Mechanically-Bent Graphene as an Effective Piezoelectric

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C: Energy Conversion and Storage; Energy and Charge Transport

Mechanically-Bent Graphene as an Effective Piezoelectric Nanogenerator Lars Duggen, Morten Willatzen, and Zhong Lin Wang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b05246 • Publication Date (Web): 14 Aug 2018 Downloaded from http://pubs.acs.org on August 17, 2018

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The Journal of Physical Chemistry

Mechanically-Bent Graphene as an Eective Piezoelectric Nanogenerator †

L. Duggen,

∗,‡,¶

M. Willatzen,

†Mads

and Z. L. Wang

‡,¶,§

Clausen Institute

University of Southern Denmark, DK-6400 Sønderborg, Denmark

‡CAS

Center for Excellence in Nanoscience, Beijing Key Laboratory of Micro-nano Energy and Sensor,

Beijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Sciences, Beijing 100083, P. R. China

¶School

of Nanoscience and Technology,

University of Chinese Academy of Sciences, Beijing 100049, P. R. China

§School

of Materials Science and Engineering,

Georgia Institute of Technology, Atlanta, GA 30332-0245, USA

E-mail: [email protected]

1

ACS Paragon Plus Environment

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Abstract Recent DFT calculations have demonstrated the potential of mechanically-bent graphene as a piezoelectric energy harvesting material. We develop a 2D model of hexagonal materials and demonstrate quantitatively the potential of a single layer of graphene to function as an eective piezoelectric nanogenerator. The piezoelectricity of graphene stems from a dynamically generated surface charge density proportional to the local curvature of the graphene layer, and the proportionality constant is found from DFT calculations on a single layer of bent graphene. By virtue of dierent tailored mechanical and electrical loadings, explored in this work, it is demonstrated that graphene can be as eective an energy harvester as a single at layer of 2D MoS2 , which is strongly piezoelectric due to its inversion-asymmetric unit cell. Demonstrations are carried out for graphene and MoS2 using a 2D nite element model to determine the generated voltage, current and power density.

Introduction In recent years, much attention has been brought to the development of nanogenerators to harvest energy, such as acoustic noise, from ambient surroundings. Nanogenerators can consist of dierent design principles, terials.

48

13

one of which is the deformation of piezoelectric ma-

In this respect, MoS2 and other 2D piezoelectric materials,

911

have attracted

interest as they both oer higher piezoelectric coecients and can sustain, thermodynamically reversibly, much larger strain values (more than

10%)

than their 3D counterparts.

The most well-known 2D material, graphene, has an inversion-symmetric unit cell and, therefore, intrinsically is not piezoelectric.

However, graphene can be made piezoelectric

in dierent ways such as mechanical prebending, combining graphene with other materials forming inversion-asymmetric heterostructures, or through the application of external elds. Calculations based on density functional theory piezoelectric with piezoelectric constants up to

2

12

have shown that graphene can be made

e = 1 C/m2 , which is comparable to standard


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piezoelectric materials, eected by prebending the graphene sheet. The bending destroys inversion symmetry and induces a dipole moment in the out-of-plane direction hence yielding an electrical response based on a mechanical deformation. In this work, we will examine, using the nite element method, the potential of graphene as a piezoelectric nanogenerator by applying mechanical prebending or through electric excitation. A mathematical model is constructed that captures the polarization dynamics of graphene subject to external bending. We assume the generated surface charge density is proportional to the local curvature of the graphene layer and compute the output current and voltage as a function of a dynamic mechanical load. Comparison is nally made between monolayer graphene and MoS2 for their potential as nanogenerator materials in harvesting vibrational energy. It is demonstrated that prebent graphene can be as eective a nanogenerator as a at sheet of 2D MoS2 .

Methods The piezoelectric eect is the eect of generating or changing electric dipole moments by mechanical deformation and vice versa (direct and converse eect). This is expressed by the piezoelectric constitutive relations

where

T = cS − eE,

(1)

D = eS + E + Psp ,

(2)

T, S denote stress and strain, respectively, D, E denote electric displacement eld and

electric eld, respectively and are the stiness tensor

c,

Psp

is the spontaneous polarization. The material parameters

the permittivity

,

and the piezoelectric stress tensor

e.

When at graphene is subject to bending, inversion symmetry is broken and electric charges of opposite sign are generated on the two surfaces normal to the bending direction.

