Jhao-Ying Wu*a,
Chiun-Yan Lina,
Godfrey Gumbsb and
Ming-Fa Lina
aDepartment of Physics, National Cheng Kung University, Tainan, Taiwan 701. E-mail: yarst5@gmail.com; mflin@mail.ncku.edu.tw
bDepartment of Physics and Astronomy, Hunter College at the City University of New York, 695 Park Avenue, New York, New York 10065, USA. E-mail: ggumbs@hunter.cuny.edu
First published on 28th May 2015
We use the tight-binding model and the random-phase approximation to investigate the intrinsic plasmon in silicene. At finite temperatures, an undamped plasmon is generated from the interplay between the intraband and the interband-gap transitions. The extent of the plasmon existence range in terms of momentum and temperature, which is dependent on the size of the single-particle-excitation gap, is further tuned by applying a perpendicular electric field. The plasmon becomes damped in the interband-excitation region. A low damped zone is created by the field-induced spin split. The field-dependent plasmon spectrum shows a strong tunability in the plasmon intensity and spectral bandwidth. This could make silicene a very suitable candidate for plasmonic applications.
The out-of-plane buckling suggests that an on-site potential difference between the A and B sublattices is tunable under a perpendicular electric field.19,20 This allows for the control of the band gap, along with the spin and valley polarized states. Accordingly, many spin and valleytronics, like the quantum spin Hall effect21–23 and the optically-generated spin-valley-polarized charge carriers,24–27 may be investigated. Additionally, when the electric potential exceeds the magnitude of the SOI, 2D silicene is predicted to undergo a topological phase transition from a topological insulator to a band insulator, a property that should be important in electronics and optics.
Collective-Coulomb excitations, dominated by electron–electron interactions, are useful for understanding the behavior of electrons in a material, due to the involvement of screening effects. This basic property has been widely studied in various low-dimensional systems, e.g., carbon nanotubes,28–30 graphene layers,31–37 and other novel materials.38–40 In monolayer graphene, plasmons (quantized collective-excitation modes) hardly exist at low frequencies due to the lack of free carriers. This may be improved by doping or insertion of a gate to increase the free-charge density,41,42 i.e., changing the Fermi level in an extrinsic condition. Alternatively, an intrinsic low-frequency plasmon (with zero Fermi energy) may be induced by increasing the thermally excited electrons and holes in the conduction and valence bands.43 This is based on the strong dependence of the carrier density on the temperature (n ∝ T2). A temperature-induced plasmon is also predicted to exist in monolayer silicene.44 The difference is that the temperature-induced plasmon in graphene is always located in the interband region and suffers the Landau damping.43 However, in silicene, the temperature-induced plasmon could be in a single-particle-excitation (SPE) gap arising from the spin–orbit interaction.44 The opening of the SPE gap makes the plasmon undamped. In this paper, we investigate the effects of a perpendicular electric field on the T-induced plasmon in silicene. The energy-loss function is used to derive the plasmon spectrum. Three plasmon modes with distinct degrees of Landau damping are found in different momentum, temperature, and field strength ranges. The occurrences of the plasmons and their discontinuous dispersions are associated with special structures in the dielectric functions.
![]() | (1) |
When the system is perturbed by a time-dependent Coulomb potential, the electron–electron interactions would induce a charge density fluctuation which acts to screen the external perturbation. This dielectric screening determines the normal modes of the charge density oscillations. Electronic excitations are characterized by the transferred momentum q and the excitation frequency ω, which determine the dielectric function. Within the random-phase approximation, the dielectric function can be expressed as:
ε(q,ω) = ε0 − vqχ0(q,ω), | (2) |
![]() | (3) |
Each Bloch state is labeled by the Bloch wave vector k and the band index n. En is the corresponding eigenvalue. The Fermi-Dirac distribution is f(En) = 1/{1 + exp[(En − μ)/kBT], where kB is the Boltzmann constant and μ is the chemical potential. μ is set to zero over the range of the symmetric conduction (c) and valence (v) bands under investigation. Γ is the energy broadening which results from various de-excitation mechanisms, e.g., the optical transitions between the valence- and conduction-band states. It is usually set low in the low-frequency region. Changing Γ would not alter the peak positions in the energy-loss function (Fig. 4), but the peak intensities and widths would be affected.
The details of the calculation of the Coulomb matrix elements are shown below.
