AIP Advances 7, 035108 (2017); https://doi.org/10.1063/1.4977827 7, 035108
© 2017 Author(s).
In silico modeling and investigation of
self-heating effects in composite nano
cantilever biosensors with integrated
piezoresistors
Cite as: AIP Advances 7, 035108 (2017); https://doi.org/10.1063/1.4977827
Submitted: 25 October 2016 • Accepted: 16 February 2017 • Published Online: 07 March 2017
Ribu Mathew and A. Ravi Sankar
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AIP ADVANCES 7, 035108 (2017)
In silico modeling and investigation of self-heating effects
in composite nano cantilever biosensors with integrated
piezoresistors
Ribu Mathew
a
and A. Ravi Sankar
b
School of Electronics Engineering (SENSE), VIT Chennai, Chennai 600 127, India
(Received 25 October 2016; accepted 16 February 2017; published online 7 March 2017)
Over the years, piezoresistive nano cantilever sensors have been extensively investi-
gated for various biological sensing applications. Piezoresistive cantilever sensor is a
composite structure with different materials constituting its various layers. Design and
modeling of such sensors become challenging since their response is governed by the
interplay between their geometrical and constituent material parameters. Even though,
piezoresistive nano cantilever biosensors have several advantages, they suffer from a
limitation in the form of self-heating induced inaccuracy which is seldom considered
in design stages. Although, a few simplified mathematical models have been reported
which incorporate the self-heating effect, several assumptions made in the model-
ing stages result in inaccuracy in predicting sensor terminal response. In this paper,
we model and investigate the effect of self-heating on the thermo-electro-mechanical
response of piezoresistive cantilever sensors as a function of the relative geometries
of the piezoresistor and the cantilever platform. Finite element method (FEM) based
numerical computations are used to model the target-receptor interactions induced
surface stress response in steady state and maximize the electrical sensitivity to ther-
mal sensitivity ratio of the sensor. Simulation results show that the conduction mode
of heat transfer is the dominant heat transfer mechanism. Furthermore, the isolation
and immobilization layers play a critical role in determining the thermal sensitivity
of the sensor. It is found that the shorter and wider cantilever platforms are more
suitable to reduce self-heating induced inaccuracies. In addition, results depict that
the piezoresistor width plays a more dominant role in determining the thermal drift
induced inaccuracies compared to the piezoresistor length. It is found that for sur-
face stress sensors at large piezoresistor width, the electrical sensitivity to thermal
sensitivity ratio improves. © 2017 Author(s). All article content, except where oth-
erwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(
http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4977827]
I. INTRODUCTION
In the last decade, nano-electro-mechanical systems (NEMS) based cantilever platform sensors
have been utilized as investigation tools for in-situ explorations ranging from measurements at micro
gram (µg) mass
1
to space applications.
2
However, in recent times much focus has been on develop-
ing piezoresistive cantilever sensors for biological sensing applications. Compared to conventional
clinical diagnostic techniques like lateral flow assays (LFAs) and enzyme-linked immunosorbent
assays (ELISAs), cantilever biosensors not only have the advantage of lower footprint but also have
an edge due to their capability to perform real time and fast detections with lower detection limits.
3
In
contrast to other sensing techniques like optical,
4
piezoelectric,
5
capacitive,
6
piezoresistive readout
technique has several advantages in terms of label free detection, lower footprint, freedom of on
or off-chip signal processing circuitry, large dynamic range, independence of operational medium,
a
E-mail: ribumathew88@gmail.com
b
a.ravishan@gmail.com
2158-3226/2017/7(3)/035108/19 7, 035108-1 © Author(s) 2017
035108-2 R. Mathew and A. R. Sankar AIP Advances 7, 035108 (2017)
cost-effectiveness due to batch fabrication, etc. The cantilever sensor can be operated in either static
or dynamic mode (where in the former relative change in cantilever deflection is measured, whereas
in the later change in resonant frequency of the cantilever is gauged). Typical applications of sensing
using cantilevers in dynamic mode include detection of volatile organic compound (VOC),
7
DNA,
8
airborne nanoparticles
9
to cite a few. Although, dynamic mode of operation has numerous advantages,
its effectiveness curtails in liquid medium due to fluid damping effect induced reduction in sensitivity
and dependence of change in resonant frequency on the position of target-receptor interactions on
the cantilever. In static mode of operation, cantilever end point deflection due to the target-receptor
interactions induced differential stress on the opposite faces of the cantilever is measured. Typi-
cal applications of static mode operated piezoresistive nano cantilever platform biosensors include
detection of cancer tissues,
10
viruses,
11
cardiac disease markers
12
and DNA sequencing
13
to mention
a few.
Even though, piezoresistive nano cantilever sensors have numerous advantages, they suffer
from a major limitation in the form of thermal drift in their output characteristics. This thermal
drift in the sensor also leads to invalid detection in nano cantilever biosensors.
