The suggested circulator is formed by a concave pattern graphene junction and three waveguides symmetrically connected to it. The graphene is supported by layers. The circulation behavior is based on the nonsymmetry of the graphene conductivity tensor which appears due to magnetization by a DC magnetic field applied normally to the graphene plane. The symmetrical mode propagating in the nonmagnetized graphene waveguide, is transformed in magnetized region to an edge-guided one providing the propagation from one port to another port and isolating the third port. The device characteristics depend on the physical parameters of the graphene junction, its dimensions and parameters of the substrate. We discuss a choice of these parameters to maximize the frequency band and isolation level and to minimize the losses and the applied DC magnetic field. The theoretical arguments are confirmed by full-wave computations. In an example, we demonstrate that the circulator can have the frequency band of 42% (from to ), with the isolation higher than 17 dB and the insertion losses better that , provided by the biasing DC magnetic field and the chemical potential of graphene .
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Circulators are the most popular nonreciprocal components in microwave technology . At lower frequencies in MHz frequency region, the so-called lumped elements circulators are used . The circulators for optical region have been discussed in many papers [3–5]. The circulators can be used for different purposes but their main function is a protection of the source from harmful reflections in the load.
The principal physical effects in the circulator design are phase and polarization nonreciprocity , resonances in ferrite elements  and edge-guided modes . Nonreciprocal properties of magnetized ferrites and some other magnetooptical materials are defined by nonsymmetry of the permeability or permittivity tensors.
The typical requirements to the circulators are large frequency band, high isolation level, low insertion losses and low applied DC magnetic field. The last requirement is stipulated by dimensions of the biasing magnetic field system. The microwave microstrip ferrite circulators with edge-guided modes are characterized by a very large frequency band which can achieve octave and more [7, 8]. For millimeter-wave edge-guided circulators, some semiconductors were suggested to use as a gyrotropic material [9, 10].
Graphene is the first 2D material obtained experimentally . The electronic characteristics of the graphene can be controlled dynamically by changing the charge density of the material and, consequently, its optical conductivity. It is achieved by chemical potential variation due to a bias voltage applied between the graphene layer and the substrate. Besides, the conductivity of graphene can be modified by magnetic field . The magnetized graphene possesses a nonsymmetrical conductivity tensor analogous to the permeability tensor of the magnetized ferrites and the permittivity tensor of the magnetized semiconductors. Due to its unique properties, graphene has shown to be a promising candidate for development of plasmonic components operating in terahertz (THz) region including nonreciprocal ones.
One can find in the literature several publications concerning nonreciprocal devices based on graphene, comprising both guided-wave components  and metasurfaces for free-space applications . Fundamental limits and near-optimal design of graphene-based devices are discussed in . Experimental confirmation of graphene-based Faraday rotation isolator realizability is presented in [16, 17].
In , the authors showed that tunable unidirectional surface plasmons can be supported in magnetized graphene. Two nonreciprocal devices with edge-guided modes for the THz and near-infrared region are discussed in this paper. The frequency band of the devices does not exceed 10%. A graphene isolator and a circulator based on the nonreciprocal coupling between double graphene layer are described in . A three-port THz graphene circulator was proposed and analyzed in our previous paper . The presented there results show a frequency band of about at the central frequency with isolation of and insertion losses of . The frequency band can be deslocated by changing the chemeical potential. Another version of THz three-port circulator was discussed in . It consists of three waveguides coupled to a cavity resonator in the center. For low temperature regime , the high BW (bandwidth) of the circualtor of 29% with the central frequency 17 THz is obtained.
In this work we propose and analyze an ultrawideband THz circulator based on edge-guided waves in graphene. Some preliminary results of this work was presented in . The suggested circulator is composed of a single layer of graphene placed on a dielectric substrate. The device has a very simple structure and does not contain any additional metal elements. We will show that at room temperature conditions, the BW of the circulator can be of 42%, and even can reach octave, i.e. it is much broader than those published in the literature. We discuss also the physical and geometrical requirements allowing one to achieve such a BW. Finally, we present two examples of projected circulators by using full-wave numerical computations.
2. Geometry of circulator and its scattering matrix
The geometry of the device and the used coordinate system are shown in Fig. 1. The graphene structure consists of the central region and the three waveguides symmetrically connected to it. The width of the waveguide strips is denoted by w and R is the radius of the circumference that defines the curvature in the central region. The thickness of the graphene layer is d, of the silica () is h1 and of the silicon (Si) is h2. The permittivity of the silica is 2.09 and of the silicon is 11.9. The uniform DC magnetic field B0 is oriented normally to the graphene layer. The magnetized part of the junction in Fig. 1 is shaded. The length of the arc Lm is defined by . Thus, R can serve as a measure of the magnetized area. Notice, that the geometry of the circulator is similar to that used in microwave microstrip ferrite circulators based on edge guided modes [7, 8].
The circulator is described by the following scattering matrix:
3. Numerical modeling of graphene
In the following, the electric conductivity tensor of graphene will be modeled by using the semiclassical approach, based on the Boltzmann transport equation. This model is described by simple equations and fits well in the THz frequency region. This approach is discussed in .
The graphene conductivity tensor can be written as follows:
The parameters of the conductivity tensor are given in :24].
The parameters and define losses in graphene. The gyrotropic activity of graphene can be described by the factor
This parameter is analogous to the gyrotropy used in ferrite technology , where k and μ are the diagonal and the off-diagonal elements of the permeability tensor. For , Eq. (6) can be simplified to , i. e. the parameter g is inversely proportional to ω. In the octave frequency band, the value of g is reduced by twice, i. e. in our case from 0.6 at the frequency to 0.3 at the frequency .
As an example, we shall consider a possibility to realize a ultrawideband circulator in the frequency region THz. Figures 2 and 3 present the frequency behaviour of the electrical conductivity tensor components of the graphene for different values of magnetic field and chemical potential, respectively. The peaks of losses corresponding to cyclotron resonance are shifted for higher frequencies with the increase of the magnetic field and with the decrease of the chemical potential in accordance with Eq. (5).
Using these graphics, it is possible to define the parameters of graphene allowing to minimize the losses and the applied DC magnetic field in the circulator. For the discussed frequency region, the gyrotropy g is shown in Fig. 4. The dependence of g on relaxation time τ in the frequency region of interest is relatively low.
4. Graphene waveguide characteristics
The numerical calculus were performed by full-wave finite element method using the commercial softwere Comsol Multiphysics . For the atomic thickness of the graphene layer, electromagnetic solvers in general do not allow to model the surface conductivity, described by Eq. (2). For this reason, an artificial thickness of the graphene was used in calculus  in such a way, that the volumetric conductivity acquires the form , where is the adopted thickness of the graphene sheet. The tensor parameter in this 3D representation of graphene was chosen to be .
It is known that transverse magnetic (TM) surface plasmon polariton (SPP) waves, which are used usually for waveguiding purposes in graphene plasmonics, in an infinite magnetized graphene layer can exist only for the frequencies , where the imaginary part of the conductivity is negative . In the waveguide with finite-width graphene, the hybrid waves propagate along the guiding strip.
The quality of SPP waveguides is usually described by two competing parameters: the propagation length and the degree of SPP confinement. The propagation length in nonmagnetized case depends on the type of guide mode, on dimensions of the waveguide andon the figure-of-merit of graphene . The FOM depends, in particular, on chemical potential μc . We shall not discuss further this problem and adopt a certain fixed value of the chemical potential.
Taking as a starting point the magnetic field and the chemical potential , one comes to the cyclotronic frequency . Therefore, to avoid high losses, one should work at the frequencies . Using the quality factor of the cyclotronic resonance , where is a full width at half maximum of the resonance, one can choose the left limit of the frequency band. In our case, it is . With , one comes to .
The two lowest plasmonic waves in graphene strip are the symmetric and antisymmetric edge mode . Figures 5(a)-5(d) depict the Ez component of these mode calculated for graphene waveguide with and without DC magnetic field at the frequency . One can see a high dislocation of the field to one edge of the guide for magnetized structure.
