**6. Plasmons in** *α***-**T **<sup>3</sup> layer coupled to conducting substrate**

In the last part of THIS CHAPTER, WE WOULD LIKE TO FOCUS ON finite-*T* plasmons in a so-called nanoscale-hybrid structure consisting of a 2D layer, such as, graphene, silicene or a dice lattice, which is Coulomb-coupled to a large, conducting material. Physically, the Coulomb coupling between the 2D layer and the conductor results in a strong hybridization of graphene plasmon and localized surfaceplasmon modes. This structure, which is referred to as an *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 K *r*, *r*<sup>0</sup> ð Þ ; *ω* , as demonstrated in Refs. [53, 54]. This nonlocal inverse dielectric function is connected to a dielectric function ϵ *r*, *r*<sup>0</sup> ð Þ ;*ω* in Eq. (4) by

$$\int d^3r' \mathcal{K}(r, r'; \alpha) \,\epsilon(r', r''; \alpha) = \delta(r - r''),\tag{45}$$

and the resonances in K *r*, *r*<sup>0</sup> ð Þ ; *ω* reveal the nonlocal hybridized plasmon modes supported by both 2D layer and the conducting surface as a single quantum system.

By using the Drude model for metallic substrate, the dielectric function can be written as <sup>ϵ</sup>*B*ð Þ¼ *<sup>ω</sup>* <sup>1</sup> � <sup>Ω</sup><sup>2</sup> *<sup>p</sup>=ω*2, where <sup>Ω</sup>*<sup>p</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *n*0*e*<sup>2</sup>*=*ϵ0ϵ*bm*<sup>∗</sup> p is the bulk-plasma frequency for the conductor, *n*<sup>0</sup> electron concentration and *m*<sup>∗</sup> is the effective mass of electrons. Drude model describes electron screening in the long-wavelength limit. Based on the previously developed mean-field theory [53, 55, 56], we are able to

#### **Figure 8.**

*Schematics for a silicene-based open system and numerical results for the two plasmon branches and their damping in this system with* Δ*SO* ¼ 0*:*3E0 *and* 0*:*1E0*, where* E0 ¼ *ħ*vF ffiffiffiffiffiffiffiffi *π*n0 <sup>p</sup> <sup>¼</sup> <sup>54</sup>*:*<sup>6</sup> *meV.*

calculate plasmon dispersions in this 2D open system. For this, the plasmon dispersions are obtained from the zeros of the so-called *dispersion factor C*ð Þ *q*,*ω* , instead of the dielectric function in Eq. (4). *C*ð Þ *q*,*ω* for this open system is given by [25, 54, 57]

which leads to a bi-quadratic equation

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

� 1 þ Ξ *EF*, Δ*<sup>β</sup>*

Eq. (51) gives rise to two solutions

*<sup>ω</sup><sup>p</sup>*,þð Þ*<sup>q</sup>* <sup>≃</sup> <sup>Ω</sup>*<sup>p</sup>*

*<sup>ω</sup><sup>p</sup>*,�ð Þ*<sup>q</sup>* <sup>≃</sup>*<sup>q</sup>* ffiffiffiffiffiffiffiffi

damping-free regions at the same time.

2

surface-plasmon energy *ħ*Ω*p=* ffiffi

branch from Ω*p=* ffiffi

**197**

*<sup>p</sup>*,� ¼ 1 þ Ξ *EF*, Δ*<sup>β</sup>*

�

� � 2*q*

" # *<sup>ω</sup>*<sup>2</sup>

� � 2*q*

1 þ Ξ *EF*, Δ*<sup>β</sup>*

ffiffi 2 p þ

!

Ω2 *p*

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

Ω2 *p*

� � 2*q*

!<sup>2</sup>

respectively. Two hybrid plasmon modes in Eq. (52) become

Ξ ffiffi 2 <sup>p</sup> <sup>Ω</sup>*<sup>p</sup>*

<sup>2</sup>*a*<sup>Ξ</sup> <sup>p</sup> � <sup>Ξ</sup> ffiffiffiffiffiffiffiffiffi

Ω2 *p*

where � terms correspond to in-phase and out-of-phase plasmon modes,

*q* �

<sup>2</sup>*a*<sup>Ξ</sup> <sup>p</sup> Ω2 *p*

In Eq. (53), both plasmon branches contain a linear � *q* term, and *ω<sup>p</sup>*,þð Þ*q* approaches a constant as *q* ! 0, i.e., an optical mode for plasmons. Two independent bandgaps, Δ*SO* and Δ*z*, together with doping density *n*, play a crucial role on shaping the plasmon dispersions, as well as the particle-hole mode damping regions. The outer boundaries of a particle-hole mode region specify an area within the *q*-*ω* plane in which the plasmon modes become damping free and are solely determined by Δ<sup>&</sup>lt; , while the group velocity of plasmon mode depends on both Δ<sup>&</sup>lt; and Δ<sup>&</sup>gt; . Since each bandgap could be experimentally tuned by applying a perpendicular electric field, we acquire a full control of both plasmon dispersions and their

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 *q* with a finite slope as *q* ≪ *k*0. The acoustic-plasmon branch starts from the origin, while the optical-plasmon

*<sup>ϕ</sup>* <sup>¼</sup> tan �<sup>1</sup>*α*), which is observed for the upper branch, as shown in (a) and (c) of **Figure 9**. In addition, we see a much smaller slope for the lower plasmon branch in **Figure 9(b)** and **(d)** due to enhanced Coulomb coupling with a reduced separation *a*. The finite-temperature upper plasmon branches in **Figure 9(e)** and **(f)** are shifted up greatly, as it is expected to be true for all finite-temperature plasmons, which is further accompanied by enhanced damping below the diagonal as seen in **Figure 6(c)**. Meanwhile, the lower plasmon branch seems much less affected by finite temperatures, as demonstrated by both upper and lower rows of **Figure 9** for different separations, except for enhanced damping in **Figure 9(f)** below the

2 <sup>p</sup> . Ω2 *p* þ Ξ *EF*, Δ*<sup>β</sup>* � � *q*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� <sup>8</sup>*q*<sup>Ξ</sup> *EF*, <sup>Δ</sup>*<sup>β</sup>* � � Ω2 *p*

Ξ Ξ <sup>þ</sup> <sup>4</sup>*a*Ω<sup>2</sup>

2 ffiffi 2 <sup>p</sup> <sup>Ω</sup><sup>3</sup> *p*

*<sup>q</sup>*<sup>2</sup> þ O *<sup>q</sup>*<sup>3</sup> � �*:*

<sup>p</sup> . The dispersion of each branch also varies with parameter *<sup>α</sup>* (or

� �

*p*

vuut , (52)

Ω2 *p* ½1 � exp ð Þ �2*qa* � ¼ 0*:*

½ � 1 � exp ð Þ �2*qa*

*<sup>q</sup>*<sup>2</sup> þ O *<sup>q</sup>*<sup>3</sup> � �,

(51)

(53)

!

<sup>2</sup> *<sup>ω</sup>*<sup>2</sup> Ω2 *p*

4 Ω2 *p ω*2

!<sup>2</sup>

$$\mathbb{S}\_{\rm C}(q,\alpha|E\_F) = 1 - \frac{2\pi a\_r}{q} \Pi\_0(q,\alpha|E\_F) \left[ \mathbf{1} + \frac{\boldsymbol{\Omega}\_p^2}{2\alpha^2 - \boldsymbol{\Omega}\_p^2} \exp\left(-2qa\right) \right],\tag{46}$$

where *a* is the separation between the 2D layer and the conducting surface. Most important, we should emphasize that the second term in Eq. (46) does not have a full analogy with polarization function of an isolated layer, and the resulting plasmon dispersions in open system represents a hybridized plasmon mode with the environment. Therefore, these plasmon dispersions are expected to be sensitive to Coulomb coupling to electrons in the conducting substrate through a factor � exp ð Þ �2*qa* in Eq. (46), similarly to what we have found for coupled double graphene layers [16]. The strong Coulomb coupling leads to a *linear dispersion* of plasmon in this open system [54, 58], which is in contrast with well-known � ffiffiffi *q* p dependence in all 2D materials.

As a special example, let us consider a silicene 2D layer with two bandgaps Δ<,<sup>&</sup>gt; and an electron doping density *n*. We start with seeking for a non-interacting polarization function in the long-wave limit *q* ≪ *k*<,<sup>&</sup>gt; *<sup>F</sup>* for doping density *n* and assume a high-enough *n* to keep the Fermi level *EF* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ħvFk<sup>β</sup> F* � �<sup>2</sup> <sup>þ</sup> <sup>Δ</sup><sup>2</sup> *β* r > Δ<sup>&</sup>gt; above the large bandgap. Under this assumption, we get

