Scattering of electromagnetic waves from a graphene-coated thin cylinder of left-handed metamaterial
Hamid Pashaeiadl Mahin Naserpour Carlos J. Zapata-Rodriguez
physics.optics

	




label1]Hamid Pashaeiadl
label1,label2]Mahin Naserpour
label1]Carlos J. Zapata-Rodríguezcor*
[cor*]Corresponding author:
carlos.zapata@uv.es
[label1]Department of Optics and Optometry and Vision Science, University of Valencia, Dr. Moliner 50, Burjassot 46100, Spain
[label2]Physics Department, College of Sciences, Shiraz University, Shiraz 71946-84795, Iran




	In this paper we explored the scattering behavior of thin cylinders made of LHM and coated by a monoatomic graphene layer.
	A spectral tunability of the resonance peaks is evidenced by altering the chemical potential of the graphene coating, a fact that occurs at any state of polarization of the incident plane wave in opposition to the case of scatterers of dielectric core.
	On the contrary, no invisibility condition can be satisfied for dielectric environments.
	A singular performance is also found for cylinders with permittivity and permeability near zero. 
	Practical implementations of our results can be carried out in sensing and wave manipulation driven by metamaterials.



	Scattering graphene left-handed metamaterials.








§ INTRODUCTION


Fabrication of metal-dielectric structures exhibiting exotic electromagnetic properties ranging from microwaves to visible wavelenghts has recently been enabled by current micro- and nano-technology <cit.>. 
For terahertz frequencies, homogeneous and isotropic artificial materials have effectively been manufactured showing negative permittivity and permeability simultaneously <cit.>. 
Such terahertz metamaterials were proposed for a great variety of applications including polarizers <cit.>, subwavelength imaging <cit.>, far-field focusing <cit.>, and cloaking <cit.>, to mention a few.

The scattering properties of bodies made of metamaterials including negative-index media have been studied considering different shapes like cylinders <cit.>, tubes <cit.>, spheres <cit.>, and also as coatings over dielectrics and metals <cit.>. 

Tunability of such properties are subject to modification of its internal geometrical configuration, which on the other hand is not always easily accessible.

The advent of graphene technology involving for instance the creation of widely tunable devices in the THz regime by simply incorporating atomically-thin layers is currently in fast development <cit.>.
Thus, particles coated by monoatomic films have been proposed for electrochemical sensing, invisible cloaking <cit.>, and improved biomedical applications based on the photothermal effect <cit.>. 
Particularly interesting results a recently published paper concerning a graphene-coated sphere in the case that the sphere is of negative refractive index <cit.>.
Here we investigate the resonant behavior of subwavelenght left-handed cylinders coated by graphene in the terahertz regime, which exhibit an anisotropic response on the polarization of the incident wave field. 
By means of numerical simulations we will show how alterations of the chemical potential in graphene by simply applying a gate voltage can serve to dynamically control the resonant peaks of the resultant scattering cross section of the Mie scatterers.





§ THEORETICAL ANALYSIS




Let us consider a cylindrical dielectric nanowire of radius R and relative permittivity ϵ_d and magnetic permeability μ_d covered by an atomically thin graphene monolayer, and oriented along the z axis, as illustrated in Fig. 1. 
The nanowire is assumed to be placed in an environment medium of the relative permittivity ϵ and magnetic permeability μ; for simplicity, here we assume that the environment medium is lossless.
The wave vector of the incident radiation is directed along the x axis, and in this case two fundamental polarization configurations can be studied: 1) the electric field vector E_in lying in the xy plane (TE^z-polarized plane wave or E_z=0 modes), and 2) the magnetic field H_in lying in the xy plane (TM^z-polarized plane wave or H_z=0 modes). 
The graphene conductivity was described according to the local random phase approximation of the Kubo formula <cit.> which can be expressed as σ=σ_intra+σ_inter, where the intraband contribution can be written as 
 
    σ_intra=2ie^2k_BTπℏ^2(ω+iΓ)ln[ 2 cosh( μ_ch2k_BT) ],

and the interband contribution is expressed as
 
    σ_inter   =   e^24ℏ[ 12 +  1πarctan( ℏω-2 μ_ch2k_BT) ] 
       -   e^24ℏ[ i2πln(ℏω + 2 μ_ch)^2(ℏω-2 μ_ch)^2+(2k_BT)^2].

