Magneto-plasmons of germanene nanoribbons
Introduction
After the discovery of graphene in 2004 [1], it has spurred great interest in explorations of novel 2D materials with graphene-like structures made of other group IV elements. Following the previous efforts, germanene [2], [3], [4], [5] and silicene [6], [7], [8] have been recently synthesized on distinct metal substrates. As compared to a flat single-layer of graphene, germanene and silicene, due to the larger ionic size, possess the low-buckled structure causing much stronger spin–orbit coupling (SOC) than that of graphene. The spin–orbit gaps of germanene and silicene are predicted to be 23.9 and 1.55 , respectively. They are much larger than the spin–orbit gap of graphene ( ) [9]. The salient differences including the buckled structure and stronger SOC will play an important role for inducing novel physical properties.
In addition to 2D silicene, 1D silicene nanoribbons (SiNRs) like graphene nanoribbons (GNRs), have successfully grown on the metal surfaces [10], [11], [12]. Since germanene and silinene have similar structure, the fabrication of germanene nanoribbons (GeNRs) in experiments will be worth waiting for. GeNRs are theoretically predicted to have the stronger buckling and SOC so that the interplay between the quantum confinement and SOC will reveal richer physical properties than SiNRs. The theoretical studies in electronic and magnetic properties of GeNRs showed strong dependence on nanoribbon’s edges and doping atoms [13], [14]. While edges of GeNRs exposed to exchange fields, a quantum-spin-Hall phase transition was induced that could give applications in giant magneto-resistance and spin filter [15], [16]. For ZGeNRs, asymmetric edge termination and temperature difference between source and drain electrodes could induce spin-polarized current that will be beneficial to spin caloritronics [17], [18]. Besides, temperature-dependences of electronic specific heat of GeNRs were strongly determined by geometric structures [19].
Both GeNRs and SiNRs have the hexagonal plane structure, perpendicular magnetic field could induce extra magnetic phase leading to an effective confinement for a cyclotron motion of electrons. While the radius of cyclotron motion starts smaller than nanoribbon’s width, spin-polarized Landau levels are exhibited which could make great modulations on electronic and optical properties [20], [21]. It is believed that the unusual SOC and buckling of GeNRs, in magnetic field, will bright more novel properties than GNRs and SiNRs.
Plasmons, quantized collective oscillation of charge density, are excited by irradiation of photon or electron beam, giving potential applications owing to the capability for confining electromagnetic energy on nano-scaled space. Plasmons found in metals or bulk materials show a great applicability in improving the efficiency of photovoltaic device [22], solar cell [23], [24], chemical sensing [25], and photodetection [26]. While materials are reduced into low-dimensional regime, electrons are restricted to limited degree of freedom leading to stronger light–mater interaction. For example, surface plasmons (SPs) in graphene [26], [27] are confined at sub-wavelength scale with smaller penetration depth in THz region and long optical relaxation time resulting in a low-loss radiation [28] and slower dissipation [29], [30], [31], [32]. Moreover, the plasmon-enhanced interaction between graphene and photons shows low-loss and effective wave-localization up to mid-infrared range, enabling the broadband applications [33], [34], [35], [36], [37], [38]. Such a unique capability of supporting and modulating SPs could bright many benefits to graphene-based nano-optical elements [39], [40], [41], [42], [43], [44]. While graphene is tailored into GNRs, their various boundaries could give rise to feature-rich collective excitations. Plasmon in GNRs with a certain propagating direction could serve as a waveguide [45], [46], [47]; moreover, its localization also enhances optical absorption. With decreasing width of GNRs, the wider band-gap gives a hope to find high frequency low-loss SPs and various edges can make plasmons split into several resonance modes. The effect from armchair and zigzag edges is very remarkable; especially, zigzag edge in electric field could induce higher-frequency and broader plasmons [48], [49].
