One important property of graphene22.214.171.124.6.7.–8 is that all its Dirac electrons, irrespective of their energy, have the same speed; as energy versus momentum relation is linear. When a transverse magnetic field is applied, the electrons are localized in circular orbits and graphene supports a variety of manetoplasmonic modes. Roldan et al.9,10 have developed a formalism of these two-dimensional (2-D) Bernstein modes based on Dirac equation using random phase approximation at intense magnetic fields, where Landau levels have energies comparable to Fermi energy.
Basov et al.1 have discussed plasmons in graphene in the presence of a transverse magnetic field in the limit of small Larmor radius. Yan et al.11,12 have detected bulk like and edge like magnetoplasmon modes in graphene discs. Crassee et al.13 have observed terahertz magnetoplasmons in graphene, mounted on substrate terrace or wrinkles, through a sharp dip in the transmission coefficient of normally impinged radiation. It is believed that at the dip frequency, laser undergoes linear mode conversion into magnetoplasmons. They have also observed Faraday rotation (FR) of an optical wave passing through single layer graphene. Martinez and Jalil14 have developed quantum formalism of FR. In the limit of (where is the electron cyclotron frequency, is the Fermi energy and is the reduced Planck’s constant), the effect of Landau quantization is unimportant.
Theoretical studies on linear mode conversion of laser into surface plasmons are largely limited to unmagnetized case. Mikhailov15 has developed a formal analytical theory of laser mode coupling to a plasmonic mode in an unmagnetized 2-D structure. Lee and Degertekin16 have developed a coupled wave analysis of multilayered grating structures, but the results are obtained numerically for reflected diffraction efficiency. Gao et al.17 have carried numerical simulations of linear mode conversion.
In this paper, we develop Vlasov formalism of magnetoplasmons in graphene mounted on a dielectric placed in a transverse static magnetic field and study their excitation via linear mode conversion. Our treatment is restricted to moderate magnetic fields such that the Landau level energy separation is smaller than Fermi energy. We find that the finite gyroradius effects give rise to multiple magnetoplasmonic Bernstein modes, similar to the ones predicted by Roldan et al. at intense magnetic fields. A suitable grating built in the dielectric substrate or an areal density ripple, facilitates linear mode conversion. The grating creates a large wave number Fourier component of the laser field that creates an electron density perturbation, driving the plasmonic wave. Finer grating would generate Bernstein modes. One may mention that though the electron dynamics in graphene is strongly correlated and one normally uses Dirac theory to deduce optical conductivity, Boltzmann’s equation reasonably describes transport properties.2,18 Furthermore, we may add that graphene plasmons can also be excited by electron beams.19,20 Batrakov and Maksimenkov20 have studied theoretically the excitation of terahertz surface wave over a system of unmagnetized graphene layers by a nonrelativistic electron beam. They obtain spatial growth rate of the order of at 30 THz in eight layered graphene, using a 10-keV electron beam. The frequency of the wave can be tuned by beam energy as well as by sheet separation and doping.
In Sec. 2, we study magnetoplasmonic modes of graphene in a transverse magnetic field including finite gyroradius effects. In Sec. 3, we study the linear mode conversion of radiation into magnetoplasmons in graphene with periodic ribbons. In Sec. 4, we study the mode conversion by a grating. In Sec. 5, we discuss the results.
Consider a graphene film mounted on a dielectric of relative permittivity (cf., Fig. 1). There exists a transverse static magnetic field . The graphene is characterized by free electrons of areal density , energy-momentum relation , velocity , and 2-D equilibrium distribution function
We perturb the equilibrium by a space charge mode of potential
The jump condition on the normal component of displacement vector at1 For , Eq. (14) reduces to the usual unmagnetized plasmon dispersion relation [cf. Eq. (18), Ref. 21], where frequency scales as and .
