advances.sciencemag.org/cgi/content/full/4/3/eaao6653/DC1

Supplementary Materials for

Electrically tunable single- and few-layer MoS2 nanoelectromechanical

systems with broad dynamic range

Jaesung Lee, Zenghui Wang, Keliang He, Rui Yang, Jie Shan, Philip X.-L. Feng

Published 30 March 2018, Sci. Adv. 4, eaao6653 (2018)

DOI: 10.1126/sciadv.aao6653

This PDF file includes:

section S1. Device fabrication

section S2. Optical interferometry measurement system

section S3. Electrical tuning of device resonance

section S4. Power handling, mass sensitivity, and frequency stability

section S5. Nanomechanical tuning and sensing of device strain and bandgap

section S6. Measuring nonlinearity and estimating critical amplitude

section S7. Translation of voltage fluctuations into frequency instability

section S8. Comparison of DR in 1D and 2D NEMS resonators

fig. S1. Calculated displacement versus reflectance values for 1L to 3L MoS2

resonators.

fig. S2. Calculated displacement-to-reflectance responsivity (ℜRef) and measured

displacement-to-voltage responsivity (ℜV) values for 1L to 3L MoS2 resonators.

fig. S3. Thermomechanical resonance with qualify (Q) factor exceeding 1000.

fig. S4. Thermomechanical vibrations with distinct signatures of digitized

thicknesses (number of layers) as a function of fres and Q.

fig. S5. Electrical gate turning of higher-mode resonances.

fig. S6. FOM for frequency tuning: Comparison across reported 2D NEMS

devices.

fig. S7. Schematic for calculating the total surface area on a deformed membrane.

fig. S8. Measured DR in 1D and 2D resonators operated at room temperature.

fig. S9. Resonance frequency scaling with device diameter D and MoS2

thickness t.

table S1. FOM for frequency tuning.

table S2. List of devices with measured nonlinear characteristics.

table S3. DRs measured in 1D and 2D resonators.

References (68–80)

section S1. Device fabrication

MoS2 resonators are fabricated by mechanically exfoliating MoS2 flakes onto predefined

microtrenches and cavities of various shapes on the substrate. We start with silicon (Si) wafers

with 290nm thermal SiO2 on top. We pattern circular microcavities with diameters D 0.5m,

0.75m, 1m, 1.25m and 1.5m using photolithography, and etch the exposed SiO2. We then

clean the substrate using Piranha, a mixture of sulfuric acid (H2SO4) and hydrogen peroxide

(H2O2), before exfoliation of MoS2.

section S2. Optical interferometry measurement system

The resonant motion of the MoS2 resonators is measured by using a custom-built optical

interferometry system. A 633nm He-Ne laser is focused on the oscillating MoS2 diaphragm using

a microscope objective (50×, NA=0.5). To achieve small spot size, the laser beam goes through a

beam expander before reaching the objective. The calculated on-device spot size is ~1m. To

minimize laser heating, the laser power is adjusted to be below 350W using a neutral density

(ND) filter (37). Interferometry is formed between reflected light form the MoS2 membrane-

vacuum and vacuum-substrate interfaces, and the interference condition changes as the vacuum

gap depth between MoS2 and the bottom of cavity (substrate) varies due to device motion. The

resulting change in the reflected light intensity is detected by a photodetector and recorded using

a spectrum analyzer. Without any external drive, this highly sensitive measurement apparatus is

capable of detecting the intrinsic thermomechanical motion of the MoS2 membranes.

Besides the undriven thermomechanical resonance, we have also measured device resonance

under different driving schemes. Note that the DRlinear,achieved values are affected by the actuation

scheme, which may introduce additional nonlinearity (e.g., capacitive softening) in device

response (23, 68, 69), and are limited by excess electronic noise in the measurement system,

which increases the noise floor (70).

S2.1 Optical Excitation of Device Resonance Motion

An RF-modulated 405nm diode laser is used for optical excitation of device resonance motion.

The modulation frequency and amplitude are controlled by the output of a network analyzer. The

405nm laser is focused onto the substrate next to the device (~5m apart), with an estimated spot

size ~5m. On-substrate power of the 405nm laser is limited to be below 250W. Such

arrangement of the spot size and low laser power ensures minimal heating effect from the 405nm

laser. A long-pass filter is installed in front of the photodetector to remove the signal from the

405nm laser.

S2.2 Electrical Excitation of Device Resonance Motion

Electrodes are made by evaporating nickel (Ni) onto the anchored regions of already suspended

MoS2 devices through a carefully aligned stencil mask. This method does not require any

chemical processing and the MoS2 membrane remains in its pristine condition. The electrical

actuation is performed by applying an AC+DC signal generated by a network analyzer and a DC

power supply through a bias-T on the highly doped p++ conducting substrate (Si) while

grounding the MoS2 flakes.

The evaporated Ni on the non-suspended part of the MoS2 resonators forms a metal-

semiconductor contact with the MoS2 flake, and can yield a Schottky barrier that might affect the

electrical performance of the device. However, we have carefully characterized such contacts in

other MoS2 devices (field effect transistors) made by the same process, where Ohmic behavior

has been repeatedly observed, with minimal effect from the Schottky barrier. Also, this work

focuses on the mechanical responses of the MoS2 drumheads with electrostatic gate tuning,

without electrical transport measurement of the samples involved. Thanks to the accurate and

precise alignment of the stencil mask, the electrode is only deposited on the corner of the

substrate-supported MoS2 while the suspended region remains pristine, thus the mechanical

properties of the MoS2 resonator is not affected. Further, in all measurements, the metal contact

is connected to the ground and all the MoS2 flakes are held at 0V.