In graphene-related systems, effects of magnetic field on SPs are valid; for example, THz excitation and tunable optical response can be theoretically realized [50], [51], [52]. In experiments, plasmon excitation modifies magneto-optical responses [53] and plasmon life time can be tuned by varying magnetic field [54], predicting a lot of potential for graphene as tunable THz magneto-optcal devices. For armchair graphene nanoribbons (AGNRs), the theoretical study shows magneto-plasmons strongly depend on the number of dimer lines [55]. However, zigzag graphene nanoribbons (ZGNRs) with partial flat bands around the Fermi energy support a broad continuum of electron–hole excitations suppressing low-frequency magneto-plasmons [56]. In contrast to GNRs, magnetic field could more effectively modulate energy bands of GeNRs due to the stronger SOC; meanwhile, the buckling would make the Coulomb interactions very different. Thus, the study attempts to explore modulations of magnetic field on low-frequency plasmons of GeNRs at finite temperature that has not been studied before. The calculated results are predicted to display richer magneto-plasmon spectra than those in GNRs, and could be further verified by measurements of electron energy loss spectroscopy (EELS) [57], [58].
In this work, low-energy magneto-electronic properties of germanene nanoribbons are first studied within the -orbital tight-binding model. Next, the dependence of collective electronic excitations on geometric structures, temperature, transferred momentum, and magnetic field are investigated. In perpendicular magnetic field, modulations of low-energy dispersions and band-gaps of are profoundly associated with boundary geometry. At zero field, only small-gapped AGeNRs exhibit low-frequency plasmons. As magnetic field increases, frequency and strength of plasmons are reduced. For large-gapped AGeNRs, the initiation of low-frequency plasmons must depends on magnetic field. The behaviors of magneto-plasmon dispersion relations are profoundly related to types of AGeNRs. On the other hand, at zero temperature ZGeNRs could not exhibit low-frequency plasmons with/without magnetic field. While temperature increases, low-frequency magneto-plasmons are enhanced due to intraband transitions. The dependence of magneto-plasmons on temperature is sensitive to different geometric structures.
Section snippets
Modulations of magnetic field on electronic properties of germanene nanoribbons
The armchair/zigzag germanene nanoribbon (AGeNR/ZGeNR), with armchair/zigzig dimer lines contains two sublattices consisting of Ge atoms at A and B sites along the -axis; thus there are Ge atoms are in a primitive unit cell. The distance between two nearest-neighbor atoms and the lattice constant, in – plane, are Å and Å, respectively [59]. The perpendicular distance between two sublattices due to the buckled structure is Å. GeNR is assumed to have an
The dielectric function and low-frequency plasmons
The -electronic excitations induced by one-dimensional -band excitations are described by the transferred momentum (unit Å , henceforth) along direction and the excitation energy . The dielectric function, at arbitrary temperature (unit K, henceforth), could be evaluated by the random-phase-approximation (RPA) [60], [61]. where is the background dielectric constant and is
Conclusion
The interplay between the SOC and magnetic field causes spin splitting that significantly modulate energy dispersions and band-gaps. The continuously increasing magnetic field reduces band-gap of AGeNRs; however, it is opposite for ZGeNRs. At zero temperature, low-frequency plasmon without magnetic field only exists in small-gapped AGeNRs. It vanishes at smaller or larger transferred momentum due to enhanced Landau damping. Plasmon frequency and strength are reduced with increasing magnetic
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (66)
- et al.
Physica E
(2017) - et al.
Physica E
(2017) - et al.
Appl. Surf. Sci.
(2017) Physica E
(2019)Physica E
(2017)Physica E
(2017)- et al.
Renew Sust. Energ. Rev.
(2018) Physica E
(2020)Physica E
(2021)- et al.
Science
(2004)
Adv. Mater
New J. Phys.
Nano Lett.
Phys. Rev. Mater.
Phys. Rev. Lett.
Y. Wang, L. Zhang, S. Du, R. Wu, L. Li, Y. Zhang, G. Li, H. Zhou, W.A. Hofer, H.J. Gao, Nano Lett
Phys. Rev. Lett.
Phys. Rev. Lett.
Appl. Phys. Lett.
Appl. Phys. Lett.
Appl. Phys. Lett.
Phys. Rev. Lett.
Phys. Rev. B
Phys. Rev. B
Nat. Mater.
J. Phys. Chem. C
Biosensors
Nanophotonics
Nat. Nanotechnol.
Opt. Express
ACS Nano.
Appl. Phys. Lett.
Phys. Rev. B
Cited by (1)
THz plasmonics and electronics in germanene nanostrips
2023, Journal of Semiconductors