Letter pubs.acs.org/NanoLett
Effective Nanometer Airgap of NEMS Devices Using Negative Capacitance of Ferroelectric Materials Muhammad Masuduzzaman* and Muhammad Ashraful Alam* School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, United States S Supporting Information *
ABSTRACT: Nanoelectromechnical system (NEMS) is seen as one of the most promising candidates for next generation extreme low power electronics that can operate as a versatile switch/memory/sensor/display element. One of the main challenges toward this goal lies in the fabrication difficulties of ultrascaled NEMS required for high density integrated circuits. It is generally understood that fabricating and operating a NEMS with an airgap below a few nanometer will be extremely challenging due to surface roughness, nonideal forces, tunneling, etc. Here, we show that by cascading a NEMS with a ferroelectric capacitor, operating in the negative capacitance regime, the effective airgap can be reduced by almost an order of magnitude, without the need to reduce the airgap physically. This would not only reduce the pull-in voltage to sub-1 V regime, but also would offer a set of characteristics which are difficult/impossible to achieve otherwise. For example, one can reduce/ increase the classical travel range, flip the traditional stable-unstable regime of the electrode, get a negative pull-out voltage, and thus, center the hysteresis around zero volt. Moreover, one can also operate the combination as an effective ferroelectric memory with much reduced switching voltages. These characteristics promise dramatic saving in power for NEMS-based switching, memory, and other related applications. KEYWORDS: NEMS, MEMS, negative capacitance, ferroelectrics
A
electrical scaling through an innovative use of a negative capacitor in series with the NEMS capacitor. Recently, ferroelectric (FE) negative capacitor9 connected in series with the classical gate oxide has drawn considerable attention for its ability to amplify the gate voltage and thereby to reduce the subthreshold swing below the 60 mV/decade limit. In this letter, we show that, in a similar manner, an FE capacitorcascaded in series with a NEMS and biased in the negative capacitance modecan reduce the “ef fective” airgap of a NEMS device to nanometer scale and the actuation voltage below 1 V. Since this approach does not require physical reduction of the NEMS airgap, it avoids the fabrication, reliability, and the nonideality issues. A closer inspection reveals that such cascading has far more significance, and we can achieve many other desirable characteristics. For example, we show that the physical travel range of the NEMS electrode can be modified below or above the classical 1/3 range, in principle, to any arbitrary limit. Second, we can make the pull-out voltage negative, a feature extremely useful in memory applications, because it avoids the use of an extra charge layer to center the hysteresis at 0 V.3 Finally, in certain operating mode, we can flip the stable-unstable regime of the NEMS which can be a complementary way of extending the travel range of a NEMS electrode.
micro/nano electromechanical system (M/NEMS) consists of a pair of electrodesone fixed and the other movableseparated by an airgap; see Figure 1a. In the ITRS roadmap, the NEMS devices have often been mentioned as one of the most promising candidates for a number of emerging device categories.1 The effective zero standby power dissipation and steep switching characteristics make it an ideal candidate for a logic switch (NEMFET),2 its intrinsic hysteresis suggests potential as a nonvolatile memory element (NEMM),3 and its high quality factor and sensitivity to changes in mass/stiffness promise applications as RF resonators4 or nanobiosensors.5 However, the most difficult challenge inhibiting the integration of these devices in next generation chips has been the reliable scalability of NEMS devices.1 For example, a low power memory/ switch requires sub-1 V actuation (VPI), which can be achieved only if the airgap is scaled to a few nanometer range.6 Such an extreme scaling poses a difficult fabrication challenge. Moreover, the reduced airgap leads to the introduction of many nonideal physical effects, such as the tunneling current (for less than 2 nm) degrading the subthreshold characteristics, the surface forces7 (van der Waals/Casimir) causing stiction, etc. Therefore, although NEMS switches with 4 nm airgap (sub-1V VPI) using a pipe-clip structure have been reported,6 the most advanced devices based on conventional geometry rely on an airgap of 15 nm (and pull-in voltage, VPI = 13V).8 In this letter, we provide a fundamentally different route to NEMS airgap scaling that obviates the physical scaling in favor of © 2014 American Chemical Society
Received: February 3, 2014 Revised: April 28, 2014 Published: May 5, 2014 3160
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Figure 1. (a) Schematic of a parallel plate NEMS capacitor in series with a ferroelectric (FE) capacitor. (b) Energy landscape at zero applied voltage is shown for a NEMS, an FE, and the combined series structure. All the energy landscapes show three possible equilibrium locations ((dU/dQ) = 0). For NEMS, the stable equilibrium solution lies at the center, whereas for FE, those lie at two opposite charge locations. Once they are combined in series, the net system could have a NEMS-like or a ferroelectric-like landscape (the former is shown). (c, and magnified in d) NEMS and FE energy landscape show that individually the former is not in a pull-in condition. Yet, the total energy reaches the pull-in (see part d) due to the energy contribution of the FE capacitor.
