Dark materials based on graphene sheet stacksAlon Ludwig and Kevin J. Webb*

School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA*Corresponding author: webb@purdue.edu

Received July 27, 2010; revised November 19, 2010; accepted November 21, 2010;posted December 1, 2010 (Doc. ID 132426); published January 5, 2011

The effective medium properties of graphene sheet stacks are calculated, and it is shown that such stacks can havevery low reflectivity and high absorbance. These properties make graphene-sheet-stack-based materials darker thanrecently studied carbon nanotube materials. Graphene stacks thus hold promise for realizing lower reflectivity coat-ings and enhanced photodetectors. The bounds of the effective medium approximation and the possible benefits ofusing graphene sheet stacks in a regime where this approximation does not hold are discussed. © 2011 OpticalSociety of AmericaOCIS codes: 260.2065, 160.3918.

Dark materials play a crucial role in the improvement ofthermal detectors and the control of reflectivity in gen-eral. While black paint or bulk graphite coatings are lim-ited by their surface reflectivity (5%–10%), it was recentlyshown that coatings composed of arrays of carbonnanotubes (CNTs) have both high absorbance and lowreflection at normal incidence and can be used to im-prove detector performance [1–3]. These studies followtheoretical work that used bulk graphite permittivities tocalculate the effective medium parameters (EMPs) of atwo-dimensional array of CNTs [4,5].Recent developments in the theory andmeasurement of

the properties of graphene sheets in the linear-responseregime [6–8] indicate that, throughout a substantial por-tion of the optical regime (250–6000 nm), the surface con-ductivity of a single graphene sheet is very close to theuniversal value of σ0 ¼ πe2=2h, where e is the electroncharge and h is Planck’s constant. Relying on this result,we calculate here theEMPs for a one-dimensional stack ofgraphene sheets separated by an arbitrary substance (seeFig. 1). In the following, z is the graphene sheet stack(GSS) axis and the temporal dependency is expð−iωtÞ.We begin with the amplitudes of the reflection, r, andtransmission, t, of a TM-polarized plane wave (see Fig. 2)impinging at an incident angle θ on a single graphenesheet immersed in a medium of relative permittivity ϵ,which are [7]

r ¼ ~α cos θ=ð ffiffiffiϵ

p þ ~α cos θÞ; t ¼ ffiffiffiϵ

p=ð ffiffiffi

ϵp þ ~α cos θÞ; ð1Þ

where ~α ¼ πα=2 and α ¼ σ0=πϵ0c is the fine-structure con-stant [6–8]. Here, c ¼ 1=

ffiffiffiffiffiffiffiffiffiϵ0μ0p

is the speed of light invacuum, and ϵ0 and μ0 are the free-space permittivityand permeability, respectively. Substituting Eq. (1) intothe translation matrix T, relating the complex amplitudesof the incident and reflected electric fields (the compo-nent parallel to the sheets) of two adjacent GSS unit cells(depicted in Fig. 2), we find

T ¼

264

ffiffiϵ

p−~α cos θffiffi

ϵp ei

ffiffiϵ

pk0d cos θ

−

~α cos θffiffiϵ

p~α cos θffiffi

ϵp

ffiffiϵ

p þ~α cos θffiffiϵ

p e−iffiffiϵ

pk0d cos θ

375; ð2Þ

where d is the intersheet distance (and unit cell size) andk0 ¼ ω=c is the free-space wave number. The elements of

Eq. (2) can be used to calculate the EMPs of the GSS in amanner similar to that used for TE plane waves impingingon an effective medium slab surrounded by free space [9].For the specific TM-polarized case studied here, the EMPsare related to the translation matrix elements by

�ϵt ¼ �β=ð�Zϵ0Þ; �μt − ðϵ sin2 θÞ=�ϵn ¼ �β�Z=μ0; ð3Þ

where �ϵt and �ϵn are the effective relative transverse andnormal permittivities of the GSS, respectively, �μt is the ef-fective relative transverse permeability of the GSS,

�Z ¼ffiffiffiffiffiffiffiμ0ϵϵ0

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðT22 þ T21Þ2 − 1

ðT22 − T21Þ2 − 1

scos θ ð4Þ

is the effective z-directed impedance of the GSS, and

�β ¼ arccosf½1þ ðT222 − T2

21Þ�=2T22g=d ð5Þ

is the effective z-directed propagation constant of theGSS. In Eqs. (4) and (5), T22 and T21 are elements ofthe translation matrix T . It is important to note that theuse of a unit-cell translation matrix for the extraction ofthe EMPs is possible because, in one-dimensional config-urations, the number of unit cells in the GSS does notaffect the extracted EMPs [9].