S2.3 Responsivity Analysis

Here we calculate the displacement-to-reflectance responsivity for devices with different number

of layers. As laser is focused on the device, the incident light undergoes multiple reflections at all

the interfaces and the device essentially becomes an interferometer. The reflectance (total

fraction of light reflected) is (37)

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

2

interferometry 1 2 3 1 2 3

incident 1 2 1 3 2 3

i i i i

i i i i

I e e e eR

I e e e e

r r r r r r

r r r r r r

(S1)

where r1, r2, and r3 are the reflection coefficients at the vacuum-MoS2, MoS2-vacuum, and

vacuum-Si (substrate) interfaces, respectively, and 1, 2 are the corresponding phase shifts

2

2

vacuum MoS

1

vacuum MoS

n

n

nr

n,

2

2

MoS vacuum

2

MoS vacuum

n

n

nr

n, vacuum Si

3

vacuum Si

n

n

nr

n (S2)

2MoS

1

2 t

n , vacuum

2

2 n d

(S3)

where t is the MoS2 thickness, d is the vacuum gap (cavity depth), and is laser wavelength.

nvacuum=1, nMoS2=5.263-1.140i (71) and nSi=3.881-0.019i are the indices of refraction. For the

He-Ne laser we use =633nm. t=0.7, 1.4, and 2.1nm for 1L, 2L, and 3L MoS2.

Figure S1 shows the calculated reflectance of 1-3L MoS2 as a function of vacuum gap cavity

depth d. Since the device motion leads to the oscillation of d value around 290nm, the slopes at

d=290nm represent the displacement-to-reflectance responsivity (ℜRef). From calculation we

obtain responsivity Ref R d -0.047, -0.087, and -0.121 %/nm, for 1-3L MoS2 devices,

respectively. We note that the calculated responsivity is roughly proportional to the number of

layers.

Experimentally we calibrate the displacement-to-voltage responsivity (ℜV) by measuring

thermomechanical noise. The measured total noise spectral density consists of noise floor from

the measurement system1/2

,sysvS , and contribution from the thermomechanical motion of the MoS2

resonators1/2

,thvS . As the noise processes are uncorrelated, 1 2

1/2

,total ,th ,sysv v vS S S . The system noise

1/2

,thvS determines the off-resonance noise level or the background of the noise spectrum (where

1/2

,thvS ≈ 0), and 1/2

,thvS is related to the thermomechanical motion of the resonator by responsivity

1/2 1/2

V ,th ,thv xS S , where

1 2

1 2 B 0,th 2

2 2 2 2 2eff

0 0

4 1x

k TS

M Q Q

(S4)

is the displacement domain thermomechanical noise spectral density (72) with kB being the

Boltzmann constant. Note that Eq. S4 degenerates to the expression in the Main Text at ω=ω0.

The total measured noise is then (72)

1/21/2 2 2 0 B,total V ,th ,sys V ,sys2 22 2

eff0 0

4 1v x v v

k TS S S S

QM Q

(S5)

The measured displacement-to-voltage responsivity V can therefore be extracted from the

fitting parameters for the thermomechanical resonances. Figure S2 plots the measured values of

V for multiple 1L and 3L devices, demonstrating proportionality to the number of layers, in

very good agreement with the calculation. Accordingly, we use these measured values plus

interpolation to determine V for all the 1-3L devices with the same dimensions (diameter and

vacuum gap depth), including those whose thermomechanical resonances cannot be directly

measured due to smaller signal and larger noise background because of their higher resonance

frequency (fres).

fig. S1. Calculated displacement versus reflectance values for 1L to 3L MoS2 resonators. The clear

distinction between the reflectance of 1-3L MoS2 devices are due to the discreteness and difference in

their thicknesses (integer numbers of atomic layers). Slopes of the curves at 290nm vacuum gap depth

represent the displacement-to-reflectance responsivity (ℜRef) values under small motion amplitude for our

1-3L devices.

1 2 30.00

0.05

0.10

|R

ef|

(%

/nm

)

Number of Layers

0.0

0.1

0.2

0.3|

V| (

mV

/nm

)

fig. S2. Calculated displacement-to-reflectance responsivity (ℜRef) and measured displacement-to-

voltage responsivity (ℜV) values for 1L to 3L MoS2 resonators. The blue spheres show calculated

responsivities (left y axis) and the red diamonds represent measured values (right y axis) from devices’

thermomechanical noise. Good linear dependence on number of layers is predicted and observed,

showing good agreement between calculations and measured data.

280 290 300

0.38

0.40

0.42

0.44

Vacuum Gap (nm)

Re

fle

cta

nce

(%

)

3L

2L

1L

fig. S3. Thermomechanical resonance with qualify (Q) factor exceeding 1000. Fitting to the

thermomechanical resonance noise spectrum measured from a 4L MoS2 device (D1 m and t=2.8 nm)

shows Q≈1050. Inset shows the optical image of the device. Scale bar: 1 m.

fig. S4. Thermomechanical vibrations with distinct signatures of digitized thicknesses (number of

layers) as a function of fres and Q. Green, magenta, and cyan surfaces show calculated displacement-

domain thermomechanical resonance for 1L, 2L, and 3L circular drumhead resonators (diameter

D1.5μm), respectively. Spherical symbols represent the measured devices.

61.0 61.2 61.4

0.2

0.3

0.4

0.5

S1/2

V (

V/H

z1/2)

Frequency (MHz)

Q~1050

section S3. Electrical tuning of device resonance

S3.1 Gate Tuning of Resonance Frequency

Mechanical response of the circular drumhead resonator under capacitive coupling can be

understood by considering both elastic energy in the suspended membrane and electrostatic

energy stored in the capacitor formed by the 2D flake and its gate. The elastic energy is, as a

function of displacement W(r,θ) (expressed in polar coordinates for a circular membrane) (73)

2 2 22π

0,Y Yelastic 0, 2 20 0 2 8 1

a

r

E t EW W t WU d rdr

r r r

(S6)

where EY, , 0,r 0,, t are Young’s modulus, Poisson’s ratio, initial radial strain, initial tangential

strain, and thickness. Assuming the radial profile of the membrane deformed by electrostatic

force is parabolic, W(r,)=z(1r2/a2), where z is defection at its center, Eq. S6 can be rewritten as

4

2 2 2Yelastic 0, 0,2 2

π 2 11

1 3 2r r

E t zU z a

a

(S7)

Here, the third term represents the energy related to the initial strain 0,r in the device.

The electrostatic energy is

2

electrostatic g g

1

2U C V (S8)

where Cg is capacitance of the device. Note that in the question of interest, we are focusing on

the change in energy due to device motion (change in capacitance), so the reference point

(Uelectrostatic=0) is chosen to be where the 2D flake is infinitely far away from the gate (Cg=0).

