**1. Introduction**

Two-dimensional materials are characterized by excellent structural, mechanical, and physical properties, making them suitable for basic science and engineering applications because of their superb properties [1]. Throughout recent years, 2D nanomaterials have been the subject of extensive studies, resulting in massive interest in their applications in novel nanodevices with unique functions. A growing interest in understanding the thermal properties of 2D materials has been observed over the last few years. In addition, in nanomaterials, thermal transport has revealed many unique phenomena, which, when understood, will open up new possibilities for the development of new nanotechnologies in thermal management. New technologies have become increasingly dependent on thermal conductivity as an essential parameter. Many benefits can be derived from nanoelectronic devices using 2D materials and they may potentially extend electronics into new fields of application.

A temperature increase occurs when advanced materials are used in some electronic applications. Increased thermal conductivity allows heat to diffuse faster and prevents large overheating, which can result in premature degradation. The majority of these unique phenomena are due to nanomaterials' notable properties. In fabrication and application, chemical functionalization, strain, and structural interruptions can alter their atomic structures, affecting their properties. Research on the micro/nano components of two-dimensional material has recently focused on their electrical, mechanical, and optical properties. It should be noted that for any micro/nano component, whether it is an electronic component or an optoelectronic component, the heat dissipation problem determines the device's performance and stability. High-density components will generate a lot of heat during high-speed operation. If the heat cannot be eliminated in time, it will cause the components to be too high in local and performance degradation, or even burnout. How to conduct immense heat away so that the components work in a relatively low-temperature environment becomes a common issue in the modern semiconductor industry [2–6].

Most microelectronic devices are combined with semiconductors and metals so the contact interface between semiconductors and metals can be seen everywhere. Microelectronic

**Citation:** Kalantari, M.H.; Zhang, X. Thermal Transport in 2D Materials. *Nanomaterials* **2023**, *13*, 117. https:// doi.org/10.3390/nano13010117

Academic Editor: Gyaneshwar P. Srivastava

Received: 2 December 2022 Revised: 20 December 2022 Accepted: 22 December 2022 Published: 26 December 2022

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

devices' heat dissipation problem involves the following physical issues: (1) How does heat transfer in micro/nano-scale materials? (2) How does heat pass through various interfaces? The most dominant heat carriers in semiconductor materials are phonons, which are found in micro/nano electronic devices. Consequently, the following questions arise: (1) How do phonons travel in micro/nano-scale semiconductor materials? (2) How do phonons pass through various interfaces? The study of these two problems enables us to solve heat dissipation problems related to micro/nano devices and thermal conductivity control.

An electronic device can generate heat in many different ways, such as by Joule heating, solar flux, or exothermic reactions. High-power density electronics such as integrated circuits, supercapacitors, LEDs, and lasers are notorious for localized Joule heating. Nanoscale devices have a higher power density, but a reduced amount of heat can be extracted as their dimensions decrease. Nanostructured solar cells and concentrated solar cells share a similar challenge of reducing efficiency with increased temperatures. For a final example, batteries can experience exothermic reactions and Joule heating, which may cause unwanted chemical reactions and device failure. Materials that must minimize heat transfer are at the other extreme. To reduce heat transfer across each leg, thermoelectric devices require materials with a low thermal conductivity. It creates a design conflict when thermoelectric materials must also be good electrical conductors. To prevent heat from reaching critical parts, thermal insulation is designed.

Advances in the electronics industry have fueled an enormous demand for pioneering thermal managemen<sup>t</sup> strategies to enhance the performance and reliability of devices by controlling energy dissipation generated in the devices. In nanoelectronics, where heat dissipation is a vital factor in the performance of high-density nanoscale circuits, or in thermoelectric materials, where low thermal conductivity is desired, controlling heat diffusion by controlling the phononic properties of fundamental components is a major interest. For instance, among the promising candidates for field effect transistor applications with a high on/off ratio and high mobility, single-layer MoS2 is a semiconductor with a large bandgap of ∼1.1–2 eV [7–9]. Low-temperature carrier mobility can be significantly enhanced by improving sample quality and using appropriate electrode materials [10]. Following these recent breakthroughs, it is highly expected that 2D materials will be used in integrated circuits (ICs) in the near future. Another example is stanene, a single-layer buckled honeycomb structure of tin atoms that exhibits near-room temperature quantum anomalous Hall effects [11] and ultra-low thermal conductivity [12], which makes it ideal for thermoelectric applications. This article discusses in detail the thermal conductivity of 2D materials.

#### **2. Two-Dimensional (2D) Materials**

Understanding material systems is at the core of the technology. Each application requires specific material properties. For example, circuits are built with copper because of their electrical conductivity, skyscrapers are constructed with concrete because of their compressive strength, and car tires are constructed with vulcanized rubber, which is pliable and durable. Technology can advance further as we gain a deeper understanding of a material's properties. Nanomaterials refer to materials with a dimension of at least one nanometer in size. Qualitative changes in physicochemical properties and reactivity are related to the number of atoms or molecules determining the material in this scale. To illustrate, the surface plasmon resonance of metal nanoparticles and quantum confinement of semiconductor particles can be observed as size-effect properties. Recent years have seen an increased interest in two-dimensional (2D) nanomaterials due to their unique properties. Furthermore, two-dimensional nanomaterials bridge the gap between one-dimensional (1D) nanomaterials and three-dimensional (3D) bulk materials, raising new fundamental problems related to low-dimensional materials that can lead to a host of new applications. As these nanodevices become more widely available, the need to understand their thermal properties has increased. Two-dimensional nanomaterials are briefly introduced in this section.