A meaningful set of EMPs is independent of the inci-dent angle of the plane wave illuminating the sample thatis used for the extraction process. A common a priori

Fig. 1. Graphene sheet stack illuminated by a plane wave.

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condition for obtaining such meaningful EMPs is that theunit-cell dimensions are small compared with the free-space wavelength, i.e., d=λ0 ≪ 1, where λ0 ¼ 2π=k0. Tobound the EMPs in the limit d=λ0 → 0, we rewrite the ele-ments of T in Eq. (2) using ~α ¼ γd=λ0 and obtain theEMPs using Eq. (3). This results in an anisotropic effec-tive relative permittivity for the GSS given by

��ϵt ¼ ϵþ iγ=π�ϵn ¼ ϵ ; where γ ¼ πσ0

cϵ0k0d: ð6Þ

In the same manner, �μt ¼ 1 is obtained. The effective re-lative normal permeability �μn ¼ 1 is obtained using a si-milar derivation based on TE incidence. The values of theother two EMPs obtained from a TE analysis, �μt and �ϵt,coincide with those obtained from the TM analysis. Theanisotropic imaginary part of the transverse permittivityin Eq. (6) can be easily understood by noting that theelectric fields normal to the sheets does not induce cur-rents, and thus only the transverse component of the per-mittivity is affected by the existence of the sheets. Therelations given in Eq. (6) were obtained in the limit whereboth ~α and d=λ0 approach zero, while the ratio betweenthem, γ, is finite. Thus, to use those relations for graphenesheets with surface conductivity σ0, where ~α ≈ 0:0116, itis important to bound the distance between the sheets dto be as small as possible compared to the wavelength λ0,but not substantially smaller than 0:01λ0. Except forthese conditions, the distance between the graphenesheets must also be large enough compared to the intera-tomic distance (∼0:3 nm) to avoid mutual effects, caus-ing a change in the electronic structure of the sheets.Under the above conditions, the effective medium ap-proximation (EMA) holds, and the EMPs can also be ob-tained via a Maxwell–Garnett (MG) approach based on avolume filling fraction by modeling the sheets as having avanishingly small fictitious thickness and a correspond-ingly large volume conductivity. However, this simple ap-proach does not allow the study of the EMP accuracy.Figure 3 shows the reflected and transmitted power

(normalized to the incident power) for a GSS set in freespace and illuminated by a normally incident plane waveat a frequency corresponding to red light (λ ¼ 630 nm) asa function of the GSS thickness,D (see Fig. 2 for the exactdefinition ofD), for GSSs of different intersheet distances,d. The plots in Fig. 3 are obtained using Eq. (6), assumingthat the EMA holds. It is evident from this figure that, asthe thickness of the GSS increases, the dominant portionof the power not absorbed in theGSS is due to reflection atits interface. The reflectivity of the GSS can be reduced by

increasing the intersheet distance. However, this has thedetrimental effect of decreasing the effective absorptioncoefficient �α ¼ 2Imð ffiffiffiffi

�ϵtp Þk0, as can be seen from the

slopes of the transmission power curves in Fig. 3. Thus,as long as the homogenization is valid, by increasingthe intersheet distance, together with the entire thicknessof theGSS, one can increase the blackness of the resultingGSS. When deviating from normal incidence, the reflec-tion and absorption of TE and TM polarizations behavedifferently. For TE polarization, the reflection from theGSS, like the reflection from most substances, increaseswith incident angle, reaching a limit of total reflection andno absorption at grazing incidence. For TM polarization,on the other hand, due to the transverse nature of the con-ductivity of the sheets, reflection and absorption are re-duced with an increase in incident angle, reaching alimit of no reflection and no absorption at grazing inci-dence. Both increased reflection and reduced absorptionhave a detrimental effect on the blackness of the GSS.

The performance of the GSS is exemplified by compar-ison with theoretical results for a medium comprising anarray of aligned CNTs of diameter 2R ¼ 10 nm that are setin free space with a two-dimensional array spacing of d ¼50 nm [3]. The array is illuminated by p-polarized (electricfield perpendicular to the tube axis) incident planewaves.The choice of incidence stems from the large reflection inthe case of s-polarized incidence (electric field parallel tothe tube axis), making it a substantially less suitablechoice for a darkmaterial. To achieve the same low reflec-tivity (0.02%) obtained for the CNT array using a GSS, weneed to place the sheets at a distance of d ¼ 41:2 nm, as-suming free space background as in the CNT array, andthat the GSS thickness, D, is large. This yields a GSS ab-sorption coefficient of �α ¼ 5:64 μm−1, roughly 5 times lar-ger than that of the CNT array. Such a large absorptioncoefficient implies thatmuch thinner coatings can be usedto achieve properties similar to those of the CNT array. Amore comprehensive comparison with CNT array-baseddark materials is given in Fig. 4, where the deviation ofthe real part of the effective refractive index from unity,Reð�n − 1Þ, with �n ¼ ffiffiffiffiffiffiffiffi

�μt�ϵtp

, is given as a function of the ef-fective absorption constant, �α, for GSSs and for two-dimensional arrays of CNTs, again assuming free-space

Fig. 2. Side view of a GSS illuminated by a TM-polarized planewave, to accompany the derivation given. The derivation for TEillumination is similar and is not dwelt upon.