Under any given gate bias Vg, the closer the flake moves towards the gate (the greater the Cg

value), the lower the electrostatic energy. This configuration gives the minus sign in the

expression of Uelectrostatic. Cg can be approximated by

0

g 2 21

єC rdrd

d z r a

(S9)

which can be further expanded in terms of the static deflection z

20 0 0g 2 32 3

є A є A є AC z z

d d d (S10)

where 0є is the vacuum permittivity, d is the vacuum gap depth, and A is the area of the device.

The equilibrium position of static displacement z can be found by minimizing the total energy

(balancing the elastic and electrostatic forces)

2

g gelastic electrostatic 3 2 2Ytot Y 0 g g22 2

8π 1 12π 0

2 23 1

C CU U E tF z E t V z V

z z za

(S11)

By solving Eq. S11, we obtain 2

20g2

Y 08

є az V

E t d for a given Vg.

The resonance frequency of the NEMS device is given by res ef f eff 2πf k M . Here, the spring

constant of the device keff can be obtained by second order differentiation of the total potential

energy. By inserting into Eq. S6 the mode shape of the circular membrane in its fundamental

resonance mode, mode 0 01

rW zJ

a

(J0: the 0th-order Bessel function J; 01 2.405 ), together

with the static deflection from Eq. S11, one can obtain an approximated expression of keff

22

gelastic electrostatic 2 2Yeff Y 0 g2 22 2

8π 14.9

21

CU U E tk z E t V

z za

(S12)

By substituting z and

2

g

2

C

z

into Eq. S12, we have

2 224 20 0

eff g Y 0 g4 32 2

Y 0

π π4.9

38 1

є є aak V E t V

d dE t

(S13)

Thus resonance frequency tuning by gate voltage can then be expressed as

2 224 20 0

g Y 0 g4 32 2

Y 0

res

eff

π π4.9

38 11

2π

є є aaV E t V

d dE tf

M

(S14)

This describes the observed “W” shaped curve. We note that under very large |Vg|, both the static

deformation and resonant mode shape of the membrane will deviate from the expressions above,

and thus the frequency response will deviate from Eq. S14.

Alternatively, one can also qualitatively understand the observed gate tuning effect in fres by

considering two main effects of |Vg| on fres separately: capacitive softening and electrostatic

stiffening.

In the case of capacitive softening, the electrostatic force Felectrostatic increases as the membrane

moves closer to the gate (the capacitance changes with the vacuum gap depth), producing a non-

zero force gradient. This is equivalent to an additional spring constant term keff that is capacitive

in origin: 2 2 2

eff g g1 2k F d C d V , with Cg and d being the capacitance and spacing

between the gate electrode and the 2D NEMS membrane. This term is always softening in

nature. For the fundamental out-of-plane mode, this effect dominates when 2D NEMS devices

have large strain (see Eq. S14). The Vg dependence of fres for the device in Fig. 3C of the Main

Text (D1.5m, 2L, 0=0.2N/m, 0=0.071%) very well fits this scenario (magenta curve in the

Vg–f plane). This effect has been observed in a number of NEMS (19, 23, 74), including

graphene resonators (45, 51, 75).

In the case of electrostatic stiffening, a larger |Vg| creates a greater attractive force between the

gate and the membrane, introducing additional strain to the device (75, 76). This effect takes

over when |Vg| is sufficiently large, or when device initial strain is sufficiently small. The

stiffening effect has been observed in many 1D and 2D NEMS (12, 23, 47, 50, 75, 77, 78).

The competition between these two effects results in non-monotonic dependence of fres on |Vg|,

usually in the form of a “W” shaped curve (Fig. 3E of the Main Text). For devices with same

dimensions, larger initial strain leads to smaller electrostatic stiffening effect, and increases the

threshold voltage for the electrostatic stiffening to become dominant (fres increases with |Vg|). The

devices in Fig. 3C of the Main Text has higher initial strain (higher fres at Vg=0), and therefore

higher threshold voltage (outside the measurement range), thus only the middle segment of the

“W” shaped curve can be observed. On the other hand, the device (D1.5μm, 2L, 0=0.15N/m,

0=0.054%) in Fig. 3E of the Main Text has lower initial strain and shows a larger portion of the

“W” shaped curve within the same range. Further, for the device with much lower strain

(D1.5μm, 4L, 0=0.11N/m, 0=0.020%), we have observed the “U” shaped frequency tuning

curve (fig. S5B). Similar effect has been observed in other NEMS resonators (12, 51, 75, 77, 78).

fig. S5. Electrical gate turning of higher-mode resonances. (A) A wide range frequency scan shows 3

resonance modes in a D1.5μm 4L MoS2 resonator. (B) Gate tuning of the two higher modes, showing

clear frequency upshift from electrostatic stiffening, with little noticeable effect from capacitive softening.

S3.1.1 Gate Tuning of Higher Resonance Modes

Figure S5 shows an example of gate tuning of higher mode resonances. The device (D1.5μm,

4L) exhibit 3 resonance modes up to 140MHz at Vg=0 (fig. S5B top). In each of the two higher

modes, fres clearly increases in with |Vg| almost monotonically (fig. S5B bottom), exhibiting

negligible capacitive softening. It shows that fres can be tuned up by at least 25% (73MHz –

93.6MHz) in the range of ±20V of Vg through electrostatic tensioning (stiffening) effects,

demonstrating electrical tuning of device tension.

S3.1.2 Tradeoff between Tuning and Initial Strain

While the capacitive softening effect is determined by device geometry, the gate voltage induced

stiffening (or hardening) effect depends on a number of factors, including the initial tension, and

therefore the strain, elastic modulus, and device dimensions. Larger initial strain leads to smaller

displacement upon application of DC gate voltage, less stretching of the membrane, and thus less

hardening effect. A tradeoff between the tuning range and the initial strain therefore exists and

limits the tunability of devices with high levels of initial strain. On the other hand, high strain in

the membrane leads to high operating frequency, which is desirable for radio frequency (RF)

20 40 60 80 100 120 1400

50

100

Frequency (MHz)

Sig

na

l A

mp

litu

de

(

V)

-20 -10 0 10 20

80

100

120

Fre

qu

ency f

(M

Hz)

Gate Voltage Vg (V)

A

B

applications. Accordingly, we define a figure of merit (FOM) for resonance tuning by

considering this tradeoff, FOMtuning= fres0/ℰ (: tuning range in %; 0: initial strain; ℰ : applied

electrical field).

