22 August 2017 Kinetic magnetoplasmons in graphene and their excitation by laser
Author Affiliations +
Abstract
A transverse magnetic field in graphene, together with the high speed of Dirac electrons moving with Fermi velocity, gives rise to a set of collective modes, viz., kinetic magnetoplasmonic modes, two-dimensional equivalent of Bernstein modes, with frequencies in between the harmonics of electron cyclotron frequency. We develop a Vlasov theory of these modes in a moderate magnetic field, including finite gyroradius effects, and study their excitation by laser through linear mode conversion, facilitated by grating or periodic ribbons. At kρ→0 (where k is the wave number and ρ is the gyroradius of electrons), the magnetoplasmonic modes have frequencies near the harmonics of electron cyclotron frequency. The frequencies rise with wave number, attain maxima in the vicinity of the next cyclotron harmonic, and then fall off. In high-mobility graphene, with ribbons or grating of appropriate ripple wave number, a normally impinged laser coverts a significant fraction of its power into magnetoplasmons, reducing the laser transmissivity as observed in experiments.
Liu and Tripathi: Kinetic magnetoplasmons in graphene and their excitation by laser

1.

Introduction

One important property of graphene12.3.4.5.6.7.8 is that all its Dirac electrons, irrespective of their energy, have the same speed; vF=108  cm/s 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 ωcϵF (where ωc is the electron cyclotron frequency, ϵF 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 0.2  cm1 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.

2.

Magnetoplasmons

Consider a graphene film mounted on a dielectric of relative permittivity ϵg (cf., Fig. 1). There exists a transverse static magnetic field Bsx^. The graphene is characterized by free electrons of areal density N00, energy-momentum relation ϵ=vFp, velocity ν=ϵ/p=vFp/p, and 2-D equilibrium distribution function

(1)

f00=1/(2π22)e(ϵϵF)/T+1,
where ϵF is the Fermi energy, νF is the Fermi velocity (108  cm/s), T is the temperature in energy units, is the reduced Planck’s constant. At a low temperature to which we confine here, ϵF/T1, N00=2π0f00pdp=ϵF2/2π2νF2.

Fig. 1

(a) Schematic of graphene mounted on a dielectric slab. A transverse static magnetic field BS is applied along x^. Magnetoplasmonic mode propagates along z^ while its amplitude falls off with |x|. (b) Graphene is embedded with periodic ribbons. A laser is normally impinged on graphene.

JNP_11_3_036015_f001.png

We perturb the equilibrium by a space charge mode of potential

(2)

ϕ=Aekzxei(ωtkzz)forx>0,ϕ=Aekzxei(ωtkzz)forx<0,
in compliance with the Poisson’s equation 2ϕ=0 (valid everywhere except in the graphene layer) and the continuity of ϕ at x=0. To incorporate the effect of graphene layer, we write the 2-D Vlasov equation for free electrons

(3)

ft+v.fe(E+v×B).fp=0,
where e is the electron charge and E, B are the electric and magnetic fields. In the presence of the space charge mode, we write f=f00+fωL and linearize the Vlasov equation
fωLt+v.fωLev×Bs.fωLpdfωLdt=eϕ.f00p,
to obtain the linear perturbation fωL

(4)

fωL=ieA1pf00ptk.pei(ωtkzz)dt,
where z, p refer to position and momentum of electron at time t and the integration is to be carried over the unperturbed trajectory of electron in the static magnetic field, governed by the equation of motion

(5)

dpdt=ev×Bs=evFpp×Bs.
Under the conditions that at t=t, electron has position and momentum y=y, z=z, pz=pz=pcosθ, py=py=psinθ (where θ is the gyrophase angle), Eq. (5) gives

(6)

pz=pcos[ωc(tt)+θ],py=psin[ωc(tt)+θ],z=z+vFωc{sin[ωc(tt)+θ]sinθ},y=y+vFωc{cos[ωc(tt)+θ]cosθ},
where ωc=eBsvF/p. Using Eq. (6) and employing the Bessel function identity, eiαsinθ=lJl(α)eilθ, we obtain from Eq. (4)

(7)

fωL=eϕvFf00plllωcωlωcJl(kzvFωc)Jl(kzvFωc)ei(ll)θ,
leading to areal density and velocity perturbations

(8)

NωL=002πfωLdθpdp=χeϵ0ekzϕ,

(9)

vzω,q=1N00002πvzfωLdθpdp=eϕωm*vF2kzS,

(10)

χe=N00e2kzm*ϵ0vF2S,S=4ll2ωc2Jl2(kzvFωc)ω2l2ωc2,
where ωc=eBs/m*, m*=ϵF/vF2, χe is the electron susceptibility.

