Pade approximant spectral fit for FDTDsimulation of graphene in the near

infrared

Adam Mock*ET 100 Central Michigan University,

Mount Pleasant, MI 48859, USA

*mock1ap@cmich.edu

http://people.cst.cmich.edu/mock1ap/

Abstract: A parameterization of the dispersive conductivity ofhighly-doped graphene has been developed and is presented for use infinite-difference time-domain simulation of near infrared graphene-basedphotonic and plasmonic devices. The parameterization is based on fittinga Pade approximant to the conductivity arising from interband electronictransitions. The resulting parameterization provides an accurate spectralrepresentation of the conductivity in the wavelength range 1.3 − 2.3μmwhich is important for near infrared graphene plasmonics. Finite-differencetime-domain simulations of straight graphene plasmonic waveguides ofinfinite and finite width are presented.

© 2012 Optical Society of America

OCIS codes: (000.4430) Numerical approximation and analysis; (160.4236) Nanomaterials;(230.7370) Waveguides; (240.6680) Surface plasmons; (240.6690) Surface waves; (350.4238)Nanophotonics and photonic crystals.

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1. Introduction

Graphene is an exciting new material that is receiving significant attention in the research com-munity as a building block for both flexible and inherently novel devices. Graphene is a two-dimensional monoatomic layer of carbon arranged in a honeycomb lattice [1,2]. It has a numberof interesting features of interest from both a fundamental physics standpoint as well as an en-gineering one. The physics of graphene is novel in that it represents one of the only examplesof a truly two-dimensional solid that can be completely isolated from a substrate or supportingstructure. Due to graphene’s particular crystal structure, the electrons in graphene lose theireffective mass and behave like Dirac fermions [3] with velocities approaching 0.01c. Grapheneis a zero gap semiconductor, and its electronic band structure is conical about its K points.

Already the interesting photonic properties of graphene have been explored for a range of ap-plications. Demonstrations of graphene-based photodetectors capable of operating frequenciesas high as 40 GHz at a wavelength of 1.55μm have been carried out, and the authors estimatethat the peak bandwidth of a graphene photodetector could be as large as 500 GHz [4]. Subse-quent work demonstrated error free detection at 10 Gbps [5]. Fabrizio et al. have explored theeffect of a graphene layer on a photonic metamaterial which resulted in resonance shifts andtransmission enhancements when compared to devices without the graphene [6]. Theoreticalproposals have been published for modulators operating at 10μm whose maximum speed islimited by a 0.1 ps carrier relaxation time [7] and graphene-based parallel plate waveguides de-signed to operate at terahertz frequencies [8]. Blake et al. demonstrated liquid crystal switchingdevices using graphene as a transparent electrode [9]. The nonlinear saturable absorption prop-erties of graphene have been intensely studied recently in demonstrations of passively mode-locked lasers emitting around 1.5 μm with pulse durations shorter than 1 ps [10–13].

Recently Vakil and Engheta discussed metamaterials and transformation optics in which thesurface plasmon polariton (SPP) properties of highly doped graphene play a key role [14]. Us-ing graphene for SPP devices instead of noble metals such as gold or silver is advantageousbecause the material properties of graphene are tunable via external electric or magnetic field,and it supports SPPs with longer propagation lengths. Metamaterials, plasmonics and integratedphotonics deal with electromagnetic behavior associated with micrometer and nanometer geo-metrical features. Typically finding the electromagnetic fields in these subwavelength geome-tries requires numerical approaches to solve Maxwell’s equations. One of the most popularnumerical techniques is the finite difference time domain (FDTD) method [15, 16] which is anexplicit discretization of Maxwell’s equations in space and time. FDTD is attractive due to itsgenerality, ease of implementation, linear scaling in execution time with problem size, straightforward parallelizability, favorable speedup with parallelization and ability to handle dispersiveand nonlinear materials. It is often the default method when modeling large three-dimensionalirregular geometries.

