A novel structure for tunable terahertz absorber based on graphene

A novel structure for tunable terahertz absorber based on graphene Bing–zheng Xu,* Chang-qing Gu, Zhuo Li, and Zhen-yi Niu College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China * [email protected]

Abstract: Graphene can be used as a platform for tunable absorbers for its tunability of conductivity. In this paper, we proposed an “uneven dielectric slab structure” for the terahertz (THz) tunable absorber based on graphene. The absorber consists of graphene-dielectric stacks and an electric conductor layer, which is easy to fabricate in the manufacturing technique. Fine tuning of the absorption resonances can be conveniently achieved by adjusting the bias voltage. Both narrowband and broadband tunable absorbers made of this structure are demonstrated without using a patterned graphene. In addition, this type of graphene-based absorber exhibits stable resonances with a wide range angles of obliquely incident electromagnetic waves. © 2013 Optical Society of America OCIS codes: (050.6624) Subwavelength structures; (230.4170) Multilayers; (240.0310) Thin films.

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#192964 - $15.00 USD Received 26 Jun 2013; revised 12 Aug 2013; accepted 29 Aug 2013; published 30 Sep 2013 (C) 2013 OSA 7 October 2013 | Vol. 21, No. 20 | DOI:10.1364/OE.21.023803 | OPTICS EXPRESS 23803

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1.

Introduction

As a two-dimension substance composed of a single layer carbon atom arranged in a honey comb lattice, graphene has attracted potential applications in many branches of engineering due to its unique properties, such as optical transparency, flexibility and extraordinary electrical properties [1–5]. One novel area where the properties of graphene layer may be influential is the ability of supporting surface plasmon polariton (SPP) waves in the terahertz and infrared ranges [6–11]. Thanks to the tunability of graphene conductivity, one can control the electromagnetic waves propagation conveniently, which is also the basis of our proposed tunable absorber in the terahertz frequency range in this paper. In the last decade, various types of terahertz (THz) absorbers have attracted an explosion of


research interest in many scopes, like spectroscopy, medical imaging, communications and so on [12]. Due to the short history of the THz technology, new material is on demand rapidly. A practical high-efficiency tunable absorber would find applications in above areas [13–16], but the energy of the new material is still far from bringing into full play. The monolayer sheet of graphene has particularly interesting and unique properties of low-loss surface reactance at terahertz and far-infrared frequencies. Over the past two years, ultrathin surfaces using graphene is used to control the transmission and absorption of electromagnetic wave [3,17–25]. Graphene acts like an absorptive dielectric with the absorption A ≈ 2.3% [26–30]. The tunable transmission of electromagnetic wave through a stack of monolayer graphene sheet (not patterned graphene) separated by dielectric slabs is reported in [21]. And in [3], by changing the various bias voltages, the absorption energy of the absorber (which contains several layers of graphene patches separated by dielectric slabs) is controlled over a relatively large interval in a narrow frequency band. But the fact of the narrow frequency band limits the use in real applications. Not long after that, both narrowband and broadband tunable graphene-based perfect absorbers have been investigated [18]. Two layers of graphene separated by the dielectric were considered, that they have electrical contacts for voltage gating. In order to tune the conductivity of graphene monolayer conveniently, the monolayer should be electrically connected.

 E  H

 θ k

z y

x

Vg silicon dioxide t1 silicon t2

heet ene s graph ctr ic d ie le

t3

PEC

(a)

(b)

Fig. 1. Geometry of a stack of periodic graphene patches separated by dielectric slabs with a plane wave of incidence, the thickness of the graphene sheet is neglected. (a) 3D view and (b) cross-section view.

