**3. Mechanical properties**

As mentioned earlier, 2D materials possess high anisotropy between the in-plane and out-ofplane mechanical properties. In pristine-layered 2D materials, the nearby atoms in the same plane are bonded covalently with low defects density resulting in strong in-plane mechanical properties. While the interlayers are stacked together via weak van der Waals interactions, allowing layers to slide easily when shear stress is applied, the effect can give rise to lubrication properties. In this section, the experimental methods used to characterize the mechanical properties of 2D materials are introduced, and the corresponding empirical results are summarized.

### **3.1. In-plane properties**

The in-plane mechanical properties (including the in-plane Young's modulus, pretension, and breaking strength/strain) of 2D materials have been studied extensively in bending experi‐ ments on suspended 2D sheets. In the bending experiments, atomic force microscope (AFM) is used widely to characterize the deformation of the suspended sheets under a certain amount of force. The force applied during the experiments can be divided into two categories: concentrated force and distributed force. In the following subsections, all the mechanical properties mentioned indicate the in-plane properties, unless stated otherwise.

#### *3.1.1. Applying concentrated force*

by combining conventional lithography, lift-off of deposited metal and transfer printing/ stamping of 2D materials, one can realize the fabrication of suspended 2D materials supported

The schematic of the second approach of fabricating a suspended 2D material structure is presented in **Figure 1(c)**. After the transfer of 2D material onto substrate and metal contacts deposition, the 2D material is suspended by etching the underlying sacrificial layer with the predeposited metal contacts acting as the etching mask and clamping of the 2D materials. In this method, SiO2 has been used widely as the sacrificial layer, which is removed commonly by anisotropic wet etching with buffered hydrofluoric acid (BOE). In order to prevent the 2D materials from collapsing due to the surface tension between the 2D materials and BOE, the drying process is operated normally in a critical point dryer (CPD) [25, 29, 30]. In addition, the etching time needs to be adjusted carefully to control the undercut of SiO2 beneath the metal contacts to prevent the metal from collapsing. Although a more complex structure can be fabricated with this approach, as shown in **Figure 2(b)**, the wet etching involved in the fabrication process may introduce some contamination in the 2D materials, which may degrade the performance of the devices. Moreover, the acids used in this process are not suitable for some 2D materials, such as Bi2Se3 [33] and metals (e.g., Ti, Al). Thus, to avoid acid etching in the fabrication process, photoresist can be used as the sacrificial layer instead of

As mentioned earlier, 2D materials possess high anisotropy between the in-plane and out-ofplane mechanical properties. In pristine-layered 2D materials, the nearby atoms in the same plane are bonded covalently with low defects density resulting in strong in-plane mechanical properties. While the interlayers are stacked together via weak van der Waals interactions, allowing layers to slide easily when shear stress is applied, the effect can give rise to lubrication properties. In this section, the experimental methods used to characterize the mechanical properties of 2D materials are introduced, and the corresponding empirical results are

The in-plane mechanical properties (including the in-plane Young's modulus, pretension, and breaking strength/strain) of 2D materials have been studied extensively in bending experi‐ ments on suspended 2D sheets. In the bending experiments, atomic force microscope (AFM) is used widely to characterize the deformation of the suspended sheets under a certain amount of force. The force applied during the experiments can be divided into two categories: concentrated force and distributed force. In the following subsections, all the mechanical

properties mentioned indicate the in-plane properties, unless stated otherwise.

on patterned metal contacts, as depicted in **Figures 1(b)** and **2(b)**.

222 Two-dimensional Materials - Synthesis, Characterization and Potential Applications

SiO2 which can be removed with photoresist developers [33, 34].

**3. Mechanical properties**

summarized.

**3.1. In-plane properties**

The indentation experiment conducted under AFM is one of the most popular methods for measuring the mechanical properties of 2D materials. In most cases, an AFM probe indents toward the center of the circular suspended 2D materials, as shown in **Figure 3(a)**. During the indentation, the displacement of piezoelectric scanner ΔZ (when the AFM probe starts to contact with the membrane) and the deflection of AFM probe *d* are recorded. The indentation depth at the center of a membrane can be determined by *δ* = Δ*Z* – *d*, and the force applied from the AFM tip onto the membrane can be derived from *F* = *k* × *d*, where *k* is the spring constant of the AFM probe, which can be calibrated via a reference cantilever [36] or calculated using the Sader method [37, 38]. Then, the force *F* versus deformation *δ* curves of a suspended 2D material can be extracted, as shown in **Figure 3(b)**. When the radius of the AFM tip *r*tip is far smaller than that of the hole *r* and the bending stiffness of the measured 2D material is negligible (monolayer or few-layer), the *F* − *δ* curves can be approximated using the Schweringtype solution, as Eq. (1) [3, 39, 40]:

$$F = \left(\sigma\_0^{2D}\pi\right)\mathcal{S} + \left(E^{2D}\frac{q^3}{r^2}\right)\delta^3,\tag{1}$$