So

ften

ing

Bo

th

Hard

enin

g

Ha

rde

nin

g

Ha

rde

nin

g

Ha

rdenin

g

Ha

rde

nin

g

Hard

enin

g

Ha

rde

nin

g

Both

Bo

th

So

ften

ing

So

ftenin

g

Both

Fig

3C

Fig

3E

Fig

S5

low

er

Fig

S5

uppe

r

Re

f. 1

2 F

ig 3

a r

ed

Re

f. 1

2

Fig

3a

blu

e

Re

f. 7

5 F

ig 1

e top

Re

f. 7

5 F

ig 1

e b

otto

m

Re

f. 7

8 F

ig 4

c

Re

f. 7

7 F

ig 2

a

Re

f. 7

7 F

ig 3

a

Re

f. 5

1 F

ig 2

Re

f. 5

1 F

ig 3

Re

f. 5

1 F

ig 4

1x10-1

1x100

1x101

1x102

FO

Mtu

nin

g (

Hz

m/V

)

fig. S6. FOM for frequency tuning: Comparison across reported 2D NEMS devices. The blue bars

represent devices in this study, and red bars are for data taken from references.

We calculate the FOMtuning for our devices and other 2D NEMS devices in the literature (12, 51,

75, 77, 78), with the values listed in table S1. Strain values are calculated based on the data given

in the references (for Ref. 75 we assume pristine single layer graphene in the calculation). The

resonance frequency is chosen to be the lowest fres on each mode, and the fractional increase

from the lowest fres to the highest fres gives the tuning range in %. The electrical field is

calculated from the applied voltage at maximum tuning and the device geometry (vacuum gap

depth and SiO2 thickness).

The calculated FOMtuning values are plotted in fig. S6 for comparison. It clearly shows that the

MoS2 NEMS devices in this work are among the best in all the reported 2D NEMS resonators.

S3.1.3 Estimation of Tunability

The electronic tunability of resonance frequency for the MoS2 resonators can be estimated by

calculating the capacitance of the device. Take the bottom (second harmonic) resonance mode in

fig. S5B (green data points) for example. The fractional frequency shift between Vg=19V and

Vg=20V is

res res

91.6MHz 88.8MHz

88.8MHz3.15% / V

20V 19V

f f

V

(S15)

We calculate the gate capacitance by assuming parallel plate capacitor geometry. The capacitor

consists of two dielectric layers: 40nm SiO2 and 250nm vacuum. The resulting capacitance is

therefore

2

2

17g

SiOvac

0 r,SiO 0

16.01 10 FC

dd

є A є є A

(S16)

where dvac, 2SiOd , A, 0є , and

2r,SiOє are vacuum gap depth, SiO2 thickness after patterning

microtrenches, area of the device, vacuum permittivity, and SiO2 relative permittivity (3.9),

respectively.

Using the calculated capacitance, the frequency shift associated with 1 Coulomb (1C) of charge

in the device is

res res res res

17 17g g

3.15% 3.15%

6.01 10 F 1V 6.01 10 C

f f f f

q C V

(S17)

Using the electron charge 1.610-19C, we estimate the frequency shift from a single electron as

19res resres res 1711

3.15%1.60 10 C 83.9ppm

6.01 10 Cee

f ff f q

q

(S18)

S3.2 Gate Tuning of Quality (Q) Factor

The measured data of Q dependence on sweeping |Vg| (see Fig. 3F in the Main Text) suggests

that in both electrical and optical drive schemes, even for optical detection, varying |Vg| could be

associated with a loaded-Q effect due to dissipative processes in the capacitive coupling during

sweeping |Vg|. Here we briefly analyze this effect. As the 2D NEMS device vibrates, the gate

capacitance oscillates. At the same time, there is an oscillating current going through the gate

capacitor, which is given by first-order derivative of the charge q=CV. Let Cg,ac(t) be the time

varying part of capacitance C, with g g,ac g g sinC C C t C C t , and V the voltage

difference between the gate and the 2D flake, composed of both DC and AC components (in case

of electrical drive), g g sinV V V t . The oscillating current is thus

g g g g g g, δ cos δ cosd CV

I t V V C t C V tdt

(S19)

Assuming the contact resistance and the resistance of the Si substrate are negligible, the device

resistance is mainly determined by the MoS2 channel resistance, d g

g 0

1R V

V G

, where is

a parameter depending on the channel mobility, capacitance and dimensions of devices, and G0 is

the channel conductance when Vg=0. Hence the energy dissipated per cycle due to the current

flow is

2π 2 2

2 2 2

elec,loss g d g g g g g g g g0

g 0

π2E I R dt V C C V C V C V

V G

(S20)

Therefore, this capacitive gate oscillating current induced dissipation can lead to a Vg dependent

loaded-Q effect, which can be expressed by

2 2

2 2

g g g g g g g g

stored g 0L, g int

1 1 1 π2

V

V C C V C V C VQ Q E V G

(S21)

where (1/Q)L,Vg, (1/Q)int, and Estored are the resultant dissipation due to this loaded-Q effect,

intrinsic device dissipation, and stored mechanical energy in the device. As Eq. S21 indicates, a

large |Vg| leads to more dissipation and thus lower observed Q, and Q remains roughly constant

for small |Vg| in the optical driving scheme (as δVg=0). Such loaded-Q effect is clearly observed

in both electrical and optical driving schemes (Fig. 3F in the Main Text).

