Supplemental Material:
Dynamic Range Tuning of Graphene Nanoresonators
Marsha. M. Parmar1, P.R. Yasasvi Gangavarapu2 and A. K. Naik2
1
Department of Physics, Indian Institute of Science, Bangalore, 560012, India
2
Centre for Nano Science and Engineering, Indian Institute of Science, Bangalore, 560012, India
A. Measurement and detection technique:
We have used frequency mixed down technique to read out the electrical response of graphene
resonator1. A dc gate voltage along with RF signal of frequency f+Δf is applied at gate terminal to
produce static deflection and actuate the device. Another RF frequency signal at f is supplied at drain
terminal. The mechanical motion is detected as a mixed down current ImixΔf which results from the
change in conductance of graphene due to change in the capacitance between the gate and the
graphene flake. Figure S1 shows the circuit used in our measurements. The 50 Ω resistors were used for
impedance matching, whereas the 100pF capacitors were used to perform dc as well as ac
measurements simultaneously2.
Figure S1: Schematics of the measurement circuit
The frequency response curves at different drives were fitted to a Lorentzian function and parameters
such as resonant frequency, quality factor and peak amplitude were extracted. The mixed down current
is given by1,
∆𝑤
𝐼𝑚𝑖𝑥
= 𝐴 + 𝐵𝜔 +
𝜔02 − 𝜔2
𝐻 cos (arctan ( 𝑄𝜔𝜔
) + ∆𝜑)
0
(S1)
2
2
2
√(1 − ( 𝜔 ) ) + ( 𝜔 )
𝜔0
𝑄𝜔0
where, A and B represent background and slope respectively. H is the value of mixed down current at
resonance, 𝜔0 is the resonant frequency, 𝑄 is the quality factor and ∆𝜑 is the phase difference due to
the background. Figure S2(a) shows the fitting of in-phase (x) and out of phase (y) components of mixed
down current. A proper fit to the circle of in-phase (x) versus out of phase (y) components of the current
indicates that the resonator is being operated in linear regime (Figure S2(b)). As mentioned in the main
text the onset of nonlinearity was determined with the hysteretic behavior between the forward and
backward frequency sweeps. Figure S2(c) shows one such behavior at VgDC = -9V, VsdAC = 22.5mV, VgAC =
4mV at 10K.
S2(a)
S2(b)
S2(c)
Figure S2: (a) Shows the fitting of the resonance curve to eq. (S1) above. The blue circles
show the in-phase component (x) and the red circles show the out of phase (y) component
𝐴𝐶
at 𝑉𝑔𝐷𝐶 = 8V, 𝑉𝑠𝑑
= 5.8mV and 𝑉𝑔𝐴𝐶 = 3.6mV. (b) Shows the data plotted in xy-plane which
indicates that the frequency response curve is linear. (c) Forward and reverse frequency
sweeps at -9V back-gate DC voltage at 10K. Hysteretic behaviour between forward and
reverse sweeps indicates the onset of nonlinearity.
B. Strain vs DC back-gate voltage:
The strain and mass loading in graphene devices were determined by fitting the variation of resonant
frequency with DC gate voltage (Figure 2(a) and 2(b))3. Strain values of 0.06% and 0.78% were observed
at 300K and 10K respectively. Figure S3 shows the fitted curves along with the fitting parameters (strain
𝛤(𝑇, 0) and mass loading (𝜌). Application of back-gate voltage (𝑉𝑔𝐷𝐶 ) also changes strain in these
devices. However, the relative change [(𝛤(𝑇, 𝑉𝑔 ) − 𝛤(𝑇, 0)]/𝛤(𝑇, 𝑉𝑔 ) in the strain due to the back-gate
voltage is fairly modest. The applied strain at different 𝑉𝑔𝐷𝐶 is calculated using eq. (S2).
𝛤(𝑇, 𝑉𝑔 ) =
𝐿 16𝛤(𝑇, 0) 256
(
+ 3 𝐴𝐷𝐶 2 )
𝜋2
3𝐿
3𝐿
(S2)
Plot of total strain vs back-gate voltage at different temperatures is shown in figure S4. At room
temperature, strain in graphene sheet changes from 6x10-4 at VgDC =0 to 10-3 at 𝑉𝑔𝐷𝐶 = 10𝑉, whereas at
10K the relative change in the strain values with back-gate dc voltage are negligible as shown in Figure
S4. Note that in all the calculations above, we have neglected the contribution to electrostatic force due
to the AC voltages applied to the device. Since the applied AC voltages are in mV and DC gate voltage is
in volts, the effect of these AC terms is two to four orders of magnitude smaller and thus can be
neglected.
