Open access peer-reviewed chapter

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

Written By

Andrii Iurov, Godfrey Gumbs and Danhong Huang

Reviewed: December 16th, 2019 Published: January 22nd, 2020

DOI: 10.5772/intechopen.90870

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The purpose of this chapter is to review some important, recent theoretical discoveries regarding the effect of temperature on the property of plasmons. These include their dispersion relations and Landau damping rates, and the explicit dependence of plasmon frequency on chemical potential at finite temperatures for a diverse group of recently discovered Dirac-cone materials. These novel materials cover gapped graphene, buckled howycomb lattices (such as silicene and germanene), molybdenum disulfide and other transition-metal dichalcogenides, especially the newest dice and α-T3 materials. The most crucial part of this review is a set of implicit analytical expressions about the exact chemical potential for each of considered materials, which greatly affects the plasmon dispersions and a lot of many-body quantum-statistical properties. We have also obtained the nonlocal plasmon modes of graphene which are further Coulomb-coupled to the surface of a thick conducting substrate, while the whole system is kept at a finite temperature. An especially rich physics feature is found for α-T3 materials, where each of the above-mentioned properties depends on both the hopping parameter α and temperature as well.


  • finite temperature effects
  • plasmon dispersion
  • 2D materials

1. Introduction

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

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

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

Figure 1.

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

Plasmons, or self-sustained collective excitations of interacting electrons in such low-dimensional materials, are especially important, since they serve as the basics for a number of novel devices and their applications [9, 10] in almost all fields of modern science, emerging nanofabrication and nanotechnology. Propagation and detection of plasmonic excitation in hybrid nanoscale devices can convert to or modify existing electromagnetic field or radiation [11, 12, 13, 14]. Localized surface plasmons are particularly of special interest considering their interactions with other plasmon modes in closely-located optoelectronic device as well as with imposed electromagnetic radiation [15].

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

At the same time, the damping rate, or broadening of the frequency, of such thermal plasmons varies as 1/T, which means that we are dealing with a long-lived plasmon with low or even nearly zero Landau damping. Both plasmon frequency and the corresponding damping rate at finite temperature could be adjusted by electron doping, and could also be determined for an intrinsic material, where the chemical potential at T=0K is located exactly at the Dirac point, while for zero temperature, intrinsic plasmons in graphene do not exist.

In this chapter, we will consider thermal behavior of plasmons, their dispersions and damping rates. By equipping with this information, it is possible to predict in advance the thermal properties of an electronic device designed for a particular temperature range. In spite of a number of reported theoretical studies on this subject [16, 17, 18], there is still a gap on demonstrating experimentally these unique thermal collective features of 2D materials. Therefore, our review can serve as an incentive to address this issue.


2. Novel two-dimensional materials beyond graphene

All the novel 2D materials considered here could be effectively assigned to an individual category based on their existing (or broken) symmetries and degeneracy in their low-energy band structure. We started with graphene having a bandgap Δ0 and single-particle energy bands ε±k=±ħvFk2+Δ02, which are symmetric with respect to the Dirac point. Moreover, there is a total spin-valley degeneracy g=gsgv=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 doubly-degenerate. The electron-hole symmetry is broken for molybdenum disulfide and other transition-metal dichalcogenides (TMDC’s). For these situations, even though there exists a single electron-hole index γ=±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.

Figure 2.

Energy dispersions and density of states (DOS) ρdE of silicene [(a) and (b)] and molybdenum disulfide MoS2 [(c) and (d)], where EF0 and kF0 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.

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=0.5×106 m/s and are given by


where γ=±1 labels symmetric electron and hole states. Here, two bandgaps [19, 20] Δ<=ΔSOΔz and Δ>=ΔSO+Δz are attributed to an intrinsic spin-orbit gap ΔSO=0.53.5 meV [21, 22, 23, 24] and a tunable asymmetry bandgap Δz proportional to applied electric field Ez. The band structure, however, depends only on one composite index ν=σξ, a product of spin σ and valley ξ index. At Ez=0, two gaps become the same. As Ez0, Δ< and Δ> change in opposite ways, and electrons stay in a topological insulator (TI) state. Additionally, Δ< decreases with Ez until reaching zero, corresponding to a new valley-spin polarized metal. On the other hand, if Ez further increases, both Δ< and Δ> will be enhanced, leading to a 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<,>, 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]


where θk=tan1ky/kx and Δ0ξ,σ=ξσΔzΔSO.