Monolayer graphene flakes were isolated from bulk graphite by mechanical exfoliation, launching the field of two-dimensional (2D) materials [13]. There have also been numerous discoveries of 2D materials since then, including transition metal dichalcogenides (TMDs, e.g., MoS2), hexagonal boron nitride (h-BN), and black phosphorus (BP) (or phosphorene). There is a wide range of physical properties available within the 2D materials family, from conducting graphene to semiconducting MoS2 to insulating h-BN. As an added advantage, 2D crystal structures exhibit superior mechanical properties, exhibiting a high in-plane stiffness and strength, as well as an extremely low flexural rigidity. Together, the 2D materials have a wide range of potential applications [14,15].

Van der Waals forces or weak covalent bonds that hold together material layers can be mechanically or chemically exfoliated down to an in-plane, covalently bonded single layer. A 2D materials history dates back to the 1960s. As early as 1980, graphene, a one-atom thick graphite layer, was isolated and studied extensively as a monolayer. Novoselov and Geim introduced 2D materials by studying graphene under electric and magnetic fields [16]. Because of the high quality of the crystals and their ease of obtaining them, many researchers have developed more complicated 2D electron gas materials for graphene. A number of graphene effects were determined as a result. It has also been shown that other layered materials are known to mechanically exfoliate. Over a thousand materials have been identified by 2020 [17]. As layers often significantly influence the electrical, optical, and thermal properties of materials, this large number of materials can enable a plethora of novel physics. For example, by changing the gap energy, MoS2 transitions from an indirect to a direct bandgap as the monolayer limit is reached.

Material properties are often thought to be determined solely by their material composition. Electricity is conducted by metals because they contain metallic bonds between their atoms, allowing electrons to drift freely throughout the material. The strength of concrete comes from the cement that rigidly locks incompressible sand and gravel together. Vulcanized rubber is made of flexible polymer chains that are firmly linked together, making it both pliable and durable. The size of a material, however, can influence its behavior. Materials with nanoscale dimensions (i.e., whose sizes can be expressed in nanometers) are particularly susceptible to this. The nanoscale can affect electrical conductivity, chemical reactivity, mechanical properties, and even how a material interacts with light. A fascinating and unexpected new property of nanomaterials is being revealed as we become more adept at creating and studying nanomaterials. With this advancement, future technologies that rely on both bulk properties and material size have opened up entirely new opportunities. A new era of nanotechnology has begun. Periodic tables of 2D materials are currently being worked on and may offer a new form of chemistry using layers rather than atoms (Figure 1). In this article, the thermal properties of 2D materials are discussed, following many excellent literature reviews on the subject.

The first novel 2D material introduced was graphene in 2004. Since then, many other 2D materials have been proposed [17] with an extensive range of properties. In this section, several of the researchers' materials of interest are presented briefly. The two types of 2D materials are single-element 2D materials (such as graphene, black phosphorus (BP), silicene, germanene, etc.) and compound 2D materials (such as TMDs, h-BN, TMCs, III– V group elements, compound semiconductors, etc.). Figure 2 below illustrates some of these types.

**Figure 1.** 2D materials family [18]. Reprinted with permission from ref. [18]. Springer Nature and Copyright Clearance Center.

**Figure 2.** Some of the introduced 2D materials.

Graphene is a semimetal that consists of a covalently bonded hexagonal lattice of carbon atoms. In most cases, it is only one-atom thick (about 0.14 nm). The distinctive band structure of graphene enables electrons to move rapidly at speeds close to 1/300 the speed of light, resulting in its excellent thermal conductivity and high tensile strength. A single monolayer of graphene could support the weight of an entire football [19].

Hexagonal boron nitride (h-BN) is an isomorph of graphene (behaves similarly in terms of its crystallographic properties), except instead of carbon, boron and nitrogen atoms make up the structure. It is a wide-bandgap insulator, in contrast to graphene.

TMDCs are transition metal dichalcogenides with the chemical formula MX2, where M is a transition metal (such as tungsten (W) or molybdenum (Mo)), and X represents a chalcogen (like selenium (Se), sulfur (S), or tellurium (Te)). A TMDC is made up of a metal layer sandwiched between two chalcogenide layers, with each layer being three atoms thick. The crystal structure of TDMCs can vary. A 2H-phase with trigonal symmetry is the most common, resulting in semiconducting properties like MoS2, WS2, and MoSe2. The bulk form of these semiconductors has an indirect bandgap. It is interesting for optoelectronics to use monolayers because their bandgap becomes direct and visible. The metallic 1T phase, the most stable polymorph of WTe2, is another example of such structures.

A single layer of black phosphorus (BP) is called phosphorene, which is a stable allotrope of elemental phosphorus. This semiconductor has a puckered honeycomb structure with a direct bandgap. Layers can be stacked on top of each other to tune the bandgap throughout the visible region. Due to this, these materials are appropriate for transistors and optoelectronic devices. The corrugated structure of phosphorene causes its properties to vary noticeably depending on its measurement direction.

MXenes are monolayers of tin (stanene), germanium (germanene), and silicon (silicene). Similar 2D materials have also been developed, such as antimony [20] and bismuth [21]. Bismuth is found to have the potential for magneto-electronic applications [22].

#### **3. Thermal Conductivity**

A greater understanding of the thermal properties of 2D nanomaterials is required due to their rapid development. The main factors contributing to this demand are as follows. First, electronic devices are subjected to ever-increasing thermal loads due to continuous miniaturization and component density increases. Electronic device components are getting smaller and smaller every year, according to Moore's law. One of the crucial components of electronics is the field-effect transistor, which has now reached a channel length of 100 nm, and a 50 nm channel length is on the horizon. Thermal design has become an essential part of electronic device development at the nanoscale, as controlling heat is critical for reliability and performance.