In the frequency domain the relationship between the electric field and current density forisotropic materials is given by �J(ω) = σ(ω)�E(ω). Because FDTD solves Maxwell’s equa-tions in the time domain, incorporating dispersive materials requires special handling. Oneof the most efficient approaches uses auxilary differential equations (ADE) [16–18] to model

#164810 - $15.00 USD Received 15 Mar 2012; revised 23 Apr 2012; accepted 26 Apr 2012; published 7 May 2012(C) 2012 OSA 1 June 2012 / Vol. 2, No. 6 / OPTICAL MATERIALS EXPRESS 773

the dispersive material properties in the time domain. The ADE method requires that the dis-persive conductivity be represented as a ratio of polynomials in ω . Popular forms include theDrude [19], Debye [17], Lorentz [18] and Cole [20] models. While the intraband relaxation con-tribution to graphene’s conductivity can be represented using the Drude model alone, the inter-band transition component results in a dispersive conductivity function that cannot be directlyincorporated into FDTD using the ADE method. In this work a Pade approximant [16,21–23] isused to represent the interband term in a form that can be incorporated into FDTD. The generalrecipe is described for performing the fit, fitting parameters are given and modeling results ofSPPs in graphene are presented.

Recent work has modeled an infinite graphene sheet using FDTD [24]; however, only theintraband (Drude) conductivity was used. While neglecting the interband term may providegood results for long wavelengths, it must be included when considering optical behavior withoptical energies near the chemical potential. This work presents a form for the interband con-ductivity term that is especially important for near infrared SPP modeling in graphene. FDTD isan extremely powerful tool for nanophotonic modeling [25–28], and the results presented herewill further enable discovery and design of photonic devices that exploit the exciting propertiesof graphene.

2. Graphene conductivity

Graphene has a frequency-dependent complex-valued conductivity which is shown in Fig. 1 forvalues of chemical potential ranging from 0.1 to 0.6 eV. The chemical potential is a functionof applied static electric field, applied magnetic field and chemical doping [24, 29–31]. Ex-perimental measurements of room temperature, undoped graphene have confirmed that in thefrequency regime dominated by interband transitions, the real part of the optical conductivityis approximately constant with a value of e2/4h [32–34].

For photon energies below two times the chemical potential, the imaginary part of the opticalconductivity is negative opening the possibility for transverse magnetic SPP waves [14,31,35].From the time-harmonic Maxwell equations, the relationship between conductivity and electricpermittivity is given by

ε(ω) = εr(ω)− iσ(ω)/ω (1)

where εr(ω) is the real part of the electric permittivity and −σ(ω)/ω is the imaginary part [36].However, if the conductivity σ(ω) = σr(ω)+ iσi(ω) is complex, then the electric permittivityis more clearly written by performing the complex arithmetic as in

ε(ω) = εr(ω)+σi(ω)/ω − iσr(ω)/ω. (2)

This demonstrates that a negative imaginary part of the conductivity σi(ω) will result in a neg-ative real part of the electric permittivity when εr(ω)+σi(ω)/ω < 0. In this case the graphenelayer behaves like a thin metal capable of supporting SPP waves.

In the absence of an applied electric or magnetic field, the conductivity is associated withelectronic intraband relaxation and interband transitions according to σ = σintra +σinter. Fol-lowing the notation of [24, 31], an expression for the conductivity as a function of frequency(ω), chemical potential (μc), carrier scattering rate (Γ) and temperature (T ) is given by

σ(ω,μc,Γ,T ) =je2/π h2

ω − j2Γ

∫ ∞

0ε(

∂ fd(ε)∂ε

− ∂ fd(−ε)∂ε

)dε +

+ je2/π h2(ω − j2Γ)∫ ∞

0

fd(ε)− fd(−ε)(ω − j2Γ)2 −4(ε/h)2 dε (3)

#164810 - $15.00 USD Received 15 Mar 2012; revised 23 Apr 2012; accepted 26 Apr 2012; published 7 May 2012(C) 2012 OSA 1 June 2012 / Vol. 2, No. 6 / OPTICAL MATERIALS EXPRESS 774

where fd(ε) = 1/(e(ε−μc)/kBT +1) is the Fermi-Dirac distribution function and kB is the Boltz-mann constant. The first integral describes the intraband carrier relaxation contribution, and thesecond integral describes the interband carrier transition contribution. The first integral can beevaluated resulting in

