Abstract
To investigate the frequency shift phenomenon by inserting graphene, a generalized lossy transmission-line model and the related electrical parameter-extraction theory are proposed in this paper. Three kinds of graphene-based transmission lines with attenuation phenomenon including microstrip line, double-side parallel strip line and uniplanar coplanar waveguide are analyzed under the common conditions where different chemical potentials are loaded on graphene. The values of attenuation constant and phase constant and the real and imaginary parts of the characteristic impedance of transmission lines are extracted to analyze in details. When the attenuation constant and the reactance part of the characteristic impedance are approximately equal to zero, this kind of transmission line has low or zero insertion loss. On the contrary, the transmission line is under the radiation mode with obvious insertion loss. The phase constant changes linearly under the transmission mode and can be varied with changing of chemical potentials which attributes to the property of frequency tunability. Furthermore, a bandwidth reconfigurable uniplanar coplanar waveguide power divider is simulated to demonstrate that this theory can be applied to the design of three-port devices. In summary, this work provides a strong potential approach and design theory to help design other kinds of terahertz and mid-infrared reconfigurable devices.
Similar content being viewed by others
Introduction
Graphene is a promising candidate for nanoelectronics and nanophotonics devices due to its remarkable electro-optical property1,2. Graphene has aroused great interest and its potential applications for various fields have been explored recently3,4,5,6,7,8,9,10,11. For example, the reported graphene-based devices include absorbers3, electro-optical switches4, field-effect transistors5, amplifiers6, diodes7, terahertz antennas8, filters9, mixers10, plasmonic bragg reflectors11 and so on. The richness of optical and electronic properties makes the graphene have multiple attractive features. A multimode interferometer12 in mid-infrared spectrum based on a dielectric-loaded graphene plasmonic waveguide has been designed. This waveguide has extremely large modal effective indexes and modal effective index contrast, which can shrink the multimode interferometer greatly. The transparent graphene is considered as a front electrode in Organic/GaAs hybrid photovoltaic cells13 and the power conversion efficiency can be greatly enhanced. In addition, the graphene is applied in a copper-graphene-based photonic crystal fiber plasmonic biosensor14 to prevent Cu oxidation and enhance sensing performance.
The surface complex conductivity of the graphene can be dynamically controlled by the applied voltage, which makes it possess unprecedented opportunities for reconfigurable plasmonic devices at terahertz and mid-infrared frequencies. The operating bandwidth of the broadband absorber based on graphene can be dynamically tuned by varying the bias voltage of the graphene3. A novel reconfigurable terahertz graphene-exploited leaky-wave antenna can allow both frequency tuning and beam steering by adjusting the graphene conductivity8. The design of the tunable optical delay line is continuously tunable due to the graphene sandwiched between the stacked ring resonators15. Tuning of the delay time could be achieved by varying the voltage across the graphene layers.
However, few researches have investigated the fundamental theory for explaining the frequency shift characteristic of graphene-loaded devices and providing the design guides. To our best knowledge, the generalized lossy transmission-line model and electrical parameter-extraction (T-LMEP-E) theory are proposed for the first time in this paper to design frequency tunable graphene-based transmission lines. Three kinds of practical and common transmission lines including microstrip, double-side parallel strip line and uniplanar coplanar waveguide are chosen and investigated. The calculated results based on the proposed T-LMEP-E theory show that the introduction of graphene in transmission line greatly affects the phase constant modeled as a function of frequency. In the linear changing range of the phase constant at a certain frequency point, different chemical potentials loaded in graphene correspond to different phase constant. Thus the changing of chemical potential of the graphene could alter the phase constant, so as to vary the resonant frequency of the transmission line. When the attenuation and the reactance part of the characteristic-impedance approximately equal to zero, the transmission line has low or zero insertion loss. Otherwise, the transmission line exhibits a large insertion loss under the radiation mode. In order to verify the application of the proposed T-LMEP-E theory on the three-port devices, a bandwidth reconfigurable uniplanar coplanar waveguide power divider is simulated. Finally, these results show that the proposed T-LMEP-E theory would provide an effective and clear design guide for reconfigurable terahertz and mid-infrared graphene-based devices.
