Supplementary Information
Silicon nanowires: where mechanics and optics meet at
the nanoscale
Daniel Ramos, Eduardo Gil-Santos, Oscar Malvar, Jose M. Llorens, Valerio
Pini, Alvaro San Paulo, Montserrat Calleja and Javier Tamayo*
Supplementary Information
S1. Device Fabrication
Silicon nanowires (SiNWs) were grown via the vapor-liquid-solid
mechanism in an atmospheric-pressure chemical vapor deposition (CVD) reactor
at 800 °C with 10% H2/Ar as both diluent and carrier gasS1. Flow rates of 270 and
60-70 s.c.c.m. were used, respectively. The carrier gas was passed through a liquid
SiCl4 bubbler maintained at 0 °C to keep a constant vapor pressure.
The nanowires were horizontally grown on [111] oriented sidewalls of
microtrenches fabricated in [110] oriented silicon on insulator (SOI) substrates by
photolithography and reactive ion etching. The size of the gold nanoparticles
(British Biocell) used as growth catalyst determines the nanowire diameters at
their bases. The vertical separation between the nanowire and the substrate
underneath ranges from 1.0 to 1.3 µm. The devices used in this work were grown
under particular conditions to obtain that the nanowire diameter linearly
decreases from the clamped to the free end. Tapering during growth occurs via
two possible mechanisms: on the one hand, gradual size reduction of the metal
catalyst gold nanoparticle during growth by either diffusion, evaporation or
chemical reactionsS2 results in a time-varying nanowire growth diameter; on the
other hand, dissociative adsorption on the gas/solid interface produces a
progressive thickening in the radial directionS3. The length of the nanowires ranges
8-16 µm and the diameter 40-260 nm.
S2. Experimental Set-up
The experimental measurements of the mechanical resonant frequencies of
the tapered nanowires were obtained using a homemade hybrid interferometricoptical systemS4 with a He-Ne laser (5 mW, 633 nm, Thorlabs, Inc) depicted in Fig.
S1. The laser beam is focused on the nanowire by means of a long working distance
Mitutoyo objective (50x, N.A. = 0.55) that produces a spot size of about 0.7-1.5 µm.
The sample is maintaining in a high vacuum environment (10-6 mbar). In order to
reach those environmental conditions, a high vacuum chamber is pumped down by
means of a rotatory and a turbomolecular pump (Varian Inc.). A Faraday isolator is
used to avoid backscattering reflections that could damage the laser cavity. By
using a polarizing plate after a quarter-wave plate, the laser beam is linearly
polarized with the desired angle. A collimated LED white source is used to image
the laser spot on the chip by means of a CCD camera. A non-polarizing beam
splitter is used to collect 50% of the reflected laser beam into a silicon amplifiedphotodetector (Thorlabs, Inc.) The position of the chip is controlled by a threedimensional closed loop nanopositioning stage (Attocube Systems, AG). The signal
from the photodetector is acquired by a high speed digitized (National Instruments
Corp.) and finally analyzed.
S1
Supplementary Information
Fig. S1. Block diagram of the experimental set-up used in the experiments
S3. Scattering Efficiency Calculation
As pointed out in the main text, when the tapered nanowires are visualized
by dark-field microscopy, we observe a color variation between the clamped and
free end (Fig. 1(b) in the main text, top). We find that the colors exhibited by the
silicon nanowires are related to the diameter of the nanowire (measured by
scanning electron microscopy). This finding can be explained on the basis of the
Mie scattering theory S5, S6. Figure S2 shows the theoretical calculation of the
scattering efficiency (Qsca) spectra for several nanowires with diameters from 80 to
160 nm. The shown analytical calculations demonstrate that the dark-field color is
correlated to the diameter.
As shown in Fig S2, the dominant wavelengths of the light scattered by the
nanowires obtained by dark-field microscopy follow the expected values from the
Mie theory calculations. For example, the scattering efficiency for a nanowire
region with a diameter around 80 nm is higher for the blue region of the visible
spectrum and thus it appears bluish in the corresponding optical dark-field image.
However, regions of nanowires with a diameter of about 120 nm appear reddish in
the optical dark-field microscopy according to a scattering efficiency peak at a
longer wavelength (~630 nm).
S1
Supplementary Information
14
100 nm
Qsca unpolarized
12
10
120 nm 140 nm
160 nm
80 nm
8
6
4
2
0
400
500
600
700
800
900
Wavelength (nm)
Fig. S2. Scattering efficiency as a function of wavelength for nanowires with different
diameters. The illumination is non-polarized and it is supposed a normal incidence to the
substrate.
In Fig. S3 we show several dark-field images of different silicon nanowires.
Since the color of the scattered light depends on the diameter of the nanowire, we
can use the different colors along the nanowire long axis as a direct measurement
of the diameter and tapering degree.