Furthermore, at the nanoscale, electronic devices exhibit thermal transport characteristics that are dramatically different from those observed at the macroscale. The electrical −thermal design of the electronic device should also take these features into account. As the use and requirement of energy sources increases, practical and efficient solutions are required for energy generation, consumption, and recycling. Heat managemen<sup>t</sup> can be improved by utilizing recently developed nanotechnologies and nanomaterials. In some cases, for efficient thermal dissipation, nanomaterials with high thermal conductivities are employed in nanoscale electronics. A low thermal conductivity is required to increase thermal conversion or preservation in other cases, such as in thermoelectric devices and thermal barrier coatings, nanomaterials, or nanoparticles.

Heat is carried primarily by phonons in the same way electricity is carried by electrons. A recent study demonstrated that phonons can carry and process information as well [23]. A variety of types of thermal logic devices have been developed theoretically and even experimentally, such as thermal rectifiers [24,25], thermal transistors [26], thermal logic gates [27], and thermal memory cells [28]. In a similar manner to electronic circuits, thermal circuits can be fabricated using these basic thermal components.

Two-dimensional nanomaterials have different thermal properties than bulk materials due to their atomic structures. For example, graphene as a 2D material has a thermal conductivity as high as ~2000 W/m·K [29], or even higher [30]. This is comparable to the highest thermal conductivity material found in nature, which is diamond. As a result, high-power electronics could potentially benefit from its use in thermal management. h-BN also has high mechanical strength and good thermal properties. High-quality bulk h-BN samples could reach a thermal conductivity of ~390 W/m·K, indicating its potential as a current generation dielectric material [31]. In a study by Joe et al., the thermal conductivity of an 11-layer sample was found to reach about ~360 W/m·K [32]. For silicene, different MD simulations calculated thermal conductivities ranging from 5 to 50 W/m·K [33,34]. TMDs show different thermal conductivity. As an illustration, it has been estimated that MoS2 has a thermal conductivity of about 26 W/m·K, according to Wei et al. [35]. WS2 CVD-grown monolayer and bilayer thermal conductivity was determined by Peimyoo

et al. For monolayer and bilayer WS2, the measured values are 32 and 53 W/m·K, respectively [36]. Theoretically, the thermal conductivity of phosphorene indicates low thermal conductivity. For instance, Qin et al. studied the simulated thermal conductivity along zigzag (ZZ) and armchair (AC) directions and found it to be 15.33 and 4.59 W/m·K, respectively [37]. However, during the manufacturing of 2D nanomaterials, structural defects such as voids, grain boundaries, and dislocations could be formed. A brief review of the thermal conductivity concept is provided in this section.

We review alternative approaches to determining temperatures and heat rates for a two-dimensional, steady-state conduction in the first step. A wide range of approaches are being used, which range from exact solutions that can be obtained for idealized conditions to approximate methods that vary in complexity and accuracy. In the following section, we consider some of the mathematical issues associated with obtaining an exact solution [38].

#### *3.1. General Considerations and Solution Techniques*

A heat flux equation, according to Fourier's law, can be expressed as follows [38]:

$$\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial y}\right) + \frac{\partial}{\partial z}\left(k\frac{\partial T}{\partial z}\right) + \dot{q} = \rho c\_p \frac{\partial T}{\partial t} \tag{1}$$

Here, . *q* is the rate of energy generated per unit volume of the medium (W/m3) and *k* is the thermal conductivity (W/m·K). In Cartesian coordinates, Equation (1) is the general form of heat diffusion. There are two primary objectives that are usually attached to any conduction analysis. Known as the heat equation, it provides the basic tool for analyzing heat conduction. *T* (*<sup>x</sup>*, *y*, *z*) can be calculated as a function of time from its solution. For the present problem, the first objective is to detect the distribution of temperature in the medium and in order to do so, it is necessary to determine *T* (*<sup>x</sup>*, *y*). Solving the heat equation in the appropriate form is the key to obtaining this objective. It is found that in two-dimensional steady-state conditions with no generation and constant thermal conductivity, this form can be calculated from Equation (1) as follows:

$$\frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial y^2} = 0 \tag{2}$$

Equation (2) can be solved analytically by an exact mathematical solution. However, despite the fact that there are several techniques that can be used to solve these equations, the solutions usually involve complicated mathematical functions and series, and only a limited number of simple geometries and boundary conditions can be used. As a result of the dependent variable *T* being a continuous function of the independent variables (*<sup>x</sup>*, *y*), the solutions are valuable. Therefore, this solution can be used to calculate the temperature at any point within the medium.

A method of separation of variables is also used to compute an exact solution to Equation (2) in order to illustrate the nature and importance of analytical methods. For typical geometries that are usually existing in engineering practice, conduction shape factors and dimensionless conduction heat rates are sets of existing solutions. Graphical and numerical methods, on the other hand, can produce approximate results at discrete points, as opposed to analytical methods, which deliver exact results at any point. Although computer solutions based on numerical methods have replaced the graphical or fluxplotting methods, they can still be used to estimate temperature distribution quickly. Generally, it is used for two-dimensional problems with adiabatic and isothermal boundary conditions. A numerical method, on the other hand, can be used to obtain accurate results for complex, three-dimensional geometries involving an array of boundary conditions, in contrast to analytical or graphical approaches.

#### *3.2. The Method of Separation of Variables*

Two-dimensional conduction problems can be solved with the separation of variables method by applying the boundary conditions to a thin rectangular plate, a long rectangular rod, or any other shape that can be described by boundary conditions. By solving the heat equation, the temperature T corresponding value, heat flux, and heat flow lines can be determined, but this method is limited, complicated, and time-consuming [38]. For a variety of other geometries and boundary conditions, including cylindrical and spherical systems, exact solutions were obtained [39,40].

