Phononic frequency comb in carbon nanotube mechanical resonators at very high frequency band

Abstract

As a typical optical measurement technique, the optical frequency comb plays an irreplaceable role in spectroscopy and precision measurement. Recently, the concept of frequency combs has been adapted to the phononic domain, leading to the development of phononic frequency combs (PFCs), which have been utilized in various micro-mechanical systems. However, the realization of PFCs in flexural vibration resonators within the very high frequency (VHF) band − crucial for applications in communications and information processing − remains unachieved. In this study, we report the realization of PFC in carbon nanotube (CNT) mechanical resonators operating within the VHF band for the first time. Additionally, we observe that the system exhibits novel frequency combs and nonlinear enhancement in a two-mode mechanical resonator. Due to the broadband operation, tunable modulation depth, as well as easy fabrication and integration of one-dimensional carbon nanotubes, our investigation into PFCs within the VHF band holds promise for advancing classical and quantum precision measurement techniques, while also deepening our comprehension of nonlinear physics.

Graphical abstract

Keywords

phononic frequency combs / carbon nanotubes / very high frequency

1 Introduction

The study of frequency combs, composed of equidistant discrete spectral lines, has been widely addressed in nonlinear optics [1-8]. Optical frequency combs encompass a wide range of frequency spectrum and have gradually found utility in optical frequency precision measurement, atomic/ion transition level measurement, remote signal clock synchronization, and satellite navigation. With the progress of micro/nanotechnology, frequency combs have recently been realized in the phononic domain [9-12], and extensively studied in different materials and mechanisms. Similar to optical resonators, mechanical resonators have also been shown to produce oscillating frequencies at equal intervals due to mechanical mixing and mode coupling. In the micro-/nano-mechanical resonators used to generate PFCs, devices can be categorized into two groups based on whether they undergo deformation. For example, devices such as silicon-based interdigital transducers and quartz crystals do not undergo flexural vibration during the formation of PFCs [13-16]. However, most mechanical resonators exhibit flexural vibration when driven to produce PFCs. These devices mainly include beams (AlN [17-19], SiN [20, 21], Si [22], GaAs [23]) and membranes (graphene [24, 25], MoS{}_2 [26], AlN [27], SiN [28]). Up to now, PFCs in micro-/nano-mechanical resonators have been limited to frequencies below 25 MHz, while PFCs in the VHF range from 30 MHz to 300 MHz remain unexplored. The development of PFCs would be highly desirable for preparing nonlinear phonon states, excitation and tracking of resonance frequencies and various related applications.

Due to the nonlinear phonon dispersion, the formation mechanism of PFC is implemented in a Fermi−Pasta−Ulam (FPU) phonon chain by using parametric resonance principle [29-34]. In order to realize a phononic frequency comb based on FPU theory, micro/nanomechanical resonators are widely used due to their advantages of being multi-mode, strong coupled and easily controlled. Recently, CNTs with a rich band structure have been employed in mechanical resonators. CNTs demonstrate a super-high Young’s modulus (E = 1.2 TPa) [35, 36], low density per unit length (5 × 10⁻²¹ kg/μm) [36], and good electrical conductivity [37]. Because they are easy to manufacture and integrate, the scale production of phonon frequency comb devices can be greatly accelerated.

In this paper, we present a nonlinear FPU system based on a gated CNT mechanical resonator with a microwave drive. The nonlinear resonance of the system occurs under strong driving, leading to the generation of a PFC. The gate voltage in CNT can be finely adjusted over a wide range, so that the tunability of the system is enhanced and the operating frequency range is expanded. By tuning the gate voltage, we were capable to obtain a highly sensitive and tunable PFC with frequency ranging up to 88 MHz. In addition to demonstration of the very-high single-mode PFC, the studied mechanical resonator can also operate as a two-mode PFC between adjacent modes of the spectrum. The resulting spectrum diagram exhibits novel characteristics distinct from those of conventional single-mode PFC. The study contributes to advancing our understanding of nonlinear phononics.