Germanene, another representative of buckled honeycomb lattices [26, 27, 28, 29, 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 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=±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.,


where Δ=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 λ0=0.042Δ, t0a0=4.95×1029 J m is a Dirac-cone term, where t0=0.884Δ is the electron hopping parameter while a0=1.843 Å is the lattice constant. The t0a0 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=σξ. There are also other less important but still non-negligible k2 mass terms with α=2.21=5.140β, and me represents the mass of a free electron.

In practical, we will neglect the k4 terms, t1a02 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., εγνk=E0νk+γΔ0νk2+t0a0k2, where Δ0νk=ħ2k2β/4me+Δ/2νλ0/2 is a k-dependent “gap term” and the band shift is E0νk=ħ2k2α/4me+νλ0/2. 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=1), corresponding to ν=±1, will be lifted and two subband will be separated by λ079.8 meV. However, this is not the case for two corresponding electron states (γ=1). Consequently, the electron-hole asymmetry exists even at k=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 ϵTqω, [19, 33] given by


where vq=2παr/qe2/2ϵ0ϵrq is the 2D Fourier-transformed Coulomb potential, αr=e2/4πϵ0ϵr, and ϵr 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, ΠTqωμT, which is, in turn, related to its zero-temperature counterpart, Π0qωEF, by an integral convolution with respect to different Fermi energies [34], given by


where the integration variable η stands for the electron Fermi energy at T=0. This equation is derived for electron doping with η=EF>0. We note that, in order to evaluate this integral, one needs to know in advance how the chemical potential μT 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=gvgs=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


where Θ0x stands for a unit-step function, and γ=±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=±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


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 nFεγνkμT. It changes to the unit-step functions Θ0EFεγνk at T=0, as used in Eq. (6). As the temperature T increases from zero, the distribution function in Eq. (6) evolves into [35, 36]


For accessible temperatures, the energy dispersions εγνk, 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=0 at T=0. In this case, μT also remains at the Dirac point for any temperature T. As T increases to kBTEF>0, on the other hand, Π0qωEF for gapless graphene gives rise to a plasmon dispersion ωpqT in the long-wave limit and the damping rate is q3/2/T. As a result, the plasmon mode becomes well defined [6] for q<16ϵ0ϵrkBT/πe2.

Additionally, finite-T polarization function ΠTqωμTΔβ of a 2D material is directly related to its optical conductivity σOTωμTΔβ through [19]


where we introduce the notation μTμT, ΠTqωμTΔβq2 as q0 for each of our considered 2D materials, regardless of their band structure, as given by Eq. (47) for T=0. This conclusion holds true even for finite T and makes the optical conductivity independent of q, and therefore the q0 limit in Eq. (9) becomes finite.

From Eq. (9), we find explicitly that


Here, the state-blocking effect due to Pauli exclusion principle directly results in the diminishing of the real part of the optical conductivity at zero temperature for ħω<2EF. However, if T>0, such state-blocking effect will not exist [37, 38, 39, 40] due to


Furthermore, for gapless (Δβ=0) but doped (EF>0) graphene in the high-T limit we obtain from Eq. (10)


where we have used the high-T limit [17] for the chemical potential μTEF2/4ln2kBT. In either case above, we have to present analytical expression for μT as a function of T so as to gain the explicit T dependence of optical conductivity. From Eq. (12) we conclude that ImσOTωμTΔβ=0 depends weakly on EF.

On the other hand, for gapped (Δβ=Δ0) but undoped (EF=0) graphene at high T (kBTΔ0 and ħω), we get its optical conductivity [41]


where the constant C00.79 appears due to a finite bandgap. [36].


4. Chemical potential at finite temperatures

As we have seen from Section 3, we need know μT 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 μT, is defined as


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


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 Θ0EΔ< and Θ0EΔ>.

Finally, the chemical potential μT can be calculated using the conservation of the difference of electron and hole densities, [17] neT and nhT, for all temperatures including T=0, i.e.,


where fγ=1ET=1+expEμT/kBT1 is the Fermi function for electrons in thermal equilibrium. The hole distribution function is just fγ=1E<0T=1fγ=1ET.

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


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=2ΔSOΔz/πħ2vF2.