We shall treat bent graphene as a material obeying the same elastic equations as

3



at graphene. The bending, however, modies the boundary conditions, since the bendinginduced surface charges generate an electric polarization that changes in the presence of an external electric eld or mechanical stress. tric material.

6/mmm the

6mm

Thus, bent graphene is eectively a piezoelec-

While at graphene belongs to the inversion-symmetric (non-piezoelectric)

point group, bent graphene is inversion asymmetric (piezoelectric) and belongs to point group.

Surface-generated charges in a bent hexagonal structure The bending of a graphene sheet induces a dipole moment due to the motion of charges away from the center. The motion of charges still upholds a zero net charge over the full structure. We shall assume that the generated surface charge density is proportional to the local curvature

κ

of the structure corresponding to the situation shown in Figure 1.

Figure 1: Single sheet of graphene with the hexagonal crystal structure indicated. Panel (a) shows the generated deformation due to stress by an applied electric eld while panel (b) shows the change in polarization due to an applied stress.

In order to analyze the piezoelectric properties of the structure, we must solve for the strain as a function of applied electric eld

Eapp .

Subsequently, we add traction loads to the

surfaces in order to obtain the symmetry of the piezoelectric constants. Since graphene is a hexagonal structure, the stiness and permittivity matrices take the

4


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form

 0 c11 c12 c13 0  c 0  12 c11 c13 0   c13 c13 c33 0 0 c=  0 0 0 c44 0   0 0 0 0 c44   0 0 0 0 0

 0    0    0  ,   0    0   1 (c − c12 ) 2 11

  k 0 0     =  0 k 0  .   0 0 ⊥

(3)

(4)

The mechanical problem with no body forces or free charges in static conditions is generally solved by

∇ · T = 0.

(5)

Since the starting point is a at sheet of graphene, the material is not piezoelectric, and we can solve the problem subject to stress-loaded boundary conditions. We write the stress tensor as a function of displacement

u = ux x ˆ + uy y ˆ + uz ˆ z and

  ∂uy ∂uz ∂ux c11 ∂x + c12 ∂y + c13 ∂z    c ∂ux + c ∂uy + c ∂uz  12 11 13  ∂x ∂y ∂z     ∂ux ∂uy ∂uz  c13 ∂x + c13 ∂y + c33 ∂z   T=   ∂uy ∂uz   c44 ∂z + ∂y       ∂ux ∂uz c + 44   ∂z ∂x   ∂u c11 −c12 ∂ux + ∂xy 2 ∂y 5


nd

(6)


The static elasticity equations,

∇ · T = 0,

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give

∂ 2 uy ∂ 2 uz ∂ 2 ux ∂ 2 uz ∂ 2 ux c11 − c12 ∂ 2 ux c11 − c12 ∂ 2 uy + c + c + + c + c + = 0, (7) 12 44 13 44 ∂x2 ∂y∂x ∂z∂x ∂z 2 ∂x∂z 2 ∂y 2 2 ∂x∂y ∂ 2 uy ∂ 2 uy ∂ 2 uz ∂ 2 uz ∂ 2 ux c11 − c12 ∂ 2 ux c11 − c12 ∂ 2 uy + c + c = 0, (8) c12 + c11 + c + + 13 44 44 ∂x∂y ∂y 2 ∂z∂y ∂z 2 ∂y∂z 2 ∂x∂y 2 ∂x2 ∂ 2 uy ∂ 2 uy ∂ 2 ux ∂ 2 uz ∂ 2 uz ∂ 2 ux ∂ 2 uz c13 + c13 + c33 + c + c + c + c = 0. (9) 44 44 44 44 ∂x∂z ∂y∂z ∂z 2 ∂z∂y ∂y 2 ∂z∂x ∂x2

c11

These equations are generally not separable and nding a general analytical solution is not possible.

For our situation, however, we only need the inhomogeneous solution, for

which constant strains will suce, satisfying the corresponding boundary conditions. the solutions for

u

be of the form

ux = Ax x + Bx y + Cx z + Dx ,

(10)

uy = Ay x + By y + Cy z + Dy ,

(11)

uz = Az x + Bz y + Cz z + Dz .

(12)

Inserting into equations (7)-(9) veries that all strains are constant). conditions.

