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Panels (a) and (b) present density plots of the imaginary and real parts of Π(0)(q,ω), respectively, at T = 0 as functions of q/kF and ħω/EF(0) for ΔSO=0.7 EF(0) and Δz=0.1 EF(0) (both spin-dependent subbands are doped and have energy gaps). Plots (c) and (d) show similar results at T=1.5TF(0)≡1.5EF(0)/kB. Panel (e) displays the real part of δΠ(0)(q,iω) due to doping, as defined in equation (11), as a function of q at T = 0 for various iω values, while plot (f) exhibits the real part of δΠ(0)(q,iω) as a function of T for q=0.2 kF (kF=EF(0)/ħvF) and various ω values. All the results are scaled by the graphene density of states DoS(EF(0))=2EF(0)/πħ2vF2 at the Fermi energy.
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We have numerically calculated electron exchange and correlation energies and dynamical polarization functions for recently discovered silicene, germanene and other buckled honeycomb lattices at various temperatures. We have compared the dependence of these energies on the chemical potential, field-induced gap and temperature and we have concluded...
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Citations
... On the other hand, the considerable spin-orbit coupling along the buckling structure creates another finite bandgap (2Δ SO ) in the energy structure of silicene. Hence, although silicene is viewed as a graphene-like material, the controllable nonzero bandgap in the energy structure of silicene, compared to a zero bandgap in graphene, makes this structure become an excellent candidate for transistor applications, in comparison with graphene [10][11][12][13][14][15][16][17][18][19]. ...
... In the case of silicene, the electric and plasmonic collective properties have been intensively investigated with analytical detailed calculations. The collective excitations in single-layer silicene (SLS) and double-layer silicene, with and without temperature effects, have been investigated in detail [10,12,13,[15][16][17][56][57][58][59][60][61][62][63][64][65]. To our knowledge, no calculations on plasmonic properties of multilayer silicene structures have been done although these might bring some useful and interesting features. ...
... In low-energy approximation, the Hamiltonian of silicene with a spin-orbit gap 2Δ SO under the effects of an out-of-plane electric field is expressed as [10,12,16,[56][57][58]60] ...
We investigate the plasmon properties in N-layer silicene systems consisting of N, up to 6, parallel single-layer silicene under the application of an out-of-plane electric field, taking into account the spin-orbit coupling within the random-phase approximation. Numerical calculations demonstrate that N undamped plasmon modes, including one in-phase optical and (N-1) out-of-phase acoustic modes, continue mainly outside the single-particle excitation area of the system. As the number of layers increases, the frequencies of plasmonic collective excitations increase and can become much larger than that in single layer silicene, more significant for high-frequency modes. The optical (acoustic) plasmon mode(s) noticeably (slightly) decreases with the increase in the bandgap and weakly depends on the number of layers. We observe that the phase transition of the system weakly affects the plasmon properties, and as the bandgap caused by the spin-orbit coupling equal that caused by the external electric field, the plasmonic collective excitations and their broadening function in multilayer silicene behave similarly to those in multilayer gapless graphene structures. Our investigations show that plasmon curves in the system move toward that in single layer silicene as the separation increases, and the impacts of this factor can be raised by a large number of layers in the system. Finally, we find that the imbalanced carrier density between silicene layers significantly decreases plasmon frequencies, depending on the number of layers.
... In recent years, by using the low-energy Dirac Hamiltonian [4], we have extensively explored varieties of dynamical properties of electrons in graphene and other two-dimensional materials, including Landau quantization [18,[31][32][33][34][35], many-body optical effects [36][37][38][39][40][41], band and tunneling transports [42][43][44][45][46][47][48][49][50], etc. In this paper, we particularly focus on the application of computed electronic states and band structures from a tightbinding model to the calculations of Coulomb and impurity scatterings of electrons in graphene on the basis of a many-body theory [3,4], where the former and latter determine the lineshape [1] of an absorption peak and the transport mobility [44], respectively. ...
