Tuning mechanical modes and influence of charge screening in nanowire resonators
ABSTRACT We probe electromechanical properties of InAs nanowire (diameter ∼100 nm) resonators where the suspended nanowire is also the active channel of a field-effect transistor. We observe and explain the nonmonotonic dispersion of the resonant frequency with dc gate voltage (Vgdc). The effect of electronic screening on the properties of the resonator can be seen in the amplitude. We observe the mixing of mechanical modes with Vgdc. We also experimentally probe and quantitatively explain the hysteretic nonlinear properties, as a function of Vgdc, of the resonator using the Duffing equation.
arXiv:1001.2882v1 [cond-mat.mes-hall] 18 Jan 2010
Tuning mechanical modes and influence of charge screening in
Hari S. Solanki,1Shamashis Sengupta,1Sajal Dhara,1Vibhor Singh,1Sunil Patil,1
Rohan Dhall,1Jeevak Parpia,2Arnab Bhattacharya,1and Mandar M. Deshmukh1, ∗
1Department of Condensed Matter Physics and Materials Science,
Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India
2LASSP, Cornell University, Ithaca NY 14853
We probe electro-mechanical properties of InAs nanowire (diameter ∼100 nm) resonators where
the suspended nanowire (NW) is also the active channel of a field effect transistor (FET). We
observe and explain the non-monotonic dispersion of the resonant frequency with DC gate voltage
). The effect of electronic screening on the properties of the resonator can be seen in the
amplitude.We observe the mixing of mechanical modes with VDC
. We also experimentally
probe and quantitatively explain the hysteretic non-linear properties, as a function of VDC
, of the
resonator using the Duffing equation.
Nano electro mechanical systems (NEMS) , are being used extensively to study small
displacements , mass sensing [3–6], spin-torque effect , charge sensing , Casimir force
 and potential quantum mechanical devices [10, 11]. A variety of NEMS devices, fab-
ricated using carbon nanotubes [12–16], graphene [17–19], nanowires (NWs) of silicon 
and by micromachining bulk silicon , have been used to probe the underlying physics
at nano scale. In this work we study the electromechanical properties of doubly clamped
suspended n-type InAs NWs. In our suspended NW FET, the gate electrode serves three
purposes: first, to modify the tension in the NW, second, to actuate the mechanical motion
of the resonator and third, enabling us to systematically study the coupling of mechanical
properties to the tunable electron density. As we will show, a tunable electron density leads
to a variable screening length of the order of the nanowire’s cross sectional dimensions. Thus
the electro-mechanical properties enter into a mesoscale regime. Such a variable electron
density is not accessible in carbon nanotubes; the screening length cannot be tuned contin-
uously – relative to the diameter of the carbon nanotube – as easily. However, the physics
of charge screening in nanoscale capacitors  and ferroelectric devices  is intimately
connected to that in our NEMS devices. Taken together, these observations suggest that
a geometrical interpretation of capacitance is inadequate at the nano scale. Additionally,
the gate allows us to tune the resonant frequency non-monotonically due to the competition
between the electrostatic force and the mechanical stiffness (∼ 1 N/m) of the nanowire,
a feature expected (but heretofore not studied in detail) for all electrostatically actuated
NEMS. In this article, we demonstrate that the InAs semiconducting nanowire system man-
ifests this transition from softening to hardening as the gate voltage is varied. In addition,
mixing of the natural mechanical modes as a function of VDC
can be understood in terms
of the structural asymmetries in the resonator. In the non-linear regime we study in de-
tail hysteretic behavior as a function of VDC
(unlike the commonly studied response as a
function of drive frequency) and we show that this can be understood by using the Duffing
equation incorporating the effect of gate voltage. The observed hysteretic response with
is an alternate knob for tuning the nonlinear response of our NEMS devices and can
be used for charge detection . Our work provides further understanding of the unique
characteristics of NEMS devices operating at room temperature. The observed behavior can
provide information on the nanomechanics of other systems whose electron density, stiffness
or screening length cannot be so readily tuned.
