SPECTRUM of electromagnetic radiation from a type-ii

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1 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 19, NO. 4, AUGUST A Superconductor THz Modulator Based on Vortex Flux Flow Farrokh Sarreshtedari, Mehdi Hosseini, Hamid Reza Chalabi, Ali Moftakharzadeh, Hesam Zandi, Sina Khorasani, Member, IEEE, and Mehdi Fardmanesh, Senior Member, IEEE Abstract A terahertz modulator based on the Type-II superconductor flux flow oscillator has been proposed. Analytical calculations are presented and the effects of intrinsic and extrinsic parameters such as disorder strength of crystal, penetration depth, frequency, and amplitude of the modulated current on the radiation power spectrum have been studied. The proposed structure also exhibits a mixer-like behavior, in the sense that its output harmonics range from the washboard frequency up to the superconductor gap frequency, so the input signal is practically mixed with the washboard frequency and its harmonics. The modulation index for each harmonic of this modulator has also been investigated. This well-featured modulator has a potential to be used in nextgeneration terahertz integrated transceivers. Index Terms Mixer, modulator, superconducting devices, vortex flux flow. I. INTRODUCTION SPECTRUM of electromagnetic radiation from a type-ii flux flow oscillator (FFO) has been derived previously by Bulaevskii and Chudnovsky [1]. Using an analytical approach and considering assumptions such as laminar flow of vortices [2], ignoring surface Meissner current and the back effect of the radiation on lattice motion, they have found the radiation power for different lattice parameters. It is shown that for a type-ii superconductor in vortex flux flow regime, as vortices come to the surface periodically with the frequency, the system radiates with harmonics of this frequency (washboard frequency) up to the superconducting gap. References [3] and [4] have studied the effect of current modulation on the motion characteristics of moving vortex lattice and washboard frequency for specific materials. In our work, based on the linear relation between and, which is satisfied in a pin-free approximation (when the Lorentz force is so strong that it exceeds the pinning force [5]), the effect of current modulation on the spectrum of the radiation power is analytically derived. We have considered the case of square vortex lattice in a type-ii isotropic superconductor and the dependence of output frequency spectrum and modulation index on different system parameters are investigated. The proposed device has the potential ability to be used as the mixer/modulator block of submillimeter integrated receivers, where FFO devices are used Manuscript received November 26, 2008; revised December 24, First published April 17, 2009; current version published July 29, This paper was recommended by Associate Editor N. F. Pedersen. The authors are with the Superconductive Electronics Research Laboratory, School of Electrical Engineering, Sharif University of Technology, Tehran, Iran( fardmanesh@sharif.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TASC Fig. 1. Schematic of the mixer/modulator. as the base device. Other technologies of existing mixers/modulators include superconductor insulator superconductor devices, Schottky Barrier diodes, hot electron bolometers, and metamaterials [6] [9]. The frequency spectrum in the output of this device, like all nonlinear mixers, consists of sum and difference harmonics of its input signals and the base frequency (washboard frequency). When all harmonics, expect the lowest order mixing products are filtered out, it may also be exploited as a terahertz (THz) amplitude modulator, since the input signal modulates the washboard frequency as the carrier. Fig. 1 shows a typical mixer/modulator and the corresponding input and output signals. As it is shown in Fig. 1, the carrier frequency (which is in fact fed from the washboard oscillation) is determined by external bias parameters of the system, such as perpendicular magnetic field, bias current, and characteristic parameters of the superconductor material used. For a vortex lattice moving at velocity and lattice constant, the carrier frequency is given by ; however, due to the nonlinear nature of FFO radiations, harmonics of this frequency exist up to the superconductor gap frequency. We also have calculated the depth of modulation in the AM modulator configuration, which is the ratio of the modulated component amplitude to the amplitude of the carrier. For simplicity, we have used the geometry and notations of [1] in derivation of equations. II. THEORETICAL MODEL For studying the electromagnetic radiation from a type-ii superconductor, the geometry of Fig. 2 is chosen. The superconductor is located within, and the modulated radiation power is then found for and. The applied magnetic field along the z-axis is assumed to satisfy, with and being the first and the second critical fields of the type-ii superconductor. By using a quasi-static approximation and Ginzburg Landau relations in the presence of magnetic field, one can find the fields /$ IEEE

2 3654 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 19, NO. 4, AUGUST 2009 Then, the associated electric fields and will be obtained by (2) (6) The homogenous parts of and are determined by the continuity of the fields at the boundaries of the superconductor [1], [10]. Using (3) and assuming, the step function can be approximated by (7) Fig. 2. Type-II superconductor with J = J + J, magnetic field H, and the moving vortex lattice with velocity v. Simplifying the expression in (1) yields inside the superconductor through solving the following equations: where is the unit vector along the z-axis, is the speed of light, and are, respectively, the magnetic and electric fields, is the unit step function, is the London penetration length, and is the position of the center of the th vortex line in the vortex lattice, where with and being integers. Referring to Fig. 2 and assuming a modulated current, the velocity of a specific vortex is denoted by (1) (2) Using (8), (1), and identity can be found as (8) (9) By integration, we obtain the coordinates as (3) (4) (5) Here, accounts for disorder in the position of vortices along y-axis. We now proceed to evaluate the electromagnetic fields and. The solution of (1) can be constructed by finding the homogenous solution and the particular solution corresponding to the inhomogeneous equation. (10)

