**3.2. Out-of-plane properties**

The research on the out-of-plane mechanical properties of 2D materials includes characterizing the elasticity perpendicular to the plane's direction and the interlayer shear force constant/ strength parallel to the plane's direction. Unlike the experiments already conducted in characterizing the in-plane mechanical properties of 2D materials, experimental investigations on quantifying the out-of-plane properties are still quite scarce, mainly because of the technical difficulties in characterization [61]. Overall, out-of-plane properties can be explored directly by applying normal/shear force to 2D materials or indirectly via Raman spectroscopy. This section introduces the various experiments conducted thus far related to the measurements of out-of-plane properties.

#### *3.2.1. Direct characterization*

#### *3.2.1.1. Perpendicular-to-plane elasticity*

Direct investigation of the perpendicular-to-plane elasticity of few-layer 2D materials remains challenging, because extremely small indentations need to be conducted on supported 2D sheets. Since the interlayer distance of 2D materials is so small (<1 nm), the maximum inden‐ tation depth should be only a few angstroms (smaller than the interlayer distance) [62].

An unconventional AFM-based method (modulated nanoindentation, as shown in **Figure 5(a)**) with a high indentation depth resolution of 0.1 Å [63, 64] has been employed to measure the perpendicular-to-plane elasticity of highly oriented pyrolytic graphite (HOPG), epitaxial graphene (EG), epitaxial graphene oxide (EGO), and conventional GO successfully [62]. During the indentation, the AFM tip oscillates at 1 kHz frequency with an amplitude of approximately 0.1 Å (Δ*Z*piezo) controlled by a piezoelectric tube. The AFM feedback loop sets a normal force *F*<sup>z</sup> applied on the 2D materials from the AFM tip by setting the position of the piezoelectric tube vertically. A tip oscillation with an amplitude of Δ*Z*piezo results in a variation of the normal force Δ*F*z monitored via the deflection of a cantilever. At a certain normal force *F*z, the tip–2D material contact stiffness *k*contact can be obtained via the expression below:

$$\frac{\Delta F\_{\rm z}}{\Delta Z\_{\rm piezo}} = \left(\frac{1}{k\_{\rm lever}} + \frac{1}{k\_{\rm contact} \left(F\_{\rm z}\right)}\right)^{-1},\tag{6}$$

**Figure 5.** (a) Schematic of modulated nanoindentation on 2D materials. (b) Force–indentation curves for SiC, 10-layer EG, and 10-layer EGO extracted from the nanoindentation. (c) Force–indentation curve for HOPG and the Hertzian fitting. Adapted from Gao et al. [62].

where *k*lever is the spring constant of the AFM cantilever. Then, the force *F*z versus indentation depth *Z*indent curves (as shown in **Figure 5(b)**) can be derived by integrating *dF*z = *k*contact (*F*z) *dZ*indent as

Mechanical Properties and Applications of Two-Dimensional Materials http://dx.doi.org/10.5772/104209 231

$$Z\_{\rm indent} = \int\_0^{F\_Z} \frac{dF\_Z}{k\_{\rm contact} \left(F\_Z\right)}. \tag{7}$$

When the indentation depth is in the subnanometer regime, the perpendicular Young's modulus can be extracted by fitting the *F*z – *Z*indent curves with the Hertz model (as shown in **Figure 5(c)**):

$$F = \frac{4}{3} E \,\mathrm{"{r}^{1/2}Z\_{\mathrm{intent}}\,}^{1/2},\tag{8}$$

where *E* \* =(1−(*υ* sample)<sup>2</sup> / *E*<sup>Y</sup> sample) + (1−(*υ* tip)<sup>2</sup> / *E*<sup>Y</sup> tip), with *<sup>υ</sup>* sample, *<sup>υ</sup>* tip, *<sup>E</sup>*<sup>Y</sup> sample, and *E*<sup>Y</sup> tip being the Poisson's ratio and Young's modulus of the measured 2D material and AFM tip, respectively. With this approach, the perpendicular Young's modulus of HOPG, EG, EGO, and conventional GO is measured to be 33 ± 3 GPa, 36 ± 3 GPa, 23 ± 4 GPa, and 35 ± 10 GPa, respectively, which is far smaller than the in-plane Young's modulus. In addition, the intercalated water between GO layers can affect the perpendicular Young's modulus significantly. This method is very sensitive to the 2D material/substrate interaction and the number of layers of 2D material, and thus is useful for investigating 2D material/substrate interaction [62].

#### *3.2.1.2. Shear force constant/strength*

section introduces the various experiments conducted thus far related to the measurements of

Direct investigation of the perpendicular-to-plane elasticity of few-layer 2D materials remains challenging, because extremely small indentations need to be conducted on supported 2D sheets. Since the interlayer distance of 2D materials is so small (<1 nm), the maximum inden‐ tation depth should be only a few angstroms (smaller than the interlayer distance) [62].

