Numerical study of vortex effects in superconducting microstrip lines

Physica C 471 (2011) 338–343

Contents lists available at ScienceDirect

Physica C journal homepage: www.elsevier.com/locate/physc

Numerical study of vortex effects in superconducting microstrip lines Jolly Andrews a, Vincent Mathew b,⇑ a b

Research and Development Centre, Bharathiar University, Coimbatore-641 045 and Christ College, Irinjalakuda, Thrissur, Kerala 680 125, India Research and Postgraduate Department of Physics, St. Thomas College, Palai 686 574, India

a r t i c l e

i n f o

Article history: Received 19 June 2010 Received in revised form 1 February 2011 Accepted 13 March 2011 Available online 21 March 2011 Keywords: Vortex effects Superconducting microstrip lines Coffey–Clem model

a b s t r a c t A high transition temperature superconducting microstrip structure is modeled using the theoretical approach developed by Coffey and Clem for elucidating the vortex effects in propagation. Impedance type Green’s functions are derived for the electric field around the strip and the propagation characteristics are computed for a wide range of applied field, reduced temperature and superconducting strip thickness in a Galerkin procedure. The increase of static field and temperature result in increased vortex motion, which in turn, causes a corresponding change in the propagation characteristics of the transmission line. Numerical results are presented for propagation parameters and quality factor. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction In many of the practically important situations, a high transition temperature superconducting (HTS) microstrip line is operated under a static magnetic field in mixed state [1–4]. Hence it is important to study the microwave response of HTS film in mixed state. Both from the material characterization and device application points of view, this requires studying the propagation characteristics of HTS microstrip lines under an applied magnetic field. It is also very important to develop simulation tools to calculate the microwave propagation characteristics of such structures, which incorporates the vortex dynamics of magnetic flux in the film. Many attempts were made to model the complex conductivity and surface impedance of the type-II superconducting materials as a function of temperature, magnetic field and other parameters. Although the two fluid description with Gorter–Casimir type dependence on temperature remains the major working rule for calculating the superconducting parameters for HTS materials [5], several modifications of the same have appeared in the literature [6–15]. The problem is to a large extent complicated due to the lack of an accomplished theory of the high Tc superconductivity. The modified two fluid empirical model proposed by Vendik et al. [10] largely portraits many of the properties of high Tc materials and is currently considered as a standard model for most of the temperature-dependent properties. The above discussed models [6–14], however, fail to describe vortex related effects of superconducting materials. It is observed that the conductivity and surface impedance of the high Tc superconducting materials in mixed state widely vary

⇑ Corresponding author. Tel.: +91 9447137629. E-mail address: [email protected] (V. Mathew). 0921-4534/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2011.03.006

as the temperature and the external magnetic field are gradually increased. The microwave losses induced by the vortex motion has been a subject of very active study [4,16–23]. Golosovsky et al. [24] has presented a survey of the various attempts made in the direction of modeling vortex dependent parameters of HTS materials. Microwave properties of YBCO, DBCO, Niobium, Rhenium, Aluminium and MgB2 thin films in dc magnetic field have also been studied and losses are computed in terms of surface resistance and reactance [23,25–34]. Clem and his group have elaborately studied various flux related aspects in superconducting thin films [35–38]. However, few studies were seen in the literature on the propagation parameters of superconducting microstrip lines in the vortex state. In order to theoretically analyze HTS microstrip lines one needs to take into account the combined effects of the applied static magnetic field, the field produced by the microwave radiation and the added field effects due to the moving vortices. The phenomenologically unified theory proposed by Coffey and Clem [39,40] takes care of the vortex dynamics including both the influence of vortex pinning and flux creep in a self consistent manner. Such an approach is extremely important when we investigate the propagation characteristics of the superconducting microstrip line, where the temperature and applied magnetic field vary. In a recent work Wu [41] has applied this formalism to the case of high Tc superconducting parallel plate waveguides. Coffey has also discussed the effect of static field on the microwave propagation in superconducting transmission lines in mixed state [42]. In this paper we study the effect of applied magnetic field on the propagation characteristics of a superconducting microstrip line in mixed state. We consider a long microstrip line of high Tc material, like YBCO, printed on a typical substrates like sapphire, where a static magnetic field is applied in the plane of the strip, which will establish vortices in it. Using the rigorous full wave