The InAs NWs used for this work were grown using metal organic vapor phase epitaxy
(MOVPE) [25, 26]. The NWs are oriented in the < 111 > direction and are 80 to 120 nm in
diameter with a length of several micrometers. The substrate used for making the devices is a
degenerately doped silicon wafer with 300 nm thick SiO2. We have fabricated suspended InAs
NW devices by sandwiching the NWs between two layers of electron-beam resist and then
using electron beam lithography to define the electrodes and suspend them by depositing
∼150 nm Cr and ∼250 nm Au after development followed by in situ plasma cleaning .
The ohmic contacts also serve as mechanical supports for the NW suspended ∼200 nm
above the surface of SiO2. Fig.1a shows the SEM image of a resonator device and scheme
for actuating and detecting the motion of the resonator. All the measurements were done
at 300 K and pressure less than 1 mBar.
To actuate and detect the resonance we use the device as a heterodyne mixer [6, 12, 18, 27–
29]. We use electrostatic interaction between the wire and gate to actuate the motion in a
plane perpendicular to the substrate. We apply a radio frequency (RF) signal Vg(ω) and a
DC voltage VDC
at the gate terminal using bias-tee. Another RF signal VSD(ω + ∆ω) is
applied to the source electrode (Fig.1a). The RF signal applied at the gate Vg(ω) modulates
the gap between NW and substrate at angular frequency ω, and VDC
alters the overall
tension in the NW. The amplitude of the current through the NW at the difference frequency
(∆ω), also called the mixing current Imix(∆ω), can be written as 
Imix(∆ω) = Ioscn(∆ω) + Iback(∆ω)
where G is the conductance of the NW, q is the charge induced by the gate, Cg
is the gate capacitance, z(ω) is the amplitude of oscillation at the driving frequency ω
and z axis is perpendicular to the substrate. The term Iback(∆ω) =
the background mixing current which is independent of the oscillation of the NW and
VSD depends on the amplitude of oscillation.Measuring
Imix(∆ω) using a lock-in allows us to monitor the resonance of the NW as the frequency
is swept. Fig.1b shows Imix(∆ω) as a function of ω for VDC
= ± 36.4 V. The sharp fea-
ture corresponds to the mechanical resonance of the NW. We address the asymmetry of
the mixing current signal for ±VDC
later. Fig.1c shows the plot of conductance (G) as a
function of VDC
. The variation of G with VDC
is very critical for this scheme of heterodyne
mixing to work as it controls the overall amplitude Imix(∆ω). The hysteresis observed in
the measurement of conductance is typical for our suspended devices and is associated with
charge trap states with dipolar nature on surface of the nanowire [30, 31].
We can connect the resonant frequency of the fundamental mode, f0, of a doubly clamped
beam at zero VDC
, to the material properties of the beam as f0= C0
is the velocity of sound; r is the radius of the beam, l is the length of the beam, E is the
Young’s modulus, ρ is the density of the material and C0= 1.78. Fig.1d shows a plot for Vs
that does not vary much from the bulk value  (dashed line) for eight different devices.