In this paper, the idea of a “graphene-uneven dielectric” layered scheme (with the geometry shown in Fig. 1) is proposed, and the structure shows tunable absorption for both polarizations over a wide THz frequency band. The absorption energy from the graphene-based absorber can be controlled via the chemical potential over a relatively large interval. This paper is organized as follows: the conductivity model of graphene for the analysis of the absorbers is presented in Section II. The simulation based on circuit model and CST Microwave Studio is used to explore the physical mechanism and effect of incident directions of incident transverse magnetic (TM) and transverse electric (TE) wave on the absorption characteristics of the absorber. With the graphene sheet characterized by a complex surface impedance which can be adjusted to change the gate voltage, the “graphene-uneven dielectric” structures for tunable absorber design are given is Section III. Finally, some conclusion remarks are drawn in Section IV.


2.

Graphene-based absorber and conductivity model

It is well known that the graphene monolayer can be electrically modeled as an infinitesimally thin conductive layer characterized by a complex-valued surface conductivity σgr (ω , τ , T, μc ), where ω is the radian frequency of the plane wave, τ is a relaxation time which is assumed to be independent of energy and T is the room temperature, μc is the chemical potential related to the electrostatic biasing, which quantifies the electronic transport properties. The conductivity value can be theoretically calculated using the Kubo formula, which yields the following equation: (time-harmonic fields of the form e jω t is assumed ) [31,32]:

σgr (ω , τ , T, μc ) = σgr + jσgr =

σ0 , 1 + jωτ

(1)

− μc e2 τ kB T + 1)), ( μ + 2k T ln(e (2) c B π h¯ 2 where kB is Boltzmann’s constant, −e is the charge of an electron, h¯ = h/2π is the reduced Planck’s constant. Throughout this work, we assume T = 300 K, τ = 1 ps, which corresponds a mean-free path of several hundred nanometers [20]. The value of the scattering time of graphene monolayer is similar to that measured in [33] (1.1 ps). And we set 0.0 – 0.5 eV as the chemical potential scope for discussion. Equation (1) corresponds to the surface impedance of a mono layer graphene sheet Zgr = Zgr + Zgr = 1/σgr [20,21]. Figures 2(a) and 2(b) show the variation of the real and imaginary parts of the surface impedance with frequency for different chemical potential at low-terahertz frequencies. It is shown that the real part of the surface impedance does not change significantly with frequency, and becomes smaller as the chemical potential gets bigger. As the frequency increases, the imaginary part of the graphene surface impedance is becoming the major section of the surface impedance, resembling a typical R-L model. It is observed that tuning the chemical potential (bias voltage) can directly affect the conductivity and impedance of graphene.

σ0 =

(b) 5 200

4 3

150

2

100

1

50

0 0

0.5 Chemical potential μ (eV) c

1

Frequency (THz)

Frequency (THz)

(a) 5

6000

4 3

4000

2 2000

1 0 0

0.5 Chemical potential μ (eV)

1

0

c

Fig. 2. (a) Real and (b) imaginary parts of the graphene surface impedance (Ω) in terms of frequency for various bias chemical potential (from μc = 0 to 1.0 eV) in the terahertz frequency regime. T = 300 K, τ = 1 ps is considered throughout the paper.

The absorbing structure is composed of a set of staked monolayer graphene sheets printed on dielectric slabs and a ground plane placed at the bottom interface (Fig. 1). According to [34,35], graphene-SiO2 -Si layers structure is considered in our simulation. In details, graphene is supported by a dielectric film made of silicon oxide SiO2 . The thickness of the silicon oxide and silicon film is t1 and t2 . If the case of multi-structure is considered, each of the 3-layer structure is separated by a dielectric slab with the permittivity εr =10.2 and thickness t3 , which is referred to [20,21] in the terahertz frequency. We can also choose a low dielectric constant materials to replace it. For an isolate graphene monolayer, the chemical potential is determined by the carrier density [4,5]:


ns =

2 π h¯ 2 νF2

∞ 0

ε [ fd (ε ) − fd (ε + 2μc )]d ε ,

(3)

where νF ≈ 1.0 × 106 m/s is the Fermi velocity, fd (ε ) = (1 + e(ε −μc )/kB T )−1 is the FermiDirac Equation distribution. In this simulation, the silicon substrate acts as a back gate, which produces a surface charge density as by an another expression : ns = εSiO2 ε0Vdc /et1 .

