**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 Δ<sup>0</sup> and single-particle energy bands *ε*�ð Þ¼� *k* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>ħ</sup>vFk* <sup>2</sup> <sup>þ</sup> <sup>Δ</sup><sup>2</sup> 0 q , which are symmetric with respect to the Dirac point. Moreover, there is a total spin-valley degeneracy *g* ¼ *gs gv* ¼ 4 for electrons and holes in each band.

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 anisotropy of its electronic states and band structures.

#### **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 *vF* <sup>¼</sup> <sup>0</sup>*:*<sup>5</sup> � <sup>10</sup><sup>6</sup> m/s and are given by

$$
\epsilon\_{\xi,\sigma}^{\mathcal{I}}(k) = \chi \sqrt{\left(\xi \sigma \Delta\_{\mathfrak{x}} - \Delta\_{\mathrm{SO}}\right)^2 + \left(\hbar v\_F k\right)^2}, \tag{1}
$$

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

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

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

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

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

At the same time, the damping rate, or broadening of the frequency, of such

long-lived plasmon with low or even nearly zero Landau damping. Both plasmon

*T*

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

*qkBT* p , where *q* is the wave vector. This dispersion relation

<sup>p</sup> , which means that we are dealing with a

some crucial publications on that subject [7, 8].

imposed electromagnetic radiation [15].

plasmon is given as � ffiffiffiffiffiffiffiffiffiffiffi

*Nanoplasmonics*

imaging and spectroscopy.

**Figure 1.**

**180**

thermal plasmons varies as ≃1*=* ffiffiffi

T refers to a transition-metal atom, such as Mo or W, while C corresponds to a

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

through the thermal convolution of the corresponding quantum states.

r

*α* ¼ 2*:*21 ¼ 5*:*140*β*, and *me* represents the mass of a free electron.

In practical, we will neglect the ≃*k*<sup>4</sup> terms, ≃ *t*1*a*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>0</sup>ð Þ*<sup>k</sup>* � �<sup>2</sup> <sup>þ</sup> ð Þ *<sup>t</sup>*0*a*<sup>0</sup> *<sup>k</sup>* <sup>2</sup>

difference by comparing **Figure 2(b)** with **Figure 2(d)**.

**3. Thermal plasmons in graphene and other materials**

from the zeros of a dielectric function ϵ*T*ð Þ *q*,*ω* , [19, 33] given by

Δ*ν*

a *k*-dependent "gap term" and the band shift is *<sup>ν</sup>*

q

ð Þ 2*t*0*a*<sup>0</sup>

where Δ ¼ 1*:*9 eV is a gap parameter leading to an extremely large direct bandgap ≃1*:*7 eV. There is also substantial internal spin-orbit coupling *λ*<sup>0</sup> ¼ <sup>0</sup>*:*042Δ, *<sup>t</sup>*0*a*<sup>0</sup> <sup>¼</sup> <sup>4</sup>*:*<sup>95</sup> � <sup>10</sup>�<sup>29</sup> J m is a Dirac-cone term, where *<sup>t</sup>*<sup>0</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>884</sup> <sup>Δ</sup> is the electron hopping parameter while *a*<sup>0</sup> ¼ 1*:*843 Å is the lattice constant. The *t*0*a*<sup>0</sup> term is ≈0*:*47 compared to *ħvF* in graphene. Similarly to silicene and germanene, the energy dispersions of TMDC's depend on one composite valley-spin index *ν* ¼ *σξ*. There are also other less important but still non-negligible � *<sup>k</sup>*<sup>2</sup> mass terms with

anisotropy, which indeed have tiny or no effect on the density of states of the considered material, but would make our model much more complicated. The above dispersions could be presented in a form similarly to those for gapped graphene, i.e.,

of somewhat cumbersome analytical expressions for the components of the wave functions corresponding to dispersions in Eq. (3) can be found from Ref. [25].

corresponding to *ν* ¼ �1, will be lifted and two subband will be separated by *λ*<sup>0</sup> ≃79*:*8 meV. However, this is not the case for two corresponding electron states (*γ* ¼ 1). Consequently, the electron-hole asymmetry exists even at *k* ¼ 0 and becomes even more pronounced at finite *k* values. One can clearly see this

, where Δ*<sup>ν</sup>*

Using Eq. (3), we can verify that the degeneracy of two hole subbands (*γ* ¼ �1),

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 *ω* on wave number *q*. Physically, these complex relations can be determined

where *v q*ð Þ¼ <sup>2</sup>*παr=<sup>q</sup>* � *<sup>e</sup>*<sup>2</sup>*=*2ϵ0ϵ*rq* is the 2D Fourier-transformed Coulomb potential, *<sup>α</sup><sup>r</sup>* <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>*=*4*π*ϵ0ϵ*r*, and <sup>ϵ</sup>*<sup>r</sup>* represents the dielectric constant of the host material. The dielectric function introduced in Eq. (4) is determined directly by the finite-temperature *polarization function*, or *polarizability*, Π*T*ð Þ *q*, *ω* j *μ*ð Þ *T* , which is,

<sup>0</sup>ð Þ¼ *<sup>k</sup> <sup>ħ</sup>*<sup>2</sup>

<sup>0</sup>ð Þ¼ *<sup>k</sup> <sup>ħ</sup>*<sup>2</sup>

ϵ*T*ð Þ¼ *q*,*ω* 1 � *v q*ð ÞΠ*T*ð Þ¼ *q*,*ω*j*μ*ð Þ *T* 0, (4)

*k*2

*k*2

MoS2 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 *γ* ¼ �1. The absence of this particle-hole symmetry is expected to have a considerable effect on the plasmon branches at both zero and finite temperatures

Specifically, the energy bands of MoS2 can be described by a *two-band* model, i.e.,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*=me* h i*k*<sup>2</sup> <sup>þ</sup> <sup>Δ</sup> � *ξσλ*<sup>0</sup> ð Þ<sup>2</sup>

<sup>0</sup> trigonal warping term and

*β=*4*me* þ Δ*=*2 � *νλ*0*=*2 is

*α=*4*me* þ *νλ*0*=*2. A set

,

(3)

<sup>2</sup> <sup>þ</sup> <sup>Δ</sup> � *ξσλ*<sup>0</sup> ð Þ*βħ*<sup>2</sup>

chalcogen atom (S, Se or Te).

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

<sup>2</sup> *ξσλ*<sup>0</sup> <sup>þ</sup> *<sup>α</sup>ħ*<sup>2</sup>

4*me*

*<sup>k</sup>*<sup>2</sup> <sup>þ</sup> *<sup>γ</sup>* 2

*ε<sup>ξ</sup>*,*<sup>σ</sup> <sup>γ</sup>* ð Þ*<sup>k</sup>* <sup>≃</sup> <sup>1</sup>

*εν*

**183**

*<sup>γ</sup>* ð Þ¼ *<sup>k</sup> <sup>ν</sup>*

<sup>0</sup>ð Þþ *k γ*

**Figure 2.**

*Energy dispersions and density of states (DOS) ρ*dð Þ *of silicene [(a) and (b)] and molybdenum disulfide MoS2 [(c) and (d)], where* Eð Þ <sup>0</sup> <sup>F</sup> *and* <sup>k</sup>ð Þ <sup>0</sup> <sup>F</sup> *are Fermi energy and wave number, respectively. For MoS2, its dispersions and DOS, corresponding to parabolic band approximation in Eq. (23), are also shown for comparison.*

standard band insulator (BI) state for electrons. As we will see below, all of the single electronic and collective properties of buckled honeycomb lattices depend on both bandgaps Δ<, <sup>&</sup>gt; , and therefore could be tuned by varying a perpendicular electric field to create various types of functional electronic devices.

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

$$\begin{aligned} \Psi^{\eta}(k) &= \begin{bmatrix} \Psi^{\eta}\_{\xi=1, \sigma=+1}(k) \\ \Psi^{\eta}\_{\xi=1, \sigma=-1}(k) \end{bmatrix}, \\\Psi^{\eta}\_{\xi, \sigma}(k) &= \sqrt{\frac{\mathcal{Y}}{2e\_{\xi, \sigma}^{\mathcal{I}}(k)}} \begin{bmatrix} \sqrt{|e\_{\xi, \sigma}^{\mathcal{I}}(k) + \Delta\_0^{\xi, \sigma}|} \\\\ \mathcal{I}\sqrt{|e\_{\xi, \sigma}^{\mathcal{I}}(k) - \Delta\_0^{\xi, \sigma}|}e^{i\theta\_k} \end{bmatrix}, \end{aligned} \tag{2}$$

where *<sup>θ</sup><sup>k</sup>* <sup>¼</sup> tan �<sup>1</sup> *ky=kx* � � and Δ*<sup>ξ</sup>*,*<sup>σ</sup>* <sup>0</sup> ¼ ∣*ξσ*Δ*<sup>z</sup>* � Δ*SO*∣.

Germanene, another representative of buckled honeycomb lattices [26–30], demonstrates substantially higher Fermi velocities and an enhanced intrinsic bandgap � 23 meV. For a free-standing germanene, first-principles studies have revealed a buckling distances � 0*:*640*:*74 Å [31, 32].

#### **2.2 Molybdenum disulfide and transition-metal dichalcogenides**

MoS2 is a typical representative of transition-metal dichalcogenide (TMDC) monolayers. TMDC's are semiconductors with the composition of TC2 type, where *Thermal Collective Excitations in Novel Two-Dimensional Dirac-Cone Materials DOI: http://dx.doi.org/10.5772/intechopen.90870*

T refers to a transition-metal atom, such as Mo or W, while C corresponds to a chalcogen atom (S, Se or Te).

MoS2 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 *γ* ¼ �1. The absence of this particle-hole symmetry is expected to have a considerable effect on the plasmon branches at both zero and finite temperatures through the thermal convolution of the corresponding quantum states.

Specifically, the energy bands of MoS2 can be described by a *two-band* model, i.e.,

$$\epsilon\_{\gamma}^{\xi,\sigma}(\mathbf{k}) \simeq \frac{1}{2} \xi \sigma \lambda\_0 + \frac{a\hbar^2}{4m\_\epsilon} k^2 + \frac{\chi}{2} \sqrt{\left[ (2t\_0 a\_0)^2 + (\Delta - \xi \sigma \lambda\_0) \theta \hbar^2 / m\_\epsilon \right]} k^2 + \left( \Delta - \xi \sigma \lambda\_0 \right)^2,\tag{3}$$

where Δ ¼ 1*:*9 eV is a gap parameter leading to an extremely large direct bandgap ≃1*:*7 eV. There is also substantial internal spin-orbit coupling *λ*<sup>0</sup> ¼ <sup>0</sup>*:*042Δ, *<sup>t</sup>*0*a*<sup>0</sup> <sup>¼</sup> <sup>4</sup>*:*<sup>95</sup> � <sup>10</sup>�<sup>29</sup> J m is a Dirac-cone term, where *<sup>t</sup>*<sup>0</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>884</sup> <sup>Δ</sup> is the electron hopping parameter while *a*<sup>0</sup> ¼ 1*:*843 Å is the lattice constant. The *t*0*a*<sup>0</sup> term is ≈0*:*47 compared to *ħvF* in graphene. Similarly to silicene and germanene, the energy dispersions of TMDC's depend on one composite valley-spin index *ν* ¼ *σξ*. There are also other less important but still non-negligible � *<sup>k</sup>*<sup>2</sup> mass terms with *α* ¼ 2*:*21 ¼ 5*:*140*β*, and *me* represents the mass of a free electron.