In this expression, -e is the charge of an electron, ℏ is the reduced Plank's constant, k_B is Boltzmann's constant, T is the temperature, Γ is the charge carriers scattering rate, μ_ch is the chemical potential and ℏω is the photon energy. 
Also, an exp(-iω t) time-dependence is implicit throughout the paper, where ω is the angular frequency, t the time coordinate, and i=√(-1). 

We followed the Lorenz-Mie scattering method, which is given in detail for instance in Refs. <cit.>, to analytically estimate the scattering efficiency of the coated nanowire. 
In the case of a TE^z-polarized plane wave, as illustrated in Fig. <ref>, the electromagnetic field of the incident plane wave may be set as 
 
    H_in = ẑ H_0 exp(ikx) = ẑ H_0∑^+∞_n=-∞ i^nJ_n(kr) exp(inϕ) ,

where r and ϕ are the radial and azimuthal cylindrical coordinates, respectively, H_0 is a constant amplitude, J_n(.) is the Bessel function of the first kind and order n, k_0=ω/c is the wavenumber in the vacuum, and k=k_0√(ϵ)√(μ). 
In Eq. (<ref>) we used the Jacobi-Anger expansion of a plane wave in series of cylindrical waves. 
Since

    E_in = ▽×H_in-iωϵϵ_0 ,
  
the electric field can be read as
 
    E_in = - E_0 ∑^+∞_n=-∞[ r̂i^nnJ_n(kr)r + ϕ̂ k i^n+1 J^'_n(kr)] ]  exp(inϕ) ,

where E_0 = H_0 / ωϵϵ_0.
Here the prime appearing in J^'_n(α) denotes derivative with respect to the variable α. 

The scattered electromagnetic field in the environment medium, r>R, may be set as
 
    H_sca=ẑ H_0 ∑^+∞_n=-∞ a_n i^n H^(1)_n(kr) exp(inϕ),

and
 
    E_sca= - E_0 ∑^+∞_n=-∞ a_n [ r̂ i^nnrH^(1)_n(kr)+ϕ̂ k i^n+1 H^'(1)_n(kr) ] exp(inϕ) ,

where H^(1)_n(.)=J_n(.)+iY_n(.) is the Hankel function of the first kind and order n, and the coefficients a_n must be determined. 
The total magnetic field in the environment medium is simply 

    H_tot=H_in+H_sca .

Finally, the electromagnetic field in the core of the nanowire r<R is expressed as
 
    H_d = ẑ H_0 ∑^+∞_n=-∞ b_n i^n J_n(k_dr) exp(inϕ),

and
 
    E_d = - E_d0∑^+∞_n=-∞ b_n [ r̂ i^n n J_n(k_dr)r + ϕ̂ k_d i^n+1 J^'_n(k_dr) ] exp(inϕ)

where E_d0 = H_0 / ωϵ_dϵ_0, the wavenumber k_d=k_0√(ϵ_d)√(μ_d). 
Also, the size parameter of the cylinder is defined as the product k R throughout the paper.

The Lorenz-Mie scattering coefficients a_n and b_n are determined by means of the proper boundary conditions. 
Due to the existence of graphene surface conductivity, the boundary conditions at r=R for the case of TE^z polarized waves are
 
