A conduction–convection model for self-heating in piezoresistive microcantilever biosensors

Sensors and Actuators A 175 (2012) 19–27

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A conduction–convection model for self-heating in piezoresistive microcantilever biosensors Mohd. Zahid Ansari, Chongdu Cho ∗ Department of Mechanical Engineering, Inha University, 253 Yonghyun-dong, Nam-Ku, Incheon 402-751, Republic of Korea

a r t i c l e

i n f o

Article history: Received 9 May 2011 Received in revised form 8 December 2011 Accepted 8 December 2011 Available online 16 December 2011 Keywords: Thermal drifting Self-heating Piezoresistive microcantilever Conduction Convection Sensitivity

a b s t r a c t Thermal drifting caused by the self-heating in piezoresistive microcantilever biosensors is a major source of inaccuracy. To this use, the present study derives a simple and accurate conduction–convection model to predict the temperature distribution in p-doped piezoresistive microcantilevers because of self-heating. The model is applied to a 4-layer gold-coated piezoresistive silicon dioxide microcantilever biosensor with u-shaped silicon piezoresistor. The analytical results are compared to the numerical results. The effect of convective heat transfer coefficient on temperature profile is also studied. The comparison results show the analytical and numerical results are accurate within 4%. The sensitivity results showed that the resistance change produced by thermal drifting is about five-to-eight times that by the surface stress. Finally, the cantilever temperature profile is found to be strongly affected by the piezoresistor size and the convective heat transfer coefficient. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Microcantilever-based piezoresistive sensors have been successfully used in many novel applications, including thermal imaging [1], flow sensor [2], calorimetry and mass detection [3], virus detection [4], trinitrotoluene vapour sensor [5], force sensor [6], protein sensor [7], shear stress sensor [8], atomic force microscopy [9] and DNA sequencing [10]. The diversity in applications of piezoresistor microcantilever sensors underscores its universal sensor characteristics. These sensors convert the external stimulus like temperature and force into the resistance change of the cantilever, which is normally measured using balanced Wheatstone bridge. Though the optical read-out type microcantilever sensors have higher sensitivity and resolution than the piezoresistive ones, their use in certain applications, such as biosensors, becomes restrictive due to well-known issues like optical properties of the medium, frequent calibrations and external power supply for read-out. In such applications, the piezoresistive type microcantilevers become more attractive. The self-sensing characteristics of piezoresistive microcantilevers have numerous advantages over optical ones in being robust, compact, portable and low power devices. These advantages become more obvious when microcantilevers are used as chemical gas sensors, vapour sensors or biosensors.

∗ Corresponding author. Tel.: +82 32 860 7321; fax: +82 32 868 1716. E-mail address: [email protected] (C. Cho). 0924-4247/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2011.12.014

In general, microcantilever-type biosensors have a major shortcoming in form of thermal drifting. This problem is more associated with piezoresistive types because of their self-heating characteristics. The electrical current passing through the piezoresistor element of the microcantilever generates heat due to resistance heating. This increases the temperature of cantilever and results in severe thermal drifting. Thermal drifting is caused by the bimetallic bending action of the cantilever that arises due to different thermo-mechanical properties of the constituent layers of the cantilever. The bending produces strain in the cantilever and changes its electrical characteristics because of piezoresistivity. The other major source of thermal drifting is the variation in temperature-dependent properties like electrical resistance and piezoresistance of the piezoresistor. The temperature coefficient of resistance (TCR) and piezoresistance (TCP) can be positive or negative depending on the dopant type and its concentration in the piezoresistor. The present study assumed the piezoresistor is made of p-doped silicon with positive TCR and negative TCP. Therefore, the electrical resistance of the piezoresistor will increases with temperature, but the piezoresistance will decreases. Thaysen et al. [11] reported that the fractional resistance of AFM piezoresistor microcantilever increased by about 2% when the temperature is increased to 110 ◦ C. However, the self-heating characteristics have also been used in novel applications. Chui et al. [12] and Binnig et al. [13] showed use of self-heating and self-sensing characteristics of piezoresistive microcantilevers for ultra-high density AFM data storage. King [14] proposed heated atomic force microscope cantilevers for nanotopography measurements. King et al.

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Fig. 1. Piezoresistive microcantilever biosensor with u-shaped piezoresistor.