An unconventional AFM-based method (modulated nanoindentation, as shown in **Figure 5(a)**) with a high indentation depth resolution of 0.1 Å [63, 64] has been employed to measure the perpendicular-to-plane elasticity of highly oriented pyrolytic graphite (HOPG), epitaxial graphene (EG), epitaxial graphene oxide (EGO), and conventional GO successfully [62]. During the indentation, the AFM tip oscillates at 1 kHz frequency with an amplitude of approximately 0.1 Å (Δ*Z*piezo) controlled by a piezoelectric tube. The AFM feedback loop sets a normal force *F*<sup>z</sup> applied on the 2D materials from the AFM tip by setting the position of the piezoelectric tube vertically. A tip oscillation with an amplitude of Δ*Z*piezo results in a variation of the normal force Δ*F*z monitored via the deflection of a cantilever. At a certain normal force *F*z, the tip–2D material contact stiffness *k*contact can be obtained via the expression below:

( )

piezo lever contact Z 1 1 , *<sup>F</sup> Z kkF*

**Figure 5.** (a) Schematic of modulated nanoindentation on 2D materials. (b) Force–indentation curves for SiC, 10-layer EG, and 10-layer EGO extracted from the nanoindentation. (c) Force–indentation curve for HOPG and the Hertzian

where *k*lever is the spring constant of the AFM cantilever. Then, the force *F*z versus indentation depth *Z*indent curves (as shown in **Figure 5(b)**) can be derived by integrating *dF*z = *k*contact (*F*z)

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Z

1

(6)


out-of-plane properties.

*3.2.1. Direct characterization*

fitting. Adapted from Gao et al. [62].

*dZ*indent as

*3.2.1.1. Perpendicular-to-plane elasticity*

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

To measure the interlayer shear force constant/strength, proper shear stress should be applied to the interlayer interface of a 2D material using a probe. Oviedo et al. [61] have measured the interlayer shear strength of MoS2 with a shearing strength test under in situ transmission electron microscopy (TEM) characterization, as shown in **Figure 6**. During the test, a multilayer MoS2 flake sandwiched between a 3.5 μm thick focused ion beam (FIB)-deposited platinum (Pt) cap and a SiO2/Si substrate has been attached to a piezoelectric manipulator, as shown in **Figure 6(a)**. Then, the sample has been moved toward a static indenter probe (attached to a force sensor) to apply force to the side of the Pt cap, thus creating shear stress in the MoS2 flake (**Figure 6(b)**). During the test, the force versus distance plot has been recorded, as depicted in **Figure 6(c)**. With the force triggering the shear *F* = 498.8 ± 1.6 μN and the sheared area *A* = 19.7 ± 0.5 μm2 (inset of **Figure 6(c)**), the shear strength of MoS2 is calculated to be 25.3 ± 0.6 MPa, about 0.1% of in-plane Young's modulus (~260 GPa).

Another method to apply shear stress is to conduct friction force microscopy (FFM) measure‐ ments. In contrast to the approach mentioned earlier, the probe, applying normal force to the planes, is placed on the top surface of the 2D material sheets. Only when the probe–layer interactions are stronger than the interlayer interactions, shear stress can be applied in the 2D material by moving the probe laterally. The challenge of this approach is whether shear stress can be transferred from the probe to the interlayer interface of the measured samples effi‐ ciently. In addition, this method is not suitable to measure the shear strength with zero normal load. With this approach, shear strengths of graphite have been measured to be 0.27–0.75 MPa depending on the sliding direction [65]. Meanwhile, the self-retracting motion of graphite, when the probe is removed away after loading, has been observed (shown in **Figure 7(a)** and **(b)**). Moreover, a set of lock-in states has been observed at certain rotation angles with 60° intervals, which requires an external force to unlock a lock-in state [66], as shown in **Figure 7(c)** and **(d)**. The interlayer shear strength of graphite where the lock-in appears is measured to be approximately 0.14 GPa [67].

**Figure 6.** (a) Schematic of the in situ TEM shearing test. (b) Low-magnification TEM image of the indenter tip pointing to the test sample. Inset: High-magnification TEM image of the test sample. (c) Force versus distance plot recorded during the test. Inset: Top-view SEM image of the sheared surface. Adapted from Oviedo et al. [61].

**Figure 7.** (a, b) Motion of a graphite flake that self-retracts after unloading. (c, d) Motion of a graphite that is in a lockin state [66].

#### *3.2.2. Raman spectroscopy*

Furthermore, the interlayer interaction of 2D materials can be investigated using Raman spectroscopy. By probing the interlayer phonons modes, both the parallel-to-plane (shear) and perpendicular-to-plane (breathing) interlayer force constants can be extracted from the Raman spectrum. Since interlayer vibrational modes are usually in the low-frequency regime, due to the weak interlayer van der Waals restoring force, a special filter in Raman spectroscopy needs to be used to suppress the Rayleigh scattering background [68, 69]. Alternatively, the interlayer interaction can be investigated from the Raman spectrum of folded 2D sheets with enhanced interlayer vibrational modes response [70]. The interlayer breathing mode or shear mode force constants can be obtained by fitting the experimental frequency of the *i*th vibrational mode, with the expression below [68]:

$$\alpha o\_i = \sqrt{\frac{k}{2\mu\pi^2c^2} \left(1 - \cos\left(\frac{(i-1)\pi}{N}\right)\right)},\tag{9}$$

where *k* is the breathing/shear mode force constant per unit area; *c* is the speed of light; *μ* and *N* are the mass per unit area and the number of layers of the 2D material, respectively. **Table 2** summarizes the shear and breathing mode force constants measured with this method. Generally, breathing mode force constant is about two to three times larger than that of shear mode, which is possibly the reason why shear exfoliation can enhance the exfoliation efficiency significantly compared with conventional exfoliation methods [71]. The interlayer interaction of multilayer graphene is reported to be the weakest so far. On the other hand, the large difference in the shear elastic modulus along two different in-plane directions reflects the strong anisotropic elastic properties of BP [72].


**Table 2.** Interlayer shear/breathing mode force constants extracted from Raman spectroscopy.