In practical, we will neglect the ≃*k*<sup>4</sup> terms, ≃ *t*1*a*<sup>2</sup> <sup>0</sup> trigonal warping term and anisotropy, which indeed have tiny or no effect on the density of states of the considered material, but would make our model much more complicated. The above dispersions could be presented in a form similarly to those for gapped graphene, i.e., *εν <sup>γ</sup>* ð Þ¼ *<sup>k</sup> <sup>ν</sup>* <sup>0</sup>ð Þþ *k γ* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ*ν* <sup>0</sup>ð Þ*<sup>k</sup>* � �<sup>2</sup> <sup>þ</sup> ð Þ *<sup>t</sup>*0*a*<sup>0</sup> *<sup>k</sup>* <sup>2</sup> q , where Δ*<sup>ν</sup>* <sup>0</sup>ð Þ¼ *<sup>k</sup> <sup>ħ</sup>*<sup>2</sup> *k*2 *β=*4*me* þ Δ*=*2 � *νλ*0*=*2 is a *k*-dependent "gap term" and the band shift is *<sup>ν</sup>* <sup>0</sup>ð Þ¼ *<sup>k</sup> <sup>ħ</sup>*<sup>2</sup> *k*2 *α=*4*me* þ *νλ*0*=*2. A set of somewhat cumbersome analytical expressions for the components of the wave functions corresponding to dispersions in Eq. (3) can be found from Ref. [25].

Using Eq. (3), we can verify that the degeneracy of two hole subbands (*γ* ¼ �1), corresponding to *ν* ¼ �1, will be lifted and two subband will be separated by *λ*<sup>0</sup> ≃79*:*8 meV. However, this is not the case for two corresponding electron states (*γ* ¼ 1). Consequently, the electron-hole asymmetry exists even at *k* ¼ 0 and becomes even more pronounced at finite *k* values. One can clearly see this difference by comparing **Figure 2(b)** with **Figure 2(d)**.

### **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 *ω* on wave number *q*. Physically, these complex relations can be determined from the zeros of a dielectric function ϵ*T*ð Þ *q*,*ω* , [19, 33] given by

$$\varepsilon\_T(q,\alpha) = \mathbf{1} - \nu(q)\Pi\_T(q,\alpha|\mu(T)) = \mathbf{0},\tag{4}$$

where *v q*ð Þ¼ <sup>2</sup>*παr=<sup>q</sup>* � *<sup>e</sup>*<sup>2</sup>*=*2ϵ0ϵ*rq* is the 2D Fourier-transformed Coulomb potential, *<sup>α</sup><sup>r</sup>* <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>*=*4*π*ϵ0ϵ*r*, and <sup>ϵ</sup>*<sup>r</sup>* represents the dielectric constant of the host material.

The dielectric function introduced in Eq. (4) is determined directly by the finite-temperature *polarization function*, or *polarizability*, Π*T*ð Þ *q*, *ω* j *μ*ð Þ *T* , which is,

standard band insulator (BI) state for electrons. As we will see below, all of the single electronic and collective properties of buckled honeycomb lattices depend on both bandgaps Δ<, <sup>&</sup>gt; , and therefore could be tuned by varying a perpendicular

*and DOS, corresponding to parabolic band approximation in Eq. (23), are also shown for comparison.*

*Energy dispersions and density of states (DOS) ρ*dð Þ *of silicene [(a) and (b)] and molybdenum disulfide MoS2*

The wave function of silicene, corresponding to eigenvalue equation in Eq. (1),

,

q

*γ*

MoS2 is a typical representative of transition-metal dichalcogenide (TMDC) monolayers. TMDC's are semiconductors with the composition of TC2 type, where

q

∣*ε γ*

∣*ε γ*

<sup>0</sup> ¼ ∣*ξσ*Δ*<sup>z</sup>* � Δ*SO*∣. Germanene, another representative of buckled honeycomb lattices [26–30], demonstrates substantially higher Fermi velocities and an enhanced intrinsic bandgap � 23 meV. For a free-standing germanene, first-principles studies have

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>ξ</sup>*,*<sup>σ</sup>*ð Þ� *<sup>k</sup>* <sup>Δ</sup>*<sup>ξ</sup>*,*<sup>σ</sup>*

*<sup>ξ</sup>*,*<sup>σ</sup>*ð Þþ *<sup>k</sup>* <sup>Δ</sup>*<sup>ξ</sup>*,*<sup>σ</sup>*

<sup>F</sup> *are Fermi energy and wave number, respectively. For MoS2, its dispersions*

0 ∣

<sup>0</sup> ∣

*e<sup>i</sup>θ<sup>k</sup>*

(2)

electric field to create various types of functional electronic devices.

*<sup>ξ</sup>*¼1,*σ*¼þ<sup>1</sup>ð Þ*<sup>k</sup>*

" #

*<sup>ξ</sup>*¼1,*σ*¼�<sup>1</sup>ð Þ*<sup>k</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *γ* 2*ε γ <sup>ξ</sup>*,*<sup>σ</sup>*ð Þ*k*

**2.2 Molybdenum disulfide and transition-metal dichalcogenides**

Ψ*γ*

Ψ*γ*

r

� � and Δ*<sup>ξ</sup>*,*<sup>σ</sup>*

revealed a buckling distances � 0*:*640*:*74 Å [31, 32].

takes the form [25]

*[(c) and (d)], where* Eð Þ <sup>0</sup>

**Figure 2.**

*Nanoplasmonics*

**182**

Ψ*γ* ð Þ¼ *k*

<sup>F</sup> *and* <sup>k</sup>ð Þ <sup>0</sup>

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

where *<sup>θ</sup><sup>k</sup>* <sup>¼</sup> tan �<sup>1</sup> *ky=kx*

in turn, related to its *zero-temperature counterpart*, Π0ð Þ *q*, *ω* j*EF* , by an integral convolution with respect to different Fermi energies [34], given by

$$\Pi\_T(q,\,\boldsymbol{\alpha}\,\,\vert\,\boldsymbol{\mu}(\boldsymbol{T})) = \frac{1}{2k\_B T} \int\_0^\infty d\eta \,\frac{\Pi\_0(q,\,\boldsymbol{\alpha}\,\,\vert\,\boldsymbol{\eta})}{1 + \cosh\left[ (\mu - \eta)/k\_B T \right]},\tag{5}$$

For accessible temperatures, the energy dispersions *ε<sup>ν</sup>*

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

numerator of Eq. (6).

Im *σ*ð Þ <sup>0</sup>

Re *σ*ð Þ <sup>0</sup>

*<sup>O</sup> ω*j*EF*, Δ*<sup>β</sup>* � � h i <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>

*<sup>O</sup> ω*j*EF*, Δ*<sup>β</sup>* � � h i <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>

limit we obtain from Eq. (10)

*<sup>O</sup> <sup>ω</sup>*j*μT*, <sup>Δ</sup>*<sup>β</sup>* <sup>¼</sup> <sup>0</sup> � �<sup>≃</sup> *<sup>e</sup>*<sup>2</sup>

*σ*ð Þ *<sup>T</sup>*

*μ<sup>T</sup>* ≈ *E*<sup>2</sup>

**185**

weakly on *EF*.

functions and their overlap factors are all temperature independent. As a result, the polarization function is expected to be modified by the same integral transformation, or a convolution, as each of the Fermi-Dirac distribution function in the

We first look at intrinsic plasmons with *EF* ¼ 0 at *T* ¼ 0. In this case, *μ*ð Þ *T* also remains at the Dirac point for any temperature *T*. As *T* increases to *kBT* ≫ *EF* >0, on the other hand, Π0ð Þ *q*, *ω*j*EF* for gapless graphene gives rise to a plasmon disper-

> <sup>2</sup> lim *q*!0

each of our considered 2D materials, regardless of their band structure, as given by Eq. (47) for *T* ¼ 0. This conclusion holds true even for finite *T* and makes the optical conductivity independent of *q*, and therefore the *q* ! 0 limit in Eq. (9) becomes finite.

*<sup>ħ</sup><sup>ω</sup>* <sup>1</sup> � <sup>Δ</sup>*<sup>β</sup>*

X *β*¼�1

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 *ħω*<2*EF*. However, if *T* > 0, such state-blocking effect will not exist [37–40] due to

Furthermore, for gapless (Δ*<sup>β</sup>* ¼ 0) but doped (*EF* > 0) graphene in the high-*T*

1 2

*ħω* <sup>16</sup>*kBT* <sup>1</sup> � <sup>1</sup>

2 ln 2*kBT*

where we have used the high-*T* limit [17] for the chemical potential

for *μ*ð Þ *T* as a function of *T* so as to gain the explicit *T* dependence of optical

� �<sup>2</sup> ( " #

*EF*

1 þ

<sup>1</sup> � tanh *<sup>ε</sup>*

3

*<sup>π</sup>ħ<sup>ω</sup>* <sup>1</sup> <sup>þ</sup> 2 ln 2 *EF*

� �<sup>2</sup> ( " #

*<sup>F</sup>=*4 ln 2*kBT* � �. In either case above, we have to present analytical expression

*<sup>O</sup> ω*j*μT*, Δ*<sup>β</sup>*

� � through [19]

Π*<sup>T</sup> q*,*ω*j*μT*, Δ*<sup>β</sup>* � �

þ 1 þ

2Δ*<sup>β</sup> ħω* � �<sup>2</sup> " #*:*

> *γ <sup>β</sup>*ð Þ� *k μ<sup>T</sup>* 2*kBT*

*ħω* 4*kBT*

� � � � *:* (11)

4 ln 2*kBT* � �<sup>4</sup> " #),

*<sup>O</sup> <sup>ω</sup>*j*μT*, <sup>Δ</sup>*<sup>β</sup>* <sup>¼</sup> <sup>0</sup> � � h i depends

sion *<sup>ω</sup><sup>p</sup>* <sup>≃</sup>*qT* in the long-wave limit and the damping rate is � *<sup>q</sup>*3*=*2*<sup>=</sup>* ffiffiffi

the plasmon mode becomes well defined [6] for *q* <16ϵ0ϵ*<sup>r</sup> kBT=πe*2. Additionally, finite-*T* polarization function Π*<sup>T</sup> q*,*ω*j*μT*, Δ*<sup>β</sup>*

� � <sup>¼</sup> *<sup>i</sup>ω<sup>e</sup>*

where we introduce the notation *μ<sup>T</sup>* � *μ*ð Þ *T* , Π*<sup>T</sup> q*, *ω*j*μT*, Δ*<sup>β</sup>*