The jump condition on the normal component of displacement vector at x=0

(11)

ϕx]0+ϵgϕx]0=eϵ0NωL
on using NωL from Eq. (8) gives the dispersion relation for magnetoplasmons

(12)

1+ϵg+χe=0,or1+ϵg=4N00e2m*ϵ0kzvF2ll2ωc2Jl2(kzvFωc)ω2l2ωc2.
This equation for one wave number offers many values of frequency, each corresponding to a magnetoplasmonic mode. The frequency of the l’th plasmonic mode lies between lωc and (l+1)ωc, where l1. There is no mode at ω<ωc. For the first mode at long wavelengths, kzvF/ωc1 (small gyroradius limit) only l=1, 1 terms are important and Eq. (12) gives

(13)

ω2=ωc2+N00e2kzm*ϵ0(1+ϵg).
This is equivalent of an upper hybrid wave in a plasma.1 For ωc=0, Eq. (14) reduces to the usual unmagnetized plasmon dispersion relation [cf. Eq. (18), Ref. 21], where frequency scales as kz1/2 and (N00)1/4.

We have solved the dispersion relation, Eq. (12), numerically for the following parameters: ϵg=3, GN00e2/m*ϵ0vFωc=22 (corresponding to N00=1011  cm2, Bs=1  Tesla or any multiple of this areal density when the magnetic field is reduced by the same factor, keeping N00/Bs constant). In Fig. 2, we have plotted the normalized frequency as a function of normalized wavenumber for four magnetoplasmonic modes. At kz0, the modes have frequencies near the harmonics of cyclotron frequency. As kzvF/ωc rises, the frequencies rise. For the first mode, the frequency rises to a maximum of ω=1.7ωc at kzvF/ωc1 and then declines, asymptotically to ωc at large kzvF/ωc. For the second mode, ω rises from 2ωc at kz=0 to 2.7ωc at kzvF/ωc1.8 and then falls off. For the third mode, ω rises from 3ωc, initially very gradually and then faster, attains a maximum ω=3.9ωc at kzvF/ωc2.5 and then falls off. For the fourth plasmonic mode, maximum ω=4.9ωc occurs at kzvF/ωc3.1. In Fig. 3, we have plotted the dispersion curves for G=110 (corresponding to N00=1012  cm2, Bs=2  Tesla). At this higher N00/Bs ratio, the mode frequencies rise more rapidly with kzvF/ωc and attain maximum values closer to the next harmonic. At kzvF/ωcl, the l’th harmonic Bernstein mode has

(14)

ωlωc[1+2N00e2Jl2(kzvFωc)m*ϵ0kzvF2(1+ϵg)].

Fig. 2

Normalized frequency versus normalized wave number for magnetoplasmonic modes, originating at integer multiples of electron cyclotron frequency in graphene mounted on dielectric. The parameters are G=N00e2/m*ϵ0υFωc=22 (N00=1011  cm2, Bs=1  Tesla), ϵg=3.

JNP_11_3_036015_f002.png

Fig. 3

Normalized frequency versus normalized wave number for magnetoplasmonic modes, originating at harmonics of cyclotron frequency in graphene for G=110 (N00=1012  cm2, Bs=2  Tesla), ϵg=3.

JNP_11_3_036015_f003.png

3.

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

(15)

N0T0=N00+Nq,Nq=Nq0eiqz.
Practically ribbon structures are step wise periodic, e.g., N0T0=N00+δN for nλ<z<nλ+a and N0T0=N00 for nλ+a<z<(n+1)λ, where n is an integer, a is the width of a ribbon, and λ is the spatial periodicity. On carrying out Fourier series expansion of areal density in spatial harmonics of wave number q=2π/λ, one obtains the amplitude of the first harmonic Nq0=(2/π)N1sin(qa/2). Higher harmonics have diminishingly smaller amplitudes.