σintra(ω,μc,Γ,T ) =− j8σ0kBT/hω − j2Γ

[μc

kBT+2ln(e−μc/kBT +1)

](4)

where σ0 =πe2

2h . For the case kBT � |μc|, hω , the second integral may be approximated by

σinter(ω,μc,Γ,T )≈− jσ0

πln

[2|μc|− (ω − j2Γ)h2|μc|+(ω − j2Γ)h

]. (5)

In this work, highly doped graphene is considered with μc = 0.6 eV, so the condition kBT �|μc|, hω is satisfied even at room temperature T = 300 K. A useful property of graphene is thetunability of its chemical potential with an external DC electric or magnetic field. However, inthis work chemically doped graphene is considered where the relationship between the carrierdensity n and chemical potential is given by

n =2

π h2v2F

∫ ∞

0ε[ fd(ε)− fd(ε +2μc)]dε (6)

where the Fermi velocity vF has a value around 106 m/s. In [31, 35, 37], values of the chemicalpotential less than or equal to 0.2 eV were discussed in the context of practical graphene-basedelectromagnetic devices. These small values of the chemical potential are consistent with lowto moderate levels of chemical doping in graphene. Because the imaginary part of the conduc-tivity is negative for optical excitations less than two times the chemical potential, these SPPdevices are expected to function only for wavelengths longer than 3μm. This work is moti-vated by integration of graphene into standard fiber optic components operating near 1.55μm.Therefore, in order for SPP waves to exist in these wavelength ranges, the chemical potentialof the graphene will need to be at least 0.5 eV. Zhao et al. reported one example of how sucha large doping level may be achieved in graphene [38]. Graphene flakes were submerged insolutions of 14 molar concentration of H2SO4 and the doping concentration was monitored insitu using Raman spectroscopy. They achieved carrier concentrations of 0.2−0.4×1013 cm−2

in bilayer graphene which is the predicted carrier concentration necessary for near infraredgraphene plasmonics [35].

Figure 1(a) displays the real and imaginary parts of the conductivity of graphene normalizedto σ0 for different values of the chemical potential (μc) ranging from 0.1 eV to 0.6 eV. Theimaginary part of the conductivity passes from positive to negative when hω < 2μc. As μc isincreased to values exceeding 0.5 eV, the imaginary part of σ becomes negative for wavelengthslonger than 1.5 μm.

3. Pade fit to interband conductivity term

A Pade approximant fits a rational polynomial of the form

a0 +a1ω + . . .+aMωM

1+b1ω + . . .+bNωN = σinter(ω) (7)

to a set of data points or samples [16, 21–23]. In this work a polynomial of the form

a0 +a1 jω +a2( jω)2

1+b1 jω +b2( jω)2 = σinter(ω) (8)

#164810 - $15.00 USD Received 15 Mar 2012; revised 23 Apr 2012; accepted 26 Apr 2012; published 7 May 2012(C) 2012 OSA 1 June 2012 / Vol. 2, No. 6 / OPTICAL MATERIALS EXPRESS 775

Fig. 1. (a) Plot of the real and imaginary part of the graphene conductivity calculated usingEq. (3) for different values of μc. Other physical values are set at T = 300 K and Γ =11meV/h. (b) Plot of the real and imaginary part of the intraband conductivity term forμc = 0.6 eV. (b) Plot of the real and imaginary part of the interband conductivity term forμc = 0.6 eV.

is used. Substituting jω for ω preserves the required conjugate symmetry (σ(ω) = σ∗(−ω)) inthe Pade fit assuming the ai and bi are real. Polynomials of order M = N = 2 in the numeratorand denominator lead to straight forward and efficient FDTD implementation. Use of higherorder polynomials did not result in significantly improved fits. The fitting procedure consists ofmoving the denominator polynomial to the right side and then moving all terms containing ωback to the left as shown below.