14
Thermal drift in
piezoresistive nano cantilever biosensors occur due to joule heating of the dc-excited piezoresistor.
Typically, the piezoresistor is placed near the central base region of the cantilever
15
which results
in a non-uniform temperature profile. Joule heating induced self-heating of piezoresistive cantilever
biosensors become significant due to (i) the lower thermal mass of the cantilever, and (ii) temper-
ature dependence of the constituent material properties of the sensor. Unlike piezoresistive inertia
sensors which have higher thermal mass,
16
the lower thermal mass of piezoresistive cantilever sen-
sors result in temperature induced cantilever deflection. This deflection is due to the difference in
the temperature coefficient of expansion (TCE) of the constituent materials. Other parameters that
contribute to the thermal drift are (i) temperature coefficient of resistance (TCR), and (ii) temper-
ature coefficient of piezoresistance (TCP) of the doped resistor. As a result of the aforementioned
factors, terminal characteristics of the sensor change even without the target-receptor interactions.
Thus, the self-heating phenomenon of cantilever platform induces inaccuracy in measurement and
thereby results in reliability issues. Therefore, to ensure reliable detection of target molecules by
piezoresistive cantilever based sensors, it becomes imperative to understand the self-heating induced
inaccuracies.
The magnitude of temperature and its spatial variation on the cantilever are a function of both
the internal and external factors. The internal factors include the material and geometrical parame-
ters of the piezoresistor and the cantilever, whereas the external factors include (i) the heat transfer
mechanism i.e. conduction and convection mode, (ii) external ambient temperature, and (iii) magni-
tude of dc-voltage supply. Treatise encompasses a few examples where researchers have investigated
the thermal drift in piezoresistive cantilever sensors through theoretical modeling
17–19
and experi-
mental studies.
20–25
Theoretical studies have primarily focused on the impact of the piezoresistor
dimensions and the external voltage supply on the thermal drift. Moreover, reported mathematical
models have not only neglected the influence of cantilever dimensions and the constituent layers
but also overlooked the interdependence of electrical, mechanical and thermal design parameters in
determining the performance of the sensors. Similarly, the reported experimental results have mainly
considered a fixed piezoresistor and cantilever sensor geometry to investigate its term inal charac-
teristics as a function of either dc-excitation voltage and/or operational ambient. Therefore, there is
a dearth of in-depth investigation which portraits the dependence of the magnitude of temperature
and its spatial profile as a function of the relative dimensions of the piezoresistor and the cantilever
platform.
In this paper, a systematic investigation is performed to understand the influence of relative
geometries of the piezoresistor and the cantilever on the thermo-electro-mechanical response of
piezoresistive cantilever biosensors. In the present study, we have considered a silicon dioxide (SiO
2
)
cantilever with a p-type single crystalline silicon (SCS) as the piezoresistor. The sensor is virtually fab-
ricated with computer aided design (CAD) multi-physics numerical simulation software IntelliSuite
r
to perform coupled thermal, electrical and mechanical investigation of the sensor response. The main
focus of the work includes (i) investigation of the thermal behavior of the sensor as a function of
relative geometries of the piezoresistor and the cantilever, and (ii) optimization of the sensitivity
035108-3 R. Mathew and A. R. Sankar AIP Advances 7, 035108 (2017)
ratio υ = (∆R/R
|
σ
s
)/(∆R/R
|
T
), where, ∆R/R|σ
s
and ∆R/R|
T
represents the relative change in the
nominal resistance of the piezoresistor due to surface stress (σ
s
) and temperature (T ) induced effects
respectively.
II. DEVICE DETAILS
The sensor considered in the present work consists of the following layers (from the bottom):
(i) a structural layer, (ii) a piezoresistor, (iii) an isolation layer, and (iv) an immobilization layer. A
top and a cross-sectional view of the piezoresistive nano cantilever sensor under investigation are
shown in Fig.
1. When exposed to target molecules, the target-receptor binding induces change in σ
s
of the cantilever surface, which results in cantilever deflection. This deflection is converted into an
equivalent electrical signal by the integrated piezoresistor.
In the present study, gold (Au) is considered as the immobilization surface, since it supports a
stable alkane-thiol based immobilization protocol,
26
typically used for the immobilization of anti-
bodies (receptors). Translation of mechanical deflection of the cantilever into an equivalent electrical
signal is a function of the structural and material parameters of the sensor. The structural parameters
include the cantilever shape, lateral dimensions and thickness of the constituent layers. In the case of
composite piezoresistive cantilevers, the relative distance between the mid-plane of the piezoresistor
(Z
R
) and neutral plane of the cantilever (Z
N
) plays a critical role in governing the electrical sensitivity.