[15] showed thermal cantilevers have better characteristics than piezoresistive cantilevers in improving the sensitivity and resolution of AFM in topology measurements. More recently, Bashir, King and co-workers [16] successfully used self-heating characteristics of piezoresistive microcantilever to study thermal cell lysis. These applications use the electrical resistivity rather than the piezoresistivity of the cantilevers, and require accurate prediction of temperature generated due to self-heating. Most of the heat produced by self-heating in the cantilever is absorbed into the cantilever base by thermal conduction and dissipated to the ambient by convection. Heat losses by radiation are generally negligible. Since the cantilever base has much larger thermal mass than the cantilever, except for the immediate vicinity of the cantilever fixed-end, the average increase in base temperature is negligible. Due to different temperature boundary conditions at the cantilever base and the ambient temperature, the entire cantilever is not heated to the same uniform temperature. The cantilever section containing the piezoresistor is normally at a higher temperature than the rest. This is undesirable because temperature gradient along the cantilever length can affect the temperature-sensitive biological or bio-chemical interactions occurring on the cantilever surface. Ideally, there should be uniform and large temperature area on the cantilever surface to provide a controlled micro-environment for analyte–receptor binding reactions to occur. Hence, for proper operation of the biosensor the temperature and its distribution on the cantilever should be known accurately. The studies on temperatures generated in the piezoresistive microcantilevers have been mostly experimental and numerical. There is scarcity of analytical relations predicting the temperature distribution in such cantilevers. On analytical approach, Choudhury et al. [17] derived a convective heat transfer model for predicting self-heating in a piezoresistive cantilever under sinusoidal input voltage. Yang and Yin [18] derived a conduction model for piezoresistive microcantilever biosensor. Using the concept of effective thermal conductivity, Ansari and Cho [19] derived a conduction model for 4-layerd piezoresistive microcantilever and showed that silicon dioxide substrate cantilevers produce higher temperature than silicon substrate ones. More recently, Loui et al. [20] proposed a comprehensive analytical model for self-heating in piezoresistive microcantilever gas sensor coated with polymer film. They found that radiation effects are negligible compared to conduction and convection heat losses. The present work derives a simple conduction–convection model for predicting the temperature distribution in piezoresistive microcantilevers. The radiation losses are neglected. The model is also applicable to piezoresistive microcantilevers with short and

narrow piezoresistor. The model establishes the relation between applied voltage, piezoresistor size, convective heat transfer coefficient and the thermo-mechanical properties of the cantilever. The cantilever is a multi-layered structure made of silicon dioxide with a u-shaped silicon piezoresistor embedded. To verify the model, analytical and numerical studies are performed on a gold-coated piezoresistive microcantilever commonly used in biosensor applications. The analytical results are compared against the numerical results obtained from finite element analysis software ANSYS. After verification, the thermal drifting phenomena and the temperature distribution for different convective heat transfer coefficients is investigated. Finally, the sensitivity of the piezoresistive microcantilever biosensor to thermal drifting and to the surface stress change due to analyte–receptor binding is studied.

2. Theory and modelling In microcantilever biosensors, the surface stress variation on the cantilever surface due to analyte–receptor binding produces transverse deflection. In piezoresistive microcantilever biosensors, the deflection induces stress in the piezoresistor element of the cantilever and changes its electrical resistance. By measuring the resistance change, the analyte and its concentration is determined. The higher the resistance change, the greater the sensitivity of the biosensor will be. Fig. 1 shows the schematic of piezoresistive microcantilever biosensor containing u-shaped piezoresistor. The piezoresistor element is fully encapsulated in the substrate, and normally a bias voltage between 5 and 10 V is applied across it. A thin film of gold is normally applied at top surface of the cantilever to help in formation of the monolayer of receptor molecules during functionalisation of the sensor. Nevertheless, the film is also a major contributor to thermal drifting in the sensor because of bimetallic effect. The analytical model describing the temperature distribution in the piezoresistive microcantilever is derived in two steps. In the first step, the heat diffusion equation is used to derive the heat conduction model to predict the temperature distribution in the cantilever due to self-heating. This model considers heat conduction as the only mode of heat transfer and neglects convection effects. In the second step, a conduction–convection equation is used to derive the final form of the analytical model predicting the temperature in the cantilever. This is achieved by also including the convective heat transfer loss into the conduction model derived in the first step. The most general form of differential-volume thermal energy conservation equation relating heat conduction, convection and

