LETTER pubs.acs.org/NanoLett
Parametric Resonance in Nanoelectromechanical Single Electron Transistors Daniel Midtvedt,* Yury Tarakanov, and Jari Kinaret Department of Applied Physics, Chalmers University of Technology, SE-412 96 G€oteborg, Sweden ABSTRACT: We show that the coupling between singleelectron charging and mechanical motion in a nanoelectromechanical single-electron transistor can be utilized in a novel parametric actuation scheme. This scheme, which relies on a periodic modulation of the mechanical resonance frequency through an alternating source-drain voltage, leads to a parametric instability and emergence of mechanical vibrations in a limited range of modulation amplitudes. Remarkably, the frequency range where instability occurs and the maximum oscillation amplitude, depend weakly on the damping in the system. We also show that a weak parametric modulation increases the effective quality factor and amplifies the system’s response to the conventional actuation that exploits an AC gate signal. KEYWORDS: Carbon nanotubes, nanoelectromechanical systems, parametric instability, parametric resonance, single-electron transistor
C
arbon nanotube (CNT)-based nanoelectromechanical systems (NEMS) hold promise, for example, for tunable radio frequency (RF) electronics due to ultralow masses and high mechanical stiffnesses that enable operating frequencies up to gigahertz range. Mechanical motion in these structures can be efficiently actuated electrostatically, and the resonance frequency can be tuned over wide ranges by changing the gate bias.1-7 Moreover, low mechanical losses corresponding to quality factors up to 105 have been shown in suspended CNT-devices,7,8 which makes them compatible with the established RF technologies. However, a reliable operation of CNT-based RF devices is hindered by difficulties in controlling the damping in the system during fabrication. As a result, the bandwidth and the attenuation factor, or output amplitude, vary unpredictably from device to device. One way to overcome this uncertainty is a setup where the losses can be actively tuned. Such a setup has recently been demonstrated by Lassagne et al.6 The current through a quantum dot residing in the suspended CNT, and hence the electrical losses, were shown to be directly coupled to the oscillation of the CNT’s electrochemical potential. A quality factor change by up to 75% was reported. The parametric resonance, an effect utilized in microelectromechanics,9-11 is another strategy to alleviate the role of mechanical losses. A parametric instability that results from a periodic modulation of the spring constant leads to oscillations in a narrow frequency band. The amplitude of the oscillations is set by nonlinearities in the system and does not depend on mechanical losses. The bandwidth is determined by the amplitude of the modulation of the spring constant and can be controlled. In this Letter, we propose a parametric resonance-based actuation scheme for nanoelectromechanical single-electron r 2011 American Chemical Society
transistors (NEM-SETs) where the mechanically active structure is a nanoscale beam such as a suspended CNT. The modulation of the spring constant is achieved through a periodic modulation of the voltage across the SET, which results in a parametric instability and hence mechanical vibrations. In contrast to other devices employing parametric resonance, the instability is limited to a finite range of modulation amplitudes. The frequency range in which the instability takes place and the maximum oscillation amplitude depend strongly on the modulation voltage but only weakly on mechanical damping. These properties dramatically reduce the requirement on losses control in the fabrication stage of NEM-SET beam resonators. Further, we show that for sufficiently low modulation amplitudes, no parametric instability takes place but the mechanical quality factor can be effectively increased as a manifestation of parametric amplification, allowing a flexible quality factor tuning over a wide range. The attractiveness of the parametric amplification due to low added losses has been recognized.12 The device we investigate is schematically depicted in Figure 1; a single-walled CNT is suspended above a gate electrode and contacted by source and gate electrodes, and a quantum dot (QD) is defined by potential barriers or defects in the tube. The structure is similar to the one studied experimentally by Lassagne et al.6 For concreteness, in our modeling we have used the device parameters for the structure studied experimentally: resonance frequency ω0/2π = 50 MHz, quality factor due to external losses Q0 = 400, ratio of gate capacitance to the spring constant C2g/K = 13 (aF)2nm/pN, charging energy Ech = 3 meV and corresponding total capacitance Cdot = 59 aF, and tunneling rates for both barriers Γ0 = 1.28 1010s-1. Received: October 19, 2010 Revised: January 10, 2011 Published: March 04, 2011 1439
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F = -[(dCg)/(du)]Vg,0ep/Cdot, where Vg,0 is the gate bias voltage and p is the dot occupation probability that obeys the rate equation p_ ¼ ðΓin, s þ Γin, d Þ - 2Γ0 p
ð2Þ
with the tunneling rate from the lead into the dot given by μ - μlead Γin, lead ¼ Γ0 f dot ð3Þ kB T Figure 1. Illustration of the system at hand. A suspended carbon nanotube with a quantum dot formed in it is connected to source and drain via tunnel barriers. Beneath the tube there is a gate electrode that can be used to shift the chemical potential of the quantum dot.