In this paper, by introducing a generalized quantum-kinetic model which is coupled self-consistently with Maxwell and Boltzmann transport equations, we elucidate the significance of using input from first-principles band-structure computations for an accurate description of ultra-fast dephasing and scattering dynamics of electrons in graphene. In particular, we start with the tight-binding model (TBM) for calculating band structures of solid covalent crystals based on localized Wannier orbital functions, where the employed hopping integrals in TBM have been parameterized for various covalent bonds. After that, the general TBM formalism has been applied to graphene to obtain both band structures and wave functions of electrons beyond the regime of effective low-energy theory. As a specific example, these calculated eigenvalues and eigen vectors have been further utilized to compute the Bloch-function form factors and intrinsic Coulomb diagonal-dephasing rates for induced optical coherence of electron-hole pairs in spectral and polarization functions, as well as the energy-relaxation time from extrinsic impurity scattering of electrons for non-equilibrium occupation in band transport.
... The reason behind why plasmons in these materials appeal to us is their very wide frequency coverage up to the terahertz limit within the Coulomb-coupled system comprising a two-dimensional layer and a semi-infinite conductor, or the so-called open systems [54]. Specifically, a fair amount of work has been done on the finite-temperature behavior of plasmons [55][56][57][58][59][60][61][62][63][64], their damping [65], and plasmon-polaritons [66] since each of these properties could be varied independently with temperature, and then the undamped plasmon branch could extend over an even higher energy range. ...
An extensive analytical and numerical investigation has been carried out to examine the role played by many-body effects on various $\alpha$-$\mathcal{T}_3$ materials under an off-resonance optical dressing field. Additionally, we explore its dependence on the hopping parameter $\alpha$ as well as the electron-light coupling strength $\lambda_0$. The obtained dressed states due to mutual interaction between Dirac electrons and incident light are shown to demonstrate rather different electronic and optical properties in comparison with those in the absence of incident light. Specifically, various collective transport and optical properties of these electron dressed states are discussed in detail and compared for both single- and double layer $\alpha$-$\mathcal{T}_3$ lattices. All of these novel properties are due to the presence of a middle flat band and the interband transitions between it and an upper conduction band. Also, coupled plasmon dispersions for interacting double layer $\alpha$-$\mathcal{T}_3$ lattices are calculated, revealing a lower acoustic-like plasmon branch with tunable group velocity determined by both the layer separation and Fermi energy of each layer. Finally, a many-body theory is presented within the random-phase approximation for calculating the optical absorbance of doped multi-layered $\alpha$-$\mathcal{T}_3$ lattices in a linearly-polarized light field. We anticipate that the discoveries reported here could impact the design of the next-generation nano-optical and nano-plasmonic devices.
... As we have seen from Section 3, we need know μ T ðÞ 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]. ...
... Lurov et al performed extensive numerical calculation about exchange and correlation energies in silicene at various temperatures and doping concentrations [216]. They found that these energies increase with concentration of doping and decrease with temperature. ...
Spintronics addresses the spin degree of freedom of electrons as an analog of the traditional information technology approach toward the charge degree of freedom. Two-dimensional (2D) materials have provided a new platform for spintronics since the successful isolation of graphene in 2004. In this review, we discuss the recent progress of chromium tri-iodide (CrI3), phosphorene, silicene, germanene, stanene, animonene, and borophene in spintronics. Among these 2D materials, CrI3 has intrinsic magnetism, while the others are single-element 2D materials (Xenes) without intrinsic magnetism. A review is presented of works based on different research methods, including experimental, theoretical, and simulation studies. Various morphologies of 2D materials are included, such as monolayers, bilayers, nanoribbons, heterojunctions, and nanoflakes of different shapes. The effect of atomic doping, vacancies, line defects, strain, and external fields is also summarized. Among the properties and applications of these materials, the spin polarization, spin/valley-dependent resistance, various Hall effects, RKKY interaction, Rashba effect, and topological phase are the most widely studied topics. Over the past two years, various novel spintronics devices have been designed based on 2D materials, and numerous fundamental functions in spintronics have been realized. Thus, we also summarize the emerging novel devices in this field. Finally, we present a summary of the remaining challenges and possible future directions.