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where Apzr and lpzr are the cross-sectional area and the electric current-carrying length of the u-shaped piezoresistor and V is volume of the cantilever section containing the piezoresistor. keff is theeffective thermal conductivity of the cantilever given as keff = mi ni ki , where i is the number of cantilever layer materials, and m and n are the layer-to-cantilever ratios for width and thickness, respectively, for each cantilever material. It should be noted that if the piezoresistor material is same as the cantilever substrate material, the effective thermal conductivity will be same as that of the substrate material. Due to its very large thermal capacity, the increase in average temperature of the cantilever base can be neglected, and the adiabatic boundary condition can be applied to the cantilever fixed-end (Fig. 2). Therefore, by applying the initial boundary condition T(x = 0) = Tb and the adiabatic heat condition dT/dx (x = 0) = 0, Eq. (3) can be solved, similar in [19], to obtain the temperature distribution in the piezoresistor section of the cantilever as: Fig. 2. Simplified cantilever model for deriving analytical relations.

Tg (x) = Tb cosh

radiation to heat storage/dissipation and energy conversion can be expressed as [21]:

∇ · q ≡ ∇ · (−k∇ T + cp Tu + qr ) = −

∂cp T  + s˙ i ∂t

(1)

i

where q is heat flux, k is thermal conductivity, T is temperature,  is mass density, Cp is heat capacity, u is fluid flow speed, qr is radiation heat flux and t is time. The last term in above equation includes the energy conversions due to change in chemical- and physical-bond energy, electromagnetic and mechanical characteristics. Self-heating or electrical-resistance heating is an energy dissipation phenomenon that converts irreversibly the electrical energy into thermal energy. Most of the thermal energy is generated due to the loss of kinetic energy of the current carrying electrons by collisions among themselves and with the lattice atoms. The volumetric rate of self-heating can be given as S˙ = ϕ2 /e L2 , where ϕ is applied electrical potential, e is electrical resistivity and L is length of the current carrying element. Electrical resistivity is a function of temperature, given as e (T) = e,0 (1 + (T − T0 )), where  is TCR and e,0 is the resistivity at the reference temperature T0 . Fig. 2 presents the simplified form of the piezoresistive microcantilever used to derive the analytical relations. Owing to its very small size and therefore very high volumetric rate of heat energy generation, the heat diffusion within the microcantilever structure via conduction will dominate the heat transfer via convection or radiation. Therefore, by neglecting the heat loss due to convection and radiation and considering self-heating as the only energy conversion, the steady-state form of Eq. (1) can be given as:

∇ · k∇ T = s˙

(2)

This equation is commonly known as heat diffusion equation with heat generation. Since the cantilever is very thin, the heat diffusion in thickness direction will be much faster than that along the other dimensions. Thus, we can assume the entire cantilever volume containing the piezoresistor is generating heat energy. Assuming thermal conductivity is constant and the heat conduction is along the cantilever length, Eq. (2) can be expressed as: ϕ2 Apzr 1 d2 T = 2 k dx eff lpzr V e (T )

(3)



C1 x +

1 C2 x2 , x ≤ l 2

(4)

where C1 = (ϕ2 Apzr )/(e,0 keff lpzr V) and C2 = ((1 − T0 )ϕ2 Apzr )/(e,0 keff lpzr V). Eq. (4) is the expression for predicting the temperature distribution in the piezoresistor section of the cantilever due to self-heating. It neglects the convection and radiation losses. The expression includes the temperature-dependency of the resistance by including TCR. If convection losses are negligible, Eq. (4) is sufficient to predict the temperature distribution in the cantilever. In the second step of derivation, the convective heat loss term is incorporated into Eq. (4). The heat loss due to convection depends on the surface area exposed to ambient, the temperature difference between cantilever and ambient air or liquid environment and the convective heat transfer coefficient. Assuming thermal conductivity is constant and the heat conduction is along the cantilever length (i.e., one-dimensional), the conduction–convection equation without heat generation can be expressed as [21]: keff Ac

d2 T − hP(T − Tf ) = 0 dx2

(5)

where Tf and h are the temperature and the convective heat transfer coefficient of the ambient fluid, and Ac and P are the cross-sectional area and the perimeter of the cantilever. By applying the initial boundary condition on Tg (x) as Tg (x = 0) = Tb and the insulated tip condition dT/dx (x = l) = 0, Eq. (5) can be solved to give the conduction–convection equation for predicting temperature distribution in the piezoresistor section of the cantilever as: Tl (x) = Tf + [Tg (x) − Tf ]

