**2.2. Thermal Properties**

The heat flow direction in a two dimensional graphene can be divided into in-plane and outof-plane directions. In-plane heat flow is greater than out-of-plane one and is developed due to covalent sp2 bonding between carbon atoms, while the latter is emanated from weak van der Waals coupling.

Graphene transistors and interconnectors are beneficiaries of in-plane heat flow depending on a certain channel length. Although thermal coupling with substrate materials is miserly, it is a prominent reason for the dissipation of heat flow. We can regulate the heat flow by phonon scattering, edges, or interfaces. Ultimately, the unusual thermal properties of graphene stem from its 2D nature, forming a rich playground for new discoveries of heat-flow physics and potentially leading to novel thermal management applications.

talline state than when produced as a single crystal. In this experiment, they used a tip of atomic force microscopy (AFM) to exert a concentrated load on a polycrystalline graphene membrane, and it was observed that the force needed to cause a tear in the graphene along GBs is an amount of 100 nN [44], this is while the force for tearing a single-crystal exfoliated graphene

Vargas et al. [45] did similar test using atomic force microscopy and molecular dynamics simulations to study the mechanical characteristics of a graphene with polycrystalline structure that is obtained by chemical vapor deposition. They used nanoindentation meas‐ urements and found that out-of-plane ripples effectively decrease in-plane stiffness in the mentioned graphene. They also found that GBs effectively decrease the breaking strength of the graphene. Molecular dynamics showed them voids can significantly weaken the graphene membranes. In fact, GB is a place where amorphous carbon and iron oxide nanoparticles are

Reprinted with permission from D. Sen, K.S. Novoselov, P. Reis, M.J. Buehler, Small, Volume 6, 1108,2010.

**Figure 2.** Schematic diagram of the setup for the tearing studies of graphene: side and top views. The inset shows the sheet orientation. An initial flap of 8 nm in width is cut in the sheet, folded back, and moved at a constant speed.

The heat flow direction in a two dimensional graphene can be divided into in-plane and outof-plane directions. In-plane heat flow is greater than out-of-plane one and is developed due

bonding between carbon atoms, while the latter is emanated from weak van

is not more than 1.7 mN [37].

6 Graphene - New Trends and Developments

absorbed [45].

**2.2. Thermal Properties**

to covalent sp2

der Waals coupling.

By reviewing thermal properties, with no respect to material, one that should be considered is the specific heat. This is a quantity that implies two things: first, the thermal energy that a body is capable to store, and second, the rate of cooling and heating that a body will experiment. The later can be modeled by the thermal time constant *τ* ≈ RCV, where *τ* is the thermal time constant, *R* is the thermal resistance for heat dissipation (the inverse of conductance, *R=1/G*) and *V* is the volume of the body. Thermal time constants can be varied from 0.1 ns for a single graphene sheet or carbon nanotube (CNT) and 10 ns for nanoscale transistors to 1 ps for the relaxation of individual phonon modes [47 -49].

The specific heat of graphene is divided into phonons (lattice vibrations) and free electrons contributions, *C=Cp+Ce,* Knowing that phonon contributions are dominant [50 -52]. Phonon specific heat as a dominant coefficient becomes constant at very high temperature near inplane Debye temperature *Θ*D ≈2100 K, At this time, we have Cp=3*NAkB* ≈25 J mol –1K–1 ≈ 2.1 J g– 1 K–1, also known as the Dulong–Petit limit, where *NA* is the Avogadro' number and *k*B is the Boltzmann constant. This property belongs in a classical solid behavior when six atomic degrees of motion are entirely exited and each carries 1/2 *k*BT energy.

In heat transport exploration, it is assumed that the thickness of a graphene monolayer is about the graphite interlayer spacing *h* ≈3.35 *A*°' . Graphene has one of the highest in-plane thermal conductivities at room temperature, about 2000–4000 W m–1 K–1, when it was found in freely suspended samples [53, 54]. This range corresponds with values between isotopically purified samples (0.01% C instead of 1.1% natural abundance) as a right end with large grains [55] and isotopically mixed samples or those with smaller grain sizes as a left end.

Disorders with no respect of their source introduce more phonon scattering, and this results in a descendant of conductivity lower than the mentioned range [56]. Figure 3a compares the thermal conductivity of natural diamond (about 2200 W m–1 K–1) with those of other related materials at room temperature [57, 58]. Figure 3b exhibits the thermal conductivity of materials in Figure 3a with respect to lack of disorders.

Heat flow is strongly limited by weak van der Waals interaction in both of directions: crossplane (along the *c* axis) and perpendicular to a graphene sheet, knowing that the van der Waals interaction in the perpendicular direction is between graphene and adjacent substrates, such as Sio2. As we can see in Figure 3a, the thermal conductivity along the *c* axis of pyrolytic graphite is just ~6 W m-1 K-1 at room temperature. The relevant metric for heat flow across such interfaces is the thermal conductance per unit area, *G*″ = *Q*″ / Δ *T* ≈ 50 MW m –2 K –1 at room temperature [60 -62]. This is approximately equivalent to the thermal resistance of an ∼ 25-nm layer of SiO2 [59] and could become a limiting dissipation bottleneck in highly scaled graphene devices and interconnects [63], as discussed in a later section. When we have a few layers of graphene (from 1 to 10 layers), it can be expertized that interlayer resistance, 1/ *G*″, remains almost constant and pretty smaller than the resistance between graphene and its nongraphene environment [61]. Indeed, the interlayer thermal conductance of bulk graphite is ∼ 24 GW m –2 K –1 if the typical 3.35- *A*° spacing and the *c* axis thermal conductivity are assumed.

It must be remarked that surface effects are able to decrease the thermal conductivity of graphene because of the sensitivity of phonon propagation to surface or edge perturbation, and as a result of this, the in-plane thermal conductivity of freely suspended graphene is drastically lower than a graphene nanoribbon or a graphene contacted with a substrate.

**Figure 3.** (a) Thermal conductivity *κ* as a function of temperature for representative data of suspended graphene [55], SiO2-supported graphene [64], ~20-nm-wide graphene nanoribbons (GNRs)[63], suspended single-walled CNTs (SWCNTs)[66], multi-walled CNTs (MWCNTs)[67], type IIa diamond, graphite in-plane and out-of-plane. Additional data for graphene and related materials are also summarized in Ref.[54]. (b) Room temperature ranges of thermal con‐ ductivity data κ for diamond [57], graphite (in-plane) [54], carbon nanotubes (CNTs) [54], suspended graphene [54, 55], SiO2-supported graphene [64], SiO2-encased graphene [65], and GNRs [63].

It has been seen that the in-plane thermal conductance *G* of graphene can reach a significant fraction of the theoretical ballistic limit in sub-micrometer samples, owing to the large phonon mean free path (*λ* ≈ 100 to 600 nm in supported and suspended samples, respectively). However, thermal properties of graphene could be highly tunable, so that makes it useful for heat- sinking applications when we regulate it in ultra-high thermal conductivity, and it is useful for thermoelectric applications when it is regulated for ultra-low thermal conductivity.