S3(a)
S3(b)
Figure S3: (a) and (b) show the fitting to the frequency tuning curve at 300K and 10K
respectively. The strain values and mass loading are indicated in each figure.
Figure S4: The variation of strain with back-gate voltage at 300K and 10K. The strain values at a
particular temperature do not change significantly with DC back-gate voltage. The change in strain
values at 300K is two times but the change in strain from 300K to 200K is more than 10 times.
C. Calculation of static displacement:
It has been reported earlier that suspended graphene sheet can be treated as one dimensional string
with negligible bending rigidity2,4. Under this assumption the static displacement from equilibrium
position is given as 2,
2 1⁄3
(3) 𝛽
𝐴𝐷𝐶 = −
(−9𝛼 2 𝛾
+
√3√4𝛼 3 𝛽 3
+
1⁄3
27𝛼 4 𝛾 2 )
+
(−9𝛼 2 𝛾 + √3√4𝛼 3 𝛽 3 + 27𝛼 4 𝛾 2 )
1⁄3
(S3)
21⁄3 32⁄3 𝛼
Where,
𝛼=
256𝐸𝑆
16𝐸𝑆𝛤(𝑇, 0) 1 ′′ 𝐷𝐶 2
1
2
,𝛽 =
− 𝐶𝑔 𝑉𝑔 , 𝛾 = − 𝐶𝑔 ′ 𝑉𝑔𝐷𝐶
3
9𝐿
3𝐿
2
2
(S4)
The calculated values of static displacement are plotted in Figure 2(c) of main text.
D. Calculation of dynamic displacement:
In frequency mixed down technique the current at resonance is given by equation,1
∆𝜔
𝐼𝑚𝑖𝑥
=
𝑉𝑔𝐷𝐶 𝑑𝐶𝑔
𝑑𝐺
𝐴𝐶
(𝑉𝑔𝐴𝐶 +
𝛿𝑧(𝜔)) 𝑉𝑠𝑑
𝑑𝑉𝑔
𝐶𝑔 𝑑𝑧
(S5)
Where, G is the conductance, 𝑉𝑔𝐴𝐶 is applied AC voltage at gate, 𝑉𝑔𝐷𝐶 is applied DC back-gate voltage,
𝐴𝐶
𝑉𝑠𝑑
is applied AC source drain voltage, 𝐶𝑔 is the gate capacitance and 𝛿𝑧 is the dynamic displacement
from equilibrium position.
The first term in equation (S5) is the background current 𝐼𝐵𝐺 and the second term 𝐼𝑟𝑒𝑠 is due to
mechanical resonance. At resonance the displacement from equilibrium position is given as,
𝐼𝑟𝑒𝑠 𝑉𝑔𝐴𝐶
𝛿𝑧 =
×
×𝑑
𝐼𝐵𝐺 𝑉𝑔𝐷𝐶
(S6)
where, 𝑑 is the distance between the graphene sheet and substrate. The calculated amplitude of motion
for resonance curve in figure 1(b) is 0.4nm (with d = 290nm, 𝐼𝐵𝐺 = 0.44nA, 𝑉𝑔𝐴𝐶 = 3.5mV and 𝑉𝑔𝐷𝐶 = 8V).
At amplitude of 1.7nm the device was driven into nonlinear regime.
Equation (S5) shows that the output current is proportional to applied DC back-gate voltage and applied
AC source drain voltage. The transduction gain in these devices is defined as the ratio of peak current at
resonance to deflection of graphene sheet at resonance as
𝑡𝑟𝑎𝑛𝑠𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑔𝑎𝑖𝑛 =
𝐼𝐵𝐺 × 𝑉𝑔𝐷𝐶 × 𝑑
𝐼𝑟𝑒𝑠
=
𝛿𝑧
𝑉𝑔𝐴𝐶
(S7)
At higher DC gate voltage the transduction gain is better.
E. Dynamic range:
The dynamic range in our devices was calculated from the ratio of experimentally observed noise floor
to the onset of nonlinearity. Although it is the thermomechanical noise which sets the lower limit of
dynamic range, we were not able to observe the thermomechanical noise floor due to large background.
Hence in our calculations, the noise floor was determined by taking the standard deviation of the
frequency response curve obtained at the ac drive levels where we could not observe any resonance
peak. The onset of nonlinearity was determined by the hysteretic behavior of frequency response curve
as shown in figure S2(c).
