APPLIED PHYSICS LETTERS 92, 184103 共2008兲
Silicon depletion layer actuators
J. H. T. Ransley, A. Aziz,a兲 C. Durkan, and A. A. Seshiab兲
Nanoscience Centre, University of Cambridge, 11 JJ Thomson Avenue, Cambridge
CB3 0FF, United Kingdom
共Received 14 March 2008; accepted 16 April 2008; published online 9 May 2008兲
The uncompensated donor or acceptor atoms present within the depletion layer of a diode can be
employed in an electrostatic actuator, which utilizes the force between opposing charges on either
side of the semiconductor junction. We describe the theory of this actuator and demonstrate its
application for the case of a diode on the top surface of a silicon cantilever. The Schottky diodes
fabricated on the top surface of cantilevers were used to drive them into resonance. As the actuator
driving voltages are varied, the amplitude of vibration of the cantilevers changes, which is in
agreement with the theoretical predictions. © 2008 American Institute of Physics.
关DOI: 10.1063/1.2920440兴
Semiconductor junctions in piezoelectric III-V and II-VI
semiconductors have been employed as actuators for a number of years1 and there has recently been renewed interest in
their application as actuators for nanomechanical resonators.2
Although silicon diodes are highly sensitive to both hydrostatic3 and anisotropic4,5 stresses 共leading to applications of
semiconductor junctions in stress sensors6–8兲, silicon does
not exhibit a significant piezoelectric effect and the use of
silicon diodes as actuators has not been reported to date.
However, diodes in silicon can be employed as actuators
although the physical mechanism by which the actuator operates is different from that of piezoelectric semiconductors.
Actuation results from the forces between the oppositely
charged regions in the depletion layer rather than the interaction of this electric field with the material in the insulating
carrier depleted region. In contrast to the conventional capacitive gap actuators widely employed in the microelectromechanical systems industry, such actuators are internal to
the structure itself. There has recently been a significant interest on the use of internal actuators based on dielectric
filled capacitors as a method to enhance the voltage-force
transduction factor.9–11. The use of semiconductor junctions
instead of these capacitors opens the possibility of fabricating higher quality resonators due to high crystallinity and
low number of defects in the silicon wafer and the absence of
additional acoustically mismatched materials in its construction. The actuator technology also has good compatibility
with integrated circuit processing. In this letter, we present
results obtained from using these actuators to drive atomic
force microscopy 共AFM兲 cantilevers which validate a model
we have previously proposed.12
The carrier depleted layer present in reverse-biased
Schottky diodes and p-n junctions acts as the dielectric
in widely employed variable capacitor or varactor devices.13.
In contrast to a conventional capacitor with a distinct insulating dielectric layer, the charge is stored over the whole
volume of the insulating region in the form of uncompensated donors or acceptors embedded within the lattice. For
the case of abrupt junctions, the electric field linearly varies
a兲
Also at Department of Materials Science and Metallurgy, University of
Cambridge, Cambridge, CB2 3QZ, United Kingdom.
Author to whom correspondence should be addressed. Electronic mail:
aas41@cam.ac.uk.
b兲
across the depletion region13 and, consequently, a linearly
varying force acts on the charged dopants within the depletion layer 关see Fig. 1共a兲兴. These forces produce a net compression of the depletion layer. This compression can be
modulated by the application of an ac voltage Vac with a dc
offset Vdc 共to maintain a reverse bias兲 across the junction. If
the diode is placed on the top surface of a cantilever, this
compression in the direction normal to the cantilever surface
produces an extension parallel to the surface via Poisson’s
ratio and results in the bending of the cantilever. We have
shown12 that if a cantilever is driven at resonance by a diode
actuator on its top surface, its amplitude of vibration is given
by
ares = 1.890
L2Q␯⬜关Nde␧共Vdc + Vbi兲兴1/2Vac
,
h 2E
共1兲
where L is the length of the cantilever, h is its thickness, ␯⬜
is Poisson’s ratio of silicon, E is Young’s Modulus of the
silicon, Nd is the dopant density, e is the electron charge, ␧ is
the dielectric constant of carrier depleted silicon, Vbi is the
turn on voltage of the diode, and Q is the quality factor of the
resonance.
Gold-silicon Schottky diodes were prepared on uncoated
␮Masch AFM cantilevers. Three cantilevers were studied,
cantilevers 1 and 2 being on the same substrate. Initially
chrome-gold counterelectrodes of several square millimeters
were evaporated onto the sample 共10 nm Cr, 30 nm Au兲
through a shadow mask. The cantilever’s native oxide layer
was then removed by a HF etch. The sample was rinsed
in water and rapidly transferred to a vacuum system for
the deposition of gold—ensuring that oxide regrowth was
minimal. 30 nm of gold was evaporated through a second
shadow mask—ensuring a small junction area compared
to that of the counter electrode. A gap between these two
sets of electrodes results in a significant resistance in series
with the diode 共see below兲. Electrical connections were
made to the gold by using silver loaded epoxy and gold wire.
Figure 1共b兲 shows a scanning electron micrograph of cantilevers 1 and 2. The dimensions and physical properties of
the three cantilevers used in this study are given in Table I
along with the other parameters used in the modeling of the
devices.
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92, 184103-1
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184103-2
Appl. Phys. Lett. 92, 184103 共2008兲
Ransley et al.