 ˇl =

hP keff

cosh ˇl (l − x) , cosh ˇl l

0
 Ac

(6)

And the maximum temperature can be given as: Tl (x = l) = Tf + [Tg (l) − Tf ]

1 cosh ˇl l

(7)

Eq. (6) is the expression for temperature distribution in the piezoresistor section of the cantilever subjected to both conduction and convections heat losses. It can be seen from the equation that if convective heat losses are negligible (i.e., h = 0), Eq. (6) will reduce to Eq. (4) which predicts conduction temperatures only. If the piezoresistor length is less than the cantilever length, Eq. (6) should be modified (Fig. 2). Using a similar approach described above, the expression for temperature distribution in the cantilever

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Table 1 The geometric properties of the piezoresistive microcantilever (␮m). Length of cantilever, L Width of cantilever, W Length of piezoresistor, l Width of piezoresistor, b Thickness of gold film, t1 Thickness of insulation, t2 Thickness of piezoresistor, t3 Thickness of substrate, t4

Table 2 Thermo-electric properties of the piezoresistive microcantilever (␮MKS unit).

200 100 45, 90, 180 15, 30, 45 0.05 0.1 0.1 0.4

Parameter

Au

SiO2

Si

Thermal conductivity (×106 pW/␮m ◦ C) Thermal expansion coefficient (×10−6 /◦ C) Heat capacity (×1012 pJ/kg ◦ C) Elastic modulus (×103 MPa) Poisson’s ratio Mass density (×10−15 kg/␮m3 ) Electrical resistivity (T ␮m) Temperature coefficient of resistivity (◦ C)

317 14.2 129 80 0.42 19.3 – –

1.38 0.5 745 70 0.20 2.22 – –

150 2.8 712 160 0.23 2.32 10−9 10−4

section without the piezoresistor can be given as:



TL (x) = Tf + Tl (l) − Tf

ˇL =

cosh ˇL (L − x) cosh ˇL (L − l)

,

hP kAc

l
(8)

And the tip temperature can be given as: TL (x = L) = Tf + [Tl (l) − Tf ]

1 cosh ˇL (L − l)

(9)

Here, k is the thermal conductivity of the cantilever without the piezoresistor and can be calculated similar to the effective thermal conductivity given before. Moreover, since the piezoresistor does not extend into this region, conductivity depends only on the substrate properties (Fig. 2). Eqs. (8) and (9) are useful for predicting temperatures in piezoresistive microcantilevers when the length of piezoresistor is less than the cantilever (i.e., l < L). However, if the length of cantilever and piezoresistor are similar (i.e., l ≈ L), Eqs. (6) and (7) are sufficient to predict the temperature profile. The above relations were derived under the assumptions that adiabatic condition exists at the base and the average temperature of the base is unchanged. It further assumed that there is no thermal contact resistance between the layers and there is perfect bonding between the layers. 3. Numerical analysis To validate the conduction–convection model derived in this study, FE analysis using a commercial software ANSYS Multiphysics v.12 was performed. The boundary conditions in the numerical analysis were identical to those used in deriving the analytical model. To save the computational time, only the cantilever was modelled and its temperature and deflection results were obtained. Since the cantilever is extremely thin structure, heat loss from the cantilever edges was neglected. Thus, only the top and bottom surfaces of the cantilever were exposed to convection heat transfer. The total size of cantilever was fixed to 200 ␮m × 100 ␮m × 0.65 ␮m. The length of piezoresistor was changed to 45, 90 and 180 ␮m, and the width to 15, 30 and 45 ␮m. The piezoresistor thickness was kept constant at 0.1 ␮m. The piezoresistor element was assumed to be made of p-doped silicon with boron doping concentration about 1 × 1019 /cm−3 . For this dopant type and concentration, the piezoresistor has a positive TCR value of about 1 × 10−4 /◦ C and a negative TCP value of about −27 × 10−4 /◦ C [22,23]. The applied voltage was increased from 5 to 10 V. The FE model was meshed by 3-D coupled field 8-node scalar SOLID5 elements. These elements have the capability to perform coupled problems involving mechanical, thermal, electrical and piezoresistive effects. About 100,000 elements were used in each analysis. Mesh convergence was performed for a numbers of cases to confirm the validity of numerical results. All the models were solved under steady state conditions. Table 1 lists the geometric properties of the 4-layer piezoresistive microcantilever model shown in Fig. 1.