S3.3 Gate Tuning of Vibration Amplitude

The different Vg dependence of the peak amplitude (green solid circles on the amplitude-Vg

planes in Fig. 3C and 3E in the Main Text) results from the electrostatic coupling of the MoS2

nanosheet to the gate electrode. When the device is electrically excited, the magnitude of the

electrostatic force on the resonator is

2g g 2 2

g g g g g g

1 1cos 2 cos

2 2

C CF V V t V V V t O V

z z

(S22)

Therefore, the magnitude of the drive (at frequency f=ω/2) is approximately proportional to |Vg|

for |Vg|<|Vg|, which is the case of our measurements (Vg=115mV in Fig. 3E in the Main text).

This effect is clearly observed for the electrically driven device in Fig. 3E in the Main Text

(green lines in the vertical plane). At higher |Vg| values, the lower Qs (see S3.2 above) contribute

to causing the peak amplitude to decrease and fall below the linear trend. In the optical drive

scheme (Fig. 3C in the Main Text), on the other hand, the photothermal driving force is set by

the 405nm laser power and its RF modulation signal (these are unchanged during the |Vg| sweep),

which do not depend on the gate voltage |Vg|. In this scheme, as |Vg| increases, at least the loaded

Q effect (see S3.2 above) is present and contributes to lowering the Q, which in turn reduces the

measured resonance amplitude.

section S4. Power handling, mass sensitivity, and frequency stability

The critical amplitude ac determines the maximum oscillating power a resonator can handle

before going into its nonlinear regime. This critical power can be calculated using 0c cP E

Q

,

where 2

c eff c

1

2E k a is the critical energy stored in the resonator with 2

eff 0 effk M is the effective

spring constant, and ac is the critical amplitude (see section S6 for more details).

Using the 2L device with fres=89.9MHz and Q=38 (Fig. 4B in the Main Text and device F in

table S2) as an example, we calculate keff =1.1N/m, Ec=85.8aJ, and Pc=1.3nW.

The intrinsic resonant mass sensitivity of the resonator is limited by its thermal fluctuations (79);

for a given Q, the greater the DR, the better the sensitivity

1 1 1

2 2 2B 20

eff eff

c 0 0

2 =2 10DR

k T f fM M M

E Q Q

(S23)

where f is the measurement bandwidth. For this device we calculate a resonant mass sensitivity

of M = 0.57zg (using f=1Hz).

Such mass sensitivity (~10-22g) is much better than that those in previous NEMS resonators (~10-

20g) nanomachined from conventional 3D crystals, and is comparable to that from nanotube

resonators at room temperature (also ~10-22g) (80) while providing a much greater active sensing

area and much broader frequency tunability. It is expected that the performance will further

improve at cryogenic temperatures, as demonstrated in nanotube NEMS resonators (47).

The ultimate limit of phase noise is also limited by the thermomechanical noise (79)

2

resB

2

c

( ) 10log2

fk Tf

PQ f

L (S24)

Consider an offset frequency f = 1kHz, we calculate L (f) = 50.4dBc/Hz for this device.

One can also express this in terms of frequency stability, the average fractional frequency

fluctuation over an averaging time τ

1

2res B

res c

1f k T

f Q P

(S25)

For this device we calculate8

res res/ 8 10f f

with τ=1s at room temperature.

Table S2 summarizes and lists the calculated results on these relevant metrics, for all the devices

with nonlinearity measured.

section S5. Nanomechanical tuning and sensing of device strain and bandgap

S5.1 Vibration Induced Strain

As the MoS2 membrane vibrates, the displacement introduces additional strain in the device. For

simplicity, we assume the mode shape of the drumhead resonator remains unchanged for any

given resonance amplitude, and the vibration induced additional tension is equally distributed

throughout the entire membrane.

The normalized mode shape of a circular membrane in its fundamental resonance mode is

0 01( ') 'Z r J r (S26)

where J0 is the 0th-order Bessel function J, and 405.201 . r’=r/a is the normalized radial

distance from the center (r’[0,1]), with a=0.75μm being the actual radius of the device used in

the following analysis. The physical out-of-plane displacement is then

eff

ModeShape

( ) ( ')Z

H r Z r

(S27)

Here, H(r) describes the physical displacement for a given point (at radial distance r) on the

membrane, Zeff is the effective (rms) displacement of the resonator (measured by using the laser

interferometry technique), and 1

2

ModeShape0

1( ') 2 ' ' 0.2695Z r r dr

is the squared ratio

between the effective displacement of the entire membrane to the displacement at its center.

We estimate the strain level by comparing the total membrane area at maximum displacement

during oscillation versus the area when the membrane has zero displacement. Here we assume

zero initial strain in the membrane. Figure S7 shows a schematic drawing for computing the total

area, which is

12 2

2

0 0

( )1

a dH rA dA dr d r

dr

(S28)

As an example, here we perform calculations for the 1L device in Fig. 4A of the Main Text

(device A in table S2). The critical amplitude (0.745ac) in the displacement domain is

Zeff,critial5nm. The total surface area of the membrane is

12 2

0.75μm2critial

critical0

( )2 1 1.767373μm

dH rA r dr

dr

(S29)

while the initial area of the membrane is0.75

2

device0

2 1.767146m

A rdr m

.

The motion-induced additional strain is then (at the limit of small initial strain)

critial

device

1.7673731 1 0.000064 64ppm

1.767146

A

A (S30)

Similarly, for the largest displacement (~10nm) shown in Fig. 4A, the additional strain can be

calculated as shown in the following

10nm

ModeShape

10nm( ) ( )H r Z r

(S31)

20.75

2

10nm0

( )2 1 1.768054

m dH rA r dr m

dr

(S32)

10nm10nm

device

1.7680541 1 0.000257 257ppm

1.767146

A

A (S33)

The resulting change in bandgap can be estimated using the bandgap-strain responsivity of

70meV/% strain for the direct bandgap (33): g 257ppm 70meV/%=1.80meVE .

We also estimate the maximum possible displacement at the strain limit (~20%) for this device

to be 325nm. To verify

325nm

ModeShape

325nm( ) ( )H r Z r

(S34)

20.75μm

2

325nm0

( )2 1 2.542357μm

dH rA r dr

dr

(S35)

325nm

device

2.5423571 1 0.20

1.767146

A

A (S36)

which is the intrinsic strain limit for MoS2 (28). Here we assume this strain limit value applies to

1L- and few-layer devices. We use the displacement-domain value (325nm) in calculating the

dynamic range for nonlinear operation (from onset of Duffing to fracture) in Fig. 4C and 4D in

the Main Text, and for all the devices in table S2.