339

J. Andrews, V. Mathew / Physica C 471 (2011) 338–343

analysis employing spectral domain Green’s functions, a dispersion relation for microwave propagation is numerically established in a Galerkin’s procedure [43]. Coffey–Clem model has been used to determine the surface impedance, which forms the complex resistive boundary condition [44], of the microstip line as a function of temperature, magnetic field and frequency. This surface impedance has been used to write down impedance type Green’s function for the field around the strip. The anisotropic nature of the flux line system has been avoided for simplicity of discussion. The changes in superconducting strip thickness is also incorporated and the corresponding changes in propagation are noticed. It was observed that a minimum strip thickness is required for the propagation of the microwave without significant dissipation. This paper is organized as follows: A brief description of the Coffey–Clem model is attempted in the next section, followed by an outline of the method of analysis. The third section discusses the numerical results. 2. Theoretical formulation The Coffey and Clem model of type-II superconductors in the mixed state takes into account the fluxon motion, the pinning effects and the flux-creep in the two fluid framework of the superconductor in a self consistent manner. The results are expressed in terms of a general complex-valued dynamic mobility for fluxon motion which self-consistently include the coupling effect of supercurrent density and vortex displacements. The dynamical flux-creep effects, which play an important role in high temperature, are accounted. The response of normal-like, quasiparticle excitations, are also included in our calculations [39,40]. The self-consistently determined penetration depth ~ k¼~ kðx; B; TÞ which accounts for the fluxon motion is given in terms of the normal fluid skin depth dnf and the complex effective skin depth ~ dvc as [39,41]

~kðx; B; TÞ ¼

k2  ði=2Þ~d2vc 1 þ 2ik2 d2 nf

!1=2 ð1Þ

The static field and temperature-dependent penetration depth k is given by k(B, T) = k(0, T)/[1  B/Bc2(T)]1/2, where k(0, T) = k0/[1  (T/ Tc)c]1/2 and the temperature-dependent upper critical field is

Bc2(T) = Bc2(0)[1  (T/Tc)2][1 + (T/Tc)2]1. In the light of [10], the value of c has been set to 2. The lengths associated with the normal fluid and vortex responses, respectively, are given in terms of the corresponding resistivities as d2nf ðx; B; TÞ ¼ 2qnf =l0 x and ~ ~ m =l0 x, d2vc ðx; B; TÞ ¼ 2q where the effective resistivity ~ m ðx; B; TÞ and l ~ m ðx; B; TÞ is the complex dynamic q~ m ðx; B; TÞ ¼ B/0 l vortex mobility given by

l~m ðx; B; TÞ ¼

0 1@

g

ixg 1 1þ þ njq I20 ðmÞ  1

!1 11 A

ð2Þ

Here g is the viscous drag coefficient defined by g = B/0/qf and, where qf = qnB/Bc2(T) is the Bardeen–Stephen (BS) flux flow resistivity. The force constant of the pinning potential well appearing in the Eq. (2) is given by jp = jp0[1  (T/Tc2)2], where Tc2 is the temperature at which B = Bc2(T). We take n = I1(m)/I0(m), where I0 and I1 are the modified Bessel functions of the first kind of order zero and one, respectively, and the argument is defined by m = U0(B, T)/2kBT. The temperature and field dependent barrier height of the periodic potential is given by U0(B, T) = U[1  (T/Tc2)3/2]B1. The normal fluid resistivity, qnf is expressed in the terms of the normal-state resistivity qn as qnf = qn/f(T, B), where f(T, B) = 1  [1  (T/Tc)c][1  B/Bc2(T)]. ~E The rf complex conductivity appearing in the expression J ¼ r ~ ¼ i=l0 xk~2 . The complex surface impedance of is related to ~ k via r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ . In the thin strip the superconductor Zs is given by Z s ¼ ixl=r 1 case [44], Zs = [h(rn  irsc)] , where h is the superconducting strip thickness which is also varied to find the microwave loss. In order to construct the dispersion relation, we first write down the field components of the propagating fields in the Fourier transform domain with respect to the transverse variable (here x). The direction of microwave propagation is along Z axis. Applying the boundary conditions of transverse electrical and magnetic fields at the interface containing superconducting strip, and incorporating the complex surface impedance of the superconductor, we derive the dyadic Green’s function which is represented in the Fourier transformed domain as [43,44]:

e Ez e Ex

! ¼

e zz  Z s Z e xz Z

e zx Z e Z xx  Z s

!

eJ z eJ x

! ð3Þ

Ex,z represent the electric field components at the interface. The tilde indicates the Fourier transformed quantity with respect to x.

Fig. 1. The variation of the phase constant with the reduced temperature for the superconducting strip thickness of h = 0.2 lm in the fields of 2.8, 5.6, 8.4 T (b = B/ Bc2(0) = 0.025, 0.05, 0.075) for the frequency 23.2 GHz.