A laser is normally incident on graphene from top

(16)

E0=z^A0ei(ωt+ωx/c),B0=y^(A0/c)ei(ωt+ωx/c).

We choose q such that it equals the magnetoplasmon wave number kz at frequency ω [cf., Eq. (12)]; qω/c. The reflected and transmitted fields are

(17)

E0R=z^A0Rei(ωtωx/c),B0R=y^(A0R/c)ei(ωtωx/c),E0T=z^A0Tei(ωt+ωηgx/c),B0T=y^(A0Tηg/c)ei(ωt+ωηgx/c),
where ηg=ϵg1/2 and we have presumed that the FR of polarization is small. The laser gives rise to perturbation in the electron distribution function f0L, which on solving the linearized Vlasov equation can be written as

(18)

f0L=eA0T1pf00ptpzeiωtdt=ieE0z2f00p[eiθωωc+eiθω+ωc],
giving the drift velocity and surface current density

(19)

v0z=1N00002πvzf0Ldθpdp=ieE0zωm*(ω2ωc2),K0zL=iN00e2E0zωm*(ω2ωc2).
Since the laser wave vector has no component in the plane of electron gyration, the Larmor radius effects do not appear in electron response. The oscillatory velocity v0z beats with the density ripple to produce an areal density perturbation, which on solving the equation of continuity, Nω,qNL/t+(1/2)/z(Nqv0z)=0, turns out to be

(20)

Nω,qNL=q2ωNqv0z.
Here, we have used the complex number identity ReA.ReB=(1/2)Re[A.B+A.B*], where Re stands for the real part of the quantity and * denotes the complex conjugate. This density perturbation acts as driver for the plasmonic wave. Let the potential of the plasmonic wave be ϕ, given by Eq. (2). It creates linear density and velocity perturbations NωL, vω,q given by Eqs. (8) and (9).

Using Nω,qL, Nω,qNL in the jump condition, Eq. (11) with NωL replaced by NωL+Nω,qNL, one obtains

(21)

DA=iNq0e2A0T2m*ϵ0(ω2ωc2),D=1+ϵg+χe.

The oscillatory velocity due to the magnetoplasmonic wave vzω,q beats with the density ripple to produce a surface current density

(22)

K0NL=12Nq*evω,q,
which acts as a source to influence the optical wave. Using this in the jump condition on the magnetic field

(23)

B0y|0+B0y|0=μ0(K0zL+K0zNL),
and employing the continuity of E0z at x=0 one obtains

(24)

A0+A0R=A0T,

(25)

A0A0R=ηgA0T+iN00e2ωm*ϵ0c(ω2ωc2)A0T+Nq0*e2ωS2m*ϵ0cqvF2A.
Equations (23), (25), and (26) yield the amplitude of the transmitted laser field

(26)

A0T=2A01+ηg+iN00e2ωm*ϵ0c(ω2ωc2)+i(Nq0*e2m*ϵ0)2ωS4cqvF2D(ω2ωc2).
The third term in the denominator corresponds to optical conductivity of graphene at the laser frequency. At frequencies one percent away from the cyclotron resonance, (ωωc0.01  ω) this term is insignificant as compared to the first two terms. At cyclotron resonance, it is masked by collisions. However, around that point, the FR of polarization is significant and present formalism is not applicable.

The last term in the denominator of Eq. (26) corresponds to mode coupling of laser to magnetoplasmons. At exact phase matching [i.e., when q equals the wave number of the plasmonic wave given by Eq. (12)], D 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 ν(ff00), where ν is the collision frequency. In the case of the upper hybrid mode, this leads to replacing, in the dispersion relation, ω2 by ω(ω+iν) leading to damping rate ωi=ν/2. For the Bernstein modes, we may replace, in the expression for susceptibility or S, ω2 by ω(ω+iν), giving an imaginary part to D, D=Dr+iDi. At resonance, Dr vanishes and Di can be written as

Di4N00e2νωm*ϵ0kzvF2ll2ωc2Jl2(kzvFωc)(ω2l2ωc2)2.

