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NEMS thermal switch operating based on thermal expansion of carbon nanotubes
Physica E 59 (2014) 210–217
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NEMS thermal switch operating based on thermal expansion of carbon nanotubes M. Rasekh, S.E. Khadem n, A. Toghraee Department of Mechanical Engineering, Tarbiat Modares University, PO Box 14115-177, Tehran, Iran
H I G H L I G H T S
Design and simulation of a carbon nanotube-based thermal switch is reported. Thermal expansibility of the carbon nanotube is used to design a thermal switch. Electrostatic load is used to adjust the thermal switch trigger level. Temperature rising causes the pull-in instability and triggers the nano-switch.
art ic l e i nf o
a b s t r a c t
Article history: Received 27 April 2013 Received in revised form 28 December 2013 Accepted 7 January 2014 Available online 15 January 2014
In this paper, design and simulation of carbon nanotube-based thermal switches is reported. A carbon nanotube placed over a ground electrode represents the switch. The response of the nanoswitch based on beam theory is studied. When a nanotube has fixed–fixed boundary conditions, rise in temperature and thermal expansion of the nanotube can cause compressive axial load. This axial load will result in change of the pull-in voltage. Considering this fact, a thermal switch is designed for specific ambient temperature range and limited temperature rise range. When the temperature rises, the nanotube deflects more and approaches its pull-in instability. If temperature exceeds the threshold, pull-in occurs and the switch is triggered. Applying different voltages can provide different temperature thresholds. Utilizing this feature, the corresponding adjusting voltages required for actuating the switch by different rises in temperature are obtained. & 2014 Elsevier B.V. All rights reserved.
Keywords: Nano-switch Carbon nanotube Thermal expansion Pull-in instability Thermal switch
1. Introduction The rapid development of micro/nano-scale fabrication technologies in recent years has led to invention of various micro/ nanoelectromechanical systems (MEMS/NEMS). Several NEMS have already been demonstrated, such as logical nanodevices [1], nanotweezers [2], random access memories [3], sensors [4–10] and nano-switches [11–16]. Recently, there has been massive interest in designing and fabrication of new nanodevices with well-defined sizes, configurations and characteristics, which determine their final applications. The current challenge is to introduce methods to effectively design nanostructures for practical applications. Even though NEMS can be designed using a number of materials, carbon nanotube based NEMS are among the best candidates with regard to their remarkable properties, including small size, low density, high stiffness, flexibility and strength, as well as excellent electronic properties and unique coupled electromechanical behaviors [17].
n
Corresponding author. E-mail address: [email protected] (S.E. Khadem).
1386-9477/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2014.01.004
Most potential applications of CNTs are highly dependent on understanding their mechanical behavior. Thus, mechanical behavior, buckling and pull-in instability of CNTs under electrostatic loads have been the subjects of numerous recent studies. Pull-in phenomenon plays a significant role in the designing of nanoswitches and has been studied effectively. Khadem et al. [11] proposed a carbon nanotube-based switch as an adjustable shock switch. Fu and Zhang [12] studied size-dependent pull-in phenomena in electrically actuated nanobeams. Dequesnes et al. [13] studied the pull-in instability in the carbon nanotube bridges using the molecular dynamic method. Rasekh et al. [14,15] investigated the dynamic behavior of carbon nanotube based switches and presented the noise immunity property of the proposed device. Rasekh and Khadem [16] presented a comprehensive model of a nano-cantilever with nonlinearity in curvature, inertia and electrostatic force. They studied the effects of relatively large electrostatic gaps on the pull-in features of a nano-cantilever and showed when the nonlinear formulation needs to be taken into account. Effects of axial load on the behavior of the carbon nanotubes have also been investigated in various studies. Zhang et al. [18] have studied the effect of axial loads on vibrations of multi-walled
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nanotubes (MWNTs), and investigated shifts in frequencies. They have considered inter-atomic interactions between layers of MWNTs in their modeling. Ru et al. [19,20] reported the instability in nanotubes due to axial loads originating from the conveying fluid inside them, and calculated the critical velocity of fluid that results in instability. Rasekh and Khadem [21] studied the nonlinear behavior of nanotubes conveying fluid under axial loads accounting for nonlinear amplitude–frequency relationship. Ru et al. in another work [22] investigated the buckling of multiwall nanotubes under axial compressing loads and external pressure. A number of studies investigated mechanical buckling of nanotubes considering the temperature effect using continuum models [23–26]. Yan et al. investigate buckling of triple-walled carbon nanotubes under temperature rise [23]. Zhang et al. [24] reported influence of temperature change on column buckling of multiwall carbon nanotubes (MWCNTs) and concluded that thermal effect on the buckling strain becomes more significant with the increase of temperature change and aspect ratio (length/diameter). Xiaohu and Qiang [25,26] studied buckling analysis of MWCNTs under torsional and axial load coupling with temperature change. The results show that at low and room temperature the critical axial load for infinitesimal buckling of a multi-walled carbon nanotube increases as the value of temperature change increases, while at high temperature the critical axial load for infinitesimal buckling of a multi-walled carbon nanotube decreases as the value of temperature change increases. This phenomenon can be attributed to change of coefficient of thermal expansion of CNTs at different temperatures [27]. In addition to continuum models (beams and shells) other methods, including molecular mechanics and experimental tests, have been used to study mechanical behavior of nanotubes [28,29]. In an experimental study by Carpick et al. [28], it has been shown that the buckling of individual MWCNTs of a wide range of aspect ratios under axial compression can be captured by AFM. Here we propose an application of carbon nanotubes in design of thermal nanoswitches. The switch comprises a single-wall carbon nanotube with fixed–fixed boundary conditions suspended over a graphite ground electrode plate. Operation of the switch is based on shift in pull-in voltage and pull-in instability due to axial load which originates from the temperature rise. As the nanotube and ground plate build up a capacitor configuration, by applying potential difference between the nanotubes and the ground electrode, the electrostatic force makes these two parts attract each other and the nanotube as the movable part deflects toward the graphite plate. When the applied voltage increases to a certain value, i.e. pull-in voltage, the nanotube sticks to the ground. On adjusting the voltage, the nanotube can approach its pull-in state. In this case, any external disturbance, here the temperature rise, can bring the tube to pull-in state and turns the switch ON. By applying different adjusting voltages, various temperature rise thresholds can be achieved. Results show that there is a linear relation between adjusting voltage and temperature rise threshold.
2. Modeling A single-wall carbon nanotube (SWCNT) over a ground graphite plate is considered. The CNT is subjected to the axial compressive force, electrostatic force, and van der Waals force (Fig. 1). When a nanotube has fixed–fixed boundary conditions, rise in the temperature leads to axial compressive load. In addition, stretching term due to elongation of CNT should be taken in to account due to these boundary conditions. The Euler–Bernoulli beam theory is employed to model the vibrational behavior of the SWCNT: EI
∂4 w ∂w ∂2 w þ c þ ρA 2 ¼ q 4 ∂t ∂x ∂t
ð1Þ
211
Fig. 1. Schematic model of the thermal nanoswitch.