The numerical analyses involved thermo-electric, thermal and surface-stress characterisation of the microcantilever with different piezoresistor sizes. The cantilever biosensor was assumed to be operated in air with h = 200 W/m2 ◦ C [24]. In the thermo-electric analysis, the cantilever was subjected to different bias voltages. The temperature of the cantilever base and the ambient air was assumed 25 ◦ C and the applied voltages were increased from 5 to 10 V. In the thermal analysis, the cantilever and the base were subjected to the same temperature of 25 ◦ C. No voltages were applied in this case. The third and final analysis investigated the effect of surface stress on maximum deflection produced in the cantilevers. The cantilever was subjected to a constant surface stress of 1 N/m on its top surface. The surface stress was modelled as in-plane tensile force applied on the top cantilever surface. No voltage and temperature load conditions were applied in this case. Typical material properties of the cantilever are listed in Table 2. 4. Results and discussion In order to ascertain the accuracy the thermal models presented in this work, the analytical results are compared against the experimental infrared microscopy results of Lee and King [25] on microcantilever hotplates. Except for the presence of a rectangular hole at the cantilever base and an extended base, the cantilever model B in their work is very similar to the one studied here. The effect of extended base on the temperatures produced in the cantilever can be approximated by now using the extended base temperature as the base temperature in our model. In their work, the silicon hotplate was heated by cantilever power 10 mW and produced a maximum temperature of about 245 ◦ C. The total cantilever size was about 250 ␮m × 50 ␮m × 1 ␮m. The presence of rectangular hole at the cantilever base however reduced the actual cantilever volume generating the heat. Adopting the extended base temperature value from [25] as Tb = 60 ◦ C and assuming k = 150 W/m ◦ C, the conduction-only temperature in the cantilever can be calculated using Eq. (4) as 239.59 ◦ C. And, under the assumption the hotplate was operated in air with h = 200 W/m2 ◦ C and Tf = 25 ◦ C, the conduction–convection temperature, calculated using (7), can be given as 222.90 ◦ C. The analytical value of 222.90 ◦ C is comparable to the experimental value of 245 ◦ C, with a deviation of about 9%. The deviation can be attributed to the approximation made regarding the cantilever model and the assumptions about its thermal conductivity and convective coefficient which can be different in the actual experiment than the ones used in our analytical equations. Thus, we can conclude that the thermal models presented in this work are sufficiently accurate and can be used to predict the self-heating temperatures in u-shaped element microcantilevers. Fig. 3 presents the comparison between maximum temperature values obtained from analytical and numerical results for h = 200 W/m2 ◦ C and Tb = Tf = 25 ◦ C. The analytical results were calculated using Eq. (7) and the numerical results were obtained from FEA. The analytical results are shown by solid lines. The heat is generated in the cantilever section containing the piezoresistor

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Fig. 3. Comparison between analytical and numerical results for maximum temperatures in the piezoresistive cantilevers. Symbols represent numerical data.

and diffuses to the rest of the cantilever to achieve the thermal equilibrium. The convective heat losses occur from the top and bottom surfaces of the cantilever. The maximum temperature values obtained from FEA for piezoresistor lengths 45, 90 and 180 ␮m are 36.98, 47.86 and 58.01 ◦ C, and 38.06, 48.79 and 58.12 ◦ C and 38.07, 47.90 and 55.74 ◦ C, respectively, for widths 15, 30 and 45 ␮m. The comparison analysis shows a good correlation between the analytical and numerical results and therefore indicates the accuracy of the analytical model derived in this study. The maximum absolute deviation between the two results is about 4%, 3% and 2% for the piezoresistor lengths 45, 90 and 180 ␮m, respectively. The decrease in deviation with the increase in piezoresistor length suggests the model is more accurate for predicting temperatures in the microcantilevers with long piezoresistor. The deviations observed in Fig. 3 can be attributed to the assumptions made in deriving equation (6). The main factor behind the deviation is the assumption that heat flow is one-dimensional. In deriving equation (6), we assumed the entire cantilever volume containing the piezoresistor generates the heat energy and the heat flow is along the cantilever length only, i.e., one-dimensional heat transfer. However, in the numerical analyses, only the piezoresistor is generating the heat and the heat flow is three-dimensional, because we used full threedimensional cantilever model. Another source for deviation can be the use of effective thermal conductivity in Eq. (6) which assumed the piezoresistor section of the cantilever has a single thermal