... The plasma oscillations of a single-layer silicene (SLS) have already been calculated both at zero temperature and finite temperature with and without a perpendicularly applied electric field. [39][40][41][42] It has been shown that, in an extrinsic SLS, the plasma oscillations follow a √ q behavior in the long-wavelength limit at zero temperature for all values of electric field (i.e. all different phases). ...
... Moreover, the presence of a small gap in SLGG and various combinations of two gaps in SLS leads to the splitting of the plasmon branch and the appearance of new undamped plasmon modes. 41,44 On the other hand, in the case of SLG, plasmons decay into particle-hole pairs in the interband portion of the SPE region, while for SLS both intra-and interband dampings can be occurred by tuning the external electric field. 40 The extrinsic (doped) finite-temperature plasmons of SLS have also been studied: as temperature increases from T = 0, the plasmon energy first lowers, reaching to a minimum value and then bounces back in both cases of SLS and SLGG. ...
... 56 Recently, it has been shown that the temperature-dependent plasmons of SLS, SLG and singlelayer n-doped and p-doped molybdenum disulfide can also exhibit such a behavior. 41 In Fig. 9(c), for ∆ z /∆ SO = 1, the optical branch follows a similar temperature trend as the other phases, but Fig. 10(c) shows that there is almost no gap left for the undamped plamons to reside. However, for T = 0 and low temperatures, SPE Inter boundary sits higher than the approximate interband SPE boundary. ...
We explore the temperature-dependent plasmonic modes of an n-doped double-layer silicene system which is composed of two spatially separated single layers of silicene with a distance large enough to prevent the interlayer electron tunneling. By applying an externally applied electric field, we numerically obtain the poles of the loss function within the so-called random phase approximation, to investigate the effects of temperature and geometry on the plasmon branches in three different regimes: topological insulator, valley-spin polarized metal, and band insulator. Also, we present the finite-temperature numerical results along with the zero-temperature analytical ones to support a discussion of the distinct effects of the external electric field and temperature on the plasmon dispersion. Our results show that at zero temperature both the acoustic and optical modes decrease by increasing the applied electric field and experience a discontinuity at the valley-spin polarized metal phase as the system transitions from a topological insulator to a band insulator. At finite temperature, the optical plasmons are damped around this discontinuity and the acoustic modes may exhibit a continuous transition. Moreover, while the optical branch of plasmons changes non-monotonically and noticeably with temperature, the acoustic branch dispersion displays a negligible growth with temperature for all phases of silicene. Furthermore, our finite-temperature results indicate that the dependency of two plasmonic branches on the interlayer separation is not affected by temperature at long wavelengths; the acoustic mode energy varies slightly with increasing the interlayer distance, whereas the optical mode remains unchanged.
... Other interesting studies of the 2D Xenes include nonlocal plasmon modes in open systems [19], chemical potentials calculated without approximations for silicene and germanene [20], exchange-correlation energies due to the Coulomb and spin-orbit interactions in monolayer Si, calculated in the presence of circularly polarized radiation [21], including temperature dependence [22]. ...
We study the binding energies and optical properties of direct and indirect excitons in monolayers and double-layer heterostructures of Xenes: silicene, germanene, and stanene. It is demonstrated that an external electric field can be used to tune the eigenenergies and optical properties of excitons by changing the effective mass of charge carriers. Our calculations show that the existence of the excitonic insulator phase is plausible in freestanding Xenes at small electric fields. The Schrödinger equation with field-dependent exciton reduced mass is solved by using the Rytova-Keldysh (RK) potential for direct excitons, while both the RK and Coulomb potentials are used for indirect excitons. It is shown that for indirect excitons, the choice of interaction potential can cause huge differences in the eigenenergies at large electric fields and significant differences even at small electric fields. Furthermore, our calculations show that the choice of material parameters has a significant effect on the binding energies and optical properties of direct and indirect excitons. These calculations contribute to the rapidly growing body of research regarding the excitonic and optical properties of this new class of two-dimensional semiconductors.
... where (x) is the Heaviside step function, accounting for unavailable electronic states within the lower gap, as illustrated schematically in Fig. 6(a). Equation (2) can be employed to determine E F at zero temperature (T = 0) with a fixed carrier density n c , leading to [50,54] 2π ...