Figures 5(e)-5(f) depict the effective index of these modes for the frequencies ranging from to for waveguides with the width and . The effective index is defined as , where k0 is the free space wave number and β is the propagation constant of the mode . For the frequencies less than , the real part effective index of mode 1 is small and the imaginary part is high. Therefore, the frequency can be chosen as the left boundary. The cutoff frequency of mode 2 is for the guide width , and for the guide width .
Figure 6 depicts the electric field in graphene-dielectric interfaces for the circulator described below. The exponential decay of the curve confirms that the device is based on plasmonic waves. The field confinement increases at higher frequencies. These results also show that for the chosen thickness of the spacer h2, the most part of electromagnetic energy propagates in the spacer, minimizing thus the ohmic losses in the highly doped Si substrate.
Now we shall consider the field structure in the device. The electric field components are shown in two different regions of the circulator. In Figs. 7(a), 8(a) and 9(a), the calculated components of the electric field E are shown along the line AB in the nonmagnetized guide. One can see that the structure of the electromagnetic field in the nonmagnetized waveguide is slightly nonsymmetric with respect to the waveguide plane of symmetry. This is because the symmetrical excitation in the input waveguide sums up with a small reflected wave from the magnetized part of the junction where a nonsymmetrical field distribution exists. In Figs. 7(b), 8(b) and 9(b), the components are calculated along the line CD close to the center of the magnetized region. The wave is now guided predominantly along the right edge.
5. Circulator bandwidth limitations
Theoretically, the maximum possible frequency band of the circulator can be defined as follows. From the lower frequencies, the band is limited by cyclotron frequency ωc. As it was descussed in Section 4, in order to avoid high losses in the vicinity of the cyclotronic resonance, the low frequency bound ωl should be defined as .
The upper bound of the circulator is defined by several factors. Firstly, the waveguides must provide one-mode regime of propagation. The principal mode does not have cutoff . At the frequencies higher than the cutoff frequency ω2 of the second mode, it starts to propagate in the waveguides (see Fig. 5(e)). In the vicinity of the cutoff, the losses of this mode are very high, and this can lead to deterioration of the circulator characteristics. This frequency is the principal factor which defines the upper frequency bound. Thus, potentially available frequency region for the design of the edge-guided circulator is . Analyzing Figs. 5(e) and 5(f) one can see that the upper bound of the single mode regime depends on the width w of the waveguides.
Secondly, when the frequency is much higher than the cyclotron frequency ωc, the gyrotropic activity of the graphene defined by the parameter g reduces significantly as it can be seen in Fig. 4, and this can lead to a weaker edge effect. As a consequence, the isolation can be small.
Thirdly, the magnetized part of the circulator depends on the radius R (see Fig. 11 below). This parameter also can limit the bandwidth of the circualtor. Small radius R of the circulator permits to decrease the in-plane dimensions of the magnetized part of the circulator and consequently, to reduce its insertion losses. However, the small R reduces the isolation level, because the edge-mode is not able to formed along the short length Lm. On the other hand, high radius R increasing the isolation, increases also insertion losses because of the enlarged length Lm. Therefore, the radius should be also optimized in accordance with the requirements of the circulator design.
6. Circulator design
Below, we consider separately two cases of the circulators with the width of the graphene strip and .
6.1. Structure of electric field in circulator,
The presence of the magnetic field causes the wave to be shifted predominantly to one side of the strip decaying exponentially along the width of the strip, thereby directing the wave to the desired port. In order to evaluate this effect, we have calculated the structure of the electric field in three frequency points: in the central region of BW at and at the left and right bounds of it at and , respectively (see Figs. 7,8,9). The fields have a similar structure. At higher frequencies, the edge effect is more pronounced.
