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JOURNAL OF APPLIED PHYSICS 99, 024305 共2006兲
Nanogap capacitors: Sensitivity to sample permittivity changes
C. Ionescu-Zanetti, J. T. Nevill, D. Di Carlo, K. H. Jeong, and L. P. Leea兲
Biomolecular Nanotechnology Center, Berkeley Sensor and Actuator Center, Department of Bioengineering,
University of California Berkeley, California, 94720
共Received 3 August 2005; accepted 5 December 2005; published online 23 January 2006兲
Sensors based on nanogap capacitance changes are being developed for genomic and proteomic
applications because they offer label-free detection on platforms amenable to high throughput
configurations. We compare impedance spectroscopy measurements with a rigorously characterized
model that predicts the impedance spectrum of these devices based on geometry. Sensitivity to
permittivity changes is also predicted by the model and compared to the measured values in
the frequency range from 1 Hz to 3 MHz. The lowest detection limit for the magnitude
of the impedance 共兩Z 兩兲 is in the region of 100– 0.2 MHz, and was measured to be a 1.7%
change in permittivity across different devices. © 2006 American Institute of Physics.
关DOI: 10.1063/1.2161818兴
I. INTRODUCTION
Dielectric spectroscopy, a type of impedance spectroscopy, measures the dielectric properties of a medium as a
function of frequency. It is based on the interaction of an
external field with the electric dipole moment of the sample,
and it is gaining importance as label-free detection tool for
biomolecular structure and binding events.1–6 Binding event
studies are often performed by immobilizing the recognition
element at the electrode surface, thereby amplifying the
signal from small analyte concentrations in the bulk
solution.2,7,8 In these cases the signal measured after the recognition event is a modification of the electrical double layer
capacitance at the interface.
Shrinking these systems to the nanoscale will provide
the important advantage of having the volume of the electrical double layers occupy a significant fraction of the dielectric sample volume, thus amplifying the effect of binding
events by reducing the contribution of the bulk solution
impedance.9–11 Such devices for nanoscale dielectric spectroscopy have been fabricated in our laboratory, and experiments suggest their ability to detect changes in DNA and
protein samples.12,13 These devices are fabricated using conventional
complementary
metal-oxide-semiconductor
共CMOS兲 integrated circuit 共IC兲 technology, which has numerous advantages, including large volume manufacturing
and easy integration with control/sensor circuitry. However,
due to the semiconductor materials used 共i.e., doped single
crystal silicon, polysilicon, and silicon dioxides兲, these
nanoscale dielectric sensors exhibit complex behavior that
requires attention. In order to extract biophysically relevant
dielectric parameters for the biomolecules of interest, the development of bottom up models that describe the dependence
of measured signals on sample permittivity is required. In
addition, modeling can aid in sensor geometry optimization
and determination of which biomolecular events are within
the sensor’s detection limits.
a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
0021-8979/2006/99共2兲/024305/5/$23.00
In this article, detailed models for a number of nanogap
capacitor geometries are presented, and their agreement with
impedance spectroscopy data is evaluated. The detection
limit is investigated using a technique for the modification of
inter-electrode permittivity. Precise changes in permittivity
of the sample region are introduced by timed etching of the
capacitor spacer and measured in order to determine the system’s sensitivity to changes in sample permittivity. We have
focused on a frequency range far below the relaxation frequencies of the dielectric materials used and we were concerned solely with the real part of the complex permittivity
of the materials used 关共␻兲兴. The models developed here can
be used to determine the sensitivity of nanogap capacitors to
dielectric changes of biomolecular materials present in the
sample region. Additionally, it is demonstrated that such devices could function as metrology tools for monitoring the
rate of removal/deposition of material in nanocavities, thus
aiding in fabrication accuracy. The sensitivity of the measured parameters 兩Z共␻兲兩共impedance magnitude兲 and ⌽共␻兲
共phase shift兲 to permittivity changes is measured and compared to model predictions. In conjunction with standard deviation of 兩Z共␻兲兩 and ⌽共␻兲 data over a number of devices
共3–6兲, sensitivity values can be used to determine detection
limits for such sensors. These validated models will be useful
to researchers using nanogap-based sensors, and will enable
the optimization of such devices as they are developed into
genomic or proteomic sensor arrays.