Methods
Lossy Transmission-Line Model and Electrical Parameter-Extraction (T-LMEP-E) Theory
In this work, we propose the generalized transmission-line concept and the corresponding mathematical model for accurately characterizing the two-port network parameters of the microstrip line, the double-side parallel strip line and the uniplanar coplanar waveguide. The proposed transmission-line concept and model especially for lossy feature are very convenient and effective to predicate the attenuation phenomenon of three kinds of transmission lines. Unlike the lossless transmission-line theory, which is widely applied in microwave engineering16, the lossy transmission-line model requires complex propagation constant and complex characteristic impedance in its mathematical expressions. As shown in Fig. 1, the complex propagation constant γ = (α + jβ) is defined, where α and β are the attenuation and phase constants, respectively. Similarly, the complex characteristic impedance is defined as Zc = Rc + jXc, where Rc and Xc are the real and imaginary parts, respectively. If the physical length of the transmission line is L and the terminated impedance is ZL, the input impedance Zin can be calculated by
where
In order to achieve the analytical and frequency-dependent scattering parameters, the ABCD matrix of this proposed transmission-line model shown in Fig. 1 can be expressed by
When the port impedances of both Ports 1 and 2 are equal to , the ABCD matrix of this proposed transmission-line model can be determined by the final scattering parameters (for both the simulation and the measurement results)17. The mathematical expressions are
where Sij(=1,2) can be arbitrary simulated or measured results. After combing (1–4), we can obtain the following equations to calculate the complex propagation constant and the characteristic impedance:
Since the scattering parameters Sij( = 1,2) are the functions of the operating frequency f, namely, frequency-dependent feature, the complex propagation constant γ and the characteristic impedance Zc are also varied with the operating frequency f. The graphene can be modeled as a two-dimensional surface with complex conductivity σ due to its property of an atom thickness18. The intra-band and inter-band conductivities defined by Kubo’s formulas are given as19,
where e is the electron charge, ω is the angular frequency, kB is the Boltzmann constant, is the reduced Plank constant, T is the temperature in Kelvin, τ is the electron relaxation time and Г = 1/(2τ) is the electron scattering rate. In this paper, the common condition that T = 300 K and τ = 1 ps is assumed20 and only the intra-band contribution is considered due to the slight influence of the inter-band conductivity. It should be noted that the technology of the hybrid graphene-metal implementation has been validated in the experiment21.
Numerical calculations
All models in this paper are simulated using the full-wave electromagnetic simulator software based on the finite element method (FEM). The determined substrate is constructed by using SiO2 with a dielectric constant of 3.8 and a thickness of 1 μm. The hybrid graphene-metal implementation is a novel method21,22,23,24 to make the whole transmission line have dynamic equivalent surface conductivity, which can be tuned by changing the chemical potential.
Results
Three kinds of transmission lines and the graphene-based power divider using uniplanar coplanar waveguide transmission line
The three kinds of transmission lines are analyzed by using the generalized lossy transmission-line concept and the corresponding mathematical model. The general propagation constant γ = (α + jβ) and the characteristic impedance Zc = Rc + jXc are both complex values, where α and β are the attenuation and phase constants while Rc and Xc are the real and imaginary parts of the characteristic impedance, respectively. The parameters of α, β, Rc and Xc of the three kinds of transmission lines are extracted to model as a function of frequency. The scattering parameters of the three kinds of transmission lines and the uniplanar coplanar waveguide (strip) power divider are simulated by the full-wave electromagnetic simulation software. Three-dimensional views of the graphene-based microstrip, the double-side parallel strip and the uniplanar coplanar waveguide transmission lines are presented in Fig. 2 while the uniplanar coplanar waveguide (strip) power divider based on graphene is shown in Fig. 3.