The scatter around the Vscalculated using bulk values could be due to the relative volume
fraction contribution of the amorphous layer around the NWs; this needs further detailed
Fig.2a and 2b show the colorscale plot of Imix(∆ω) as a function of VDC
and ω on a
logscale spanning more than 3 decades. Data is taken by sweeping VDC
for each value of
frequency. In the data from both the devices we see a symmetric evolution of the resonant
frequency as a function of VDC
. The parabolic behavior is expected as the attractive force
exerted by the gate on the wire is given by FDC =
)2∇Cg. An increase in VDC
enhances the tension in the NW. We now discuss the particular W-shaped dispersion of
modes as a function of VDC
. As |VDC
| is increased, initially the mode disperses negatively
and after a certain threshold voltage Vth
it disperses with a positive slope. Although purely
negatively and positively dispersing modes have been studied in detail before by Kozinsky et
al. , we observe the crossover regime where these interactions compete. This particular
dispersion can be understood using a toy-model in which a wire is suspended from a spring
of force constant Ki above the substrate. The wire and the substrate make up the two
electrodes of a capacitor. There are two consequences of increasing |VDC
|: first, it changes
the equilibrium position by moving the NW closer to the substrate and second, it makes
the local potential asymmetric and less steep. The effective force constant, Keff, is reduced,
resulting in negative dispersion of the mode for |VDC
| < |Vth
g|. If the intrinsic force constant
Kihad been independent of VDC
, the modes would always disperse negatively when motion
occurs perpendicular to the gate plane. However, in general Ki can be written as k +
)); where k, α, and β are constants. With VDC
then, to a first approximation, Keff= Ki−1
dz2. We find that the W-shape of the
dispersion curve can be explained only if one considers the case where β > 0. The result of
such calculations (Fig.2c) quantitatively explains the experimental observations. Vth
value at which the crossover from negative to positive dispersion occurs is a function of the
dimensions of the NW and the capacitive geometry of the device. The effect of the device
dimension is clearly seen in a larger value of Vth
in device-2 shown in Fig.2b where the
doubly clamped beam is 120 nm thick, as against 100 nm for device-1 in Fig.2a (the lengths
differ by 200 nm).
Next, we consider another complementary aspect of the data – the amplitude of the
mixing current. Fig.2d shows Imix(∆ω) (log-scale) as a function of VDC
. We see that the
amplitude of the mixing current for the negative values of VDC
are significantly larger than
those for positive VDC
for the same mechanical mode; this is also seen in Fig.1b. We now
try to understand this asymmetry as our InAs NW are n-type semiconductors  as seen in
Fig.1c. To understand this asymmetry we have carried out detailed fits of the experimental
data for the amplitude of mixing current as a function of frequency using Eqn.1. The
amplitude z(ω) of oscillation at frequency ω is given by
cos(∆φ + arctan(ω2
(1 − (ω
ampis the amplitude at resonant frequency ω0, Q is the quality factor and ∆φ is
the relative phase difference between the ω and ω + ∆ω signals that depends on the device
parameters like the contact resistance. Fitting from Eqns.1 and 2 allow us to extract the
variation of Q as a function of VDC
as shown in Fig.2e (). We have also estimated the
amplitude of oscillation using Eqns.1 and 2 by examining the ratio
A plot of the calculated amplitude (zreso
amp) is seen in Fig.2f. We see that as the |VDC
increased Q and zreso
ampare observed to decrease. We also observe that there are noticeable
differences in Q and zreso
ampfor positive and negative values of VDC
. The values of Q and zreso
are larger for positive VDC
. One of the possible mechanisms that can explain this behavior
in the amplitude is that with increasing VDC
one increases the density of electrons in the
NW leading to reduction in the screening length. This implies that the simple geometrical
capacitance is inaccurate particularly since the screening length can be comparable to NW
diameter at low densities (at negative VDC
in our case). In our device geometry, using
Thomas-Fermi approximation , screening length is around 20-40 nm (diameter of our
devices are 100 nm) and the distance between the nanowires and gate oxide 200nm. So,
the screening length is a significant fraction of the diameter and the suspension distance -
this plays a critical role in observing the effect of density gradients within the cross-section
of the nanowire. In case of single walled carbon nanotube, diameter is 1-2 nm and height
of suspension is typically 100nm or more [13–16]. Additionally the screening length in
carbon nanotubes is typically several multiples of the nanotube diameter , so due to an
increased ratio of suspension distance to diameter and the large screening length compared
to the diameter it is very difficult to observe the physics we discuss for the case of nanowires
in carbon nanotubes. We would like to point out that this is not a peculiarity of the InAs
nanowires and should be seen in other semiconducting nanowire devices as well with similar
dimensions. Additionally, if one considers the realistic case with non-uniform density of
carriers in the NW due to the device geometry  the NW will have a gradient of dielectric
constant . A gradient of dielectric constant  alongwith a change in capacitance as a
function of the density changes the capacitive coupling of the NW to the gate. This results
in differing amplitudes for two different electron densities. Our device geometry with NW
diameter comparable to the gap accentuates this effect.