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 μT, that is


where Li2x is a polylogarithm or dilogarithm function, defined mathematically by


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


as well as a well-known analytical expression for μ0T of 2D electron gas with Schrödinger-based electron dynamics


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 μT of silicene are presented in Figure 3. In all cases, μT decreases with increasing T from zero. However, it is very important to notice that μT 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< and Δ>. The special case with Δ<=Δ> corresponds to gapped graphene, for which plasmon modes at T=0 were studied in Ref. [33]. The graphene Fermi wave number is kF=πn, irrelevant to its bandgap. The general relation between kF and n in 2D materials is given by 2π2n=gπkF2. The experimentally allowable electron (or hole) doping is within the range of n=10101012 cm2, leading to kF=106107 cm1. 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 μT 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 μT can be zero and change its sign as T increases, in contrast to the results in Figure 3.

Figure 3.

Temperature dependence of the chemical potential μT for silicene with two inequivalent energy subbands with various bandgaps and a fixed doping density n=1×1011cm2. Panel (a) highlights the situation close to T=0, while panel (b) shows the whole temperature range. Here, E0 is the Fermi energy of graphene.

Figure 4.

Temperature dependence of the chemical potential μT for molybdenum disulfide for cases of electron (a) and hole (b) doping with various doping densities. μT 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.

Eq. (18) could also be applied to a wide range of 2D materials if its DOS has a linear dependence on energy E. 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=0. As an estimation, for k/k05.0, the correction term βΔk4 must be taken into account. Meanwhile, the highest accessible doping n=1013 cm2 only gives rise to a Fermi energy EFλ0, comparable to spin-orbit coupling.

Now, we turn to calculate μT as a function of T for MoS2 with a much more complicated band structure. After taking into account the k2 mass terms, we are able to write down [18]


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


From Eq. (22), we further seek an explicit expression for DOS in the form a piecewise-linear function of energy E: ρdE=Ai+BiE. 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 EΔ/2λ0, yielding


or numerically,


The calculated numerical results for DOS in all regions are listed in Table 1. All introduced coefficients c0i,c1i for i=1,2,3 can be deduced from the calculated parameters Ai and Bi using a similar correspondence as in Eq. (24).

Range indexEnergy rangeγνAi[1/t0a02]Bi[1/t0a02]

Table 1.

Linearized density of states (DOS) ρdE=Ai+BiE of MoS2 for all three energy regions. Here, the DOS within the gap region, Δ/2+λ0<E<Δ/2, is zero.

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


Therefore, for most experimentally accessible densities n1013 cm2, the lower hole subband still could not be populated at T=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 EFe is determined by the following relation


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


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


where c12<0 and E<Δ/2+λ0. From Eq. (29) we easily find the doping density


The right-hand side of Eq. (16) for TMDC’s could be expressed as a combination of electron and hole contributions IeΔTIhΔλ0T. Here, we will introduce two self-defined functions


so that


For convenience, we introduce another function RpTX


where ξ=EΔ</kBT. Consequently, we are able to rewrite Eq. (32) as


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


Using these self-defined functions and their notations, we finally arrive at the “hole termIhΔλ0T in Eq. (16)


where δci1=ci1ci2 for i=0,1, and


Here, both IeΔT in Eq. (34) and IhΔλ0T in Eq. (36) comprise a finite-temperature 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, μT of TMDC’s could be computed from a transcendental equation in Eq. (18), similarly to finding μT for silicene.

By using the calculated μT, 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 μT. The doping 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 μT with various doping densities or Fermi energies. Furthermore, the plasmon energy in (c) is pushed up slightly by increasing doping density at finite T.

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 ImΠ0(qωμT) at zero and finite T, respectively, while plot (c) presents the finite-T plasmon branch with μT calculated from Eq. (16).


5. Dice lattice and α-T3 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 T3 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-T3 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-T3 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 ε0k0. The other two bands are equivalent to Dirac cone ε±k=±ħvFk in graphene with the same Fermi velocity vF=108 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-T3 model was initially considered only as a theoretical contraption, an interpolation between graphene and a dice lattice. As parameter α0, 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-T3 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-T3 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-T3 rests on the availability of materials with various α values, i.e., with small and large rim-hub hopping coefficients, ranging from α=0 for graphene up to α=1 for a dice lattice.

The low-energy Dirac-Weyl Hamiltonian for the α-T3 model is [45]


where k=kxky is the electron wave vector, k±ξ=ξkx±iky, ξ=±1 corresponds to two different valleys, and vF is the Fermi velocity. Here, the parameter α is related to the geometry phase ϕ=tan1α 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-T3 model. In particular, for α=1 or ϕ=π/4 we get a dice lattice with its Hamiltonian given by [50].