The simulated results for the three kinds of transmission lines are shown in Figs 4, 5, 6. It can be seen from Figs 4(a), 5(a) and 6(a) that when the value of |S21| is almost equal to zero, the attenuation α is close to zero. While the value of |S21| is obviously not equal to zero, the attenuation α is not zero as well. The energy can be transmitted from the input port to the output port of the transmission line with a very low loss within the frequency range where the |S21| is nearly equal to 0 dB. Thus, the transmission lines are under the transmission mode in this situation. Once the |S21| is not zero and the attenuation α has a large value, the energy will be dissipated in the transmission line to a great degree. In this case, the transmission lines are not under the transmission mode but the radiation mode. From Figs 4(a), 5(a) and 6(a), it also can be seen clearly that the frequency is tunable with different chemical potentials loaded on the graphene under the transmission mode. As the chemical potential changes from 0 eV to 0.015 eV, the frequency where meets the condition initially of |S21| = −3 dB and |S11| < −10 dB increases.
From the proposed T-LMEP-E theory, the complex characteristic impedance Zc = Rc + jXc is extracted and plotted in Figs 4(b), 5(b) and 6(b) for all the three kinds of transmission lines. When the values of Rc and Xc are in flat area which means the values of them almost maintain the same with the changing of frequencies, the attenuation α is close to 0. At the same time, the impedance Rc is about 100 Ω, 55 Ω and 50 Ω extracted from microstrip transmission line, double-side parallel strip transmission line and uniplanar coplanar waveguide transmission line, respectively. And all of the reactance Xc are approximately equal to zero in the three situations, which means the transmission line is under the transmission mode. It would become easy to achieve acceptable impedance transformation and good power transmission characteristics for matching circuits/systems because the characteristic impedance is real. When the Rc and Xc have variable values, the corresponding attenuation α has a considerable value. The transmission line is under the radiation mode and the energy is easily dissipated because the reactance has considerable value. The values of the reactance Xc are related to impedance matching which has an influence on transmission characteristics. Thus the proposed T-LMEP-E theory can be used to extract the parameters of attenuation constant and phase constant, the real and imaginary parts of the characteristic impedance and guide the design of graphene-based devices.
The Figs 4(c), 5(c) and 6(c) show that the calculated attenuation constant α and phase constant β versus the frequency extracted from the proposed lossy transmission-line model and its related theory. Under the transmission mode, the phase constants almost have a linear relationship with frequency. The values of phase constant (β = 2π/λg, where λg is the equivalent wavelength) are different when the chemical potential is altered. Under the transmission mode, the equivalent wavelengths λg (λg = vg/fg) are unequal when the phase constants are not the same. Therefore, it finally verifies that the different chemical potentials of the graphene can make the graphene-based devices/systems achieve the reconfigurable frequency property.
The lossy transmission-line model and electrical parameter-extraction theory can be also applied to three-port devices. According to similar uniplanar power divider circuit structures25, a power divider utilizing graphene-metal uniplanar coplanar waveguide (strip) transmission line is proposed. Figure 3 shows the three-dimensional views of the proposed graphene-based uniplanar power divider. The corresponding scattering parameters are presented in Fig. 7(a–c). The power divider can achieve bandwidth reconfigurable function due to the introduction of the graphene with tunable chemical potentials. When |S21| is near to −3 dB, the energy can be divided equally into two output ports with extremely low insertion loss level. From Fig. 7(a), the initial frequencies where the magnitude |S21| is close to −3 dB and |S11| below −10 dB are 1.31 THz, 1.55 THz, 1.94 THz and 2.14 THz with the chemical potential changing from 0 eV to 0.015 eV. The initial frequencies increase as the chemical potential increasing. The tunable bandwidth function can be observed visibly from Fig. 7(b–c). The operating frequencies which |S22| below −10 dB move to higher frequency bands with the increasing chemical potential. The factor that greatly restricts the bandwidth of the proposed power divider is the isolation (|S23|) between two output ports. In order to make the proposed graphene-based power divider have reliable performance, |S23| should be below −15 dB. Thus the function of tunable bandwidth can be achieved when it meets the condition of |S22| < −10 dB, |S23| < −15 dB, |S11| < −10 dB and |S21| ≈ −3 dB. The detailed information is also tabulated in the Table 1.
