I. THE HIGH-Q MODE OF THE DOUBLE-PADDLE OSCILLATOR

The detailed mode shape of the high-Q mode used in this work is shown in the movie,

measured by the laser Doppler vibromtry technique [1].

II. FINITE ELEMENT FOR THE IDENTIFICATION OF THE FLEXURAL

STIFFNESS IN THE VIBRATION MODES

Norris and Photiadis [2] calculated the �exural participation factor pf = Ef/E, which is

the fraction Ef of the total energy E of a resonant mode n which is �exural, for an arbitrary

mode of an arbitrarily-shaped plate resonator to be:

pf =IE

E0(1− ν2)

∫dA(trκ)2, (1)

where the curvature κ is de�ned by κij = −∂2w/∂xi∂xj if w is the z displacement describing

the mode. Here, ν is Poisson's ratio, I = t3/12 is the appropriate moment of inertia (t being

thickness), E is the Young's modulus, E0 is the total mode energy, and the overbar represents

the time average through one complete cycle of oscillation. For the DPO as a whole, �nite

element analysis [3] shows that pf,DPO = 0.05704.

However the integrand of Eq. (1), and thus the �exure, is strikingly nonuniform across

the surface of the DPO, and indeed, much of the �exure is found in the head and wings. This

is illustrated in Fig. 1. The graphene �lms, deposited onto the neck, experience a di�erent

fraction of �exural energy than a �lm deposited over the entire resonator would. Thus we

seek the �exural participation factor of the neck, pf, neck = Ef, neck/Eneck.

We �rst use the Finite Element Analysis to integrate the strain energy, both over of the

whole resonator and over the neck area. Two neck area de�nitions were used, corresponding

to typical �lm placements, and the results were averaged. The neck de�nitions are illustrated

in Fig.1. Taking the ratio r of these calculations eliminates any arbitrary normalization

introduced by the FEM software. Then we have Eneck = rE. Likewise, since the �exural

energy is proportional to the integrand of Eq (1), the ratio s of this quantity integrated over

the neck to that integrated over the whole resonator gives Ef, neck = sEf . Then pf, neck =

pfs/r. Calculated values are shown in Table II, the average of which gives pf = 0.0086.

1

Quantity Symbol Small Large

Flexure s 0.0359 0.1113

Strain Energy r 0.4355 0.5094

Participation Factor pf, neck 0.00470 .01246

Average pf, neck 0.0086

Table I: Results of Finite Element Calculation of relevant parameters for the �exural participation

factor in the neck.

III. ERROR ANALYSIS IN FREQUENCY DIFFERENCE

In the �rst-order, lumped-element calculation of the DPO frequency,

fosc =

√κneck/Ihead

2π, (2)

where κneck is the torsional spring constant of the neck, and Ihead is the moment of inertia of

the head. Mass loading would manifest itself through the lumped-element moment of inertia.

From Eq. (2) we get the equation for frequency shift that results from the deposition of a

�lm over the entire DPO surface:

∆fosc

fosc

=3Gfilmtfilm

2Gsubtsub

− ρfilmtfilm

2ρsubtsub

(3)

Strain Energy, Log Scale Flexure, Log Scale Small Neck Area Large Neck Area

Figure 1: Far left: color scale represents 6 highest orders of magnitude of strain energy, shown on

log scale; deep red is highest, deep blue is lowest, arbitrary units. Second to left: highest 7 orders

of magnitude of �exural energy, same color scheme as strain energy. Second to right and far right

illustrate areas used for pf calculations.

2

=∆fosc

fosc

∣∣∣∣∣elastic

+∆fosc

fosc

∣∣∣∣∣mass

. (4)

where ρ is density, G is shear modulus, and t is thickness.

Here we estimate the contribution to the frequency shift that might be made by any

residual PMMA on the DPO neck. We will estimate each term of Eq. (3) and compare their

magnitudes. If we take Gfilmtfilm = 100 GPa nm, then the frequency shift from the spring

constant of the �lm will be ∆f/f |elastic ≈ 8× 10−6.