    ϕ̂·E_d = ϕ̂· (E_sca + E_in)

and
 
    ẑ·H_d = ẑ· (H_sca+H_in) + ϕ̂·E_dσ,

to ensure the continuity of the tangential components of the electromagnetic fields at the graphene layer. 
Since the electric field is set in the xy plane, only the ϕ-component of the effective conductivity must be taken into account. 
These two equations given above can be set as
 
    k_dϵ_db_nJ^'_n(k_dR) = kϵ[ J^'_n(kR)+a_nH^'(1)_n(kR) ],

and
 
    b_n J_n(k_dR) = J_n(kR) + a_nH^(1)_n(kR)-σik_dωϵ_dϵ_0b_nJ^'_n(k_dR),

respectively, where the coefficients of different index n can be treated separately due to the linear independence of the multipolar components of the wave field. 
Finally, we obtain the analytical expression of the scattering coefficients
  
    a_n    =   { -k ωϵ_d ϵ_0 J_n(k_dR) J^'_n(kR) .   
          . + k_d J^'_n(k_dR) [ ωϵϵ_0J_n(kR)-ikσ J^'_n(kR) ] } /       
          { -k_dωϵϵ_0H^(1)_n(kR)J^'_n(k_dR) .    
          . + kH^'(1)_n(kR)[ωϵ_dϵ_0J_n(k_dR)+ik_dσ J^'_n(k_dR)] },

and
  
    b_n    =    k ωϵ_d ϵ_0 [ H^'(1)_n(kR) J_n(kR) - H^(1)_n(kR) J^'_n(kR) ]  /   
          { -k_dωϵϵ_0H^(1)_n(kR)J^'_n(k_dR) .   
          . + kH^'(1)_n(kR) [ ωϵ_dϵ_0J_n(k_dR) + i k_d σ J^'_n(k_dR) ] } .


As seen above, it is possible to evaluate analytically the scattering coefficients a_n given in Eq. (<ref>), which enable the estimation of the scattering efficiency as <cit.>
   
    Q_sca=2kR∑^-∞_n=+∞|a_n|^2.




 §.§ Equations for TM^z polarziation


Let us point out that the case of TM^z incident wave fields can be simply determined by means of the duality principle <cit.>. 
In this case, we may right

    E_in = ẑ E_0 exp(ikx) = ẑ E_0∑^+∞_n=-∞ i^nJ_n(kr) exp(inϕ) ,

for the incident plane wave, 

    E_sca=ẑ E_0 ∑^+∞_n=-∞ c_n i^n H^(1)_n(kr) exp(inϕ),

for the scattered field, and

    E_d = ẑ E_0 ∑^+∞_n=-∞ d_n i^n J_n(k_dr) exp(inϕ),

for the electric field inside the LHM core.

The boundary conditions at r=R for TM^z-polarized waves reads as

    ẑ·E_d=ẑ· (E_sca+E_in) ,

and

    ϕ̂·H_d=ϕ̂· (H_sca+H_in)-ẑ·E_dσ .

Therefore, the analytical expression of the scattering coefficients can be deduced as

    c_n    =   { -k_dμ J^'_n(k_dR)J_n(kR) .     
          . + μ_dJ_n(k_dR) [ kJ^'_n(kR) + iωμμ_0σ J_n(kR) ] } /    
          {-kμ_dH^'(1)_n(kR)J_n(k_dR)  .    
          . + μ H^(1)_n(kR) [ k_dJ^'_n(k_dR)-iωμ_0μ_dσ J_n(k_dR) ] },

and

    d_n    =   { kμ_d [ H^(1)_n(kR)J^'_n(kR)-H^'(1)_n(kR)J_n(kR) ] } /     
          { -kμ_dH^'(1)_n(kR)J_n(k_dR) .         
          . + μ H^(1)_n(kR) [ k_dJ^'_n(k_dR)-iωμ_0μ_dσ J_n(k_dR) ] }.

Finally, the scattering efficiency can be obtained by using <cit.>

    Q_sca=2kR∑^-∞_n=+∞|c_n|^2.