voltage across the NEMS is VN = Q/CN, where CN = (ϵ0AN)/(x0 − x). Substituting the value of x from eq 1, and defining positive quantities αN ≡ x0/(ϵ0AN) and βN ≡ (1/(2KN(ϵ0AN)2)), we deduce the equation of a NEMS as follows:
It is well-known that an FE capacitor is characterized by a negative capacitance around zero charge, but the capacitor−by itself−cannot be operated at this unstable point. Operation of an FE capacitor around this negative capacitance regime is possible only if the overall system is stabilized by adding a series capacitor.9 The stability comes from the fact that the charge state of the combined structure is determined by, not the FE alone, but the energetics of the overall structure. An elementary circuit analysis shows that, in the negative capacitance regime, the voltage across the FE capacitor is negative, even when a positive voltage is applied across the combined structure. Since the total applied voltage is constant, therefore, the voltage across the remaining series capacitor is higher than the applied voltage. Such voltage amplification is the basis of sub-60 mV/decade operation of the negative capacitance field effect transistor (NCFET) where the gate stack contains the FE film, and the remaining gate and the channel capacitance act as the series capacitor that stabilizes the FE film.9−13 Since a NEMS can be viewed as a special type of parallel plate capacitor, in an FENEMS cascade, the NEMS acts as the stabilizing series capacitor. Likewise, the NEMS experiences the voltage amplification that results in lower applied voltage required to pull-in the NEMS. In addition to the pull-in voltage, the FE-NEMS cascade also affects the travel range, pull-out, and the stability of the NEMS, often in a positive way, as discussed in the following sections. Unlike a simple capacitor, however, a NEMS electrode is not fixed and moves with the voltage. In fact, similar to an FE capacitor, a NEMS also has an unstable regime of operation, defined by negative capacitance (dQ/dV < 0). In certain mode, the role of FE and NEMS can be reversed that can lead to reduced switching (coercive) voltage for the FE capacitor. Q−V Characteristics of a NEMS Capacitor. To explain the operation of an FE-NEMS cascade, we will first derive the charge−voltage (Q−V) relationship for a NEMS capacitor. When a charge Q is applied on a NEMS capacitor, the movable electrode experiences an electrostatic force Q2/(2ϵ0AN), and is displaced by a distance x (Figure 1a). Here, ϵ0 is the permittivity of air and AN is the area of the electrode. Using the simple onedimensional spring-mass model,14 the displacement is limited by the spring force KNx, where KN is the spring constant of the electrode. At equilibrium, the net force on the movable electrode of the NEMS is zero, i.e., Fnet = −KNx + (Q2/(2ϵ0AN)) = 0, giving
VN = αN Q − βN Q 3
The state of the NEMS at any instant is defined by the electrode charge, Q(VN) (Figure 2a, gray line), as found from eq 2 for an applied bias, VN. The electrode location corresponding to the charge (at stable or unstable equilibrium) can be found from eq 1, and is shown in Figure 2b (gray line). A direct experimental demonstration of Figure 2, parts a and b, can be found in ref 15. The system energy is given by UN = (1/2)αNQ2 − (1/4)βNQ4 − VNQ (Figure 1b, NEMS) and the equilibrium solutions lie in the crest and valley locations (dUN/dQ = 0). The voltage at which the stable equilibrium solution (valley) disappears (at the critical charge, Qcr = (αN/3βN)1/2 as shown in Figure 2a, (diamond), see Supporting Information for derivation) defines the pull-in voltage, and is given by VPI =
Q 2KN ϵ0AN
2αN 3 3
αN = βN
8(KN )(x0)3 27ϵ0AN
(3)