#### *3.3.* The Conduction Shape Factor

The process of finding an analytical solution to a two-dimensional or three-dimensional heat equation can be time-consuming and even impossible in some cases. This leads to the consideration of a different approach. For example, the heat diffusion equation can be quickly solved in many examples by employing existing solutions to it to solve twoor three-dimensional conduction problems. Shape factor *S* or steady-state dimensionless conduction heat rates *q*<sup>∗</sup>*ss* are used to present these solutions [38].

In some cases, it may be possible to provide accurate mathematical solutions to steady, two-dimensional conduction problems by using analytical methods, as outlined above. For a variety of simple geometry and boundary conditions, these solutions have been generated [41–43]. In spite of this, there are many two-dimensional problems that do not involve simple geometries and boundary conditions that would allow them to be solved with such solutions. A numerical approach such as finite-difference, finite-element, or boundary element may be the best choice in these cases in order to solve the problem.

Although the corresponding κ may differ significantly between 2D materials (such as graphene, molybdenum sulfide, and black phosphorus) and their bulk counterparts, taking advantage of these differences can lead to new possibilities in a variety of applications, such as thermal managemen<sup>t</sup> and energy conversion. It is necessary to study the microscopic picture of two-dimensional materials in order to understand the heat transfer properties of these materials. This section will discuss some general fundamentals and concepts of phonon thermal transport before discussing details of 2D materials.

#### *3.4. Thermal Transport at the Nanoscale*

A brief description of thermal transport at the microscale is necessary before discussing different effects in 2D materials. In order to transfer energy from one region of space to another region of space, the transportation or conduction of thermal energy requires the use of carriers, such as particles or waves. Except for alloys with extremely low electrical conductivity, metals conduct thermal energy mainly through electrons [44]. As shown in Figure 3, from a microscopic perspective, in dielectrics and semiconductors, it can be seen that heat is primarily carried by phonons or quantized vibrations of atoms in the lattice that can function as a particle to represent the phonon wave packets, according to the quantum state during their production. A phonon, when viewed from the angle of energy, will behave as a particle and collide with other phonon particles, as well as impurities and boundaries around it.

Firstly, before starting the detailed analysis, it is crucial that the length scales be clarified in advance. The figure shows a structure with a size of *L* (in 2D materials, *L* can be considered as the material's thickness) and a wavelength of *λ* for phonon wave packets. It is noted that the phonon is considered to be a particle in the spherical regime (gray regime). Phonons collide with other phonons, impurities, and boundaries when they move within solids. A phonon mean free path Λ is a distance between two collisions, which is an incredibly significant concept in the field of thermal transport. At room temperature, the mean free path Λ typically ranges from nanometers to tens of micrometers. It can be much longer at low temperatures. Generally, transport properties are discussed primarily on length scales larger than phonon wavelengths *λ* and comparable to or smaller than mean free paths Λ. When size *L* exceeds the mean free path Λ, the size effect will not be taken

into consideration, which echoes the bulk of classical Fourier's law. A smaller size *L* than the mean free path Λ will result in phonons scattering on the boundaries before further phonon-to-phonon scattering. Due to these extra scatterings on the boundary, heat transfer will then be constrained by the boundaries. A primary cause of the reduction in thermal conductivity of 2D materials can be attributed to this phenomenon. It is known as the classical size effect when this type of effect occurs. In cases where the size *L* is smaller than the wavelength *λ*, we will encounter a quantum size effect as a result. A 2D material with a thickness of *L* usually exceeds the wavelength of the phonons unless the temperature is very low [44].

**Figure 3.** The schematic for the phonon particle picture [44]. Reprinted with permission from ref. [44]. Elsevier and Copyright Clearance Center.

Materials, even crystals, do not have infinite thermal conductivity because phonons are scattered with one another. As result of this scattering, it is known as phonon−phonon scattering. High temperatures lead to stronger scattering and shorter mean free paths Λ. There is a grea<sup>t</sup> deal of complexity involved when it comes to phonon–phonon scattering, and it can also be very challenging to determine the phonon–phonon scattering time. As a result of the advancement of computation algorithms and the availability of more powerful computation capabilities, there have been noticeable advances in the calculation of the phonon–phonon scattering rate through the first principle. The following section provides an overview of some of these methods.

#### **4. Simulation Methods**

Advancement of computers and technology resulted in the development of atomistic models that have become so precise that they can typically be compared with experimental results. The use of atomic simulations has therefore become more common in nanomaterials research in the past few decades.

#### *4.1.* Atomistic Simulations of Thermal Transport

The thermal transport properties of a material can be predicted using atomic simulations by understanding its atomic structure and interatomic interactions. A variety of atomistic simulation approaches have been developed to study nanomaterial thermal transport properties.

To study the thermal transport properties of 2D materials, various theoretical methods have been introduced, including molecular dynamics simulations (MD), Boltzmann transport equations (BTE), and atomistic Green's functions (AGF).

#### *4.2. Introduction to Simulation Approaches*

In Figure 4, a variety of simulation techniques are presented for the study of material thermal properties [45]. The Boltzmann transport equation and the non-equilibrium Green's function are examples of first-class approaches. In each of these methods, the thermal properties are predicted by solving the lattice dynamics equations based on understanding the fundamentals of phonon properties. Direct MD simulation is used in the second class for calculating thermal properties, such as the equilibrium Green−Kubo approach and the non-equilibrium MD method (also called the direct approach).

**Figure 4.** Materials thermal property classifications based on atomistic simulations [45]. Reprinted with permission from ref. [45]. Taylor & Francis Group LLC—Books.

In the case of first-class methods, such as Monte Carlo (MC), during the calculation, phonon transport and scattering are taken into consideration. As a result, prior knowledge of phonon transport is needed. Obtaining such a requirement is easy if the material has a simple lattice structure. On the other hand, phonon transport is usually difficult to predict when there are structural disruptions, such as sharp interfaces. Contrary to the first-class approaches, the MD simulation in the second class uses Newton's equations of motion in terms of time for a group of atoms interacting with each other via potential empirical functions. Furthermore, MD simulations are capable of modeling both small and large systems to calculate material thermal properties using phonon properties and solving lattice dynamics equation simulations, while a system's size can be modeled. The following brief overview presents three of the more commonly used simulation methods for predicting the thermal conductivity of nanostructured materials.

