Opto-thermally excited multimode parametric resonance in graphene membranes

In the field of nanomechanics, parametric excitations are of interest since they can greatly enhance sensing capabilities and eliminate cross-talk. Above a certain threshold of the parametric pump, the mechanical resonator can be brought into parametric resonance. Here we demonstrate parametric resonance of suspended single-layer graphene membranes by an efficient opto-thermal drive that modulates the intrinsic spring constant. With a large amplitude of the optical drive, a record number of 14 mechanical modes can be brought into parametric resonance by modulating a single parameter: the pre-tension. A detailed analysis of the parametric resonance allows us to study nonlinear dynamics and the loss tangent of graphene resonators. It is found that nonlinear damping, of the van der Pol type, is essential to describe the high amplitude parametric resonance response in atomically thin membranes.


S1: Complete datasets for the analysis of mechanical nonlinearities
The frequency conversion option on the vector network analyzer loses information on the phase at which the resonator is oscillating. To show that the parametrically excited resonance has two stable phases separated by 180 degrees, the experiment was repeated by using the frequency doubler in the circuit used for the parametric amplification experiment ( Fig. 1(c) in the main text). Using this, the VNA does not require to perform a frequency conversion and phase information is preserved. This results in the mechanical responses shown in Fig. S4.

S3: Parametric amplification
Here we investigate the effects of parametric drive at low driving levels (δ < δ t ) by examining parametric amplification of the directly driven resonance. To measure parametric amplification, it is required to simultaneously drive the system at f and 2 f (where f is near the resonance frequency f 0 ). This is realized by splitting the driving circuit connected to the diode laser into two parts. One path provides a small direct drive that excites the primary resonance of the membrane in the linear regime. The second path contains a frequency doubler, amplifier and phase shifter to enable parametric driving with controllable phase and gain with respect to the direct drive. A harmonic oscillator model is fitted to the response to extract the amplitude and the effective quality factor. The relation between amplitude gain G, parametric drive amplitude δ and phase shift φ of the direct drive is given by 1, 2 : (S1) First, the amplification effect as function of parametric pumping amplitude in Fig. S5(a) was examined by keeping the phase φ fixed at φ = -45 degrees. Increasing the amplitude of parametric drive increases the amplitude at resonance by a factor of 3-4 ( Fig. S5(b)) and the effective quality factor of resonance by almost a factor of 3 ( Fig. S5(c)). Figure S5(d) shows that shifting the phase of the parametric drive significantly changes the amplitude of harmonic resonance. Figure S5(e)-(f) shows that the gain G and effective Q-factor Q eff depend strongly on the phase of the parametric drive with respect to the direct drive. Fits of the data in Fig. S5(b), (e) show that the drive and phase-dependence of the parametric amplification is in accordance with theory.

S4: Additional discussion: mechanism for direct and parametric driving
Opto-thermal driving leads to two mechanisms that can excite the resonance in the graphene resonators. Parametric drive (Fig.  S6a) occurs due to the modulation of pretension n 0 (t) in the membrane via laser heating and thermal expansion, since the stiffness term for the out-of-plane deflection field w of the membrane is determined by the pre-tension. Parametric driving will only activate the parametric resonance if the modulation of the blue laser is near twice the mechanical resonance frequency.
As demonstrated in the main text (Figs. 3, 4), the experiments also show a direct driving component. This can be explained 3 by assuming a small initial membrane displacement w 0 from equilibrium (Fig. S6b). In graphene resonators rippling, wall adhesion or out-of-plane crumples could lie at the root of such an initial displacement.
In order to analyze the data, we will derive the equations of motion (Eq. S11) using a Lagrangian approach by including this initial deflection field. In this manner, the equations are reduced to a single-degree-of-freedom (s-dof) model that can be used to fit to the data, significantly simplifying the analysis. The derivation of this s-dof model is shown below in section S4. (e) Amplitude of resonance as function of phase φ , the red line is a fit using eq. S1. (f), Effective quality factor as function of phase φ .