§ SIMULATIONS AND RESULTS




 §.§ TE^z polarized waves




In Fig. <ref> we show the scattering efficiency spectrum of an infinitely-long cylinder illuminated by a TE^z-polarized plane wave, that is the electric field lies in the xy plane transversally to the cylinder axis, the latter made of a material of relative permittivity ϵ_d and permeability μ_d.
In the simulations, R = 0.5 μm is the radius of the cylinder, which in addition is wrapped by a graphene monolayer of chemical potential μ_ch = 0.5eV. 
As assumed from here on in all calculations, the temperature is T = 300K and the charge carriers scattering rate of graphene is Γ = 0.1meV.
As the most characteristic case in practical implementations, we will consider an environment medium with ϵ = 1 and μ = 1.
For purely-dielectric cylinders, a set of a peak and a minimum in the scattering-efficiency pattern is observed, which are closely located <cit.>.
Note that the minimum has been proposed as an effective mechanism of invisibility <cit.>.
However, when both ϵ_d and μ_d are simultaneously negative, as occurs in LHM, the invisibility condition cannot be found.
As a result, a single resonant peak dominates the scattering efficiency spectrum, appearing also non-neglecting secondary peaks related with higher-order resonances.
In general, the full width at half maximum of the main resonances are notably higher for cylinders made of LHM in comparison with scatterers having a dielectric core.
Interestingly, no resonance can be identified for the case ϵ_d = -2 and μ_d = -2.





The resonance peak can be significantly modulated by means of the chemical potential in the graphene coating, an effect that can be achieved for instance by means of an applied gate voltage.
Figure <ref> shows the spectrum of the scattering efficiency in the case of a graphene-coated cylinder made of a LHM of ϵ_d = -0.3 and μ_d = -1, varying the chemical potential μ_ch from 0.1eV to 2.0eV (the latter probably far from realistic setups), in terms of the size paramenter k R.
An increasing value of μ_ch leads to a significant blue shift of the resonant peak frequency; note that the size parameter is directly proportional to the radiation frequency.
It is worth highlighting that such resonances become extinct in the case of a bare cylinder (not shown in the figure).
The mentioned blue-shift at increasing μ_ch occurs not only to the main peak but also to higher-order multipolar resonances.

In order to further analyze the dependence of the resonances upon the cylinder radius, we evaluated the scattering efficiency spectra for a graphene-coated cylinder for a varying parameter R, when the core permittivity ϵ_d = -0.3, permeability μ_d = -1, and chemical potential of graphene μ_ch = 0.5eV are kept fixed. 
In general, the cylinder radius can also modify the frequency of the resonances substantially; a longer radius R displaces the resonant frequency to lower values, affecting not only to the main peak but also to secondary resonances, as shown in Fig. <ref>.
This fact has also been observed in graphene-coated dielectric cylinders <cit.>.



In Fig. <ref>(a) we represent the scattering efficiency evaluated in the ϵ_d μ_d plane for a cylinder coated by a graphene monolayer, maintaining fixed its radius R = 0.5 μm, chemical potential of graphene μ_ch = 0.5eV, and its wavelength λ = 14.48 μ m.
The main peak resonance is nearly mirrored both for dielectrics where ϵ_d and μ_d are positive, and for LHM where permittivities and permeabilities remain negative simultaneously.
However, this is no longer true for near-zero values of ϵ_d and μ_d.
In such case, a resonance arises at ϵ_d = -0.3 independently from the (moderate) value of μ_d, as shown in Fig. <ref>(b).





 §.§ TM^z polarized waves




Now let us analyze the scattering behavior of the graphene-coated cylinder when it is illuminated by a TM^z-polarized plane wave, that is the electric field is parallel to the cylinder axis.
In Fig. <ref> we show Q_sca for a cylinder made of a material of relative permittivity ϵ_d and permeability μ_d taking some different values; in the simulations, its radius is R = 0.5 μm and the chemical potential of graphene is μ_ch = 0.5eV. 
For such polarization, resonances cannot surge when the cylindrical core is purely dielectric, but a minimum of efficiency features the pattern at low frequencies <cit.>.
On the other hand, some resonance peaks (such that of the lowest frequency located at ≈ 80THz) are evident for ϵ_d = -0.3 and μ_d = -1.
Such resonances not always emerge in LHM cylinders.
For instance, it does not occur for both near zero (and negative) ϵ_d and μ_d when the size parameter remains below the unity.
By the way, the efficiency spectra are now practically identical for near-zero metamaterial cores.
Furthermore, the invisibility condition cannot be found when both ϵ_d and μ_d are simultaneously negative.
Finally, note that curves drawn for different radii R are not matching, even when using the size parameter in the representation of Q_sca, due to the dispersive character of the graphene surface conductivity.