4*EF*

directly related to its optical conductivity *σ*ð Þ *<sup>T</sup>*

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

*σ*ð Þ *<sup>T</sup>*

From Eq. (9), we find explicitly that

4*πħ*

Θ *EF* � *ε*

*γ <sup>β</sup>*ð Þ*k* � � )

*ħ*

þ *i*

conductivity. From Eq. (12) we conclude that Im *σ*ð Þ *<sup>T</sup>*

X *β*¼�1

<sup>4</sup>*<sup>ħ</sup>* <sup>Θ</sup>ð Þ *<sup>ħ</sup><sup>ω</sup>* � <sup>2</sup>*EF*

*<sup>O</sup> ω*j*μT*, Δ*<sup>β</sup>*

*<sup>γ</sup>* ð Þ*k* , corresponding wave

*T*

� � of a 2D material is

*<sup>q</sup>*<sup>2</sup> , (9)

� � � *<sup>q</sup>*<sup>2</sup> as *<sup>q</sup>* ! 0 for

2Δ*<sup>β</sup> ħω*

� �<sup>2</sup> " # ln <sup>2</sup>*EF* � *<sup>ħ</sup><sup>ω</sup>*

� � � �

<sup>p</sup> . As a result,

2*EF* þ *ħω*

� � � �

(10)

(12)

) ,

where the integration variable *η* stands for the electron Fermi energy at *T* ¼ 0. This equation is derived for electron doping with *η* ¼ *EF* > 0. We note that, in order to evaluate this integral, one needs to know in advance how the chemical potential *μ*ð Þ *T* varies with temperature *T*. Such a unique *T* dependence reflects a specific selection of a convolution path for a particular material band structure, which we will discuss in Section 4.

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 *g* ¼ *gvgs* ¼ 4 for graphene with either finite or zero bandgap. We begin with the expression of the partial polarization function with two inequivalent doubly-degenerate pairs of subbands labeled by a composite index *ν*

$$\begin{split} \Pi\_{0}^{(\nu)}(q,\boldsymbol{\varrho}\mid E\_{F}) &= \frac{1}{4\pi^{2}} \Bigg\{ d^{2}\mathbf{k} \sum\_{\boldsymbol{\gamma},\boldsymbol{\tau}'=\pm 1} \Bigg\{ \mathbbm{1} + \gamma\boldsymbol{\gamma}' \frac{\mathbf{k}\cdot(\mathbf{k}+\boldsymbol{\varrho}\mid +\boldsymbol{\Delta}\_{\boldsymbol{\nu}}^{2})}{\varepsilon\_{\boldsymbol{\gamma}}^{\nu}(\mathbf{k})\varepsilon\_{\boldsymbol{\gamma}'}^{\nu}(|\mathbf{k}+\boldsymbol{\varrho}\mid)} \Bigg\} \\ &\times \frac{\Theta\_{0}\left[E\_{F}-\boldsymbol{\varepsilon}\_{\boldsymbol{\gamma}}^{\nu}(\mathbf{k})\right]-\Theta\_{0}\left[E\_{F}-\boldsymbol{\varepsilon}\_{\boldsymbol{\gamma}'}^{\nu}(|\mathbf{k}+\boldsymbol{\varrho}\mid)\right]}{\hbar(\boldsymbol{\alpha}+i\mathbf{0}^{+})+\boldsymbol{\varepsilon}\_{\boldsymbol{\gamma}}^{\nu}(\mathbf{k})-\boldsymbol{\varepsilon}\_{\boldsymbol{\gamma}'}^{\nu}(|\mathbf{k}+\boldsymbol{\varrho}\mid)}, \end{split} \tag{6}$$

where Θ0ð Þ *x* stands for a unit-step function, and *γ* ¼ �1 stands for the electron or hole state with energy dispersions above or below the Dirac point. Moreover, the index *ν*, which equals to *σξ* ¼ �1 for buckled honeycomb lattices or molybdenum disulfide, specifies two different pairs of degenerate subbands from Eq. (1) or Eq. (3).

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

$$\Pi\_0(q, a|E\_F) = \sum\_{\nu=\pm 1} \Pi\_0^{(\nu)}(q, a|E\_F). \tag{7}$$

If the dispersions of low-energy subbands do not depend on the valley or spin indices *ξ* and *σ*, summation in Eq. (7) simply gives rise to a factor of two, as we have obtained for graphene.

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 *nF ε<sup>ν</sup> <sup>γ</sup>* ð Þ� *k μ*ð Þ *T* h i. It changes to the unit-step functions <sup>Θ</sup><sup>0</sup> *EF* � *εν <sup>γ</sup>* ð Þ*k* h i at *<sup>T</sup>* <sup>¼</sup> 0, as used in Eq. (6). As the temperature *T* increases from zero, the distribution function in Eq. (6) evolves into [35, 36]

$$\begin{split} m\_{F} \left[ \boldsymbol{\varepsilon}\_{\gamma}^{\boldsymbol{\nu}}(\boldsymbol{k}) - \boldsymbol{\mu}(\boldsymbol{T}) \right] &= \frac{\mathbf{1}}{2} \Big[ \mathbf{1} - \tanh \left( \frac{\boldsymbol{\varepsilon}\_{\gamma}^{\boldsymbol{\nu}}(\boldsymbol{k}) - \boldsymbol{\mu}(\boldsymbol{T})}{2k\_{B}T} \right) \Big] \\ &= \int\_{0}^{\infty} d\eta \, \frac{\Theta\_{0} \left[ \boldsymbol{\mu}(\boldsymbol{T}) - \boldsymbol{\varepsilon}\_{\gamma}^{\boldsymbol{\nu}}(\boldsymbol{k}) \right]}{4k\_{B}T \cosh^{2} \{ [\boldsymbol{\mu}(\boldsymbol{T}) - \boldsymbol{\eta}]/2k\_{B}T \}}. \end{split} \tag{8}$$

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

For accessible temperatures, the energy dispersions *ε<sup>ν</sup> <sup>γ</sup>* ð Þ*k* , corresponding wave functions and their overlap factors are all temperature independent. As a result, the polarization function is expected to be modified by the same integral transformation, or a convolution, as each of the Fermi-Dirac distribution function in the numerator of Eq. (6).

We first look at intrinsic plasmons with *EF* ¼ 0 at *T* ¼ 0. In this case, *μ*ð Þ *T* also remains at the Dirac point for any temperature *T*. As *T* increases to *kBT* ≫ *EF* >0, on the other hand, Π0ð Þ *q*, *ω*j*EF* for gapless graphene gives rise to a plasmon dispersion *<sup>ω</sup><sup>p</sup>* <sup>≃</sup>*qT* in the long-wave limit and the damping rate is � *<sup>q</sup>*3*=*2*<sup>=</sup>* ffiffiffi *T* <sup>p</sup> . As a result, the plasmon mode becomes well defined [6] for *q* <16ϵ0ϵ*<sup>r</sup> kBT=πe*2.

Additionally, finite-*T* polarization function Π*<sup>T</sup> q*,*ω*j*μT*, Δ*<sup>β</sup>* � � of a 2D material is directly related to its optical conductivity *σ*ð Þ *<sup>T</sup> <sup>O</sup> ω*j*μT*, Δ*<sup>β</sup>* � � through [19]

$$\sigma\_O^{(T)}\left(o|\mu\_T,\Delta\_\beta\right) = i o e^2 \lim\_{q \to 0} \frac{\Pi\_T\left(q,o|\mu\_T,\Delta\_\beta\right)}{q^2},\tag{9}$$

where we introduce the notation *μ<sup>T</sup>* � *μ*ð Þ *T* , Π*<sup>T</sup> q*, *ω*j*μT*, Δ*<sup>β</sup>* � � � *<sup>q</sup>*<sup>2</sup> as *<sup>q</sup>* ! 0 for each of our considered 2D materials, regardless of their band structure, as given by Eq. (47) for *T* ¼ 0. This conclusion holds true even for finite *T* and makes the optical conductivity independent of *q*, and therefore the *q* ! 0 limit in Eq. (9) becomes finite.

From Eq. (9), we find explicitly that

$$\begin{split} \mathrm{Im}\left[\sigma\_{O}^{(0)}\left(\boldsymbol{\mu}\mid\boldsymbol{E}\_{F},\Delta\_{\beta}\right)\right] &= \frac{e^{2}}{4\pi\hbar} \sum\_{\beta=\pm 1} \left\{ \frac{4E\_{F}}{\hbar\alpha} \left[1 - \left(\frac{\Delta\_{\beta}}{E\_{F}}\right)^{2} \right] + \left[1 + \left(\frac{2\Delta\_{\beta}}{\hbar\alpha}\right)^{2} \right] \ln \left|\frac{2E\_{F} - \hbar\alpha}{2E\_{F} + \hbar\alpha}\right| \right\}, \\ \mathrm{Re}\left[\sigma\_{O}^{(0)}\left(\boldsymbol{\mu}\mid\boldsymbol{E}\_{F},\Delta\_{\beta}\right)\right] &= \frac{e^{2}}{4\hbar} \Theta(\hbar\alpha - 2E\_{F}) \sum\_{\beta=\pm 1} \left[1 + \left(\frac{2\Delta\_{\beta}}{\hbar\alpha}\right)^{2} \right]. \end{split} \tag{10}$$

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 *ħω*<2*EF*. However, if *T* > 0, such state-blocking effect will not exist [37–40] due to

$$\Theta\left(E\_F - \varepsilon\_\beta^\prime(\mathbb{k})\right) \Rightarrow \frac{1}{2}\left\{\mathbf{1} - \tanh\left[\frac{\varepsilon\_\beta^\prime(\mathbb{k}) - \mu\_T}{2k\_BT}\right]\right\}.\tag{11}$$

Furthermore, for gapless (Δ*<sup>β</sup>* ¼ 0) but doped (*EF* > 0) graphene in the high-*T* limit we obtain from Eq. (10)

$$\begin{split} \sigma\_{O}^{(T)}\left(w \mid \mu\_{T}, \Delta\_{\beta} = 0\right) &\simeq \frac{e^{2}}{\hbar} \left\{ \frac{\hbar \alpha}{16k\_{B}T} \left[ 1 - \frac{1}{3} \left( \frac{\hbar \alpha}{4k\_{B}T} \right)^{2} \right] \\ &\quad + i \frac{2 \ln 2k\_{B}T}{\pi \hbar \alpha} \left[ 1 + 2 \ln 2 \left( \frac{E\_{F}}{4 \ln 2k\_{B}T} \right)^{4} \right] \right\}, \end{split} \tag{12}$$

where we have used the high-*T* limit [17] for the chemical potential *μ<sup>T</sup>* ≈ *E*<sup>2</sup> *<sup>F</sup>=*4 ln 2*kBT* � �. In either case above, we have to present analytical expression for *μ*ð Þ *T* as a function of *T* so as to gain the explicit *T* dependence of optical conductivity. From Eq. (12) we conclude that Im *σ*ð Þ *<sup>T</sup> <sup>O</sup> <sup>ω</sup>*j*μT*, <sup>Δ</sup>*<sup>β</sup>* <sup>¼</sup> <sup>0</sup> � � h i depends weakly on *EF*.