In order to better understand the effect of the gate voltage in tuning the spatial charge
density across the cross-section of the nanowire we selfconsistently solve three dimensional
Poisson’s equation using finite element method for our device geometry. We use approach
followed by Khanal et al.  by solving ∇·ǫd∇Φ(x,y,z) = ρ(x,y,z), throughout the space
of the nanowire and its dielectric environment (here ǫdis dielectric constant, Φ is the local
electrostatic potential in the system due to applied gate voltage, and ρ is space charge density
inside the NW). The geometry consists of a 100 nm diameter and 1.5 µm long InAs nanowire
clamped by metallic electrodes. The wire is suspended 100 nm above a 300 nm thick silicon-
oxide dielectric on the gate electrode. Inside the nanowire the ρ(x,y,z) = e(Nd−n(Φ)+p(Φ))
where e is the charge of an electron, Ndis the density of the n-type dopants (∼ 1016cm−3),
n(Φ) and p(Φ) are the densities of electrons and holes. The unintentional source of n-type
dopants in our growth is Si and C, from the metal organic precursors and are assumed to
be uniformly distributed throughout the nanowire. A selfconsistent calculation gives us the
distribution of potential throughout the space and space charge density in the nanowire.
Fig.3a and 3b show the colourscale distribution of potential when the VDC
= −5 V. In
order to model the consequences of modifying gate voltage we calculated the distribution of
space charge density through the nanowire for VDC
= −25,−5 and 5V. Fig.3c, 3d, 3e, and 3f
show the result of such a calculation for varying VDC
for space charge density in the center
of the NW. For positive voltages the Fermi energy is very close to the conduction band of
the InAs and as a result the charge density is very uniform while behaving as a metal like
system. This simple modeling supports our arguments that a gradient of electron density
can modify the capacitive coupling and the resulting amplitude. Further analysis is needed
to solve self consistent solutions to the Poisson’s equation where the dielectric constant is
itself a function of the density and the quantitative variation of Q and amplitude with sign
. Our measurements suggest a way to tune the efficiency of actuation by tuning the
density of carriers.
We next consider three other features of our data – first, the presence of other mechanical
modes near the fundamental mode; second, the mixing of modes as a function of the VDC
and third, the non-linear properties of NW oscillators driven to large amplitudes. Fig.4a
and 4b show the plot of Imix(∆ω) as a function of VDC
and ω for device-3 and device-2. It is
well known that for a doubly clamped beam with no tension, fn= Cn
ρwith C0= 1.78,
C1= 4.90 and C2= 9.63 for the transverse modes. It is clear that if the fundamental mode
(f0) is described by a mode with zero nodes and moving in a plane perpendicular to the
substrate, the other observed modes, in the frequency range near f0(Fig.4a, 4b), cannot
be defined by f1 and f2. We have observed the anticipated f1 and f2 modes at higher
frequencies. The other modes beside the fundamental in Fig.4a, 4b, are explained due to
geometrical asymmetry along the diameter in the NW (Fig.4c, details of this calculation
provided in Supporting Information). These are the modes involving motion in a plane
that is not perpendicular to the substrate. This would explain the less steep slope of the
dispersion as a function of VDC
Fig.4a and 4b also show the mixing of the modes as a function of VDC
. The mode mixing
can be seen clearly in Fig.4d which shows a close-up of the data in Fig.4b. Displacement
along the transverse direction y (perpendicular to z) will weakly affect the capacitance
because of any slight asymmetry in the physical structure of the NW. The coupling coefficient
Information). For the device shown in Fig.4b the minimum frequency gap in the region of
)2appearing in the potential energy gives rise to mode-mixing (see Supporting
level repulsion is 0.25 ± 0.02 MHz. The asymmetry in the amplitude of the modes away
from the region of mixing can also be understood within this model.
