Design of ultra-broadband graphene absorberusing circuit theoryAMIN KHAVASI

Electrical Engineering Department, Sharif University of Technology, Tehran 11155-4363, Iran (khavasi@sharif.edu)

Received 17 April 2015; revised 22 July 2015; accepted 23 July 2015; posted 24 July 2015 (Doc. ID 238348); published 19 August 2015

I propose a novel method for designing a broadband THz absorber by using periodic arrays of graphene ribbonson a Salisbury-screen-like structure. The recently proposed analytical circuit model of graphene arrays is used forobtaining analytical expressions for the input admittance of the proposed device. The input admittance is thenadjusted to be closely matched to the free space in a wide frequency range. Consequently, it is demonstrated that abandwidth of 90% absorption can be extended up to 100% of the central frequency with only one layer ofpatterned graphene. © 2015 Optical Society of America

OCIS codes: (310.3915) Metallic, opaque, and absorbing coatings; (250.5403) Plasmonics; (310.6628) Subwavelength structures,

nanostructures.

http://dx.doi.org/10.1364/JOSAB.32.001941

1. INTRODUCTION

Graphene, a two-dimensional (2D) layer of carbon atomsarranged in a honeycomb lattice, has attracted a great dealof interest [1]. Quantum hall effect [2], gate-variable opticalconductivity [3], controllable plasmonic properties [4] andhigh-speed operation [5] are some of the exceptional electricaland optical properties of graphene. Moreover, it has been dem-onstrated that graphene is one of the best materials for design-ing THz-wave absorbers because of its capability of surfaceplasmon polariton-based absorption at THz frequencies[6–12]. Electromagnetic wave absorbers play a central role inmany devices utilized in biosensing, imaging, and communica-tions [13]. The absorbers based on graphene can be tunable [6]and broadband (in a THz regime) [7] which are importantproperties in some applications.

The perfect absorption can be explained by means of circuittheory: a graphene layer is periodically patterned on a properlydesigned substrate, and the whole structure is impedancematched to the incident medium. Therefore, the structurehas no reflection, and if the transmission channel is also closed,a perfect absorber will be realized. Some authors have used cir-cuit theory and the impedance matching concept to design per-fect absorbers [6,12,14]. However, in these works, the circuitmodel that restricts the efficiency of the designing process hasbeen obtained numerically. As a result, the best achieved band-width of 90% absorption (normalized to the central frequency)is 58% [12]. More recently, a bandwidth of 84% has beendemonstrated in [7], but the use of three layers of patternedgraphene in its proposed structure makes its fabrication morechallenging.

In this work, an ultra-broadband absorber is designed inwhich a periodic array of graphene ribbons (PAGR) is placeda quarter wavelength above a reflecting surface. I use the recentlyproposed analytical circuit model of PAGR [15] for designingthe device. The admittance of the structure is designed to beclose to the free-space admittance. Moreover, the variations ofthe admittance around the central frequency are carefullyadjusted by applying an additional condition on the derivativeof the admittance. As a result, the reflection remains less than10% in a wide frequency range, and the normalized bandwidthof 90% absorption can reach up to 100%, which is significantlylarger than the previous record of 58%. Furthermore, all param-eters of the absorber are given by closed form expressions.

However, it should be mentioned that the central frequencyof the absorber must fall within a specific range of frequencies.This range is determined by imposing two essential constraints:(1) the width of ribbons has to be smaller than the period ofPAGR, and (2) the structure must be a subwavelength to guar-antee the validity of the circuit model. The above-mentionedfrequency range is expressed explicitly as a function of the elec-tron mobility of graphene. It is shown that, for realistic valuesof electron mobility, this range falls into the THz regime.

The paper is organized as follows: In Section 2, I present thecircuit model of the proposed device, and then a design schemeis proposed leading to an ultra-broadband absorber. InSection 3, the limitations of the proposed design are investi-gated, and the allowed frequency range for the central fre-quency of the absorber is obtained. Finally, conclusions aredrawn in Section 4. A time dependence of the form ejωt isassumed and suppressed throughout this study.