The corresponding electrode location defines the travel range, xTR = x0/3 (Figure 2b), and can be obtained by inserting Qcr in eq 1a well-known result for isolated parallel plate M/NEMS.14 Q−V Characteristics of an FE Capacitor. The electric fieldpolarization (EF − P) characteristics of a ferroelectric capacitor can be expressed as16 EF = α′P + β′P3 + γ′P5 + .... Although additional terms may be necessary to characterize the double well energy landscape (Figure 1b, FE) with greater accuracy, a two term approximation (with α′ < 0, β′ > 0, and γ′ = 0) provides the simplest representation of the same, and is adopted here to demonstrate the concept analytically. The additional terms would not change the essence of the following argument. Since EF = VF/t1, where VF is the voltage and t1 is the thickness of the dielectric, and the charge across the ferroelectric capacitor is given by, Q = PAF, where AF is the area of the FE capacitor, we can deduce the Q−V relationship (Figure 2c) for the ferroelectric capacitor as VF = −αFQ + βF Q 3
(4)
β′t1/AF3
where the coefficients αF ≡ −α′t1/AF and βF ≡ are both positive quantities. Note that, for the ferroelectric capacitor, the first term is negative (compare, for a simple capacitor, V = +(1/C)Q), a reason why a ferroelectric capacitor is sometimes referred as a
2
x=
(2)
(1)
Equation 1 defines the location of the electrode (x < x0) for any arbitrary charge Q, where x0 is the nominal airgap. Now, the 3161
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Q−V Characteristics of an FE-NEMS Cascade. When the two capacitors are connected in series, the net voltage (VFN) can be found by adding eqs 2 and 4 as VFN = (αN − αF )Q − (βN − βF )Q 3 ≡ αNEff Q − βNEff Q 3 ≡ rαNαN Q − rβNβN Q 3
(5)
f where the factors rαN ≡ (αEf N /αN) = 1 − (t1/x0)(AN/AF)(|α′|ϵ0) Ef f and rβN ≡ (βN /βN) = 1 − (2β′KNϵ02)((t1AN2)/AF3) depends on the dimensions and the material parameters of both the FE and the NEMS capacitors. Such an FE-NEMS combination reduces f the effective charge coefficients (e.g., αEf N = αN − αF, in eq 5), and correspondingly, flattens the energy landscape (Figure 1b) of the combined system. This flattening of the overall energy landscape def ines the essential innovation of the work, which, owing to the opposing characteristics of the FE and the NEMS energy landscapes (Figure 1b), is valid regardless of the complexity of the models used to represent the FE or the NEMS capacitor (e.g., Euler−Bernoulli equations17 instead of 1D spring mass model for NEMS). A series combination of similar capacitors (e.g., two classical, ferroelectrics, or NEMS capacitors) cannot achieve such a cancellation of charge coefficients. Now, depending on the sign f Ef f of the effective charge coefficients (αEf N and βN ), the combined structure could have a NEMS or an FE characteristics. Effective NEMS Mode. One may operate the FE-NEMS cascade in an effective NEMS mode (compare eq 5 and eq 2) by f Ef f ensuring that both αEf N and βN are positive (or, 0 < (rαN,rβN) < 1). Because of the change in the charge coefficients, the critical charge for pull-in for the effective structure has changed to (see Figure 2d, “+” and “×”)
Q cr , Eff =
αNEff 3βNEff
=
rαN rβN
Q cr
Using Qcr,Ef f in eq 1, the travel range is found as ⎛ r ⎞⎛ ⎞ 1 α xTR , Eff = ⎜⎜ N ⎟⎟⎜ ⎟x0 ⎝ r ⎝ βN ⎠ 3 ⎠
(6)
which is different from the classical 1/3 range. To understand physically why the travel range changes, Figure 1c (magnified in Figure 1d) shows the individual (UN and UF) as well as the combined energy landscapes (U = UN + UF) of the FE-NEMS combination at the pull-in voltage (see methods in the Supporting Information). Note that although the NEMS itself is “not ready” to pull-in (Figure 1d), the energy landscape of the ferroelectric is such that the overall energy of the system warrants pull-in condition. The new pull-in voltage (VPI,Ef f) of the series structure compares with the original VPI of the NEMS (see eq 3) as-
Figure 2. (a) Q−V characteristics, and (b) the corresponding electrode displacement of a NEMS are shown in gray lines. When operating alone, the NEMS is stable below Qcr (above 1/3 in part b). (c) FE hysteresis shown in gray lines. (d) When a NEMS and an FE are combined in series, the new Q−V characteristics (green-stable, and red-unstable) has a different critical charge Qcr,Eff, and travel range xTR,Ef f, which is mapped in parts a and b, respectively (+). The component voltages for both stable and unstable solutions in part d are plotted in part e. The same in parts a−e is repeated in parts f−j, respectively. This time, the overall system has a net ferroelectric behavior (compare part i with part d).