2 Experimental setup

The sample structure is depicted in Fig.1(a). A few layers CNT with diameter \sim3 nm is suspended over a trench (1.5 μm wide and 220 nm deep) between two metal (Ti/Au) electrodes. We construct a mechanical oscillator with a resonant frequency of {\omega }_0 using CNTs. To realize the PFC, we first apply a strong microwave signal {\omega }_{\mathrm{D}} through the bottom electrode of the CNT near the resonant frequency to excite the signal at {\omega }_{\mathrm{D}}, as shown in the upper pannel of Fig.1(b). The 2:1 parametric resonance occurs under the strong drive, and we get an equally spaced phonon frequency comb near the resonant frequency {\omega }_0, as shown in the lower pannel of Fig.1(b). All measurements were conducted in dilution refrigerator at a base temperature of approximately 20 mK and a pressure below 2\times {10}^{-6} Torr. To characterize the spectral characteristics of our resonator, we apply a driving tone of frequency {\omega }_{\mathrm{D}} to the bottom gate. Then we measure the rectified current through the source and drain using a precision multimeter. Fig.1(c) shows the dependence of differential conductance dI_{\mathrm{s}\mathrm{d}}/dV_{\mathrm{s}\mathrm{d}} as a function of the source-drain bias V_{\mathrm{s}\mathrm{d}} and the gate voltage V_{\mathrm{g}}. When we utilize the gate electrode to modulate the chemical potential of the CNT, we can construct a quantum dot in the system. This is evidenced by the Coulomb diamonds in Fig.1(c). It provides a highly sensitive and straightforward DC detection method for the measuring transport properties [38]. By varying the bottom gate voltage, we can tune the CNT resonant frequency over a wide range, resulting in a flexible VHF mechanical resonator as shown in Fig.1(d). The quality factor of our mechanical resonator can reach as high as 1700.

In this paper, we consider an FPU chain with two phonon modes as the theoretical model of the studied nonlinear system. In the three-wave-mixing system, we refer to the first-order nonlinear resonance as the direct nonlinear resonance (DNR). The two-site FPU chain with a monochromatic driving force of strength F_{\mathrm{D}} applied to the first site serves as a driver for both phonon modes. Here, we describe the dynamics of the driven-damped FPU system as follows [10]:

(1)\begin{array}{rl}& {\ddot{Q}}_1+{\omega }_1^2{Q}_1+2{\zeta }_1{\dot{Q}}_1+{\alpha }_{\mathrm{F}\mathrm{P}\mathrm{U}}{A}_{2,2}{Q}_2^2\\ & +{\beta }_{\mathrm{F}\mathrm{P}\mathrm{U}}{A}_{1,1,1}{Q}_1^3=P_{\mathrm{D}}\text{cos}\left({\omega }_{\mathrm{D}}t\right),\end{array}

(2)\begin{array}{rl}& {\ddot{Q}}_2+{\omega }_2^2{Q}_2+2{\zeta }_2{\dot{Q}}_2+{\alpha }_{\mathrm{F}\mathrm{P}\mathrm{U}}{A}_{1,2}{Q}_1{Q}_2\\ & +{\beta }_{\mathrm{F}\mathrm{P}\mathrm{U}}{A}_{2,2,2}{Q}_2^3=0.\end{array}

In both equations, Q_{1(2)} represents the canonical coordinate of the phonon, and {\omega }_{1(2)} denotes the intrinsic frequency of Q_{1(2)}. The term 2{\zeta }_1{\dot{Q}}_1, representing damping, and the term P_{\mathrm{D}}\text{cos}\left({\omega }_{\mathrm{D}}t\right), representing driving, are also included. {\alpha }_{\mathrm{F}\mathrm{P}\mathrm{U}} denotes the quadratic coupling coefficients, and {\beta }_{\mathrm{F}\mathrm{P}\mathrm{U}} denotes the cubic nonlinear terms.

The solution Q_i=A_0\cos \left({\omega }_{i}t\right)+A_p \text{cos}\left({\omega }_i+p\mathrm{\Delta }\omega \right)t can be obtained using the Poincaré−Lindstedt (PL) perturbation method. The solution demonstrates that a set of equally spaced frequency spectral lines appears near {\omega }_1 with spacing \mathrm{\Delta } \omega. These equidistant lines are the PFCs that are to be studied. Additionally, the eigenfrequencies of mode Q_1 ({\omega }_1) and mode Q_2 ({\omega }_2) satisfy the relationship of 2:1. In general, when the driving frequency \left({\omega }_{\mathrm{D}}\cong {\omega }_1\right)\gg {\omega }_2, the excitation amplitude of mode Q_2 is ignored, and only mode Q_1 is significantly excited. When the strong driving frequency matches the resonant frequency, mode Q_1 is sufficiently excited, and mode Q_2 reappears without additional spectral lines. When the driving power is sufficiently high, mode Q_2 is triggered automatically at {\omega }_{\mathrm{D}} /2. Therefore, we refer to mode Q_2 as parametric resonance, which is indirectly generated by the driving mode Q_1. Finally, we successfully obtain the novel PFC without a second frequency line.