Research Article Vol. 32, No. 9 / September 2015 / Journal of the Optical Society of America B 1941

0740-3224/15/091941-06$15/0$15.00 © 2015 Optical Society of America

2. BROADBAND ABSORBER DESIGN

A schematic of the proposed device is shown in Fig. 1(a). APAGR is placed a quarter wavelength from a reflecting surface,and geometrically and electronically tuned to mimic a surfacewith an impedance closely matched to free space. The reflectingsurface is a good conductor (Au) which can be considered to bean approximate short circuit, and it is transformed to be anapproximate open circuit by a λ∕4 transformer. Therefore,the admittance of the graphene array is seen as the input ad-mittance of the structure, and it can be tuned to be matched tothe free-space admittance near its resonant frequency [14].

A Fourier modal method with adaptive spatial resolution isused in this paper for numerical simulations [16,17]. Themethod solves Maxwell’s equations for periodic structuresrigorously. The graphene layer is modeled numerically by a thinlayer (with thickness Δ � 1 nm) of permittivity εGR �ε0 − jσGR∕�ωΔ�, with σGR as the surface conductivity of gra-phene which can be derived using the well-known Kuboformula [18] and written as

σGR � 2e2kBTπℏ2

jjτ−1 − ω

ln�2 cosh�EF∕2kBT ��

−je2

4πℏln

�2EF − ℏ�ω − jτ−1�2EF � ℏ�ω − jτ−1�

�;

where e is the electron charge, kB is the Boltzmann constant, ℏis the Planck constant, T is the temperature (fixed to 300 K),EF is the Fermi energy, ω is the angular frequency, and τ is therelaxation time.

The equivalent circuit of the device at normal incidence forTM polarization (magnetic field along the y direction) is de-picted in Fig. 1(b). Homogenous regions are modeled withtransmission lines and the PAGR with a shunt admittance

Y G . The admittance of transmission lines corresponding tohomogenous regions is Y � n∕η0 and β � ω∕c, where n isthe refractive index of the region, η0 ≈ 120π is the free-spaceimpedance, and c � 3 × 108∕n is the speed of light in the re-gion. The transmission line corresponding to the gold reflectingsurface can simply be replaced by an admittance, because theincoming wave cannot transmit through this layer. Throughoutthis paper, a refractive index of ns � 1.5 for the dielectricsubstrate and conductivity of σ � 4 × 107 S∕m for the Aureflector are assumed.

It has been shown that the admittance of PAGR, Y G , iscomposed of an infinite number of parallel R-L-C circuits, eachrepresenting a mode of the graphene ribbon [15]. These modesare Fabry–Perot resonances of a graphene plasmon trapped inthe finite width of graphene ribbons. I design the proposed de-vice near the first resonance frequency of PAGR; thus, the ef-fect of higher-order modes can be neglected. Hence, the PAGRcan be modeled with a series RLC circuit corresponding to thefirst mode, as shown in Fig. 1(b). At sufficiently low frequencies(where the interband term of conductivity, the second term inthe above equation, is negligible) and for EF ≫ kBT , σGR willbe of the Drude form:

σGR � e2EF

πℏ2

jjτ−1 − ω

:

Thus, the following simple expressions are obtained [15]:

R1 �DS21

πℏ2

e2EF τ; (1a)

L1 �DS21

πℏ2

e2EF; (1b)

C1 �S21D

2εeffq1

; (1c)

and

S21 ≅8

9W ;

where D and W are the period and the width of the grapheneribbons, respectively; εeff � ε0�1� n2s �∕2 is the average per-mittivity of the mediums surrounding the graphene ribbons;and q1 is the first eigenvalue of the equation governing the cur-rent on the PAGR. (Values of q1 for differentW ∕D have beengiven in Table 1.)

It should be noted that, although the design is performed bytwo simplifying assumptions, neglecting the effect of higher-or-der modes and using the Drude form expression for the surfaceconductivity, the final results will be obtained without theseassumptions.

The conditions for perfect absorption are Re�Y in� � Y 0 �1∕η0 and Im�Y in� � 0, where

Y in � Y trAu � Y G; (2a)

Y G � 1

R1 � jωL1 � 1jωC1

; (2b)

Y trAu � Y s

Y Au � jY s tan βsdY s � jY Au tan βsd

. (2c)Fig. 1. (a) Schematic of the absorber. The lines on the substrate arethe graphene ribbons. (b) Equivalent circuit of the absorber.