Three energy bands from Hamiltonian in Eq. (38) or Eq. (39) are ε0γk=γħvFk for valence (γ=1), conduction (γ=+1) and flat (γ=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


where tanθk=ky/kx. Meanwhile, for the flat band, we find


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-T3 materials, including plasmon dispersion.

Now we turn to deriving plasmon branches and their damping rates at finite T in α-T3 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-T3 model, the finite-T polarization function ΠTqωμT can be obtained by an integral transformation of its zero-temperature counterpart Π0qωEF, as presented in Eq. (5). In this case, the zero-T counterpart Π0qωEF is calculated as


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=0 so that the summation index runs over ±1 and 0 instead of two. On the other hand, the overall expression for Π0qωEF 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] Oγ,γξkkϕλ0 with respect to the initial Ψγξkλ0 and the final Ψγξkλ0 states with a momentum transfer q=kk, i.e.,


where βk,k=θkθk is the scattering angle between two electronic states and k=k2+q2+2kqcosβk,k. Moreover, we find from Eq. (43) [52]


for an arbitrary value of ϕ or α. It is easy to verify the known results 1+cosβk,k/2 for graphene and 1+cosβk,k2/4 for a dice lattice as two limiting cases of our general result in Eq. (44) as α0 or α1, 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 Oγ,γξkk+qϕλ0 persists if α-T3 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 ImΠO(qωμT)0 is presented in Figure 6, where we find plasmon branch will be completely free from damping within the region determined by ħω/E01 and q/k01, independent of geometry phase ϕ. On the other hand, another region with ħωħvFq (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).

Figure 6.

Particle-hole modes, determined by non-zero ImΠO(qωμT) within the q-ω plane, for an α-T3 layer with ϕ=π/10 (in (a), (c)) and ϕ=π/7 in (b). Panels (a) and (b) correspond to T=0, while plot (c) is for kBT=1.0EF.

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=0 in (a), (b) and kBT=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 high-frequency region beyond ħω=EF. To visualize a full plasmon dispersion clearly, we also include damped counterpart in Figure 7(b) and (d) at T=0 and kBT=EF, respectively, where a significant enhancement of plasmon energy appears for large q values, moving upwards from the diagonal.

Figure 7.

Plasmon branches for an isolated α-T3 layer with ϕ=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=0, while kBT=1.0EF for right panels (c) and (d).


6. Plasmons in α-T3 layer coupled to conducting substrate

In the last part of THIS CHAPTER, WE WOULD LIKE TO FOCUS ON finite-T plasmons in a so-called nanoscale-hybrid structure consisting of a 2D layer, such as, graphene, silicene or a dice lattice, which is Coulomb-coupled to a large, conducting material. Physically, the Coulomb coupling between the 2D layer and the conductor results in a strong hybridization of graphene plasmon and localized surface-plasmon modes. This structure, which is referred to as an open system, could be realized experimentally or even by a device fabrication.

Our schematics for an open system is shown in Figure 8. The dynamical screening to the Coulomb interaction between electrons in a 2D layer and in metallic substrate is taken into account by a nonlocal and dynamical inverse dielectric function Krrω, as demonstrated in Refs. [53, 54]. This nonlocal inverse dielectric function is connected to a dielectric function ϵrrω in Eq. (4) by

Figure 8.

Schematics for a silicene-based open system and numerical results for the two plasmon branches and their damping in this system with ΔSO=0.3E0 and 0.1E0, where E0=ħvFπn0=54.6 meV.


and the resonances in Krrω reveal the nonlocal hybridized plasmon modes supported by both 2D layer and the conducting surface as a single quantum system.

By using the Drude model for metallic substrate, the dielectric function can be written as ϵBω=1Ωp2/ω2, where Ωp=n0e2/ϵ0ϵbm is the bulk-plasma frequency for the conductor, n0 electron concentration and m 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 calculate plasmon dispersions in this 2D open system. For this, the plasmon dispersions are obtained from the zeros of the so-called dispersion factorSCqω, instead of the dielectric function in Eq. (4). SCqω for this open system is given by [25, 54, 57]


where a is the separation between the 2D layer and the conducting surface. Most important, we should emphasize that the second term in Eq. (46) does not have a full analogy with polarization function of an isolated layer, and the resulting plasmon dispersions in open system represents a hybridized plasmon mode with the environment. Therefore, these plasmon dispersions are expected to be sensitive to Coulomb coupling to electrons in the conducting substrate through a factor exp2qa in Eq. (46), similarly to what we have found for coupled double graphene layers [16]. The strong Coulomb coupling leads to a linear dispersion of plasmon in this open system [54, 58], which is in contrast with well-known q dependence in all 2D materials.