340

J. Andrews, V. Mathew / Physica C 471 (2011) 338–343

The unknown strip current Jz and Jx are expanded in terms of known basis functions Jzm and Jxm [43,44] as

Jz ¼

N X

cm J zm ðxÞ

m¼1

Jx ¼

M X

dm J xm ðxÞ

m¼1

Now taking the scalar product with the same set of basis functions will lead to an equation of the form [G][C] = 0, where [C] is the column matrix of the expansion coefficients. The equation det [G] = 0 will give the dispersion relation. In computation M and N can be kept minimum by selecting suitable basis functions [43]. 3. Numerical results and discussion Based on the above theory we have performed numerical computations for evaluating the dispersion. The material parameters

used are the typical values of high Tc superconducting system of YBCO. Tc = 92 K, k0 = 140 nm, qn(T) = 1.1  108T + 2  106 Xm, U = 0.15 eV, jp0 = 2.1  104 N/m and Bc2 = 112 T [39,40]. The real and the imaginary parts of the rf conductivity with the reduced field b = B/Bc2(0) = 0 recovers the two fluid results [45]. On the other hand, the real and the imaginary parts of the conductivity in applied field show a remarkable effect of moving vortices below Tc2 as illustrated by Coffey and Clem [39] which will have obvious influence on surface impedance and hence on dispersion. The parameters of a typical superconducting microstrip line are taken as following. The thickness of the superconducting strip h is varied from 0.15 lm to 0.45 lm, below the static field and temperature-dependent penetration depth k(B, T) [10], i.e., h < k(B, T). The constraint that the superconducting strip thickness h  a0, where a0 is the inter-vortex spacing [40], is fairly satisfied at the lowest strip thickness of h = 0.15 lm which is used in the study, for when B = 2.8 T we have a0 = ( hp/eB)1/2 0.027 lm. The value of

Fig. 2. The variation of the attenuation constant with the reduced temperature for the superconducting strip thickness of h = 0.2 lm in fields of 2.8, 5.6, 8.4 T (b = B/Bc2(0) = 0.025, 0.05, 0.075) for the frequency 23.2 GHz.

Fig. 3. The variation of the phase constant with the superconducting strip thickness for T = 82.8 K (t = 0.9) in fields of 2.8, 5.6, 8.4 T (b = B/Bc2(0) = 0.025, 0.05, 0.075) for the frequency 23.2 GHz.

J. Andrews, V. Mathew / Physica C 471 (2011) 338–343

 = 1.054  1034 and e is the electronic charge. For higher field h values, we have lower values for a0. The temperature variation graphs (Figs. 1, 2, 5) are studied in a reduced temperature of t = 0.6–0.9 at h = 0.2 lm. For simplicity we assume B as uniform. The highest strip thickness taken in the study is limited to 0.45 lm, and it is done at a high temperature t = 0.9. Thus h variation graphs (Figs. 3, 4, 6) will not violate the condition h < k(B, T). As for the case of Fig. 7 where the study is done in a low temperature, t = 0.1, we have limited the strip thickness from 0.15 lm to 0.25 lm. The substrate thickness is taken as 0.5 mm with permittivity 9.8, typical of sapphire. The width of the superconducting strip is 0.19 mm. The applied magnetic field B is varied between 2.8, 5.6 and 8.4 T (b = B/Bc2(0) = 0.025, 0.05, 0.075) and is applied tangential to the plane of the strip. The frequency used in the study is 23.2 GHz.

341

The dispersion relation has been solved for various magnetic fields, temperatures and strip thickness. We first study the microwave propagation for a low strip thickness of 0.2 lm. In Fig. 1 and 2 we present the phase constant and the attenuation constant as a function of reduced temperature for the various field strengths of b = 0.025, 0.05 and 0.075. For a fixed strip thickness we find an increase of attenuation constant and phase constant with the increase of temperature and static field. This is because of the inclusion of more vortices in the strip. A similar behavior is also seen in the case of superconducting parallel plate waveguides [41]. At lower temperature each vortex will be confined to its own pinning potential well. But when temperature is increased the vortex may diffuse out of its pinning potential well from its initial position before the restoring forces bring it back. This oscillatory motion of vortices at higher temperatures will result in

Fig. 4. The variation of the attenuation constant with the superconducting strip thickness for T = 82.8 K (t = 0.9) in fields of 2.8, 5.6, 8.4 T (b = B/Bc2(0) = 0.025, 0.05, 0.075) for the frequency 23.2 GHz.

Fig. 5. The variation of the Q value with the reduced temperature for the magnetic fields of 2.8, 5.6, 8.4 T (b = B/Bc2(0) = 0.025, 0.05, 0.075) for the superconducting strip thickness h = 0.2 lm.

342

J. Andrews, V. Mathew / Physica C 471 (2011) 338–343

Fig. 6. The variation of the Q value with the superconducting strip thickness for T = 82.8 K (t = 0.9) in fields of 2.8, 5.6, 8.4 T (b = B/Bc2(0) = 0.025, 0.05, 0.075) for the frequency 23.2 GHz.