Fig. 2. SWCNT van der Waals interaction over a graphite electrode.
where E is Young0 s modulus of CNT, I is the moment of inertia of the cross section of CNT, c is the damping coefficient, A is the area of the cross section of the CNT; and ρ is the density of CNT; x, t, and w are coordinates along the length of nanotubes, time, and transverse displacement, respectively; q is the total applied force per unit length of the nanotube. In order to simulate the response of a nano-switch, mathematical models and relations for these forces are required. The thermal expansion coefficient of the CNT will be considered as a function of the temperature, T 0 . 2.1. Electrostatic force The combination of the nanotube and the ground forms a capacitor, so the electrostatic forces can be obtained by finding the relation for capacitance of the capacitor; the capacitance per unit length for a cylinder over a conductive ground plate is given by CðrÞ ¼
2πε qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln f1 þr=R þ ðr=R þ 1Þ2 1g
ð2Þ
where ε0 is the permittivity of vacuum, r is the gap between the CNT and the plate and R is the diameter of the carbon nanotubes (Fig. 2). By differentiating the energy stored in the capacitor with respect to the displacement, the electrostatic force per unit length is obtained [13] επV 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qelec ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R rðr þ 2RÞ=R2 log ½1 þ ðr=RÞ þ rðr þ 2RÞ=R2
ð3Þ
where V is the applied voltage. 2.2. Axial forces Two sources for axial forces are considered in this study. First, the term due to elongation of the beam (stretching term) and the second, external axial force. The stretching term can be obtained by calculating the axial strain and multiplying it by the axial
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stiffness (EA). Considering small displacements and using Taylor expansion, the axial strain and the corresponding stretching axial force would be 1 εStretching ¼ 2
RL 0
ð∂w=∂xÞ2 dx L
) P Stretching ¼ εStretching
AE AE ¼ 2L
Z 0
L
2 ∂w dx ∂x
ð4Þ The external axial forces can be exerted directly (P Ext ) and/or caused by thermal stresses (P Thermal ). Using the thermal strain, the axial thermal force would be εThermal ¼ αðT 0 Þ ΔT
)
P Thermal ¼ AE α ðT 0 ÞΔT
ð5Þ
where αðT 0 Þis the coefficient of thermal expansion of the CNT and ΔT is the rise in temperature with respect to ambient temperature, T 0 . Considering all axial forces, the normal component of the axial force effective in transverse vibration of the nanotube is ∂ ∂w ð6Þ qN axial ¼ ðP Stretching þ P Thermal þ P Ext Þ ∂x ∂x 2.3. Van der Waals forces The van der Waals forces between the ground and nanotube can be obtained from the Lennard–Jones potential which describes the potential between two atoms as follows [13]: c12 c6 ϕij ¼ 12 6 r ij r ij
ð7Þ
where rij is the distance between atoms i and j, c6 , c12 are van der Waals constants. For carbon–carbon interaction these values are c6 ¼ 15.2 eV Å6 and c12 ¼2.42 keV Å12 [13]. These interactions are between atoms of carbon nanotube and layers of graphite in the ground plate. Consequently, to have the total van der Waals (vdW) forces acting on tube, interactions between the nanotube and each layer should be added together. It should be mentioned that when the air gap is bigger than 3 nm (r 43 nm), the effect of van der Waals force becomes very weak and can be ignored compared to the electrostatic force. Differentiating Eq. (7) with respect to r and integrating over the whole graphite plate, the applied vdW force on an atom on the carbon nanotube and a single graphene layer is calculated as c c6 12 F graphite ¼ 2π s 11 5 ð8Þ a a where a is the distance between the atom on the nanotube and the graphene layer and s ¼ 38 nm 2 [13] is the graphene surface density. Relation (8) gives the interaction between one atom and one layer. By integrating Eq. (8) over a ring of the nanotube, and considering N layers of graphite (Fig. 2), the distributed van der Waals load can be obtained. N
qvdW ¼ ∑
n¼1
Z
π π
2π s
c12 11
½ðn 1Þd þ r þ Rþ R sin θ
c6 ½ðn 1Þd þ r þ R þ R sin θ5
sR dθ
ð9Þ
3. Solving equations Considering the electrostatic, stretching and van der Waals distributed loads and rewriting Eq. (1), the transverse displacement of nanotubes is modeled by a well-known nonlinear beam equation: EI
" Z # ∂4 w ∂w ∂2 w EA L ∂w 2 ∂2 w þ ρA þ c ¼ dx þ AEαðT ÞΔT þ P þ qelec þ qvdW Ext 0 ∂t 2L 0 ∂x ∂x4 ∂x2 ∂t 2
ð10Þ where the terms in the brackets are components of axial forces.
Table 1 SWCNT properties. SWCNT properties
Symbol
Value
Unit
Density Cross sectional area
ρ A
1330
Radius
R
0:68 10 9
kg=m3 m2 m
Moment of inertia
I
2:134 10 37
Young0 s modulus
E
1:024 10 18
1054 10
9
m4 Pa
For better understanding and reducing computational errors, nondimensional variables have been used as follows: sffiffiffiffiffiffiffiffiffiffi ρAL4 x t w ^ ¼ t~ ¼ ð11Þ ; x^ ¼ ; t^ ¼ ; w L r EI t~ Next, using Galerkin0 s method, a reduced-order model is formed by discretizing Eq. (10) into a finite-degree-of-freedom system including ordinary-differential equations in time. To this end, the deflection of nanobeam is described using functions in time and space as follows: n
wðx; tÞ ¼ ∑ qi ðtÞ ϕi ðxÞ
ð12Þ
i¼1
where qi(t) is the ith generalized coordinate and ϕi ðxÞ is the ith mode shape of the linear free vibration of the nanobeam with fixed–fixed boundary conditions. Substituting Eq. (12) into Eq. (10), multiplying by ϕi ðxÞ, and integrating over the length of the nanotube (x¼0–L), yields the reduced-order model. The response of the nanobeam can be simulated by solving the set of ordinary differential equations of the reduced-order model in time. Numerical integration is used which provides the advantage of using five modes (and more) and also reduces the run-time to a very short and desirable one. In this study, up to five mode shapes are used to simulate behavior of nanotubes. It should be noted that considering the geometrical symmetry and physics of the system only odd (symmetric) mode shapes (n ¼1, 3, 5,…) have been employed. Results have been computed using the physical properties for a single-wall carbon nanotube presented in Table 1 [14].
4. Results 4.1. Effect of axial load on the pull-in voltage In the first step, the effect of external axial load on the pull-in voltage of nanotubes is studied. In this case, we assume no temperature rise and consequently no thermal axial load. It is assumed that the magnitude of the applied external axial load is proportional to critical buckling load for a fixed–fixed beam and the coefficient of proportion isλ: P cr ¼
4πEI L2
; λ¼
P Ext P cr
ð13Þ
Another parameter which is used to present the results is η ¼ V pi =V pi0 which expresses the ratio of the pull-in voltage in the presence of axial load, Vpi, to the pull-in voltage in the absence of any axial load, Vpi0. As depicted in Figs. 3 and 4, in the presence of axial load, the pull-in voltage (Vpi) is less than its value when there is no axial load (Vpi0) or in other words, η o 1 and this value decreases as the axial load increases. From Fig. 4, when λ 4 1, even for low applied voltages the nanotube has large deflection and for greater axial loads a small disturbance (applied voltage) results in the pull-in state. In lower air gaps, the pull-in voltage ratio, η, is almost the same for different lengths of the nanotube but this ratio changes a bit for different lengths as the air gap increases.