$$\Pi\_0(q,\alpha|E\_F) = \frac{q^2}{\pi\hbar^2\alpha^2} \sum\_{\beta=>\ \ ,<} k\_F^{\beta} \left| \frac{\partial \mathbb{E}\_{\beta}(k)}{\partial k} \right|\_{k=k\_F^{\beta}} = \frac{E\_F}{\pi} \left( 2 - \frac{\Delta\_<^2}{E\_F^2} - \frac{\Delta\_>^2}{E\_F^2} \right) \frac{q^2}{\hbar^2\alpha^2}, \tag{47}$$

where *k<sup>β</sup> <sup>F</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi 2*πn<sup>β</sup>* p are two different Fermi wave numbers associated with a single Fermi energy *EF*, and *n<sup>β</sup>* is the electron density for each subband satisfying *n* ¼ *n* <sup>&</sup>lt; þ *n* <sup>&</sup>gt; .

In the limit of *a* ! ∞, the plasmon branch of an isolated silicene layer can be recovered from Eq. (46), yielding

$$
\alpha\_p^2(q) = \frac{4\alpha\_r}{\hbar^2 E\_F} \left( E\_F^2 - \frac{\Delta\_{>}^2 + \Delta\_{<}^2}{2} \right) q \equiv \Xi q,\tag{48}
$$

where for convenience we introduced a coefficient

$$\Xi(E\_F, \Delta\_\beta) = \frac{4\alpha\_r}{\hbar^2 E\_F} \left( E\_F^2 - \frac{\Delta\_\lambda^2 + \Delta\_\zeta^2}{2} \right). \tag{49}$$

We notice from Eq. (48) that *ωp*ð Þ� *q* ffiffiffi *q* p , disregarding of the energy bandgaps Δ<sup>&</sup>lt;,<sup>&</sup>gt; or doping density *n*. On the other hand, the Fermi energy *EF* for silicene is given by Eq. (17).

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

$$\left[\mathbf{1} - \Xi(E\_F, \Delta\_\beta) \frac{q}{a^2} \left[\mathbf{1} + \frac{\Omega\_p^2}{2a^2 - \Omega\_p^2} \exp\left(-2qa\right)\right] = \mathbf{0},\tag{50}$$

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

which leads to a bi-quadratic equation

calculate plasmon dispersions in this 2D open system. For this, the plasmon dispersions are obtained from the zeros of the so-called *dispersion factor C*ð Þ *q*,*ω* , instead of the dielectric function in Eq. (4). *C*ð Þ *q*,*ω* for this open system is given by

Π0ð Þ *q*, *ω*j*EF* 1 þ

where *a* is the separation between the 2D layer and the conducting surface. Most important, we should emphasize that the second term in Eq. (46) does not have a full analogy with polarization function of an isolated layer, and the resulting plasmon dispersions in open system represents a hybridized plasmon mode with the environment. Therefore, these plasmon dispersions are expected to be sensitive to Coulomb coupling to electrons in the conducting substrate through a factor � exp ð Þ �2*qa* in Eq. (46), similarly to what we have found for coupled double graphene layers [16]. The strong Coulomb coupling leads to a *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 Δ<,<sup>&</sup>gt;

and an electron doping density *n*. We start with seeking for a non-interacting

Ω2 *p* <sup>2</sup>*ω*<sup>2</sup> � <sup>Ω</sup><sup>2</sup> *p*

" #

exp ð Þ �2*qa*

*<sup>F</sup>* for doping density *n* and

<sup>þ</sup> <sup>Δ</sup><sup>2</sup> *β*

*ħ*2

*q* � Ξ*q*, (48)

*q* p , disregarding of the energy bandgaps

*:* (49)

¼ 0, (50)

> Δ<sup>&</sup>gt; above

*<sup>ω</sup>*<sup>2</sup> , (47)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ħvFk<sup>β</sup> F* � �<sup>2</sup>

r

*<sup>π</sup>* <sup>2</sup> � <sup>Δ</sup><sup>2</sup>

<sup>&</sup>gt; <sup>þ</sup> <sup>Δ</sup><sup>2</sup> < 2 � �

exp ð Þ �2*qa*

< *E*2 *F*

� Δ2 > *E*2 *F*

� � *q*<sup>2</sup>

, (46)

[25, 54, 57]

*Nanoplasmonics*

� ffiffiffi

Π0ð Þ¼ *q*,*ω*j*EF*

where *k<sup>β</sup>*

*n* ¼ *n* <sup>&</sup>lt; þ *n* <sup>&</sup>gt; .

given by Eq. (17).