To obtain maximum electrical sensitivity, the distance between the piezoresistor and neutral plane
should be more. More specifics on the structural parameters and their optimization can be found in.
27
Material parameters which determine the electrical sensitivity include the piezoresistor gauge factor
(G) and the Young’s modulus (E) of the structural layer material. For a fixed cantilever geometry,
electrical sensitivity depends on the ratio of the piezoresistor gauge factor (G) to the Young’s modulus
(E) of the structural layer i.e. G/E.
27
In the case of solid-state semiconductors, the combination of
doped single crystalline silicon (SCS) piezoresistor and SiO
2
structural layer provides the highest
G/E ratio. Therefore, in this work, we have chosen doped SCS as the piezoresistor and SiO
2
as the
structural layer. To accomplish insulation of the piezoresistor from external environment, the piezore-
sistor is protected with a thin isolation layer. Here, SiO
2
is chosen as the isolation layer material due
to its excellent electrical insulating properties and lower E.
FIG. 1. A top view (without the immobilization and isolation layers) and a cross-sectional view (across AA’) of the composite
piezoresistive cantilever sensor with a diffused U-shaped piezoresistor. The symbols L
C
and W
C
represent the cantilever length
and width respectively, whereas the symbols L
P
, W
P
, and W
S
depict the piezoresistor length, width and leg space respectively.
This graphic is not drawn up to the scale.
035108-4 R. Mathew and A. R. Sankar AIP Advances 7, 035108 (2017)
TABLE I. Dimensional details of the composite piezoresistive cantilever sensor.
Parameter Value
Cantilever length (L
C
) 200 µm
Cantilever width (W
C
) 100 µm
Piezoresistor length (L
P
) 60 µm
Piezoresistor width (W
P
) 35 µm
Piezoresistor leg space (W
S
) 30 µm
Thickness of structural SiO
2
layer 500 nm
Junction depth of the boron doped piezoresistor (t
P
) 100 nm
Thickness of isolation SiO
2
layer 100 nm
Thickness of immobilization Au layer 50 nm
The device is designed in (100) silicon-on-insulator (SOI) wafers with both the cantilever and the
piezoresistor length aligned along the <110> direction. Geometrical dimensions of the cantilever and
the piezoresistor are determined by (i) the mechanical stability, and (ii) the electrical sensitivity of the
sensor. Details of the parameters that influence the mechanical stability and the electrical sensitivity
are explained in detail in our previous work.
28
Design specifications of the sensor investigated in this
work include: (i) electrical sensitivity (∆R/R)/σ
s
(m/N) > 1 E-2, (ii) resonant frequency (f
0
) (Hz)
> 5 E3, (iii) spring constant (k
s
) (N/m): 100 E-3 < k
s
< 10, and (iv) measurand: surface stress (σ
s
)
(N/m) = 0-100 E-3. These specifications are typical for a piezoresistive cantilever biosensor used
for antigen-antibody detection applications. The initial device dimensions (mentioned in Table I) are
chosen by analytical models
27,29
to satisfy the aforementioned specifications.
III. THEORY AND MODELING
A. Thermal model of the sensor
In piezoresistive cantilever sensors, there are mainly three modes of heat dissipation: (i) conduc-
tion, (ii) convection and (iii) radiation. The generalized thermal energy conversion equation for such
a system is given by
19
∇.q ≡ ∇.(−k∇T + ρsTu + q
r
) (1)
where, the symbols q, k, T and s represents the heat flux, thermal conductivity, temperature and heat
capacity respectively. Similarly, the symbols ρ, u and q
r
represents the mass density, fluid flow speed
and radiation heat flux respectively. Among the three heat dissipation modes, the radiation loss from
a cantilever surface is negligible, since it contributes less than 1% to the total heat dissipation even
when the cantilever is heated more than 500 K.
The heat flux is generated by the dc-biased U-shaped piezoresistor integrated within the cantilever
stack. The volumetric rate of heat generated in the U-shaped piezoresistor is given as
Q =
V
b
2
ρ
e
(2L
p
+ W
s
)
2
(2)
where, the symbols V
b
and ρ
e
represents the dc-voltage and electrical resistivity of the material
respectively. The 1D conduction-convection model for piezoresistive cantilever sensor is given by
19
λ
eff
A
c
d
2
T
dx
2
− hP(T − T
0
) = 0 (3)
where, λ
eff
and x are the effective thermal conductivity of the cantilever stack and longitudinal
dimension of the cantilever. The symbols A
c
, P, h and T
0
represent the cross-sectional area, cantilever
perimeter, heat convection coefficient and ambient temperature respectively.
The 1D temperature profile of the cantilever section with the piezoresistor is given as
T
L
p
(x) = T
0
+ [T
g
(x) − T
0
]
cosh β
L
p
(L
p
− x)
cosh β
L
p
L
p
, 0 < x ≤ L
p
(4)