Magnetic damping of a carbon nanotube NEMS resonator

A suspended, doubly clamped single wall carbon nanotube is characterized at cryogenic temperatures. We observe specific switching effects in dc-current spectroscopy of the embedded quantum dot. These have been identified previously as nano-electromechanical self-excitation of the system, where positive feedback from single electron tunneling drives mechanical motion. A magnetic field suppresses this effect, by providing an additional damping mechanism. This is modeled by eddy current damping, and confirmed by measuring the resonance quality factor of the rf-driven nano-electromechanical resonator in an increasing magnetic field.

Nano-electromechanical resonator systems provide an intriguing field of research, where both technical applications and fundamental insights into the limits of mechanical motion are possible. Among these systems, carbon nanotubes provide the ultimate electromechanical beam resonator [1,2,3], because of their stiffness, low mass, and high aspect ration. At the same time, they are an outstanding material for transport spectroscopy of quantum dots at cryogenic temperatures [4,5]. Chemical vapour deposition (CVD) has been shown to produce on chip defect-free single wall carbon nanotubes [6]. By performing this growth process as last chip fabrication step, suspended defect-and contamination-free macromolecules can be integrated into electrode structures and characterized. On the electronic side, this has led to many valuable insights into, e.g., the physics of spatially confined few-carrier systems [7,8,9]. In terms of nano-electromechanical systems, these ultra-clean nanotubes have shown exceedingly high mechanical quality factors at cryogenic temperatures [10]. This has allowed for the observation of direct interaction between single electron tunneling and mechanical motion [11,12,13].
In this article, we report on low temperature transport spectroscopy measurements on a suspended, doubly clamped carbon nanotube, as displayed in figure 1(a). The carbon nanotube acts as an ultra-clean quantum dot as well as a nano-electromechanical transversal resonator. Figure 1(b) shows a typical chip electrode structure including dimensions. On a highly p+ doped Si substrate with ∼ 300 nm thermally grown SiO 2 on top, contact patterns are defined via electron beam lithography and evaporation of 40 nm rhenium. This metal layer directly serves as etch mask for subsequent anisotropic dry etching of the oxide, generating deep trenches between the electrodes. As last fabrication step, CVD growth catalyst is locally deposited at the center of each contact electrode structure and the nanotube growth is performed [6].
Electronic transport measurements were conducted in a 3 He evaporation cooling system at T3 He = 290 mK, and in a dilution refrigerator at T mc,base = 25 mK. The electronic measurement setup, as sketched in figure 1(a), closely follows Refs. [10,11]. A gate voltage V g is applied to the substrate as back gate, a bias voltage V sd across the device. The resulting dc current through the device is measured via a preamplifier, as required for Coulomb blockade transport spectroscopy [14]. An antenna suspended close to the chip provides means to apply a radio-frequency signal contact-free.
At first, we characterize the basic electronic and electromechanical properties of the device. As can be seen from the dc current measurement in figure 1(c), our device exhibits the typical electronical behavior of a very clean and regular small band gap nanotube. The measurement displays the dc current I dc as function of the applied gate voltage V g , for a low constant dc bias voltage V sd = 0.2 mV. For V g < 0.5 V, highly transparent contacts in hole conduction lead to a rapid transition from Coulomb blockade to the Fabry-Perot interference regime [15]. Around V g 0.75 V, current is suppressed as the electrochemical potential is located within the semiconducting band gap. For V g > 1 V, electron conduction becomes visible through sharp, well-defined Coulomb blockade oscillations with the characteristic four-fold pattern of the carbon nanotube level structure [16]. Regular Kondo conductance enhancement [17] emerges for N e − > 15, again confirming the presence of a defect-free single wall carbon nanotube.
When a radiofrequency signal is applied at mechanical resonance, the nanotube vibrates, leading to a change in detected, time-averaged dc current [10]. This signal can be identified via its characteristic dependence on the back gate voltage V g : electrostatical forces on the influenced charge on the nanotube lead to mechanical tension, and thereby an increase in resonance frequency of the transversal vibration mode. Figure 1(d) shows a map of such resonance positions, displaying the resonance frequency as function of back gate voltage V g . It thus characterizes the basic electromechanical properties of our device. Among several other weaker features, four clear structures, plotted in figure 1(d) and labelled A, B, A', and B', can be seen in the observed frequency range, with the overall gate voltage dependence typical for the mechanical response of a tensioned carbon nanotube resonator [1,2,10].