in turn, related to its *zero-temperature counterpart*, Π0ð Þ *q*, *ω* j*EF* , by an integral

∞ð

Π0ð Þ *q*, *ω* j *η*

<sup>1</sup> <sup>þ</sup> *γγ*<sup>0</sup> *<sup>k</sup>* � *<sup>k</sup>* <sup>þ</sup> *<sup>q</sup>* � � <sup>þ</sup> <sup>Δ</sup><sup>2</sup>

( )

*εν <sup>γ</sup>* ð Þ*<sup>k</sup> <sup>ε</sup><sup>ν</sup>*

� <sup>Θ</sup><sup>0</sup> *EF* � *εν*

*<sup>γ</sup>* ð Þ� *<sup>k</sup> <sup>ε</sup><sup>ν</sup>*

*ν*

*<sup>γ</sup>*<sup>0</sup> <sup>j</sup>*<sup>k</sup>* <sup>þ</sup> *<sup>q</sup>* <sup>j</sup> � � , (6)

. It changes to the

*<sup>γ</sup>*<sup>0</sup> <sup>j</sup>*<sup>k</sup>* <sup>þ</sup> *<sup>q</sup>* <sup>j</sup> � �

<sup>0</sup> ð Þ *q*, *ω*j*EF :* (7)

*<sup>γ</sup>* ð Þ� *k μ*ð Þ *T* h i

*<sup>γ</sup>* ð Þ� *k μ*ð Þ *T* 2*kBT*

� � � �

<sup>Θ</sup><sup>0</sup> *<sup>μ</sup>*ð Þ� *<sup>T</sup> εν*

at *T* ¼ 0, as used in Eq. (6). As the temperature

*<sup>γ</sup>* ð Þ*k* h i

f g ½ � *<sup>μ</sup>*ð Þ� *<sup>T</sup> <sup>η</sup> <sup>=</sup>*2*kBT :* (8)

*<sup>γ</sup>*<sup>0</sup> <sup>j</sup> *<sup>k</sup>* <sup>þ</sup> *<sup>q</sup>* <sup>j</sup> � � h i

<sup>1</sup> <sup>þ</sup> cosh ½ � ð Þ *<sup>μ</sup>* � *<sup>η</sup> <sup>=</sup>kBT* , (5)

0 *dη*

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 *g* ¼ *gvgs* ¼ 4 for graphene with either finite or zero bandgap. We begin with the expression of the partial polarization function with two inequivalent doubly-degenerate pairs of

<sup>Θ</sup><sup>0</sup> *EF* � *<sup>ε</sup><sup>ν</sup>*

*<sup>γ</sup>* ð Þ*k* h i

*ħ ω* þ *i*0<sup>þ</sup> ð Þþ *εν*

where Θ0ð Þ *x* stands for a unit-step function, and *γ* ¼ �1 stands for the electron or hole state with energy dispersions above or below the Dirac point. Moreover, the index *ν*, which equals to *σξ* ¼ �1 for buckled honeycomb lattices or molybdenum disulfide, specifies two different pairs of degenerate subbands from Eq. (1) or Eq. (3). Finally, the full polarization function at zero temperature is obtained as

> X *ν*¼�1

If the dispersions of low-energy subbands do not depend on the valley or spin indices *ξ* and *σ*, summation in Eq. (7) simply gives rise to a factor of two, as we have

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

*T* increases from zero, the distribution function in Eq. (6) evolves into [35, 36]

<sup>1</sup> � tanh *<sup>ε</sup><sup>ν</sup>*

4*kBT* cosh <sup>2</sup>

Πð Þ*<sup>ν</sup>*

where the integration variable *η* stands for the electron Fermi energy at *T* ¼ 0. This equation is derived for electron doping with *η* ¼ *EF* > 0. We note that, in order to evaluate this integral, one needs to know in advance how the chemical potential *μ*ð Þ *T* varies with temperature *T*. Such a unique *T* dependence reflects a specific selection of a convolution path for a particular material band structure, which we

convolution with respect to different Fermi energies [34], given by

2*kBT*

<sup>Π</sup>*T*ð Þ¼ *<sup>q</sup>*,*<sup>ω</sup>* <sup>j</sup> *<sup>μ</sup>*ð Þ *<sup>T</sup>* <sup>1</sup>

will discuss in Section 4.

*Nanoplasmonics*

Πð Þ*<sup>ν</sup>*

obtained for graphene.

unit-step functions <sup>Θ</sup><sup>0</sup> *EF* � *εν*

*nF ε<sup>ν</sup>*

**184**

subbands labeled by a composite index *ν*

<sup>0</sup> ð Þ¼ *q*, *ω* j *EF*

1 4*π*<sup>2</sup> ð *d*2 *k* X *γ*, *γ*<sup>0</sup> ¼�1

�

Π0ð Þ¼ *q*,*ω*j*EF*

Eq. (6) is the Fermi-Dirac distribution function *nF ε<sup>ν</sup>*

*<sup>γ</sup>* ð Þ� *k μ*ð Þ *T* h i

*<sup>γ</sup>* ð Þ*k* h i

> ¼ 1 2

¼ ð ∞

0 *dη*

On the other hand, for gapped (Δ*<sup>β</sup>* ¼ Δ0) but undoped (*EF* ¼ 0) graphene at high *T* (*kBT* ≫ Δ<sup>0</sup> and *ħω*), we get its optical conductivity [41]

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

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

where Li2ð Þ *x* is a polylogarithm or dilogarithm function, defined mathematically

ð*z*

0

Interestingly, the right-hand side of Eq. (18) contains terms corresponding to

as well as a well-known analytical expression for *μ*0ð Þ *T* of 2D electron gas with

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

Numerical results for *μ*ð Þ *T* of silicene are presented in **Figure 3**. In all cases, *μ*ð Þ *T* decreases with increasing *T* from zero. However, it is very important to notice that

*Temperature dependence of the chemical potential μ*ð Þ T *for silicene with two inequivalent energy subbands with various bandgaps and a fixed doping density* <sup>n</sup> <sup>¼</sup> <sup>1</sup> � <sup>10</sup><sup>11</sup> *cm*�<sup>2</sup>*. Panel (a) highlights the situation close to* T ¼ 0*, while panel (b) shows the whole temperature range. Here,* E<sup>0</sup> *is the Fermi energy of graphene.*

*<sup>μ</sup>*0ð Þ¼ *<sup>T</sup> kBT* log 1 <sup>þ</sup> exp *<sup>π</sup>ħ*<sup>2</sup>

�Li2 � exp

*dt* ln 1ð Þ � *<sup>t</sup> t*

*γ*Li2 � exp

*γ μ*ð Þ� *T* Δ*<sup>i</sup> kBT*

� �*:* (19)

� � � � , (20)

*:* (21)

*γ μ*ð Þ *T kBT*

*n*0 *m*<sup>∗</sup> *kBT*

! " #

� � ���, (18)

� � � � �

*γ μ*ð Þ� *T* Δ*<sup>i</sup> kBT*

*v*2 *F*.

density required to occupy the upper subband of silicene is *nc* <sup>¼</sup> <sup>2</sup>Δ*SO*Δ*z=πħ*<sup>2</sup>

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

*γ* X *i*¼ <, >

Li2ð Þ¼� *z*

both pristine and gapless graphene, using which we find from Ref. [17].

*<sup>F</sup>* ¼ �<sup>X</sup> *γ*¼�1

*kBT* ln 1 <sup>þ</sup> exp

for *μ*ð Þ *T* , that is

by

**Figure 3.**

**187**

*ħvF kBT* � �<sup>2</sup>

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

*<sup>n</sup>* <sup>¼</sup> <sup>1</sup> *π* X *γ*¼�1

1 <sup>2</sup>ð Þ *kBT* <sup>2</sup> *<sup>E</sup>*<sup>2</sup>

Schrödinger-based electron dynamics

solution near specific temperature assigned.

þ Δ*i*

$$\begin{aligned} \mathrm{Re}\left[\sigma\_O^{(T)}(\boldsymbol{\omega}|\boldsymbol{\mu}\_T, \boldsymbol{\Delta}\_0)\right]\_{E\_F=0} &= \frac{e^2}{16\hbar} \left(\frac{\hbar\boldsymbol{\alpha}}{k\_B T}\right) \left(1 - \frac{\boldsymbol{\Delta}\_0}{\hbar\boldsymbol{\alpha}}\right), \\\\ \mathrm{Im}\left[\sigma\_O^{(T)}(\boldsymbol{\omega}|\boldsymbol{\mu}\_T, \boldsymbol{\Delta}\_0)\right]\_{E\_F=0} &= \frac{4e^2}{\pi\hbar} \left(\frac{k\_B T}{\hbar\boldsymbol{\alpha}}\right) \left\{2\ln 2 - \left(\frac{\boldsymbol{\Delta}\_0}{k\_B T}\right)^2 \left[\mathbb{C}\_0 - \ln\left(\frac{\boldsymbol{\Delta}\_0}{2k\_B T}\right)\right]\right\}, \end{aligned} \tag{13}$$

where the constant <sup>0</sup> ≃0*:*79 appears due to a finite bandgap. [36].

### **4. Chemical potential at finite temperatures**

As we have seen from Section 3, we need know *μ*ð Þ *T* as a function of *T* explicitly so as to gain *T* dependence of polarization function, plasmon, transport and optical conductivities, or any other quantities related to collective behaviors of 2D materials at finite temperatures [42].

The density of states (DOS), which plays an important tool in calculating electron (or hole) Fermi energy *EF* and chemical potential *μ*ð Þ *T* , is defined as

$$\rho\_d(\mathbb{E}) = \sum\_{\mathbf{y} = \pm 1} \sum\_{\xi, \sigma = \pm 1} \int \frac{d^2 \mathbf{k}}{(2\pi)^2} \,\delta(\mathbb{E} - \varepsilon\_{\xi, \sigma}^{\mathbf{y}}(\mathbf{k})),\tag{14}$$

where *δ*ð Þ *x* is Dirac delta function. Using Eq. (14), we immediately obtain a piece-wise linear function for silicene [43]

$$\rho\_d(\mathbb{E}) = \frac{1}{\pi} \sum\_{\chi=\pm 1} \frac{\chi \mathbb{E}}{\hbar^2 \nu\_F^2} \sum\_{i=<, >} \Theta\_0 \left( \frac{\mathbb{E}}{\chi} - \Delta\_i \right). \tag{15}$$

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 Θ0ð Þ jj�Δ<sup>&</sup>lt; and Θ0ð Þ jj�Δ<sup>&</sup>gt; .