S5.2 Electrostatic Tensioning Induced Strain

We calculate the DC gate voltage induced strain by assuming a uniform electrostatic force

applied to the entire MoS2 membrane. The resulting deformation of MoS2 introduces additional

tension to the device and thus shifts its resonance frequency. If one assumes the frequency shift

results from electrostatic stiffening only (see section S3 for detailed discussion), the additional

tension γ and additional strain can be estimated by using the equation res

2.40482 f

a t

for circular membrane resonator

2

1 0

Y

2

2.4048

a f ft

E t

(S37)

where a is the device radius, is the (3D) mass density, t is thickness, f1 and f0 are fres with and

without tensioning, and EY is Young’s modulus. In our measurements we observe electrostatic

frequency tuning of as much as 25% (see section S3.1.1 and fig. S5 for example). If one applies

Eq. S37 to the 1L device shown in Fig. 4A of the Main Text, a 25% increase in the fres (from

28.47MHz to 35.63MHz) gives an increase in tension of γ=0.00632N/m (from 0.01068 to

0.01700N/m). This corresponds to an stain increase of =45ppm (from 76.29ppm to

121.43ppm), which translates to a change in bandgap of 45ppm 70meV/%=0.32meVgE .

The static spatial displacement can also be estimated. Here the deformed membrane takes form

of a spherical cap (by assuming the out-of-plane displacement being much smaller than the

radius, which we confirm below). The surface area of the deflected MoS2 membrane is then

2

10

2 sinA l d

(S38)

Here, 2 2

2

a zl

z

is radius of sphere, a is radius of circular membrane, and z is displacement at

center of membrane. is the angle of the membrane radius to the center of the sphere:

arctana

l z

. The gating induced strain is given by

1

device

1A

A (S39)

Accordingly, for the same 1L device an additional strain level =45ppm corresponds to a

displacement (at the center of the membrane) of z=7.13nm, much smaller than the device radius.

fig. S7. Schematic for calculating the total surface area on a deformed membrane. We use cylindrical

coordinate system (r, ) in the calculation. dA represents an infinitesimal area, and the integral is over the

entire membrane.

rd

dr

r

rd

dr((dH/dr)2 +1)

1/2

rd dA

S5.3 Strain Sensitivity

The resonance frequency’s strong dependence on the strain in the membrane makes these

devices highly sensitive strain gauges. Here we used the 1L device with fres=28.5MHz and Q=82

(Fig. 4A in the Main Text and device A in table S2) as an example. The relation between fres of

the fundamental mode and the membrane tension is (without applying gate voltage, Vg)

2

res

0

2

2.4048

a ft

(S40)

From measured fres, we determine the initial tension in the device as 0.01068N/m.

The ultimate limit on resolving resonance frequency shift is set by the device’s

thermomechanical fluctuations (55)

1

2res B

res c

1f k T

f Q P

(S41)

and with τ=1s, the frequency resolution is fres45Hz for this device. This translates into a

change in tension of γ=3.510-8N/m, corresponding to a strain variation of 0.25ppb. From

this we estimate an excellent strain-to-fres responsivity, ℜstrain=45Hz/0.25ppb=0.18MHz/ppm.

A much more conservative and modest way of estimation uses fres≈ fres/Q =0.347MHz as the

frequency resolution, which gives a strain resolution of 0.347MHz÷0.18MHz/ppm=1.93ppm.

With a 25% increase in resonance frequency (corresponding to 45ppm strain increase;

calculations detailed in section S5.2 above), this gives a measurement range of ~102.

Here we compare our strain resolution with photoluminescence (PL) and Raman techniques. For

PL measurements, it is very challenging to achieve 1meV resolution even at low temperature.

Using this optimistic value and the responsivity of 70meV/(% strain) for the direct bandgap (30)

we obtain a strain resolution of 140ppm. Similarly, with an optimistic Raman resolution of

0.1cm-1 and a strain responsivity in Raman, 4.1cm-1/(% strain) (Ref. 34), we obtain 220ppm as

strain resolution for Raman measurements.

section S6. Measuring nonlinearity and estimating critical amplitude

A resonator with nonlinear spring constant is governed by the following equation (up to the

cubic term)

2 30 3 ext1 2

eff eff eff eff

k Fk kz z z z z

Q M M M M

(S42)

where z, ω0, Q, Meff, Fext are the displacement, resonance (angular) frequency, quality (Q) factor,

effective mass of the resonator, and external driving force, respectively. k1, k2 and k3 are the

effective linear, quadratic, and cubic (Duffing) spring constant, with k1=Meffω02 , and k3/k1

measures the (relative) degree of Duffing nonlinearity of the system. Note that with small static

deflection (which is the case here, as all nonlinear resonant responses are measured at Vg=0V)

the cubic term dominates the nonlinear response and Eq. S42 degenerates to a standard Duffing

equation. The solution to the Duffing equation (under different Fext) forms a family of frequency-

response curves (the set of colored dashed lines in Fig. 4A in the Main Text). The amplitude

remains a single-valued function of driving frequency f until the driving force reaches a critical

value (corresponding to the red dashed curve in Fig. 4A in the Main Text), beyond which the

response becomes multi-valued, leading to the bifurcation phenomena responsible for the

observed hysteresis (the solid olive and magenta curves following respective arrows in Fig. 4A

of the Main Text). Under the critical drive, the point on the frequency response curve where the

slope is infinity (red solid circle in Fig. 4A in the Main Text) determines the critical amplitude

(ac), which represents the onset of bistability in the response. This value can be experimentally

determined by fitting the peak position in the measurement to the backbone curve 2

peak peak

0

3

8a

(blue long-dashed curve in Fig. 4A and 4b of the Main Text), where σpeak is the frequency

detuning from the (angular) resonance frequency ω0, and 3 effk M is the nonlinear coefficient.