**Figure 3.** (a) Side view schematic of the indentation experiment on a suspended 2D membrane. (b) Representative force–deformation curves for multilayer WSe2. The fitted curves using the Schwering-type solution agree well with the experimental results [42].

where *σ*<sup>0</sup> <sup>2</sup>*<sup>D</sup>* is the pretension, *E*<sup>2</sup>*<sup>D</sup>* is the 2D elastic modulus, *υ* is the Poisson's ratio, and *q* is a dimensionless constant determined by *q* = 1/(1.05 – 0.15*υ* – 0.16*υ*<sup>2</sup> ). By fitting the experimental curves with Eq. (1), the pretension *σ*<sup>0</sup> <sup>2</sup>*<sup>D</sup>* and 2D elastic modulus *E*<sup>2</sup>*<sup>D</sup>* of the membranes can be derived. Due to the increasing thickness of 2D material (>15 nm) [41], the mechanical behavior of 2D materials undergoes a membrane-to-plate regime transition, and therefore, the bending stiffness should be taken into consideration by adding another term into Eq. (1), thus, forming a modified model [6] which gives a better estimation of the pretension *σ*<sup>0</sup> <sup>2</sup>*<sup>D</sup>*:

$$F = \left[\frac{4\pi E^{2D}}{3(1-\nu^2)}\cdot \left(\frac{t}{r}\right)^2\right]\delta + \left(\sigma\_0^{2D}\pi\right)\delta + \left(E^{2D}\frac{q^3}{r^2}\right)\delta^3,\tag{2}$$

where *t* is the thickness of the measured 2D material.

During an indentation experiment with a spherical indenter, the maximum stress for a circular and linear elastic membrane as a function of the applied force *F* can be derived with the expression as follows [43]:

$$
\sigma\_{\text{max}}^{2D} = \sqrt{\frac{F E^{2D}}{4 \pi r\_{\text{tip}}}},
\tag{3}
$$

where *σ*max <sup>2</sup>*<sup>D</sup>* is the maximum stress at the center of the film (under the AFM tip). Thus, the breaking stress of the 2D material can be estimated by acquiring the force which breaks the 2D material during the indentation. Assuming the stress of the 2D material has a linear relation‐ ship with its strain, the breaking strain can be predicted by *ε*max =*σ*max <sup>2</sup>*<sup>D</sup>* / *E* <sup>2</sup>*<sup>D</sup>*.

Apart from the circular membrane, the indentation experiment can be operated also at the center of a beam-structured 2D material with two ends fixed [4, 44, 45]. In this case, the relation between applied force *F* and deformation at the center of the 2D material *δ* can be modeled with the expression [46]:

$$F = \frac{16E^{2D}\mathcal{W}^2}{l^3}\mathcal{S} + \sigma\_0^{2D}\mathcal{S} + \frac{8\nu E^{2D}}{3l^3}\mathcal{S}^3,\tag{4}$$

where *l* and *w* are the length and width of the suspended beam, respectively.

It is worth noting that it is extremely important to identify the zero displacement/force point precisely for nanoindentation experiments [47]. An inaccuracy of 2–5 nm in determining this point may lead to a 10% error in the extracted *E*<sup>2</sup>*<sup>D</sup>* [35]. In order to compare the elastic properties of a particular 2D material with its bulk counterpart as well as other 2D materials, the 2D elastic modulus *E*<sup>2</sup>*<sup>D</sup>* sometimes needs to be converted to the normal 3D Young's modulus *E*<sup>y</sup> by dividing the 2D value by the thickness of the 2D material *t*.

From the models of Eqs. (1), (2), and (4), we can see that the applied load has an approximate linear relationship with the indentation depth when the membrane deformation is small, and significantly follows a cubic relationship under large deformation. Thus, in the linear regime (small deformation of membrane), the effective spring constant of circular and beam-struc‐ tured membrane can be extracted as

$$\begin{aligned} k\_{\text{circular}} &= \frac{4\pi r^3}{3(1-\nu^2)r^2} E\_\text{Y} + \sigma\_0^{2D} \pi, \\ k\_{\text{beam}} &= \frac{16\nu r^3}{l^3} E\_\text{Y} + \sigma\_0^{2D} \text{ .} \end{aligned} \tag{5}$$

By measuring the effective spring constant of the same kind of 2D material with different design, thickness *t*, or dimensions (*r, w*, or *l*), the Young's modulus *E*<sup>Y</sup> and pretension *σ*<sup>0</sup> <sup>2</sup>*<sup>D</sup>* can be extracted. Note that this method is only valid under the assumption that *E*y and *σ*<sup>0</sup> <sup>2</sup>*<sup>D</sup>* are independent on the thickness or dimension of the 2D material, assumptions of which are under debate at present [48, 49].