We next consider the non-linear response of these NEMS oscillators. Due to electrostatic
actuation, the potential energy of the oscillator is asymmetric about the equilibrium position
and has the form V (z) =1
2Keffz2+θz3+µz4+H.O.(z), where Keff, θ and µ are coefficients
. We have experimentally probed the non-linear and hysteretic responsedepending on VDC
of the device. Fig.5a, 5b, 5c, and 5d, show the experimentally measured non-linear response
of this device leading to hysteretic behavior as function of VDC
with increasing amplitude of
driving force. Here, VDC
is swept at a given drive frequency to measure the mixing current
and several such VDC
sweeps are collated (the distinct response of sweeping the frequency
at a fixed VDC
– a common mode to study non-linear response – is described in supporting
information). There are two features that we would like to point out – first, in the region
at the bottom of the W-shaped dispersion curve, two branches of the same mode merge
into one broad peak where the oscillator has large amplitude over a wide range of VDC
second, whenever during the VDC
sweep, one crosses the region with a local negative value
(here, f0is the resonant frequency at a particular VDC
), shows a curved hysteretic
j-shaped response (indicated by ⋆), seen in Fig.5b. In order to understand and explain the
experimentally observed hysteretic response as a function of VDC
, we have used the Duffing
equation  for our resonator. The result of such a calculation for amplitude is seen in
Fig.5e, 5f, 5g, and 5h, with increasing amplitude of driving force. To calculate the amplitude
we have only used the observed dispersion relation (W shape) as input for Duffing equation.
There is a qualitative agreement between the experimentally measured data shown in Fig.5a-
d and results of calculation using the Duffing equation shown in Fig.5e-h. We find that for
every increase in excitation amplitude by 100 mV corresponds to an increase of a factor of
2.5 in the anharmonic component of the Duffing equation (from observing the calculated
data). Additional aspect of the nonlinearity of the oscillator is also seen in the evolution of
dispersion near VDC
= 0 as the actuation amplitude is gradually increased from 100 mV to
400 mV in Fig.5a-d. The negative dispersion is due to the softening of electrostatic force
and with larger amplitude of oscillation the effective spring constant changes, as the wire
samples a region with varying electric field; this difference is clearly seen in the dispersion
= 0 for the data shown in Fig.5 a and d.
Fig.5i, shows the line plot (along the dashed line in Fig.5f) of calculated amplitude
for different sweep direction of VDC
and the resulting hysteresis. The observed hysteretic
response, as a function of VDC
, is quite different from the hysteretic response as a function
of drive frequency ( discussed in supporting information ). This non-linear response of our
devices with VDC
may be utilized for charge detection  as near the onset of non-linearity
the change in amplitude as a function of VDC
is very large.
In summary, we have studied the electromechanical properties of doubly clamped InAs
NW resonators. Their size and tunable electron density allow us to map behavior that has
not been manifested in a single device. We have observed and quantitatively explained the
competition between the softening of stiffness of the restoring force of the resonator due to
the variation of the electrostatic force under variable gate voltage. At larger voltages, the
stretching of the nanowire leads to increased stiffness resulting in a non-monotonic dispersion
of the fundamental mode with VDC
. The screening of electric fields due to the variation in
the density of the electrons in our suspended FET devices modifies the amplitude because
the variation of the screening length spans the cross sectional dimension of our nanowire.
Further, the non-linear properties of our device can be understood qualitatively using the
Duffing equation that explains the hysteretic response of the amplitude as a function of
gate voltage. Thus in a single device, we demonstrate, separate and account for three
diverse behaviors. Our measurements indicate that measuring electromechanical response
influenced by charge screening could lead, in the future, to new ways to probe spin physics
by exploiting spin-dependent charge screening . Probing the physics by tuning electron
density in NEMS devices may help probe the role of defects  and electron hopping as one
moves from insulating to conducting regimes. Control over the non-linear dynamics may be
achievable by controlling mode mixing and DC gate voltage.