From the fitting coefficients one can obtain α and thus determine the critical amplitude using2

2 0c

8 3

9a

Q

. Figure 4B in the Main Text shows a set of experimental data for a 2L MoS2

resonator under different driving levels of modulated laser. The fitted backbone curve (blue long-

dashed line) is shown together with the extracted value of ac (horizontal red dotted line). The

1dB compression point below ac (0.745 ac) is used as the onset of nonlinearity (16), and is used

in Fig. 4C and 4D in the Main Text.

section S7. Translation of voltage fluctuations into frequency instability

Here we estimate the frequency instability of devices induced by the voltage fluctuations in the

external voltage source connected to the gate electrode. The power handling of the device at its

critical amplitude (i.e., upper limit of its linear operation regime) is given by

2

c res eff cP f k a Q (S43)

where keff is effective stiffness, fres is resonance frequency, Q is quality factor and ac is

displacement at the critical point. For the 1L device shown in Fig. 4A, without applying gate

voltage the measured critical amplitude ac is ~4.9nm, which translates into power handling of

Pc0.79pW. At T=300K, we estimate frequency stability of the device in Fig. 4A, by using Eq.

S41 with 1s averaging time (τ=1s), as 6

res res/ 1.6 10f f

and thus fres45Hz. With

frequency tuning responsivity (i.e., 0.45MHz/V at 20V) shown in fig. S5B, equivalent

voltage fluctuation is V=fres/=0.0001V and V/V=5ppm. This equivalent voltage fluctuation

is similar to that of some high-stability voltage sources (e.g., V/V=5ppm/24hours for ADCMT

6166 DC voltage/current source) and higher than that of lead battery. Therefore, by choosing

such commercially available voltage sources or appropriate batteries, it is feasible to tune the

frequency over a very large range while maintaining the frequency stability achieved at Vg=0V.

section S8. Comparison of DR in 1D and 2D NEMS resonators

Here we compare the dynamic range of 1D and 2D NEMS resonators. Qualitatively, it can be

understood by comparing the upper and lower limits of the dynamic range. Consider a 2D and a

1D NEMS resonator of identical thickness (t) and size (length L for the 1D case, diameter D for

the 2D case, L=D=2a, with a being the radius), for the motional part. The onset of mechanical

nonlinearity, which determines the upper limit of the linear dynamic range, is limited by

geometry in both 2D and 1D cases (given the same material thickness and same characteristic

length or size, L=D) and on the same order of magnitude (see next paragraph for quantitative

analysis of an example). The thermomechanical noise (lower limit of linear dynamic range),

however, is expected to be quite different between these devices. This is because the 2D device

has a higher stiffness k: consider that the structure is composed of point masses (atoms or

molecules) connected by infinitesimal springs; given a displacement at one point mass (e.g., the

one at the geometric center), the restoring force, in the 2D case, comes from the connected

neighboring masses in all directions, while in the 1D case it only includes contribution from just

two nearest axial neighbors. Therefore, given the identical thickness and same lateral extent, the

2D device has higher frequency than its 1D counterpart (i.e., enhanced stiffness due to extrusion

from 1D to 2D). Meanwhile, the effective mass for fundamental mode in 2D circular drumhead

membrane is Meff,2D=0.270M2D, and for 1D doubly clamped ribbon it is Meff,1D=0.5M1D, with

their actual masses M2D >> M1D (M2D/M1D L/w or D/w >>1, where w is width of 1D doubly

clamped ribbon) due to extrusion. Given the expression for thermomechanical noise on

resonance 1/2 3

,th 0 B 0 eff4xS k TQ M , the 2D device will have lower thermomechanical

noise, and thus greater linear dynamic range (see next paragraph for quantitative analysis for an

example).

As a quantitative example, here we examine 1D nanoribbon and 2D membrane resonators with

identical dimensions for their cross section. For the 1D device in the shape of a nanoribbon, we

use t=0.7nm (1L MoS2), ribbon width w=7nm, and L=1.5m. For the 2D circular membrane, we

use t=0.7nm and diameter D=1.5m, so both resonators have the same thickness and same lateral

extent (size measure) (i.e., the characteristic length of the 1D device and diameter of the 2D

device set to be the same). We use Q=100 and strain = 0.0001 for both devices.

The critical amplitude of a 1D doubly-clamped nanoribbon resonator is given by (16)

2 2

c,1 34

2 1

33D

t La

Q

(S44)

where Q is quality factor, t is thickness, is strain and L is length of the 1D string. For a 2D

circular drumhead resonators, the critical amplitude is (46)

c,2D 2

8 3

9a

Q (S45)

Here, is Duffing-type nonlinear coefficient,

2

2

13 21 4

30 1 a

where is Poisson’s ratio of

the material, a is the radius. For the above devices, the calculated critical amplitudes are

ac,1D=0.728nm for the 1D device and ac,2D=1.36nm for the 2D one, showing that the 2D

resonator has a larger upper limit for dynamic range.

We now calculate the thermomechanical noise using 1/2 3

,th 0 B 0 eff4xS k TQ M (see

Supplemental Note S2). For 1L MoS2 resonators with parameters given above, we obtain

fundamental mode resonance frequency of 23.9MHz for the 1D device and 36.5MHz for the 2D

device, confirming that 2D structure has higher frequency. This gives thermomechanical

fluctuation amplitude (assuming 1Hz measurement bandwidth) of 5.13pm for the 1D device and

0.281pm for the 2D device, again confirming the above analysis.

Combining the upper and lower limits, we obtain DR = 37dB for the 1D device and 68dB for its

2D counterpart in this case study, clearly demonstrating that 2D devices has broader DR than 1D

devices with identical thickness, characteristic length, and strain.

fig. S8. Measured DR in 1D and 2D resonators operated at room temperature. Magenta stars show

results in this work and blue circles represent results from other 1D and 2D resonators.