5/9
Laser modulation Membrane motion  Figure S6. Explanation of the actuation mechanisms of the opto-thermal drive. For illustration of the mechanism, it is assumed the membrane motion follows the force adiabatically (phase delays are omitted). A blue membrane represents low temperature and a red membrane represents high temperature. a) Parametric excitation, this is due to the pre-tension modulation of the membrane. Each time the tension is maximum the membrane passes through its equilibrium position, leading to a period doubling. This mechanism activates the resonance if the driving frequency is twice the resonance frequency. b) Direct excitation, which exists due to a small initial deviation from equilibrium. This mechanism does not cause period doubling, but instead it activates the resonance if the driving frequency is equal to the resonance frequency.

S5: Equations of motion
A Lagrangian approach is used to obtain equations of motion of an opto-thermally excited monolayer graphene membrane. In this respect, the potential energy of the thermally actuated circular membrane is obtained as 4 : where h is the thickness, R is the radius, α is the thermal expansion coefficient, and ∆T is the temperature change in the membrane. Moreover, σ rr , σ θ θ , τ rθ , are the Kirchhoff stresses that can be obtained as follows: in which ε rr , ε θ θ , and γ rθ are the Green strains and are derived as: where u, v and w are the radial, tangential and transverse displacements, respectively. Moreover, w 0 is the deviation of the membrane from flat configuration, E is the Young's modulus and ν is the Poisson's ratio. The temperature difference ∆T can be obtained by solving the following heat conduction equation: in which P abs is the power absorbed by the membrane, τ is the thermal time constant 5 , C t is the thermal capacitance, andt represents the time variable. For a membrane with fixed edges u and w shall vanish at r = R. Moreover, u should be zero at r = 0 for continuity and symmetry. Furthermore, assuming only axisymmetric vibrations (v = 0 and ∂ u/∂ θ = ∂ v/∂ θ = ∂ w/∂ θ = 0), the solution can be approximated as 6 : Here it should be noted that for axisymmetric vibrations the shear strain γ rθ would become zero. In equation (S6), x(t) is the generalized coordinate associated with the fundamental mode of vibration. Furthermore, in equation (S7), q k (t)'s are the generalized coordinates associated with the radial motion. Moreover, J 0 is the zeroth order Bessel function of the first kind and α 0 = 2.40483. In addition,N is the number of necessary terms in the expansion of radial displacement and u 0 is the initial displacement due to pre-tension n 0 that is obtained from the initial stress σ 0 = n 0 /h as follows : The kinetic energy of the membrane neglecting in-plane inertia, is given by: The Lagrange equations of motion are given by: and q=[x(t),q k (t)], k = 1, . . . ,N is the vector containing all the generalized coordinates. Equation (S10) leads to a system of nonlinear equations comprising of a single differential equation associated with the generalized coordinate x(t) andN algebraic equations in terms of q k (t) . By solving theN algebraic equations it is possible to determine q k (t) in terms of x(t) 6 . This will reduce theN+1 set of nonlinear equations to the following Duffing-Matthieu-Hill equation: where( •) represents derivative with respect to timet and m is the mass. c 1 and c 2 are the linear viscous damping coefficient and nonlinear material damping coefficient, respectively 7,8 . They are added to the equation of motion explicitly to introduce dissipation. k 1 represents the linear stiffness term dominated by the pre-tension n 0 and F p is the amplitude of parametric drive resulting from temperature variation ∆T . Moreover, k 2 represents the quadratic nonlinear stiffness coefficient due to imperfection w 0 and k 3 denotes the cubic nonlinear stiffness coefficient arising from geometric nonlinearity. Finally, F d is the amplitude of direct drive term due to the presence of imperfection w 0 , and ω is the excitation frequency. Indeed for a flat membrane, k 2 = F d = 0.

Definition
Scaled direct excitation amplitude Table S3. Normalized parameter definitions Here it should be noted that, mass m of the single layer graphene membrane is unknown. Without the exact mass value, optical transduction factors present between the voltage signal measured by the VNA during the experiment and the actual motion of the membrane in physical units cannot be calibrated. Thus, the normalized coefficients shown in table S3 include a linear transduction factor 'κ' for the oscillation amplitude (x = κV 1 ), η for the parametric drive amplitude (F p = ηV 2 ) and λ for the direct drive amplitude (F d = λV 3 ). Where V 1 ,V 2 and V 3 are voltage signals measured in the experiment.
Finally, the equation (S13) is simulated using a pseudo arc length continuation and collocation technique 10 to detect bifurcations and obtain periodic solutions. The simulations are performed as follows: