Linearized density of states (DOS) _{2} for all three energy regions. Here, the DOS within the gap region,

## Abstract

The purpose of this chapter is to review some important, recent theoretical discoveries regarding the effect of temperature on the property of plasmons. These include their dispersion relations and Landau damping rates, and the explicit dependence of plasmon frequency on chemical potential at finite temperatures for a diverse group of recently discovered Dirac-cone materials. These novel materials cover gapped graphene, buckled howycomb lattices (such as silicene and germanene), molybdenum disulfide and other transition-metal dichalcogenides, especially the newest dice and α - T 3 materials. The most crucial part of this review is a set of implicit analytical expressions about the exact chemical potential for each of considered materials, which greatly affects the plasmon dispersions and a lot of many-body quantum-statistical properties. We have also obtained the nonlocal plasmon modes of graphene which are further Coulomb-coupled to the surface of a thick conducting substrate, while the whole system is kept at a finite temperature. An especially rich physics feature is found for α - T 3 materials, where each of the above-mentioned properties depends on both the hopping parameter α and temperature as well.

### Keywords

- finite temperature effects
- plasmon dispersion
- 2D materials

## 1. Introduction

Graphene, a two-dimensional (2D) carbon layer with a hexagonal atomic structure [1, 2, 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 well-known 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].

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, 12, 13, 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

At the same time, the damping rate, or broadening of the frequency, of such thermal plasmons varies as

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, 17, 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.

## 2. Novel two-dimensional materials beyond graphene

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

Silicene and germanene, which represent buckled honeycomb lattices, possess subbands depending on valley and spin indices, and therefore are only doubly-degenerate. 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

### 2.1 Buckled honeycomb lattices

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

where

The wave function of silicene, corresponding to eigenvalue equation in Eq. (1), takes the form [25]

where

Germanene, another representative of buckled honeycomb lattices [26, 27, 28, 29, 30], demonstrates substantially higher Fermi velocities and an enhanced intrinsic bandgap

### 2.2 Molybdenum disulfide and transition-metal dichalcogenides

MoS_{2} is a typical representative of transition-metal dichalcogenide (TMDC) monolayers. TMDC’s are semiconductors with the composition of TC_{2} type, where T refers to a transition-metal atom, such as Mo or W, while C corresponds to a chalcogen atom (S, Se or Te).

MoS_{2} displays broken inversion symmetry and direct bandgaps. Its most crucial distinction from the discussed buckled honeycomb lattices is its broken symmetry between the electrons and holes so that the corresponding energy bands are no longer symmetric with respect to the Dirac point, but could still be classified by a single index

Specifically, the energy bands of MoS_{2} can be described by a *two-band* model, i.e.,

where

In practical, we will neglect the

Using Eq. (3), we can verify that the degeneracy of two hole subbands (

## 3. Thermal plasmons in graphene and other materials

One of the most important features in connection with plasmons at zero and finite temperatures is its dispersion relations, i.e., dependence of the plasmon frequency

where

The dielectric function introduced in Eq. (4) is determined directly by the finite-temperature *polarization function*, or *polarizability*, *zero-temperature counterpart*,

where the integration variable

The zero-temperature polarizability, which is employed in Eq. (5), is quite similar for all 2D materials considered here. The only difference originates from the degeneracy level of the low-energy band structure, such as

where

Finally, the full polarization function at zero temperature is obtained as

If the dispersions of low-energy subbands do not depend on the valley or spin indices

Integral transformation in Eq. (5), which is used to obtain the finite-temperature polarization function from its zero-temperature counterparts with different Fermi energies, was first introduced in Ref. [34]. It could be derived in a straightforward way by noting that the only quantity which substantially depends on temperature in Eq. (6) is the Fermi-Dirac distribution function

For accessible temperatures, the energy dispersions

We first look at intrinsic plasmons with

Additionally, finite-

where we introduce the notation

From Eq. (9), we find explicitly that

Here, the *state-blocking effect* due to Pauli exclusion principle directly results in the diminishing of the real part of the optical conductivity at zero temperature for

Furthermore, for gapless (

where we have used the high-

On the other hand, for gapped (

where the constant

## 4. Chemical potential at finite temperatures

As we have seen from Section 3, we need know

The density of states (DOS), which plays an important tool in calculating electron (or hole) Fermi energy

where

This result is equivalent to the DOS of graphene except that there are no states within the bandgap region, as demonstrated by two unit-step functions

Finally, the chemical potential *conservation of the difference of electron and hole densities*, [17]

where

At

where we have assumed that both subbands are occupied for simplicity. The discussions of other cases can be found from Ref. [18]. Consequently, minimum electron density required to occupy the upper subband of silicene is

On the other hand, by applying Eq. (16), in combination with DOS in Eq. (15), for silicene, a transcendental (non-algebraic) equation [43, 44] could be obtained for

where

Interestingly, the right-hand side of Eq. (18) contains terms corresponding to both pristine and gapless graphene, using which we find from Ref. [17].

as well as a well-known analytical expression for

An advantage of Eq. (18) is that it could be solved even without taking an actual integration. In fact, one can either readily solve it numerically using some standard computational algorithms, or introduce an analytical approximation to the sought solution near specific temperature assigned.