3 Results and discussion

3.1 Single-mode PFC

To explore the process of this interesting PFC, we conduct systematic experiments on a CNT mechanical resonator under various driving conditions. The frequency spectrum of the resulting PFC was thoroughly investigated and analyzed. When a strong drive is applied to the system near the resonant frequency, the system exhibits parametric coupling, and a PFC with equally spacing \mathrm{\Delta }\omega =300\mathrm{k}\mathrm{H}\mathrm{z} can be expanded in a specific frequency window {\omega }_{\mathrm{D}} /(2\pi )\in [87.586 MHz, 87.864 MHz] near the resonance frequency, as shown in Fig.2(a) and (b). In the highly tunable gated CNT mechanical resonator, the spectrum of the frequency comb can be further modulated across a broad frequency range through adjustments in the gate voltage, as shown in Fig.2(c) and (d).

As can be seen from Fig.2(a), the PFC is only generated near the resonant frequency in a certain frequency range when only the drive frequency is changed. However, it is notable that the CNT mechanical resonator can generate a PFC when driven by strong microwaves, but its second frequency line disappears due to being in the opposite phase compared to the driving frequency. In Fig.2(b), maintaining constant driving frequency and gate voltage constant, we examine the dependence of PFC on the driving signal S_{\mathrm{i}\mathrm{n}} for {\omega }_{\mathrm{D}} /(2\pi )=87.8\mathrm{M}\mathrm{H}\mathrm{z}. When , the driving power is at a weak level. In this case, the PFC can remain linear. As S_{\mathrm{i}\mathrm{n}} continues to increase, the spacing of the frequency comb becomes larger, and the frequency spectrum transitions from linear to nonlinear shown in Fig. S3. In the suspended carbon nanotube system, the resonant frequency of the mechanical oscillator can be flexibly varied with the electrode voltage. As can be seen in Fig.2(c), only the gate voltage changes, and the frequency and power of the drive signal remain constant. Although the PFC can exist in a specific voltage range, the spacing of the phonon frequency comb with the increasing of electrode voltage no longer remains unchanged. It decreases first and then increases. The spacing reaches the minimum around V_{\mathrm{g}}= 4.05\mathrm{V} where the resonant frequency is very close to the driving frequency. In Fig.2(d), the driving frequency varies with the gate voltage. A linear relationship between frequency and voltage can be extracted from the resonance curve. And we can get a PFC without frequency window limit. The horizontal axis of the four subfigures is the changing parameters, and the vertical axis is the scanning signal.

3.2 Two-mode PFC

The above are the experimental results of the PFC formed when there is only one eigenmode in the carbon nanotube mechanical resonator. However, when we examine a mechanical resonator with multiple eigenmodes, the situation differs regarding the experimental results of the PFC under strong driving conditions. Here, we take a resonator with two eigenmodes as an example to discuss novel PFCs with different mechanisms.

In a system with two adjacent eigenmodes, unlike the conventional phonon frequency comb mechanism, direct nonlinear coupling between the two neighboring modes occurs under strong driving conditions. As depicted in Fig.3(a), the positive diagonal direction in the spectrum diagram exhibits a clear formation of a frequency comb, whereas the reverse diagonal direction only displays a broad spreading, attributable to excessive nonlinearity.