3 SARRESHTEDARI et al.: SUPERCONDUCTOR THz MODULATOR BASED ON VORTEX FLUX FLOW 3655 And after some simplifications, we find the expression for magnetic flux density as is negligible [1]. Ignoring the relatively small terms and simplifications gives (11) By integration over component of the wave vector and considering a specific harmonic with, we get The power radiation can be equivalently rewritten as (16) (17) de- Using the same approach adopted in [1], the parameter fined as the ratio of electrical to magnetic field, is given by (12) where, and the structural factor is given by (18) (13) Using the relation, Maxwell equations, and (6), we can express the total field only by and as (14) Evaluating the radiation power at frequency, outside the superconductor on as in Fig. 1, results in (15) It should be noted that there is a small difference between right-propagating and left-propagating radiation powers, which III. RESULTS AND DISCUSSIONS We have obtained the frequency contents of the modulator through approximation of the radiated power (17) for different limits as introduced in [1]. Different cases are investigated in three sections, according to the magnitude of dimensionless transversal and longitudinal correlation lengths for moving vortex lattice which are expressed in units of the intervortex spacing a [1] and [11]. These three approximation regions are considered as weak disorder and large crystal, weak disorder and small crystal, and strong disorder. All calculations are based on the numerical data given in Table I. A. Weak Disorder and Large Crystal For the case of weak disorder, we assume and neglect the variation of denominator with respect to. By using the identities (19) (20)

4 3656 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 19, NO. 4, AUGUST 2009 TABLE I NUMERICAL VALUES OF MODULATOR Fig. 4. Modulation index versus angular frequency for different carrier frequencies. Fig. 3. Normalized power versus angular frequency. The modulator radiation power spectrum is found as Fig. 5. Schematic of normalized power of different carrier frequencies together with their corresponding harmonics. (21) In Fig. 3, we have shown the power of harmonics versus angular frequency of each harmonic order. It can be inferred that the carrier component has a constant power versus frequency and the modulated components show a parabolic behavior, as given by (21). B. Weak Disorder and Small Crystal Assuming weak disorder and small crystal dimensions, the radiation power can be found as (22) C. Strong Disorder Finally, in the case of stronger disorder with smaller correlation lengths, the radiation power is simplified as (23) It can be inferred from (22) and (23) that the behavior of these two cases is roughly the same, but the amplitudes would be different. In Fig. 4, the frequency contents of the modulator for cases B and C is shown for the first ten harmonics of a typical high-tc superconductor for the proposed modulator. It is evident in Fig. 4 that the power increases quadratically with the harmonic order number. Although the variation of frequency contents of modulator for case A is essentially different from the other, the modulation index was observed to be the same, as illustrated in Fig. 5.