... Here, it is important to point out that T (q, ω | μ T , β ) ∼ q 2 as q → 0 for all 2D systems, regardless of the band gaps [59] and temperatures [54,57,72]. Therefore, the optical conductivity in Eq. (9) becomes independent of q. ...
Thermal and dynamical properties of optical and transport conductivities in doped buckled honeycomb lattices are studied for various doping densities and band gaps. At finite temperatures, a thermally convoluted polarization function is calculated by employing analytically derived temperature-dependent chemical potentials. With this finite-temperature polarization function, the optical conductivity, originating from an induced polarization current, is obtained in the long-wavelength limit, where both steps and negative peaks are shown as a function of frequency in its real and imaginary parts, respectively. Such spectral features can be used for analyzing plasmon dampings in silicene and ultrafast light modulations based on field-tunable band gaps. Additionally, in the presence of static screening, derived with the aid of the polarization function, for impurity elastic scattering, the transport conductivities are calculated for different doping densities and band gaps within the second-order Born approximation. The enhanced transport conductivity with a smaller band gap at intermediate temperatures will lead to high-mobility electron transistors for ultrafast electronics.
... Physically, once the μ(T ) dependence becomes known, one can obtain the finite-T dynamical polarization function from Eq. (18), which is a key component for all relevant many-body calculations. These include optical absorption, electronic transport, plasmon excitations as well as electron exchange and correlation energies [83]. Here, we focus on finite-T plasmon excitations, demonstrating doping effects on modifications of plasmonenergy dispersions at intermediate T . ...
Analytic expressions for chemical potentials without any approximations are derived for all types of extrinsic (doped) gapped Dirac-cone materials including gapped graphene, silicene, germanene, and single-layer transition-metal dichalcogenides. In setting up our derivations, a reliable piecewise-linear model has been established for calculating the density of states in molybdenum disulfide, showing good agreement with previously obtained numerical results. For spin- and valley-resolved band structures, a decrease of chemical potential with increasing temperature is found as a result of enhanced thermal populations of an upper subband. Due to the broken symmetry with respect to electron and hole states in MoS2, the chemical potential is shown to cross a zero-energy point at sufficiently high temperatures. It is important to mention that the chemical potential at a fixed temperature can still be tuned by varying the doping density and band structure of a system with an external electric or strain field. Since a thermal-convolution path (or a chemical-potential-dependent response function for the thermal convolution of fermions) starting from zero temperature must be selected in advance before obtaining finite-temperature properties of any collective quantities, e.g., polarizability, plasmon modes, and damping, a control of their thermal dependence within a certain temperature range is expected for field-tunable extrinsic gapped Dirac-cone materials.
... The Hamiltonian, energy dispersion, and related electronic properties of germanene are qualitatively similar to those of silicene although the Fermi velocity, the bandgap induced by spin-orbit coupling, and the buckling height in germanene are still different in magnitudes. We, therefore, believe that our previous theory 49 for silicene can equally be applicable to Ge-based hybrid structures. ...
Full ranges of both hybrid plasmon-mode dispersions and their damping are studied systematically by our recently developed mean-field theory in open systems involving a conducting substrate and a two-dimensional (2D) material with a buckled honeycomb lattice, such as silicene, germanene, and a group \rom{4} dichalcogenide as well. In this hybrid system, the single plasmon mode for a free-standing 2D layer is split into one acoustic-like and one optical-like mode, leading to a dramatic change in the damping of plasmon modes. In comparison with gapped graphene, critical features associated with plasmon modes and damping in silicene and molybdenum disulfide are found with various spin-orbit and lattice asymmetry energy bandgaps, doping types and levels, and coupling strengths between 2D materials and the conducting substrate. The obtained damping dependence on both spin and valley degrees of freedom is expected to facilitate measuring the open-system dielectric property and the spin-orbit coupling strength of individual 2D materials. The unique linear dispersion of the acoustic-like plasmon mode introduces additional damping from the intraband particle-hole modes which is absent for a free-standing 2D material layer, and the use of molybdenum disulfide with a large bandgap simultaneously suppresses the strong damping from the interband particle-hole modes.