Traces A' and B' coincide over a wide range with double the frequency of traces A and B (plotted in figure 1(d) as thin black lines). It appears unlikely that these represent higher mechanical modes, since in the low tension limit an exact frequency doubling is not expected [18]. Instead, A' and B' can represent different driving mechanisms for the modes of A and B. In literature, e.g., parametric resonance has been demonstrated in measurements on nanotube resonators [19,20]. The observation of the two modes A and B is consistent  Figure 2. Feedback effects in a non-driven resonator (all measurements at T mc,base ). (a) Differential conductance dI/dV sd (V g ,V sd ) (no rf signal applied; lock-in measurement with an excitation of V sd,ac = 5 µV RMS at 137 Hz) of the carbon nanotube quantum dot at zero magnetic field, displaying four-fold shell filling combined with Kondo effect and traces of superconductivity in the metallic leads (see text). At finite bias, strong switching effects attributable to mechanical self-excitation become visible, indicated by white arrows [11,21]. (b) Detail of (a) for increasing magnetic field, this time plotting the numerical derivative of the simultaneously measured dc current I dc (V g ,V sd ). Already at B = 0.8 T, the selfdriving effects are completely suppressed. (c) dc current I dc (V g ) along the trace of constant V sd = −1.15 mV, as marked in (b) with a dotted line, and (d) corresponding differential conductance dI/dV sd (V g ).
with mechanical motion of two adjacent suspended nanotube segments of different length, as visible in the chip geometry of figure 1(b). Assuming the minimum resonance frequency close to charge neutrality to be the case of vanishing mechanical tension, the ratio of the minimum frequencies of A and B, f min,A / f min,B = 253 MHz/182 MHz 1.39 agrees very well with the expectation from the different trench widths ( B / A ) 2 = (700 nm/600 nm) 2 1.36. The detailed mechanism leading to the signal contribution of the second nanotube segment next to the contacted 700 nm gap is still under investigation.
In the following we return to measurements without any applied rf driving signal. Figure 2(a) displays a lock-in measurement of the differential conductance of the suspended carbon nanotube quantum dot, as function of gate voltage V g and bias voltage V sd . A positive gate voltage is used to tune the quantum dot into the regime with an electron number of 38 ≤ N e − ≤ 42, where it is strongly tunnel-coupled to the contact electrodes. Several features on the plot can immediately be identified and are well understood. The narrow, approximately gate voltage independent conductance minimum around V sd = 0 in figure 2(a) is caused by superconductivity of the metallic rhenium leads; two Kondo ridges of enhanced lowbias conductance become clearly visible around V g = 3.84 V (N e − = 39) and V g = 4.0 V (N e − = 41).
In addition, the differential conductance signal from figure 2(a) exhibits sharply delineated regions of modified signal level, often accompanied by switching behavior, see white arrows in the figure. This has already been observed previously in clean suspended carbon nanotube quantum dots [11]. As predicted in Refs. [21,22] and confirmed in Ref. [11], in these parameter regions single electron tunneling from dc current alone suffices to coherently drive the mechanical motion via a positive feedback mechanism. In turn, this becomes visible in the recorded current or conductance signal as well.
The panels of figure 2(b) display a detail enlargement of the parameter region of figure 2(a), this time plotting as differential conductance the numerical derivative of the dc current recorded simultaneously with the lock-in signal. Although this value is affected by a larger noise level, it reproduces more faithfully one-time switching events while sweeping the bias voltage, which delimit the feedback regions. A clear substructure emerges inside the feedback region, which so far has not found any equivalent in theoretical considerations. In addition, when increasing an externally applied magnetic field perpendicular to the chip surface, the parameter regions of positive feedback shrink. As can be seen from the panels of figure 2(b), applying a magnetic field of B = 0.8 T already completely suppresses the selfdriving effect within the observed region. This is further illustrated by the line traces of figure 2(c) and (d), displaying the dc current (c) and the differential conductance (d) as function of gate voltage V g at constant V sd = −1.15 mV across the parameter regions of figure 2(b). While no significant changes take place outside the positive feedback region, the discontinuous behavior at zero field becomes smooth, and only slight fluctuations remain at B = 1 T. In particular the current agrees very well with the prediction of Refs. [21,22] in the cases of present and suppressed feedback.
While it has been shown in Ref. [11] that large electronic tunnel rates are an important prerequisite for self-excitation, here the conductance remains unchanged outside the feedback-dominated regions. The magnetic field does not significantly influence the electronic tunnel rates, excluding such a mechanism for the suppression of the selfexcitation. A second prerequisite is a high mechanical quality factor [21,22], since the feedback mechanism has to compensate and overcome damping of the mechanical oscillation. Consequently, the suppression of self-driving indicates a magnetic field induced additional damping mechanism.
To verify this conclusion from the pure dc measurements, we measure the frequency dependence of the radio frequency-driven resonator response. An additional damping mechanism in a magnetic field should here become visible as a resonance peak broadening, i.e. a decrease in effective quality factor Q. A constant positive gate voltage V g = 3.91 V is used to tune the quantum dot into the Coulomb blockade region with electron number N e − = 40. Because of the transparent tunnel barriers to the leads, significant cotunneling conductance on the order of G cot 0.4 e 2 /h can still be observed in this parameter region, enabling the detection of the mechanical resonance in dc current. Extending the mechanical resonance detection setup of Refs. [10,11] to increase sensitivity, the applied radio frequency signal is amplitude-modulated at a low frequency f am = 137 Hz, such that the period 1/ f am 7 ms is larger than the oscillation decay timescale ∼ Q/ f expected from literature [10]. The corresponding low-frequency modulation of the current signal is recorded by a lock-in amplifier. In addition, we drive at double frequency (A' in figure 1(d)) as this results in a stronger resonance signal. Figure 3(a) displays a typical resulting in-phase (x), out-of-phase (y), and phase angle (φ ) amplitude modulation response signal as function of the driving frequency f . In the in-phase (x) response, a multi-peak structure emerges. Indications of this multi-peak shape (see arrows in figure 3(a)) remain visible even at lowermost driving power and suggest a more complex coupling of the radiofrequency driving signal into the electromechanical system than only actuation via electrostatical force [10,20,19]. As can be seen from both the y response and the phase angle, in spite of the low amplitude modulation frequency, a distinct phase shift of the response on resonance is still visible. The shift of approximately ∆φ = 0.2 rad corresponds to a delay time of ∆t = 0.35 ms, on the order of 10 5 mechanical oscillation cycles. Given that previously observed nanotube resonators [10,20] have exhibited quality factors on that order of magnitude, this is consistent with mechanical storage of vibration energy and later release within one amplitude modulation cycle, leading to a delayed driving response.
To avoid fitting of the multi-peak structure in the in-phase (x) signal, we focus in the following on the out-of-phase (y) signal induced by the phase shift. Figure 3(b) shows selected frequency response traces of the mechanical resonance, recorded at an external magnetic field of B = 1 T and B = 3 T, respectively, and pointing towards a slight broadening of the peak structure at higher magnetic field. Evaluation of many similar curves, including repeated measurements at the same magnetic field value and over a large driving power range, leads to the plot of figure 3(d). Here, the width of the resonance peaks is plotted in terms of an experimentally observed quality factor Q as function of the magnetic field B. Indeed, the measured peak width increases (and Q decreases) significantly above B = 1 T. A straightforward circuit model sketched in figure 3(c) can be used to describe the magnetic field dependence. A vibration component perpendicular to the magnetic field leads to an induced ac voltage across the resonator. We assume the carbon nanotube resonator to be partially electrically shortened in the rf signal frequency range via an Ohmic resistance R and a large parasitic capacitance. For simplicity, we do not take into account the deflection shape but assume a uniform beam deflection along the entire nanotube of length L and mass m to estimate the magnetic flux modulation. As a result, eddy currents lead to a damping of the mechanical motion corresponding to as both induced voltage and resulting eddy current are proportional to B. Assuming an additional magnetic field independent resonator damping, which determines the zero external field quality factor Q 0 , we obtain the expression The solid line in figure 3(d) provides a best fit of this function to the data, using Q 0 and q as free parameters and resulting in the values q = 5.381 × 10 5 T 2 and Q 0 = 25020. As visible in figure 3 this model describes our measurement results well. This thereby confirms the presence of a magnetic-field induced damping mechanism. Using the resonance frequency f and estimating L 700 nm and m 1.3 · 10 −21 kg, we obtain a value for the Ohmic resistance of R 200 kΩ in the replacement circuit of figure 3(c). As a last remark, using the fit function of figure 3(d) one obtains Q(0.8 T)/Q(0 T) 0.97, i.e. only a very small decrease of the effective quality factor within the magnetic field range covered in figure 2. A likely reason for this is that the resonance peak widths evaluated in figure 3(d) do not solely correspond to the device quality factor entering the self-excitation, but are broadened by additional mechanisms, leading to an underestimation of the low-field quality factor Q 0 .
Summarizing, we characterize a quantum dot in a suspended ultra-clean single wall carbon nanotube, which also acts as nano-electromecanical resonator. We observe how feedback and self-driving effects, where only dc current is sufficient to drive resonator motion, are suppressed in a finite magnetic field. The conclusion that the magnetic field induces an additional damping mechanism is confirmed by tracing the driven resonator response as a function of magnetic field. We model the decrease of the mechanical quality factor successfully using eddy current damping.