#### *4.3. A Mento Carlo Simulation Method*

Boltzmann's equation for the transport of phonons usually forms the basis of a theoretical analysis of phonon transport [46]. To solve this equation in a closed-form manner, many critical assumptions and simplifications must be made, which can cause huge deviations from experimental observations, especially for materials whose geometrical and lattice structure are relatively complicated and whose defects are multiple types. The MC approach was originally developed as a numerical solution to the Boltzmann equation in the context of electron transport [46]. The method has been widely applied since then to manage the transport problems of particles. In bulk materials [47], thin films [48,49], nanowires [50], and nanocomposites [51], the MC simulation has been successfully applied to determine thermal transport properties.

In the MC method, phonons are treated as random particles drifting in space when solving the transport problem. While MC simulations are typically used to predict thermal properties using lattice dynamics, i.e., phonons, this approach lacks the disadvantage that a numerical expression of the phonon dispersion is required to obtain reasonable phonon numbers and distributions in both the spatial and spectral spaces. A clear understanding of the scattering rate *τ*<sup>−</sup><sup>1</sup>(*ω*) resulting from different scattering mechanisms should also be provided in order to address scattering events. For GE-based nanomaterials, these two requirements necessitate a lot of considerations, including strain, chemical functionalization, interface, and defects.

#### *4.4. First Principles Method*

Using the Boltzmann transport equation (BTE) in conjunction with the Schrödinger equation, first principles calculations can be performed on thermal transport. For the first principles calculation, no fitting parameters are required, as opposed to the traditional

method of extracting phonon scattering times. The following steps are involved in the first principles of the thermal transport-based method. As a first step, first principles simulation refers to solving the Schrödinger equation numerically. Numerical computation is performed to calculate the atomic potential force constant. Based on these force constants, the anharmonic lattice dynamics will be used to extract the phonon dispersion relation and scattering rate. Then, the Boltzmann transport equation (BTE) can be linearized and solved numerically as well. As a result of this process, both the dispersion relation and the phonon scattering rate (or the relaxation time) for each phonon mode are calculated. Lastly, in order to calculate thermal conductivity, the lattice thermal connectivity can be extracted.

A wide range of 2D materials have been studied by using the first principles method since its development, including graphene [52,53], phosphorene [54,55], molybdenum disulfide (MoS2) [56,57], and silicene [34,58]. More detailed explanations of this method can be found in a number of outstanding review papers [59,60] or books [61].

#### *4.5. Molecular Dynamics Simulations Method*

There is also another widely accepted method for thermal transport in 2D materials known as Molecular Dynamics (MD), which relies on Newton's law of motion as its physical foundation. Starting with the atomic potential between atoms, the process begins. As a result, the force acting on each atom can be calculated, as well as its velocity at any given moment. In the modern era of supercomputers, it is possible to determine the location of every atom at any time. Then, based on statistical mechanics principles, it is possible to study the expected macroscale properties.

To calculate the material's thermal conductivity directly during MD simulation, nonequilibrium MD and equilibrium Green −Kubo approaches are most commonly used. A non-equilibrium MD approach is similar to the experimental measurement of thermal conductivity. As a result of this method, one region of the simulation cell is heated up, and another region that is situated at a distance is cooled down. As soon as the system reaches a stable state, it is possible to extract the temperature profile between the hot and cold regions, from which the temperature gradient between the two regions can be determined. It is a direct approach that can be easily implemented in MD simulation since it relies only on classical quantities such as force, velocity, and position to compute the temperature and heat flux. Furthermore, the calculation of the thermal properties does not require any significant assumptions to be made.

A note should be made here that the accuracy of classical MD calculations is highly dependent on the quality of the interatomic potentials. Recent studies have focused on constructing a reliable potential function from first principles calculations [62]. To obtain the thermal properties of 2D materials, equilibrium MD (EMD) and non-equilibrium MD (NEMD) simulations are modified. The Green −Kubo formula or the Einstein Relation ratio is used to calculate thermal conductivity in EMD [63,64]. The hot and cold reservoirs are connected to each side of a sample for NEMD simulations. To calculate thermal conductivity, the stationary-state heat flux and temperature are extracted after sufficient run-time in terms of Fourier's law of heat conduction.

A major advantage of MD is that it is of atomic scale, ye<sup>t</sup> it can be applied to large structures. Moreover, the computational cost can be significantly reduced when compared to the first principles approach. Therefore, the MD method is able to simulate systems that are several orders of magnitude larger than a first principle. The simulation can include millions, or even billions, of atoms. A 2D material can benefit from this since one direction is isolated. For example, recent research has modeled the thermal rectification device employing different graphene geometries [65]. This method can only be applied to systems with well-known potentials, such as carbon-based materials, due to the limited number of potentials identified and verified. Machine learning algorithms have become extremely useful methods of calculating atomic potentials over the past few decades [59]. We can expect more reliable potentials as these algorithms develop and computational power increases.

The simulation of nanoscale objects, such as GE, is usually carried out by either first principles simulations or molecular dynamics simulations (MD). MD simulations are a suitable alternative to first principles simulations because they allow the system size to ge<sup>t</sup> relatively large compared to first principles simulations. A study of nanomaterials' thermal properties can thus be facilitated by this method.