here, f(x) = 1/(1 þ ex) is the Fermi function, μdot - μlead = eVgC0 g/Cdotu - eVlead, and lead = s,d stands for source and drain electrodes, respectively. In the limit of fast tunneling rates, Γ0 . ω0, the occupation probability for the dot is p = Γin/2Γ0, and the current through the dot is given by14 I = Γin,s - Γin,d. When symmetric oscillating voltages Vsd = ( (1/2)VsdAC cos(ωt) are applied to the source and drain contacts, the spring constant shift δk = -dF/du becomes dCg e 2 1 0 eVsd f V0 cosðωtÞ δkðt, Vsd Þ ¼ du Cdot 2kT 2kT eVsd þf 0 cosðωtÞ ð4Þ 2kT which is a periodic function of time. In the Fourier expansion of eq 4, only cosine-terms at even multiples of ω are nonzero,
Figure 2. Contour plot of the deflection amplitude (in nanometers) at the driving frequency ω ≈ ω1 when the dot is biased at the degeneracy point by gate voltage. Note the boomerang shape of the region where the excitation occurs. The solid black line shows the boundaries of the instability evaluated using eq 7, while the dashed line gives the renormalized resonance frequency.
The shift of the spring constant, and hence of the resonance frequency, arises from the dependence of the electrostatic force on the tube position. This shift peaks when the chemical potential of the tube is aligned with the chemical potential of source and drain. When an oscillating voltage Vsd(t) is applied between the source and drain electrodes and the gate voltage is kept constant, the alignment occurs twice each period, leading to a double frequency modulation of the spring constant, characteristic for a parametric oscillator. Using this method of actuation, the mechanical motion is induced purely by the dynamical mismatch between dot and control charges, that is, by the coupling between electrical and mechanical properties of the tube. To describe the mechanical vibrations of the CNT coupled to the single-electron dynamics of the QD, we use the model described in ref 6. We model the fundamental mode oscillations of the suspended CNT with the harmonic oscillator equation (we disregard mechanical nonlinearities, that is, dependencies of the mechanical resonance frequency ω0 on deflection u) ::u þ β u_ þ ω2 u ¼ F=m 0 0
ð1Þ
where m is the mass of the oscillator and F is the force acting on the oscillator. The quantum dot is assumed to be semiconducting with level spacing exceeding the source-drain voltage and the temperature. Specifically, we consider tunneling into and out of only a single level of the dot. In capacitive terms,13 the electrostatic force arising from the single-electron tunneling is then
n¼¥
δkðt, Vsd Þ ¼ m
∑
n¼0
h2n ðVsd Þcosð2ωtÞ
ð5Þ
Keeping only the lowest frequency terms in the expansion, we can rewrite the oscillator equation as ::u þ β_u þ ω2 ½1 þ ~h ðV Þcosð2ωtÞu ¼ 0 ð6Þ 1
2
sd
where ω1 (ω02 - h0)1/2 and β β0 þ h0/Γ are the renormalized resonance frequency and damping coefficient, respectively, and ~h2(Vsd) = h2/(ω20 - h0) is the amplitude of the parametric modulation. Equation 6 is recognized as the Mathieu equation.15,16 At low ~h2, or if the driving frequency ω is far from ω1, u 0 is the only solution. However, if the condition ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1~ jω - ω1 j < ð7Þ h2 ω1 - β2 2 is met, this solution is unstable. Any disturbance in initial conditions leads to a parametric instability, the amplitude phase-locks to the driving signal and increases exponentially in time. In real systems, the amplitude saturates at a value set by nonlinearities. To study the effect of a parametric resonance on our system, we integrate numerically eqs 1 and 2 until transient solutions vanish. The resulting deflection amplitude is plotted in Figure 2 as a function of amplitude and frequency of the driving voltage. We see that there is a well-defined, boomerang-shaped region in which this internal driving occurs, and there are no oscillations outside this region. Its finite extension in driving voltages is due to a nonmonotonic dependence of the modulation ~h2 on the source-drain voltage amplitude. Using eq 7 we can estimate the boundaries of the actuation region as well as the resonance frequency of the oscillator. These are superimposed in Figure 2. There is a good agreement between this estimate and the full 1440