6.2. Frequency responses of circulator,
Bandwidth was calculated as an intersection of frequency bands corresponding to the reflection which is higher than , the transmission which is better than and the isolation higher than . From Fig. 10 one can see that, in the band of THz, the circulator has a good matching where S11 is better than , the level of isolation is better than , and the insertion loss is better than . Thus, the resultant bandwidth is 42%.
We show in Fig. 10 the frequency responses for two values of the graphene relaxation time and . One can see that the main difference in the responses for is slightly higher insertion losses.
The dependence of the circulator BW on the radius R is shown in Fig. 11. The BW is about 40% for the interval of R between and with the optimum value of 42% for .
At the frequency the length Lm is approximately , and at it is , where λ0 is the operating wavelength in the free space. Thus, the circulator has a very small footprint.
6.3. Frequency responses of circulator,
In order to verify our assumption about the high frequency limit of the BW, we calculated the circulator with the strip width while keeping all the other parameters fixed as in case of . These results are presented in Fig. 12. The lower value of w shifts the cutoff frequency ω2 of the second mode to high frequencies, in accordance to Fig. 5. As a result, the upper frequency boundary of the circulator with displaces to higher frequencies. However, at low frequencies the isolation level becomes somewhat lower and, besides, a peak of slightly enlarged insertion losses at appears at the upper bound.
The above arguments allow one to choose roughly the parameters of the circulator for ultrawideband regime. To improve the circulator characteristics and to avoid try-and-cut procedure, as a next step one can use a fine multiparametric optimization of the circulator in terms of w, R, B0 and μc.
7. Discussion of results
One of the main problem in the graphene photonic technology is relatively high losses. Some comments concerning this problem in the circualtor design will be made below. The principal source of the losses is intrinsic physical processes in graphene. The losses are usually defined by the relaxation time τ. In this work, we used the value of this parameter ps. This value corresponds to a free-standing graphene. The presence of a substrate reduces relaxation time to approximately ps. However, improvement of the quality of the graphene and the use of some special materials for substrate, for example, hexagonal boron nitride , are expected to enlarge the parameter τ.
The losses depend also on the length Lm. In general, smaller footprint of the junction provides lower insertion losses. Comparing with the published in  analogous devices with the in-plane dimensions of about , one can see that the suggested above circulator has the dimensions of about which are one order lower ones.
Another possible source of losses is electromagnetic radiation from junction. These losses appear mostly due to discontinuities in the circuit. For a well projected circulator with small reflections corresponding to small parameter S11, the radiation losses can be reduced to very low values.
One more comment is related to the biasing field. We believe that the circulator can be projected for higher frequencies. In accordance to the analysis of Section 3, the higher frequency operation region will require a larger DC magnetic field. In circulator design, it is always desirable to work with biasing magnetic field B0 as low as possible because this field defines the weight and dimensions of the necessary magnetic system. This specification can be included in the optimization process.
In this work, a three-port graphene-based circulator operating in the THz region is proposed and analyzed. The device has a very simple geometry, consisting of one-layer graphene junction of three waveguides placed on a dielectric substrate. For modelingthe electric conductivity of the graphene, the semi-classical model was used.
The components of the electric field in the circulator have been calculated in two different planes. The first one is located in the guide without magnetization and another plane is in the magnetized region. The results show the field displacement to one the edges of the junction for the magnetized case, confirming the edge guiding principle of the circulator functioning.
We have shown that the low-frequency boundary of the BW of the circulator is defined by the cyclotronic resonance, and the high frequency boundary by the cut-off frequency of the second mode. The realizability of the circulator depends on the correct choice of its physical and geometrical parameters. The principal parameters defining the BW are the DC magnetic field, the chemical potential, the width of the graphene strips and the radius of the circumference that defines the curvature in the central region.
By full-wave numerical simulations we have demonstrated that the suggested circulator possesses very high bandwidth which can achieve octave with good transmission, isolation and low reflection characteristics. Therefore, it can be a good candidate foruse in integrated THz and infrared circuits.
Brazilian Agency National Counsel of Technological and Scientific Development (CNPq).
This work was supported by Pro-Rectory of Research and Post-Graduation - PROPESP/UFPA.
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