Finally, the chemical potential *μ*ð Þ *T* can be calculated using the *conservation of the difference of electron and hole densities*, [17] *ne*ð Þ *T* and *nh*ð Þ *T* , for all temperatures including *T* ¼ 0, i.e.,

$$n = n\_e(T) - n\_h(T) = \int\_0^\bullet d\mathbb{E} \rho\_d(\mathbb{E}) f\_{\gamma=1}(\mathbb{E}, T) - \int\_{-\infty}^0 d\mathbb{E} \rho\_d(\mathbb{E}) \left[ \mathbf{1} - f\_{\gamma=1}(\mathbb{E}, \mathbb{T}) \right], \tag{16}$$

where *<sup>f</sup> <sup>γ</sup>*¼<sup>1</sup>ð Þ¼ , *<sup>T</sup>* f g <sup>1</sup> <sup>þ</sup> exp ½ � ð Þ � *<sup>μ</sup>*ð Þ *<sup>T</sup> <sup>=</sup>kBT* �<sup>1</sup> is the Fermi function for electrons in thermal equilibrium. The hole distribution function is just *<sup>f</sup> <sup>γ</sup>*¼�<sup>1</sup>ð Þ¼ <0, *<sup>T</sup>* <sup>1</sup> � *<sup>f</sup> <sup>γ</sup>*¼<sup>1</sup>ð Þ , *<sup>T</sup>* .

At *T* ¼ 0, it is straightforward to get the Fermi energy *EF* from Eq. (16) for silicene

$$E\_F^2 - \frac{1}{2} \left(\Delta\_{>}^2 + \Delta\_{<}^2\right) = \left(\hbar v\_F\right)^2 \pi n. \tag{17}$$

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

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 *nc* <sup>¼</sup> <sup>2</sup>Δ*SO*Δ*z=πħ*<sup>2</sup> *v*2 *F*.

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 *μ*ð Þ *T* , that is

$$\begin{split} \left(\frac{\hbar \nu\_{F}}{k\_{B}T}\right)^{2} & n = \frac{1}{\pi} \sum\_{\gamma=\pm 1} \gamma \sum\_{i=<\cdot,>} \left\{-\text{Li}\_{2}\left(-\exp\left[\frac{\gamma\mu(T)-\Delta\_{i}}{k\_{B}T}\right]\right) \right. \\ & \left. + \frac{\Delta\_{i}}{k\_{B}T} \ln\left(1+\exp\left[\frac{\gamma\mu(T)-\Delta\_{i}}{k\_{B}T}\right]\right) \right\}, \end{split} \tag{18}$$

where Li2ð Þ *x* is a polylogarithm or dilogarithm function, defined mathematically by

$$\text{Li}\_2(z) = -\int\_0^z dt \left[ \frac{\ln \left( 1 - t \right)}{t} \right]. \tag{19}$$

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

$$\frac{1}{2(k\_B T)^2} E\_F^2 = -\sum\_{\gamma=\pm 1} \gamma \operatorname{Li}\_2 \left\{ -\exp\left[\frac{\gamma \mu(T)}{k\_B T}\right] \right\},\tag{20}$$

as well as a well-known analytical expression for *μ*0ð Þ *T* of 2D electron gas with Schrödinger-based electron dynamics

$$
\mu\_0(T) = k\_B T \log \left( 1 + \exp \left[ \frac{\pi \hbar^2 n\_0}{m^\* \, k\_B T} \right] \right). \tag{21}
$$

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 *μ*ð Þ *T* of silicene are presented in **Figure 3**. In all cases, *μ*ð Þ *T* decreases with increasing *T* from zero. However, it is very important to notice that

**Figure 3.**

On the other hand, for gapped (Δ*<sup>β</sup>* ¼ Δ0) but undoped (*EF* ¼ 0) graphene at

<sup>1</sup> � <sup>Δ</sup><sup>0</sup> *ħω* � �

2 ln 2 � <sup>Δ</sup><sup>0</sup>

,

*kBT* � �<sup>2</sup>

*γ <sup>ξ</sup>*,*<sup>σ</sup>*ð Þ*k* � �

<sup>0</sup> � ln <sup>Δ</sup><sup>0</sup>

( ) � � � �

2*kBT*

, (14)

*:* (15)

,

(13)

*ħω kBT* � �

*kBT ħω* � �

where the constant <sup>0</sup> ≃0*:*79 appears due to a finite bandgap. [36].

conductivities, or any other quantities related to collective behaviors of 2D

X *ξ*, *σ*¼�1

The density of states (DOS), which plays an important tool in calculating electron (or hole) Fermi energy *EF* and chemical potential *μ*ð Þ *T* , is defined as

> ð *d*<sup>2</sup> *k* ð Þ <sup>2</sup>*<sup>π</sup>* <sup>2</sup> *<sup>δ</sup>* � *<sup>ε</sup>*

where *δ*ð Þ *x* is Dirac delta function. Using Eq. (14), we immediately obtain a

X *i*¼ <, >

This result is equivalent to the DOS of graphene except that there are no states

Finally, the chemical potential *μ*ð Þ *T* can be calculated using the *conservation of the*

ð 0

*<sup>d</sup>ρd*ð Þ <sup>1</sup> � *<sup>f</sup> <sup>γ</sup>*¼<sup>1</sup>ð Þ ,

h i

<sup>2</sup> *π n:* (17)

, (16)

�∞

*difference of electron and hole densities*, [17] *ne*ð Þ *T* and *nh*ð Þ *T* , for all temperatures

where *<sup>f</sup> <sup>γ</sup>*¼<sup>1</sup>ð Þ¼ , *<sup>T</sup>* f g <sup>1</sup> <sup>þ</sup> exp ½ � ð Þ � *<sup>μ</sup>*ð Þ *<sup>T</sup> <sup>=</sup>kBT* �<sup>1</sup> is the Fermi function for

At *T* ¼ 0, it is straightforward to get the Fermi energy *EF* from Eq. (16) for

Θ<sup>0</sup> *<sup>γ</sup>* � <sup>Δ</sup>*<sup>i</sup>* � �

*γ ħ*2 *v*2 *F*

within the bandgap region, as demonstrated by two unit-step functions

*<sup>d</sup>ρd*ð Þ *<sup>f</sup> <sup>γ</sup>*¼<sup>1</sup>ð Þ� , *<sup>T</sup>*

electrons in thermal equilibrium. The hole distribution function is just

As we have seen from Section 3, we need know *μ*ð Þ *T* as a function of *T* explicitly so as to gain *T* dependence of polarization function, plasmon, transport and optical

high *T* (*kBT* ≫ Δ<sup>0</sup> and *ħω*), we get its optical conductivity [41]

*EF*¼<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*<sup>2</sup> 16*ħ*

*EF*¼<sup>0</sup> <sup>¼</sup> <sup>4</sup>*e*<sup>2</sup> *πħ*

**4. Chemical potential at finite temperatures**

*<sup>ρ</sup>d*ð Þ¼ <sup>X</sup>

*<sup>ρ</sup>d*ð Þ¼ <sup>1</sup>

∞ð

0

*E*2 *<sup>F</sup>* � <sup>1</sup> 2 Δ2 <sup>&</sup>gt; <sup>þ</sup> <sup>Δ</sup><sup>2</sup> < � � <sup>¼</sup> ð Þ *<sup>ħ</sup>vF*

*γ*¼�1

*π* X *γ*¼�1

materials at finite temperatures [42].

piece-wise linear function for silicene [43]

Θ0ð Þ jj�Δ<sup>&</sup>lt; and Θ0ð Þ jj�Δ<sup>&</sup>gt; .

including *T* ¼ 0, i.e.,

silicene

**186**

*n* ¼ *ne*ð Þ� *T nh*ð Þ¼ *T*

*<sup>f</sup> <sup>γ</sup>*¼�<sup>1</sup>ð Þ¼ <0, *<sup>T</sup>* <sup>1</sup> � *<sup>f</sup> <sup>γ</sup>*¼<sup>1</sup>ð Þ , *<sup>T</sup>* .

Re *σ*ð Þ *<sup>T</sup>*

*Nanoplasmonics*

Im *σ*ð Þ *<sup>T</sup>*

*<sup>O</sup> ω*j*μ<sup>T</sup>* ð Þ , Δ<sup>0</sup> h i

*<sup>O</sup> ω*j*μ<sup>T</sup>* ð Þ , Δ<sup>0</sup> h i

*Temperature dependence of the chemical potential μ*ð Þ T *for silicene with two inequivalent energy subbands with various bandgaps and a fixed doping density* <sup>n</sup> <sup>¼</sup> <sup>1</sup> � <sup>10</sup><sup>11</sup> *cm*�<sup>2</sup>*. Panel (a) highlights the situation close to* T ¼ 0*, while panel (b) shows the whole temperature range. Here,* E<sup>0</sup> *is the Fermi energy of graphene.*

*μ*ð Þ *T* never reaches zero or changes its sign in the systems with an electron–hole symmetry due to increasing contribution from holes in Eq. (16). All of displayed results in **Figure 3** depend on individual bandgaps Δ<sup>&</sup>lt; and Δ<sup>&</sup>gt; . The special case with Δ<sup>&</sup>lt; ¼ Δ<sup>&</sup>gt; corresponds to gapped graphene, for which plasmon modes at *<sup>T</sup>* <sup>¼</sup> 0 were studied in Ref. [33]. The graphene Fermi wave number is *kF* <sup>¼</sup> ffiffiffiffiffi *πn* p , irrelevant to its bandgap. The general relation between *kF* and *n* in 2D materials is given by 2ð Þ *<sup>π</sup>* <sup>2</sup> *<sup>n</sup>* <sup>¼</sup> *<sup>g</sup> <sup>π</sup>k*<sup>2</sup> *<sup>F</sup>*. The experimentally allowable electron (or hole) doping is within the range of *<sup>n</sup>* <sup>¼</sup> <sup>10</sup><sup>10</sup> � <sup>10</sup><sup>12</sup> cm�2, leading to *kF* <sup>¼</sup> <sup>10</sup><sup>6</sup> � <sup>10</sup><sup>7</sup> cm�1. For two pair of inequivalent subbands, such as in silicene or MoS2, there are two different Fermi wave numbers for these subbands. Moreover, the numerically calculated *μ*ð Þ *T* as functions of *T* for electron and hole doping are presented in **Figure 4**. In this case, however, there exists no electron-hole symmetry, and therefore the resulting *μ*ð Þ *T* can be zero and change its sign as *T* increases, in contrast to the results in **Figure 3**.