Let

∇·T=0

We can now determine the

(as is also evident from the fact that

12

constants by imposing boundary

Since strains are constant, so are the stresses.

Hence the traction forces on

opposite sides of the cube must be equal with opposite signs. This yields

9

conditions (3

for each independent side of the cube). Further we have three conditions by choosing the

6


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reference point:

u(0, 0, 0) = 0,

yielding

ρ

The other conditions read

T1 (x = 0) = c11 Ax + c12 By + c13 Cz = 0,

(13)

T2 (y = 0) = c12 Ax + c11 By + c13 Cz = 0

(14)

T3 (z = 0) = c13 Ax + c13 By + c33 Cz = ρEz,app ,

(15)

T4 (z = 0) = c44 Bz + c44 Cy = ρEy,app ,

(16)

T5 (z = 0) = c44 Cx + c44 Az = ρEx,app ,

(17)

T6 (x = 0) =

where

Dx = Dy = Dz = 0.

c11 − c12 c11 − c12 Bx + Ay = 0, 2 2

(18)

is the surface charge density and three conditions are omitted as they coincide

with the given conditions (T4 (y coecients

Bz , Cy , Cx , Az , Bx , Ay

= 0) = T4 (z = 0)

etc.). A consequence of this is that the

can only be determined up to an unknown constant - this

is the well known principle of free choice of shear strains

13

). Note that in writing the stress

boundary conditions we have neglected a minor nonlinear contribution from the Coulomb attraction between opposite surfaces. The solutions are now

Ax =

−2c213

−c13 ρ Ez,app , + c11 c33 + c12 c13

By = Ax , Cz =

(19)

(20)

(c11 + c12 )ρ Ez,app , + c11 c33 + c12 c13

−2c213

ρ Ey,app , c44 ρ Cx + A z = Ex,app , c44 Bz + Cy =

Bx + Ay = 0.

(21)

(22)

(23)

(24)

7



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and we nd for the strain, in the absence of an externally applied stress,





0 d31  0   0  0 d 31       0 0 d33    E, S=   0 d15 0      d  0 0  15    0 0 0

(25)

where

−c13 ρ , + c11 c33 + c12 c13 (c11 + c12 )ρ , = 2 −2c13 + c11 c33 + c12 c13 ρ = . c44

d31 = d33 d15

(26)

−2c213

(27)

(28)

The above result shows that the presence of a bending-induced surface charge non-zero piezoelectric coecients coecients from

e = dc.

d

ρ

leads to

which can be expressed in terms of the piezoelectric

e

Finally we check for piezoelectric symmetry, i.e., that the polariza-

tion due to an applied stress obeys the Onsager symmetry rule as for ordinary piezoelectric materials. Here, we can use the same solutions as before except now the applied stress is given by

Tz,app .

For an applied

uz =

−2c213

Tz,app

we obtain

c11 + c12 1 1 Tzy,app y + Tzx,app x, Tzz,app z + + c11 c33 + c12 c33 2c44 2c44

where we have chosen the strain reference

Az = C x

and

Bz = Cy .

(29)

The change in polarization

is then

1 δPz = Adz

Z

1 ρuz (x, y, z = dz)dxdy − Adz A 8

Z ρuz (x, y, z = 0)dxdy = d33 Tzz,app . A


(30)

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We note that since the coecients

Az , Bz

x

and

y

uz

dependence of

is the same for both

z=0

and

z = dz ,

the

do not inuence the overall polarization.

A similar analysis can be carried out for the other polarization directions. Eventually, we obtain

 0 0 0 d15 0  δP =  0 0 d15 0 0  d31 d31 d33 0 0 and the piezoelectric strain matrix shows the required are directly proportional to the charge density calculating the piezoelectric stress tensor

e,

ρ

 0  0  Tapp ,  0

(31)

6mm symmetry where the parameters

and are mutually connected. In fact, when

we nd:

  0 0 0 0 ρ 0    e = dc =  0 0 0 ρ 0 0 ,   0 0 ρ 0 0 0

(32)

δP = eS,

(33)

in agreement with

as the polarization change is simply a change in the charge location which again is proportional to the applied strain. In order to evaluate the piezoelectric constants of bent graphene, we use the induced charges taken from Figure

14

atom induced by bending is

in Kundalwal,

0.2

12

which shows that the dipole moment per

Debye at a radius of curvature

R = 12

Å. The charge

density is given by

ρ = ακ,

9


(34)


where

κ

is the curvature given by

d2 y dx2

κ= q

1+

Since

P = ρdz ,

where

dz

−1

3.