II. EXPERIMENT
Nanocavities are defined by a sacrificial layer 共i.e., silicon dioxide insulating spacer兲 between two conductive
doped silicon capacitor plates, as described previously.13
Briefly, a polysilicon upper electrode is patterned using standard photolithography on top of a doped silicon wafer that is
insulated by a nanoscale film of thermally grown silicon dioxide. This oxide layer is partially etched to undercut the
upper electrode and expose a nanoscale cavity. The right side
of Fig. 1共a兲 shows a cross section of a typical nanogap device. In order to maximize the nanocavity volume, a number
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J. Appl. Phys. 99, 024305 共2006兲
Ionescu-Zanetti et al.
FIG. 1. Device geometry and model description. 共a兲 A top-view optical
image of the polysilicon top electrode containing a contact pad and sensing
fingers illustrates the general device geometry 共left兲. The detailed morphology of the 10-␮m-wide fingers is shown in a cross-sectional scanning electron micrograph 共right兲. The gap size shown was ⬃300 nm for clarity, while
data were reported from devices with 91 nm gaps. 共b兲 Schematic representation of materials used in the device fabrication process 共not a physical
cross section兲. Dielectric measurements are taken using probes between the
top polysilicon electrode 共left兲 and the bottom conductive wafer 共right兲. The
measured space consists of a silicon oxide spacer as well as an under etched
nanocavity for sample introduction. 共c兲 A detailed electrical model of the
nanocapacitor system. Aside from permittivity of the sample region, the
response is also influenced by contact resistances, the capacitance of the
poly Si pad, and distributed resistance along the top electrode, which is
7 mm long and 300 nm high. Cable theory was applied by dividing the
sensing capacitor into n RC elements as shown. The cable models used n
= 100 RC elements.
of “comb” geometries were used, as well as a “serpentine”
geometry. The left side of Fig. 1共a兲 shows such a device with
10-␮m-wide, 7000-␮m-long “fingers.” Contact to the macroscopic world is made through probe contacts to gold pads
deposited both on the top poly Si and on the bottom doped Si
wafer. Gold pads are beneficial in reducing the contact resistance to the probe tips. A simplified schematic of the device
geometry elements is shown in Fig. 1共b兲.
The devices tested had four different geometries for the
top electrode, as depicted in Fig. 2共a兲. Device A has five
fingers each 10 m wide 共circle兲, devices B and C have five
fingers each 5 and 4 m wide 共square and triangle兲, and device D has a 10-m-wide serpentine geometry 共star兲.
The sacrificial oxide layer, which acts as a spacer, was
selectively etched out from under the upper electrode to create a cavity with one nanoscale dimension, and this was performed in three sequential 500 nm steps to alter the permittivity of the nanogap. Impedance spectra were recorded at
each point and compared to the model in order to determine
device sensitivity and detection limits. The modeling and
parameter optimization was accomplished using MATLAB
共Mathworks, MA兲 scripts. Measurements of impedance and
FIG. 2. 共Color online兲 Model predictions compared to data for various device geometries of unreleased devices. 共a兲 Depiction of the four different
device geometries. Three devices have a comb geometry with various finger
widths, from 10 ␮m 共circle兲 down to 5 ␮m 共square兲 and 4 ␮m 共triangle兲.
The fourth device has a serpentine geometry with a single 10-␮m-wide
finger 共star兲. 共b, d兲 Model predictions 共lines兲 for the magnitude of the impedance 共兩Z 兩兲 and phase shift 共⌽兲 are compared to data 共scatter兲 from the
same four devices. 共c, e兲 The percent difference between the predicted and
measured values for both 兩Z兩 and ⌽ are plotted to illustrate quality of fit
across the frequency spectrum.
phase shift were carried out with an Alpha-N high resolution
dielectric analyzer 共Novocontrol, Germany兲.