From Eq. (22), we further seek an explicit expression for DOS in the form a piecewise-linear function of energy : *ρd*ð Þ¼ *Ai* þ *Bi*. A complete set of expressions for DOS of MoS2 has been reported in Ref. [18]. Here, we merely provide and discuss these DOS expression around the lower hole subband with ≈ � Δ*=*2 � *λ*0,

<sup>0</sup> <sup>þ</sup> � <sup>Δ</sup>

h i<sup>2</sup>

*t*0*a*<sup>2</sup> 0

1 ð Þ *t*0*a*<sup>0</sup>

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

The critical doping density which is required to populate the lower hole subband

Therefore, for most experimentally accessible densities *n* ≤10<sup>13</sup> cm�2, the lower

Next, we would evaluate both sides of Eq. (16) for MoS2. As an example, we

4

þ *c* ð Þ3 <sup>0</sup> *Ee*

*<sup>F</sup>* � <sup>Δ</sup> 2

**0**

consider electron doping with density *ne* > 0. The electron Fermi energy *Ee*

� �

From Eq. (27), we can easily find the electron Fermi energy *EF* >0 as

*i* ¼ 1 < � Δ*=*2 � *λ*<sup>0</sup> �1 þ1 0*:*0174 �0*:*169 *i* ¼ 2 ∣ þ Δ*=*2∣<*λ*<sup>0</sup> �1 �1 0*:*043 �0*:*308 *i* ¼ 3 > Δ*=*2 þ1 0*:*078 þ0*:*179

*Linearized density of states (DOS) ρd*ð Þ¼ *Ai* þ *Bi of MoS2 for all three energy regions. Here, the DOS*

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

**Range index Energy range** *γ ν Ai***[1***= t***0***a***<sup>2</sup>**

ð Þ *a*0*t*<sup>0</sup>

<sup>2</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>0</sup> � � � � *<sup>c</sup>*

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>β</sup>* � *<sup>α</sup>* ð Þ <sup>Δ</sup> � *νλ*<sup>0</sup>

*E*0 ð Þ *ħvF* 2 ,

<sup>1</sup> for *i* ¼ 1, 2, 3 can be deduced from the calculated

1 ð Þ *ħvF*

<sup>2</sup> *:* (25)

*:* (26)

� �*:* (27)

� �**]** *Bi***[1***=*ð Þ *<sup>t</sup>***0***a***<sup>0</sup> <sup>2</sup>**

*<sup>F</sup>* is

**]**

Δ � *νλ*<sup>0</sup>

*β=*ð Þ 4*me* ð Þ Δ � *νλ*<sup>0</sup>

¼ 15*:*17

<sup>2</sup> ¼ �2*:*077

<sup>2</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>0</sup> � <sup>10</sup><sup>14</sup> cm�<sup>2</sup>

*=*ð Þ 4*me* ½ � ð Þ *β* � *α* ð Þ Δ � *νλ*<sup>0</sup> n o<sup>3</sup> <sup>&</sup>lt; 0, (24)

ð Þ1 1 ,

,

*δε*

*ρd*ð Þ¼ *c*

ð Þ *a*<sup>0</sup> *t*<sup>0</sup>

ð Þ *a*<sup>0</sup> *t*<sup>0</sup>

*c* ð Þ1

ð Þ*i* <sup>0</sup> ,*c* ð Þ*i*

*nc* <sup>¼</sup> <sup>2</sup> *π*

hole subband still could not be populated at *T* ¼ 0.

*ne* <sup>¼</sup> *<sup>c</sup>* ð Þ3 1 2

determined by the following relation

*within the gap region,* �Δ*=*2 þ *λ*<sup>0</sup> << Δ*=*2*, is zero.*

<sup>1</sup> ¼ �0*:*458

*c* ð Þ1

*c* ð Þ1 <sup>0</sup> <sup>¼</sup> <sup>1</sup> 2*π* X *ν*¼�1

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

ð Þ1

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

<sup>2</sup> <sup>þ</sup> *<sup>ħ</sup>*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *<sup>ħ</sup>*<sup>2</sup>

<sup>0</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>233</sup> <sup>1</sup>

parameters *Ai* and *Bi* using a similar correspondence as in Eq. (24).

*λ*0Δ ð Þ *t*0*a*<sup>0</sup>

yielding

*c* ð Þ1 <sup>1</sup> <sup>¼</sup> <sup>1</sup> *π* X *ν*¼�1

or numerically,

introduced coefficients *c*

in MoS2 is found to be

**Table 1.**

**189**

Eq. (18) could also be applied to a wide range of 2D materials if its DOS has a linear dependence on energy . Particularly, it is valid for calculating the finite-*T* chemical potential of TMDC's with an energy dispersion presented in Eq. (3). However, we are aware that some terms in Eq. (3) for TMDC's, which might be insignificant for dispersions of other 2D materials, become essential in DOS because of very large bandgap and mass terms around *k* ¼ 0. As an estimation, for *k=k*<sup>0</sup> ≈ 5*:*0, the correction term ≃*β*Δ*k*<sup>4</sup> must be taken into account. Meanwhile, the highest accessible doping *<sup>n</sup>* <sup>¼</sup> <sup>10</sup><sup>13</sup> cm�<sup>2</sup> only gives rise to a Fermi energy *EF* � *<sup>λ</sup>*0, comparable to spin-orbit coupling.

Now, we turn to calculate *μ*ð Þ *T* as a function of *T* for MoS2 with a much more complicated band structure. After taking into account the � *<sup>k</sup>*<sup>2</sup> mass terms, we are able to write down [18]

$$\rho\_d(\mathbb{E}) = \frac{1}{2\pi\hbar^2} \sum\_{\chi,\nu=\pm 1} \left| \frac{a + \chi\beta}{4m\_\varepsilon} + \frac{\chi\left(t\_0 a\_0\right)^2}{\hbar^2 \left(\Delta - \nu\lambda\_0\right)} \right|^{-1} \Theta\_0 \left(\chi \left[\mathbb{E} - \frac{\nu\lambda\_0}{2}\right] - \frac{1}{2} (\Delta - \nu\lambda\_0)\right), \tag{22}$$

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

$$\epsilon\_{\gamma}^{\nu}(k) = \frac{1}{2} [\nu \lambda\_0 (1 - \gamma) + \gamma \Delta] + \left[ \frac{\hbar^2}{4m\_e} (a + \gamma \beta) + \frac{\gamma \left( t\_0 a\_0 \right)^2}{\Delta - \nu \lambda\_0} \right] k^2. \tag{23}$$

#### **Figure 4.**

*Temperature dependence of the chemical potential μ*ð Þ T *for molybdenum disulfide for cases of electron (a) and hole (b) doping with various doping densities. μ*ð Þ T *might change its sign in contrast to the previously considered graphene and silicene. The two insets demonstrate how the Fermi energy depends on the electron and hole doping densities, correspondingly.*

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

From Eq. (22), we further seek an explicit expression for DOS in the form a piecewise-linear function of energy : *ρd*ð Þ¼ *Ai* þ *Bi*. A complete set of expressions for DOS of MoS2 has been reported in Ref. [18]. Here, we merely provide and discuss these DOS expression around the lower hole subband with ≈ � Δ*=*2 � *λ*0, yielding

$$\rho\_d(\mathbb{E}) = c\_0^{(1)} + \left[\mathbb{E} - \left(\frac{\Delta}{2} + \lambda\_0\right)\right] c\_1^{(1)},$$

$$c\_0^{(1)} = \frac{1}{2\pi} \sum\_{\nu=\pm 1} \frac{\Delta - \nu \lambda\_0}{(a\_0 t\_0)^2 + (\beta - a)(\Delta - \nu \lambda\_0)},$$

$$c\_1^{(1)} = \frac{1}{\pi} \sum\_{\nu=\pm 1} \frac{\left[\left(a\_0 t\_0\right)^2 + \hbar^2 \beta / (4m\_\epsilon)\left(\Delta - \nu \lambda\_0\right)\right]^2 \delta \varepsilon}{\left\{\left(a\_0 t\_0\right)^2 + \hbar^2 / (4m\_\epsilon)\left[(\beta - a)(\Delta - \nu \lambda\_0)\right]\right\}^3} < 0,\tag{24}$$

or numerically,

*μ*ð Þ *T* never reaches zero or changes its sign in the systems with an electron–hole symmetry due to increasing contribution from holes in Eq. (16). All of displayed results in **Figure 3** depend on individual bandgaps Δ<sup>&</sup>lt; and Δ<sup>&</sup>gt; . The special case with Δ<sup>&</sup>lt; ¼ Δ<sup>&</sup>gt; corresponds to gapped graphene, for which plasmon modes at *<sup>T</sup>* <sup>¼</sup> 0 were studied in Ref. [33]. The graphene Fermi wave number is *kF* <sup>¼</sup> ffiffiffiffiffi

irrelevant to its bandgap. The general relation between *kF* and *n* in 2D materials is

within the range of *<sup>n</sup>* <sup>¼</sup> <sup>10</sup><sup>10</sup> � <sup>10</sup><sup>12</sup> cm�2, leading to *kF* <sup>¼</sup> <sup>10</sup><sup>6</sup> � <sup>10</sup><sup>7</sup> cm�1. For two pair of inequivalent subbands, such as in silicene or MoS2, there are two different Fermi wave numbers for these subbands. Moreover, the numerically calculated *μ*ð Þ *T* as functions of *T* for electron and hole doping are presented in **Figure 4**. In this case, however, there exists no electron-hole symmetry, and therefore the resulting *μ*ð Þ *T* can be zero and change its sign as *T* increases, in contrast to the results in

Eq. (18) could also be applied to a wide range of 2D materials if its DOS has a linear dependence on energy . Particularly, it is valid for calculating the finite-*T* chemical potential of TMDC's with an energy dispersion presented in Eq. (3). However, we are aware that some terms in Eq. (3) for TMDC's, which might be insignificant for dispersions of other 2D materials, become essential in DOS because

*k=k*<sup>0</sup> ≈ 5*:*0, the correction term ≃*β*Δ*k*<sup>4</sup> must be taken into account. Meanwhile, the highest accessible doping *<sup>n</sup>* <sup>¼</sup> <sup>10</sup><sup>13</sup> cm�<sup>2</sup> only gives rise to a Fermi energy *EF* � *<sup>λ</sup>*0,

Now, we turn to calculate *μ*ð Þ *T* as a function of *T* for MoS2 with a much more complicated band structure. After taking into account the � *<sup>k</sup>*<sup>2</sup> mass terms, we are

2

� � � � �

�1

<sup>Θ</sup><sup>0</sup> *<sup>γ</sup>* � *νλ*<sup>0</sup>

ð Þþ *<sup>α</sup>* <sup>þ</sup> *γβ <sup>γ</sup>* ð Þ *<sup>t</sup>*0*a*<sup>0</sup>

" #

2 � �

� 1 2

� �

2

*k*2

Δ � *νλ*<sup>0</sup>

ð Þ Δ � *νλ*<sup>0</sup>

*:* (23)

,

(22)

ð Þ Δ � *νλ*<sup>0</sup>

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

4*me*

*Temperature dependence of the chemical potential μ*ð Þ T *for molybdenum disulfide for cases of electron (a) and hole (b) doping with various doping densities. μ*ð Þ T *might change its sign in contrast to the previously considered graphene and silicene. The two insets demonstrate how the Fermi energy depends on the electron*

of very large bandgap and mass terms around *k* ¼ 0. As an estimation, for

<sup>þ</sup> *<sup>γ</sup>* ð Þ *<sup>t</sup>*0*a*<sup>0</sup>

*ħ*2

<sup>½</sup>*νλ*0ð Þþ <sup>1</sup> � *<sup>γ</sup> <sup>γ</sup>*Δ� þ *<sup>ħ</sup>*<sup>2</sup>

*<sup>F</sup>*. The experimentally allowable electron (or hole) doping is

given by 2ð Þ *<sup>π</sup>* <sup>2</sup>

*Nanoplasmonics*

**Figure 3**.

*<sup>n</sup>* <sup>¼</sup> *<sup>g</sup> <sup>π</sup>k*<sup>2</sup>

comparable to spin-orbit coupling.

able to write down [18]

2*πħ*<sup>2</sup>

*and hole doping densities, correspondingly.*

X *γ*, *ν*¼�1

� � � � �

*α* þ *γβ* 4*me*

*<sup>ρ</sup>d*ð Þ¼ <sup>1</sup>

*εν <sup>γ</sup>* ð Þ¼ *<sup>k</sup>* <sup>1</sup> 2

**Figure 4.**

**188**

*πn* p ,

$$c\_0^{(1)} = 0.233 \frac{1}{t\_0 a\_0^2} = 15.17 \frac{E\_0}{\left(\hbar v\_F\right)^2},$$

$$c\_1^{(1)} = -0.458 \frac{1}{\left(t\_0 a\_0\right)^2} = -2.077 \frac{1}{\left(\hbar v\_F\right)^2}.\tag{25}$$

The calculated numerical results for DOS in all regions are listed in **Table 1**. All introduced coefficients *c* ð Þ*i* <sup>0</sup> ,*c* ð Þ*i* <sup>1</sup> for *i* ¼ 1, 2, 3 can be deduced from the calculated parameters *Ai* and *Bi* using a similar correspondence as in Eq. (24).