We have solved the dispersion relation, Eq. (12), numerically for the following parameters: , (corresponding to , or any multiple of this areal density when the magnetic field is reduced by the same factor, keeping constant). In Fig. 2, we have plotted the normalized frequency as a function of normalized wavenumber for four magnetoplasmonic modes. At , the modes have frequencies near the harmonics of cyclotron frequency. As rises, the frequencies rise. For the first mode, the frequency rises to a maximum of at and then declines, asymptotically to at large . For the second mode, rises from at to at and then falls off. For the third mode, rises from , initially very gradually and then faster, attains a maximum at and then falls off. For the fourth plasmonic mode, maximum occurs at . In Fig. 3, we have plotted the dispersion curves for (corresponding to , ). At this higher ratio, the mode frequencies rise more rapidly with and attain maximum values closer to the next harmonic. At , the ’th harmonic Bernstein mode has
Mode Conversion in Graphene Ribbons
We introduce a ribbon structure on graphene, effectively making the areal electron density to have a ripple. The total electron density is thus
A laser is normally incident on graphene from top
We choose such that it equals the magnetoplasmon wave number at frequency [cf., Eq. (12)]; . The reflected and transmitted fields are
Using , in the jump condition, Eq. (11) with replaced by , one obtains
The oscillatory velocity due to the magnetoplasmonic wave beats with the density ripple to produce a surface current density
The last term in the denominator of Eq. (26) corresponds to mode coupling of laser to magnetoplasmons. At exact phase matching [i.e., when equals the wave number of the plasmonic wave given by Eq. (12)], vanishes and the last term in the denominator of Eq. (26) overflows. However, in the present formalism, we have neglected damping of the plasmonic mode that may arise due to collisions. The damping of plasmons would arrest the resonance. As a phenomenological model of collisional effects, the right-hand side of the Vlasov equation may be replaced by , where is the collision frequency. In the case of the upper hybrid mode, this leads to replacing, in the dispersion relation, by leading to damping rate . For the Bernstein modes, we may replace, in the expression for susceptibility or , by , giving an imaginary part to , . At resonance, vanishes and can be written as
From Eqs. (21) and (26), one may write the normalized plasmonic field amplitude and laser intensity transmission coefficient at the mode conversion point asFig. 4 the intensity transmission coefficient of laser through graphene as a function of normalized laser frequency. At the plasmon resonance, corresponding to the excitation of the first magnetoplasmonic mode of frequency , the transmission coefficient falls to 42%. At the second plasmonic resonance corresponding to the excitation of the second plasmonic mode, the transmission coefficient is 52%. At the third plasmon resonance, it is 60%.
Mode Conversion in Dielectric Grating
Consider a dielectric grating of thickness sandwiched between graphene and the substrate. The relative permittivity of the grating region is
A laser is normally impinged on graphene with incident, reflected, and transmitted fields given by Eqs. (16) and (17). The driven plasmonic wave has potential given by Eq. (2). In the grating region, the Poisson equation can be written as
The transmission coefficient is given by Eq. (27) with given by
A transverse magnetic field opens up a variety of magnetoplasmonic modes in graphene. The lowest frequency mode, at wavelengths longer than the electron gyroradius, is sort of an upper hybrid mode; however, its frequency variation with wave number and electron density is very different from plasmas. Other modes have frequencies close to harmonics of cyclotron frequency. At shorter wavelengths, kinetic effects become important and mode frequencies rise with wave number, acquire maxima, and then fall off gradually. This behavior is similar to the one reported by Roldan et al. at high magnetic fields when quantum effects are important.
The magnetoplasmonic modes can be excited by laser via linear mode conversion, when the graphene layer is embedded with ribbons or a dielectric grating is employed between the substrate and graphene. For a given ripple wave number, the magnetic field helps tuning the process of laser mode conversion. Crassee et al.13 carried experiments with wrinkled surfaces and -type graphene. They observed a dip in the laser transmission coefficient of magnitude comparable to what we get in Fig. 4. However, their variation of transmission coefficient with magnetic field does not reveal the discrete character as one would expect from mode conversion to Bernstein modes. The experiments need to minimize collisional damping of the modes.
The wide range of magnetoplasmonic modes offer new opportunities for active plasmonic devices at terahertz frequencies. One may envisage stimulated excitation of these modes by electron hole recombination under conditions of population inversion. Their frequencies can be tuned by magnetic field. The application of a surface ripple would convert the plasmonic modes into terahertz radiation emission.
The present treatment of mode conversion is limited to a low depth of amplitude modulation of ribbons, i.e., otherwise, a large number of spatial harmonics of plasmons are excited.22
Chuan Sheng Liu is a plasma physicist credited with groundbreaking contributions to theory of laser plasma interaction and promoting international collaboration in higher education. He joined the University of Maryland in 1975 as a professor of physics and served twice as a department chair. He also served as a vice president for research and dean of the graduate school. From 2003 to 2006, he was a president of National Central University, Taiwan. Currently, he is now Master Emeritus of Chao Kuang Piu College.
Vipin K. Tripathi received his master’s degree in physics from Agra University in 1967 and a PhD from IIT Delhi in 1971. In 1972, he joined the IIT faculty. In 1976, he moved to the University of Maryland and worked in the area of thermonuclear fusion, developing nonlinear theories of RF heating and current drive in tokamak. In 1983, he joined IIT Delhi as a professor of physics. He established a leading group in free electron laser and laser plasma interaction.