A number of quantities were measured independent of
the nanogap permittivity as follows: Parasitic capacitance
and inductance were measured for the probes/leads in absence of the device 共Cpar = 0.6 pF, Lpar = 1.5 H兲. Probe contact
resistance was averaged for a number of repeated probe contact experiments 共Rc1 = Rc2 = 0.6 ⍀兲, while the contact resistance between the Au pads to poly Si and to the doped silicon were measured using test structures 共RcSi = 61 ⍀,
RcWafer = 17 ⍀兲. For both the top poly Si fingers and the
bottom wafer, sheet resistance was measured by using
appropriate test structures 共␳sheet Poly = 17 ⍀ / square, ␳sheet Si
= 110 ⍀ / square兲.
III. MODEL
The model used to predict device behavior includes both
the nanoscale device geometry and the characteristics of the
interface to the macro world 共Fig. 1兲. The circuit elements
were either extracted from geometrical parameters, measured
individually using test structures, or optimized for best fit. At
the heart of the model is a cable theory 共also called transmission line theory兲 treatment of the resistive/capacitive elements along each finger. The equivalent circuit is presented
in Fig. 1共c兲. This model was utilized because of significant
resistance along the top poly Si capacitor plate 共␳sheet Si
= 110 ⍀ / square兲. Each finger was divided into N elements
consisting of a top access resistance 共Rt共i兲 = ␳sheetPolyx 共finger
length兲/共finger width兲/N兲, a bottom access resistance 共Rb共i兲兲
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024305-3
Ionescu-Zanetti et al.
and a nanogap capacitive element Cg共i兲 = Atop / dgapx⑀device / N,
where ⑀device is the average nanocapacitor permittivity 共see
below兲. Three parameters were optimized: ⑀SiO2, the dielectric constant of SiO2, Rleak the leak resistance between the
top and bottom plates, and the silicon dioxide etch rate. The
parameters were found to be in agreement with literature,
and the same values were used for all model results. Therefore, we perform a direct comparison of model predictions to
dielectric data across four different geometries.
We chose to output model results for 兩Z兩 and ⌽ data 共as
opposed to real and imaginary impedance兲 because these
quantities are measured directly and offer an unambiguous
relationship to the frequency dependent impedance of a system. Purely capacitive elements will exhibit an impedance
that decreases linearly 共on a log-log plot兲 with increasing
frequency and a phase angle of −90°, while the impedance of
a purely resistive element will have a slope of zero and a
phase angle of zero degrees.
IV. RESULTS AND DISCUSSION
The model was tested against the measured impedance
response of nanoscale capacitors fabricated by the method
described in the experiment section. In order to achieve a
high enough capacitance from the underetched region, a geometry exhibiting high perimeter to area ratios is required of
the top electrode surface. The generality of the model presented follows from the need for any CMOS IC compatible
nanocapacitive sensor to follow the design constraints outlined above. We chose to satisfy these constraints by using
comb finger and serpentine geometries, which have high perimeter to surface area ratios.
The functionality of the model was first tested and optimized by comparing predictions for the four different unreleased device geometries with spectroscopic data 共Fig. 2兲.
Impedance magnitude 共兩Z 兩兲 and phase shift 共⌽兲 were measured as a function of frequency from 1 to 3 ⫻ 106 Hz, with
the material inside the sensing region being filled with silicon dioxide 共no undercut兲. Model predictions for each device
共lines兲 are compared to 兩Z兩 and ⌽ data in Figs. 2共b兲 and 2共d兲
共scatter兲. As one moves across the frequency spectrum, the
total impedance is dominated by various elements of the circuit model. For most devices, the response is purely capacitive in the region from 10 Hz to 10 kHz. At lower frequencies, the high impedance leak starts to play a role, leading to
a slight departure from the capacitive behavior. At high frequency the sheet resistance along the polysilicon top electrode leads to a reduction in 兩Z兩 slope and changes in ⌽ from
−90° to around −60°. This effect is more important in the
serpentine geometry 共blue data兲, where the polysilicon resistance is higher along a single long finger. Finally, in the MHz
range, parasitic capacitance of the macro contacts starts to
dominate, moving back toward a capacitive response. The
quality of the fit is shown in Figs. 2共c兲 and 2共e兲, where the
percent difference between model predictions and data is
plotted.
Most model parameters were measured using test structures 共see Sec. II兲. However, two of the model parameters
were not measured independently, and had to be optimized.