(47) that

**196**

*C*ð Þ¼ *<sup>q</sup>*, *<sup>ω</sup>*j*EF* <sup>1</sup> � <sup>2</sup>*πα<sup>r</sup>*

*q* p dependence in all 2D materials.

*q*2 *πħ*<sup>2</sup> *ω*2

*<sup>F</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi 2*πn<sup>β</sup>*

recovered from Eq. (46), yielding

*ω*2 *<sup>p</sup>*ð Þ¼ *q*

*q*

polarization function in the long-wave limit *q* ≪ *k*<,<sup>&</sup>gt;

assume a high-enough *n* to keep the Fermi level *EF* ¼

X *β*¼ >, < *kβ F* � � � �

4*α<sup>r</sup> ħ*2 *EF*

� � <sup>¼</sup> <sup>4</sup>*α<sup>r</sup>*

*<sup>ω</sup>*<sup>2</sup> <sup>1</sup> <sup>þ</sup>

where for convenience we introduced a coefficient

Ξ *EF*, Δ*<sup>β</sup>*

We notice from Eq. (48) that *ωp*ð Þ� *q* ffiffiffi

1 � Ξ *EF*, Δ*<sup>β</sup>*

� � *q*

*<sup>∂</sup>β*ð Þ*<sup>k</sup> ∂k*

� � � � *<sup>k</sup>*¼*k<sup>β</sup> F*

single Fermi energy *EF*, and *n<sup>β</sup>* is the electron density for each subband satisfying

In the limit of *a* ! ∞, the plasmon branch of an isolated silicene layer can be

*E*2 *<sup>F</sup>* � <sup>Δ</sup><sup>2</sup>

*ħ*2 *EF*

Δ<sup>&</sup>lt;,<sup>&</sup>gt; or doping density *n*. On the other hand, the Fermi energy *EF* for silicene is

<sup>¼</sup> *EF*

p are two different Fermi wave numbers associated with a

<sup>&</sup>gt; <sup>þ</sup> <sup>Δ</sup><sup>2</sup> < 2 � �

> *E*2 *<sup>F</sup>* � <sup>Δ</sup><sup>2</sup>

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

Ω2 *p* <sup>2</sup>*ω*<sup>2</sup> � <sup>Ω</sup><sup>2</sup> *p*

" #

the large bandgap. Under this assumption, we get

$$2\left(\frac{\alpha^2}{\Omega\_p^2}\right)^2 - \left[1 + \Xi(E\_F, \Delta\_\beta)\frac{2q}{\Omega\_p^2}\right] \left(\frac{\alpha^2}{\Omega\_p^2}\right) + \Xi(E\_F, \Delta\_\beta)\frac{q}{\Omega\_p^2} [1 - \exp\left(-2qa\right)] = 0. \tag{51}$$

Eq. (51) gives rise to two solutions

$$\begin{split} \frac{4}{\Omega\_p^2} \boldsymbol{\alpha}\_{p,\pm}^2 &= \left( \mathbf{1} + \boldsymbol{\Xi} \left( \mathbf{E}\_F, \Delta\_\beta \right) \frac{2q}{\Omega\_p^2} \right) \\ &\pm \sqrt{\left( \mathbf{1} + \boldsymbol{\Xi} \left( \mathbf{E}\_F, \Delta\_\beta \right) \frac{2q}{\Omega\_p^2} \right)^2 - \frac{8q \boldsymbol{\Xi} \left( \mathbf{E}\_F, \Delta\_\beta \right)}{\Omega\_p^2} \left[ \mathbf{1} - \exp \left( -2qa \right) \right]}} \end{split} \tag{52}$$

where � terms correspond to in-phase and out-of-phase plasmon modes, respectively. Two hybrid plasmon modes in Eq. (52) become

$$\begin{split} \omega\_{p,+}(q) &\simeq \frac{\Omega\_p}{\sqrt{2}} + \frac{\Xi}{\sqrt{2}\Omega\_p} q - \frac{\Xi\left(\Xi + 4a\Omega\_p^2\right)}{2\sqrt{2}\Omega\_p^3} q^2 + \mathcal{O}(q^3), \\\\ \omega\_{p,-}(q) &\simeq q\sqrt{2a\Xi} - \frac{\Xi\sqrt{2a\Xi}}{\Omega\_p^2} q^2 + \mathcal{O}(q^3). \end{split} \tag{53}$$