(35)

0.2 · 3.33 · 10−30 , 8.3 · 108 · 3.35 · 10−10

where the prime indicates "per atom", the m

dy 2 dx

is the distance between the plates, we obtain

α0 =

8.3 · 108

Page 10 of 25

is the curvature (equal to

charges. Using an atom density of

3.33 · 10−30

1/R)

and

1.42

is conversion from Debye to C·m,

3.35 · 10−10

2/Auc = 3.9 · 1019 m−2 ,

area calculated from the C-C bond distance of

(36)

where

m is the distance between

Auc = 5.1

Å

2

is the unit cell

Å, we obtain

α = 9.3 · 10−11 C/m.

Hence the piezoelectric stress constant

e

(37)

is given by

e33 = e24 = e15 = 9.3 · 10−11

C/m

· κ.

(38)

Hence, the induced piezoelectric coecient of graphene scales linearly with the curvature

κ.

We point to that the only way to increase the eective piezoelectric constant of graphene is to increase the bending of the graphene layer.

It can be seen that for relatively large

curvatures, the piezoelectric constant becomes fairly large; e.g. bent graphene sheet with a radius of curvature of application of

uapp

Tapp

constant

P0 = eS0 .

C/m

2

for a

nm. It is important to note that the

as prebending to the sides of the structure induces a strain

associated piezoelectric polarization from

1

e33 = 0.093

S0

and the

Above that, the additional stress loading

induces a second piezoelectric eect from which we determine the piezoelectric

e33 = 0.093

2 C/m .

In the following, we shall demonstrate that a single layer of

10


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prebent graphene can generate electrical power comparable to a single layer of MoS2 .

Results In this section, we present numerical results using the nite element method. We use the developed model to assess quantitatively piezoelectric eects in bent graphene.

COMSOL model of a bent 2D structure To extend the previous analysis, we now consider the structure shown in Figure 2. Mechanical loads on the side (deformation

uapp )

and stress on the top and bottom surfaces (Tapp ) lead

to repositioning of surface electric charges and modify the polarization.

This is another

manifestation of the piezoelectric nature of bent graphene. Here we apply the same principle as with the at sheet but solve for the curvature as a function of space. This is done by invoking a generated surface charge according to the already mentioned relation

ρ = ακ.

(39)

As the curvature is measured on the top and bottom surfaces, they are generally not equal. Thus, in order to maintain total charge neutrality in the modeling, the charge densities are scaled according to

R ρu →

where subscript

u

ρu −

ακdA + su

R

ακdA sl

2

denotes upper surface values and

l

! ,

denotes lower surface values.

(40)

The

lower charge density is adjusted correspondingly. The right and left surfaces are considered

R 14 ) as indicated by the cyan

mechanically rigid (mechanically rigidly connected in COMSOL

11



R

Figure 2: Bending analysis performed in COMSOL . (a) denotes the initial situation, (b)

uapp , (c) indicates the maximum uapp without other external loading, maximum uapp with external stress loading.

indicates an increase in and (d) indicates

12


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arrows for

uapp .

The mathematical conditions for the connector are

u = uapp + (R − I)(X − Xc ),

where

Xc

is the centroid of the corresponding surface and

impose a motion in

x-direction

by making

ucx

(41)

R is the rotation matrix.

a function of time while

uapp,y = 0

We then

to restrict

vertical body translation. The rotation matrix is given by





cos(θ) − sin(θ) R= , sin(θ) cos(θ), where

θ

(42)

is the angle of rotation of the rigid boundary. The angle

θ

introduces another degree

of freedom to the system, since it is determined by invoking moment of force equilibrium around the centroid, i.e.,

Z (Tn) × r dA = 0,

(43)

A

where

n

is the surface normal vector and

is a function of

uapp ,

θ,

this fully determines

θ

the connector boundary condition.

r

is the point vector from the centroid. Since

T

at the boundary and, together with the prescribed It should be noted that we have used a slightly

prebent structure so as to avoid singularities from buckling of a straight beam. In analyzing the piezoelectric behavior, we simulate the structure by using a quasi-static approach. First we prebend the structure by moving the rigid connector by an amount of

0.2

Å inwards during the rst

surface starting at

0.25

seconds, followed by applying a traction force to the

t = 0.5 s reaching 10 MPa as t = 1.5 s.