The critical doping density which is required to populate the lower hole subband in MoS2 is found to be

$$m\_c = \frac{2}{\pi} \frac{\lambda\_0 \Delta}{\left(t\_0 a\_0\right)^2} = 1.0 \times 10^{14} \text{ cm}^{-2}.\tag{26}$$

Therefore, for most experimentally accessible densities *n* ≤10<sup>13</sup> cm�2, the lower hole subband still could not be populated at *T* ¼ 0.

Next, we would evaluate both sides of Eq. (16) for MoS2. As an example, we consider electron doping with density *ne* > 0. The electron Fermi energy *Ee <sup>F</sup>* is determined by the following relation

$$m\_e = \frac{c\_1^{(3)}}{2} \left[ \left( E\_F^{\epsilon} \right)^2 - \frac{\Delta^2}{4} \right] + c\_0^{(3)} \left( E\_F^{\epsilon} - \frac{\Delta}{2} \right). \tag{27}$$

From Eq. (27), we can easily find the electron Fermi energy *EF* >0 as


**Table 1.**

*Linearized density of states (DOS) ρd*ð Þ¼ *Ai* þ *Bi of MoS2 for all three energy regions. Here, the DOS within the gap region,* �Δ*=*2 þ *λ*<sup>0</sup> << Δ*=*2*, is zero.*

*Nanoplasmonics*

$$E\_F^\epsilon = \frac{1}{c\_1^{(3)}} \left[ -c\_0^{(3)} + \sqrt{\left(c\_0^{(3)} + c\_1^{(3)}\frac{\Delta}{2}\right)^2 + 2n\_\epsilon c\_1^{(3)}} \right]. \tag{28}$$

Using these self-defined functions and their notations, we finally arrive at the

*μ*ð Þ� *T kBT*

� � � � �<sup>1</sup>

*μ*ð Þ� *T kBT*

(36)

(37)

� � � � �<sup>1</sup>

*d c* ð Þ2 <sup>0</sup> � *c* ð Þ2 <sup>1</sup> � � <sup>1</sup> <sup>þ</sup> exp

*d δc* ð Þ1 <sup>0</sup> � *δc* ð Þ1 <sup>1</sup> � � <sup>1</sup> <sup>þ</sup> exp

*<sup>h</sup>* ð Þ Δ, *λ*<sup>0</sup> j*T* ,

ð Þ 0 2

<sup>2</sup> ð Þ *kBT* <sup>2</sup> <sup>R</sup>1ð Þ *<sup>T</sup>*, �½ � *<sup>μ</sup>*ð Þþ *<sup>T</sup>* <sup>Δ</sup>*=*<sup>2</sup> � *<sup>λ</sup>*<sup>0</sup> ,

ð Þ 0 1

Here, both I*e*ð Þ Δj*T* in Eq. (34) and I*h*ð Þ Δ, *λ*<sup>0</sup> j*T* in Eq. (36) comprise a finitetemperature part for the right-hand side of Eq. (16). Its left-hand side has already been given by Eq. (27). From these results, it is clear that there exists no symmetry between the electron and hole states at either zero or finite *T*. Finally, *μ*ð Þ *T* of TMDC's could be computed from a transcendental equation in Eq. (18), similarly to

By using the calculated *μ*ð Þ *T* , the plasmon dispersions and their Landau damping, determined from Eqs. (4) and (5), are displayed in **Figure 5** for silicene at different *T*. Comparison of panels (a) and (b) indicates that the *T* dependence of plasmon damping is not uniform even on a fixed convolution path *μ*ð Þ *T* . The doping

*Particle-hole modes and plasmon branch for extrinsic (doped) silicene layer at a finite temperature. Panels (a) and (b) show two comparative graphs for* Im½Π0ð � q,*ω*j*μ*ð ÞÞ T *at zero and finite* T*, respectively, while plot (c)*

*presents the finite-*T *plasmon branch with μ*ð Þ T *calculated from Eq. (16).*

R1ð Þ *T*, �½ � *μ*ð Þþ *T* Δ*=*2 þ *λ*<sup>0</sup> *:*

R0ð Þ *T*, �½ � *μ*ð Þþ *T* Δ*=*2 � *λ*<sup>0</sup> ,

R0ð Þ *T*, �½ � *μ*ð Þþ *T* Δ*=*2 þ *λ*<sup>0</sup> ,

<sup>ð</sup> �Δ*=*2þ*λ*<sup>0</sup>

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

�∞

<sup>ð</sup> �Δ*=*2�*λ*<sup>0</sup>

�∞

Ið Þ*j*

þ

� <sup>X</sup> 4

*j*¼1

*<sup>i</sup>* for *i* ¼ 0, 1, and

� �

<sup>2</sup> � *<sup>λ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>c</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup><sup>c</sup>*

� �

"*hole term*" I*h*ð Þ Δ, *λ*0j*T* in Eq. (16)

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

*<sup>d</sup>ρd*ð Þ <sup>j</sup><sup>j</sup> <sup>1</sup> � *<sup>f</sup>* <sup>1</sup>ð Þ , � � <sup>¼</sup>

ð 0

�∞

where *δc*

Ið Þ1

Ið Þ2

Ið Þ3

Ið Þ 4

**Figure 5.**

**191**

ð Þ1 *<sup>i</sup>* ¼ *c* ð Þ1 *<sup>i</sup>* � *c* ð Þ2

*<sup>h</sup>* ð Þ¼ <sup>Δ</sup>, *<sup>λ</sup>*<sup>0</sup> <sup>j</sup>*<sup>T</sup> kBT* <sup>Δ</sup>

*<sup>h</sup>* ð Þ¼ <sup>Δ</sup>, *<sup>λ</sup>*<sup>0</sup> <sup>j</sup>*<sup>T</sup> kBT* <sup>Δ</sup>

ð Þ1

ð Þ1 <sup>1</sup> ð Þ *kBT* <sup>2</sup>

*<sup>h</sup>* ð Þ¼ Δ, *λ*<sup>0</sup> j*T c*

*<sup>h</sup>* ð Þ¼ Δ, *λ*<sup>0</sup> j*T δc*

finding *μ*ð Þ *T* for silicene.

In a similar way, for hole doping with density *nh* and the Fermi energy *Eh F* located between two hole subbands (region 2), we find

$$|E\_F^h| = \frac{1}{c\_1^{(2)}} \left\{ \sqrt{\left[c\_0^{(2)} - c\_1^{(2)}\left(\frac{\Delta}{2} - \lambda\_0\right)\right]^2 - 2n\_h c\_1^{(2)}} - c\_0^{(2)} \right\},\tag{29}$$

where *c* ð Þ2 <sup>1</sup> < 0 and < � Δ*=*2 þ *λ*0. From Eq. (29) we easily find the doping density

$$m\_h = \left(\frac{\Delta}{2} + E\_F^h - \lambda\_0\right) \left[c\_0^{(2)} + \frac{c\_1^{(2)}}{2} \left(\frac{\Delta}{2} - \left(E\_F^h + \lambda\_0\right)\right)\right].\tag{30}$$

The right-hand side of Eq. (16) for TMDC's could be expressed as a combination of electron and hole contributions I*e*ð Þ�I Δj*T <sup>h</sup>*ð Þ Δ, *λ*0j*T* . Here, we will introduce two self-defined functions

$$\begin{aligned} \mathcal{A}\_0(\mathbb{E}, T) &= \left[ \mathbf{1} + \exp\left( \frac{\mathbb{E} - \mu(T)}{k\_B T} \right) \right]^{-1}, \\\mathcal{A}\_1(\mathbb{E}, T) &= \mathbb{E} \mathcal{A}\_0(\mathbb{E}, T) = \mathbb{E} \left[ \mathbf{1} + \exp\left( \frac{\mathbb{E} - \mu(T)}{k\_B T} \right) \right]^{-1}, \end{aligned} \tag{31}$$

so that

$$\mathcal{I}\_{\epsilon}(\Delta|T) = \sum\_{j=0}^{1} c\_{j}^{(3)} \int\_{\Delta/2}^{\infty} d\mathbb{E} \, \mathcal{A}\_{j}(\mathbb{E}, T) . \tag{32}$$

For convenience, we introduce another function R*p*ð Þ *T*,*X*

$$\mathcal{R}\_p(T, X) = \bigcap\_{0}^{\infty} d\xi \frac{\xi^p}{1 + \exp\left(\xi - X/k\_B T\right)},\tag{33}$$

where *ξ* ¼ ð Þ � Δ<sup>&</sup>lt; *=kBT*. Consequently, we are able to rewrite Eq. (32) as

$$\mathcal{I}\_{\varepsilon}(\Delta \mid T) = k\_B T \left( c\_0^{(3)} + \frac{\Delta}{2} \right) \mathcal{R}o \left( T, \mu(T) - \frac{\Delta}{2} \right) + c\_1^{(3)} \left( k\_B T \right)^2 \mathcal{R}\_1 \left( T, \mu(T) - \frac{\Delta}{2} \right), \tag{34}$$

where two terms with *p* ¼ 0, 1 are physically related to a 2D electron gas. Explicitly, Eq. (33) leads to

$$\begin{aligned} \mathcal{R}\_0(T, X) &= \ln\left(1 + \exp\left(\frac{X}{k\_B T}\right)\right), \\ \mathcal{R}\_1(T, X) &= -\text{Li}\_2\left(-\exp\left(\frac{X}{k\_B T}\right)\right), \end{aligned} \tag{35}$$