VPI , Eff =
2αNEff 3 3
αNEff βNEff
⎛ rαN ⎞ ⎟V = ⎜⎜rαN ⎟ PI r βN ⎠ ⎝
⎛ rαN ⎞ ⎛ 8(KN )(x0)3 ⎞ ⎟ ⎜ ⎟ = ⎜⎜rαN ⎟ r βN ⎠ ⎝ 27ϵ0AN ⎠ ⎝
negative capacitor for charge magnitude close to zero.9 Also note that both eqs 2 and 4 have the same form except that the signs of the coeff icients are opposite, which is also reflected in the inverted energy landscape (UF = −(1/2)αFQ2 + (1/4)βFQ4 − VFQ) and the two stable minima of the ferroelectric capacitor (Figure 1b), as compared to a single minima for a NEMS capacitor.
(7)
Note that, if one would start with an standalone NEMS with scaled parameters xEf f and KEf f, where xEff = rαNx0 and KEff = KN /rβN 3162
(8)
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Figure 3. Energy landscapes of the FE-NEMS system are shown for the pull-in condition (a) and for different pull-out conditions (b−d). (a) After pullin, the charge is limited by the pull-in capacitance (vertical line). (b−d) During pull-out, this charge (vertical line) coincides with the unstable peak on the right, representing the pull-out charge, (QPO,1−3, respectively for three different biases). An unstable equilibrium exists and the system is pulled out. (d) If the pull-out voltage is −VPI, then the system is pulled-in again with a reversed charge.
one would achieve the same Q−V relationship as the FE-NEMS cascade (eq 5), and would get a pull-in voltage as,14 VPI, Scaled = (8(KN/rβN)(rαNx0)3/(27ϵ0AN))1/2, which is same as eq 7. Thus, the series structure effectively reduces the airgap of the NEMS as in eq 8, as long as pull-in is concerned. For a quantitative example, we take a NEMS having a TiN cantilever, with Young’s modulus =450 GPa,18 width/length/ thickness =3/6/0.5 μm, and an airgap x0 = 15 nm.8 The ferroelectric capacitor is SBT (Sr0.8Bi2.2Ta2O9) (α′ = −6.5 × 107 m/F, β′ = 3.75 × 109 m5F/C2 and γ′ = 0, at room temperature19) with t1 = 500 nm and AF = 0.4 μm2. Such an FE-NEMS combination gives rαN = 0.137, rβN = 0.225. This reduces the effective airgap from 15 nm to 2 nm, the pull-in voltage from 1.8 to 0.19 V (a factor of 9.4) and the travel range from 5 to 3 nm. The dramatic reduction in VPI, Eff by about an order of magnitude can be traced to two factors: (i) the reduced Qcr, Ef f (or travel range), and (ii) the voltage amplification,9 as discussed. This can be understood from Figure 2e which shows the component voltages (VF and VN) as a function of the total applied voltage, VFN. The positive applied bias where the pull-in occurs is denoted by the symbol “+”. Note that although VFN is positive, the voltage across the FE (VF) is negative. This results in an amplification of the applied voltage across the NEMS (i.e., VN = VFN − VF = VFN + |VF|). Thus, pull-in at much lower applied voltage has been possible. Finally, combining Figure 2, parts d and e, the equilibrium charge states in the series configuration have been mapped in Figures 2a−c (green (stable) and red (unstable) regions, separated by “+” or “×”), showing the operating regimes of the NEMS and the FE. Note that, in the stable equilibrium conditions of the series configuration (green), the FE capacitor operates in the unstable negative capacitance regime (dQ/dVF < 0), however, the NEMS capacitor operates in the stable positive capacitance regime (dQ/dVN > 0), see green regions of Figure 2, parts c and a, respectively. The modified critical charge and the travel range are also evident from Figure 2a,b (symbol “+”), respectively. Pull-out. After pull-in (and as long as it remains in the pulledin state), see Figure 3a, the charge of any NEMS capacitor is determined by the thickness of the dielectric layer on the bottom electrode td, or equivalently, the pull-in capacitance (CPI = (ϵ0AN)/(td/ϵr) ≡ 1/αPI, where ϵr is the relative permittivity of the dielectric layer) as VN = αPIQ
VPO , Eff = rV (VPO , Eff )
(αNEff − rV (VPO , Eff )αPI )αPI 2 βNEff
(10)