The results are shown in Figure 3.

It is seen that the polarization increases linearly with stress up to applied stress corresponding to changes by an amount close to the value

d33 = 2 · 10−13

δSzz = 2.6·10−4 .

0.0074

C/m

2

2·10−6 C/m2 for 10 MPa

C/N. The average strain

Hence the piezoelectric constant

found as

9.3 · 10−11 · 8 · 107 13


Szz

in the material

e33 = 0.0077 C/m2 ,

2 C/m , where

8 · 107

m

−1

is the


Figure 3: Polarization

Pz

as a function of time for a transverse motion of the rigid connector

t = 0 corresponds 0 < t < 0.25 s corresponds to Figure 2(b), 0.25 s≤ t < 0.5 s corresponds to Figure 2(c) and 0.5 s≤ t ≤ 1.5 s corresponds to Figure 2(d). The red curves correspond to a maximum Tapp = 10 MPa while the blue curves are for Tapp = 0. The inset shows the relatively small linear build-up in polarization due to Tapp that sets in from t > 0.5 s. and subsequent applied normal stress to the upper and lower surface. Time to the situation in Figure 2(a),

14


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maximum curvature occurring in the middle of the sheet. Note also that the polarization changes drastically in the rst section, from to

0.0046 C/m2

0.0069 C/m2 with an applied deformation of ux,app = 0.1 Å. This corresponds to an eective

strain of

Sx = −0.02[nm]/5[nm] = −0.004 and leads to e13 = −0.0023/0.004 = −0.57 C/m2 .

It should be stressed that the latter

e13

is not a piezoelectric constant in the classical sense,

since its origin is not an Onsager reversible eect. However, as we shall demonstrate next, to employ graphene for energy harvesting in the most ecient way, the graphene sheet must be bent in a fully dynamical fashion during the energy harvesting process, i.e., without a static prebending.

Discussions We next discuss the use of bent graphene and MoS2 in nanogenerator applications.

Nanogenerator operation Having evaluated the piezoelectric coecients of bent graphene, we will now simulate the use of bent graphene for generator applications. To this purpose we are bending the graphene sheet with a time-dependent displacement thus employing the

e13

component of piezoelec-

tricity. A sketch of this setup is shown in Figure 4. A possible experimental design for using

2D

structures as nanogenerator materials can be found in the Extended Data Figure

the Methods section of Wu et al.

2a and

10

As deformation input, we impose

ux,app

10 1 −H t− = ±0.5(cos(ωt) − 1) · H t − f f

on the left and right rigid connectors, respectively. function and the angular frequency is

ω = 2πf

15

The function

rad/s with


f =1

H

[Å],

(44)

is the Heaviside step

kHz. To determine the


Figure 4: A graphene layer operated in generator mode. Panel (a) illustrates the situation where

uapp

is increased. Panel (b) shows the structure where the bending reaches maximum.

Panels (c) and (d) illustrate the situations where

16

uapp

is released again.


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electric elds we solve

∇ · D = 0,

(45)

with generated charges proportional to the top and bottom surfaces as mentioned above. The top and bottom surfaces are now constrained to be equipotential surfaces with charges

q = ±(qbending − qtrans ),

respectively, where

qbending

is the surface charge due to bending, and

R

where

V

(46)

qtrans

is given by

dqtrans = V, dt

(47)

is the potential dierence between the surfaces and

in parallel to the graphene sheet. The generated current

R is an ohmic resistance coupled

dqtrans /dt and potential dierence as

a function of time is shown in Figure 5. We have simulated the system for dierent resistance values in order to demonstrate the phase-shift eect the resistor has on the system. introduced delay in

ux,app

allows the system to discharge from the initial charge state given

that the resistor is not too high. Note that at time electrically to the resistor

The

R

t = 0,

the graphene sheet is connected

Vgen

explaining the transient behavior in

It can be seen that the smallest resistor value of

R = 1014 Ω

and

Igen

for

t > 0.

allows the initial charge to

discharge before the mechanical excitation occurs. The amplitude of the following oscillation is approximately

0.08

V with a corresponding current of

according to the resistance value.