#### *4.6. Equilibrium Green*−*Kubo Approach*

According to the equilibrium Green−Kubo approach, thermal conductivity is calculated by monitoring the dissipation time of these fluctuations. The Green−Kubo approach calculates the thermal conductivity of an isotropic material as follows [60]:

$$
\lambda = \frac{1}{3k\_B V T^2} \int\_0^\infty \langle \stackrel{\rightarrow}{f}(0) \stackrel{\rightarrow}{f}(t) \rangle dt \tag{3}
$$

where → *J* (0) → *J* (*t*) is the autocorrelation function for heat flux and the angular brackets demonstrate ensemble averages. Here, *T* is the temperature, *kB* is the Boltzmann constant, and *V* is the system volume.

The benefit of this method is that it requires significantly fewer simulation cells than non-equilibrium MD. It is also suitable for perfect crystals like Si and diamond with long phonons. However, when complex potential functions are employed, this approach lacks convergence and makes it difficult to calculate the heat flux. Furthermore, when dealing with inhomogeneous systems, this approach computes a thermal conductivity average over all the systems, i.e., an interface. As a result, the detailed behavior of phonons at the interface cannot be studied. Based on these facts, it seems that this approach cannot be used to investigate the thermal properties of 2D nanomaterials containing impurities and interfaces.

#### *4.7. Atomistic Green's Functions*

Since it is evident, phonons are wave-like particles. Wave effects on a discrete atomic lattice can be accurately modeled using Atomistic Green's Functions (AGF). Initially, this method was introduced to deal with quantum electron transport in nanostructures [61,66–71]. The approach can be applied to a variety of nanostructures by making a few careful substitutions [72–75]. It is particularly suitable for low-dimensional heterostructures such as Si/Ge [76], graphene/h-BN [77], MoS2/metal [78] interfaces, and others [79].

#### **5. Experimental Measurement**

Due to the difficulty of extracting precise temperature gradients and heat fluxes, measuring the thermal conductivity of 2D materials is challenging. These nanostructures cannot be measured with traditional tools for temperature and heat flux measurements since most nanostructures are orders of magnitude smaller than the finest thermocouples. Therefore, a variety of optical and electrical tools have been utilized to measure the thermal properties of 2D materials.

There are two experimental method types in the micro/nano-scale thermal conductivity measurements: steady-state measurement (i.e., suspended thermal bridge method, Raman method, etc.) and transient measurement (i.e., 3*ω* method, time-domain thermal reflection technique, shock optical pulse thermal measurement method, etc.). Some 2D measurement producers are briefly discussed in the following sections.

#### *5.1. Suspended Thermal Bridge Method*

The invention of the suspended thermal bridge method benefits from the advancement of micro/nano processing technology development. For the first time, the suspended thermal bridge method was used in micro/nano-scale thermal conductivity measurements in 2001. The sample measured in this experiment was a single root of multi-walled carbon

nanotubes [80]. Previously, traditional methods could only measure the overall thermal conductivity of a bundle of nanowires. The phonon scattering between nanowires (or nanotubes) makes it impossible to accurately determine the thermal conductivity of a single sample [81]. The suspended thermal bridge method is more useful for the study of low-dimensional thermoelectric materials [82,83]. A thermal bridge microdevice is made of two suspended silicon nitride membranes (SiN*x*) that are patterned with thin metal lines (Pt resistors). Figure 5 illustrates how the resistors are electrically connected to contact pads via four Pt leads and used as microheaters and thermometers, providing Joule heating and fourprobe resistance measurements, respectively. The heat transfer in the suspended sample is extracted by considering the generated Joule heating on the heated membrane and the temperature change on the sensing membrane while the sample is held between the two membranes and bonded to Pt electrodes. As a result of the high accuracy of Pt thermometers and direct temperature calibration, this method can provide a high temperature resolution of ~0.05 K in a range from 4 to 400 K [80,84]. The experimentally measured thermal conductance G and thermal conductivity *k* are calculated from the equations *G* = 1/*Rtot* and *K* = *<sup>L</sup>*/(*ARtot*), respectively. Here, *Rtot* is the total measured thermal resistance, *L* is known as the length of the sample, and *A* is the cross-sectional area of the sample. As mentioned, *Rtot* is the total thermal resistance of the entire system, including the thermal resistance of the suspended sample, the thermal resistance contribution from the membraneconnected parts of the sample, the internal thermal resistances of the two membranes, and the additional thermal resistance contribution from the part of the membranes which are linked with the heaters/thermometers.

**Figure 5.** Thermal bridge method [84]. Reprinted with permission from ref. [84]. American Society of Mechanical Engineers ASME.

In recent years, there has been a massive demand for measurement of the thermal conductivity of low thermal conductivity micro/nano-scale materials. Therefore, the Wheatstone bridge method [85] and the comparator method [86] were developed to improve the measurement. Xu Xiangfan et al. [86] used the comparator method to measure the thermal conductivity of a single polyimide nanofiber. In this experiment, the thermal conductivity of the sample is about 1.0 × 10−<sup>10</sup> W/K, which is an order of magnitude lower than the lower limit that can be measured by the ordinary thermal bridge method. It can be seen that the use of this method greatly broadens the application range of the thermal bridge method. Zheng et al. [87] used AC heating to eliminate white noise, which can further increase the measurement accuracy to about 0.25 W/K.