conductivity. In the numerical analyses, however, the cantilever is a layered structure of materials having different thermal conductivities. In addition, the analytical model did not considered the particular shape of the piezoresistor. The shape can be critical in heat generation and heat diffusion in the cantilever. In contrast, the numerical analysis used a perfect u-shaped piezoresistor. There can be non-uniform heat generation and heat flow in the cantilever. Nevertheless, the low deviation between the analytical and numerical result provides a simple and useful analytical relation for estimating the self-heating in a complicated piezoresistive Table 3 Comparison between analytical and numerical results for tip temperatures at ϕ = 10 V. Piezoresistor size (␮m)

Tip temperature (◦ C) Eq. (9)

FEA

|Error (%)|

l = 45

b = 15 b = 30 b = 45

33.34 41.07 50.49

33.90 40.64 46.10

1.65 1.05 9.52

l = 90

b = 15 b = 30 b = 45

36.67 46.51 54.89

36.28 49.99 52.01

1.07 6.96 5.54

l = 180

b = 15 b = 30 b = 45

37.44 47.86 55.46

38.07 47.89 55.74

1.65 0.06 0.50

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Fig. 4. Temperature distribution in different (l, b) piezoresistor size cantilevers.

microcantilever. Table 3 shows the comparison between tip temperatures, calculated using Eq. (9), and the numerical analysis for different (l, b) values at h = 200 W/m2 ◦ C and ϕ = 10 V. It is obvious from the table that Eq. (9) is in good accord with the numerical results in predicting the tip temperatures. Fig. 4 presents the temperature distribution in the piezoresistive microcantilevers for h = 200 W/m2 ◦ C and ϕ = 10 V. It can be seen in the figure that the maximum temperatures are produced around the tip of the piezoresistor and decrease towards the tip of the cantilever. The decrease is more pronounced in case of cantilevers with short piezoresistor. In general, the cantilever tip temperature is lower than the maximum temperature. This observation can be attributed to the relatively long heat flow path from the piezoresistor tip to the cantilever free-end (Fig. 2). In addition, the long path exposes large surface area of the cantilever to convection heat loss. These effects result in reducing the temperature at the free-end of the cantilevers. It can also be observed in Fig. 4 that the temperature distribution, which is about one-dimensional in case of long piezoresistors, transforms into two-dimensional for short piezoresistors, especially in case of cantilevers with very short and wide piezoresistor. This behaviour partly explains the deviations observed between analytical and numerical results presented in Fig. 3. In such cases, though the maximum temperature can still be predicted from Eq. (6), the

equation loses its accuracy in predicting the temperature distribution in the cantilever. Thus, the assumption of one-dimensional heat flow in Eq. (6) becomes inadequate and needs further improvement by including two-dimensional thermal model, which is our future works. Fig. 5 shows the temperature distribution in piezoresistive microcantilevers for different values of convective heat transfer coefficients at ϕ = 10 V. The present study assumed the microcantilever biosensor is operated in air. However, if the cantilever is used in liquid the value of convective heat transfer coefficient becomes much higher. In general, the convective heat transfer coefficient for water is more than 20 times of air. It is obvious from Fig. 5 that the convective heat transfer coefficient has significant effect on the maximum temperature produced and its distribution in the cantilever. For h = 0, the maximum temperature is 62.34 ◦ C, whereas, it is 36.45 ◦ C for h = 2500 W/m2 ◦ C. This observation is understandable because high convective heat transfer coefficients will faster carry the heat away from the cantilever. The higher the convective heat transfer coefficient, the lower the cantilever temperature will be. Another significant feature of Fig. 5 is the change in temperature distribution pattern in the cantilevers with increase in convective heat transfer coefficient. The non-uniformity in temperature distribution increases with convective heat transfer coefficient. This effect is more pronounced in wide and short piezoresistors (Fig. 4).

