Posted: November 26th, 2022

Understanding graphene’s thermal properties: A phonon perspective

Understanding graphene’s thermal properties: A phonon perspective

Table of Contents
Abstract 3
1. Introduction 4
2. Scope of this paper and main objectives 4
3. The structure of graphene 5
5. Heat conduction in graphene 8
7. A correlation between the 2D and 3D with respect to thermal conductivity 10
9. Conclusion 12

Abstract
The ability of a material to conduct heat is usually based on the atomic structure of the material; therefore, the thermal properties of the material may change when their structure is put on the nanometer scale. Graphene, a 2D structure that consist of a lattice of carbon atoms, has attained a great attention from various researchers and scientists in matter physics and materials science due to its extraordinary and unique thermal properties. Ultimately, the study and analysis of the properties of graphene owes great number of applications ranging from electronic and structural uses among others. However, much of the properties as well as details pertaining graphene are still unknown. Essentially, high thermal conductivity of graphene is as a result of strong bonding between the carbons atoms while thermal heat flowing out is constrained by weak forces due to van der Waals coupling. Normally, the divergence of thermal conductivity in 2D graphene occurs if and only if the anharmonicity is insufficient to restore the thermal equilibrium. Under such condition, the size of the system should be constrained or introduction of the disorder in order to achieve the finite value of thermal conductivity. This paper focuses on discussing the thermal properties of graphene paying from a phonon perspective. In addition, the paper examines the key differences between metals and graphene in terms of their thermal conductivity.
Keywords: graphene, phonon, thermal conductivity, Umklapp scattering, dispersion relation

Introduction
The elimination of heat loss in electronic devices has been a problem encountered by various researchers in an electronic industry. Therefore, the industry has put more efforts and resources in searching for better materials which dissipates less heat in order to better improve innovations of the next generations of integrated circuits (ICs) designs as well as 3D electronic devices [1]. Fundamentally, materials ability to conduct heat is based on the arrangement of atomic structure. The restructuring of atomic structure will alter the thermal properties of a material on a nanometer scale [2]. Primarily, nanowires and bulk crystals don’t allow transfer of charges or heat due to high components of photons, dispersion or scattering of photon.
Graphene, a recently 2D developed allotrope of the nanocarbon, is known as a single atomic layer of graphite [3]. The strongest bonds are a result of the in-plane covalent hexagonally sp2 between the carbon atoms which are significantly stronger than the sp3 diamond bonding. The presence of graphene planes in graphite crystals results to weak van der Waal forces between them [4]. The anisotropic nature of the crystals can be used to determine the thermal attributes of graphene. Studies shows that the K divergence in 2D crystals is due to insufficient crystal anharmonicity for restoring the thermal equilibrium thus requires the need of limiting the system size or introduction of disorder to achieve the significance of the value of K [1]. Ultimately, carbon materials, in which graphene is one of them, formed a basis of study of a variety of allotropes which have a specific value as far as thermal properties are concerned [5].
Scope of this paper and main objectives
There are number of papers in open literature regarding the study of graphene’s thermal behaviour. A great deal of them focuses on theoretical and experimental studies, based on heat conduction. However, there are very little details on the lattice dynamics of the graphene as well as their physical understandings/interpretation is not well clear. Thus, the main objective of this paper is to scientifically analyse/ investigates graphene’s thermal properties based on elaborated lattice dynamics fundamentals. As well, to analyse the most important property that distinguishes graphene from metals. In this paper, fundamental lattice dynamics and associated role in thermal conduction are discussed based on the available literature and using the knowledge learned from MSE 200 course to answer some critical questions mentioned above. As such, this paper is much focused in order to avoid survey style.
3. The structure of graphene
Since the material’s performance and properties is a function of its structure, it is very important to understand the structure of graphene in order to appreciate its high thermal conductivity. Figure 1 gives a clear understanding of the various allotropes of carbon in existence. Ideally, 2D crystal is an atomic plane with single layer while 100 layers are considered to be a thin film of 3D material. Electronic structure advances drastically with various layers approaching the 3D limit of at least 10-20 layers of graphite material [6]. Figure 2 illustrates the atomic arrangement of graphene and how one atom is bonded to the next atom.

Figure 1: Various allotropes of carbon [7]