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Fig. 3. Ratio of pull-in voltage in the presence of axial load to pull-in voltage without axial load vs. axial load ratio for various air gaps: (a) r ¼10 nm, (b) r ¼ 12 nm, and (c) r ¼15 nm.
Fig. 4. Deflection at the middle of the nanotube vs. voltage in the presence of various axial loads: (a) r¼ 3 nm, L ¼ 40 nm and (b) r ¼3 nm, L ¼20.7 nm.
In addition, in lower air gaps (r ¼10 nm), the slope of η–λ lines is more than the slopes in higher air gaps (r ¼15 nm). In other words, shift in pull-in voltage is more sensitive to axial load in lower air gaps. This can be attributed to electrostatic force which has an inverse relation to air gap (r) and consequently, makes the axial load more effective in lower air gaps. This shift in pull-in voltage can be used as a measurement for the magnitude of the axial load.
4.2. Thermal nanoswitch idea The operation of the presented thermal nanoswitch is based on the thermal expansion of a carbon nanotube with fixed–fixed boundary conditions. Rise in the temperature results in the axial load in the CNT with restricted boundary conditions. If the nanotube has a pre-deflection (due to the applied voltage), the
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temperature rise deflects the tube more and brings the tube closer to ground electrode and may result in the pull-in state. In other words, approaching the corresponding pull-in deflection due to the pre-deflection makes the switch more sensitive to the temperature rise. If the temperature exceeds a level, the pull-in occurs and the nanotube will stick to the ground and turn the thermal nanoswitch ON. This temperature level is in relation with the predeflection of the nanotube and consequently to the applied voltage. When the applied voltage is lower, the resulted deflection is lower and larger temperature rise is necessary to bring the nanotubes to the pull-in state. The same way, when the applied voltage is higher, the switch will be triggered by a lower temperature rise. In the first step, it is assumed that the nanotube is deformed to a steady-state pre-deflected position by applying levels of voltage. In this case, there is a certain distance between the tube and the ground plate. Smaller distance between the ground plate and the nanotube can detect lower temperature rise. To trigger the switch to the ON state, the remaining distance to the pull-in state must be covered by the deflection due to temperature rise. Adjusting voltage gives us the advantage to set the switch for different temperature thresholds. Increasing the adjusting voltage leads to lower temperature rise thresholds. It also should be noted that because of the dependence of thermal expansion coefficient of the CNT to the ambient temperature, in addition on the rising temperature the ambient temperature should be considered to design the trigger threshold of the switch. It is essential that each
Fig. 5. The maximum temperature rise that can be detected by the switch vs. the length of the nanoswitch, while nanotubes have a pre-deflection equal to 0.5 of the air gap.
step takes place after the nanotube reaches its steady-state. Otherwise, the switch may show different behaviors for the same temperature rise. Different parameters are effective in response and behavior of the nanoswitch. Length of the nanotube (L) has a significant role on the response since it changes the stiffness of the system. Having lower stiffness, longer nanotubes can be deflected with weaker external loads. Thus, lower voltages and temperature rise can trigger the switch. Additionally, the air gap between the nanotube and the ground is an important parameter and besides the length of nanotubes can determine the maximum temperature rise that the switches can detect. To investigate the effect of the length and the air gap on the temperature range of nanoswitch, the switch is deformed to have a pre-deflection of half of the air gap. Next, adjusting voltage is set to zero to maximize the temperature range and then the maximum temperature rise before the pull-in state is obtained. Results are shown in Fig. 5. It should be noted that Fig. 5 shows a wide temperature rise domain with respect to the length of the CNT. Due to variation of the thermal expansion coefficient, Fig. 5 presents a rough dependence of the operating range of the switch, particularly for the shorter lengths. So, achieving an accurate switch would be contingent on a narrow operating range or in other words triggering the switch by low temperature rise because low temperature rise means that the thermal expansion coefficient of the CNT could be assumed a constant. To achieve the deflection equal to 0.5 of the air gap by the maximum possible applied voltage, the potential difference is applied gradually in three steps (V1, V2, V3), as applying the total of these three voltages abruptly may result in the pull-in state. When these three voltage levels are applied and the tube attains half of the air gap, the adjusting voltage, V4, can be applied corresponding to each temperature rise. Indeed, the total voltage of these four voltages is the corresponding pull-in voltage for each temperature rise. The relation between the temperature ris and the adjusting voltage, V4, is depicted in Fig. 6 for various lengths and air gaps of the nanoswitch. It is observed from Fig. 6 that the relation between voltage and temperature rise is linear and slopes of the lines show the resolution of the switch. For example, for a nanoswitch with r ¼25 nm and length of L¼ 1250 nm the slope of the line is one degree celsius per millivolt (1 1C=mV). It means that if the designer wants to change the temperature rise threshold by 1 1C (for example from 100 1C to 101 1C) the voltage supply must be able to change the voltage level by 1 mV (for example from 80 mV to 81 mV). In other words, if the minimum possible change in the voltage level is 1 mV the minimum change in the temperature rise,
Fig. 6. Adjusting voltage (V4) vs. the temperature rise (the pre-deflection is 0.5 of the air gap). For interpretation of the references to color in this figure, the reader is referred to the web version of this article.
M. Rasekh et al. / Physica E 59 (2014) 210–217
which can be detected by switch, will be 1 1C. Consequently, if the voltage supply can change the voltage level by lower resolution (for example from 80 mV to 80.1 mV) a lower temperature rise can be detected. These slopes have been given in Table 2. It can be realized that slopes of the temperature–voltage lines are more dependent on the air gap of the nanoswitch and changes in the length do not have a dominant effect. By increasing the air gap, better resolutions can be achieved. However, as the air gap increases, higher levels of adjusting voltage are required to adjust and actuate the switch which means higher energy consumption. Adjusting voltage (V4) added to levels of voltages (V1, V2, V3) required to deflect the nanotubes to 0.5 of air gap (pre-deflection) may give the total applied voltage and a criterion for energy consumption. The total voltage required to deflect the nanotubes to 0.5 of air gap (V1 þ V2 þV3) for various lengths and air gaps of nanoswitch are shown in Fig. 7 which shows that length of the nanotube has a significant role on the applied voltage. The total voltage required to adjust the switch for different rises in
Table 2 Resolution of the thermal switch for various lengths and air gaps (1 C=mV). Air gap (nm)
r ¼ 15 r ¼ 20 r ¼ 25
Length (L) of nanotube (nm) 2000
1250
750
1.125 1.052 1.000
1.138 1.058 1.003
1.138 1.060 1.005
temperature has also been illustrated in Fig. 8. Thus, while a switch with r¼ 20 nm and L ¼750 nm (black dashed line in Fig. 6) can detect a more extended range of temperature rise (50–180 1C rise in temperature), for detecting 80 1C rise in temperature, a switch with r ¼ 25 nm and L¼ 1250 nm (green dotted line in Fig. 6) is more appropriate since it requires lower applied voltage. However, it can detect a narrower range of temperature rise (35–95 1C). 4.3. Some example cases To have a better understanding of how the designed switch works, some example cases are given and the corresponding simulated response of the thermal nanoswitch is shown. Before that, some notes should be considered.