If we take as values for a residual PMMA �lm ρfilm = 1 g/cm3 and tfilm = 5 nm we �nd

∆f/f |mass ≈ 3.6 × 10−6. However, this would be the value for mass loading from a �lm

covering the entire head. Inspection of the DPO geometry reveals that Ineck/Ihead = 0.008.

Thus we scale ∆f/f |elastic by this factor, as the relevant mass exists only on the neck, giving

us ∆f/f |elastic = 3× 10−8.

Thus the relative error would be

∆f/f |PMMA mass, neck

∆f/f |graphene, elastic

= 0.004.

IV. HYBRID ATOMISTIC - FINITE ELEMENT ANALYSIS OF SINGLE LAYER

GRAPHENE SHEETS

Models of single layer graphene sheets have been developed using a lattice representa-

tion, where the sp2 C-C bonds are simulated through the use of Timoshenko beams with

deep shear deformation capability and 6 degrees-of-freedom per atom (3 translational and 3

rotational). The equivalent material constants (Young's modulus E, Poisson's ratio ν, and

shear modulus G) required to model these beams can be calculated considering the equiva-

lence between the harmonic potential of the C-C bonds and the strain energy deformation

associated to the structural beams when small linear elastic deformations are involved [4]:

kr2

(δr)2 =EA

2L(δr)2 ;

kτ2

(δϕ)2 =GJ

2L(δϕ)2 ;

kθ2

(δθ)2 =EI

2L

4 + Φ

1 + Φ(δθ)2 . (5)

In (5), the quantities δr, δϕ and δθ are the in�nitesimal deformations of the three degrees

of freedom (bond stretching, out-of-plane bending and torsion [5, 6], and in-plane bending,

respectively). A, L, J , and I denote the cross-section area, the equilibrium length, the polar

moment of inertia, and the second moment of inertia of a graphene sheet, respectively. The

3

force constants in (5) are taken from the AMBER and linearised Morse potential. For the

AMBER model, the constants are kr = 6.52 × 10−7N nm−1, kθ = 8.76 × 10−10N nm rad−2,

and kτ = 2.78 × 10−10N nm rad−2. For linearised Morse potential model, they are kr =

8.74 × 10−7N mm−1, kθ = 9.00 × 10−10N nm rad−2, and kτ = 2.78 × 10−10N nm rad−2. The

shear deformation constant is given by [7]:

Φ =12EI

GAsL2(6)

In the Equation (6), As = A/Fs is the reduced cross section of the beam and the shear

correction term Fs is give by [8]:

Fs =6 + 12ν + 6ν2

7 + 12ν + 4ν2(7)

By substituting (6) and (7) in (5) the relation between the thickness t and the Poisson's

ratio ν of the beam becomes:

kθ =krt

2

16

4A+B

A+B(8)

Where:

A = 112L2kτ + 192L2kτν + 64L2kτν2, (9)

B = 9krt2 + 18krt

4ν + 9krt4ν2. (10)

Equation (8) is a nonlinear function of t, ν, and L. Assuming L = 0.142 nm and

an isotropic material behaviour of the C-C bond, i.e., G = E/2/ (1 + ν), the thickness is

identi�ed as t = 0.083 nm for the AMBER and t = 0.079 nm for the Morse potential. For

discussions about the thickness distribution in graphene and carbon nanotube systems one

can also refer to [9] and [10].

The e�ect of the Si substrate on single layer graphene sheets has been modelled using a

distribution on nonlinear springs simulating Lennard-Jones interactions between the C and

Si atoms, in a similar manner to LJ interlayer potentials in bilayer graphene [11�13]. The

nonlinear springs were represented by forces calculated as the derivative of the Lennard-

Jones potential versus the displacement r from the equilibrium position rmin between the i

and j atoms:

Fij = −12 ε r6min

[r6min

(1

rmin + r

)13

−(

1

rmin + r

)7]

(11)

4

In (11), the following constants for LJ potentials betwen C and Si atoms have been

adopted from the UFF model [14]: rmin = 0.373 nm, ε = 8.909 meV. No cut-o� distance

has been set for the LJ interactions, because of the small elastic deformations imposed on

the models.