1942 Vol. 32, No. 9 / September 2015 / Journal of the Optical Society of America B Research Article

It can be simply achieved for a given frequency, f 0, bysetting R1 � η0, L1C1 � 1∕�2πf 0�2, βsjf �f 0

d � π∕2, andY Au → ∞. The physics behind this complete absorption canbe described from another point of view. The incident photons,whose frequency is matched to that of the first localized modeof graphene ribbons, excite this mode. Therefore, some pho-tons are absorbed, and others are reflected and transmitted.Transmitted photons are reflected back by the Au reflector,and some of them come out of the structure with a phase differ-ence of 2βsd � π. This process is repeated again and again, andthese partial reflections destructively interfere with each other,leading to zero reflection. On the other hand, the transmissionchannel is closed by the Au reflector; thus, the incoming wave iscompletely trapped, and, finally, it is absorbed by the gra-phene array.

However, these conditions do not lead to broadband absorp-tion. To obtain broadband absorption (not necessarily perfect),I relax the first condition:

Re�Y in�jf �f 0� R−1

1 � α∕η0; (3)

where α is a numerical parameter larger than unity and its maxi-mum value is determined by the minimum allowedabsorption. From the circuit model, the absorption at centralfrequency can be expressed in terms of α:A � 1 − ��1 − α�∕�1� α��2. Therefore, I set α � 1.75 throughout this workto ensure absorption larger than 90%. I also apply the followingcondition:

ddf

Im�Y in�����f �f 0

� 0: (4)

The latter condition leads to a small imaginary part in a widerange of frequencies near the central frequency. On the otherhand, the real part of input admittance is smaller than α∕η0around the central frequency, so the absorption will be largerthan 90%.

Now we can start the design of the broadband absorber.First, we apply βsjf �f 0

d � π∕2 that gives

d � c4nsf 0

: (5)

As the next step, let us simplify Eq. (4) to extract moreexplicit conditions. Assuming βsd ≈ π∕2 and Y Au → ∞, itis obtained from Eqs. (2c) and (5) that

Im�Y trAu�jf ≈f 0

≅ −Y s cot βsd ≅nsη0

π

2

�ff 0

− 1

�: (6)

On the other hand, with L1C1 � 1∕�2πf 0�2, the admit-tance of PAGR for frequencies around f 0 can be expressed as

Im�Y G�jf ≈f 0≅ −

2πL1R21f

�f 2 − f 20�: (7)

Now, differentiating Eqs. (6) and (7) at f � f 0, Eq. (4) canbe written as

nsη0f 0

π

2−4πL1R21

� 0: (8)

It is obvious from Eq. (1) that L1 � τR1; thus, by substitut-ing Eq. (3) in Eq. (8) we have

τ � ns8αf 0

: (9)

The relaxation time, τ, can be tuned by the Fermi energy EFthrough the relation τ � μEF∕ev2F , where μ is the electronmobility and vF � 106 m∕s is the graphene Fermi velocity.Thus, I set

EF � τev2F∕μ: (10)

The fill factor of the graphene array,W ∕D, is then obtainedfrom Eq. (1a):

WD

� 9πℏ2

8e2EF τR1

: (11)

The width of ribbons is closely related to their resonancefrequency. Therefore, the width of ribbons can be easilydetermined by substituting L1 and C1 from Eq. (1) inω20 � 1∕L1C1,

W � r1e2EF

2ℏ2εeffω20

; (12)

where r1 � q1 W∕π is a function of the fill factor, and it can becomputed by interpolating the data given in Table 1.

Now let us design an ultra-broadband absorber for a centralfrequency of f 0 � 1 THz. The electron mobility is assumed tobe μ � 0.2 m2∕Vs (� 2000 cm2∕Vs). It should be noted thatthe electron mobility of graphene on a substrate ranges fromabout 0.1 m2∕Vs [11] to 6 m2∕Vs [19], depending on the fab-rication process. From Eq. (5), the thickness of the dielectricsubstrate is set to d � 50 μm. The relaxation time and Fermilevel are chosen as τ � 1.07 × 10−13 s and EF � 0.536 eVthat are given by Eqs. (9) and (10), respectively. Finally,from Eqs. (11) and (12), one obtains W �91.4 μm andD�118 μm. The absorption of this structure is plotted inFig. 2(a), calculated by means of the FMM (solid line) andthe circuit model (dashed line). The bandwidth of 90% absorp-tion reaches about 0.98 THz. It should be pointed out thatthe central frequency has been shifted slightly to the left(f 0 � 0.984 THz); thus, the normalized bandwidth is 100%of the central frequency. This shift results from neglecting theeffect of higher-order modes of graphene ribbons in the circuitmodel shown in Fig. 1(b), which was used in the design proc-ess. However, the effect of the higher-order modes has beenincluded in the simulations.