The switch is designed to be actuated by specific rise in the
Fig. 7. Total required voltage (V1þ V2 þV3) to give a pre-deflection equal to 0.5 of the air gap.
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temperature at specific temperature. For example, if the reference temperature is assumed to be 600 1C and the switch is supposed to be activated at 670 1C, the switch must be adjusted for ΔT ¼ 70 1C. Rise in the temperature is assumed to be linear (Fig. 9). The nano-switches are designed for specific ranges of ambient temperature; so, an average coefficient of axial thermal expansion (CTE) of α ¼ 2 10 6 =1C has been considered for a singlewall carbon nanotube. Considering this CTE, the operating temperature of the switch is approximately 700–900 1C. For simplification, it is assumed that the rise in temperature is limited; so, it does not change the properties of carbon nanotubes. Here, three steps of voltage have been used to obtain a predeflection equal to 0.5 of air gap. Since operation of the switch deals with steady states, number of steps is not important and different combinations of number of steps and amount of voltages, which give the same pre-deflection, may be used. A schematic model of stepped applied voltages (V1, V2, V3, V4) has been shown in Fig. 9.
As an example, suppose we have a nanoswitch with r ¼15, L¼ 2000 nm and the switch should be adjusted to be activated by 8 1C rise in temperature. By applying three levels of voltages, the nanotubes will be given a pre-deflection equal to 0.5 of air gap (w=r ¼ 0:5). These voltages are V1 ¼20 mV, V2¼ 20 mV and V3 ¼11 mV. Next, the adjusting voltage, V4, must be determined. From Fig. 6 for r ¼ 15 nm and L ¼2000 nm (red solid line) the adjusting voltage will be around 5 mV and by exact calculation V4 ¼5.6 mV is obtained. Fig. 10 shows the corresponding simulated
Fig. 8. Total required voltage (V1þ V2 þV3þ V4) to adjust the nanoswitch for different temperature rise. For interpretation of the references to color in this figure, the reader is referred to the web version of this article.
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Fig. 9. Schematic models for (a) rise in temperature and (b) applied voltages.
Fig. 10. Nondimensional time response of nanoswitch at x¼ L/2 for r ¼ 15 nm, L ¼ 2000 nm, and V4¼ 5.6 mV: (a) ΔT ¼ 7:81 C (switch OFF) and (b) ΔT ¼ 81 C (switch ON).
Fig. 11. Nondimensional time response of nanoswitch at x¼ L/2 for r ¼15 nm, L ¼750 nm, and V4 ¼23.8 mV: (a) ΔT ¼ 721 C (switch OFF) and (b) ΔT ¼ 731 C (switch ON).
response of nanoswitch. Five levels of deflection can be detected in the switch response. First three ones (D1, D2, D3) are due to voltages V1, V2 and V3, respectively. Next, there is deflection due to adjusting voltage (D4) and finally, deflection resulting from rise in temperature (DT). Simulation shows that if the temperature rise is below the threshold (8 1C) the switch will not be turned ON (i.e. pull-in will not occur). As another example, a switch with r ¼15 nm and L ¼750 nm is considered. The switch will be adjusted for a temperature rise equal to 73 1C. For pre-deflection voltages V1 ¼170 mV, V2 ¼170 mV and V3 ¼26 mV are applied. Then from Fig. 6 for r ¼15 nm and L¼ 750 nm (red dashed line) V4 will be estimated to
be 24 mV, and by exact calculation V4 ¼23.8 mV. Results are shown in Fig. 11. Again, when the temperature rise is less than 73 1C the switch will be in OFF state.
5. Summary and conclusions Design and simulation of a thermal nanoswitch operated based on thermal expansion of a carbon nanotube was presented. Electrostatic, van der Waals, axial and nonlinear stretching forces have been considered to fully capture the response of the nanotube and Galerkin0 s method was used to reduce the governed
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equations. Results show the axial load effects on pull-in voltage and that the compressive loads reduce the pull-in voltage. By increasing the axial load, the nanotube becomes unstable and sticks to ground. The magnitude of load required to destabilize the nanotubes is related to the applied voltage. When nanotubes have fixed–fixed boundary conditions, rise in temperature will result in compressive axial load and by adjusting the applied voltage, switch can be set to be actuated at specific temperature rise. To this end, effect of geometrical parameters, including the air gap and the length of the tube, on the required applied voltage and the maximum temperature rise limit were investigated. Then, the relation between adjusting voltage and corresponding rise in temperature was obtained for various air gaps and lengths of the nanoswitch. Results show that as the air gap increases, the nanoswitch can detect more extended ranges of temperature rise with better resolution (sensitivity); however, the required applied voltage and consequently, energy consumption of the switch will also increase. Finally, some examples were given to show how the thermal switch is adjusted and the response of the switch was simulated using those adjustments. Results illustrated that carbon nanotubes can be utilized to design thermal nanoswitches. References [1] Q. Li, S.M. Koo, M.D. Edelstein, J.S. Suehle, C.A. Richter, Nanotechnology 18 (2007) 315. [2] G.W. Wang, Y. Zhang, Y.P. Zhao, G.T. Yang, J. Micromech. Microeng. 14 (2004) 1119.