As a special example, let us consider a silicene 2D layer with two bandgaps Δ<,> and an electron doping density n. We start with seeking for a non-interacting polarization function in the long-wave limit qkF<,> for doping density n and assume a high-enough n to keep the Fermi level EF=ħvFkFβ2+Δβ2>Δ> above the large bandgap. Under this assumption, we get


where kFβ=2πnβ are two different Fermi wave numbers associated with a single Fermi energy EF, and nβ is the electron density for each subband satisfying n=n<+n>.

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


where for convenience we introduced a coefficient


We notice from Eq. (48) that ωpqq, disregarding of the energy bandgaps Δ<,> or doping density n. On the other hand, the Fermi energy EF for silicene is given by Eq. (17).

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


which leads to a bi-quadratic equation


Eq. (51) gives rise to two solutions


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


In Eq. (53), both plasmon branches contain a linear q term, and ωp,+q approaches a constant as q0, i.e., an optical mode for plasmons. Two independent bandgaps, ΔSO and Δz, together with doping density n, play a crucial role on shaping the plasmon dispersions, as well as the particle-hole mode damping regions. The outer boundaries of a particle-hole mode region specify an area within the q-ω plane in which the plasmon modes become damping free and are solely determined by Δ<, while the group velocity of plasmon mode depends on both Δ< and Δ>. Since each bandgap could be experimentally tuned by applying a perpendicular electric field, we acquire a full control of both plasmon dispersions and their damping-free regions at the same time.

Numerical results for thermal plasmons in open system are presented in Figure 9. Similarly to what we have found for graphene and silicene, there are two plasmon branches, both of which depend linearly on q with a finite slope as qk0. The acoustic-plasmon branch starts from the origin, while the optical-plasmon branch from Ωp/2. The dispersion of each branch also varies with parameter α (or ϕ=tan1α), which is observed for the upper branch, as shown in (a) and (c) of Figure 9. In addition, we see a much smaller slope for the lower plasmon branch in Figure 9(b) and (d) due to enhanced Coulomb coupling with a reduced separation a. The finite-temperature upper plasmon branches in Figure 9(e) and (f) are shifted up greatly, as it is expected to be true for all finite-temperature plasmons, which is further accompanied by enhanced damping below the diagonal as seen in Figure 6(c). Meanwhile, the lower plasmon branch seems much less affected by finite temperatures, as demonstrated by both upper and lower rows of Figure 9 for different separations, except for enhanced damping in Figure 9(f) below the surface-plasmon energy ħΩp/2.

Figure 9.

Nonlocal hybridized plasmon dispersions for α-T3 layer coupled to a closely-located surface of a semi-infinite conductor. Panels (a)–(d) are for T=0, while plots (e) and (f) for kBT=E0. All the upper-row plots correspond to the separation a=1.0kF1, and the lower-row ones to a=0.5kF1. Additionally, middle-column plots, (b) and (d), correspond to ϕ=π/7, and all other columns to ϕ=π/10.


7. Summary and remarks

In conclusion, we have developed a general theory for finite-temperature polarization function, plasmon dispersions and their damping for all known innovation 2D Dirac-cone materials with various types of symmetries and bandgaps. We have also derived a set of explicit transcendental equations determining the chemical potential as a function of temperature, which serves as a key part in calculating finite-temperature polarization function through the so-called thermal convolution path. The selection of a particular path with a specific μT could be employed for studying the temperature dependence of plasmon modes in each of the considered 2D materials. The fact that a chemical potential keeps its sign is true only for materials with symmetric energy bands of electrons and holes, but can cross the zero line for TMDC’s with asymmetric electron and hole bands.

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

We have generalized our developed theory for 2D materials further to most recently proposed α-T3 lattices, in which the characteristic parameter α is the ratio of hub-rim to hub-hub hopping coefficients and can vary from 0 to 1 continuously corresponding to different material properties. For α-T3 materials, we have obtained the hybrid plasmon modes for different α values at both zero and finite temperatures and demonstrated that the resulting hybridized plasmon dispersions could be tuned sensitively by geometry phase, temperature, and separation between α-T3 layer and conducting surface. Such tunability has a profound influence on performance of α-T3 material based quantum electronic devices.



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


Conflict of interest

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


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Written By

Andrii Iurov, Godfrey Gumbs and Danhong Huang

Reviewed: December 16th, 2019 Published: January 22nd, 2020