In our work we have shown that the linear dynamic range of graphene resonators can be increased by
increasing the strain of graphene sheets. Strain values are largely changed by changing the temperature
in our devices. Apart from change in strain at low temperatures, we have also observed a change in
quality factor of the devices.
In general, higher quality factors are desirable for resonators. E.g. the mass resolution of an
electromechanical sensor is given by, 𝛿𝑚 ≈ −2 𝑀𝑒𝑓𝑓 ⁄𝑄 10−𝐷𝑅⁄20 . Thus higher quality factor would
improve the mass resolution. However, the dynamic range is also affected by the quality factor of the
device.
The dynamic range is affected by quality factor in following two ways5:
1/2
a) Thermomechanical noise given by 𝑆𝑥
= √4𝑘𝐵 𝑇𝑄⁄𝑀𝑒𝑓𝑓 𝜔03 is higher. We would like to point
out that in the measurements reported in this manuscript we are limited by the amplifier noise.
𝜖
b) The critical amplitude at which the device response becomes nonlinear is given by 𝑎𝑐 ∝ √𝑄 ,
where 𝜖 is the strain in the device.
Thus, assuming that the noise floor is due to thermomechanical noise, the dynamic range is given by
1/2
𝐷𝑅 = 20 log(𝑎𝑐 /𝑆𝑥 ) = 20 log(𝐴/𝑄) where A is a constant.
Reducing the operating temperature of the suspended graphene device in general improves the quality
factor. For instance, we observe an improvement from Q of ~300 at room temperature to ~1000 at 10K.
This improvement in quality factor should reduce the dynamic range by about 10dB. Thus, the
improvement in dynamic range we observe with temperature would have been even larger had the
quality factor NOT improved with lowering of temperature.
It is to be noted that although reducing the quality factor improves the dynamic range, this cannot in
general be used as a strategy to improve the performance of the device. For instance, the better mass
resolution necessitates higher quality factor. Furthermore, lower quality factors make it extremely
difficult to efficiently couple the devices to the next stage of amplification.
F. Calculation of mass resolution:
The minimum measurable mass that can be detected depends upon the effective mass of the device,
quality factor and the dynamic range.
𝛿𝑚 =
−2𝑀𝑒𝑓𝑓
× 10−𝐷𝑅⁄20
𝑄
For, 𝑀𝑒𝑓𝑓 = 2.32 × 10−16 𝑔𝑚/𝜇𝑚2, 𝑄 = 1000, the calculated values for minimum
resolvable mass is plotted in figure S5.
(S8)
Figure S5: Plot of minimum mass resolution with DC back-gate voltage.
G. Equation of motion and effective nonlinear coefficient:
We approximate our device as a doubly clamped beam actuated by an electrical load and subjected to
some damping. We have followed the theoretical approach presented in “Nonlinear Oscillations” by
Nayfeh and Mook6 while deriving the effective nonlinearity coefficient. The equation of motion
considered for our system is:
𝑧̈1 (𝑡) + 2𝜇𝑧̇ + 𝜔02 𝑧1 (𝑡) + 𝛼2 𝑧12 (𝑡) + 𝛼3 𝑧13 (𝑡) = 𝐹(𝑡) = 𝐾𝑐𝑜𝑠(Ω𝑡)
(S9)
Where, α2 and α3 are quadratic and cubic nonlinearities of the system, µ the damping coefficient and F is
the forcing amplitude. Frequency of excitation Ω can be expressed in terms of a detuning parameter 𝜎
which describes the proximity of Ω to primary resonance frequency 𝜔0 as
Ω = 𝜔0 + 𝜖 2 𝜎
(S10)
An approximate solution to this system can be obtained using method of multiple scales 6 where in the
solution is expressed in different time scales as
𝑧(𝑡; 𝜖) = 𝑧0 (𝑇0 , 𝑇1 , 𝑇2 ) + 𝜖𝑧1 (𝑇0 , 𝑇1 , 𝑇2 ) + 𝜖 2 𝑧2 (𝑇0 , 𝑇1 , 𝑇2 ) + ⋯
(S11)
Where, 𝑇0 = t, 𝑇1 = 𝜖t and 𝑇2 = 𝜖 2 t . The forcing term can be written in terms of 𝑇0 and 𝑇1 as
𝐹(𝑡) = 𝜖 2 𝑘 cos(𝜔0 𝑇0 + 𝜎𝑇2 )
(S12)
Substituting these two equations (S11) and (S12) in the equation of motion (S9) gives us the
approximate solution as
𝑧 = 𝑎 cos(𝜔0 𝑡 + 𝛽)
(S13)
Where, 𝑎 and 𝛽 are given by the following equations
𝑎′ = −𝜇𝑎 +
𝑎𝛽 ′ =
𝑘
sin(𝜎𝑇2 − 𝛽)
2𝜔0
(9𝛼3 𝜔02 − 10𝛼22 ) 3
𝑘
𝑎 −
cos(𝜎𝑇2 − 𝛽)
3
2𝜔0
24𝜔0
(S14)
(S15)
Now consider a system with cubic nonlinearity.