(4)

in which εSiO2 = 3.9 and ε0 are the relative permittivity of SiO2 and air, respectively. It is easy to figure out the relationship between voltage and chemical potential according to Eqs. (3) and (4). With the graphene sheet characterized by a complex surface impedance which is adjusted to change of the gate voltage and the interaction in a graphene dielectric stack is formed by plane-wave reflection, the reflection of the graphene-dielectric stack can been obtained with the use of circuit modeling approach. The equivalent network of single-layer absorber formed by graphene sheet is shown in Fig. 3. The dimension of the unit cell is assumed to be subwavelength and we can easily find that the surface impedance based on graphene sheet Zgr is a series R-L circuit model. In addition, β0 is the propagation constant for the air space. The √ free space impedances Z0 and Zd are given by Z0 = η , Zd = η / εr (same treatment applies to ZSiO2 ). Since the silicon is assumed to be used as a gate electrode, it should have some conductivity, that would influence the properties of the absorber. Although we consider the conductivity of the silicon in this paper, the absorption results with the medium-level doped and high resistivity silicon have little difference. In other words, the absorption changes insignificantly when the conductivity of silicon is less than 103 S/m. Due to the little impact on the result, the properties of silicon in our paper is similar to that measured in [34,35]. Once the parameters of the analysis model are derived, the reflection of the structure is assessed as follows:

t2

t1

 SiO2  SiO2  Si  Si

 0  

Z gr Z0

t3

Rgr Z SiO2

 r  r

PEC

Lgr Z Si

Zd

Fig. 3. The equivalent circuit model of the absorber formed by graphene sheet.

A + B/Z0 −CZ0 − D 2 , R = |S11 |2 = A + B/Z0 +CZ0 + D

(5)

in which the terms A, B,C, D are the elements of the transmission line matrix of the composite structure, which can be evaluated as the product of several cascaded matrices: A B (6) = Dgr · DSiO2 · DSi · Dd · De , C D where the transmission line matrix can be expressed as : #192964 - $15.00 USD Received 26 Jun 2013; revised 12 Aug 2013; accepted 29 Aug 2013; published 30 Sep 2013 (C) 2013 OSA 7 October 2013 | Vol. 21, No. 20 | DOI:10.1364/OE.21.023803 | OPTICS EXPRESS 23807

Dgr = DSi =

1 σgr

cos(βSit2 ) jsin(βSit2 )/ZSi

cos(βSiO2 t1 ) jZSiO2 sin(βSiO2 t1 ) DSiO2 = cos(βSiO2 t1 ) jsin(βSiO2 t1 )/ZSiO2 cos(βr t3 ) 1 jZSi sin(βSit2 ) jZd sin(βr t3 ) Dd = De = cos(βSit2 ) cos(βr t3 ) σz jsin(βr t3 )/Zd 0 1

Note that the absorber is backed by the PEC (σz → ∞), no transmission is possible (T = 0) and thus the absorption A = 1 − R. In order to present a practical design of the structure, the predictions of our model must be checked against experimental and numerical results. The forthcoming section will give the analytical predictions and full simulation results. Results and discussions

Frequency (THz)

(a)

(b) 4 0.8 3 0.6 2

0.4

1

0

0.2 10 20 30 Dielectric thickness (μ m)

40

Frequency (THz)

3.