Fig. S3. Dark-field Microscopy image (objective 100×) of different silicon nanowires showing
different tapering degrees.
S1
Supplementary Information
S4. Far-Field FEM Calculations
By assuming that the length of the nanowires greatly exceeds their diameter
(usually 100:1 length to diameter ratio), we can reduce the dimensionality of the
problem to a cross-sectional 2D study. Finite element method (FEM) simulations
were performed in order to solve the Maxwell’s equations in the near environment
of the nanowire, showing a spatial confinement of the electric field intensity within
the dielectric cross-section. From these simulations, it is clear that the nanowire
size actively selects the amount of light confined within the nanostructure,
depending on the incident laser wavelength. The use of similar confined
electromagnetic modes to efficiently extend optomechanics to sub-wavelength
semiconductor structures was recently reported in the literatureS4. The
‘evanescent’ field emanating from these confined modes interacts with
interference field produced by the incoming and reflected waves from the silicon
substrate. To quantify how this interactions depends on the displacement of the
nanowire, we have calculated the far field collected by the objective in our
experimental set-up.
The far field scattered by the nanowire is calculated by means of the near
field propagation in free space. The Stratton-Chu formula for a 2D scattering
problem can be casted as,
,
(1)
where the calculated electric far-field
in the direction from the origin towards
point p is taken at infinity but with a well-defined angular dependency (). In this
formula, the scattering object (nanowire cross-section) is supposed to be located
at the origin of coordinates. and are the electric and magnetic fields on the
closed path S enclosing the scatterer, is the unit vector pointing from the origin
to the field point p, is the unit vector normal to the surface S,
is the
impedance referred to as a function of the permeability, , and the permittivity, ;
k and  are the wave number and the wavelength, respectively; and is the
position vector of the surface S. In this case the surface S should enclose, not only
the nanowire cross-section but also the reflecting surface, Fig. S4.
Fig. S4. Calculation scheme.
All results computed by FEM have been validated with analytical
calculations obtained by expanding the electric and magnetic fields in cylindrical
harmonics following the derivation of Videen and NgoS7.
S1
Supplementary Information
The integral of the electric field intensity over the objective lens numerical
apertureS8 was calculated as a function of the nanowire radius and the distance to
the substrate. In Fig. S5 we show the normalized resulting matrix for two TE and
TM polarizations. The lower the interaction between the laser beam and the
nanowire, the higher the signal (reddish areas).
a
b
0.75
1
0.25
0.5
0
Fig. S5. Normalized signal strength as a function of the nanowire radius and the nanowiresubstrate separation for TM (a) and TE (b) polarizations.
The derivative of the signal strength with respect to the nanowire-substrate
separation is proportional to the displacement responsivity (as defined in the main
text). Fig. S6 shows the absolute value of the corresponding normalized matrix for
TM and TE polarizations.
a
1
b
0.75
0.5
0.25
0
Fig. S6. Absolute value of the normalized gradient of the signal strength as a function of the
nanowire radius and the nanowire-substrate separation for TM (a) and TE (b) polarizations.
S5. Near-Field FEM Calculations
In order to obtain insight on how the optical resonances of the nanowires
interact with the interference field, we have calculated the near-field electric field
intensity distribution of the first optical modes for different nanowire diameters
and different separations between the nanowire and the substrate underneath.
S1
Supplementary Information
The displacement responsivity is intimately linked to the strength of the coupling
of the excited optical mode with the interference field. Figures S7 and S8 show the
electric field intensity near the nanowire for TE and TM polarizations, respectively.
TE(0,1)
TE(1,1)
TE(2,1)
TE(3,1)
Fig. S7. Near-field calculations showing the coupling of the leaky modes with the
surrounding standing wave for TE polarization (d nanowire’s diameter and D nanowire to
substrate distance)
TM(0,1)
TM(1,1)
TM(2,2)
TM(1,2)
TM(2,1)
Fig. S8. Near-field calculations showing the coupling of the leaky modes with the
surrounding standing wave for TM polarization (d nanowire’s diameter and D nanowire to
substrate distance)
S1
Supplementary Information
In Fig. S9 we show the calculation of the scattering efficiency for the optical
mode without a substrate, first row for modes TM (0,1), TM (1,1) and TE (0,1). The
TM (0,1) mode is excited for a nanowire diameter of 40 nm when there is not
substrate that induces the interference field. The interaction with the standing
wave field shifts the resonant diameter between 10 and 24 nm depending on the
separation between the substrate and the nanowire. Similar shifts can be observed
for the other modes.
Fig. S9. Displacement sensitivity as a function of the nanowire diameter for modes TM(0,1),
TM (1,1) blue curves, and TE (0,1), red curves. The substrate separation is 800 nm, 875 nm,
950 nm, 1025 nm and 1100 nm. The first row shows the scattering efficiency without a
substrate.
References
S1
Supplementary Information
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