J. Appl. Phys. 99, 024305 共2006兲
FIG. 3. Parameter optimization. Optimization of ⑀SiO2 and Rleak was performed simultaneously by averaging errors over all unreleased devices geometries. One-dimensional plots of the average model error as a function of
parameter value 共a, b兲 are shown for simplicity 共the second parameter was
fixed at its optimal value兲. The etch rate was optimized using fixed ⑀SiO2 and
Rleak from unreleased devices, and by minimizing error from partial release
experiments. The parameter values at the point of minimum error were
determined to be 4.25, 2.5⫻ 1010 ⍀, and 113 nm/ min for relative oxide
permittivity, leak resistance, and oxide etch rate, respectively.
They were the dielectric constant of the oxide 共⑀SiO2兲 and the
large leak resistance across the nanogap. The leak is most
likely due to small imperfections in the oxide layer, since it
is present even in unreleased devices. Both parameters were
set by minimizing the impedance magnitude and phase shift
mean error, which yields the best correlation between the
data and the model across all four device geometries tested
共see Fig. 3兲. Optimization was performed by varying both
parameters and minimizing over a two-dimensional parameter matrix. The optimized value of ⑀SiO2 was found to be
4.25⑀o 关Fig. 3共a兲兴, and used in all subsequent modeling. This
value agrees well with reported values for ⑀SiO2, which range
from 3.81⑀o to 5.0⑀o.14 In the same manner, the leak resistance was found to be Rleak = 26.6 G 关Fig. 3共b兲兴. Error plots
关Figs. 3共a兲 and 3共b兲兴 were presented along one dimension
only for clarity; the ⑀SiO2 plot was made for the optimal Rleak
value, and vice versa. All other model parameters were measured directly using test structures.
Determination of nanogap sensor sensitivity to changes
in the permittivity constant of the capacitor material is a
principal aim of this work. Permittivity changes were introduced by successive under etches of the SiO2 spacer that
resulted in the replacement of high dielectric material
共⑀SiO2 = 4.25⑀o兲 with lower permittivity air 共⑀air = ⑀o兲 over a
distance drel. The under etched length was determined by
timed etches and roughly confirmed using test structures.
The expected etch rate for the HF concentration used was
100 nm/ min.15 However, the partial release data are best fit
with the slightly different etch rate of 112 nm/ min 关Fig.
3共c兲兴. Etch rate was fit while all the other parameters were
fixed from the unreleased device model. Using this etch rate,
the average difference between the model and experiment is
again below 2% for the 5 ␮m comb finger devices tested
共Fig. 4兲.
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024305-4
Ionescu-Zanetti et al.
FIG. 4. 共Color online兲 Model predictions and device data in response to
varying permittivity. The impedance magnitude 共a兲 and phase shift 共c兲 are
plotted as a function of frequency for varying permittivity values 共release
amounts兲 for the 5 ␮m nanogap geometry. Model predictions 共lines兲 agree
with data 共scatter兲 within 1 standard deviation 共standard deviation not
shown兲. 关共b兲–共d兲兴 A plot of percent difference between the data and model
for 兩Z兩共b兲 and ⌽ 共d兲. Incremental permittivity changes of approximately
10% ⑀ were obtained by replacing part of the SiO2 spacer material with air
through successive etch steps.
The percentage change of measured parameters as a
function of ⑀ is plotted for three representative frequencies
for both 兩Z兩 and ⌽ 共Figs. 5共a兲 and 5共b兲兲. Average nanocapacitor permittivity was calculated by ⑀device = 共w − 2drel兲 ⑀SiO2
+ 2drelx⑀o, where w is the finger width. From the unreleased
device to the last etch step, the dielectric constant varied
from 4.25⑀o down to 2.2⑀o. Plots at different frequencies illustrate differences in sensitivity regimes and measurement
standard deviation across the frequency spectrum. In the low
frequency region, standard deviation is high due to measurement noise at very high system impedance. At high frequency, system response is dominated by parasitic capacitance and the dependence on ⑀device is greatly reduced.