![]() | (4) |
The unscreened-excitation spectrum is helpful in understanding the single-particle excitation (SPE) channels. At zero temperature (Fig. 3(a)), the interband excitations are the only available excitation channel. The threshold excitation energy is expressed as . The real (ε1) and imaginary (ε2) parts of the longitudinal dielectric function, related through the Kramers–Kronig relationship, exhibit a logarithmic-divergent peak and a step-like structure at ωinterth, respectively. The interband-gap transitions account for these prominent structures. There are no zero points in ε1. That is, for T = 0, no self-sustained plasmon excitations can exist in the low-frequency region. The intraband transitions are allowed at finite temperatures; these transitions create additional special structures in ε, i.e., an asymmetric dip and peak in ε1 and ε2, respectively, as illustrated in Fig. 3(b) for T = 50 K. At the same time, the interband-transition-related structures become weaker. The highest intraband-excitation energy is ωintraex = vFq, which is lower than ωinterth. Therefore, a SPE gap is developed in ε2. The asymmetric dip in ε1 creates two zero points. One of the zero points is located in the ε2 gap (only for the condition kBT < Eg), where an undamped plasmon mode can occur. The temperature range that allows the undamped plasmon depends on the width of the SPE gap; this width is tunable by an external E-field.
The effects of the E-field on ε are displayed in Fig. 3(c)–(e) for fixed temperature of T = 50 K and different Ez values. At Ez = 0.5Ec, shown in Fig. 3(c), the step structure (peak) in ε2(ε1) is split into two owing to the lifting of the spin degeneracy, as indicated by the pair of blue arrows. The two steps of the structure develop at ωinter1,th and ωinter2,th. Between ωinter1,th and ωinter2,th the density of the e–h pairs is half of that beyond ωinter2,th. This means the creation of a low-damped plasmon region. ωinter1,th decreases when Ez is increased from zero to Ec, i.e., the SPE gap is reduced. This may hinder the occurrence of an undamped plasmon. The opposite is true for Ez > Ec (Fig. 3(e)). At Ez > Ec (Fig. 3(d)), ωinter1,th and ωintraex merge. Therefore, both zero points of ε1 are located where ε2 is finite. Under such conditions, only damped plasmons can exist.
The energy-loss function (∝Im[−1/ε]), related to the screened excitation spectrum, is used to understand the collective-excitation mode of the electrons. The quantity can be measured from electron energy loss spectroscopy in transmission50,51,55,56 and reflection modes,57 and inelastic light scattering.58 Each prominent structure in Im[−1/ε] may be viewed as a plasmon excitation with different degrees of Landau damping. At T = 0, the low-frequency Im[−1/ε] appears as plateaus without and with an electric field, as shown in the insets of Fig. 4(a) and (b), respectively. The featureless plateau structures correspond to the interband excitations, which exhibit single-particle like behaviors. At finite temperatures, a prominent peak is induced with a frequency lower than the plateaus (Fig. 4(a)–(d) for T = 50 K). The corresponding dielectric functions reveal that this peak could arise from the interplay between the intraband and interband-gap transitions (Fig. 3(b), (c) and (e)) or merely from the sufficiently strong intraband transitions under the condition of a zero SPE gap (Fig. 3(d)). The former leads to a stronger peak intensity and a narrower peak width in the energy-loss function (Fig. 4(a), (b) and (d)), compared with that caused by the latter (Fig. 4(c)).