The real and imaginary parts of the input admittance, Y in

(normalized to the free-space admittance), are also plotted inFig. 2(b). It should be emphasized that, in these figures, a moreaccurate circuit model is used, in which the effect of higher-order modes of PAGR is included, and the Au layer is modeledwith a finite conductivity (σ � 4 × 107 S∕m). It is seen that

Table 1. Eigenvalue q1 for Different Fill Factors [15]

W ∕D 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9q1 W ∕π 0.734 0.725 0.710 0.689 0.658 0.620 0.571 0.507 0.420

Research Article Vol. 32, No. 9 / September 2015 / Journal of the Optical Society of America B 1943

the imaginary part of the input admittance is near zero, andit is very flat around the central frequency. These featureslead to wideband impedance matching and, thus, broadbandabsorption.

The effect of ribbon width variations, because of possiblefabrication errors, on the device performance is also investi-gated in Fig. 3(a). In this figure, the absorption is plottedfor three widths, W − � 0.95 W, W , and W � � 1.05 W,whereW and the other parameters of the structure are the sameas the previous example. The normalized bandwidth of the de-vice is 96%, 100%, and 103% for ribbon widths of W −, W ,and W �, respectively, and the central frequency is 0.991,0.984, and 0.980 THz. Therefore, the performance of the de-vice is not very sensitive to variations in the widths of ribbons.

The sensitivity of the device with respect to variations inFermi energy is also examined in Fig. 3(b). It is obvious fromthis figure that the absorption spectra do not change signifi-cantly for �10% variation in Fermi energy.

3. LIMITATIONS

The design algorithm presented in the previous section wasvery straightforward, and explicit expressions were obtainedfor all of the parameters. However, I show in this section that,

for a given electron mobility μ, the central frequency cannot bechosen arbitrarily and has some limitations.

The first limitation stems from Eq. (11) becauseW ∕Dmustbe smaller than unity. To be conservative, I apply W ∕D < 0.9which translates into

f 0 <0.482ffiffiffi

μp THz; (13)

which is obtained by using Eqs. (3) and (9)–(11), and withns � 1.5. On the other hand, the period of PAGR must besmaller than the wavelength; otherwise, the proposed circuitmodel would not be valid [15]. The wavelength in the freespace and the substrate dielectric is λ and λ∕ns, respectively.Therefore, the minimum wavelength is λmin � nsc∕f max, withf max � 1.5f 0 for a normalized bandwidth of 100%. By apply-ing λmin > D and using Eqs. (3) and (9)–(12), the followingcondition is numerically obtained:

f 0 >0.437ffiffiffi

μp THz: (14)

For example, if we set the electron mobility to μ �0.2 m2∕Vs (as in the numerical example of the previous sec-tion), we will obtain 0.978 THz < f 0 < 1.08 THz. Hence,

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (THz)

Abs

orpt

ion

FMM

Circuit Model

0.98 THz

0 0.5 1 1.5 2-2

-1

0

1

2

Frequency (THz)

Nor

mal

ized

Inpu

t Adm

ittan

ce

Re(Yin)

Im(Yin)

(a)

(b)

Fig. 2. (a) The absorption spectra of the designed device. (b) Realand imaginary parts of the input admittance of the designed devicenormalized to the free-space admittance.

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (THz)

Abs

orpt

ion

W-=0.95W

W

W+=1.05W

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (THz)

Abs

orpt

ion EF-=0.9EF

EF

EF+=1.1EF

(a)

(b)

Fig. 3. Absorption spectra of the device calculated by FMM for(a) three different ribbon widths and (b) three different Fermi levelsof energy.

1944 Vol. 32, No. 9 / September 2015 / Journal of the Optical Society of America B Research Article

electron mobility should be lowered to design the broadbandabsorber at higher frequencies, and vice versa.