The graphene sheets were considered as planar (graphitic state), and represented as

square assemblies of 2778 atoms. Two cases of shear loading were considered: a biaxial

shear consisting in a compressive strain εy and a tensile strain εx (Figure 2a), and a pure

shear induced by an angular strain γ (Figure 2b). The tensile and angular strains imposed

on the graphene sheets were equal to 0.01 %, to maintain the simulations within the limits

of linear elasticity. Static nonlinear problems were solved with a Newton-Raphson solver

with convergence criteria based on forces and moments. The commerical Finite Element

code ANSYS 11.0 was used for the simulations. The shear modulus for the biaxial shear

was calculated as:

Gbi =F̄x/Y − F̄y/X2 t (εx − εy)

(12)

Where X and Y are the overall dimensions of the graphene layer along the x and y

directions, and t is the thickness of the graphene sheet. The quantities F̄x and F̄y are the

summations of the atomic forces along the x and y directions on the boundaries of the

graphene. The pure shear modulus Gp was calculated as:

U =1

2Gp V γ

2, (13)

where V is the volume of the graphene sheet (considering its thickness) and U is the

strain energy associated to the pure shear deformation γ applied to the graphene layer.

Figure 2: (a) Biaxial shear; (b) pure shear

5

[1] X. Liu, S. F. Morse, J. F. Vignola, D. M. Photiadis, A. Sarkissian, M. H. Marcus, B. H.

Houston, On the modes and loss mechanisms of a high q mechanical oscillator, Applied Physics

Letters 78 (10) (2001) 1346�1348.

URL http://link.aip.org/link/?APL/78/1346/1

[2] A. N. Norris, D. M. Photiadis, Thermoelastic relaxation in elastic structures, with applications

to thin plates, Q. J. Mech. Appl. Math. 58 (2005) 143�163.

[3] Comsol AB, FEMLAB 3.1.

[4] Scarpa F, Adhikari S, Phani A S, E�ective elastic mechanical properties of single layer graphene

sheets, Nanotechnology 20 (2009) 065709.

[5] Tserpes K I, Papanikos P, Finite Element modelling of single-walled carbon nanotubes, Comp.

B 36 (2005) 468.

[6] F. Scarpa, L. Boldrin, H. X. Peng, C. D. L. Remillat, S. Adhikari, Coupled thermomechanics

of single-wall carbon nanotubes, Applied Physics Letters 97 (2010) 151903 (3 pp).

[7] Przemienicki J S, Theory of Matrix Structural Analysis, McGraw-Hill, New York, 1968.

[8] Kaneko T, On Timoshenko's correction for shear in vibrating beams, J. Phys. D: App. Phys.

8 (1974) 1927.

[9] Huang Y, Wu J, Hwang K C, Thickness of graphene and single wall carbon nanotubes, Phys.

Rev. B 74 (2006) 245413.

[10] F. Scarpa, S. Adhikari, A. J. Gil, C. D. L. Remillat, The bending of single layer graphene

sheets: the lattice versus continuum approach, Nanotechnology 21 (2010) 125702.

[11] F. Scarpa, S. Adhikari, R. Chowdhury, The transverse elasticity of bilayer graphene, Physics

Letters A 374 (19-20) (2010) 2053 � 2057.

[12] Y. Chandra, R. Chowdhury, F. Scarpa, S. Adhikaricor, Vibrational characteristics of bilayer

graphene sheets, Thin Solid Films 519 (18) (2011) 6026 � 6032.

[13] Y. Chandra, R. Chowdhury, S. Adhikari, F. Scarpa, Elastic instability of bilayer graphene

using atomistic �nite element, Physica E: Low-dimensional Systems and Nanostructures 44 (1)

(2011) 12 � 16.

[14] A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard, W. M. Ski�, U�, a full periodic

table force �eld for molecular mechanics and molecular dynamics simulations, Journal of the

6

American Chemical Society 114 (25) (1992) 10024�10035.

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