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[3] J.W. Kang, O.K. Kwon, J.H. Lee, H.J. Lee, Y.J. Song, Y.S. Yoon, H.J. Hwang, Physica E 33 (2006) 41. [4] C. Li, E.T. Thostenson, T.W. Chou, Compos. Sci. Technol. 68 (2008) 1227. [5] D. Sickert, A. Taegar, I. Kühne, M. Mertig, W. Pompe, G. Eckstein, Phys. Status Solidi B 243 (2006) 3542. [6] I. Kang, Y.Y. Heung, J.H. Kim, J.W. Lee, R. Gollapudi, S. Subramaniam, S. Narasimhadevara, D. Hurd, G.R. Kirikera, V. Shanov, M.J. Schulz, D. Shi, J. Boerio, S. Mall, M. Ruggles-Wren, Compos. Part B 37 (2006) 382. [7] M. Dequenses, S.V. Rotkin, N.R. Aluru, J. Comput. Electron 1 (2002) 313. [8] P. Dharap, Z. Li, S. Nagarajaiah, E.V. Barrera, Nanotechnology 15 (2004) 379. [9] J.H. He, S. Singamaneni, C.H. Ho, Y.H. Lin, M.E. McConney, V.V. Tsukruk, Nanotechnology 20 (2009) 065502. [10] M. Rasekh, S.E. Khadem, J. Sound Vib. 332 (2013) 6155. [11] S.E. Khadem, M. Rasekh, A. Toghraee, Appl. Math. Model. 36 (2012) 2329. [12] Y. FU, J. Zhang, Appl. Math. Model. 35 (2011) 941. [13] M. Dequesnes, Z. Tang, N.A. Aluru, J. Eng. Mater. Technol. Trans. ASME 126 (2004) 230. [14] M. Rasekh, S.E. Khadem, M. Tatari, J. Phys. D: Appl. Phys. 43 (2010) 315301. [15] M. Rasekh, S.E. Khadem, M. Tatari, ASME Eng. Syst. Des. Anal. 5 (2010) 25162. [16] M. Rasekh, S.E. Khadem, Int. J. Mech. Sci. 53 (2011) 108. [17] M.J. O’Connell, Carbon Nanotubes: Properties and Applications, CRC Press, Massachusetts, USA, 2006. [18] Y. Zhang, G. Liu, X. Han, Phys. Lett. A 340 (2005) 258. [19] J. Yoon, C.Q. Ru, A. Mioduchowski, Compos. Sci. Technol. 65 (2005) 1326. [20] J. Yoon, C.Q. Ru, A. Mioduchowski, Int. J. Solids Struct. 43 (2005) 3337. [21] M. Rasekh, S.E. Khadem., J. Phys. D: Appl. Phys. 42 (2009) 135112. [22] C.Q. Ru, J. Mech. Phys. Solids 49 (2001) 1265. [23] Y. Yan, W.Q. Wang, L.X. Zhang, Appl. Math. Model. 34 (2010) 3422. [24] Y.Q. Zhang, X. Liu, J.H. Zhao, Phys. Lett. A 372 (2008) 1676. [25] Y. Xiaohu, H. Qiang, J. Eng. Mater. Technol. 128 (2006) 419. [26] Y. Xiaohu, H. Qiang, Compos. Sci. Technol. 67 (2007) 125. [27] H. Jiang, B. Liu, Y. Huang, K.C. Hwang, J. Eng. Mater. Technol. 126 (2004) 265. [28] H.W. Yap, R.S. Lakes, R.W. Carpick, Nano Lett. 7 (2007) 1149. [29] C.L. Zhang, H.S. Shen, Carbon 44 (2006) 2608.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
NEMS thermal switch operating based on thermal expansion of carbon nanotubes M. Rasekh, S.E. Khadem n, A. Toghraee Department of Mechanical Engineering, Tarbiat Modares University, PO Box 14115-177, Tehran, Iran
H I G H L I G H T S
Design and simulation of a carbon nanotube-based thermal switch is reported. Thermal expansibility of the carbon nanotube is used to design a thermal switch. Electrostatic load is used to adjust the thermal switch trigger level. Temperature rising causes the pull-in instability and triggers the nano-switch.
art ic l e i nf o
a b s t r a c t
Article history: Received 27 April 2013 Received in revised form 28 December 2013 Accepted 7 January 2014 Available online 15 January 2014
In this paper, design and simulation of carbon nanotube-based thermal switches is reported. A carbon nanotube placed over a ground electrode represents the switch. The response of the nanoswitch based on beam theory is studied. When a nanotube has fixed–fixed boundary conditions, rise in temperature and thermal expansion of the nanotube can cause compressive axial load. This axial load will result in change of the pull-in voltage. Considering this fact, a thermal switch is designed for specific ambient temperature range and limited temperature rise range. When the temperature rises, the nanotube deflects more and approaches its pull-in instability. If temperature exceeds the threshold, pull-in occurs and the switch is triggered. Applying different voltages can provide different temperature thresholds. Utilizing this feature, the corresponding adjusting voltages required for actuating the switch by different rises in temperature are obtained. & 2014 Elsevier B.V. All rights reserved.
Keywords: Nano-switch Carbon nanotube Thermal expansion Pull-in instability Thermal switch
1. Introduction The rapid development of micro/nano-scale fabrication technologies in recent years has led to invention of various micro/ nanoelectromechanical systems (MEMS/NEMS). Several NEMS have already been demonstrated, such as logical nanodevices [1], nanotweezers [2], random access memories [3], sensors [4–10] and nano-switches [11–16]. Recently, there has been massive interest in designing and fabrication of new nanodevices with well-defined sizes, configurations and characteristics, which determine their final applications. The current challenge is to introduce methods to effectively design nanostructures for practical applications. Even though NEMS can be designed using a number of materials, carbon nanotube based NEMS are among the best candidates with regard to their remarkable properties, including small size, low density, high stiffness, flexibility and strength, as well as excellent electronic properties and unique coupled electromechanical behaviors [17].
n
Corresponding author. E-mail address: [email protected] (S.E. Khadem).
1386-9477/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2014.01.004
Most potential applications of CNTs are highly dependent on understanding their mechanical behavior. Thus, mechanical behavior, buckling and pull-in instability of CNTs under electrostatic loads have been the subjects of numerous recent studies. Pull-in phenomenon plays a significant role in the designing of nanoswitches and has been studied effectively. Khadem et al. [11] proposed a carbon nanotube-based switch as an adjustable shock switch. Fu and Zhang [12] studied size-dependent pull-in phenomena in electrically actuated nanobeams. Dequesnes et al. [13] studied the pull-in instability in the carbon nanotube bridges using the molecular dynamic method. Rasekh et al. [14,15] investigated the dynamic behavior of carbon nanotube based switches and presented the noise immunity property of the proposed device. Rasekh and Khadem [16] presented a comprehensive model of a nano-cantilever with nonlinearity in curvature, inertia and electrostatic force. They studied the effects of relatively large electrostatic gaps on the pull-in features of a nano-cantilever and showed when the nonlinear formulation needs to be taken into account. Effects of axial load on the behavior of the carbon nanotubes have also been investigated in various studies. Zhang et al. [18] have studied the effect of axial loads on vibrations of multi-walled
M. Rasekh et al. / Physica E 59 (2014) 210–217
nanotubes (MWNTs), and investigated shifts in frequencies. They have considered inter-atomic interactions between layers of MWNTs in their modeling. Ru et al. [19,20] reported the instability in nanotubes due to axial loads originating from the conveying fluid inside them, and calculated the critical velocity of fluid that results in instability. Rasekh and Khadem [21] studied the nonlinear behavior of nanotubes conveying fluid under axial loads accounting for nonlinear amplitude–frequency relationship. Ru et al. in another work [22] investigated the buckling of multiwall nanotubes under axial compressing loads and external pressure. A number of studies investigated mechanical buckling of nanotubes considering the temperature effect using continuum models [23–26]. Yan et al. investigate buckling of triple-walled carbon nanotubes under temperature rise [23]. Zhang et al. [24] reported influence of temperature change on column buckling of multiwall carbon nanotubes (MWCNTs) and concluded that thermal effect on the buckling strain becomes more significant with the increase of temperature change and aspect ratio (length/diameter). Xiaohu and Qiang [25,26] studied buckling analysis of MWCNTs under torsional and axial load coupling with temperature change. The results show that at low and room temperature the critical axial load for infinitesimal buckling of a multi-walled carbon nanotube increases as the value of temperature change increases, while at high temperature the critical axial load for infinitesimal buckling of a multi-walled carbon nanotube decreases as the value of temperature change increases. This phenomenon can be attributed to change of coefficient of thermal expansion of CNTs at different temperatures [27]. In addition to continuum models (beams and shells) other methods, including molecular mechanics and experimental tests, have been used to study mechanical behavior of nanotubes [28,29]. In an experimental study by Carpick et al. [28], it has been shown that the buckling of individual MWCNTs of a wide range of aspect ratios under axial compression can be captured by AFM. Here we propose an application of carbon nanotubes in design of thermal nanoswitches. The switch comprises a single-wall carbon nanotube with fixed–fixed boundary conditions suspended over a graphite ground electrode plate. Operation of the switch is based on shift in pull-in voltage and pull-in instability due to axial load which originates from the temperature rise. As the nanotube and ground plate build up a capacitor configuration, by applying potential difference between the nanotubes and the ground electrode, the electrostatic force makes these two parts attract each other and the nanotube as the movable part deflects toward the graphite plate. When the applied voltage increases to a certain value, i.e. pull-in voltage, the nanotube sticks to the ground. On adjusting the voltage, the nanotube can approach its pull-in state. In this case, any external disturbance, here the temperature rise, can bring the tube to pull-in state and turns the switch ON. By applying different adjusting voltages, various temperature rise thresholds can be achieved. Results show that there is a linear relation between adjusting voltage and temperature rise threshold.