4 0.8 3 0.6 2

0.4

1

0

0.2 0.2 0.4 Chemical potential (eV)

0.6

(c)

Fig. 4. (a) Absorption as a function of frequency and dielectric slab t3 for the absorber formed by graphene monolayer with μc = 0.5 eV. (b) Absorption as a function of frequency and chemical potential for the absorber with t1 =14.1 nm, t2 =1.5 μ m, t3 = 38 μ m. (c) Magnitude of the absorption versus frequency for different chemical potential are plotted. The analytical predictions (symbols) are in a good correspondence with full-wave simulation results (dash line). And the relationship between the bias voltage and the chemical potential is depicted in subfigure.

3.1.

Graphene-even dielectric slab structure and single layer case

Now consider a normal plane wave incident on the absorber formed by the “graphene-even dielectric slab” sheet. The simplified structure consists of graphene and several dielectric slabs. #192964 - $15.00 USD Received 26 Jun 2013; revised 12 Aug 2013; accepted 29 Aug 2013; published 30 Sep 2013 (C) 2013 OSA 7 October 2013 | Vol. 21, No. 20 | DOI:10.1364/OE.21.023803 | OPTICS EXPRESS 23808

0 1

.(7)

Figure 4(a) gives the absorption as function of the thickness t3 (ranging from 0 to 40 μ m) and the frequency. When t3 is upper than 30 μ m, it can be seen that the typical Fabry-P´erot resonance appears at three peaks in the range from 0 to 4 THz, one can simply say that f2 ≈ 2 f1 and f3 ≈ 3 f1 (absorption peaks from low to high is f1 , f2 and f3 ). In addition, the chemical potential of graphene has a measurable impact on the absorption. Figure 4(b) shows the absorption as a function of the chemical potential μc and the frequency. The frequency of absorption resonance decreases with the increase of μc . To obtain a good performance, the thickness of the dielectric slab and the chemical potential are fixed to t3 = 38 μ m and μc = 0.5 eV. In details, Fig. 4(c) shows the absorption coefficient plotted in terms of the excitation frequency for several chemical potentials of normal incidence. The relationship between the applied bias voltages and the chemical potential is depicted in the subfigure. It can be seen that a absorption resonance is associated with the Fabry–P´erot-type resonance of the dielectric slabs load with graphene sheets. What’s more, the proposed absorber is like a kind of Salisbury Screen absorber, which consists of a resistive sheet on a dielectric spacer above a metal-backed plane. The absorption is zero between the two absorption peaks for all selected chemical potentials, this phenomenon is similar to [36]. For the fixed length, cot(ϕ ) → ∞ (ϕ is the phase advance upon one propagation through dielectric [18]), and the zero absorption occurs, which can be hardly achieved for graphene with any chemical potential [36]. It is seen that the results calculated from transmission line model (symbols) match the CST simulation results (dash lines) well in the whole frequency region under studied. When no bias voltage is applied (μc = 0 eV), the absorption is not small enough at frequencies close to f1 . This structure needs to be improved. 3.2.

Graphene-uneven dielectric slab structure and multilayer case

For above discussions, we obtain an absorption curve under incidence with only one layer of graphene sheet and even dielectric spacer (SiO2 ). But the working bandwidth of the absorber is narrow and the state of off-state is not satisfactory (absorption A < 0.1). In the beginning, the periodical arrays of patterned graphene (common structure e.g. quare loops and patches which represented dual capacitive-inductive nature [20]) is considered for the structure which is conducive to resonance. However, the unit cell of the lattice is relatively isolated, and the objective of the electrically biased graphene surface is hardly achieved [3,18]. To solve the problem, we plan to achieve a broadband tunable absorber through a whole graphene sheet printed on the dielectric slab with different thickness, which is often used to support SPP waves [6]. The bias voltage is imposed between the graphene surface and the silicon dielectric [geometry and schematic of a unit cell is shown in Fig. 5(a)]. For the thickness t1(α ,β ,γ ) of the SiO2 is set as 508 nm, 17.2 nm, 508 nm respectively, the conductivity of graphene in loop (region sβ ) is different with the other two areas in one unit cell (region sα and sγ ). As we know, graphene can be a platform for tunable absorber based on the capability of dynamically tuning the conductivity of graphene by changing the voltage Vg . Since the “uneven ground plane” (dielectric spacer SiO2 ) and bias voltage (Vg =12 V) are fixed, each of the chemical potential (μc = 0.07, 0.45, 0.07 eV) of background graphene in separate region can be obtained from Eqs. (3) and (4). In Fig. 5(b), the magnitude of the absorption coefficient in terms of frequency is depicted for various voltage Vg . We could assure that 1.9 THz is the center resonant frequency points when Vg = 12 V (μc = 0.45 eV). Moreover, the working bandwidth with absorption A ≥ 0.9 is only 0.3 THz, with the thickness of the whole structure greater than 15 μ m. When a lower bias voltage is applied, the surface impedance in the whole region becomes bigger, and a large amount of the incoming field is reflected due to the mismatching. The structure is conducive to