Discussion
The frequency reconfigurable graphene-metal transmission line based on lossy transmission-line model and electrical parameter-extraction (T-LMEP-E) theory is proposed in this paper. Three kinds of transmission lines, including microstrip line, double-side parallel strip line and uniplanar coplanar waveguide (strip) are simulated to verify the proposed theory. The values of attenuation constant and phase constant and the real and imaginary parts of the complex the characteristic impedance are extracted to analyze in details, respectively. When the attenuation constant and the reactance part of the characteristic impedance are both close to zero, the transmission line is under the transmission mode with extremely low insertion loss. While the attenuation and the reactance part of the characteristic impedance have considerable value, the transmission line is under the radiation mode with large attenuation. The phase constant, as a function of frequency, has a linear relationship under the transmission mode. Within the range of linear relationship, the values of phase constants are different when the graphene is loaded with dynamic chemical potentials at the same frequency, indicating the tunability of the operating frequency. Thus the operating frequency can be altered with changing chemical potential μc. This theory can be applied for the typical three-port devices such as power dividers. The principle of the bandwidth reconfigurable uniplanar coplanar waveguide (strip) power divider is the same with that of transmission lines. Furthermore, this work will provide a potential approach and design theory to other kinds of microwave, terahertz and mid-infrared multi-port devices/systems requiring frequency reconfigurable properties.
Additional Information
How to cite this article: Wu, Y. et al. A Generalized Lossy Transmission-Line Model for Tunable Graphene-Based Transmission Lines with Attenuation Phenomenon. Sci. Rep. 6, 31760; doi: 10.1038/srep31760 (2016).
References
Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).
Lee, C., Wei, X., Kysar, J. W. & Hone, J. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321, 385–388 (2008).
Yao, G. et al. Dynamically electrically tunable broadband absorber based on graphene analog of electromagnetically induced transparency. IEEE Photonics Journal 8, 7800808:1-8 (2016).
Li, Y. et al. Graphene-based floating-gate nonvolatile optical switch. IEEE Photonics Technology Letters 28, 284–287 (2016).
Fahad, M. S., Srivastava, A., Sharma, A. K. & Mayberry, C. Analytical current transport modeling of graphene nanoribbon tunnel field-effect transistors for digital circuit design. IEEE Transactions on Nanotechnology 15, 39–50 (2016).
Yurchenko, S. O., Komarov, K. A. & Pustovoit, V. I. Multilayer-graphene-based amplifier of surface acoustic waves. AIP Advances 5, 057144:1-12 (2015).
Al-Dirini, F., Mohammed, M. A., Hossain, F. M., Nirmalathas, T. & Skafidas, E. All-graphene planar double-quantum-dot resonant tunneling diodes. Journal of the Electron Devices Society 4, 30–39 (2016).
Wang, X.-C., Zhao, W.-S., Hu, J. & Yin, W.-Y. Reconfigurable terahertz leaky-wave antenna using graphene-based high-impedance surface. IEEE Transaction on Nanotechnology 14, 62–69 (2015).
Correas-Serrano, D., Gomez-Diaz, J. S., Perruisseau-Carrier, J. & Álvarez-Melcón, A. Graphene-based plasmonic tunable low-pass filters in the terahertz band. IEEE Transactions on Nanotechnology 13, 1145–1153 (2014).
Mao, X. et al. Optoelectronic mixer based on graphene FET. IEEE Electron Device Letters 36, 253–255 (2015).
Wang, Y., Chen, Q. & Shen, X. Actively controlled plasmonic Bragg reflector based on a graphene parallel-plate waveguide. AIP Advances 5, 077152:1-7 (2015).