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simulations that confirms that the parametric instability is responsible for the oscillations. The slight discrepancy in the width of the actuation region can be attributed to higher order terms in the Fourier expansion of the electrostatic force. We note that higher order harmonics ~h2n give rise to additional parametric instabilities when the system is driven at frequencies ω ≈ ω1/n, n is an integer, occurring in a region similar to the leading order instability shown in Figure 2. On the other hand, higher order parametric instabilities given by higher powers of ~h2 do not arise due to high losses level in the system. We now analyze the system upon actuation of vibrational motion in the region of parametric instability in more detail. The parametric effect arises from a modulation of the spring constant whenever the chemical potential of the dot aligns itself with either, or both, of the leads. It can readily be shown that this modulation will be most efficient if the phase and amplitude of the chemical potential of the dot μdot lock to either of the leads; interestingly, this implies that the saturation amplitude depends strongly on the applied source-drain voltage, but only weakly on the mechanical losses. The mechanical saturation amplitude is shown in Figure 3A as a function of frequency and in Figure 3B as a function of the applied source-drain voltage at the resonance frequency of the tube. The parametric instability makes the system robust to variations in mechanical losses. It can also be used to amplify the response to other driving signals beyond the limits set in the linear regime by the intrinsic damping. If, in addition to the oscillating source-drain bias Vsd, a small AC signal VAC g cos(ωt) is applied to the gate, a driving term Fdr = C0 V0VAC g cos (ωt) appears in eq 6. When the parametric modulation amplitude is below the threshold for instability given by eq 7, the amplitude of deflection at resonance frequency is given by uAmp
C0 g Vg, 0 Vg, AC ω1 β
1 ~h2 ω1 2 1-
ð8Þ
2β
From this we see that the parametric pumping effectively decreases the dissipation, β f β(1 - (h2ω1/2β)2), amounting to an amplification of the external signal and a reduction of the width of the resonance curve, as illustrated in Figure 4. The saturation of deflection amplitude is the dominant limiting mechanism for the decrease in dissipation, and thus very high effective quality factors are achievable. The parametric instability can be detected by monitoring the current through the CNT. As mentioned previously, the motion of the parametrically actuated CNT is characterized by a dynamical alignment of the chemical potentials of the dot and the source or drain electrode. For a static CNT with identical source and drain junctions, the chemical potential is μdot = Static (1/2)(μs þ μd) and ΓStatic in,s = Γout,d , while for a parametrically actuated dot, μdot ≈ μs and μdot - μd ≈ eVsd . kT, which results in a current limited by the tunneling rate at the source junction Γin,s(t) ≈ (1/2)ΓStatic in,s . Hence, in the instability region the current amplitude is reduced by one-half as seen in Figure 5. Consequently, the differential AC conductance (dI)/(dVAC sd ) is negative at the onset of the parametric instability. To simplify experimental detection, the high frequency AC current can be mixed down to low frequencies by superimposing a weak test signal VAC test cos[(2ω þ δω)t] to Vsd(t). The mixing current, which arises from inherent nonlinearities of the system, is depicted in the inset
Figure 3. (A) Saturation amplitude of oscillations as a function of AC driving frequency at VAC sd = 1 mV (solid line) and Vsd = 2 mV (dashed line). (B) Oscillation amplitude of the CNT driven at ω = ω1 plotted against the chemical potential of the source for quality factors Q = 200, 400, and 800 (red, green, and light blue lines, respectively) of the suspended CNT (solid lines), and the CNT oscillation amplitude corresponding to amplitude-locking of the chemical potentials (dashed line).