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Fig. 5. The effect of convective heat transfer coefficient on temperature distribution in (180,45) cantilever.

Thus, we may conclude that the temperature distribution in the cantilever strongly depends of the convective heat transfer coefficient and the piezoresistor size. In general, piezoresistive sensors are highly temperature sensitive and require fine temperature-compensation mechanism for satisfactory performance. In case of biosensors, balanced Wheatstone bridge combined with differential read-out technique is commonly used to negate the adverse effects of temperature and other sources of drift on sensitivity and resolution of the sensor. The sensitivity of piezoresistive microcantilever is generally expressed in terms of fractional resistance change caused by the external stimulus. The expressions for temperature and deflection sensitivities for a 2-layer piezoresistive microcantilever can be given as [7]: Rtemp = T, R

Rdef R

=

3 l Ed z 2L2

(10)

1 × 10−4 ◦ C, 72 × 10−11 Pa−1 and −27 × 10−4 /◦ C, respectively. The relatively higher value of ˇ suggests the temperature increase is more detrimental to piezoresistivity than to electrical resistivity. Further, the negative coefficient of ˇ indicates the piezoresistivity decreases with increase in temperature. Fig. 6 presents the total sensitivity of the microcantilevers with different piezoresistor sizes for h = 200 W/m2 ◦ C and ϕ = 10 V. The relative contributions of TCR, surface stress and bimetallic effects to the total sensitivity are also indicated. The deflection and temperature values were adopted from the numerical results. The temperature values were averaged over the piezoresistor to achieve a uniform temperature in the whole piezoresistor. The TCR sensitivity and the bimetallic sensitivity are found to be of the same order, an observation similar to [7]. However, Loui et al. [20] found that TCR sensitivity was much higher than the bimetallic sensitivity of their cantilevers. This apparent discrepancy in the

where l is longitudinal coefficient of piezoresistivity, E is elastic modulus of piezoresistor, d is the distance between piezoresistor axis and the neutral axis of the cantilever, z is cantilever deflection and L is cantilever length. This equation can be used to calculate the sensitivity of the cantilever to both thermal- and surface stressinduced deflections. The total sensitivity of a piezoresistive microcantilever biosensor can be expressed as the sum of temperature sensitivity, bimetallic deflection sensitivity and surface-stress induced deflection sensitivity. Eq. (10) can also be applied to multi-layer cantilever structure because for a given deflection the bending stress induced at any cantilever cross-section depends on its distance from the neutral axis. Thus, assuming the deflections are in the same downward direction and TCP (ˇ) is also influential, Eq. (10) can be modified to give the total sensitivity as: R 3 l (1 + ˇT )Ed = T + (zT + zss ) R 2L2

(11)

where zT and zss are deflections due to bimetallic effect and surface-stress change, respectively. The typical values of , l and ˇ for the p-doped silicon piezoresistor used in this study are

Fig. 6. Total resistance change in piezoresistive microcantilever biosensor for different (l, b) piezoresistor size.

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Table 4 S/N ratios for different piezoresistive microcantilevers. Size (␮m)

S/N ratio

(45,15) (45,30) (45,45) (90,15) (90,30) (90,45) (180,15) (180,30) (180,45)