This work was supported by Government of India. J. Parpia was supported by DMR-
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FIG. 1: (color online) a) Tilted angle SEM image with the circuit used for actuation and detection
of resonance for an InAs NW resonator. The diameter of the wire is 100 nm and the length of NW
is 2.9 µm. The scale bar indicates a length of 200 nm. b) The mixing current (∆ω/2π = 17 KHz)
as a function of frequency for two values of VDC
(mixing current shown for +ve VDC
, is 5 times of
its original value). c) Variation of the conductance as a function of DC gate voltage. d) The plot
of sound velocity Vs=
1.78rcalculated using the measured frequency (ω/2π) of the fundamental
mode of the InAs NW resonators and geometrical values. The dashed line indicates the speed of
sound obtained for Vs=
?E/ρ using bulk values for E and ρ .
Gate Voltage (V)
-40 -20020 40
-40 -20 0
Gate Voltage (V)
Gate Voltage (V)
Gate Voltage (V)
-40 -20 0 20 40
Absolute Gate Voltage (V)
For +ve Vg
For -ve Vg
For +ve Vg
For -ve Vg
Absolute Gate Voltage (V)
FIG. 2: (color online) a) and b) Color logscale plots of mixing current as a function of VDC
ω/2π for two devices (for device-1, Fig.2a, diameter d=100 nm, length l=2.9 µm, for device-2,
Fig.2b, d=120 nm, l= 3.1 µm). c) Calculated dispersion as a function of VDC
. d) Lineplot at 29
MHz for device-1 (dashed line in Fig.2a). e) The plot of the Q as a function of VDC
f) The plot of the “amplitude” zreso
ampas a function of VDC
for device-1. (The red and blue traces
in Fig.2e, and in Fig.2f, show the data for negative and positive gate voltages).
Space Charge (C/m3)
FIG. 3: (color online) a) and b) Shows the result of a FEM based self-consistent solution of Poisson’s
equation giving the potential around a doubly clamped suspended nanowire device 100 nm in
diameter and 1.5 µm long. The separation of the NW and 300 nm thick SiO2is 100 nm. The
back plane of the nanowire is the gate held at -5V and the two terminals of the wire are grounded.
The maximum of the colorscale bar (red)is 0V and the minimum (blue) is -5 V. c) The space
distribution in the cross section of the nanowire for VDC
= -25V shows the gradient. The log-
colorscale below varies from 10 C/m3(blue) to 106C/m3(red). d) Plot of space charge density in
the vertical direction at VDC
= -25V through the middle of the wire. The asymmetry along the
vertical direction due the gate below the wire is clearly seen. e) The space distribution in the cross
section of the nanowire for VDC
= -5V shows the gradient. The log-colorscale below varies from
103C/m3(blue) to 105C/m3(red). f) The space distribution in the cross section of the nanowire
for Vg= 5V shows uniform space charge distribution as the electron density in the nanowire is
increased when the nanowire field effect transistor is turned on. The log-colorscale below varies
from 103C/m3(blue) to 105C/m3(red).
Gate Voltage (V)
-20 -510 25
Gate Voltage (V)
-40 -200 2040
Gate Voltage (V)
Gate Voltage (V)
0 20 40
) b) a
) d ) c
FIG. 4: (color online) a) and b) Shows mode mixing for device-3 and device-1 respectively (for
device-3, d=103 nm and l=3.1 µm). c) Calculated dispersion for mode mixing using asymmetry
in the wire. d) Zoomed-in view for device-1 (Fig.4b) which shows the mixing of the modes as a
function of VDC
. Minimum separation between the modes, indicated by the double-headed arrow,
is 0.25 ± 0.02 MHz.