In spite of this, some technical challenges still need to be considered. In order to measure *Rtot* accurately, it is necessary to account for the thermal contact resistance components that unavoidably contribute to them. The first component that needs to be mentioned is thermal contact resistance ( *Rc*, *f*) between the two ends of the suspended sample and the SiN*x* membranes [88,89]. Various studies have shown the need for a fin resistance model to esti-

mate this resistance [90,91]. The thermal contact resistance between the sample−membrane interface and the thermometer *Rc*,*<sup>m</sup>* is another component of *Rtot*, as it results from a nonuniform temperature distribution on the heating membrane. The *Rc*,*<sup>m</sup>* factor can be ignored if the membrane has a uniform temperature distribution, i.e., when the thermal resistance of the suspended sample is greater than the internal thermal resistance of the membrane. A high thermal conductivity material, such as graphene or carbon nanotubes, however, is not the case. By re-analyzing heat transport results in CVD single-layer graphene samples, Jo et al. concluded that these extrinsic thermal contact resistances contribute up to 20% of the measured thermal resistance [91]. Recently, several studies found that resistance line thermometers can be employed as a replacement for serpentine Pt thermometers to reduce the size of the temperature measurement resistance (between heater/sensor and contact point) [92,93]. It has been determined that by employing numerical heat conduction calculations, the contribution of *Rc*,*<sup>m</sup>* decreases to about 30–40% compared to the values that correspond to the serpentine resistance thermometer [91]. The device fabrication and sample transfer are also time-consuming and complex with this technique. In most cases, exfoliated 2D materials are transferred to the thermal bridge structure using a dry transfer method, causing polymer residues, defects, and rough edges on the sample surface that greatly affect the measured total thermal resistance [32,94]. Within the temperature range of 4 to 400 K, the suspended thermal bridge method can be applied. An advanced method based on the tunnel current in a metal−insulator−superconductor junction has been proposed for sub-Kelvin measurements [95]. This allows measurements to be made down to 1 <sup>m</sup>·K

Additionally, various materials, such as nanofilms [88,89] and 2D materials, including graphene [91,96–99], boron nitride [100], and TMDC materials [101,102], have also been measured using the thermal bridge method.

#### *5.2. Electron Beam Self-Heating Method*

In the above-mentioned suspended thermal bridge method, the thermal contact resistance between the sample and the suspended platform is one of the main faults of this process. Although there are already some methods to improve it, the effect of this defect cannot be eliminated from the experimental principle. Researchers at Li Baowen's lab developed the electron beam self-heating method based on the suspended thermal bridge method in 2010 [103,104]. This method omits the influence of the contact thermal resistance between the sample and the suspended platform on the experimental results in principle and measures the spatial distribution of the thermal resistance of the micro/nano-scale materials. A scanning electron microscope (SEM) is used to measure the electron beam self-heating method. As demonstrated in Figure 6, heat is generated by the interaction between the high-energy electron beam in the SEM and the sample. It is possible to scan (move on) the sample continuously.

**Figure 6.** Electron beam self-heating method [104]. Reprinted with permission from ref. [104]. Elsevier and Copyright Clearance Center.

A scanning electron beam is used as a heating source, while the two suspended membranes behave as temperature sensors in the electron beam self-heating method. Hot spots emerged as a result of the electrons' energy absorption along the length of the sample during the scanning of the focused electron. As heat is generated from the local spots, it flows to the two membranes where it increases their temperature. The thermal conductivity of the sample can be obtained by:

$$k = A / (d\mathbb{R} / dx) \tag{4}$$

Here, *A* is the cross-sectional area of the sample, *x* represents the distance between the membrane and the heating spot, and *R* is the calculated thermal resistance from one membrane to the heating spot.

This method has the advantage of measuring *R* by combining the diffusive thermal resistance of the suspended part ( *Rd*) and the thermal contact resistance between the suspended sample and contact electrodes ( *Rc*), as shown by Equation (5):

$$R = R\_d + R\_{\varepsilon} \tag{5}$$

In this equation, ( *Rd*) and ( *Rc*) can be calculated by solving the following equations:

$$R\_d = L/kt\mathcal{W} \tag{6}$$

$$RW = L/kt + R\_c \mathcal{W} \tag{7}$$

The thermal conductivity, length, thickness, and width of the suspended sample are represented by *k*, *L*, *t*, and *W*, respectively. According to the *Rd* formulation, it is evident that its value decreases as t increases and *L* decreases. Also, taking the limit *L*/*t* → 0 will give *Rc*. However, generally in laser-based methods, the spatial resolution is restricted by the heating volume within the sample rather than the spot size. It follows that the spatial resolution of this technique is dependent on the properties of the studied materials [105]. Recent studies have used electron beam self-heating to determine the thermal conductivity and thermal resistance of suspended Si and SiGe nanowires and MoS2 ribbons [105–107]. Also, this method has been employed to measure the interfacial thermal resistance between few-layer MoS2 and Pt electrodes [102].

Although the advantages of the electron beam self-heating technique are evident, its drawback cannot be ignored, such as the fact that it cannot apply variable temperature measurement, cannot measure materials that are sensitive to electron beams, and is susceptible to impurities on the sample surface (organic matter, etc.).

#### *5.3. Raman Method*

In 2008, the first experimental measurement of the thermal conductivity of twodimensional material, single-layer graphene in the suspended plane was the Raman method [84]. In two-dimensional materials, the Raman method has become one of the most important experimental methods for measuring thermal conduction. Several twodimensional materials have been measured successfully using this method, including boron nitride [108,109], black phosphorus [110], and molybdenum sulfide [56,111,112]. The Raman method can be used to measure the thermal conductivity of two-dimensional materials by taking into account the following two factors: (1) Raman lasers can be used as heat sources because 2D materials have an absorption effect on them; (2) The Raman spectrum absorption peak positions of two-dimensional materials and a certain linear relationship between temperature [110,112,113]; in this way, the surface temperature of the material can be determined by the Raman spectrum of the material. The thermal conductivity of a two-dimensional material can be calculated by combining the two principles mentioned above through the heat conduction model. A schematic of Raman spectroscopy is shown in Figure 7. Using Raman peaking shifting, the temperature is measured for the sample [30,114]. Obtaining the temperature can be achieved since the Raman peak is a

linear function of the temperature. Thermal conductivity can be measured based on the absorbed power and temperature.

