**5. Dice lattice and** *α***-**T **<sup>3</sup> 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 T <sup>3</sup> lattice, for which its atomic composition consists of hexagons similarly to graphene, but with an additional atom at the center of each hexagon. In a dice lattice, the bond coupling between a central site and three nearest neighbors is the same as that between atoms on corners, while for an *α*-T <sup>3</sup> model the ratio *α* between hub-rim and rim-rim hopping coefficients can vary [45, 46] within the range of 0 <*α* <1.

The low-energy electronic states of *α*-T <sup>3</sup> materials are specified by a 3 � 3 pseudospin-1 Dirac Hamiltonian, which results in three solutions for the energy dispersions and includes one completely flat and dispersionless band with *ε*0ð Þ� *k* 0. The other two bands are equivalent to Dirac cone *ε*�ð Þ¼� *k ħvFk* in graphene with the same Fermi velocity *vF* <sup>¼</sup> <sup>10</sup><sup>8</sup> cm/s. All of three bands touch at the corners of the first Brillouin zone, and therefore the band structure becomes metallic. In addition, the flat band has been shown to be stable against external perturbations, magnetic fields and structure disorders [47].

The *α*-T <sup>3</sup> model was initially considered only as a theoretical contraption, an interpolation between graphene and a dice lattice. As parameter *α* ! 0, this structure approaches graphene and a completely decoupled system of the hub atoms at the centers of each hexagon. A bit later, first evidence of really existing or fabricated materials with *α*-T <sup>3</sup> electronic structure began mounting up. This includes Josephson arrays, optical arrangement based on the laser beams, Kagome and Lieb lattices with optical waveguides, Hg1�xCdxTe for a specific electron doping density, dielectric photonic crystals having zero-refractive index and a few others [48, 49]. So far, *α*-T <sup>3</sup> model is believed to be the most promising innovative low-dimensional systems, and is one of the mostly investigated material in modern condensed matter physics. The most important technological application of *α*-T <sup>3</sup> rests on the availability of materials with various *α* values, i.e., with small and large rim-hub hopping coefficients, ranging from *α* ¼ 0 for graphene up to *α* ¼ 1 for a dice lattice.

The low-energy Dirac-Weyl Hamiltonian for the *α*-T <sup>3</sup> model is [45]

$$
\hat{\mathcal{H}}\_{\xi}^{\phi}(\mathsf{k}) = \hbar v\_{F} \begin{bmatrix} \mathbf{0} & k\_{-}^{\xi} \cos \phi & \mathbf{0} \\ k\_{+}^{\xi} \cos \phi & \mathbf{0} & k\_{-}^{\xi} \sin \phi \\ \mathbf{0} & k\_{+}^{\xi} \sin \phi & \mathbf{0} \end{bmatrix}, \tag{38}$$

where *k* ¼ *kx*, *ky* � � is the electron wave vector, *k<sup>ξ</sup>* � <sup>¼</sup> *<sup>ξ</sup>kx* � *iky*, *<sup>ξ</sup>* ¼ �1 corresponds to two different valleys, and *vF* is the Fermi velocity. Here, the parameter *α* is related to the geometry phase *<sup>ϕ</sup>* <sup>¼</sup> tan �<sup>1</sup>*<sup>α</sup>* which directly enters into the Hamiltonian in Eq. (38). The phase *ϕ* possesses a fixed, one-to-one correspondence to the Berry phase of electrons in *α*-T <sup>3</sup> model. In particular, for *α* ¼ 1 or *ϕ* ¼ *π=*4 we get a dice lattice with its Hamiltonian given by [50].

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

$$
\hat{\mathbb{H}}\_{\xi}^{d}(\mathbb{k}) = \frac{\hbar v\_{F}}{\sqrt{2}} \begin{bmatrix} \mathbf{0} & k\_{-}^{\xi} & \mathbf{0} \\ k\_{+}^{\xi} & \mathbf{0} & k\_{-}^{\xi} \\ \mathbf{0} & k\_{+}^{\xi} & \mathbf{0} \end{bmatrix} . \tag{39}
$$

Three energy bands from Hamiltonian in Eq. (38) or Eq. (39) are *ε γ* <sup>0</sup>ð Þ¼ *k γ ħvFk* for valence (*γ* ¼ �1), conduction (*γ* ¼ þ1) and flat (*γ* ¼ 0) bands. These energy bands are degenerate with respect to *ξ* and phase *ϕ*. The corresponding wave functions for the valence and conduction bands take the form

$$\Psi\_0^{\nu=\pm 1}(\mathbf{k}|\xi,\phi) = \frac{1}{\sqrt{2}} \begin{bmatrix} \xi \cos \phi \,\mathrm{e}^{-i\xi \theta\_\mathbf{k}} \\ \gamma \\ \xi \sin \phi \,\mathrm{e}^{i\xi \theta\_\mathbf{k}} \end{bmatrix},\tag{40}$$

where tan *θ***<sup>k</sup>** ¼ *ky=kx*. Meanwhile, for the flat band, we find

$$\Psi\_0^{\mathbf{y}=0}(\mathbf{k}\mid\xi,\phi) = \begin{bmatrix} \xi \sin\phi \,\mathrm{e}^{-i\xi\theta\_\mathbf{k}} \\ \mathbf{0} \\ -\xi \cos\phi \,\mathrm{e}^{i\xi\theta\_\mathbf{k}} \end{bmatrix}.\tag{41}$$

Here, the components of wave functions in Eqs. (40) and (41) depend on valley index *ξ* and phase *ϕ*, which leads to the same dependence on all collective properties of an *α*-T <sup>3</sup> materials, including plasmon dispersion.

Now we turn to deriving plasmon branches and their damping rates at finite *T* in *α*-T <sup>3</sup> model. The computation procedure is quite similar to that in the case of two non-equivalent doubly degenerate subband pairs, including silicene, germanene and MoS2 discussed in Section 4.

For *α*-T <sup>3</sup> model, the finite-*T* polarization function Π*T*ð Þ *q*, *ω*j*μ*ð Þ *T* can be obtained by an integral transformation of its zero-temperature counterpart Π0ð Þ *q*,*ω*j*EF* , as presented in Eq. (5). In this case, the zero-*T* counterpart Π0ð Þ *q*,*ω*j*EF* is calculated as

$$\begin{split} \Pi\_{0}(q,\omega|E\_{F}) &= \frac{1}{\pi^{2}} \sum\_{\boldsymbol{\gamma},\boldsymbol{\gamma}'=0,\,\pm 1} \Big[ d^{2} \operatorname{k} \mathbb{O}\_{\boldsymbol{\gamma},\boldsymbol{\gamma}'} \Big( \mathbf{k}, \mathbf{k} + \mathbf{q} \, | \boldsymbol{\phi} \Big) \\ &\times \frac{\Theta\_{0} \left( E\_{F} - \varepsilon\_{\boldsymbol{\gamma}}(\mathbf{k}) \right) - \Theta\_{0} \left( E\_{F} - \varepsilon\_{\boldsymbol{\gamma}'}(|\mathbf{k} + \mathbf{q}|) \right)}{\hbar (\boldsymbol{\alpha} + i\mathbf{0}^{+}) + \varepsilon\_{\boldsymbol{\gamma}}(\boldsymbol{k}) - \varepsilon\_{\boldsymbol{\gamma}'}(|\mathbf{k} + \mathbf{q}|)} . \end{split} \tag{42}$$

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 *γ* ¼ 0 so that the summation index runs over �1 and 0 instead of two. On the other hand, the overall expression for Π0ð Þ *q*, *ω*j*EF* in Eq. (42) is simplified because the 4-fold degeneracy of each energy band independent of valley and spin index.

Here, we would limit our consideration to the case of electron doping with *n*> 0 and apply the random-phase approximation theory only for that case. For electron doping with *n* >0, we can neglect the transitions within the valence band and also the transitions between the flat and valences bands due to full occupations of these electronic states. On the other hand, the overlap of initial and final electron transition states is defined by [51] *<sup>ξ</sup> <sup>γ</sup>*,*γ*<sup>0</sup> *k*, *k*<sup>0</sup> j*ϕ*, *λ*<sup>0</sup> � � with respect to the initial Ψ*ξ <sup>γ</sup>* ð Þ *<sup>k</sup>*, *<sup>λ</sup>*<sup>0</sup> and the final <sup>Ψ</sup>*<sup>ξ</sup> <sup>γ</sup>*<sup>0</sup> *k*<sup>0</sup> , *λ*<sup>0</sup> � � states with a momentum transfer *<sup>q</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>0</sup> � *<sup>k</sup>* , i.e.,

density, on the other hand, widens the plasmon damping-free regions. Therefore, both the thermal and doping effects are found to compete with each other in dominating the plasmon dampings through selecting different convolution paths *μ*ð Þ *T* with various doping densities or Fermi energies. Furthermore, the plasmon energy in (c) is pushed up slightly by increasing doping density at finite *T*.

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 T <sup>3</sup> lattice, for which its atomic composition consists of hexagons similarly to graphene, but with an additional atom at the center of each hexagon. In a dice lattice, the bond coupling between a central site and three nearest neighbors is the same as that between atoms on corners, while for an *α*-T <sup>3</sup> model the ratio *α* between hub-rim and rim-rim hopping coefficients can vary [45, 46] within the range of 0 <*α* <1. The low-energy electronic states of *α*-T <sup>3</sup> materials are specified by a 3 � 3

pseudospin-1 Dirac Hamiltonian, which results in three solutions for the

perturbations, magnetic fields and structure disorders [47].

energy dispersions and includes one completely flat and dispersionless band with *ε*0ð Þ� *k* 0. The other two bands are equivalent to Dirac cone *ε*�ð Þ¼� *k ħvFk* in graphene with the same Fermi velocity *vF* <sup>¼</sup> <sup>10</sup><sup>8</sup> cm/s. All of three bands touch at the corners of the first Brillouin zone, and therefore the band structure becomes metallic. In addition, the flat band has been shown to be stable against external

The *α*-T <sup>3</sup> model was initially considered only as a theoretical contraption, an

interpolation between graphene and a dice lattice. As parameter *α* ! 0, this structure approaches graphene and a completely decoupled system of the hub atoms at the centers of each hexagon. A bit later, first evidence of really existing or fabricated materials with *α*-T <sup>3</sup> electronic structure began mounting up. This includes Josephson arrays, optical arrangement based on the laser beams, Kagome and Lieb lattices with optical waveguides, Hg1�xCdxTe for a specific electron doping density, dielectric photonic crystals having zero-refractive index and a few others [48, 49]. So far, *α*-T <sup>3</sup> model is believed to be the most promising innovative low-dimensional systems, and is one of the mostly investigated material in modern condensed matter physics. The most important technological application of *α*-T <sup>3</sup> rests on the availability of materials with various *α* values, i.e., with small and large rim-hub hopping coefficients, ranging from *α* ¼ 0 for graphene up to *α* ¼ 1 for a

The low-energy Dirac-Weyl Hamiltonian for the *α*-T <sup>3</sup> model is [45]

*kξ*

� � is the electron wave vector, *k<sup>ξ</sup>*

0 *k<sup>ξ</sup>*

0 *k<sup>ξ</sup>*

sponds to two different valleys, and *vF* is the Fermi velocity. Here, the parameter *α* is related to the geometry phase *<sup>ϕ</sup>* <sup>¼</sup> tan �<sup>1</sup>*<sup>α</sup>* which directly enters into the Hamiltonian in Eq. (38). The phase *ϕ* possesses a fixed, one-to-one correspondence to the Berry phase of electrons in *α*-T <sup>3</sup> model. In particular, for *α* ¼ 1 or *ϕ* ¼ *π=*4 we get a

<sup>þ</sup> cos *<sup>ϕ</sup>* <sup>0</sup> *<sup>k</sup><sup>ξ</sup>*

� cos *<sup>ϕ</sup>* <sup>0</sup>

<sup>þ</sup> sin *<sup>ϕ</sup>* <sup>0</sup>

� sin *<sup>ϕ</sup>*

3 7 7

� <sup>¼</sup> *<sup>ξ</sup>kx* � *iky*, *<sup>ξ</sup>* ¼ �1 corre-

5, (38)

**5. Dice lattice and** *α***-**T **<sup>3</sup> materials**

*Nanoplasmonics*

dice lattice.

**192**

^ *ϕ*

where *k* ¼ *kx*, *ky*

*<sup>ξ</sup>* ð Þ¼ *k ħvF*

dice lattice with its Hamiltonian given by [50].

*Nanoplasmonics*

$$\begin{aligned} \mathbb{O}\_{\mathbb{Y},\mathbb{Y}'}^{\boldsymbol{\xi}} \left( \mathbf{k}, \mathbb{k} + \boldsymbol{\mathsf{q}} \, | \boldsymbol{\phi}, \boldsymbol{\lambda}\_{0} \right) &= \left| \mathbb{S}\_{\mathbb{Y},\mathbb{Y}'}^{\boldsymbol{\xi}} \left( \mathbb{k}, \mathbb{k} + \boldsymbol{\mathsf{q}} \, | \, \boldsymbol{\phi}, \boldsymbol{\lambda}\_{0} \right) \right|^{2}, \\ \mathbb{S}\_{\mathbb{Y},\mathbb{Y}'}^{\boldsymbol{\xi}} \left( \mathbb{k}, \mathbb{k} + \boldsymbol{\mathsf{q}} \, | \, \boldsymbol{\phi}, \boldsymbol{\lambda}\_{0} \right) &= \left\langle \, \Psi\_{\mathbb{Y}}^{\boldsymbol{\xi}}(\mathbb{k}, \mathbb{\boldsymbol{\lambda}}\_{0}) \, | \, \Psi\_{\mathbb{Y}'}^{\boldsymbol{\xi}}(\mathbb{k} + \boldsymbol{\mathsf{q}}, \boldsymbol{\lambda}\_{0}) \right\rangle, \end{aligned} \tag{43}$$

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

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

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

realized experimentally or even by a device fabrication.

*(b) corresponds to* T ¼ 0*, while* kBT ¼ 1*:*0EF *for right panels (c) and (d).*

ð *d*<sup>3</sup> *r*<sup>0</sup>

system.

**Figure 8.**

**195**

**Figure 7.**

written as <sup>ϵ</sup>*B*ð Þ¼ *<sup>ω</sup>* <sup>1</sup> � <sup>Ω</sup><sup>2</sup>

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

Our schematics for an open system is shown in **Figure 8**. The dynamical screen-

, *r*<sup>00</sup> ð Þ¼ ;*ω δ r* � *r*<sup>00</sup> ð Þ, (45)

*n*0*e*<sup>2</sup>*=*ϵ0ϵ*bm*<sup>∗</sup> p is the bulk-plasma fre-

ffiffiffiffiffiffiffiffi *π*n0 <sup>p</sup> <sup>¼</sup> <sup>54</sup>*:*<sup>6</sup> *meV.*

results in a strong hybridization of graphene plasmon and localized surfaceplasmon modes. This structure, which is referred to as an *open system*, could be

*Plasmon branches for an isolated α-*T <sup>3</sup> *layer with ϕ* ¼ *π=*10*. Panels (a) and (c) only show undamped plasmons, while (b) and (d) display full plasmon branches including damped ones. Left panels (a) and*

ing 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 dielec-

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

By using the Drude model for metallic substrate, the dielectric function can be

*<sup>p</sup>=ω*2, where <sup>Ω</sup>*<sup>p</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

quency 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

*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

tric function is connected to a dielectric function ϵ *r*, *r*<sup>0</sup> ð Þ ;*ω* in Eq. (4) by

K *r*, *r*<sup>0</sup> ð Þ ;*ω* ϵ *r*<sup>0</sup>

where *β***<sup>k</sup>**,**k**<sup>0</sup> ¼ *θ***k**<sup>0</sup> � *θ***<sup>k</sup>** is the scattering angle between two electronic states and *k*<sup>0</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>k</sup>*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*kq* cos *<sup>β</sup>***<sup>k</sup>**,**k**<sup>0</sup> q . Moreover, we find from Eq. (43) [52]