For the FE-NEMS cascade, however, eq 10 is an implicit expression, containing the voltage ratio rV(VFN) ≡ VFN/VN, and must be solved numerically (Figure 2e). In the absence of the FE capacitor, rαN = rβN = rV = 1, and eq 10 reduces to the classical formula20 VPO = (2KN(x0 − td/ϵr)(td/ϵr)2/ϵ0AN)1/2 for any given dielectric material and thickness (td/ϵr). Conversely, one can find the required dielectric thickness (or CPI) for any desirable VPO. For that, one first finds the pull-out charge (QPO) for that VPO (see Figure 3b-d), then the corresponding electrode displacement xPO from eq 1, and finally, the required capacitance as CPI = ((ϵrϵ0AN)/(x0 − xPO)) ≤ ∞. For example, for VPO = 0 (Figure 3c), one finds that QPO = (αEff N / f 1/2 βEf ) (see Supporting Information), x = (r /r )x , and N PO αN βN 0 CPI(VPO = 0V) = ((ϵrϵ0AN)/(x0 − (rαN/rβN)x0)). For a single NEMS, since rαN = rβN = 1, thus CPI(VPO = 0V) = ∞, or equivalently, the required dielectric thickness (td) is zero−a wellknown result. Since, CPI = ∞ is the maximum possible capacitance, VPO cannot be negative for an isolated NEMS. However, for the FE-NEMS structure, since (rαN, rβN) < 1, one may have negative VPO as long as CPI ≤ ∞. For example, for VPO = f Eff 1/2 −VPI (Figure 3d), one finds that QPO = 2(αEf (see N /3βN ) Supporting Information), xPO = (4/3)(rαN/rβN)x0, and CPI(VPO = −VPI) = ((ϵrϵ0AN)/(x0 − (4/3)(rαN/rβN)x0)). With (rαN/rβN) < (3/4), CPI is positive and finite, and thus, negative VPO is possible. However, it should be noted that a VPO ≤ −VPI is not desirable, as the NEMS will be pulled-in again with reversed charge (Figure 3d). To avoid this condition, the maximum pull-in capacitance is CPI(VPO = −VPI) ≤ ∞, or equivalently, the minimum dielectric thickness is td,min = (x0 − (4/3)(rαN/rβN)x0) ≥ 0. Note that, in addition to the reduction of VPI, the capability of designing a NEMS with a negative VPO is another advantage of the FE-NEMS cascade. Regarding the experimental realization of the proposed concept, we make the following observations. First, if we simply consider the reduction of VPI in the effective NEMS mode, our concept relies on the key enabling fact that an FE can act as a negative capacitor when connected in series with a simple positive capacitor. There are multiple experimental demonstration of this fact, starting from the proof-of-concept experiment21 to the demonstration of sub 60 mV/decade for transistor applications.10−13 A NEMS capacitor mathematically differs from a simple positive capacitor only by the second term of eq 2 involving βN. Apart from reducing the ferroelectric parameter βF to an ef fective βF (= βF − βN), this additional term
(9)
At the point of pull-out, the charge reaches an unstable equilibrium (QPO, see Figure 3b) and the pull-out voltage (see Supporting Information) is a subset of the solutions of eqs 9 and 5, as3163
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Interestingly, in the effective FE mode, the stable vs unstable regions of the NEMS are flipped! The NEMS electrode can now be stabilized below the critical point that separates the two regions in the airgap (Figure 2g, green (stable) and red (unstable) regions), which is opposite to any traditional NEMS where the electrode is stable only above the critical point in the airgap (Figure 2b). The flipping of the stable vs unstable regions occurs, because, in an effective FE mode, the f Eff 1/2 overall system is unstable at |Q| < Q′cr,Eff ≡ (αEf and F /3βF ) stable at |Q| > Q′cr,Ef f (Figure 2i), which is exactly opposite to a traditional NEMS operation (compare with Figure 2d). The operation of a NEMS below the critical point in the airgap can be viewed as a complementary way of extending the travel range24 and can potentially be useful where continuous modulation of airgap is required, such as the IMOD-based displays for larger color range,25,26 etc. In summary, we have shown that a NEMS capacitor connected in series with a ferroelectric capacitor operating in the negative capacitance regime can be stabilized, and effectively, acts like a NEMS with a much smaller airgap and reduced pull-in voltage. Such combination also offers tuning of the travel range by choosing ferroelectric of appropriate material and dimensions. We have also shown that, with this effective NEMS mode, the pull-out voltage can be made negative and the hysteresis can be centered around zero volt. Such a feature is not possible even for a scaled NEMS, and is extremely useful in operating the NEMS as a nonvolatile memory. In the effective FE mode, we have shown that the switching voltage of the ferroelectric capacitor is significantly reduced, and the NEMS can operate below the critical point of the airgap. All the related expressions are provided analytically (also see Supporting Information).