Another way to design a broadband absorber at higherfrequencies is by relaxing the condition expressed by Eq. (8)as follows:

βns

η0f 0

π

2� 4πL1

R21

; (15)

where β < 1. This implies that the derivative of Im�Y in� is neg-ative around the central frequency. The derivative of Im�Y in� isdesigned to be negative, because it is positive for frequencies farfrom the central frequency. Consequently, such a design resultsin limited variation of Im�Y in� around the central frequency.However, β should not be very small unless Im�Y in� variesgreatly around the central frequency, leading to large values ofIm�Y in� and a substantial decrease of the bandwidth. It shouldbe noticed that the relaxation time has to be modified as

τ � βns8αf 0

; (16)

but other relations in the previous section remain unchanged.The upper limit for the central frequency will be obtained if

one uses Eq. (15) instead of Eq. (8):

f 0 <0.482

min�β� ffiffiffiμ

p THz: (17)

As a numerical example, let us design an absorber withβ � 0.5. We again set the electron mobility to

μ � 0.2 m2∕Vs. The upper limit for central frequency is in-creased to 2.15 THz according to Eq. (17). For f 0 � 2 THz,the parameters of the desired broadband absorber ared � 25 μm, EF � 0.536 eV, D � 29.5 μm, W � 22.8 μm,and τ � 1.07 × 10−13 s that are given by Eqs. (5), (10)–(12),and (16), respectively. The absorption of the structure and thereal and imaginary parts of the input admittance are shown inFig. 4. The normalized bandwidth of the absorber is decreasedto 85.3%, but it is still larger than previous records [7,12].

It is worth noting that the bandwidth of the proposedabsorber is almost independent of the central frequency andelectron mobility. Nevertheless, it depends strongly on β, asobserved in the previous example. Figure 5 shows the normal-ized bandwidth versus β. The maximum allowed centralfrequency according to Eq. (17) is also plotted in the samefigure. (The electron mobility is μ � 0.2 m2∕Vs.) Theminimum allowed central frequency is 0.978 THz (withβ � 1). If a bandwidth of more than 50% is acceptable, onecan design the absorber for 0.978 THz < f 0 < 4.71 THzwith 0.23 < β < 1.

4. CONCLUSION

In summary, I have proposed a novel approach for designingultra-broadband absorbers by PAGRs. The design takes advan-tage of the circuit theory and the controllability of the grapheneconductivity for adjusting the real and imaginary parts of thedevice input admittance. The real part is tuned to be suffi-ciently close to the free-space admittance around the centralfrequency, and, on the other hand, the imaginary part andits variations are set to be zero. As a result, a normalized band-width of 100% is demonstrated by using only one layer ofPAGR deposited a quarter wavelength from a reflector.

I also discussed the limitations of the proposed scheme, andI presented the upper and lower bound for the central fre-quency as a function of the electron mobility of graphene. Theupper bound, however, can be increased at the cost of decreasedbandwidth, by allowing non-zero, small variations for theimaginary part of the input admittance.

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

Frequency (THz)

Abs

orpt

ion

FMMCircuit Model

1.76 THz

0 1 2 3 4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Frequency (THz)

Nor

mal

ized

Inpu

t Adm

ittan

ce

Re(Yin)

Im(Yin)

(a)

(b)

Fig. 4. (a) Absorption spectra of the designed device with β � 0.5.(b) Real and imaginary parts of the input admittance of the designeddevice normalized to the free-space admittance.

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

β

Nor

mal

ized

Ban

dwid

th (

%)

0

4

8

12

16

20

max

(f0)

(TH

z)

Fig. 5. Normalized bandwidth of the proposed absorber versus β(solid line). The upper limit for the central frequency of the absorberversus β (dashed line) is also plotted in this figure.

Research Article Vol. 32, No. 9 / September 2015 / Journal of the Optical Society of America B 1945

The device absorbs only TM polarized waves. A polarizationinsensitive absorber may be designed by 2D arrays of graphenepatches or disks instead of graphene ribbons.

The approach presented in this work is applicable to any2D material with Drude form and tunable conductivity.The electron mobility of the material determines the workingfrequency of the device.

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1946 Vol. 32, No. 9 / September 2015 / Journal of the Optical Society of America B Research Article