The q-dependent behavior of plasmons is important in understanding their characteristics. Fig. 5(a)–(c) show the results for different Ez values at a fixed temperature of T = 50 K. The color scale stands for the values of the energy-loss function. The dashed curves in each figure represent the highest intraband (white curves) and the lowest interband (blue curves) SPE energies, which obey the relationship ω = vFq and with s = ±1, respectively. The expressions are derived from the analytical solution of the single-particle spectrum near the K point.54 These boundary curves define the various regions with different degrees of Landau damping. Between the white and the blue boundary curves lies the SPE gap, where an undamped plasmon branch exists. The plasmon frequency ωp is well fitted by
in the long-wavelength limit, a feature of a 2D plasmon. For lower temperatures (Fig. 5(a) and (b)), this plasmon mode is finally damped out in the intraband SPE continuum at larger q values. The existence range of the undamped plasmon in momentum is proportional to the width of the SPE gap, which is reduced in the region of 0 < Ez < Ec (Fig. 5(b) for Ez = 0.5Ec). There are two blue boundary curves in Ez ≠ 0 (Fig. 5(b)–(d)). They are the result of the lifting of the spin degeneracy. Between the two blue curves lies a low damped region. In Fig. 5(b), a broad plasmon branch appears in the region, just above the undamped plasmon mode. The coexistence of the two collective-excitation modes can only be realized at kBT ≈ E1g. The SPE gap is closed at Ez = Ec (Fig. 5(c)). Then, the plasmon is damped at arbitrary q values. Until Ez > Ec, the undamped plasmon branch is regained (not shown). Under the above conditions, the plasmon dispersion is confined between two SPE boundary curves. However, when the temperature is sufficiently high, e.g., T = 200 K and Ez = 0.5Ec in Fig. 5(d), the plasmon branch crosses the SPE boundaries and shows abrupt changes in intensity and bandwidth. The discontinuous plasmon dispersion is more evident when the plasmon crosses the upper blue curve. After that, the plasmon enters the strong damped region. It is worth noting that the temperature-induced plasmon differs from the extrinsic plasmon discussed in doped silicene.59 The dispersion relationship of the extrinsic plasmon is decided by the Fermi energy. This is in great contrast to the intrinsic plasmon whose dispersion is dominated by the temperature. The different electron distributions result in different plasmon dispersions between the two conditions.
The effect of temperature variation on the plasmon excitation is worth further discussion. Fig. 6(a)–(c) show the T-dependent plasmon spectra at fixed q = 1 and different Ez values. The distribution range of the undamped plasmon mode (between the blue and white dashed curves) with T is reduced in the region of 0 < Ez < Ec (Fig. 6(b) for Ez = 0.5Ec) and diminished to zero at Ez = Ec (Fig. 6(c)), compared to the condition of Ez = 0 (Fig. 6(a)). The undamped plasmon is transformed to a damped one above a critical temperature (Tc ≃ 100 K for Ez = 0). The transformation in the plasmon mode during the change in T is a discontinuous process, a feature that could allow for the identification of the spin–orbit energy gap. The discontinuous dispersions of the plasmons remain when changing the background dielectric constant ε0 and energy broadening Γ. In Fig. 6(b) for Ez = 0.5Ec, two critical temperatures exist through which the plasmon spectrum is drastically altered. The two Tc values correspond to the two energy gaps opening in the system. This could be used to examine the spin degeneracy and determine Ec. Note that above the highest Tc value (≈175 K for Ez = 0.5Ec), the plasmon intensity grows monotonically with T, similar to the behavior of the intrinsic plasmon in gapless graphene.43 This indicates that at such high temperatures the SOI effect becomes negligible.
For fixed q and T, the plasmon existence and its spectral weight can be effectively controlled by varying Ez, as shown in Fig. 7. At T = 50 K and q = 1 (Fig. 7(a)), the undamped plasmon is stable for Ez < 0.5Ec due to the significantly large SPE gap. When Ez is increased to near Ec, the plasmon is damped by the interband SPEs and fades. The SPE gap is reopened in the region of Ez > Ec. After that the plasmon is intensified first with the increment of Ez, and when Ez exceeds another critical value E′c (E′c ≈ 3Ec for T = 50 K and q = 1), the plasmon is gradually damped out in the intraband SPE continuum. At T = 100 K (Fig. 7(b)), the undamped plasmon barely exists in Ez < 2Ec, because the overly strong intraband transitions make the plasmon frequency higher than the threshold interband SPE energy. Until Ez > 2Ec, the energy gap becomes larger than kBT, and the undamped plasmon emerges. The undamped plasmon at T = 100 K can survive in a larger Ez than that at T = 50 K. The stronger intraband transitions can explain this. At T = 200 K (Fig. 7(c)), the plasmon spectrum displays three distinct degrees of Landau damping in different Ez ranges. In the first region (0 < Ez < 0.7Ec), the plasmon is located beyond the two interband SPE boundaries and suffers strong Landau damping. In the second region (0.7Ec < Ez < 2.5Ec), the plasmon lies between the two interband boundaries and is weakly damped. In the third region (Ez > 2.5Ec), the plasmon lies inside the SPE gap and is undamped. Though the plasmon intensity strongly depends on Ez, the change in the plasmon frequency is insignificant. This illustrates that the appearance of the low-frequency plasmon is mainly dominated by the intraband transitions.
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Fig. 7 The Ez-dependent plasmon spectrum at fixed q = 1 and different T values: (a) T = 50 K, (b) T = 100 K, and (c) T = 200 K. The color scale stands for the values of the energy-loss function. |
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