Nonlinear Stiffness and Nonlinear Damping in Atomically Thin MoS₂ Nanomechanical Resonators

Abstract

We report on experimental measurements and quantitative analyses of nonlinear dynamic characteristics in ultimately thin nanomechanical resonators built upon single-layer, bilayer, and trilayer (1L, 2L, and 3L) molybdenum disulfide (MoS2) vibrating drumhead membranes. This synergistic study with calibrated measurements and analytical modeling on observed nonlinear responses has led to the determination of nonlinear damping and stiffness coefficients at cubic and quintic orders for these two-dimensional (2D) resonators operating in the very high frequency (VHF) band (up to ∼90 MHz). We find that the quintic force can be ∼20% of the Duffing force at larger amplitudes, and thus, it generally cannot be ignored in a nonlinear dynamics analysis. This study provides the first quantification of nonlinear damping and frequency detuning characteristics in 2D semiconductor nanomechanical resonators and elucidates their origins and dependency on engineerable parameters, setting a foundation for future exploration and utilization of the rich nonlinear dynamics in 2D nanomechanical systems.

Mode Coupling in Electromechanical Systems: Recent Advances and Applications

“…[ 104 ] Researches show that synchronization can be achieved through both the mode‐dependent dissipation mechanism, [ 102 ] such as the famous example of Huygens’ clocks, and reactive coupling [ 105 ] through nonlinear frequency pulling. [ 106 ] Synchronized states are often represented by Arnold's tongues or frequency entrainment. [ 37c ] In addition, the form of synchronization can be of unidirectional coupling (injection locking [ 107 ] ) and bidirectional coupling (subharmonic synchronization [ 99,108 ] ) types.…”

Probing Linear to Nonlinear Damping in 2D Semiconductor Nanoelectromechanical Resonators toward a Unified Quality Factor Model

“…38−40 Nonlinear damping has been demonstrated in graphene and MoS 2 NEMS resonators, and the extraction of parameters for nonlinear damping has been performed. 33,41,42 Furthermore, parametric-direct internal resonance has been employed to tune nonlinear damping in graphene NEMS resonators. 43 When the vibrational amplitude is close to the critical amplitude, the damping includes both linear and nonlinear damping.…”

“…Linear damping can be greatly reduced by properly optimizing the measurement environment or tuning strain. For example, by cooling a graphene resonator to 15 mK, Q over 1 million has been demonstrated; by pumping down the pressure from 167 Torr to 15 mTorr, Q of molybdenum disulfide (MoS 2 ) NEMS resonators can increase from 14 to 191; and by tuning the strain using the gate voltage, Q can be tuned by up to 448%. , When the vibration amplitude is high enough so that the nonlinear damping force is much larger than the linear damping force, nonlinear damping becomes dominant, with a larger vibration amplitude leading to a smaller Q or larger damping. − Nonlinear damping has been demonstrated in graphene and MoS 2 NEMS resonators, and the extraction of parameters for nonlinear damping has been performed. ,, Furthermore, parametric-direct internal resonance has been employed to tune nonlinear damping in graphene NEMS resonators . When the vibrational amplitude is close to the critical amplitude, the damping includes both linear and nonlinear damping.…”

“…Furthermore, we can model and extract both linear damping Q lin –1 and nonlinear damping Q non –1 , and we demonstrate large Q tuning ranges ΔQ / Q min up to 269% for MoS 2 resonators and 184% for MoTe 2 resonators. Previous studies focus on either linear damping or nonlinear damping; ,, however, for a broad range of device amplitude approaching the critical amplitude, the damping should include both linear damping and nonlinear damping. The time-domain measurements are critical not only for accurately measuring the Q of these NEMS resonators in the transition regime but also for studying the bistability and nonlinear coupling.…”

“…We find that the extracted η values of our 2D MoS 2 and MoTe 2 NEMS resonators are consistent with those of previously reported MoS 2 NEMS resonators with a similar circular drumhead device geometry measured at room temperature . Furthermore, the η values in this work are smaller than the η value for a graphene NEMS resonator with doubly clamped geometry measured at 4 K, because η could be related to temperature, pressure, and device geometric parameters. ,, Such reduced damping is beneficial for enhancing the Q of 2D NEMS resonators.…”