From Eqs. (21) and (26), one may write the normalized plasmonic field amplitude and laser intensity transmission coefficient at the mode conversion point as

(27)

T=|A0TA0|2ηg=4ηg[1+ηg+ψ2]2+[N00e2ωm*ϵ0c(ω2ωc2)]2,

(28)

|qAA0|=|Nq0e2qm*ϵ0Di(ω2ωc2)|(Tηg)1/2,ψ=(Nq0/N00)2G2Sω/qc4Di(ω2/ωc21),
where GN00e2/m*ϵ0vFωc. One may note that the amplitude of the plasmon electric field |qA| may far exceed the laser field due to strong localization of plasmons. The fractional laser power going into the magnetoplasmonic mode is

(29)

η=1T|A0RA0|2=4ψ(1+ηg+ψ)2.
We have carried numerical calculations for the following parameters Nq0/N0m0=0.5, ϵg=3, G=22, qvF/ωc=1, ν/ωc=2×103 corresponding to N0m0=1011  cm2, m*=1032  kg, q=1.6×105  cm1, ϵF=50  meV, Bs=1  Tesla, ωc=1.6×1013  rad/s. We have plotted in Fig. 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 ω=1.7ωc, the transmission coefficient falls to 42%. At the second plasmonic resonance ω=2.7ωc corresponding to the excitation of the second plasmonic mode, the transmission coefficient is 52%. At the third plasmon resonance, it is 60%.

Fig. 4

Intensity transmission coefficient of laser normally impinged on graphene (with periodic ribbons of wave numbers q=ωc/υF) as a function of normalized laser frequency. The parameters are: G=22 (N00=1011  cm2, Bs=1  Tesla), ϵg=3. Dips at A, B, C refer to excitations of first, second, and third kinetic (Bernstein) magnetoplasmonic modes.

JNP_11_3_036015_f004.png

4.

Mode Conversion in Dielectric Grating

Consider a dielectric grating of thickness d sandwiched between graphene and the substrate. The relative permittivity of the grating region is

(30)

ϵr=ϵg0+ϵq,ϵq=ϵq0e1qz.

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

(31)

ϵ0ϵg02ϕ=ρNL,ρNL=i2qϵ0ϵqE0z.
ρNLd can be treated as effective surface charge density to excite the plasmonic mode. Equivalent areal electron density at ω, q is

(32)

Nω,qNL=deρNL=iqd2eϵ0ϵqE0z.
From the Maxwell’s equation ×B=μ0(J+D/t), one may see that the last term, as a beat of the plasmonic wave field with the permittivity ripple, gives an effective surface current density at frequency ω and wave number kz=0

(33)

K0zNL=iωϵ0ϵq*2(iqϕ)d.
With these expressions for Nω,qNL and K0zNL, Eqs. (11) and (23) lead to
DA=idϵq2ϵ0A0T,A0A0R=ηgA0T+iN00e2ωm*ϵ0c(ω2ωc2)A0Tωϵq*qd2cA,
giving

(34)

A0T=2A01+ηg+iN0  m0e2ωm*ϵ0c(ω2ωc2)+i|ϵq|2ωqd24cD.

The transmission coefficient is given by Eq. (27) with ψ given by

ψ=|ϵq|2ωqd24cωiD/ω.
The treatment is valid for ϵq0/ϵg0<1, qd<1. The effect of the dielectric grating is very similar to that of the density ripple.

5.

Discussion

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 p-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., N1/Nq0<1/2 otherwise, a large number of spatial harmonics of plasmons are excited.22

Acknowledgments

The authors are thankful to Prof. Hao In for fruitful discussions.

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Biography

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.

© The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.
Chuan Sheng Liu, Vipin K. Tripathi, "Kinetic magnetoplasmons in graphene and their excitation by laser," Journal of Nanophotonics 11(3), 036015 (22 August 2017). https://doi.org/10.1117/1.JNP.11.036015 Submission: Received 20 April 2017; Accepted 10 July 2017
Submission: Received 20 April 2017; Accepted 10 July 2017
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