**Figure 7.** Raman spectroscopy schematic.

As an optical method, Raman studies the phonons and vibrational modes of molecular vibrations in solids. In this method, the inelastically scattered light of a monochromatic laser beam that interacts with a material is studied. An oscillating dipole moment is generated as a result of the oscillating electromagnetic field of the incident light which acts as a radiation source causing Raman scattering. Based on the nature of the chemical bonds and the crystal structure, each material or solid crystal has a characteristic set of molecular vibrations and phonons. Materials can be characterized in an elementary and structural manner using this technique. In addition, this method can be used to determine small changes in the crystal structure resulting from embedded strain, thermal expansion, sample composition, and structural disorder, impurities, and contamination, as well as pseudo-phases and deformation of the material [115–118].

With the continuous development and improvement of the Raman method, it has become one of the most accepted methods of micro/nano scale heat conduction measurement. In a recent review article, Malekpour and Balandin provide a detailed description of Raman-based techniques to measure the thermal properties of graphene and related materials [114].

There is, however, considerable uncertainty associated with this method due to the uncertainty of absorptivity. By using a method known as energy transfer state-resolved Raman (ET-Raman), the accuracy of this method can be improved in order to overcome this issue [119]. To achieve this, two steady-state lasers with different sizes are used to heat the sample and measure the temperature shifts as a result. The absorptivity term in this method will be canceled and the signal-to-noise ratio will be improved significantly with this method. In this way, the measurements made with this method are more accurate than those performed using the original optothermal Raman method.

#### *5.4. Time-Domain Thermoreflectance Method*

In 1983, Eesley applied the picosecond pulsed laser to detect the non-equilibrium heat transport process in metallic copper [120] since the time-domain thermoreflectance (TDTR) method has been formally applied to the measurement of material thermal properties. The TDTR method has been developed over a period of thirty years. It has now become one of the most widely used methods for measuring the thermal properties of materials in an unsteady state. This method is usually employed to measure the thermal conductivity and interfacial thermal resistance of materials [92,93,121]. Its basic principle is that a beam of a femtosecond pulsed laser is divided into a pump light and a probe light through a beam splitter. In this system, the pump light is used as a heat source for heating the surface of the material, and the probe light measures the change in the surface temperature of the material (the reflectivity of the material surface to the laser is related to the temperature). The displacement platform can accurately control the optical path difference between the two beams and then control the time interval between them to reach the surface of the

material, resulting in a certain time delay (*td*). A schematic is illustrated in Figure 8. The temperature change process is related to the thermophysical properties of the material.

The measurement system of the TDTR includes a femtosecond laser generator, beam splitter, displacement platform, electro-optic (acousto-optic) modulator, photodiode detector, lock-in amplifier, etc. [122]. Before measurement, generally, it is necessary to coat a metal film on the sample surface to be tested as the sensing layer because the reflectivity of the metal surface to the laser is approximately linear with the temperature under the condition of a small temperature rise, and the surface temperature can be calibrated more accurately through the above measurement process. The lock-in amplifier will output the in-phase signal ( *Vin*) and the inverted signal ( *Vout*) based on the modulation frequency, which contains the information of the temperature change of the sample surface, and then the in-phase signal and the inverted signal can be obtained. Finally, the thermal conductivity model is derived and the experimental data is fitted to extract the correlation of the thermal properties of the sample data. Measurements of the thermophysical properties of 2D materials, such as graphene [123], black phosphorus [55], molybdenum sulfide [92,93], and tungsten selenide [124], including thermal interface resistance between the graphene and SiO2 [125], have been conducted using the TDTR method.

**Figure 8.** Time-domain thermoreflectance diagram (TDTR) [125]. Reprinted with permission from ref. [125]. Copyright 2003 American Chemical Society.

Compared with the steady-state thermal measurement method, TDTR does not require measurement in a vacuum chamber. Secondly, it can be applied for ultra-fast thermal transport mechanism research (i.e., electro-acoustic interaction, etc.). Some 2D materials, especially single-layer and multi-layer materials, adsorb impurities or deposits on their surfaces in order to suppress acoustic phonons out of plane, so this method cannot accurately measure the intrinsic heat of these materials [126].

#### *5.5. Micro-Suspended-Pad Method*

The suspended-pad method, first used to measure carbon nanotubes [80] and silicon nanowires [126,127], is another method frequently used for nanostructures, nanoribbons, and 2D materials. The micro-pad devices are manufactured in batches as part of this method. A device consists of two adjacent silicon nitride membranes suspended by a silicon nitride long beam [127]. The patterned platinum heaters are manufactured on both the pads and long beams. Normally, samples are transferred using a nanomanipulator. Utilizing focused ion-beam deposition, the thermal contacts can be increased by Pt deposition, making the contacts electrically and thermally ohmic. It is, therefore, possible to ignore the thermal resistance of the junction. Various 2D materials have been measured using the suspended-pad method, such as h-BN [32], black phosphorus [128], and MoS2 [129]. Due to the accuracy of this method, the electrical signal can be very precisely derived. In addition, the interpretation of the data is very clear. The input current can accurately control the heat flux, and the temperature can be precisely measured via temperature-dependent electrical resistance. Thermal conductivity can therefore be accurately measured with the heat flux across the pads and the temperature on both pads (usually the uncertainty level is

under 5%). A few factors limit its application, despite its effectiveness in measuring the thermal transport properties of nanostructures. First, it is heavily dependent on intricate manufacturing. There is also the possibility of defects arising from sample preparation as a second factor. This includes the polymer residues on BN samples [43] or the defects in black phosphorus [128]. Before using this method, it is important to take into account these factors.

#### **6. Research Progress on 2D Thermal Conductivity**

The thermal conductivity of various two-dimensional materials has been studied, and research progress on the thermal conductivity of the most regular 2D materials is summarized here.