a0 +a1 jω +a2( jω)2 −b1 jωσinter(ω)−b2( jω)2σinter(ω) = σinter(ω) (9)

This procedure results in a linear equation in the expansion coefficients ai and bi. In order toenforce purely real expansion coefficients Eq. (9) must be separated into its real and imaginaryparts according to

a0 −a2ω2 +b1ωIm[σinter(ω)]+b2(ω)2Re[σinter(ω)] = Re[σinter(ω)] (10)

anda1ω −b1ωRe[σinter(ω)]+b2(ω)2Im[σinter(ω)] = Im[σinter(ω)]. (11)

Letting γ(ω) = Re[σinter(ω)] and η(ω) = Im[σinter(ω)], the resulting matrix equation is

⎡⎢⎢⎢⎢⎣

1 0 −ω21 ω1η(ω1) ω2

1 γ(ω1)0 ω1 0 −ω1γ(ω1) ω2

1 η(ω1)1 0 −ω2

2 ω2η(ω2) ω22 γ(ω2)

0 ω2 0 −ω2γ(ω2) ω22 η(ω2)

1 0 −ω23 ω3η(ω3) ω2

3 γ(ω3)

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

a0

a1

a2

b1

b2

⎤⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎣

γ(ω1)η(ω1)γ(ω2)η(ω2)γ(ω3)

⎤⎥⎥⎥⎥⎦

where ω1, ω2 and ω3 are three distinct frequency values at which σinter(ω) is sampled. Figure 2displays the real and imaginary parts of the resulting Pade fit to the interband conductivity termassociated with μc = 0.6 eV. The fit is very good for wavelengths longer than 1.0μm. Table 1displays the coefficients used in the fit.

Table 1. Pade Expansion Coefficients for Fit Displayed in Fig. 2

a0 a1 a2 b1 b2

-9.114e-28 1.674e-20 1.343e-36 8.082e-17 2.148e-31

#164810 - $15.00 USD Received 15 Mar 2012; revised 23 Apr 2012; accepted 26 Apr 2012; published 7 May 2012(C) 2012 OSA 1 June 2012 / Vol. 2, No. 6 / OPTICAL MATERIALS EXPRESS 776

Fig. 2. Comparison between the interband conductivity calculated using Eq. (5) and thePade approximant fit. (a) Real part, (b) imaginary part.

Previous work has determined that the dispersion relation for a transverse magnetic (TM)SPP wave on an infinite graphene sheet is given by β = k0[1+ 2/η0σ(ω)]1/2 where k0 isthe free space wave number, and η0 is the impedance of free space [14, 31]. Figure 3 dis-plays the dispersion relation of a TM SPP wave using the exact formulas (Eqs. (4) and (5))for σ = σintra +σinter as well as the Pade approximation to σinter. For comparison, the dis-persion relation using only the intraband term σ = σintra is shown to illustrate the importanceof including the interband term in the near infrared region. The shaded yellow region showsthe frequency region where a significantly improved fit is obtained. The yellow shaded re-gion also corresponds to the targeted frequency region for near infrared graphene plasmonicswhich makes this fitting procedure highly effective for this wavelength region. As expected,for shorter wavelengths above the yellow shaded region, there is less agreement between thefrequency dispersion using the Pade fit and the exact formulas.

Fig. 3. Comparison between infinite graphene sheet TM SPP dispersion relations calculatedusing the actual interband dispersion term (Eq. (5)) and the Pade fit. Agreement is very goodfor wavelengths greater than 1.3 μm. Also shown is the SPP dispersion relation calculatedusing only the intraband (Drude) term showing that including the interband term is essentialfor accurate SPP simulation.