to Figure 5 as a function of frequency for fixed VsdAC and detuning δω. The current reduction in the instability region can clearly be seen. The parametric instability can also be detected by slightly biasing the system off the degeneracy point and measuring the resulting DC-current. The DC current at the source electrode will be given by Idc ¼ ÆΓin, s æ - Γ0 Æpæ
ð9Þ
where ÆXæ denotes the time averaged value of X. Inserting the stationary solution to the rate equation into eq 9, we find μdot ðtÞ - μs ðtÞ μdot ðtÞ þ μs ðtÞ Idc ¼ Γ0 f -f kT kT ð10Þ Writing μdot = Æμdotæ þ μAC dot(t) and Taylor expanding eq 10 around Æμdotæ, we find that the lowest order contribution to the DC current is 00
Idc f ðÆμdot æÞÆμdot ðtÞμs ðtÞæ
ð11Þ
Since μdot(t) and μs(t) phase-lock, the time-averaged value will be nonzero. Current rectification in a NEMS has previously been discussed by in a different system by Ahn and co-workers.17 In conclusion, we have analyzed parametric effects in a nanoelectromechanical single electron transistor and have shown that mechanical oscillations can be excited by a periodically varying voltage applied between the source and drain electrodes. For a finite range of applied voltages and frequencies, the system experiences a parametric instability where the mechanical oscillation amplitude is limited by electrical nonlinearities. Outside the 1441
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Figure 4. Parametric amplification of an external signal. By tuning the source-drain voltage to a point slightly below the threshold for parametric instability, we can achieve substantial amplification of a small AC oscillating gate voltage. Here Vg(t) =VAC g cos(ωt) and Vsd = Vsd cos(ωt) AC AC = 1 μV and V = 0 (solid line) and V = 0.2 mV (dashed with VAC g sd sd line).
LETTER
(5) Lassagne, B; Garcia-Sanchez, D; Aguasca, A.; et al. Nano Lett. 2008, 8, 3735. (6) Lassagne, B; Tarakanov, Y; Kinaret, J.; et al. Science 2009, 325, 1107. (7) Huttel, A. K; Steele, G. A.; et al. Nano Lett. 2009, 9, 2547. (8) Steele, G. A.; Huttel, A. K.; et al. Science 2009, 325, 1103. (9) Unterreithmeier, Q. P.; Weig, E. M.; Kotthaus, J. P. Nature 2009, 458, 1001. (10) Turner, K. L.; Miller, S. A.; Hartwell, P. G.; MacDonald, N. C.; Strogatz, S. H.; Adams, S. G. Nature 1998, 396, 149. (11) Rhoads, J. F.; Shaw, S. W.; Turner, K. L.; Baskaran, L. J. Vibr. Acoust. 2005, 127, 423. (12) Bergeal, N; Vijay, R.; Manucharyan, V. E.; Siddiqi, I; Schoelkopf, R. J.; Girvin, S. M.; Devoret, M. H. Nat. Phys. 2010, 6, 296. (13) It is worth to note that the capacitive model is not strictly applicable in our case. The basic requirement is that the energy of the relevant level in the quantum dot depends on the position; in the geometry at hand, this arises from an electric field between the gate and the CNT. (14) When the position of the CNT changes, the occupation probability of the dot changes and a displacement current arises; the assumption Γ0 . ω0 allows one to neglect this displacement current on the mechanical oscillations time scale. (15) Landau, L. D.; Lifshitz, E. L. Mechanics, 3rd ed.; translation from Russian by Sykes, J. B.; Bell, J. S.; Pergamon: Oxford, 1977. (16) Nayfeh, A. H.; Nook, D. T. Nonlinear Oscillations; Wiley: NewYork, 1979. (17) Ahn, K.-H; Park, H. C.; et al. Phys. Rev. Lett. 2006, 97, No. 216804.
Figure 5. A surface plot of the amplitude of the AC current against the driving frequency f and the source-drain voltage amplitude VAC sd . The region where the current is reduced corresponds to the parametric excitation of mechanical vibrations shown in Figure 2. Inset: the mixeddown current using the method proposed in the text using a mixing frequency of ∼50 kHz at a source-drain voltage VAC sd of 2 mV.
instability region, an AC source-drain voltage can be exploited to parametrically amplify the response to a gate excitation. As direct detection of the mechanical motion of nanoscale objects is challenging, we suggest a more practical detection scheme.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We acknowledge fruitful discussions with Leonid Gorelik, Andreas Isacsson, and Adrian Bachtold. This work was supported by the Swedish Foundation for Strategic Research and the Swedish Research Council. ’ REFERENCES (1) Sazonova, V.; Yaish, Y.; et al. Nature 2004, 431, 284. (2) Witkamp, B.; Poot, M.; van der Zant, H. S. J. Nano Lett. 2006, 6, 2904. (3) Eriksson, A; Lee, S.; Sourab, A. A.; et al. Nano Lett. 2008, 8, 1224. (4) Isacsson, A; Kinaret, J. M.; Kaunisto, R. Nanotechnology 2007, 18, 195203. 1442
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