Sensitivity to permittivity changes is a critical parameter in
evaluating the possibility of using nanogap capacitors in order to map changes in protein structure, which will result in
small changes in the permittivity of sample proteins. It
should be noted here that permittivities measured were in the
range of 2 – 4.25⑀o, similar to the permittivity of low hydra-
FIG. 5. Changes in measured quantities 兩Z兩共a兲 and ⌽ 共b兲 as a function of
relative permittivity inside the nanogap. Three representative frequencies
were chosen: a low frequency 共11 Hz兲, a mid range frequency 共1.2
⫻ 104 Hz兲, and a high frequency 共0.8⫻ 106 Hz兲 are presented. The data
illustrate the frequency dependence of sensitivity for such devices.
J. Appl. Phys. 99, 024305 共2006兲
FIG. 6. Sensitivity to changes in permittivity for four different device geometries: Device A with 5 ␮m fingers 共a兲, device B with 4 ␮m fingers 共b兲,
device C with 10 ␮m fingers 共c兲 and device D with a 10 ␮m serpentine 共d兲.
兩Z兩共•兲 and ⌽ 共䉱兲 are plotted for each geometry, as well as the predicted
sensitivity based on the model 共dashed and solid lines兲. The error bars represent one standard deviation and are based on the calculated sensitivity
resulting from the three partial releases described in the experiment section.
tion protein powders which range from 1 to 5⑀o.16–18 This
suggests that similar changes in permittivity could be measured between different dehydrated organic molecule
samples. The device sensitivity was defined as the percentage change in 兩Z兩 and ⌽ over the percentage change in ⑀. The
sensitivity results are plotted in Fig. 6 for all four device
geometries. The model was also used to predict device sensitivity, and these results are plotted alongside the measured
values in Fig. 6. Note that the phase is more sensitive in the
region where the impedance sensitivity decreases. At these
higher frequencies, the system is less dominated by the capacitance, and phase sensitivity increases with the increase in
overall system resistance.
In addition to sensitivity, detection limit is another important parameter needed for device characterization. Here,
the detection limit is defined as the smallest change in
sample permittivity that results in a detectable change in the
measured quantities 兩Z兩 and ⌽, therefore equaling the standard deviation of the measurement: ⑀dlimit = d / dXxstd共X兲,
where X represents the measured quantity. The lowest detection limit for 兩Z兩 is below 2% of ⑀ in the region from 102 to
105 Hz when standard deviation is taken from one device to
another. If the same device is used repeatedly, and standard
deviation defined as the difference between measurements of
the same device on different days, the standard deviation
value is reduced by about an order of magnitude. Consequently, detection limits are also decreased by the same factor, down to below 0.2% ⑀. For ⌽, the low detection limit
range is 2 ⫻ 104 – 106 Hz and measured to be 1.5% across
multiple devices.
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Ionescu-Zanetti et al.
V. CONCLUSION
We present detailed models for a variety of nanogap sensors based on device geometry parameters and measured material properties, as well as a very resistive short contributing
to low frequency response. Four different device geometries
were fabricated and tested in response to changes in the dielectric constant of the sample space.
For both the impedance magnitude 共兩Z 兩兲 and phase shift
共⌽兲, the models agree with experimental data for a variety of
geometries and dielectric constants with an average difference of below 2% over the frequency spectrum examined
共1 Hz– 3 MHz兲. The variability of permittivity sensitivity as
a function of frequency was also measured for nanogap capacitors and shown to behave as predicted by the model. The
results show that nanogap devices can be used to detect permittivity changes of below 0.2%, giving a lower limit for the
detectable changes in the dielectric constant of biomolecular
samples. The model developed is a valuable data interpretation tool, and will provide a basis for future optimization of
nanogap devices.
ACKNOWLEDGMENTS
The authors would like to acknowledge the help of
Daniele Malleo 共visiting research student from the University of Southampton School of Electronics and Computer
Science supported by the UK Interdisciplinary Research
Centre in Bio-Nanotechnology兲 in the form of code optimization and helpful discussion, and Hugh Crenshaw, Sudhanshu Gakhar, and Daniel Hartmann from GlaxoSmithKline for
important insights. This material is based upon work sup-
ported by the National Science Foundation, a Whitaker foundation fellowship, an NDSEG fellowship, as well as funding
from GlaxoSmithKline. Two of the authors 共C.I.-Z兲共J.T.N.兲
contributed equally.
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