Relevant Properties of Graphene and Related 2D Materials

The discovery of graphene and other two-dimensional (2D) materials has triggered interest in the development of the next generation of optoelectronic devices, creating a new platform for a variety of photonic applications [1,2], including fast photodetectors [3,4], transparent electrodes in displays and photovoltaic modules [5], optical modulators [6], plasmonic devices [7], and ultrafast lasers [8].

  • 2D materials are atomically thin films, originally derived from layered crystals such as graphite, hexagonal boron nitride (h-BN), the family of transition metal dichalcogenides (TMDs, such as MoS2, WSe2, MoTe2, and others), and black phosphorus (bP). The materials have attracted significant attention, especially in the past decade, owing to their unique and distinctive physical and chemical properties:
    • • quantum confinement in the direction to the 2D plane, which leads to extraordinary electronic and optical properties, that are of benefit for light absorption,
    • • a weak stack of atomic planes on top of each other, held by van der Waals (vdW) forces, leaves no dangling bonds, which make it easy to construct vertical heterostructures and to integrate 2D materials with silicon chips,

• the atomically thin characteristic enables scaling-down to nanodevices without parasitic capacitance.

In this chapter, we briefly discuss the fundamental properties of 2D crystals. As there are many comprehensive papers focusing on 2D material synthesis, this field will not be reviewed here. Although 2D materials cover a wide range of compounds across the periodic table and cover a very wide range of the electromagnetic spectrum, only a handful of material groups have been explored for traditional optoelectronic applications. As shown in Fig. 5.1, the 2D materials range from graphene and its zero bandgap to small-/mid-bandgap materials, such as phosphorene and the TMDs, and to wide-bandgap hexagonal boron nitride (h-BN), which is used exclusively as a topologically smooth insulator. Table 5.1 lists the significant

Bandgap of the different layered semiconductors and the electromagnetic spectrum. The exact bandgap value depends on the number of layers, strain level, and chemical doping. FIR

FIGURE 5.1 Bandgap of the different layered semiconductors and the electromagnetic spectrum. The exact bandgap value depends on the number of layers, strain level, and chemical doping. FIR: far infrared; LWIR: long wavelength infrared; MWIR: mid wavelength infrared; SWIR: short wavelength infrared; NIR: near infrared; UV: ultraviolet. The atomic structures of hexagonal boron nitride (h-BN), TMDs, black phosphorus (bP), and graphene are shown at the bottom of the panel, from left to right. The crystalline directions (x and у) of anisotropic bP are indicated.

TABLE 5.1 Room Temperature Properties of Selected 2D Materials.

2D Material



Effective Mass (m„)

Device Mobility (cm2/Vs)

Saturation Velocity (m/s)


Modulus (GPa)

Thermal Conductivity** (W/mK)

CTE" (10" K-‘)




103-5 X 104

(1-5) XlO5




1L MoS2








Bulk MoS2

1.2 (I)




50(||), 4(1)


1L WSe2

1.7 (D)



4 X104




Bulk WSe,

1.2 (I)




9.7(||), 2(1)



5.9 (D)




250-360 (||), 2(1)



0.3-2 (D)












-60 (zigzag) -27 (armchair)

60-80 (zigzag) 30 (armchair)


All listed values should be considered estimates. In some cases, experimental or theoretical values are not available (NA). * I), I represent direct, indirect energy gap, respectively

’"The || symbol signifies the in-plane direction; 1 signifies the out-of-plane direction.

"'CTE, coefficient of thermal expansion

2D materials used for fabricating optoelectronic devices and a few of their electronic properties.

Relevant Graphene Properties

Graphene has been extensively and comprehensively studied since 2004, due to its unique and exceptional electronic and optical properties [10-12]. The most intriguing and fascinating electronic property of graphene is its linear dispersion relation between the energy and the wave vector where relativistic-like energy dispersion is accompanied by electrons being transported at a Fermi velocity only 100 X lower than the speed of light.

Graphene consists of sp2 hybridized carbon atoms, arranged as a hon-


eycomb, with a lattice constant a = 1.42 A. The valence and conduction bands drop at the Brillouin zone corners (Dirac points), making graphene almost a zero-bandgap semiconductor, as shown in Fig. 5.2. Due to the zero density of states at the Dirac points (the Brillouin zone corners), the conductivity is reasonably low. However, the Fermi level (EF) position can be changed and modified by doping (with electrons or holes) to create a material that is potentially better in terms of conductivity than copper at room temperature. It is commonly known that carbon atoms have a total of six electrons, two in the inner shell and four in the outer shell. The four outer shell electrons in the carbon atom are available for chemical bonding, but, in graphene, each atom is connected to three other carbon atoms on the 2D plane, leaving one electron freely available in the third dimension for electrical conduction. These ^-electrons exhibit high mobility and are located above and below the graphene sheet, where л-orbitals overlap and enhance the carbon-to-carbon bonds in graphene. Fundamentally,

The band structure of graphene in the honeycomb lattice

FIGURE 5.2 The band structure of graphene in the honeycomb lattice (a). The enlarged picture shows the energy bands close to one of the Dirac points. Schematic of electron a- and ^-orbitals of one carbon atom in graphene (b).

the electronic properties of graphene are determined by the bonding and anti-bonding (the valence and conduction bands) of the Jt-orbitals.