(a)

(b)

Fig. 5. (a) Schematic of uneven ground plane underneath the graphene layer. The gap distance between the silicon and graphene sheet is filled up with a regular dielectric space (SiO2 ). The lattice parameter is p = 5 μ m, and the gap w1 = 0.25 μ m, w2 = 1.175 μ m. The thickness of dielectric slabs is t3 = 12 μ m and t3 = 3.5 μ m. (b) Absorption of the structure is tuned by the bias voltage from 0 to 12 V.

resonance for an extraordinary conductivity in the loop part. Another situation is shown in Fig. 6(c), the two graphene-layers structure with different periodic areas sα (4.5 μ m × 4.5 μ m and 3 μ m × 3 μ m) is considered. Obviously, different resonances will widen the absorption bandwidth. Hence, the working bandwidth in this struc ture reaches 0.5 THz with the thickness t3 = 10 μ m and t3 = 4 μ m. This type of absorber has broadband absorption band. Because of the axial symmetry of the structure, the absorber is insensitive to polarization angle. And we will examine the sensitivity of absorption to oblique incidence plane wave. For simplicity, the incident angle θ is defined as the angle between the z− axis and the incident wave (see Fig. 1). Figures 6(a) and 6(b) show the calculated absorption curves varied with incident angle TM or TE waves. It can be observed that the absorber shows relatively stable absorption performance with different incidence angle θ up to 60◦ . The absorption peak monotonously decreases with the increasing incident angle. The reason for the stable resonances can be attributed to the choice of the graphene sheets printed on an electrically thin high-permittivity substrate.


(a)

(b) 4 0.8

3

0.6 2 0.4 1 0

0.2 20 40 60 80 Angle of incidence θ (o)

0

Frequency (THz)

Frequency (THz)

4

0.8

3

0.6 2 0.4 1 0

0.2 20 40 60 80 Angle of incidence θ (o)

(c)

Fig. 6. Simulation of full-wave CST results of the absorption for the graphene-based absorber at oblique angles of incidence for (a) TE and (b) TM polarization (Vg = 12 V, μc = 0.45 eV in sα , μc = 0.07 eV in sβ ). (c) Absorption of another type of structure is tuned by the bias voltage from 0 to 12 V. The lattice parameter is p = 6 μ m.

4.

Conclusions

In summary, we have designed terahertz absorbers and achieved a high absorption for a relatively wide bandwidth. Moreover, the absorber is implemented on the basis of a whole graphene monolayer, which is convenient to adjust the conductivity of graphene by changing the gate voltage. And the bandwidth of the absorption can be effectively improved by using multilayer structure. It is a promising candidate for absorbing elements in scientific and technical applications because it can achieve tunable absorption and is insensitive to the incidence angle θ . More types of the absorber can be used in “uneven dielectric slab” structure, which has potential applications in spectroscopic detection, thermal detectors, polarization sensitive sensors and tunable broadband planar filters. Acknowledgments This work has been partially supported by the priority academic program development of Jiangsu Higher Education Institutions.