#164810 - $15.00 USD Received 15 Mar 2012; revised 23 Apr 2012; accepted 26 Apr 2012; published 7 May 2012(C) 2012 OSA 1 June 2012 / Vol. 2, No. 6 / OPTICAL MATERIALS EXPRESS 777

4. Compact FDTD

To demonstrate the quality of the Pade fit to the interband conductivity, the compact FDTD(cFDTD) method is used to simulate an infinite sheet. cFDTD is a fully vectorial numericalmethod for solving Maxwell’s equations [39–45]. For geometries uniform along the z direction,the electric and magnetic fields can be written as

Fβi (x,y,z) = f β

i (x,y)exp( jβ z) (12)

where Fβi (x,y,z) refers to the ith vector component of the electric or magnetic field, z is the

propagation direction, and β is the propagation constant. If Eq. (12) is inserted into Maxwell’scurl equations in cartesian coordinates, derivatives with respect to z can be evaluated explicitlyand replaced with jβ . The FDTD method can then be used to solve for the two-dimensionalfield quantities f β

i (x,y). In this case, the problem still incorporates six vector components, butthe computational domain requires only the two-dimensional cross section of the waveguide.Examples of the FDTD update equations for Ey and Hx are shown below.

Ey|n+1i, j+ 1

2= Ey|ni, j+ 1

2− Δt

εi, j+ 12Δx

[Hz|n+12

i+ 12 , j+

12−Hz|n+

12

i− 12 , j+

12]− βΔt

εi, j+ 12

Hx|n+12

i, j+ 12− Δt

εi, j+ 12

Jx|n+12

i, j+ 12

(13)

Hx|n+12

i, j+ 12= Hx|ni, j+ 1

2− Δt

μi, j+ 12Δy

[Ez|ni, j+1 −Ez|ni, j]+βΔt

μi, j+ 12

Ey|ni, j+ 12

(14)

Transforming the relationship Jy(ω) = σ(ω)Ey(ω) to the time domain by substituting jω →∂/∂ t results in an auxilary differential equation relating Jy back to Ey. The approach allows forthe analysis of waveguide structures one user-specified β -value at a time. The method is labeled“compact” because it takes advantage of geometry uniformity along the propagation directionrequiring only a two-dimensional computational grid that represents the cross-section of thegraphene geometry. The computational domain is illustrated in Fig. 4(a). Because the graphenesheet is assumed to be infinite in both the x and z directions, derivatives with respect to x canbe set to zero. This implies a SPP wave uniform along x and propagating along z. The resultingcomputational domain is only one-dimensional.

The thin atomic monolayer of graphene is represented as a single grid point along the ydirection. It should be noted that the conductivity expressions presented in Section 2 correspondto a surface conductivity associated with the atomically thin graphene sheet. The conductivityappearing in the three-dimensional Maxwell’s equations is a volume conductivity with units ofSiemens per meter [14, 24]. In order to incorporate an effective volume conductivity, σ/Δy isused in the simulation.

The spatial discretization was no larger than Δy = 2.5nm, and a time step no larger than 0.8times the Courant stability limit is used. The spatial domain included 200 discretization pointsalong y. The boundary is truncated with 15 layers of perfectly matched layers (PMLs). TMSPP waves are excited on the graphene sheet using a broadband initial condition. The initialexcitation consists of a one time step pulse assigned to one or more field components at lowsymmetry points in close proximity to the graphene sheet. A time sequence on the order of afew hundred thousand time steps is recorded. The discrete Fourier transform is taken, and Padeinterpolation is applied to extract the propagation frequency [23, 46]. This process is repeatedfor all β values of interest. A comparison between the FDTD calculated dispersion relation andthe exact result is shown in Fig. 4(c). The agreement is good with a maximum error no largerthan 2%. Figure 5 displays the dominant electric field components of the TM SPP wave. Both

#164810 - $15.00 USD Received 15 Mar 2012; revised 23 Apr 2012; accepted 26 Apr 2012; published 7 May 2012(C) 2012 OSA 1 June 2012 / Vol. 2, No. 6 / OPTICAL MATERIALS EXPRESS 778

Fig. 4. (a) Diagram of computational domain for FDTD analysis of infinite graphene sheet.A two-dimensional region is shown, but only a one-dimensional grid is used representingthe geometry variation in the y direction. (b) Comparison between exact dispersion relation(using both the actual interband conductivity formula Eq. (5) and the Pade fit) and thedispersion calculated using the FDTD method.

