuit
(see Supplementary Note 1 and Supplementary Fig. 2 for details).
Comparing similar geometry coils with the same self-resonance
frequency, we can attribute a factor of 3.8 increase in impedance
due to fabrication on a membrane and another factor of 2 due to
a reduction of the coil pitch from 4
m
m (ref. 32) to 1
m
m.
Figure 1g displays the simulated motional capacitance and
vacuum coupling rate versus capacitor slot size
s
for both the
flexural and breathing modes of the beam resonator assuming
a coil of pitch
P
¼
1
m
m(
n
¼
42,
L
¼
68 nH,
C
l
¼
2.1 fF,
o
coil
/2
p
¼
13.68 GHz). Here,
q
C
m
/
q
u
is calculated for each
specific mechanical mode using a perturbation theory depending
on the integral of the electric field strength at the dielectric and
metallic boundaries of the vacuum gap capacitor
33
. For a gap size
of
s
¼
60 nm, the vacuum coupling rate is estimated to be
g
0
/2
p
¼
43 Hz (156 Hz for
Z
¼
1) for the flexural mode and
g
0
/2
p
¼
6 Hz (43 Hz for
Z
¼
1) for the breathing mode. It is
noteworthy that here we assume the outer electrode of the
vacuum gap capacitor extends along the entire length of beam in
the case of the flexural mode, whereas for the breathing mode we
limit the outer capacitor electrode to the central six lattice
constants of the beam where the breathing mode has significant
amplitude. In addition, for the breathing mode simulations the
two vacuum gap capacitors are assumed to be connected in
parallel, which doubles the vacuum coupling rate due to the mode
symmetry.
Fabrication of the membrane microwave circuits begins with
the LPCVD growth of 300 nm-thick stoichiometric Si
3
N
4
layers
on the top and bottom surfaces of 200
m
m-thick silicon wafer
and involves a series of electron beam lithography, dry etching,
aluminum evaporation and chemical wet etching steps. An
optical image of the fully fabricated and wirebonded chip is
shown in Fig. 2a. Zoom-in scanning electron microscope images
of the inductor coil and nanobeam regions of the device are
shown in Fig. 2b–e. The main fabrication steps are depicted in
Fig. 2f and discussed in more detail in Supplementary Note 2.
One important feature of our fabrication method is the use of the
tensile membrane stress (
s
¼
1 GPa) to fabricate capacitive slot
gaps that shrink on release of the membrane, providing a
controllable way to create ultra-small gaps. As can be seen in the
device figures of Fig. 2b,c, stress release cuts are used above and
below the nanobeam region so as to allow the membrane to relax
Coil
Beam
Coupler
C
s
C
m
fg
I
c
L
C
l
cd
a
W
y
C
m
L
,
C
l
I
c
W
x
W
W
Al
0
200
400
600
800
00.5
b
0.5
1
2
40
60
80
100
120
140
1
5
10
50
100
160
80
40
20
10
5
0.5
1
5
10
0
1
2
3
4
5
C
l
(fF)
1
10
100
1,000
Z
0
= 20 k
Ω
(4 GHz)
Z
0
= 1 k
Ω
(68.5 GHz)
g
0
/2

(Hz)
C
m
(fF)
Capacitor gap,
s
(nm)
L
(nH)
Z
0
= 5 k
Ω
(15 GHz)
Coil pitch,
p
(

m)
Coil turns,
n

m
/2

(MHz)
k
x
(2

/a)
a
e
Figure 1 | Device design.
(
a
) Schematic of the membrane electromechanical circuit. (
b
) Unit cell of the phononic crystal nanobeam. (
c
) Acoustic band
diagram of the phononic crystal nanobeam with
a
¼
W
¼
2.23
m
m,
W
x
¼
W
y
¼
1.52
m
m and
W
Al
¼
170 nm. The nitride membrane thickness and aluminum
wire thickness are
t
mem
¼
300 nm and
t
Al
¼
65 nm, respectively. The acoustic bandgap is shaded in blue, with the localized breathing mode frequency
indicated as a dashed line. (
d
) Plot of the finite-element method (FEM)-simulated breathing mode profile. Mechanical motion is indicated by an
exaggerated displacement of the beam structure and by colour, with red (blue) colour indicating regions of large (small) amplitude of the motion.
(
e
) Electrical circuit diagram, where
I
c
is the current through the reflective coupler,
L
is the coil inductance,
C
l
is the coil capacitance,
C
s
is additional
stray capacitance and
C
m
is the motional capacitance. The simulated displacement of the in-plane fundamental flexural mode of the beam is shown.
(
f
) Inductance (
L
) and capacitance (
C
l
) of a planar square coil inductor of constant area
A
coil
¼
87
m
m

