**1. Introduction**

Graphene, a two-dimensional (2D) carbon layer with a hexagonal atomic structure [1–3], has recently attracted outstanding attention from both academic scientists doing fundamental researches and engineers working on its technical applications [4]. Now, the scientific community is actively investigating the innovative semiconductors beyond graphene, with intrinsic spin-orbit interaction and tunable bandgap [5].

A remarkable feature of graphene is the absence of the bandgap in its energy dispersions. In spite of the obvious advantage of such bandstructure for novel electronic devices, electrons in graphene could not be confined due to the wellknown Klein paradox [6]. To resolve this issue, graphene may be replaced with a material with a buckled structure and substantial spin-orbit interaction, such as silicene and germanene.

A new quasi-two-dimensional structure which has recently gained popularity among device scientists, is molybdenum disulfide monolayer, a honeycomb lattice which consists of two different molybdenum and sulfur atoms. It reveals a large direct band gap, absence of inversion symmetry and a substantial spin-orbit coupling. A summary of all recently fabricated materials beyond graphene is given in **Figure 1**. The last relevant example is black phosphorous (phosphorene) with a strong anisotropy of its composition and electron energy dispersion. Even though we do not study plasmons in phosphorene in the present chapter, there have been some crucial publications on that subject [7, 8].

frequency and the corresponding damping rate at finite temperature could be adjusted by electron doping, and could also be determined for an intrinsic material, where the chemical potential at *T* ¼ 0*K* is located exactly at the Dirac point, while

In this chapter, we will consider thermal behavior of plasmons, their dispersions and damping rates. By equipping with this information, it is possible to predict in advance the thermal properties of an electronic device designed for a particular temperature range. In spite of a number of reported theoretical studies on this subject [16–18], there is still a gap on demonstrating experimentally these unique thermal collective features of 2D materials. Therefore, our review can serve as an

All the novel 2D materials considered here could be effectively assigned to an individual category based on their existing (or broken) symmetries and degeneracy in their low-energy band structure. We started with graphene having a bandgap Δ<sup>0</sup>

with respect to the Dirac point. Moreover, there is a total spin-valley degeneracy

The energy dispersions of buckled honeycomb lattices, obtained from a Kane-Mele type Hamiltonian, appear as two inequivalent doubly-degenerate pairs of subbands with the same Fermi velocity *vF* <sup>¼</sup> <sup>0</sup>*:*<sup>5</sup> � <sup>10</sup><sup>6</sup> m/s and are given by

q

ð Þ *ξσ*Δ*<sup>z</sup>* � Δ*SO*

where *γ* ¼ �1 labels symmetric electron and hole states. Here, two bandgaps [19, 20] Δ<sup>&</sup>lt; ¼ ∣Δ*SO* � Δ*z*∣ and Δ<sup>&</sup>gt; ¼ Δ*SO* þ Δ*<sup>z</sup>* are attributed to an intrinsic spinorbit gap Δ*SO* ¼ 0*:*5 � 3*:*5 meV [21–24] and a tunable asymmetry bandgap Δ*<sup>z</sup>* proportional to applied electric field E*z*. The band structure, however, depends only on one composite index *ν* ¼ *σξ*, a product of spin *σ* and valley *ξ* index. At E*<sup>z</sup>* ¼ 0, two gaps become the same. As E*<sup>z</sup>* 6¼ 0, Δ<sup>&</sup>lt; and Δ<sup>&</sup>gt; change in opposite ways, and electrons stay in a topological insulator (TI) state. Additionally, Δ<sup>&</sup>lt; decreases with E*<sup>z</sup>* until reaching zero, corresponding to a new valley-spin polarized metal. On the other hand, if E*<sup>z</sup>* further increases, both Δ<sup>&</sup>lt; and Δ<sup>&</sup>gt; will be enhanced, leading to a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>ħ</sup>vFk* <sup>2</sup>

Silicene and germanene, which represent buckled honeycomb lattices, possess subbands depending on valley and spin indices, and therefore are only doublydegenerate. The electron-hole symmetry is broken for molybdenum disulfide and other transition-metal dichalcogenides (TMDC's). For these situations, even though there exists a single electron-hole index *γ* ¼ �1, the energy of corresponding states does not have opposite values for each wave number, even at the valley point. In contrast to the electron states, the hole subbands reveal a splitting, as shown in **Figure 2**. All these partially broken symmetries strongly affect the chemical potential of 2D materials as well as its finite-temperature many-body properties. Black phosphorous, apart from all previously discussed broken symmetries, further acquires a preferred spatial direction in its atomic structure which leads to a strong

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>ħ</sup>vFk* <sup>2</sup> <sup>þ</sup> <sup>Δ</sup><sup>2</sup>

0

, which are symmetric

, (1)

for zero temperature, intrinsic plasmons in graphene do not exist.

*Thermal Collective Excitations in Novel Two-Dimensional Dirac-Cone Materials*

**2. Novel two-dimensional materials beyond graphene**

incentive to address this issue.

*DOI: http://dx.doi.org/10.5772/intechopen.90870*

and single-particle energy bands *ε*�ð Þ¼� *k*

*g* ¼ *gs gv* ¼ 4 for electrons and holes in each band.

anisotropy of its electronic states and band structures.

*ε γ <sup>ξ</sup>*,*<sup>σ</sup>*ð Þ¼ *k γ*

**2.1 Buckled honeycomb lattices**

**181**

Plasmons, or self-sustained collective excitations of interacting electrons in such low-dimensional materials, are especially important, since they serve as the basics for a number of novel devices and their applications [9, 10] in almost all fields of modern science, emerging nanofabrication and nanotechnology. Propagation and detection of plasmonic excitation in hybrid nanoscale devices can convert to or modify existing electromagnetic field or radiation [11–14]. Localized surface plasmons are particularly of special interest considering their interactions with other plasmon modes in closely-located optoelectronic device as well as with imposed electromagnetic radiation [15].

Finite-temperature plasmons are of special interest for possible device applications. Among them is the possibility to increase the frequency (or energy) of a plasmon by an order of magnitude or even more, specifically, as a consequence of the raised temperature. As it was shown in Ref. [16], the dispersion of a thermal plasmon is given as � ffiffiffiffiffiffiffiffiffiffiffi *qkBT* p , where *q* is the wave vector. This dispersion relation reveals the fact that the plasmon energy is monotonically increased with temperature and could be moved to the terahertz range and even above, which is crucial for imaging and spectroscopy.

At the same time, the damping rate, or broadening of the frequency, of such thermal plasmons varies as ≃1*=* ffiffiffi *T* <sup>p</sup> , which means that we are dealing with a long-lived plasmon with low or even nearly zero Landau damping. Both plasmon

**Figure 1.** *Recently discovered two-dimensional materials: graphene and beyond graphene.*

*Thermal Collective Excitations in Novel Two-Dimensional Dirac-Cone Materials DOI: http://dx.doi.org/10.5772/intechopen.90870*

frequency and the corresponding damping rate at finite temperature could be adjusted by electron doping, and could also be determined for an intrinsic material, where the chemical potential at *T* ¼ 0*K* is located exactly at the Dirac point, while for zero temperature, intrinsic plasmons in graphene do not exist.

In this chapter, we will consider thermal behavior of plasmons, their dispersions and damping rates. By equipping with this information, it is possible to predict in advance the thermal properties of an electronic device designed for a particular temperature range. In spite of a number of reported theoretical studies on this subject [16–18], there is still a gap on demonstrating experimentally these unique thermal collective features of 2D materials. Therefore, our review can serve as an incentive to address this issue.