Figure 2: The crystalline structure of graphene [4]
Graphene unlike other materials is uniquely identified by two characteristics. Perfect order is a distinct property of graphene, usually located in its sheets [8]. This property means no existence of atomic defects such as vacancies. Indeed, this property implies that the sheets of graphene are completely pure with traces of carbon atoms being available. Secondly, the properties of graphene correlates with the type of unbounded electrons [9]. It implies that the electrons at room temperature are highly charged and thus move faster as compared with electrons movement in other metals or semiconductors [10]. It has strength of about (~ 130 GPa) and a thermal conductivity K of (~ 5000 W/m.K) which is termed as the best; for the sake of comparison pure copper has a thermal conductivity of (~ 400 W/m.K).
In particular, the electronic structure of graphene is unique and different from metals. Graphene is a non-metal component; the electrons are basically arranged in a pi-bonds model that allows the next atoms to be interconnected as they travel from one atom to another. In contrast with metals in which electric charge is carried by free mobile electron, the graphene on the other hand engages with the lattice in a massless manner [11]. Such properties uniquely define the graphene.
The basics of heat conduction in graphene
When dealing with solid materials, heat is usually carried by acoustic phonons and electron such that: K = Kp + Ke (1)
Where Kp and Ke are the phonon and electron contributions, respectively.
Conduction of heat in carbon materials is normally dominated by phonons [8]. Conductivity is explained by the presence of a strong covalent sp2 bonding which results in more effective heat transfer by lattice vibrations. The conductivity k can be approximated as:
k≈ΣCvλ (2)
V-minimum phonon velocity
λ –mean free path
C-heat capacity
The expression outlines a relationship of definite heat with v- optimum photon velocity while λ -free path.
Graphene is strongly affected by interfacial interactions, atomic defects, and edges. The formula is mainly used if the samples size have greater than the optimum free path where by (L>λ). Presumably, the layer of the graphene is taken to be graphite interlayer spacing h ≈ 3.35 [12].

The thermal conductivity of the phonon is normally expressed as:

Kp = Σj∫Cj(ω)υj^2(ω)τj(ω)dω (3)

Where j is the phonon polarization branch, υ is the phonon group velocity, C is the heat capacity, u- Phonon group velocity, τ- Is the phonon relaxation time, ω -is the phonon frequency.
The mean-free path (Λ) of the phonon is correlated with the time of relaxation and is given as Λ = τυ. In dealing with the approximation of the time of relation, several mechanisms of scattering, limiting Λ, are usually added, τ−1 = Στi−1, where i enumerates the scattering processes [13]. This property leads to ballistic conductance at room temperature [4]. This property is normally understood in details by the examination of the structure of the bond. Indeed, when a phonon is absorbed, it can cause huge momentum change to an electron with less energy change [7].
In materials that are nanostructures, K is normally reduced though scattering from the boundaries and this can be determined by:
1/τB = (υ/D)((1−p)/(1+p)) (3)
Where τB is the phonon lifetime, 1/τB is the scattering rate of the phonon, D is the grain size, and p is the specularity parameter.

Figure 3 shows K values for two types of high-purity graphite (sp2 bonding), diamond (sp3) and amorphous carbon (disordered mixture of sp2 and sp3). Essentially, the in-plane thermal conductivity of graphene at normal room temperature is recognized to have the highest of nearly 2000-4000W. In case of any disorder or residue will induce more phonon scattering as well as lowering the values.

Figure 3: The thermal properties of carbon allotropes and their derivatives[14]
Heat conduction in graphene
Phonons components are majorly responsible for carrying heat in carbon materials. The diffusive and ballistic are two types of transport in existence [15, 16]. The thermal kind of transport is usually referred to as the diffusive when the size of the sample is seemingly greater than Λ implying that phonons is associated with various scattering incidences [17]. Likewise, when the size of the sample is much less than the free path (L< Λ) the transport is referred to as ballistic [18]. Thermal conductivity is basically termed as intrinsic if it is constrained by the crystal-lattice an-harmonicity [19]. Such concept assumes Fourier’s law of diffusive transport. When the potential energy has terms relatively higher than the 2nd order in relation to ion displacement from the equilibrium, then the crystal lattice is termed as anharmonic [20]. Principally, when the crystal is perfect, the intrinsic K limit will be reached with no impurities or defects and thus phonons will only be scattered by other photons due to an-harmonicity [11].
Engagement of the harmonic phonon leads to finite K value in 3D which is vividly described by the Umklapp processes. The crystal degree an-harmonicity is usually depicted by the Gruneissen boundary γ, which is expressed in the Umklapp scattering rates. In essence, the thermal conductivity is designated as extrinsic if and only if its constrained by extrinsic effects hence results to phonon defect scattering or phonon-rough-boundary [21].
Ideally, when compressive or tensile forces are applied alongside the temperature slope of graphene, thermal conductivity significantly reduces. In addition, graphene with zigzag boundaries alongside the slope of the temperature have higher thermal conductivity as compared with graphene with armchair boundary [22].
Thermal conductance of graphene, referring to (in-plane) reduces when a substrate get in touch with 2D or confined into graphene Nano-ribons (GNRs) [15]. In effect, the results are unforeseen owing to the presence of phonon propagation in thin atomic graphene hence causes sensitivity to edge perturbations [16] (as illustrated in Fig. 4).

Figure 4: Thermal conductivity of graphene Nano-ribbons obtained from MD simulations as a function of n showing a similar trend [23]
For the samples in which (L ≫ λ0) implies a constant thermal conductivity k, whereas thermal conductance is inversely proportional with length, G= κA/L [12]. On the other hand, quantum treatment of small amounts of graphene devices (L ≪ λ0) discloses the thermal conductance approaches a constant (Gball) which is independent of length in ballistic transport of free scattering [24]. In addition, a sample size with (L ≫ λ0) implies a constant thermal conductivity k; alternatively, the graphene ballistic thermal conductance can be numerically computed based on phonon dispersion [25].