Numerical results for _{2}, there are two different Fermi wave numbers for these subbands. Moreover, the numerically calculated

Eq. (18) could also be applied to a wide range of 2D materials if its DOS has a linear dependence on energy

Now, we turn to calculate _{2} with a much more complicated band structure. After taking into account the

where the calculation is based on a parabolic-subband approximation, i.e.,

From Eq. (22), we further seek an explicit expression for DOS in the form a piecewise-linear function of energy _{2} has been reported in Ref. [18]. Here, we merely provide and discuss these DOS expression around the lower hole subband with

or numerically,

The calculated numerical results for DOS in all regions are listed in Table 1. All introduced coefficients

Range index | Energy range | ||||
---|---|---|---|---|---|

The critical doping density which is required to populate the lower hole subband in MoS_{2} is found to be

Therefore, for most experimentally accessible densities

Next, we would evaluate both sides of Eq. (16) for MoS_{2}. As an example, we consider electron doping with density

From Eq. (27), we can easily find the electron Fermi energy

In a similar way, for hole doping with density

where

The right-hand side of Eq. (16) for TMDC’s could be expressed as a combination of electron and hole contributions

so that

For convenience, we introduce another function

where

where two terms with

Using these self-defined functions and their notations, we finally arrive at the “*hole term*”

where

Here, both

By using the calculated

## 5. Dice lattice and α -T 3 materials

In addition to graphene and silicene, another type of Dirac-cone materials is the one with fermionic states in which multiple Dirac points evolve into a middle flat band. One of the first fabricated materials with such a flat band is a dice or a

The low-energy electronic states of

The _{1−x}Cd_{x}Te for a specific electron doping density, dielectric photonic crystals having zero-refractive index and a few others [48, 49]. So far,

The low-energy Dirac-Weyl Hamiltonian for the

where

Three energy bands from Hamiltonian in Eq. (38) or Eq. (39) are

where

Here, the components of wave functions in Eqs. (40) and (41) depend on valley index

Now we turn to deriving plasmon branches and their damping rates at finite _{2} discussed in Section 4.

For

Structurally, Eq. (42) looks quite similarly to Eq. (6) for buckled honeycomb lattices and TMDC’s. The most significant difference comes as the existence of an additional flat band with

Here, we would limit our consideration to the case of electron doping with

where

for an arbitrary value of

Density plots for Landau damping with

In a correspondence to the damping of plasmons presented in Figure 6, we show in Figure 7 the density plots for plasmon dispersions at

## 6. Plasmons in α -T 3 layer coupled to conducting substrate

In the last part of THIS CHAPTER, WE WOULD LIKE TO FOCUS ON finite-*open system*, could be realized experimentally or even by a device fabrication.

Our schematics for an open system is shown in Figure 8. The dynamical screening to the Coulomb interaction between electrons in a 2D layer and in metallic substrate is taken into account by a nonlocal and dynamical inverse dielectric function

and the resonances in

By using the Drude model for metallic substrate, the dielectric function can be written as *dispersion factor*

where *linear dispersion* of plasmon in this open system [54, 58], which is in contrast with well-known

As a special example, let us consider a silicene 2D layer with two bandgaps

where

In the limit of

where for convenience we introduced a coefficient

We notice from Eq. (48) that

Furthermore, using the notation defined by Eq. (49), we get from Eqs. (46) and (47) that

which leads to a bi-quadratic equation

Eq. (51) gives rise to two solutions

where

In Eq. (53), both plasmon branches contain a linear

Numerical results for thermal plasmons in open system are presented in Figure 9. Similarly to what we have found for graphene and silicene, there are two plasmon branches, both of which depend linearly on

## 7. Summary and remarks

In conclusion, we have developed a general theory for finite-temperature polarization function, plasmon dispersions and their damping for all known innovation 2D Dirac-cone materials with various types of symmetries and bandgaps. We have also derived a set of explicit transcendental equations determining the chemical potential as a function of temperature, which serves as a key part in calculating finite-temperature polarization function through the so-called thermal convolution path. The selection of a particular path with a specific

Using the calculated finite-temperature polarization function, we have further found the dispersions of hybrid plasmon-modes in various types of open systems including a 2D material coupled to a conducting substrate. The obtained plasmon dispersions in these 2D-layer systems are crucial for measuring spin-orbit interaction strength and dynamical screening to Coulomb interaction between electrons in 2D materials, as well as for designing novel surface-plasmon based multi-functional near-field opto-electronic devices.

We have generalized our developed theory for 2D materials further to most recently proposed

## Acknowledgments

A.I. thanks Liubov Zhemchuzhna for helpful and fruitful discussions, and Drs. Armando Howard, Leon Johnson and Ms. Beverly Tarver for proofreading the manuscript and providing very useful suggestions on the style and language. G.G. would like to acknowledge the financial support from the Air Force Research Laboratory (AFRL) through grant FA9453-18-1-0100 and award FA2386-18-1-0120. D.H. thanks the supports from the Laboratory University Collaboration Initiative (LUCI) program and from the Air Force Office of Scientific Research (AFOSR).

## Conflict of interest

All the authors declare that they have no conflict of interest.