Zheng, R., Gao, D. & Dong, J. Ultra-compact broadband tunable graphene plasmonic multimode interferometer. IEEE Photonics Technology Letters 28, 645–648 (2016).
Huang, C.-H., Yu, S.-C., Lai, Y.-C., Chi, G.-C. & Yu, P. Efficiency enhancement of Organic/GaAs hybrid photovoltaic cells using transparent graphene as front electrode. IEEE Journal of Photovoltaics 6, 480–485 (2016).
Rifat, A. A. et al. Copper-graphene-based photonic crystal fiber plasmonic biosensor. IEEE Photonics Journal 8, 4800408:1-8 (2016).
Conteduca, D., Dell’Olio, F., Ciminelli, C. & Armenise, M. N. Resonant graphene-based tunable optical delay line. IEEE Photonics Journal 7, 7802409:1-9 (2015).
Basl, P. A. W., Bakr, M. H. & Nikolova, N. K. Theory of self-adjoint S-parameter sensitivities for lossless non-homogenous transmission-line modeling problems. IET Microwaves, Antennas & Propagation 2, 211–220 (2008).
Zhu, L., Sun, S. & Li, R. Microwave bandpass filters for wideband communications, Chapter 2 (2012).
Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).
Hanson, G. W. Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene. Journal of Applied Physics, 103, 064302:1-8 (2008).
Correas-Serrano, D., Gomez-Diaz, J. S., Alù, A. & Álvarez Melcón, A. Electrically and magnetically biased graphene-based cylindrical waveguides: analysis and applications as reconfigurable antennas. IEEE Transaction on Terahertz Science and Technology 5, 951–960 (2015).
Wang, L. et al. One-Dimensional Electrical Contact to a Two-Dimensional Material. Science 342(6158), 614–617 (2013).
Cheng, X. et al. Circular beam-reconfigurable antenna base on graphene-metal hybrid. Electronics Letter 52, 494–496 (2016).
Wu, Y., Qu, M., Jiao, L., Liu, Y. & Ghassemlooy, Z. Graphene-based Yagi-Uda antenna with reconfigurable radiation patterns. AIP Advances 6, 065308:1-11 (2016).
Wu, Y., Qu, M., Jiao, L. & Liu, Y. Tunable Terahertz Filter-Integrated Quasi-Yagi Antenna Based on Graphene. Plasmonics, in press doi: 10.1007/s11468-016-0328-9 (2016).
Fan, L. & Chang, K. Uniplanar power dividers using coupled CPW and asymmetrical CPS for MIC’s and MMIC’s. IEEE Transactions on Microwave Theory and Techniques 44, 2411–2420 (1996).
Acknowledgements
This work was supported by National Key Basic Research Program of China (973 Program) (No. 2014CB339900) and National Natural Science Foundations of China (No. 61422103 and No. 61327806).
Author information
Authors and Affiliations
Contributions
Y.W. conceived the idea and was responsible for theoretical analysis and the physical-structures construction of the lossy transmission-line model. M.Q. wrote the manuscript and performed the full-wave simulations. Y.L. provided various constructive comments and suggestions for improving this work. Y.W. and M.Q. contributed equally to this work. All authors contributed to this article.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Wu, Y., Qu, M. & Liu, Y. A Generalized Lossy Transmission-Line Model for Tunable Graphene-Based Transmission Lines with Attenuation Phenomenon. Sci Rep 6, 31760 (2016). https://doi.org/10.1038/srep31760
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep31760
This article is cited by
-
Bio-based Materials for Microwave Devices: A Review
Journal of Electronic Materials (2021)
-
Modeling and Analysis of Terahertz Graphene Switches for On-Wafer Coplanar Transmission Lines
Journal of Infrared, Millimeter, and Terahertz Waves (2020)
-
Single- and dual-band filtering-response power dividers embedded SIW filter with improved output isolation
Scientific Reports (2017)
-
A Broadband Graphene-Based THz Coupler with Wide-Range Tunable Power-Dividing Ratios
Plasmonics (2017)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.