10-4

10-3

10-2

10-1

100

40

60

80

100

120

Mea

sure

d D

yna

mic

Ra

ng

e

(dB

)

Device Volume (m)3

MoS2

This Work

Graphene

(Ref. 54)

Graphene

(Ref. 12)

Si

(Ref. 17)

Carbon Nanofiber

(Ref. 58)

SiC

(Ref. 23)

SiC

(Ref. 55)

AlN

(Ref. 57)

AlN

(Ref. 56)

Graphene

(Ref. 11)

Graphene

(Ref. 53)

fig. S9. Resonance frequency scaling with device diameter D and MoS2 thickness t. Theoretical

values are shown as lines. For D=0.5µm (magenta curves) and 6µm (red curves), calculations are shown

for γ=0.5 and 0.1N/m with shaded area in between. For devices with D=1.5µm (blue curves), additions

tension values of γ=0.05, 0.02, and 0.01N/m are also shown. Spherical symbols represent the measured

values for 1L (green), 2L (magenta), 3L (blue) devices and 4L (black) devices (D1.5μm). Data for

thicker devices with D6μm (red) are taken from Ref. 37.

1 10 100

10

100

1000

1L MoS2

2L MoS2

3L MoS2

4L MoS2

f res

(MH

z)

Number of Layers

table S1. FOM for frequency tuning.

Device

Number of

Layers

(#L)

Initial

Tension

0 (N/m)

Initial

Strain

0

(ppm)

fres

Range

(MHz)

Tuning Range

(fres/fres)

Vg

(V)

Electrostatic

Gap (d)

Electrical

Field

ℰ (V/m)

FOMtuning

(Hz·m/V)

MoS2

Fig. 3C 2L MoS2 0.202 721 78-87 0.115 20 250nm

+40nm SiO2 76.8 84.5

Fig. 3E 2L MoS2 0.15 536 74.2-75.3 0.0148 20 250nm

+40nm SiO2 76.8 7.67

fig. S5B

lower 4L MoS2 0.102 182 73-91.6 0.255 20

250nm

+40nm SiO2 76.8 44.1

fig. S5B

upper 4L MoS2 0.102 182 108-126 0.167 20

250nm

+40nm SiO2 76.8 42.7

Graphene

Ref. 12

Fig. 3a

red

1L

Graphene 0.0136 40 30-77 1.57 10

100nm

+200nm SiO2 66.1 28.4

Ref. 12

Fig. 3a

blue

1L

Graphene 0.068 200 42-58 0.381 10

100nm

+200nm SiO2 66.1 48.4

Ref. 75

Fig. 1e

top

1L

Graphene 0.1372 404 77-87 0.130 15

170nm +

130nm SiO2 73.8 54.7

Ref. 75

Fig. 1e

bottom

1L

Graphene 0.1372 404 62-78 0.258 15

170nm +

130nm SiO2 73.8 87.5

Ref. 78

Fig. 4c

1L

Graphene 0.00209 6.15 13.4-32.8 1.448 9 285nm 31.6 3.77

Ref. 77

Fig. 2a

1L

Graphene 0.013 38.2 3.47-4.56 0.314 12 200nm 60 0.695

Ref. 77

Fig. 3a

1L

Graphene 0.01 29.4 5.2-7.9 0.519 10 200nm 50 1.59

Ref. 51

Fig. 2

1L

Graphene 0.04 118

177.2-

177.9 0.00395 30 500nm 60 1.37

Ref. 51

Fig. 3

1L

Graphene 0.205 603 56.3-57.7 0.0249 30 500nm 60.0 14.1

Ref. 51

Fig. 4

1L

Graphene 0.00997 29.3 64-100 0.563 7 500nm 14 75.4

table S2. List of devices with measured nonlinear characteristics.

Device

Number

of

Layers

fres

(MHz) Q

Sx1/2

(fm/Hz1/2)

ac

(nm)

DRin

(dB)

DRnonlinear

(dB)

Pc

(pW)

L(f)

(dBc/Hz)

fres/fresτ (10-6)

M

(zg)

A 1 28.47 82 380 4.92 76.78 38.95 0.79 -35.00 1.57 5.28

B 1 46.23 125 220 1.59 71.44 48.78 0.23 -29.11 1.90 6.40

C 2 77.69 42 42 2.58 90.16 44.57 17 -33.85 0.655 4.42

D 2 78.44 42 42 2.86 91.19 43.67 22 -34.79 0.582 3.92

E 2 80.53 51 44 4.08 93.78 40.57 40 -38.84 0.355 2.40

F 2 89.9 38 32 17 108.86 28.20 1300 -50.41 0.0840 0.567

G 3 29.2 70 190 2.26 75.81 45.71 0.63 -32.44 2.05 20.7

H 3 34.88 70 150 2.68 79.61 44.22 1.5 -34.70 1.32 13.4

table S3. DRs measured in 1D and 2D resonators.

Ref # Device

Structure Material

DR

(dB)

fres

(MHz) Q

Temperature

(K) Dimensions

Volume

(m3)

Ref. 11

Doubly

clamped

string

Graphene 60 36 ~50 300 L=2.7m

w=630nm

t=5nm

8.510-3

Ref. 12

Doubly

clamped

string

Graphene 40 65 ~125 300

L=1.1m

w=3m

t=0.335nm

1.1110-3

Ref. 23

Doubly

clamped

beam

SiC 69-73 8.78 3000 300 L=15m

w=150nm

t=100nm

2.2510-1

Ref. 53 Circular

membrane Graphene 40 8.78 ~800 300

D=12m

t=0.335nm 3.7910-2

Ref. 54

Doubly

clamped

string

Graphene 60 1.19 1180 300

L=1.8m

w=1.2m

t=0.335nm

3.0210-4

Ref. 56

Doubly

clamped

beam

AlN 85 14.31 1220 300 L=9m

w=470nm

t=210nm

8.8810-1

Ref. 57

Doubly

clamped

beam

AlN 89 14.28 N.A. 300 L=10m

w=470nm

t=210nm

9.8710-1

Ref. 17

Doubly

clamped

beam

Si 48-56 22-96 550-1200 300 L=1.8 to 4.9m

w or t=30 to

100nm

1.2710-3

to

2.9810-2

Ref. 55

Doubly

clamped

beam

SiC 96 428 2500 22 L=1.65m

w=120nm

t=80nm

1.5410-2

Ref. 58 Cantilever Carbon

nanofiber 52 6.65 270 300

L=4.6m

w or t=65nm 1.5310-2

This

Work

Circular

membrane MoS2 71-109 28.5-96.5 40-125 300

D=1.5m

t=0.7 to 2.8nm

1.210-3

to

4.910-3