#### *3.1.2. Applying distributed force*

where *σ*<sup>0</sup>

<sup>2</sup>*<sup>D</sup>* is the pretension, *E*<sup>2</sup>*<sup>D</sup>* is the 2D elastic modulus, *υ* is the Poisson's ratio, and *q* is a

derived. Due to the increasing thickness of 2D material (>15 nm) [41], the mechanical behavior of 2D materials undergoes a membrane-to-plate regime transition, and therefore, the bending stiffness should be taken into consideration by adding another term into Eq. (1), thus, forming

> <sup>4</sup> , 3(1 ) *<sup>D</sup> E t D D*

During an indentation experiment with a spherical indenter, the maximum stress for a circular and linear elastic membrane as a function of the applied force *F* can be derived with the

2

*r*

breaking stress of the 2D material can be estimated by acquiring the force which breaks the 2D material during the indentation. Assuming the stress of the 2D material has a linear relation‐

Apart from the circular membrane, the indentation experiment can be operated also at the center of a beam-structured 2D material with two ends fixed [4, 44, 45]. In this case, the relation between applied force *F* and deformation at the center of the 2D material *δ* can be modeled

> 2 2 2 2 3 3 0 16 8 <sup>3</sup>

+

*D D E wt <sup>D</sup> wE l l*

dd

s

where *l* and *w* are the length and width of the suspended beam, respectively.

, <sup>3</sup>

 d

p

tip , <sup>4</sup> *D*

<sup>2</sup>*<sup>D</sup>* is the maximum stress at the center of the film (under the AFM tip). Thus, the

é ù æ ö æ ö ê ú × + ç ÷ ç ÷ ê ú - è ø ë û è ø

). By fitting the experimental

<sup>2</sup>*<sup>D</sup>*:

<sup>2</sup>*<sup>D</sup>* and 2D elastic modulus *E*<sup>2</sup>*<sup>D</sup>* of the membranes can be

3

 d

<sup>=</sup> (3)

<sup>2</sup>*<sup>D</sup>* / *E* <sup>2</sup>*<sup>D</sup>*.

+ (4)

*r*

2 23 0 2

= + (2)

dimensionless constant determined by *q* = 1/(1.05 – 0.15*υ* – 0.16*υ*<sup>2</sup>

224 Two-dimensional Materials - Synthesis, Characterization and Potential Applications

2

u

p

where *t* is the thickness of the measured 2D material.

a modified model [6] which gives a better estimation of the pretension *σ*<sup>0</sup>

*r*

2 max

s

ship with its strain, the breaking strain can be predicted by *ε*max =*σ*max

*F* =

*<sup>D</sup> FE*

( ) <sup>2</sup> <sup>2</sup>

*<sup>q</sup> F E*

d s pd

curves with Eq. (1), the pretension *σ*<sup>0</sup>

expression as follows [43]:

with the expression [46]:

where *σ*max

Apart from the concentrated force applied with an indenter, distributed force (such as electrostatic force [50] or pressure force [51]) can be applied on suspended 2D material to measure the mechanical properties. In order to produce an electrostatic force, metal contacts need to be made on/below 2D membrane, so that a voltage can be applied between the membrane and the back-gate electrode, as shown in **Figure 4(a)** and **(b)**. Moreover, by creating a pressure difference between the inside of microcavities covered by the 2D membrane *p*int and the outside atmosphere *p*ext, a pressure force Δ*p* can be produced, as depicted in **Figure 4(c)** and **(d)**. The deformation of a membrane under distributed force can be characterized directly via tapping mode AFM (**Figure 4(e)**). In addition, by measuring the Raman shift of 2D material under loading and without loading, the local strain of 2D material can be extracted indirectly [49]. After building a specific mechanical model which describes the relationship of the deformation of the membrane due to the voltage bias [50] or pressure difference [51, 52], the mechanical properties can be extracted by fitting the experimental results with the appropriate model.

**Figure 4.** (a) Top view optical image of a graphene suspended over hole arrays. (b) Schematic of producing electrostat‐ ic force by applying a voltage *V*S across the back gate and graphene [50]. (c) Schematic of a graphene-sealed micro‐ chamber. Inset: Optical image of a graphene membrane over a hole. (d) Side view schematic of the graphene-sealed microchamber. (e) Tapping mode AFM image of a graphene membrane with Δ*p* > 0 [52].