3.3 Nonlinear Duffing phenomena

In the system we can not only observe two PFCs but also witness the nonlinear Duffing phenomenon (DP) [39-47] when the driving power remains unchanged. It means that when the driving frequency is close to one of the resonant modes, energy transfer occurs between two modes, and it affects the other mode contributing to nonlinear Duffing phenomenon. Specifically, we examine gate-controlled CNT two-mode ({\mathrm{R}\mathrm{M}}_{\mathrm{a}} and {\mathrm{R}\mathrm{M}}_{\mathrm{b}}) mechanical resonator device with a frequency difference of about 300 kHz [{\omega }_{\mathrm{a}} /(2\pi )=158.8\mathrm{M}\mathrm{H}\mathrm{z}, {\omega }_{\mathrm{b}} /(2\pi )=159.1\mathrm{M}\mathrm{H}\mathrm{z}]. When we apply external drive from the bottom electrode and gradually increase the driving frequency, we can form not only a PFC near mode {\mathrm{R}\mathrm{M}}_{\mathrm{a}} and {\mathrm{R}\mathrm{M}}_{\mathrm{b}} respectively, but also affect the coupling between {\mathrm{R}\mathrm{M}}_{\mathrm{a}} and {\mathrm{R}\mathrm{M}}_{\mathrm{b}}. The new results obtained from our experiments are shown in Fig.3(a). Initially, the driving frequency gradually increases from small to nearly the resonant frequency of {\mathrm{R}\mathrm{M}}_{\mathrm{a}}, and the first PFC is formed near {\mathrm{R}\mathrm{M}}_{\mathrm{a}}. The frequency comb is marked with white dotted lines. Concurrently, the nonlinearity of the adjacent {\mathrm{R}\mathrm{M}}_{\mathrm{b}} is significantly increased. And the nonlinear expansion of mode {\mathrm{R}\mathrm{M}}_{\mathrm{b}} is obvious. As the driving frequency continues to increase, it moves out of the frequency window and lies between {\mathrm{R}\mathrm{M}}_{\mathrm{a}} and {\mathrm{R}\mathrm{M}}_{\mathrm{b}}. During the range of frequency, the novel nonlinear phenomena of {\mathrm{R}\mathrm{M}}_{\mathrm{a}} and {\mathrm{R}\mathrm{M}}_{\mathrm{b}} disappear. When the driving power continues to increase and approaches the mode {\mathrm{R}\mathrm{M}}_{\mathrm{b}}, a second PFC can be obtained with white dotted lines. Under the interaction of the two modes, another nonlinear broadened resonance peak can be observed near {\mathrm{R}\mathrm{M}}_{\mathrm{a}}. The sketch of the interaction between two adjacent resonant modes under microwave drive is shown in Fig.3(b). In Fig.3(b), the upper panel shows the frequency distribution that produces two PFCs. The lower panel shows the physical machanism of two adjacent resonant modes driven by microwave and influencing each other.

Next, we continue to analyze the nonlinear Duffing phenomenon of two adjacent resonance modes in a mechanical resonator. Specifically, it involves the energy transfer between modes. When the driving frequency is close to one of the resonant modes, the PFC are generated. Additionally, the nonlinear Duffing phenomenon appeared around the other mode. The nonlinear expansion exhibits a trend of initially increasing and then decreasing with the increase of driving frequency, as shown in Fig.3(c). When the driving frequency moves away from the frequency window generated by the comb, the strong nonlinear Duffing phenomenon of the other mode also disappears with the disappearance of the comb. In Fig.3(d), we extract the data from the nonlinear resonance curves and then fit them to find that the nonlinear coefficient \alpha is about 3.38\times {10}^{33}{\mathrm{m}}^{-2}\cdot {\mathrm{s}}^{-2}. Here, we have obtained a novel spectrum diagram when one of both modes of a CNT mechanical resonators was driven. A pair of PFCs are formed on the diagonal of the spectrum. One can conclude that with parametric resonance, although only one of the modes is driven, the adjacent resonance modes still influence each other. These new observations provide valuable insights. The nonlinearity of the frequency comb can be more easily analyzed through the nonlinearity of adjacent modes. Alternatively, a new nonlinear enhancement mechanism at extremely low temperatures is constructed by utilizing a two-mode system. Unlike the feedback oscillator-based approach, we can achieve high-stability measurement of the resonant frequency when exploring PFC-based resonant tracking, thus expanding the application in timing and sensing.

To our knowledge, compared with previous PFCs in various materials, such as AlN, SiN, graphene, GaAs, Si, and different structures such as drum membrane, trampoline, nanobeam, and string mechanical resonators [17-28], our PFC with CNT has the smallest material area and the highest frequency. The small area of our device is conducive to large-scale integration in the future. The large resonant frequencies make our device capable of achieving frequency combs in the VHF range, as shown in Fig.4. These advantages of our device may find various applications in sensing, communications, and quantum information sciences.

Conclusion

In summary, this paper reports the first experimental demonstration of a PFC based on a gated CNT mechanical resonator at VHF band. The rich band structure of CNTs and the excellent tunability through gating make the PFC system highly adjustable, greatly improving the frequency range of previous realizations. Besides the single-mode PFC, we have also demonstrated the two-mode PFC in the CNT mechanical resonator, revealing numerous emerging novel phenomena. The possibility of enhancing the nonlinearity in this system is also discussed. Our model has the clear link with other nonlinear systems in both quantum and classical domains [48-52]. The VHF comb based on CNT mechanical resonator opens new horizon for dealing with high frequencies in quantum metrology. Moreover, the ease of fabrication and large-scale integration of carbon nanotube devices can greatly accelerate the development of precision nanomechanical resonance sensors and other instrumentation for precision measurement applications.

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