$$\mathbb{G}\_{\gamma,\mathbf{y}'}^{\xi} \left( \mathbf{k}, \mathbf{k} + \mathbf{q} | \phi, \lambda\_0 \right) = \frac{1}{4} \left[ \left( \mathbf{1} + \cos \beta\_{\mathbf{k},\mathbf{k}'} \right)^2 + \cos^2(2\phi) \sin^2 \beta\_{\mathbf{k},\mathbf{k}'} \right] \tag{44}$$

for an arbitrary value of *ϕ* or *α*. It is easy to verify the known results 1 þ cos *β***<sup>k</sup>**,**k**<sup>0</sup> � �*=*2 for graphene and 1 <sup>þ</sup> cos *<sup>β</sup>***<sup>k</sup>**,**k**<sup>0</sup> � �<sup>2</sup> *=*4 for a dice lattice as two limiting cases of our general result in Eq. (44) as *α* ! 0 or *α* ! 1, respectively. Furthermore, we find from Eq. (44) that the overlap does not depend on valley index *ξ*, even though individual wave function does, and then this index can be dropped. However, the valley-dependence in *<sup>ξ</sup> <sup>γ</sup>*,*γ*<sup>0</sup> *k*, *k* þ *q* j*ϕ*, *λ*<sup>0</sup> � � persists if *<sup>α</sup>*-<sup>T</sup> <sup>3</sup> material is irradiated by circularly- or elliptically-polarized light. This incident radiation permits creating an valleytronic filter or any other types of valleytronic electron device.

Density plots for Landau damping with Im½Π*O*ð � *q*,*ω*j*μ*ð ÞÞ *T* 6¼ 0 is presented in **Figure 6**, where we find plasmon branch will be completely free from damping within the region determined by *ħω=E*<sup>0</sup> ≤ 1 and *q=k*<sup>0</sup> ≤1, independent of geometry phase *ϕ*. On the other hand, another region with *ħω*≤ *ħvFq* (below the diagonal) becomes always Landau damped. Increasing *T* is able to increase greatly the damping in the region below the diagonal, as seen in **Figure 6(c)**.

In a correspondence to the damping of plasmons presented in **Figure 6**, we show in **Figure 7** the density plots for plasmon dispersions at *T* ¼ 0 in (a), (b) and *kBT* ¼ *EF* in (c), (d). Comparing **Figure 7(a)** with **Figure 7(c)** we have clearly seen the thermal suppression of Landau damping for plasmon mode entering into a highfrequency region beyond *ħω* ¼ *EF*. To visualize a full plasmon dispersion clearly, we also include damped counterpart in **Figure 7(b)** and **(d)** at *T* ¼ 0 and *kBT* ¼ *EF*, respectively, where a significant enhancement of plasmon energy appears for large *q* values, moving upwards from the diagonal.

#### **Figure 6.**

*Particle-hole modes, determined by non-zero* Im½ΠOð � q,*ω*j*μ*ð ÞÞ T *within the* q*-ω plane, for an α-*T <sup>3</sup> *layer with ϕ* ¼ *π=*10 *(in (a), (c)) and ϕ* ¼ *π=*7 *in (b). Panels (a) and (b) correspond to* T ¼ 0*, while plot (c) is for* kBT ¼ 1*:*0EF*.*

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

#### **Figure 7.**

*ξ*

*ξ*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>k</sup>*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*kq* cos *<sup>β</sup>***<sup>k</sup>**,**k**<sup>0</sup>

*<sup>γ</sup>*,*γ*<sup>0</sup> *k*, *k* þ *q*j*ϕ*, *λ*<sup>0</sup> � � <sup>¼</sup> <sup>1</sup>

*k*<sup>0</sup> ¼

q

*Nanoplasmonics*

*ξ*

1 þ cos *β***<sup>k</sup>**,**k**<sup>0</sup>

electron device.

**Figure 6.**

**194**

kBT ¼ 1*:*0EF*.*

*<sup>γ</sup>*,*γ*<sup>0</sup> *k*, *k* þ *q*j*ϕ*, *λ*<sup>0</sup>

*<sup>γ</sup>*,*γ*<sup>0</sup> *k*, *k* þ *q*j*ϕ*, *λ*<sup>0</sup>

� � <sup>¼</sup> <sup>∣</sup>*<sup>ξ</sup>*

� � <sup>¼</sup> <sup>Ψ</sup>*<sup>ξ</sup>*

4

� �*=*2 for graphene and 1 <sup>þ</sup> cos *<sup>β</sup>***<sup>k</sup>**,**k**<sup>0</sup>

dropped. However, the valley-dependence in *<sup>ξ</sup>*

*q* values, moving upwards from the diagonal.