#### *3.1.3. Results summary*

**Table 1** summarizes the mechanical properties of 2D material families ranging from conduc‐ tors (graphene), semiconductors (semiconducting TMDCs and BP), to dielectrics (graphene oxide [GO], mica, and h-BN). Overall, the Young's modulus of 2D materials is larger than that of the corresponding bulk materials, due to the lower crystal defects and interlayer stacking faults in 2D materials [6].



**Figure 4.** (a) Top view optical image of a graphene suspended over hole arrays. (b) Schematic of producing electrostat‐ ic force by applying a voltage *V*S across the back gate and graphene [50]. (c) Schematic of a graphene-sealed micro‐ chamber. Inset: Optical image of a graphene membrane over a hole. (d) Side view schematic of the graphene-sealed

**Table 1** summarizes the mechanical properties of 2D material families ranging from conduc‐ tors (graphene), semiconductors (semiconducting TMDCs and BP), to dielectrics (graphene oxide [GO], mica, and h-BN). Overall, the Young's modulus of 2D materials is larger than that of the corresponding bulk materials, due to the lower crystal defects and interlayer stacking

1 1000 ± 100 70–740 130 ± 10 ~12 Indentation on

4 930 ± 48 N/A N/A N/A Pressurizing

1–5 1000 ± 31 N/A N/A N/A Pressurizing

**Breaking stress (GPa)**

23–43 ~1000 N/A N/A N/A Electrostatic force [50]

**Breaking strain (%)** **Characterization method**

circular membrane

membranes

membranes

**Ref.**

[3]

[52]

[51]

**Pretension (mN/m)**

microchamber. (e) Tapping mode AFM image of a graphene membrane with Δ*p* > 0 [52].

226 Two-dimensional Materials - Synthesis, Characterization and Potential Applications

**Young's modulus**

**(GPa)**

*3.1.3. Results summary*

faults in 2D materials [6].

**Material Number of**

Graphene (Mechanical exfoliated )

**layers**


**Table 1.** Summary of the in-plane mechanical properties of 2D materials measured from experiments.

#### *3.1.3.1. Young's modulus*

Pristine monolayer graphene (prepared by mechanical exfoliation from bulk graphite) is reported to be the stiffest 2D material on earth so far with a Young's modulus of approximately 1 TPa [3, 49, 51], because of the strong in-plane covalent carbon–carbon bonds. For 2D TMDCs––MX2 (M = Mo, W; X = S, Se) with the same crystal structure (chalcogen atoms in two hexagonal planes separated by a plane of transition metal atoms) [11], a smaller Young's modulus of WSe2 has been observed compared with MoS2 and WS2 [42]; due to a decrease in the charge transfer and an increase in the lattice constant, resulting in a weakened binding between the metal and chalcogen [53], as M changes from Mo to W and X changes from S to Se.

Meanwhile, the Young's modulus of some 2D materials (e.g., MoS2, BP, and h-BN) [2, 7, 41, 48] have been found to decrease with an increase in their thickness (number of layers), which is caused mainly by interlayer stacking errors. The occurrence of interlayer sliding in multi‐ layer 2D materials during indentation is also a factor for underestimating the intrinsic Young's modulus [7]. However, the Young's modulus of WSe2 remains unchanged statistically with increasing number of layers, which possibly results from the strong interlayer interaction in WSe2 [42]. As stated earlier, for 2D materials with thickness-dependent Young's modulus, precaution needs to be taken when using model Eq. (5) to derive the Young's modulus. Furthermore, the highly anisotropic atomic structure in 2D materials, such as BP, presents an anisotropic Young's modulus along the different crystal orientations [54].

In addition, the mechanical properties of 2D materials largely depend on the density of crystal defects and thus are related to the preparation methods. For instance, the larger number of vacancy defects in the GO-reduced graphene and the existence of voids at the grain boundaries, together with wrinkles in polycrystalline graphene prepared by the CVD method, can contribute to the weaker mechanical properties [4, 55]. In addition, the presence of a larger number of grain boundaries can affect the Young's modulus of 2D materials negatively [56]. By optimizing the processing steps of suspended 2D materials fabrication, the quantity of crystal defects and wrinkles in 2D materials can be reduced, thus leading to an improvement of the mechanical properties [56]. Research has shown that the elastic properties can be recovered by flattening the wrinkles in CVD graphene with a small prestretch [47]. The mechanical properties of 2D materials can be improved also by introducing controlled density of defects, such as Ar+ plasma irradiation [35].