Fig. 3. (Color online) Reflected (R) and transmitted (T) power(normalized to the incident power) for a GSS illuminated by anormally incident plane wave at a wavelength λ ¼ 630 nm as afunction of the GSS thickness, D, for different intersheet dis-tances, d.

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background for both structures. The curves for both theGSS and the CNT array are obtained by variation of theunit cell dimension, d (intersheet distances for theGSS), in a rangewhere the EMA is reasonable (d < 0:16λ),and the MG approach used for the derivation of EMPs ofthe CNT arrays [4] is valid (i.e., the inclusions have a smallfilling factor f ¼ πR2=d2 < 0:09, where the influence ofthe inner radius of the CNT is ignored, as in [4]). The threedifferent curves that are shown in Fig. 4 for both the GSSand the CNT-based structures correspond to three differ-ent wavelengths of operation. The curves for the GSSs areobtained using Eq. (6), assuming that the EMA holds. Theshaded area below those lines shows results obtained di-rectly from the translation matrix T for all incident anglesand both TM and TE incidence. The widening of theshaded area as d increases indicates the beginning of abreakdown of the EMA, and the width gives a rough esti-mate of the error introduced by this approximation. Notethat the logarithmic scale of the ordinate plays a factor inthe widening of the shaded area as well. While numericalerror estimates for the curves describing the CNT arraysare not shown, a breakdown of the EMA at the left-handside of those curves is expected, as well. All curves showthe trade-off of reduced refractive index resulting in re-duced absorption. However, when comparing the curvesof the CNT arrays to those for the GSSs, it is evident thatGSSs provide a substantially better trade-off. This conclu-sion is invariant to change in the outer radius of the tubes,R, because the MG mixing formula is dependent only onthe filling factor, which is proportional to ðR=dÞ2, and,thus, the shape of the CNT array curves is retained. Note

that, within the wavelength range where graphene can bedescribed by a frequency-independent surface conductiv-ity, the scaling principle can be used to apply the resultsshown in Fig. 4 to other wavelengths.

Once the intersheet medium of the GSS is no longer thefree-space background, the curves in Fig. 4 exhibit a lar-ger Reð�n − 1Þ, and the performance of the GSS as a blackmaterial deteriorates. It was recently shown that gra-phene films can be transferred to almost any material[10,11], making it plausible to coat the films with low per-mittivity material and pattern it to obtain pillars. The for-mation of a graphene stack with a low permittivity spacerthus becomes possible. If the spacer is made of a matrixof SiO2 pillars with a diameter of 0:1d and a separation ofd, the effective permittivity of this medium is estimatedvia MG formulation to be 1.008, under normal incidence,and Reð�n − 1Þ of the curves is below 4:6 × 10−3, still mak-ing such a GSS substantially darker than the CNT array.

By the measures used (reflectivity and absorption perunit length), the GSS appears to be the blackest broad-bandmaterial currently available, making thismetamater-ial interesting for applications such as sensitive thermaldetection and solar energy harvesting. Outside the EMA,where we suspect that the reflectivity continues todecrease as the GSSs become sparser, the material black-ness could be further increased by reducing the reflectiv-ity at the first few layers of theGSS and absorbing the lightin the bulk of the GSS. This can be done by continuouslyreducing d with spatial location and optimizing the func-tional change in d to improve the blackness.

This work was supported by the National ScienceFoundation (NSF, grants 0524442 and 0901383), theU.S. Department of Energy (DOE, grant DE-FG52-06NA27505), the U.S. Army Research Office (USARO,grant W911NF-10-1-0492), and the Technion ViterbiFamily Foundation.

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Fig. 4. (Color online) The deviation of the real part of the ef-fective refractive index from unity Reð�n − 1Þ versus the effec-tive absorption constant �α. Results are shown for GSSs ofdifferent intersheet distances, d (unit cell dimension), and atdifferent wavelengths of operation, λ, and compared with re-sults for two-dimensional arrays of CNTs [5] for the same unitcell dimensions and wavelengths. Results for the curves of theGSSs are obtained from Eq. (6), while results for the shadedregions below these curves are obtained directly fromEqs. (2)–(5) for all incident angles and both TM and TEincidence.

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