does not fundamentally affect the voltage amplification mechanism of a negative FE capacitor as have been demonstrated in the above experiments. For a complementary proof, a MEMS with a series positive capacitor (as opposed to our proposed negative capacitor) has been experimentally demonstrated to effectively increase the travel range so that the NEMS operates within the unstable 2/3 regime.22 As our theory would also have predicted, both the effective airgap and the VPI of the MEMS have been experimentally shown to increase in such configuration.22 The proposed use of negative capacitance to reduce the airgap, and thereby VPI, is conceptually unique but mathematically not different from the above case. Therefore, our proposed approach is both conceptually and experimentally viable, from both ferroelectric and NEMS perspectives. In fact, the concept of the combination of materials with positive and negative characteristics is even more general, and there have been several recent examples in other fields as well. For example, one can combine materials of negative stiffness (naturally unstable) with that of positive stiffness to achieve higher stiffness property of the effective material.23 The essential idea in all these examples (including the one proposed here) is that combining a stable system with an otherwise unstable system can produce an overall system which is stable and have improved characteristics. Before concluding, we will present a final example of such stable-unstable cascading. This will demonstrate that using the same FE-NEMS cascade, one can also achieve a stable effective ferroelectric capacitor, as described in the following (see Figure 2f−j): Eff Effective FE Mode. In eq 5, both αEff N and βN (or rαN and rβN) can be made negative with appropriate choice of parameters. If Eff Eff Ef f Ef f we define positive quantities αEff F ≡ −αN , βF ≡ −βN , rαF ≡ αF / αF = −rαN/(1 − rαN) and rβF ≡ βEff F /βF = −rβN/(1 − rβN), where 0 < (rαF, rβF) < 1, then eq 5 can be written as,
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* Supporting Information Calculation methods, derivation of different critical charges for pull-in and pull-out, derivation of VPO, and a table summarizing the equations. This material is available free of charge via the Internet at http://pubs.acs.org.
VFN = −(αF − αN )Q + (βF − βN )Q 3 ≡ −αFEff Q + βFEff Q 3 ≡ −rαFαFQ + rβFβF Q 3
■
(11)
VC , Eff
αFEff βFEff
⎛ rαF ⎞ ⎜ ⎟V = ⎜rαF C rβF ⎟⎠ ⎝
AUTHOR INFORMATION
Corresponding Authors
This represents an ef fective FE mode (compare eq 11 with eq 4), where the Q−V is similar to a ferroelectric capacitor (note the difference between Figure 2i and Figure 2d). As before, the voltages across the two components are shown in Figure 2j and the equilibrium charge states and electrode locations are mapped in Figure 2f−h to show the operating regimes of the NEMS and the ferroelectric capacitor in the FENEMS cascade. The switching (coercive) voltage of the effective FE (VC, Ef f) is modified from that of the single FE (VC) as 2α Eff = F 3 3
ASSOCIATED CONTENT
S
*(M.M.) E-mail:
[email protected]. *(M.A.A.) E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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(12)
Note that VC, Eff could be reduced by choosing appropriate values of rαF and rβF. For example, in the previous example, if the NEMS cantilever has width/length/thickness =3.3/6/0.5 μm, the ferroelectric capacitor has a thickness of 600 nm, and if everything else remains the same, then the FE-NEMS cascade operates in the effective FE mode with rαF = 0.122 and rβF = 0.192, which reduces the coercive voltage to 9.8% of the original value (VC). 3164
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