5 SARRESHTEDARI et al.: SUPERCONDUCTOR THz MODULATOR BASED ON VORTEX FLUX FLOW 3657 IV. SUMMARY AND CONCLUSIONS We have shown wideband THz mixing and modulation effects, through modulating the bias current of a type-ii superconductor FFO by a small ac signal. Calculations reveal that the radiated power from vortex flow has two modulated components accompanied by harmonics of washboard frequency up to the superconducting gap. By changing either the bias current, or the perpendicular magnetic field, one can tune the washboard frequency of moving vortex lattice, and therefore, the carrier frequency of proposed modulator. We have found that in the weak disorder and large crystal approximation, the radiated power has constant carrier amplitude and parabolic behavior for modulated components, while the assumptions of weak disorder and small crystal dimensions, or strong disorder result in a parabolic carrier amplitude and quadratic dependence of the modulation components on the harmonic frequency. REFERENCES [1] L. N. Bulaevskii and E. M. Chudnovsky, Electromagnetic radiation from vortex flow in type-ii superconductors, Phys. Rev. Lett., vol. 97, pp. (197002)1 4, [2] A. E. Koshelev and V. M. Vinokur, Dynamic melting of the vortex lattice, Phys. Rev. Lett., vol. 73, pp , [3] A. Schmid and W. Hauger, On the theory of vortex motion in an inhomogeneous superconducting film, J. Low Temp. Phys., vol. 11, pp , [4] J. M. Harris, N. P. Ong, R. Gagnon, and L. Taillefer, Washboard frequency of the moving vortex lattice in YBa Cu O detected by ac dc interference, Phys. Rev. Lett., vol. 74, pp , [5] K. Fossheim and A. Sudbø, Superconductivity: Physics and Applications. Hoboken, NJ: Wiley, [6] V. P. Koshelets and S. V. Shitov, Integrated superconducting receivers, Supercond. Sci. Technol., vol. 13, pp. R53 R69, [7] V. P. Koshelets, P. N. Dmitriev, A. B. Ermakov, A. S. Sobolev, M. Y. Torgashin, V. V. Kurin, A. L. Pankratov, and J. Mygind, Optimization of the phase-locked flux-flow oscillator for the submm integrated receiver, IEEE Trans. Appl. Supercond., vol. 15, no. 2, pt. 1, pp , Jun [8] A. M. Barychev, Superconductor-Insulator-Superconductor THz Mixer Integrated With a Superconducting Flux-Flow Oscillator, Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, [9] H. T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, A metamaterial solid-state terahertz phase modulator, Nature Photon., vol. 3, pp , [10] L. N. Bulaevskii and A. E. Koshelev, J. Supercond. Novel Magnetism, vol. 19, pp. 3 5, [11] G. Blatter, M. V. Feigel man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, Vortics in high-temperature superconductors, Rev. Mod. Phys., vol. 66, pp , Farrokh Sarreshtedari was born in Iran in He received the B.S. and M.S. degrees in electrical engineering from Sharif University of Technology, Tehran, Iran, in He is currently working toward the Ph.D. degree in electrical engineering from Sharif University of Technology. His research interests include superconductivity, SQUID devices, SQUID-based systems, and solid-state and photonic devices. Mehdi Hosseini was born in Iran in He received the B.S. and M.S. degrees in physics from Sharif University of Technology, Tehran, Iran, in 2005 and 2007, respectively. He is currently working toward the Ph.D. degree from Sharif University of Technology. His research interests include statistical physics, solid-state physics, and superconductivity. Hamid Reza Chalabi was born in Yazd, Iran, in He is currently working toward the B.Sc. degrees in electrical engineering and physics from Sharif University of Technology, Tehran, Iran. His research interests include photonic devices, solid-state physics, superconductivity, and high-frequency microelectronics. Ali Moftakharzadeh was born in Yazd in October, He received the B.Sc. degree in electrical engineering from K. N. Toosi University of Technology and the M.Sc. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in He is currently working toward the Ph.D. degree in electrical engineering from Sharif University of Technology. His research interests include cryogenic system design, Josephson-junction-based devices, quantum computation, quantum optics, and superconductor bolometers. Hesam Zandi was born in Tehran, Iran, in June, He received the B.Sc. and M.Sc. degrees in electrical engineering from Sharif University of Technology, Tehran, in He is currently working toward the Ph.D. degree in electrical engineering from Sharif University of Technology, Tehran, Iran. His research interests include photonics, photonic devices, solid-state lasers, plasmons, quantum optics, and especially quantum bits. Sina Khorasani (M 06) was born in Tehran, Iran, on November 25, He received the B.Sc. degree from Abadan Institute of Technology, in 1995, and the M.Sc. and Ph.D. degrees from Sharif University of Technology, Tehran, in 1996 and 2001, respectively, all in electrical engineering. After spending a two year term as a postdoctoral fellow at the School of Electrical and Computer Engineering, Georgia Institute of Technology, he returned to Sharif University of Technology where he is now an Assistant Professor of Electrical Engineering. He has authored or coauthored more than 140 contributions in journals and conferences in various fields of optics and photonics, plasma physics, and solid-state electronics. Mehdi Fardmanesh (S 91 M 97 SM 02) was born in Tehran, Iran, in He received the B.S. degree from Tehran Polytechnic University and the M.S. and Ph.D. degrees from Drexel University, PA, all in electrical engineering, in 1987, 1991, and 1993, respectively. In 1989, he joined Drexel University, and until 1993, he conducted research in development of the thin- and thick-film high-temperature superconducting materials, devices, and development of ultralow noise cryogenic characterization systems, where he was awarded a research fellowship by the Ben Franklin Superconductivity Center in From 1994 to 1996, he was Principal Manager for R&D and the Director of a private sector research electrophysics laboratory, while also teaching at the Departments of Electrical Engineering and Physics at Sharif University of Technology, Tehran. In 1996, he joined the Electrical and Electronics Engineering Department at Bilkent University in Ankara, teaching in the areas of solid-state and electronics, while also supervising his established Superconductivity Research Laboratory. In 1998 and 1999, he was invited to ISI-Forschungszentrum Juelich in Germany where he pursued the development of low-noise High-Tc rf-squid-based magnetic sensors. In 2000, he established an international collaboration between Bilkent University and Juelich Research Center in Germany in the field of superconductivity, and from 2000 to 2004, he was the director of the joint project for development of High Resolution High-Tc SQUID-based magnetic imaging system. Since 2000, he also reestablished his activities at the Electrical Engineering Department of Sharif University, where he is presently the head of the Electronics Department. He set up a Superconductor Electronics Research Laboratory (SERL) at Sharif University of Technology in 2003, which he has directed since then. His research interest has mainly focused on the design, fabrication, and modeling of high-temperature superconducting devices and circuits such as bolometers, microwave filters and resonators, Josephson Junctions, and SQUID-based systems, in the areas of which he holds several international patents.