In Eq. (53), both plasmon branches contain a linear � *q* term, and *ω<sup>p</sup>*,þð Þ*q* approaches a constant as *q* ! 0, i.e., an optical mode for plasmons. Two independent bandgaps, Δ*SO* and Δ*z*, together with doping density *n*, play a crucial role on shaping the plasmon dispersions, as well as the particle-hole mode damping regions. The outer boundaries of a particle-hole mode region specify an area within the *q*-*ω* plane in which the plasmon modes become damping free and are solely determined by Δ<sup>&</sup>lt; , while the group velocity of plasmon mode depends on both Δ<sup>&</sup>lt; and Δ<sup>&</sup>gt; . Since each bandgap could be experimentally tuned by applying a perpendicular electric field, we acquire a full control of both plasmon dispersions and their damping-free regions at the same time.

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 *q* with a finite slope as *q* ≪ *k*0. The acoustic-plasmon branch starts from the origin, while the optical-plasmon branch from Ω*p=* ffiffi 2 <sup>p</sup> . The dispersion of each branch also varies with parameter *<sup>α</sup>* (or *<sup>ϕ</sup>* <sup>¼</sup> tan �<sup>1</sup>*α*), which is observed for the upper branch, as shown in (a) and (c) of **Figure 9**. In addition, we see a much smaller slope for the lower plasmon branch in **Figure 9(b)** and **(d)** due to enhanced Coulomb coupling with a reduced separation *a*. The finite-temperature upper plasmon branches in **Figure 9(e)** and **(f)** are shifted up greatly, as it is expected to be true for all finite-temperature plasmons, which is further accompanied by enhanced damping below the diagonal as seen in **Figure 6(c)**. Meanwhile, the lower plasmon branch seems much less affected by finite temperatures, as demonstrated by both upper and lower rows of **Figure 9** for different separations, except for enhanced damping in **Figure 9(f)** below the surface-plasmon energy *ħ*Ω*p=* ffiffi 2 <sup>p</sup> .

could be tuned sensitively by geometry phase, temperature, and separation between *α*-T <sup>3</sup> layer and conducting surface. Such tunability has a profound influence on

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).

performance of *α*-T <sup>3</sup> material based quantum electronic devices.

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

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

\*, Godfrey Gumbs2,3 and Danhong Huang<sup>4</sup>

University of New York, Brooklyn, NY, USA

\*Address all correspondence to: aiurov@mec.cuny.edu;

1 Department of Physics and Computer Science, Medgar Evers College of the City

2 Department of Physics and Astronomy, Hunter College of the City University of

3 Donostia International Physics Center (DIPC), San Sebastian, Basque Country,

4 Air Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Acknowledgements**

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

**Conflict of interest**

**Author details**

New York, NY, USA

Base, NM, USA

theorist.physics@gmail.com

provided the original work is properly cited.

Andrii Iurov<sup>1</sup>

Spain

**199**

#### **Figure 9.**

*Nonlocal hybridized plasmon dispersions for α-*T <sup>3</sup> *layer coupled to a closely-located surface of a semi-infinite conductor. Panels (a)–(d) are for* T ¼ 0*, while plots (e) and (f) for* kBT ¼ E0*. All the upper-row plots correspond to the separation* <sup>a</sup> <sup>¼</sup> <sup>1</sup>*:*0k�<sup>1</sup> <sup>F</sup> *, and the lower-row ones to* <sup>a</sup> <sup>¼</sup> <sup>0</sup>*:*5k�<sup>1</sup> <sup>F</sup> *. Additionally, middle-column plots, (b) and (d), correspond to ϕ* ¼ *π=*7*, and all other columns to ϕ* ¼ *π=*10*.*

## **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 *μ*ð Þ *T* could be employed for studying the temperature dependence of plasmon modes in each of the considered 2D materials. The fact that a chemical potential keeps its sign is true only for materials with symmetric energy bands of electrons and holes, but can cross the zero line for TMDC's with asymmetric electron and hole bands.

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 *α*-T <sup>3</sup> lattices, in which the characteristic parameter *α* is the ratio of hub-rim to hub-hub hopping coefficients and can vary from 0 to 1 continuously corresponding to different material properties. For *α*-T <sup>3</sup> materials, we have obtained the hybrid plasmon modes for different *α* values at both zero and finite temperatures and demonstrated that the resulting hybridized plasmon dispersions

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

could be tuned sensitively by geometry phase, temperature, and separation between *α*-T <sup>3</sup> layer and conducting surface. Such tunability has a profound influence on performance of *α*-T <sup>3</sup> material based quantum electronic devices.