𝑧̈1 (𝑡) + 2𝜇𝑧̇ + 𝜔02 𝑧1 (𝑡) + 𝛼𝑧13 (𝑡) = 𝐹(𝑡) = 𝐾𝑐𝑜𝑠(Ω𝑡)
(S16)
Frequency of excitation Ω can be expressed in terms of a detuning parameter 𝜎 which describes the
proximity of Ω to primary resonance frequency 𝜔0 as
Ω = 𝜔0 + 𝜖𝜎
(S17)
An approximate solution to this system can be obtained using method of multiple scales where in the
solution is expressed in different time scales as
𝑧(𝑡; 𝜖) = 𝑧0 (𝑇0 , 𝑇1 ) + 𝜖𝑧1 (𝑇0 , 𝑇1 ) + ⋯
(S18)
Where, 𝑇0 = t and 𝑇1 = 𝜖 t. The forcing term can be written in terms of 𝑇0 and 𝑇1 as
𝐹(𝑡) = 𝜖𝑘 cos(𝜔0 𝑇0 + 𝜎𝑇1 )
(S19)
Substituting these equations (S18) and (S19) in the equation (S16) gives us the approximate solution as
𝑧 = 𝑎 cos(𝜔0 𝑡 + 𝛽)
(S20)
Where, 𝑎 and 𝛽 are given by the following equations
𝑎′ = −𝜇𝑎 +
𝑎𝛽 ′ =
𝑘
sin(𝜎𝑇1 − 𝛽)
2𝜔0
3𝛼 3
𝑘
𝑎 −
cos(𝜎𝑇1 − 𝛽)
8𝜔0
2𝜔0
(S21)
(S22)
The solutions obtained in both the systems given by Eq.(S15) and Eq.(S22) are similar if we equate their
respective nonlinear coefficients.
3𝛼
(9𝛼3 𝜔02 − 10𝛼22 )
=
8𝜔0
24𝜔03
𝛼 = 𝛼3 −
10 2
𝛼
9𝜔02 2
(S23)
(S24)
We define this term α which includes both quadratic and cubic nonlinear effects as the “effective
nonlinearity” coefficient.
H. Taylor’s expansion of gate capacitance:
The graphene sheet suspended above the gate electrode forms a capacitance. Under the application of
DC back-gate voltage, this capacitance changes. Expanding the gate capacitance 𝐶𝑔 using Taylor’s series
as
𝐶𝑔 = 𝐶0 + 𝐶1 𝑧 + 𝐶2
𝑧2
𝑧3
𝑧4
+ 𝐶3 + 𝐶4
2
3
4
(S25)
3
1
𝐶𝑔′ = 𝐶1 + 2𝐶2 𝑧 + 𝐶3 𝑧 2 + 𝐶4 𝑧 3
2
3
where, Ci’s are the capacitance coefficients given as 𝐶1 =
𝑑𝐶𝑔
𝑑𝑧
, 𝐶2 =
(S26)
𝑑 2 𝐶𝑔
𝑑 3 𝐶𝑔
𝑑𝑧
𝑑𝑧 3
2 , 𝐶3 =
and 𝐶4 =
𝑑 4 𝐶𝑔
𝑑𝑧 4
.
In systems with electrostatic actuation, the force F is given as
1
𝐹 = 𝐶𝑔′ 𝑉 2
2
(S27)
Substituting 𝐶𝑔′ from equation (S27) gives the force F as
𝐹=
1
3
1
[𝐶1 + 2𝐶2 𝑧 + 𝐶3 𝑧 2 + 𝐶4 𝑧 3 ] 𝑉 2
2
2
3
(S28)
From the above equation (S28), the cubic nonlinearity term (coefficient of “𝑧 3 ”) depends on the the
fourth-order coefficient 𝐶4 of the Taylor expansion of the electrostatic force.
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2
C. Chen, (2013).
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6
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