2. Modeling A single-wall carbon nanotube (SWCNT) over a ground graphite plate is considered. The CNT is subjected to the axial compressive force, electrostatic force, and van der Waals force (Fig. 1). When a nanotube has fixed–fixed boundary conditions, rise in the temperature leads to axial compressive load. In addition, stretching term due to elongation of CNT should be taken in to account due to these boundary conditions. The Euler–Bernoulli beam theory is employed to model the vibrational behavior of the SWCNT: EI
∂4 w ∂w ∂2 w þ c þ ρA 2 ¼ q 4 ∂t ∂x ∂t
ð1Þ
211
Fig. 1. Schematic model of the thermal nanoswitch.
Fig. 2. SWCNT van der Waals interaction over a graphite electrode.
where E is Young0 s modulus of CNT, I is the moment of inertia of the cross section of CNT, c is the damping coefficient, A is the area of the cross section of the CNT; and ρ is the density of CNT; x, t, and w are coordinates along the length of nanotubes, time, and transverse displacement, respectively; q is the total applied force per unit length of the nanotube. In order to simulate the response of a nano-switch, mathematical models and relations for these forces are required. The thermal expansion coefficient of the CNT will be considered as a function of the temperature, T 0 . 2.1. Electrostatic force The combination of the nanotube and the ground forms a capacitor, so the electrostatic forces can be obtained by finding the relation for capacitance of the capacitor; the capacitance per unit length for a cylinder over a conductive ground plate is given by CðrÞ ¼
2πε qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln f1 þr=R þ ðr=R þ 1Þ2 1g
ð2Þ
where ε0 is the permittivity of vacuum, r is the gap between the CNT and the plate and R is the diameter of the carbon nanotubes (Fig. 2). By differentiating the energy stored in the capacitor with respect to the displacement, the electrostatic force per unit length is obtained [13] επV 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qelec ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R rðr þ 2RÞ=R2 log ½1 þ ðr=RÞ þ rðr þ 2RÞ=R2
ð3Þ
where V is the applied voltage. 2.2. Axial forces Two sources for axial forces are considered in this study. First, the term due to elongation of the beam (stretching term) and the second, external axial force. The stretching term can be obtained by calculating the axial strain and multiplying it by the axial
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stiffness (EA). Considering small displacements and using Taylor expansion, the axial strain and the corresponding stretching axial force would be 1 εStretching ¼ 2
RL 0
ð∂w=∂xÞ2 dx L
) P Stretching ¼ εStretching
AE AE ¼ 2L
Z 0
L
2 ∂w dx ∂x
ð4Þ The external axial forces can be exerted directly (P Ext ) and/or caused by thermal stresses (P Thermal ). Using the thermal strain, the axial thermal force would be εThermal ¼ αðT 0 Þ ΔT
)
P Thermal ¼ AE α ðT 0 ÞΔT
ð5Þ
where αðT 0 Þis the coefficient of thermal expansion of the CNT and ΔT is the rise in temperature with respect to ambient temperature, T 0 . Considering all axial forces, the normal component of the axial force effective in transverse vibration of the nanotube is ∂ ∂w ð6Þ qN axial ¼ ðP Stretching þ P Thermal þ P Ext Þ ∂x ∂x 2.3. Van der Waals forces The van der Waals forces between the ground and nanotube can be obtained from the Lennard–Jones potential which describes the potential between two atoms as follows [13]: c12 c6 ϕij ¼ 12 6 r ij r ij
ð7Þ
where rij is the distance between atoms i and j, c6 , c12 are van der Waals constants. For carbon–carbon interaction these values are c6 ¼ 15.2 eV Å6 and c12 ¼2.42 keV Å12 [13]. These interactions are between atoms of carbon nanotube and layers of graphite in the ground plate. Consequently, to have the total van der Waals (vdW) forces acting on tube, interactions between the nanotube and each layer should be added together. It should be mentioned that when the air gap is bigger than 3 nm (r 43 nm), the effect of van der Waals force becomes very weak and can be ignored compared to the electrostatic force. Differentiating Eq. (7) with respect to r and integrating over the whole graphite plate, the applied vdW force on an atom on the carbon nanotube and a single graphene layer is calculated as c c6 12 F graphite ¼ 2π s 11 5 ð8Þ a a where a is the distance between the atom on the nanotube and the graphene layer and s ¼ 38 nm 2 [13] is the graphene surface density. Relation (8) gives the interaction between one atom and one layer. By integrating Eq. (8) over a ring of the nanotube, and considering N layers of graphite (Fig. 2), the distributed van der Waals load can be obtained. N
qvdW ¼ ∑
n¼1
Z
π π
2π s
c12 11
½ðn 1Þd þ r þ Rþ R sin θ
c6 ½ðn 1Þd þ r þ R þ R sin θ5
sR dθ
ð9Þ
3. Solving equations Considering the electrostatic, stretching and van der Waals distributed loads and rewriting Eq. (1), the transverse displacement of nanotubes is modeled by a well-known nonlinear beam equation: EI
" Z # ∂4 w ∂w ∂2 w EA L ∂w 2 ∂2 w þ ρA þ c ¼ dx þ AEαðT ÞΔT þ P þ qelec þ qvdW Ext 0 ∂t 2L 0 ∂x ∂x4 ∂x2 ∂t 2
ð10Þ where the terms in the brackets are components of axial forces.
Table 1 SWCNT properties. SWCNT properties
Symbol
Value
Unit
Density Cross sectional area
ρ A
1330
Radius
R
0:68 10 9
kg=m3 m2 m
Moment of inertia
I
2:134 10 37
Young0 s modulus
E
1:024 10 18
1054 10
9
m4 Pa
For better understanding and reducing computational errors, nondimensional variables have been used as follows: sffiffiffiffiffiffiffiffiffiffi ρAL4 x t w ^ ¼ t~ ¼ ð11Þ ; x^ ¼ ; t^ ¼ ; w L r EI t~ Next, using Galerkin0 s method, a reduced-order model is formed by discretizing Eq. (10) into a finite-degree-of-freedom system including ordinary-differential equations in time. To this end, the deflection of nanobeam is described using functions in time and space as follows: n
wðx; tÞ ¼ ∑ qi ðtÞ ϕi ðxÞ
ð12Þ
i¼1
where qi(t) is the ith generalized coordinate and ϕi ðxÞ is the ith mode shape of the linear free vibration of the nanobeam with fixed–fixed boundary conditions. Substituting Eq. (12) into Eq. (10), multiplying by ϕi ðxÞ, and integrating over the length of the nanotube (x¼0–L), yields the reduced-order model. The response of the nanobeam can be simulated by solving the set of ordinary differential equations of the reduced-order model in time. Numerical integration is used which provides the advantage of using five modes (and more) and also reduces the run-time to a very short and desirable one. In this study, up to five mode shapes are used to simulate behavior of nanotubes. It should be noted that considering the geometrical symmetry and physics of the system only odd (symmetric) mode shapes (n ¼1, 3, 5,…) have been employed. Results have been computed using the physical properties for a single-wall carbon nanotube presented in Table 1 [14].