Using these self-defined functions and their notations, we finally arrive at the "*hole term*" I*h*ð Þ Δ, *λ*0j*T* in Eq. (16)

$$\begin{split} \int\_{-\infty}^{0} d\mathbb{E} \rho\_{d}(|\mathbb{E}|) \left[\mathbb{1} - f\_{1}(\mathbb{E}, \mathbb{T})\right] &= \int\_{-\infty}^{-\Delta/2 + l\alpha} d\mathbb{E} \left(c\_{0}^{(2)} - c\_{1}^{(2)}\mathbb{E}\right) \left[\mathbb{1} + \exp\left(\frac{\mu(T) - \mathbb{E}}{k\_{B}T}\right)\right]^{-1} \\ &+ \int\_{-\Delta/2 - l\alpha}^{-\Delta/2 - l\alpha} d\mathbb{E} \left(\delta c\_{0}^{(1)} - \delta c\_{1}^{(1)}\mathbb{E}\right) \left[\mathbb{1} + \exp\left(\frac{\mu(T) - \mathbb{E}}{k\_{B}T}\right)\right]^{-1} \\ &\equiv \sum\_{j=1}^{4} \mathcal{I}\_{h}^{(j)}(\Delta, \mathbb{A}\_{0}|T), \tag{36} \end{split} \tag{37}$$

where *δc* ð Þ1 *<sup>i</sup>* ¼ *c* ð Þ1 *<sup>i</sup>* � *c* ð Þ2 *<sup>i</sup>* for *i* ¼ 0, 1, and

*Ee <sup>F</sup>* <sup>¼</sup> <sup>1</sup> *c* ð Þ3 1

∣*Eh <sup>F</sup>*<sup>∣</sup> <sup>¼</sup> <sup>1</sup> *c* ð Þ2 1

where *c*

*Nanoplasmonics*

so that

ð Þ2

two self-defined functions

<sup>I</sup>*e*ð Þ¼ <sup>Δ</sup>j*<sup>T</sup> kBT c*ð Þ<sup>3</sup>

Explicitly, Eq. (33) leads to

**190**

�*c* ð Þ3 0 þ

> *c* ð Þ2 <sup>0</sup> � *c* ð Þ2 1

4

located between two hole subbands (region 2), we find

s

<sup>2</sup> <sup>þ</sup> *Eh*

A0ð Þ¼ , *T* 1 þ exp

*<sup>F</sup>* � *λ*<sup>0</sup> � �

A1ð Þ¼ , *T* A0ð Þ¼ , *T* 1 þ exp

<sup>I</sup>*e*ð Þ¼ <sup>Δ</sup>j*<sup>T</sup>* <sup>X</sup>

For convenience, we introduce another function R*p*ð Þ *T*,*X*

∞ð

0

R*p*ð Þ¼ *T*,*X*

<sup>0</sup> þ Δ 2 � � 1

*j*¼0 *c* ð Þ3 *j*

8 < :

*nh* <sup>¼</sup> <sup>Δ</sup>

2 s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ 2*ne c* ð Þ3 1

� 2*nh c* ð Þ2 1

3

� *c* ð Þ2 0

*<sup>F</sup>* þ *λ*<sup>0</sup>

9 = ;

5*:* (28)

*F*

, (29)

*:* (30)

(31)

� �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>1</sup> < 0 and < � Δ*=*2 þ *λ*0. From Eq. (29) we easily find the doping density

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

� � " # � �

,

 � *μ*ð Þ *T kBT*

,

*d*A*j*ð Þ , *T :* (32)

<sup>1</sup> ð Þ *kBT* <sup>2</sup> <sup>R</sup><sup>1</sup> *<sup>T</sup>*, *<sup>μ</sup>*ð Þ� *<sup>T</sup>* <sup>Δ</sup>

2

,

(34)

(35)

� �

<sup>1</sup> <sup>þ</sup> exp ð Þ *<sup>ξ</sup>* � *<sup>X</sup>=kBT* , (33)

� � � � �<sup>1</sup>

Δ <sup>2</sup> � *<sup>λ</sup>*<sup>0</sup>

� � � � <sup>2</sup>

*c* ð Þ2 <sup>0</sup> þ *c* ð Þ2 1 2

The right-hand side of Eq. (16) for TMDC's could be expressed as a combination of electron and hole contributions I*e*ð Þ�I Δj*T <sup>h</sup>*ð Þ Δ, *λ*0j*T* . Here, we will introduce

> � *μ*ð Þ *T kBT*

> > ∞ð

Δ*=*2

*<sup>d</sup><sup>ξ</sup> <sup>ξ</sup><sup>p</sup>*

2

þ *c* ð Þ3

> *X kBT*

*X kBT* ,

,

� � � �

� � � �

where *ξ* ¼ ð Þ � Δ<sup>&</sup>lt; *=kBT*. Consequently, we are able to rewrite Eq. (32) as

� �

where two terms with *p* ¼ 0, 1 are physically related to a 2D electron gas.

R0ð Þ¼ *T*, *X* ln 1 þ exp

R1ð Þ¼� *T*,*X* Li2 � exp

<sup>R</sup><sup>0</sup> *<sup>T</sup>*, *<sup>μ</sup>*ð Þ� *<sup>T</sup>* <sup>Δ</sup>

� � � � �<sup>1</sup>

*c* ð Þ3 <sup>0</sup> þ *c* ð Þ3 1 Δ 2

In a similar way, for hole doping with density *nh* and the Fermi energy *Eh*

$$\begin{split} \mathcal{Z}\_{h}^{(1)}(\Delta,\lambda\_{0}|T) &= k\_{B}T\Big(\frac{\Delta}{2} - \lambda\_{0} + c\_{2}^{(0)}\Big) \mathcal{R}\_{0}(T, -[\mu(T) + \Delta/2 - \lambda\_{0}]), \\ \mathcal{Z}\_{h}^{(2)}(\Delta,\lambda\_{0}|T) &= c\_{2}^{(1)}(k\_{B}T)^{2} \mathcal{R}\_{1}(T, -[\mu(T) + \Delta/2 - \lambda\_{0}]), \\ \mathcal{Z}\_{h}^{(3)}(\Delta,\lambda\_{0}|T) &= k\_{B}T\Big(\frac{\Delta}{2} + \lambda\_{0} + \delta c\_{1}^{(0)}\Big) \mathcal{R}\_{0}(T, -[\mu(T) + \Delta/2 + \lambda\_{0}]), \\ \mathcal{Z}\_{h}^{(4)}(\Delta,\lambda\_{0}|T) &= \delta c\_{1}^{(1)}(k\_{B}T)^{2} \mathcal{R}\_{1}(T, -[\mu(T) + \Delta/2 + \lambda\_{0}]). \end{split} \tag{37}$$

Here, both I*e*ð Þ Δj*T* in Eq. (34) and I*h*ð Þ Δ, *λ*<sup>0</sup> j*T* in Eq. (36) comprise a finitetemperature part for the right-hand side of Eq. (16). Its left-hand side has already been given by Eq. (27). From these results, it is clear that there exists no symmetry between the electron and hole states at either zero or finite *T*. Finally, *μ*ð Þ *T* of TMDC's could be computed from a transcendental equation in Eq. (18), similarly to finding *μ*ð Þ *T* for silicene.

By using the calculated *μ*ð Þ *T* , the plasmon dispersions and their Landau damping, determined from Eqs. (4) and (5), are displayed in **Figure 5** for silicene at different *T*. Comparison of panels (a) and (b) indicates that the *T* dependence of plasmon damping is not uniform even on a fixed convolution path *μ*ð Þ *T* . The doping

#### **Figure 5.**

*Particle-hole modes and plasmon branch for extrinsic (doped) silicene layer at a finite temperature. Panels (a) and (b) show two comparative graphs for* Im½Π0ð � q,*ω*j*μ*ð ÞÞ T *at zero and finite* T*, respectively, while plot (c) presents the finite-*T *plasmon branch with μ*ð Þ T *calculated from Eq. (16).*

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*.

^ *d*

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

functions for the valence and conduction bands take the form

<sup>0</sup> ð Þ¼ *<sup>k</sup>*j*ξ*, *<sup>ϕ</sup>* <sup>1</sup>

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

<sup>0</sup> ð Þ¼ *k* j*ξ*, *ϕ*

Ψ*<sup>γ</sup>*¼�<sup>1</sup>

Ψ*γ*¼<sup>0</sup>

of an *α*-T <sup>3</sup> materials, including plasmon dispersion.

1 *π*2

�

X *γ*, *γ*<sup>0</sup> ¼0, �1 ð

and MoS2 discussed in Section 4.

Π0ð Þ¼ *q*,*ω*j*EF*

transition states is defined by [51] *<sup>ξ</sup>*

*<sup>γ</sup>*<sup>0</sup> *k*<sup>0</sup> , *λ*<sup>0</sup>

*<sup>γ</sup>* ð Þ *<sup>k</sup>*, *<sup>λ</sup>*<sup>0</sup> and the final <sup>Ψ</sup>*<sup>ξ</sup>*

is calculated as

valley and spin index.

Ψ*ξ*

**193**

*<sup>ξ</sup>* ð Þ¼ *<sup>k</sup> <sup>ħ</sup>vF*

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

ffiffi 2 p

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

0 *k<sup>ξ</sup>*

0 *k<sup>ξ</sup>*

*kξ*

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

> ffiffi 2 p

> > 2 6 4

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

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

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*

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

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

j*ϕ*, *λ*<sup>0</sup>

� � states with a momentum transfer *<sup>q</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>0</sup>

*<sup>γ</sup>*,*γ*<sup>0</sup> *k*, *k*<sup>0</sup>

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

<sup>Θ</sup><sup>0</sup> *EF* � *εγ* ð Þ*<sup>k</sup>* � � � <sup>Θ</sup><sup>0</sup> *EF* � *εγ*<sup>0</sup> <sup>j</sup>*<sup>k</sup>* <sup>þ</sup> *<sup>q</sup>*<sup>j</sup> � � � �

*<sup>ħ</sup> <sup>ω</sup>* <sup>þ</sup> *<sup>i</sup>*0<sup>þ</sup> ð Þþ *εγ* ð Þ� *<sup>k</sup> εγ*<sup>0</sup> <sup>j</sup> *<sup>k</sup>* <sup>þ</sup> *<sup>q</sup>* <sup>j</sup> � � *:* (42)

� � with respect to the initial

� *k* , i.e.,

� 0

<sup>þ</sup> 0

*ξ* cos *ϕ* e�*iξθ***<sup>k</sup>** *γ ξ* sin *ϕ* e*iξθ***<sup>k</sup>**

*ξ* sin *ϕ* e�*iξθ***<sup>k</sup>** 0 �*<sup>ξ</sup>* cos *<sup>ϕ</sup>* <sup>e</sup>*<sup>i</sup>ξθ***<sup>k</sup>** 3 7 7

3 7

�

3 7 7

5*:* (39)

*γ*

5, (40)

<sup>5</sup>*:* (41)

<sup>0</sup>ð Þ¼ *k γ ħvFk*

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