Fig. 5. Cross sectional field profiles of dominant electric field components of TM SPPmode supported by an infinite graphene sheet. On the right a one-dimensional plot of thefield along the y direction is shown. For an infinite sheet, only the one-dimensional fieldsare calculated in the FDTD simulation. The two-dimensional field profiles are created byextruding the one-dimensional plot along the x direction. The wavelength is 1.82 μm.

#164810 - $15.00 USD Received 15 Mar 2012; revised 23 Apr 2012; accepted 26 Apr 2012; published 7 May 2012(C) 2012 OSA 1 June 2012 / Vol. 2, No. 6 / OPTICAL MATERIALS EXPRESS 779

fields are shown at the same moment in time. The tight field confinement to the graphene layeris clear.

The cFDTD method is capable of calculating the propagation length, and a typical calculatedvalue is 10λ or (20μm) which is consistent with [35]. These results generally agree with previ-ous theoretical work which incorporate electron-phonon losses via the relaxation time approxi-mation. However, the relaxation time can be influenced by a number of experimental parametersincluding doping mechanism and temperature and can vary depending on sample preparationmethod. However, one of the strengths of the presented fitting procedure is that it can incor-porate more realistic experimental measurements of the material properties of graphene in thenear infrared region.

5. Finite width graphene plasmonic waveguide

To conclude this work, results are presented for a graphene SPP waveguide of finite width.The SPP modes of a finite width structure do not have closed form solutions which makesa numerical calculation necessary. A complete modal analysis of the SPP modes of a finitewidth graphene waveguide is beyond the scope of the current work. However, an example ofthe dispersion and electric fields of a TM-like mode is shown in Fig. 6. The FDTD simulationparameters are the same as those in the previous section with the exception of using a two-dimensional grid with 300 grid points along x and a Δx no larger than 10 nm. As the width ofthe graphene is reduced, the dispersion relation is shifted to higher frequencies consistent withwell known waveguide theory. The field profiles display the same localized field behavior andsymmetry along the y direction as the fields for the infinite graphene sheet (Fig. 5). However,the fields along the x direction are no longer uniform and display higher order mode behavior.

Fig. 6. (a) Waveguide dispersion for an infinite graphene sheet (blue circles) and a graphenesheet 1.0 μm in width (red circles). (b) Cross sectional field profiles of dominant electricfield components of 1.0 μm wide graphene SPP mode. The wavelength is 6.1 μm (corre-sponding to left most red circle data point).

6. Conclusion

This work presents a method for introducing the graphene interband conductivity term intoFDTD simulations. The fitting procedure is based on Pade interpolation which leads to straightforward integration with FDTD via the ADE method. The approach is illustrated for highlydoped graphene which supports SPP waves in the near infrared. This work is important for

#164810 - $15.00 USD Received 15 Mar 2012; revised 23 Apr 2012; accepted 26 Apr 2012; published 7 May 2012(C) 2012 OSA 1 June 2012 / Vol. 2, No. 6 / OPTICAL MATERIALS EXPRESS 780

modeling and design of near infrared graphene plasmonic devices. FDTD modeling resultsfor relatively simple structures are presented to demonstrate the method; however, much morecomplicated SPP waveguiding structures may be modeled in future studies such as multilayerstructures or graphene layers with curved or non-planar shapes. Furthermore, these results arenot restricted to waveguiding geometries; three-dimensional structures such as curved waveg-uides or graphene-based SPP microcavities can also be investigated with this method.

Acknowledgments

This research is supported in part by Funds for Advancing Research from the College of Scienceand Technology at Central Michigan University.

#164810 - $15.00 USD Received 15 Mar 2012; revised 23 Apr 2012; accepted 26 Apr 2012; published 7 May 2012(C) 2012 OSA 1 June 2012 / Vol. 2, No. 6 / OPTICAL MATERIALS EXPRESS 781