4. Results 4.1. Effect of axial load on the pull-in voltage In the first step, the effect of external axial load on the pull-in voltage of nanotubes is studied. In this case, we assume no temperature rise and consequently no thermal axial load. It is assumed that the magnitude of the applied external axial load is proportional to critical buckling load for a fixed–fixed beam and the coefficient of proportion isλ: P cr ¼
4πEI L2
; λ¼
P Ext P cr
ð13Þ
Another parameter which is used to present the results is η ¼ V pi =V pi0 which expresses the ratio of the pull-in voltage in the presence of axial load, Vpi, to the pull-in voltage in the absence of any axial load, Vpi0. As depicted in Figs. 3 and 4, in the presence of axial load, the pull-in voltage (Vpi) is less than its value when there is no axial load (Vpi0) or in other words, η o 1 and this value decreases as the axial load increases. From Fig. 4, when λ 4 1, even for low applied voltages the nanotube has large deflection and for greater axial loads a small disturbance (applied voltage) results in the pull-in state. In lower air gaps, the pull-in voltage ratio, η, is almost the same for different lengths of the nanotube but this ratio changes a bit for different lengths as the air gap increases.
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213
Fig. 3. Ratio of pull-in voltage in the presence of axial load to pull-in voltage without axial load vs. axial load ratio for various air gaps: (a) r ¼10 nm, (b) r ¼ 12 nm, and (c) r ¼15 nm.
Fig. 4. Deflection at the middle of the nanotube vs. voltage in the presence of various axial loads: (a) r¼ 3 nm, L ¼ 40 nm and (b) r ¼3 nm, L ¼20.7 nm.
In addition, in lower air gaps (r ¼10 nm), the slope of η–λ lines is more than the slopes in higher air gaps (r ¼15 nm). In other words, shift in pull-in voltage is more sensitive to axial load in lower air gaps. This can be attributed to electrostatic force which has an inverse relation to air gap (r) and consequently, makes the axial load more effective in lower air gaps. This shift in pull-in voltage can be used as a measurement for the magnitude of the axial load.
4.2. Thermal nanoswitch idea The operation of the presented thermal nanoswitch is based on the thermal expansion of a carbon nanotube with fixed–fixed boundary conditions. Rise in the temperature results in the axial load in the CNT with restricted boundary conditions. If the nanotube has a pre-deflection (due to the applied voltage), the
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temperature rise deflects the tube more and brings the tube closer to ground electrode and may result in the pull-in state. In other words, approaching the corresponding pull-in deflection due to the pre-deflection makes the switch more sensitive to the temperature rise. If the temperature exceeds a level, the pull-in occurs and the nanotube will stick to the ground and turn the thermal nanoswitch ON. This temperature level is in relation with the predeflection of the nanotube and consequently to the applied voltage. When the applied voltage is lower, the resulted deflection is lower and larger temperature rise is necessary to bring the nanotubes to the pull-in state. The same way, when the applied voltage is higher, the switch will be triggered by a lower temperature rise. In the first step, it is assumed that the nanotube is deformed to a steady-state pre-deflected position by applying levels of voltage. In this case, there is a certain distance between the tube and the ground plate. Smaller distance between the ground plate and the nanotube can detect lower temperature rise. To trigger the switch to the ON state, the remaining distance to the pull-in state must be covered by the deflection due to temperature rise. Adjusting voltage gives us the advantage to set the switch for different temperature thresholds. Increasing the adjusting voltage leads to lower temperature rise thresholds. It also should be noted that because of the dependence of thermal expansion coefficient of the CNT to the ambient temperature, in addition on the rising temperature the ambient temperature should be considered to design the trigger threshold of the switch. It is essential that each
Fig. 5. The maximum temperature rise that can be detected by the switch vs. the length of the nanoswitch, while nanotubes have a pre-deflection equal to 0.5 of the air gap.
step takes place after the nanotube reaches its steady-state. Otherwise, the switch may show different behaviors for the same temperature rise. Different parameters are effective in response and behavior of the nanoswitch. Length of the nanotube (L) has a significant role on the response since it changes the stiffness of the system. Having lower stiffness, longer nanotubes can be deflected with weaker external loads. Thus, lower voltages and temperature rise can trigger the switch. Additionally, the air gap between the nanotube and the ground is an important parameter and besides the length of nanotubes can determine the maximum temperature rise that the switches can detect. To investigate the effect of the length and the air gap on the temperature range of nanoswitch, the switch is deformed to have a pre-deflection of half of the air gap. Next, adjusting voltage is set to zero to maximize the temperature range and then the maximum temperature rise before the pull-in state is obtained. Results are shown in Fig. 5. It should be noted that Fig. 5 shows a wide temperature rise domain with respect to the length of the CNT. Due to variation of the thermal expansion coefficient, Fig. 5 presents a rough dependence of the operating range of the switch, particularly for the shorter lengths. So, achieving an accurate switch would be contingent on a narrow operating range or in other words triggering the switch by low temperature rise because low temperature rise means that the thermal expansion coefficient of the CNT could be assumed a constant. To achieve the deflection equal to 0.5 of the air gap by the maximum possible applied voltage, the potential difference is applied gradually in three steps (V1, V2, V3), as applying the total of these three voltages abruptly may result in the pull-in state. When these three voltage levels are applied and the tube attains half of the air gap, the adjusting voltage, V4, can be applied corresponding to each temperature rise. Indeed, the total voltage of these four voltages is the corresponding pull-in voltage for each temperature rise. The relation between the temperature ris and the adjusting voltage, V4, is depicted in Fig. 6 for various lengths and air gaps of the nanoswitch. It is observed from Fig. 6 that the relation between voltage and temperature rise is linear and slopes of the lines show the resolution of the switch. For example, for a nanoswitch with r ¼25 nm and length of L¼ 1250 nm the slope of the line is one degree celsius per millivolt (1 1C=mV). It means that if the designer wants to change the temperature rise threshold by 1 1C (for example from 100 1C to 101 1C) the voltage supply must be able to change the voltage level by 1 mV (for example from 80 mV to 81 mV). In other words, if the minimum possible change in the voltage level is 1 mV the minimum change in the temperature rise,
Fig. 6. Adjusting voltage (V4) vs. the temperature rise (the pre-deflection is 0.5 of the air gap). For interpretation of the references to color in this figure, the reader is referred to the web version of this article.