0.188 0.158 0.140 0.202 0.164 0.142 0.193 0.160 0.138

relative contribution of bimetallic and TCR sensitivities to the total sensitivity can be explained by the different geometric and electromechanical properties of the cantilever and the piezoresistor used in their studies. Thus, we may conclude that the dominating factor causing thermal drifting in piezoresistive microcantilevers entirely depends on the particular cantilever design and the level of piezoresistor doping. The sensitivity of a piezoresistive microcantilever biosensor is defined by the resistance change produced entirely by the analyte–receptor interaction. Other sources of resistance change are generally considered noise. It is obvious in Fig. 6 that temperature sensitivity of the piezoresistive microcantilever biosensor, manifested in form of TCR and bimetallic effect, is much higher than the surface stress sensitivity. In other words, the cantilever deflections caused by the bimetallic effect are higher than those caused by the surface-stress variations. This suggests a low signalto-noise ratio for the cantilever, and the cantilever is more sensitive to thermal drifting than to surface stress effects. The signal-to-noise ratio (S/N) is defined here as the ratio of the resistance change caused by the surface stress to the total resistance change. Table 4 lists the S/N ratios of microcantilevers of different piezoresistor sizes for h = 200 W/m2 ◦ C and ϕ = 10 V. The values range from 0.138 for (180,45) to 0.202 for (90,15). The S/N values are highest for b = 15 ␮m and decrease with further increase in width. The piezoresistor size (90,15) is the best choice because it shows the highest S/N value. It should be noted that the practical values of the surface stress induced by analyte–receptor interactions are much lower than the representative value of 1 N/m used in this study. Typical surface stress values for antigen-antibody bindings are about 0.04 N/m [7] and 0.05 N/m [26]. Therefore, in practice, the S/N ratio will be much lower and will require sophisticated electronic filters for accurate measurements. In such a case, the choice of selecting piezoresistor size (90,15) becomes more apparent. The adverse effect of thermal drifting on piezoresistive microcantilever biosensors can be reduced by using appropriate techniques. The drifting effects can be reduced by applying low bias voltages. Low voltages produce less self-heating and result in less temperature increase. In addition, since piezoresistivity decreases with increase in temperature, low voltages will be helpful in producing relatively larger contribution of surface stress-induced resistance to the total resistance change. In other words, they will increase the S/N ratio of the biosensor. However, low applied voltages will also reduce the current flow through the piezoresistor and can make the measurements difficult. This can lead to a reduction in the resolution of the biosensor. The resolution can be improved by using suitable electronic amplifiers. The drifting effects can be controlled by using differential read-out technique with symmetrical Wheatstone bridge. 5. Conclusions The present study derived a simple and accurate model for predicting the temperature distribution in p-doped piezoresistive

microcantilever biosensors produced by the self-heating. The model included both conduction and convection modes of heat transfer. Numerical analyses were performed to assess the accuracy of the model. The analytical and numerical results were found to be accurate within 4%. The thermal structure of the cantilever was successfully modelled using the effective thermal conductivity. The model can also be used to accurately predict the tip temperature of the cantilevers with short piezoresistor element. The maximum and minimum temperature generated in the cantilevers for h = 200 W/m2 ◦ C are 58.12 ◦ C for (90,45) at ϕ = 10 V and 28 ◦ C for (45,15) at ϕ = 5 V. Results showed that the temperature and its distribution strongly depends on the convective heat transfer coefficient of the fluid. The temperature values decreased from 62.34 ◦ C for h = 0 to 36.45 ◦ C for h = 2500 W/m2 ◦ C. The increase in convective heat transfer coefficient transformed the temperature distribution from one-dimensional to two-dimensional. This is the main reason for the deviations between analytical and numerical results at high convective heat transfer coefficients. Investigations on the sensitivity of the biosensors showed that (90,15) is most suitable piezoresistor design for achieving highest signal-to-noise. The analytical model presented here can be especially useful in predicting temperatures in applications where piezoresistor microcantilevers are used as heaters and hotplates.

Acknowledgement This study was supported by Inha University.

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Biographies M.Z. Ansari received his B. Tech. degree in Mechanical Engineering from the Aligarh Muslim University, Aligarh, India in 2001, and MS and PhD degrees in Mechanical Engineering from Inha University, Incheon, Korea in 2006 and 2010, respectively. He is currently working as lecturer at the same institution. His research interests include BioMEMS, biomechanics and microcantilever biosensors. Chongdu Cho received the B.S. degree in Mechanical Engineering from the Seoul National University, Seoul, Korea in 1983 and M.S. degree in Mechanical Engineering from Korea Advanced Institute of Science and Technology, Daejeon, Korea in 1985. He received the Ph.D. degree in Mechanical Engineering and Applied Mechanics from the University of Michigan, Ann Arbor, MI, USA in 1991. He joined as assistant professor in Mechanical Engineering of Inha University, Incheon, Korea in 1992, and become full professor in 2002. His research interests include thermal stress analysis, structural analysis, mechanical modelling and mechanics of composite materials, smart materials mainly by using finite element methods.