*<sup>γ</sup>*,*γ*<sup>0</sup> *k*, *k* þ *q* j j *ϕ*, *λ*<sup>0</sup> � �<sup>2</sup>

D E

*<sup>γ</sup>* ð Þj *<sup>k</sup>*, *<sup>λ</sup>*<sup>0</sup> <sup>Ψ</sup>*<sup>ξ</sup>*

. Moreover, we find from Eq. (43) [52]

h i

where *β***<sup>k</sup>**,**k**<sup>0</sup> ¼ *θ***k**<sup>0</sup> � *θ***<sup>k</sup>** is the scattering angle between two electronic states and

� �<sup>2</sup>

1 þ cos *β***<sup>k</sup>**,**k**<sup>0</sup> � �<sup>2</sup> <sup>þ</sup> cos <sup>2</sup>

limiting cases of our general result in Eq. (44) as *α* ! 0 or *α* ! 1, respectively. Furthermore, we find from Eq. (44) that the overlap does not depend on valley index *ξ*, even though individual wave function does, and then this index can be

material is irradiated by circularly- or elliptically-polarized light. This incident radiation permits creating an valleytronic filter or any other types of valleytronic

damping in the region below the diagonal, as seen in **Figure 6(c)**.

Density plots for Landau damping with Im½Π*O*ð � *q*,*ω*j*μ*ð ÞÞ *T* 6¼ 0 is presented in **Figure 6**, where we find plasmon branch will be completely free from damping within the region determined by *ħω=E*<sup>0</sup> ≤ 1 and *q=k*<sup>0</sup> ≤1, independent of geometry phase *ϕ*. On the other hand, another region with *ħω*≤ *ħvFq* (below the diagonal) becomes always Landau damped. Increasing *T* is able to increase greatly the

In a correspondence to the damping of plasmons presented in **Figure 6**, we show

*Particle-hole modes, determined by non-zero* Im½ΠOð � q,*ω*j*μ*ð ÞÞ T *within the* q*-ω plane, for an α-*T <sup>3</sup> *layer with ϕ* ¼ *π=*10 *(in (a), (c)) and ϕ* ¼ *π=*7 *in (b). Panels (a) and (b) correspond to* T ¼ 0*, while plot (c) is for*

in **Figure 7** the density plots for plasmon dispersions at *T* ¼ 0 in (a), (b) and *kBT* ¼ *EF* in (c), (d). Comparing **Figure 7(a)** with **Figure 7(c)** we have clearly seen the thermal suppression of Landau damping for plasmon mode entering into a highfrequency region beyond *ħω* ¼ *EF*. To visualize a full plasmon dispersion clearly, we also include damped counterpart in **Figure 7(b)** and **(d)** at *T* ¼ 0 and *kBT* ¼ *EF*, respectively, where a significant enhancement of plasmon energy appears for large

for an arbitrary value of *ϕ* or *α*. It is easy to verify the known results

,

,

*β***<sup>k</sup>**,**k**<sup>0</sup>

� � persists if *<sup>α</sup>*-<sup>T</sup> <sup>3</sup>

*=*4 for a dice lattice as two

(43)

(44)

*<sup>γ</sup>*0ð*k* þ *q*, *λ*0Þ

ð Þ <sup>2</sup>*<sup>ϕ</sup>* sin <sup>2</sup>

*<sup>γ</sup>*,*γ*<sup>0</sup> *k*, *k* þ *q* j*ϕ*, *λ*<sup>0</sup>

*Plasmon branches for an isolated α-*T <sup>3</sup> *layer with ϕ* ¼ *π=*10*. Panels (a) and (c) only show undamped plasmons, while (b) and (d) display full plasmon branches including damped ones. Left panels (a) and (b) corresponds to* T ¼ 0*, while* kBT ¼ 1*:*0EF *for right panels (c) and (d).*