Thermal transportation with consideration of few-layer graphene
It is very vital to consider and examine the thermal properties of thin-layer of graphene as its thickness increases. There are two instances; thermal transport which is constrained by intrinsic properties of few-layered graphene of lattice such as the crystal anharmonicity while the second part is the extrinsic effects such as phonon-boundary or defect scattering [15]. Research indicates that the suspended unsealed few-layered graphene reduces with increase in n (number of planes) approaching the bulk limit [13]. The evolution of K was expounded by putting intrinsic quasi-2D crystal properties into consideration as outlined by the phonon Umklapp scattering. However, an increase in the few number of graphene layers available, the phonon scattering tends to transform and thus leads to a phase-space state availability for phonon scattering hence K decreases [5].
From the top to bottom, phonon scattering is restricted in suspended few layered graphene boundaries if constant n is maintained in the layer length. A small thickness of few layer graphene (n<4) implies that the phonons doesn’t have transverse parameters in the velocity group to which they belong hence leads to a weak phonon scattering from the top boundary to the bottom [23]. On the other hand, a few number of layered graphene whereby n>4 implies that the boundary scattering can increment. It is also hard to maintain the constant n throughout the whole area of few layer graphene hence a K value below the graphite limit is obtained. Graphite value recuperates thicker films [26].
A correlation between the 2D and 3D with respect to thermal conductivity
Graphene, as 2D material, is quite different from 3D materials such as graphite owing to its unique low dimensional structure as well as the strong covalent bonds hence resulting to different phonon scattering techniques which gives graphene a very high thermal conductivity as compared to other carbon allotropes [27]. Obviously, graphene possess charge carriers which can travel thousands interatomic distances without being deflected or scattered [15]. The 2D crystallites are differentiable in a metastable state. They are extracted from 3D materials while for instances of small size <<1mm and strong interatomic bonds guarantee that the thermal variations cannot result to the production of dislocations or any other defects due to high temperatures [14].
From the complementary perspective, the 2D crystals are intrinsically stable by moderate crumpling in the 3D on tangential scale of almost equal to 10nm [28]. Consequentially, 3D is experimentally observed to cause elastic energy gain which suppresses thermal vibrations (anomalously large in 2D) in which more than the optimum temperature can lead to minimization of the total free energy [15]. Phonon scattering between multi-layered graphene sheets results to a relative decline of the instinct thermal conductivity.
Some unique features of conduction of heat in 2D crystals
An investigation of the conduction of heat in graphene normally raises the issue of ambiguity in defining intrinsic thermal conductivity for 2D crystal lattices [29]. It is widely accepted that K limited by the anharmonicity of the crystal and hence are referred to as being intrinsic in nature. It has values which are finite in 3D bulk crystals.
Heat conduction in graphene is unique and can be illustrated using the expression derived by Klemens [9, 15], for the intrinsic Umklapp limited thermal conductivity of graphene:

K = (2πγ2)−1ρm(υ4/fmT) ln(fm/fB) (7)
Where fm is the upper limit of the phonon frequencies defined by the phonon dispersion, and
fB = (Mυ3fm/4πγ2kBTL)1/2 (8)
Here M is the mass of an atom, is the size-dependent on low-bound cut-off frequency for acoustic phonons, introduced by limiting the phonon mean-free path with the graphene layer size L. Klemens neglected the contributions of out-of-plane acoustic phonons because of their low group velocity and large γ.
The dispersion of phonon and γ in graphene are displayed in Fig. 5a and 5b, respectively. The longitudinal optical (LO), transverse optical (TO), out-of-plane optical (ZO), longitudinal acoustic (LA), transverse acoustic (TA) and out-of plane acoustic (ZA) phonon polarization branches can be seen clearly on the figures. The dependence of K on L is obtained from the model shown on Fig. 5c and 5d; γLA and γTA are the Gruneisen parameters that have been separately averaged for each branch of phonon [11]. The actions which results to infinite K in 1D and 2D is quite different from the ballistic heat conduction in materials with structures with small size in than the mean free path of phonon [15].

Figure 5: The dispersion of phonon and γ in graphene [25]
Conclusion
Graphene is one of the outstanding materials used in different dimensions of applications. Graphene is uniquely known for its thermal property which make it attracted for many researchers and used in many applications. Intrinsic and extrinsic thermal limits of graphene have been expounded in details in respect to phonon scattering. Particularly, phonon component is the main aspect in examining the thermal properties of graphene. Apart, graphene is distinctive from metals because of its unique atomic structure. Metals usually contain electrons that are delocalized which enable them to carry a charge from one point to another. Contrary, graphene electrons are arranged in a pi-bond model which allows interconnectivity with the neighboring atoms as they travel from one atom to another. In essence, graphene has a low dimensional structure and a strong covalent bonding which makes it comparatively different from graphite 3D. Similarly, the same property results to different phonon scattering which gives graphene high thermal conductivity. Ultimately, the interatomic travel path of graphene cannot be deflected or scattered.

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