TABLE I. Properties of the cantilevers measured in this study. Cantilevers 1
and 2 were located on the same substrate; cantilever 3 was on a different
substrate. w is the width of the cantilever, L is its length, and h is its
thickness—these parameters were measured in a scanning electron microscope 共SEM兲, with the exception of the thickness of cantilever 3—which is
estimated from the resonant frequency. Note that the cantilever thickness
was found to significantly vary along its length. Nd is its estimated dopant
density 共obtained by fitting to the observed data and within the manufacturer’s specified range of 共1 – 5兲 ⫻ 1017 cm−3兲, Q is its quality factor 共obtained
from the response curves in Fig. 2兲, f res is its measured resonant frequency,
and f pred is the predicted resonant frequency accounting for both the mass of
the deposited gold 共Ref. 14兲 共the gold was 30 nm thick with a density of
19 300 kg m−3兲 and the mass of the tip 共Ref. 15兲 共dimensions obtained from
SEM兲. All three cantilevers were fabricated from 共001兲 silicon wafers with
edges along the 具110典 directions so the silicon has a Young’s modulus,
E = 169 GPa and a 关001兴 to 关110兴 Poisson’s ratio of ␯⬜ = 0.36. The dielectric
constant of carrier depleted silicon is ␧ = 1.1⫻ 10−10 F m−1 and its density is
␳ = 2329 kg m−3.
Property
w / ␮m
L / ␮m
h / ␮m
Na / 1017 cm−3
Q
f res / kHz
f pred / kHz
FIG. 1. 共Color online兲 共a兲 Diagram showing the electric field and charges
present at the gold-silicon Schottky barrier fabricated on the surface of a
cantilever. The forces on the charges produced by the electric field result in
a net compression of the depletion layer. As a result of Poisson’s ratio of the
silicon, this compression in the z direction produces extension in the direction parallel to the surface of the cantilever and bending of the cantilever as
shown in the figure. 共b兲 Scanning electron micrograph of cantilevers 1 and
2. On the right hand side of the image, the edge of the epoxy contact can be
seen.
Figure 2 shows the current-voltage characteristic of the
devices. Both diodes have a turn on voltage of approximately
300 mV. The forward characteristics are limited by the series
resistances of 125 ⍀ 共cantilever 3兲–250 ⍀ 共cantilevers 1 and
2兲. These values are reasonable considering the device layer
doping level of 共1 – 5兲 ⫻ 1017 cm−3, which leads to a resistance per square of 330– 865 ⍀ At biases between 0 and
2 V, the leakage resistance of the diodes is greater than 5 k⍀
and its capacitance is estimated to be of order 1 nF 共which
corresponds to an impedance of order 10 k⍀ at resonant frequencies of order 20 kHz兲. This means that both dc and ac
voltages will be primarily dropped over the diode provided
that the reverse bias across the device is less than 2 V.
The deflection of each cantilever with various applied dc
and ac voltages was measured by using an AFM optical
beam-deflection setup and a lock-in amplifier. The lock-in’s
internal oscillator generated the ac signal applied to the
Cr–Au counterelectrode, while a dc reverse bias was applied
to the gold on the junction itself. After the measurements
were completed, the measured deflection was calibrated by
bending the cantilever by a known amount after bringing it
Cantilever 1
Cantilever 2
Cantilever 3
34⫾ 1
285⫾ 1
1.9⫾ 0.1
1.5
101
24.85
28⫾ 2
35⫾ 1
335⫾ 1
1.9⫾ 0.1
1.5
101
21.15
20⫾ 1
35⫾ 1
250⫾ 1
1.4
2.0
98
24.84
¯
in contact with a surface mounted on a piezoelectric stage.
Figure 3 shows the response of the three devices as the reference frequency was swept from 5 to 30 kHz.
Figure 4 shows the variation of each cantilever’s vibration amplitude with applied ac bias, at a dc reverse bias of
1 V. Figure 5 shows how the vibration amplitude varies with
applied dc reverse bias, with a fixed ac bias of amplitude
85 mV. For this measurement, the ac voltage was held at a
fixed frequency as the dc bias was varied. The lock-in output
was measured with a data acquisition card. The dc and ac
dependence of the observed amplitude are in good agreement
with the expression in Eq. 共1兲 共shown as a thin line in the
figures兲. Note that it was not possible to measure the dopant
level in these samples in a straightforward manner and the
values were assumed based on fits to the data in Figs. 4 and
5 共these values are given in Table I兲. The dopant density was
the only free parameter used in the fitting of the data—all the
FIG. 2. 共Color online兲 Current-voltage characteristics of the diodes deposited on the cantilevers measured in this study. Note that cantilevers 1 and 2
were on the same substrate and, therefore, the same continuous diode was
used to drive them. The inset shows the detail of the measurement range
used in this study, as well as the turn on.
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184103-3
Appl. Phys. Lett. 92, 184103 共2008兲
Ransley et al.
FIG. 3. 共Color online兲 Frequency response of the cantilevers measured in
this study at a dc reverse bias of 1 V and an ac amplitude of 85 mV.
other parameters were directly measured. A log-log plot of
the data in Fig. 4 共not shown兲 gives gradients of 0.4 共cantilevers 1 and 2兲 to 0.6 共cantilever 3兲 in reasonable agreement
of the predicted power dependence of 0.5.
In summary, we have presented a model for the response
of clamped-free silicon cantilevers as they are driven into
resonance by a diode actuator. A good agreement between
this model and the response of devices fabricated from silicon AFM cantilevers has been obtained. Diode actuators
based on pn junctions offer the potential to drive high Q
FIG. 4. 共Color online兲 Response of the devices as the ac signal is changed at
a fixed dc reverse bias of 1 V. The solid lines show a fit to the data based on
the model given in Eq. 共1兲 by using the parameters given in Table I.
FIG. 5. 共Color online兲 Response of the devices as the dc signal is changed
at a fixed ac signal of 85 mV amplitude. The thin solid lines show a fit to the
data based on the model given in Eq. 共1兲 by using the parameters given in
Table I.
resonators in silicon without narrow gaps or the added complexity of dielectric layers.
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