87
m
m and variable wire-to-wire pitch
p
. Wire width
and thickness are 500 and 120 nm, respectively. Method of moments
30
numerically simulated values are shown as open circles (inductance) and open
squares (capacitance). Calculations using an analytical model of the planar coil inductor
31
are shown as a solid line. Vertical lines are shown for coils with a
characteristic impedance of
Z
0
¼
1, 5 and 20 k
O
, with the coil self-resonance frequency indicated in brackets. (
g
) FEM simulations of the modulated
capacitance
C
m
(blue symbols) and the electromechanical coupling
g
0
/2
p
(red symbols) of the in-plane fundamental flexural mode (circles) and the
phononic crystal breathing mode (squares) as a function of the capacitor gap size
s
. Solid curves indicate a 1/
s
fit for
C
m
and 1/
s
d
with
d
E
1.4 for
g
0
.
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12396
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NATURE COMMUNICATIONS
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3
on either side of the beam. A simulated plot of the membrane
relaxation is shown in Fig. 2g. Comparison of the simulated slot
gap change (
d
y
) and measured slot gap change for a series of
fabricated devices with different cut angles
a
is shown in Fig. 2h,
indicating that slot gap adjustments up to 100 nm can be reliably
predicted and produced. In the measured device of this work we
use this feature to controllably close the capacitor slot
s
from an
initial slot size of
s
¼
150 nm right after dry etching, down to a
final slot size of
s
E
80 nm after membrane release.
As shown in Fig. 2b, in the device studied here each nanobeam
is coupled on one side to one coil and on the other side to another
coil. The capacitor electrodes also extend across the whole length
of the device, to maximize coupling to the low-frequency flexural
mode of the beam. The two coils have different lengths, resulting
in different LC resonant frequencies. As will be presented
elsewhere, such a double-coil geometry can be used to perform
coherent microwave frequency translation using the intermediate
nanomechanical resonator as a parametric converter
34,35
. In the
following, however, we will focus on the lower frequency circuit
(larger coil) only. The device is cooled to a fridge temperature
of
T
f
¼
11 mK using a cryogen-free dilution refrigerator and
connected to a microwave test set-up consisting of low
noise control and readout electronics for electromechanical
characterization (see Supplementary Note 3 and Supplementary
Fig. 1 for details).
Coherent electromechanical response
. Sweeping a narrowband
microwave source across the 6–12 GHz frequency range and
measuring in reflection, we find a high-
Q
, strongly coupled
microwave resonance at
o
r
/2
p
¼
7.965 GHz corresponding to the
larger coil of 42 turns. This is very close to the expected LC
resonance frequency based on the above simulations, indicating
that the stray and motional capacitance of the circuit are close to
the expected values. Using a two-tone pump and probe scheme
we are able to study the coherent interaction between the
microwave electrical circuit and the coupled nanobeam
mechanical resonator. In the driven linearized limit
11
, the circuit
electromechanical system is approximately described by an
interaction Hamiltonian
H
OM
¼
G
ð
^
a
w
^
b
þ
^
a
^
b
w
Þ
, where
^
a
(
^
a
w
)is
the microwave photon annihilation (creation) operator for the
LC
resonator mode of the circuit and
^
b
(
^
b
y
) are the phonon
Pull-in
Al
Si
3
N
4
L
,
C
l
Beam
Pull-in cuts
Coil
Si
3
N
4
Si
LPCVD
C
4
F
8
/ SF
6
Aluminum
Si-TMAH
a
b
c
d
fe
–120
–80
–40
0
SEM
Simulation
02
8
46
10
x
y
C
m
g
h
I
c
80 nm

y
(nm)

(deg)

Figure 2 | Sample fabrication.
(
a
) Optical image of the membrane microchip, which is mounted, bonded and clamped to a low loss printed circuit board
(scale bar, 1 mm). The microchip contains four sets of four membranes. In this image the Si
3
N
4
membranes of thickness 300 nm are semi-transparent
purple, the aluminum coated regions are grey and the uncoated silicon substrate is green. The two bright regions in the middle of each membrane
correspond to the two coil resonators coupled to each nanobeam resonator. (
b
) False colour scanning electron microscope (SEM) image of the centre part
of the membrane depicting two aluminum planar coils (white) coupled to two sides of a single patterned phononic crystal nanobeam with stress pull-in c
uts
(black). Scale bar, 100
m
m. (
c
) SEM image zoom-in of the spiral inductor (
p
¼
1
m
m,
n
¼
42), showing the cross-overs needed to connect the inductor coil to
the vacuum gap capacitor across the nanobeam resonator. Scale bar, 50
m
m. (
d
) SEM image zoom-in of the released centre region of the nanobeam
mechanical resonator and vacuum gap capacitors with gap size of
s
E
80 nm. Scale bar, 5
m
m. (
e
) Tilted SEM image of the capacitor gap showing the etch
profile of the nanobeam and the aluminum electrode thickness (
E
65 nm). Scale bar, 200 nm. (
f
) Schematic of the main circuit fabrication steps: (i) LPCVD
of stoichiometric Si
3
N
4
on both sides of a 200
m
m-thick silicon substrate, (ii) C
4
F
8
:SF
6
plasma etch through the nitride membrane defining the mechanical
beam resonator and pull-in cuts on the top side and membrane windows on the bottom side, (iii) electron beam lithography, aluminum deposition and
lift-off steps to pattern the microwave circuit and (iv) final release of the nitride membrane using a silicon-enriched tetramethylammonium hydroxi
de
(TMAH) solution. (
g
) Simulation of the membrane relaxation during release. The image shows the regions of positive (red) and negative (blue)
displacement,
d
y
, of the membrane. The stress release cuts (white) are shaped at an angle
a
to controllably narrow the capacitor gaps
s
during release.
The rounded shape of the pull-in cut end section has been optimized to minimize the maximal stress points to avoid membrane fracturing. (
h
) Plot of the
simulated (solid red curve) and SEM-measured (blue solid circles) change in the slot gap (
d
y
) versus slot-cut angle
a
. Error bars indicate the single s.d.
uncertainty in SEM measurements of the gap size.
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