M. Rasekh et al. / Physica E 59 (2014) 210–217
which can be detected by switch, will be 1 1C. Consequently, if the voltage supply can change the voltage level by lower resolution (for example from 80 mV to 80.1 mV) a lower temperature rise can be detected. These slopes have been given in Table 2. It can be realized that slopes of the temperature–voltage lines are more dependent on the air gap of the nanoswitch and changes in the length do not have a dominant effect. By increasing the air gap, better resolutions can be achieved. However, as the air gap increases, higher levels of adjusting voltage are required to adjust and actuate the switch which means higher energy consumption. Adjusting voltage (V4) added to levels of voltages (V1, V2, V3) required to deflect the nanotubes to 0.5 of air gap (pre-deflection) may give the total applied voltage and a criterion for energy consumption. The total voltage required to deflect the nanotubes to 0.5 of air gap (V1 þ V2 þV3) for various lengths and air gaps of nanoswitch are shown in Fig. 7 which shows that length of the nanotube has a significant role on the applied voltage. The total voltage required to adjust the switch for different rises in
Table 2 Resolution of the thermal switch for various lengths and air gaps (1 C=mV). Air gap (nm)
r ¼ 15 r ¼ 20 r ¼ 25
Length (L) of nanotube (nm) 2000
1250
750
1.125 1.052 1.000
1.138 1.058 1.003
1.138 1.060 1.005
temperature has also been illustrated in Fig. 8. Thus, while a switch with r¼ 20 nm and L ¼750 nm (black dashed line in Fig. 6) can detect a more extended range of temperature rise (50–180 1C rise in temperature), for detecting 80 1C rise in temperature, a switch with r ¼ 25 nm and L¼ 1250 nm (green dotted line in Fig. 6) is more appropriate since it requires lower applied voltage. However, it can detect a narrower range of temperature rise (35–95 1C). 4.3. Some example cases To have a better understanding of how the designed switch works, some example cases are given and the corresponding simulated response of the thermal nanoswitch is shown. Before that, some notes should be considered.
The switch is designed to be actuated by specific rise in the
Fig. 7. Total required voltage (V1þ V2 þV3) to give a pre-deflection equal to 0.5 of the air gap.
215
temperature at specific temperature. For example, if the reference temperature is assumed to be 600 1C and the switch is supposed to be activated at 670 1C, the switch must be adjusted for ΔT ¼ 70 1C. Rise in the temperature is assumed to be linear (Fig. 9). The nano-switches are designed for specific ranges of ambient temperature; so, an average coefficient of axial thermal expansion (CTE) of α ¼ 2 10 6 =1C has been considered for a singlewall carbon nanotube. Considering this CTE, the operating temperature of the switch is approximately 700–900 1C. For simplification, it is assumed that the rise in temperature is limited; so, it does not change the properties of carbon nanotubes. Here, three steps of voltage have been used to obtain a predeflection equal to 0.5 of air gap. Since operation of the switch deals with steady states, number of steps is not important and different combinations of number of steps and amount of voltages, which give the same pre-deflection, may be used. A schematic model of stepped applied voltages (V1, V2, V3, V4) has been shown in Fig. 9.
As an example, suppose we have a nanoswitch with r ¼15, L¼ 2000 nm and the switch should be adjusted to be activated by 8 1C rise in temperature. By applying three levels of voltages, the nanotubes will be given a pre-deflection equal to 0.5 of air gap (w=r ¼ 0:5). These voltages are V1 ¼20 mV, V2¼ 20 mV and V3 ¼11 mV. Next, the adjusting voltage, V4, must be determined. From Fig. 6 for r ¼ 15 nm and L ¼2000 nm (red solid line) the adjusting voltage will be around 5 mV and by exact calculation V4 ¼5.6 mV is obtained. Fig. 10 shows the corresponding simulated
Fig. 8. Total required voltage (V1þ V2 þV3þ V4) to adjust the nanoswitch for different temperature rise. For interpretation of the references to color in this figure, the reader is referred to the web version of this article.
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Fig. 9. Schematic models for (a) rise in temperature and (b) applied voltages.
Fig. 10. Nondimensional time response of nanoswitch at x¼ L/2 for r ¼ 15 nm, L ¼ 2000 nm, and V4¼ 5.6 mV: (a) ΔT ¼ 7:81 C (switch OFF) and (b) ΔT ¼ 81 C (switch ON).
Fig. 11. Nondimensional time response of nanoswitch at x¼ L/2 for r ¼15 nm, L ¼750 nm, and V4 ¼23.8 mV: (a) ΔT ¼ 721 C (switch OFF) and (b) ΔT ¼ 731 C (switch ON).
response of nanoswitch. Five levels of deflection can be detected in the switch response. First three ones (D1, D2, D3) are due to voltages V1, V2 and V3, respectively. Next, there is deflection due to adjusting voltage (D4) and finally, deflection resulting from rise in temperature (DT). Simulation shows that if the temperature rise is below the threshold (8 1C) the switch will not be turned ON (i.e. pull-in will not occur). As another example, a switch with r ¼15 nm and L ¼750 nm is considered. The switch will be adjusted for a temperature rise equal to 73 1C. For pre-deflection voltages V1 ¼170 mV, V2 ¼170 mV and V3 ¼26 mV are applied. Then from Fig. 6 for r ¼15 nm and L¼ 750 nm (red dashed line) V4 will be estimated to
be 24 mV, and by exact calculation V4 ¼23.8 mV. Results are shown in Fig. 11. Again, when the temperature rise is less than 73 1C the switch will be in OFF state.
5. Summary and conclusions Design and simulation of a thermal nanoswitch operated based on thermal expansion of a carbon nanotube was presented. Electrostatic, van der Waals, axial and nonlinear stretching forces have been considered to fully capture the response of the nanotube and Galerkin0 s method was used to reduce the governed
M. Rasekh et al. / Physica E 59 (2014) 210–217
equations. Results show the axial load effects on pull-in voltage and that the compressive loads reduce the pull-in voltage. By increasing the axial load, the nanotube becomes unstable and sticks to ground. The magnitude of load required to destabilize the nanotubes is related to the applied voltage. When nanotubes have fixed–fixed boundary conditions, rise in temperature will result in compressive axial load and by adjusting the applied voltage, switch can be set to be actuated at specific temperature rise. To this end, effect of geometrical parameters, including the air gap and the length of the tube, on the required applied voltage and the maximum temperature rise limit were investigated. Then, the relation between adjusting voltage and corresponding rise in temperature was obtained for various air gaps and lengths of the nanoswitch. Results show that as the air gap increases, the nanoswitch can detect more extended ranges of temperature rise with better resolution (sensitivity); however, the required applied voltage and consequently, energy consumption of the switch will also increase. Finally, some examples were given to show how the thermal switch is adjusted and the response of the switch was simulated using those adjustments. Results illustrated that carbon nanotubes can be utilized to design thermal nanoswitches. References [1] Q. Li, S.M. Koo, M.D. Edelstein, J.S. Suehle, C.A. Richter, Nanotechnology 18 (2007) 315. [2] G.W. Wang, Y. Zhang, Y.P. Zhao, G.T. Yang, J. Micromech. Microeng. 14 (2004) 1119.
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