Figure <ref> shows the spectrum of the scattering efficiency, in terms of the size parameter, in the case of a graphene-coated cylinder made of a LHM of ϵ_d = -0.3 and μ_d = -1, varying the chemical potential μ_ch from 0.2eV to 0.8eV.
Importantly, the set of multipolar peaks originated by polariton resonances are also strongly governed by the chemical potential of the graphene coating.
Again, higher values of μ_ch are associated with higher frequencies of a given resonance.
Finally, a bare cylinder where graphene is not coating the scatterer cannot experience such resonances at so low size parameters. 



Here, the cylinder radius also alters the frequency of the resonances significantly, as depicted in Fig. <ref> for a graphene-coated cylinder of core permittivity ϵ_d = -0.3, permeability μ_d = -1, and chemical potential of graphene μ_ch = 0.5eV. 
The longer the radius R of the scatterer the lower frequency where resonances emerge, a fact that involves the main peak and secondary resonances; note that the size parameter itself includes the parameter R and therefore a convenient interpretation of Fig. <ref> must be taken carefully.
Note also that the scattering efficiency is notably enhanced for ultra-thin cylinder with a radius below R = 0.2 μm, a fact that can be difficult to verify in practice.



Finally, the scattering efficiency is determined in the ϵ_d μ_d plane for a cylinder coated by graphene, keeping its radius R = 0.5 μm, chemical potential of graphene μ_ch = 0.5eV, and its wavelength (= 14.5 μ m) as fixed parameters.
The results are depicted in Fig. <ref>(a).
As inferred from the duality principle <cit.>, we again have the main peak resonance mirrored both in the quadrant for dielectrics and that for LHM.
For near-zero values of ϵ_d and μ_d, however, now the resonance emerges at values of μ_d in the vicinities of ≈ -1, apart from the value of ϵ_d, as evidenced in Fig. <ref>(b).





§ CONCLUSIONS


We investigated the scattering behavior of thin cylinders made of LHM and coated by a monoatomic graphene layer.
Many of the properties we found are evident at any state of polarization (we analyzed TM^z or TE^z) of the incident plane wave, as for instance: 
(1) a set of resonant peaks emerge for size parameters below the unity, except maybe for ϵ_d and μ_d near-zero. Note that a bare cylinder where graphene is not coating the scatterer cannot experience such resonances at so low size parameters; 
(2) no invisibility condition can be satisfied in opposition to the case of dielectric cores; 
(3) the resonance peaks can be significantly modulated by means of the chemical potential in the graphene coating, an effect that can be achieved for instance by means of an applied gate voltage. In this case, an increasing value of μ_ch leads to a significant blue shift of the resonant peak frequencies; 
(4) the radius of the cylinder also alters the frequency of the resonances significantly. 
The longer the radius R of the scatterer the lower frequency where resonances (both main and secondary peaks) emerge.

Finally, we studied the scattering efficiency in the ϵ_d μ_d plane.
We revealed that the main peak resonance is nearly mirrored both for dielectrics where ϵ_d and μ_d are positive, and for LHM where permittivities and permeabilities remain negative simultaneously.
However, there is an exception in the case of near-zero values of ϵ_d and μ_d; for TE^z-polarized incident plane waves, a main resonance arises for varying values of μ_d but keeping fixed ϵ_d, however if illumination is carried out by TM^z polarization, the resonance surges at values of nearly-invariant μ_d, apart from the value of ϵ_d.

To conclude, practical implementations of our results can be achieved in sensing and wave manipulation driven by metamaterials.





§ ACKNOWLEDGMENTS


This work was supported by the Spanish Ministry of Economy and Competitiveness (MINECO) (TEC2014-53727-C2-1-R). 



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