8 · 10−16

A. The other values scale

Also noticeable is the resulting phase change, i.e., the

voltage and current for the largest resistor value are almost

90◦

out of phase compared to

the case with the lowest resistor value. In order to compare with existing nanogenerators, specically MoS2 , we have performed a similar simulation for a single layer of MoS2 . The setup corresponds to the one in Ref.

17


10

where


Vgen (upper) and current Igen (lower) as function of the applied 16 15 deformation uapp (for graphene). The resistances are R = 10 Ω (red), R = 10 Ω (green), R = 1014 (blue). Tapp = 0 at all times. At t = 0 the graphene sheet is connected electrically to the resistor R explaining the initial transient behavior. Figure 5: Generated potential

18


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electrodes are placed on the sides of the MoS2 layer (see also the sketch in Figure 6) with the exception that we have adjusted dimensions and resistance parameters to be comparable to the graphene case, i.e., the ake dimensions are

5×5

2 nm and the parallel resistance is

again varied as before.

Figure 6: Setup for the generator employing MoS2 with electrode conguration similar to the case in Ref. 10.

Since MoS2 is piezoelectric by nature, we discard, for comparison purposes, curvatureinduced surface charges of MoS2 providing otherwise an additional contribution to the effective piezoelectricity of MoS2 . Results are shown in Figure 7. It is observed that there is almost no delay in the system between the applied displacement and the generated voltage in contrast to the graphene case. This is due to the fact that the electrode surfaces for the MoS2 case are much smaller than in the graphene case hence leading to storage of a smaller charge. This corresponds to evaluating the capacitance in a 1D piezoelectric model where the

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Page 20 of 25

capacitance in the MoS2 case is approximately one hundred times smaller than the graphene case. This is also evidenced by the fact that the shift in phase between applied deformation and generated voltage occurs at a resistance approximately one hundred times larger than in the graphene case. More importantly, it is observed that the generated voltages and currents are very close to the ones obtained for graphene. been

e31 = 0.67

The employed piezoelectric coecient for MoS2 has

2 15 C/m , which is given by the linear coecient divided by the single layer

eective thickness

6 Å. The similarity is also given by the generated electric power as shown

in Figure 8, where the power output for lower resistances is much higher for graphene as compared to MoS2 , while for higher resistances the MoS2 generator performs better.

Conclusions We have presented a continuum model for the piezoelectric eect generated in bent graphene. The model consists of the continuum-mechanical equations by invoking curvature-induced electric charges on the upper and lower graphene surfaces. The charges generate an electrical force due to the Couloumb interaction and a polarization.

Since the amount of charges

depends on the mechanical deformation, there is, generally, both a direct and a converse eect. We have used results from DFT calculations to determine the charge amount induced by the curvature so as to evaluate the use of graphene as a nanogenerator. We found that the power generated by a single graphene sheet is in the same order of magnitude as 2D MoS2 when imposing

1

Å deections on

5×5

2 nm sheets. Considering the possibility to induce

even larger bending into graphene sheets, this stipulates the potential in using mechanicallybent graphene, or other mechanically-bent non-piezoelectric 2D and thin-lm materials, as nanogenerators.

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Vgen (upper) and current Igen (lower) as function of the applied 17 deformation uapp for a MoS2 ake. The resistances are R = 10 Ω (cyan), R = 1016 Ω (red), R = 1015 Ω (green), R = 1014 (blue). Tapp = 0 at all times. At t = 0 the graphene sheet is connected electrically to the resistor R explaining the initial transient behavior.

Figure 7: Generated potential

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Figure 8:

Pgen of graphene (upper) and MoS2 (lower) Tapp = 0 at all times. At t = 0 the graphene R explaining the initial transient behavior.

Generated power per unit sheet area

as function of the applied deformation

uapp .

sheet is connected electrically to the resistor

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Acknowledgement LD acknowledges nancial support from Beijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Sciences.

MW acknowledges nancial support from a Talent 1000

Program for Foreign Experts, China.

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(9) Zhu, H.; Wang, Y.; Xiao, J.; Liu, M.; Xiong, S.; Wong, Z. J.; Ye, Z.; Ye, Y.; Yin, X.; Zhang, X. Observation of Piezoelectricity in Free-standing Monolayer MoS2. Nature Nanotechnology

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(11) Fei, R.;

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TOC Graphics

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Mechanically-Bent Graphene as an Effective Piezoelectric

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