Graphene exhibits potential for ballistic carrier transport, with the predicted and assessed mean free paths > 2 pm at room temperature, where carriers have been found to spread via diffraction, similarly to the light in a waveguide, rather than by carrier diffusion, as happens with a conventional semiconductor.

As mentioned before, the graphene carrier mobility and saturation velocity show potential for use in high-speed photonic devices [13]. A layered graphene structure, with long relaxation times for both electrons and holes, allows for a significant performance improvement in optoelectronic devices. Theoretically, graphene exhibits a room temperature electron mobility of 250000 cm2/Vs; however, the transport mechanism is extremely dependent on the local environment and on material processing. Vacuum-suspended graphene, fabricated by exfoliation, is characterized by extremely high carrier mobilities, >200000 cm2/Vs at room temperature. Unfortunately, these films have been reported to have a very small area (approx. 100 pm2), making them expensive for industrial applications. When placed on a substrate, the graphene mobility is reduced by both charged impurities and remote interfacial phonon scattering effects (Fig. 5.3). On Si02, interfacial phonon scattering limits graphene mobility to 40000 cm2/Vs [14]. Exposure to atmospheric conditions and processing contaminants, such as resist residue, water, and metallic impurities, also act as scattering sources, limiting mobility.

Another feature which makes graphene interesting in terms of optoelectronic devices is its high thermal conductivity (approx. 10 X copper and 2 X diamond) and high conductivity (approx. 100 X copper). Graphene is also characterized by high tensile strength (130 GPa, compared to 400 MPa for A36 structural steel).

In comparison with the metals, with large quantities of free charges, graphene should be considered to be a semimetal, where carriers can be induced, through chemical doping or electrical gating, with great flexibility, due to their 2D nature, where the doping concentration from 1012 to 1013/cm2 can easily be reached. Therefore, the semimetal nature of graphene allows for an electrical tunability, which is not feasible for conventional metals.

The optical properties of graphene are also interesting [15]. Graphenes optical conductivity is defined as я/?, where /i is equal to (IAne0)(e2!hc^,

Electron mobility in graphene at room temperature in comparison with other material systems

FIGURE 5.3 Electron mobility in graphene at room temperature in comparison with other material systems.

e is the electron charge, h is Planck’s constant, c is the speed of light, providing graphene with broadband (visible and infrared, IR) linear absorption of 2.3% per monolayer. The figure of roughly 2.3% of the incident light absorbed by a 0.33-nm graphene monolayer is 10-1000 Xhigher than for semiconductors like silicon and Gallium Arsenide (GaAs), whereas graphene also covers a much broader spectral bandwidth.

Although pristine graphene is a zero-bandgap semiconductor, one can open a bandgap in graphene, using different methods. Graphene’s band- gap structure can be modified by substitutional doping [Fig. 5.4(b)], by the addition of two layers [Fig. 5.4(c)] or by bilayer doping [Fig 5.4(d)]. The bandgap of graphene can also be opened by its patterning into a nanoribbon shape or by applying a perpendicular electric field to bilayer graphene. In fact, graphene can cover the range from 0 eV to 0.2 eV. The doping of a graphene layer can move the E/: either up or down, decreasing the mobility of both electrons and holes. Graphene’s thickness restriction creates high resistance and chemical inertness, making pure conductive applications less feasible.

Graphene was reported to exhibit the highest-reported specific interaction strength (absorption per atom of material). Silicon has typically a

Modification of the bandgap structure of graphene

FIGURE 5.4 Modification of the bandgap structure of graphene: the Dirac Fermi cone (a), substitutional doping (b), bilayer graphene (c), and doped bilayers (d).

10-|rm absorption depth, causing 2.3% light absorption in a 200-nm layer, whereas graphene reaches the same optical absorption at a much thinner (0.33-nm) layer (interplane spacing).

The absorption spectrum of graphene covers an ultrabroadband range, from visible to terahertz (THz) spectral range [2]. There are two photoexcitation modes: inter-band transition and intra-band transition. Figure 5.5 presents a typical absorption spectrum of doped graphene [16]. For visible and near-infrared (near-IR) light, electrons can be excited from the valence band to the conduction band though the inter-band transition. In the low-frequency TFIz region, the photon energy is below 2EF and the inter-band transition is prohibited, whereas the intra-band transition dominates. The absorption is due mainly due to the free-carrier (Drude) response. In doped graphene in the mid-IR region, the optical absorption is minimal, and the residual absorption is generally attributed to the disorder in imparting the momentum for the optical transition. A transition occurs close to 2EF, where direct inter-band processes lead to a universal 2.3% absorption. In THz band, the coupling of graphene and photons can be enhanced by intra-band transition, and thus it is possible to achieve sensitive THz detection.

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