Fig. 7. The variation of the Q value with superconducting film thickness for T = 9.2 K (t = 0.1) in fields of 2.8, 5.6, 8.4 T (b = B/Bc2(0) = 0.025, 0.05, 0.075) for the frequency 23.2 GHz.

increased dissipation of microwave energy. The increase of static field will also introduce more vortices in the strip and the moving vortices significantly cause more dissipation as is observed in Fig. 2. In the Figs. 3 and 4 we present the variation of phase constant and attenuation constant with the superconducting strip thickness h at high temperature (T = 82.8 K), i.e., at a reduced temp t = 0.9. For a fixed strip thickness we find a higher value of phase constant for a higher field. As the strip thickness is increased this value reduces to a minimum. Fig. 4 which plots attenuation shows a higher dissipative value as the magnetic field is increased for a fixed strip thickness. We find very strong dissipation even for a reduced field of b = 0.05. In two fluid model, the real part of the conductivity increases with the temperature, but in Coffey–Clem theory the value decreases as the temperature increases. This effect is attributed to

vortex motion which plays a significant role in the high value of attenuation at higher temperature. The value of attenuation constant attains a minimum value when the strip thickness is increased to 0.45 lm beyond which dissipation is minimum even for high temperature and a high static field. Quality factor of a microstrip conducting line is determined from the phase constant and the attenuation constant [41]. The Fig. 5 studies the variation of unloaded quality factor Q with reduced temperature at the strip thickness of 0.2 lm. As is expected the Q value decreases with increasing temperature and static field, but this is more pronounced at lower strip thickness. Figs. 6 and 7 compare the Q values at high and low temperatures. Here the variation of Q value is studied with the superconducting strip thickness. As seen in the two plots the Q value increases with an increase in strip thickness. Fig. 6 depicts that even at high temper-

J. Andrews, V. Mathew / Physica C 471 (2011) 338–343

ature, we have a comparable Q value, if the reduced magnetic field is restricted to 0.025 and the strip thickness is around 0.4 lm. Thus there is a high Q value even for a high temperature if the strip thickness is sufficient, in mixed state. The formalism presented in this paper can be applied to study the effect of various mechanisms in vortex system of superconducting strip lines. This gives facility to design HTS strip line devices in mixed state for high frequency signal transmission as well as for material investigation. 4. Conclusion The Coffey–Clem model is brought into the study of the propagation characteristics of a superconducting microstrip line. The vortex effects which come into play when the strip is under a magnetic field is thus taken into account in a self consistent manner. Spectral domain Green’s function method is used to construct the dispersion relation, which is numerically solved. Numerical results show significant dependance of propagation, attenuation and unloaded quality factor on the applied static field. Acknowledgment

[11] [12] [13] [14] [15] [16] [17] [18]

[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

The authors would like to thank UGC, Govt. of India for financial assistance through a major project (Ref. No. 38-234/2009(SR)), for supporting this work.

[30] [31] [32]

References

[33]

[1] C. Song, M.P. Defeo, K. Yu, B.L.T. Plourde, Appl. Phys. Lett. 95 (2009) 232501. [2] S.Y. Gavrilkin, O.M. Ivanenko, A.N. Lykov, K.V. Mitsen, A.Y. Tsvetkov, C. Attanasio, C. Cirillo, S.L. Prischepa, Supercond. Sci. Technol. 23 (2010) 65019. [3] K. Matsumoto, P. Mele, Supercond. Sci. Technol. 23 (2010) 14001. [4] A.A. Gallitto, S. Fricano, M.L. Vigni, Physica C 384 (2003) 11. [5] M. Tinkham, Introduction to Superconductivity, Dover Books on Physics, 2004. [6] W. Rauch, E. Gornik, A.A. Valenzuela, F. Fox, H. Behner, J. Alloys Compd. 195 (1993) 571. [7] G.F. Dionne, IEEE Trans. Appl. Supercond. 3 (1) (1993) 1465. [8] D.A. Bonn, P. Dosanjh, R. Liang, W.N. Hardy, Phys. Rev. Lett. 68 (15) (1992) 2390. [9] E.J. Pakulis, R.L. Sandstorm, P. Chaudhari, R.B. Laibowitz, Appl. Phys. Lett. 57 (9) (1990) 940. [10] O.G. Vendik, I.B. Vendik, D.I. Kaparkov, IEEE Trans. MTT 46 (1998) 469.

[34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]

343

Y. Kobayashi, T. Imai, ICICE Trans. Electron. E74-C (1991) 1986. D.S. Linden, T.P. Orlando, W.G. Lyon, IEEE Trans. Appl. Supercond. 4 (1994) 136. J.G. Ma, I. Wolff, IEEE Trans. MTT 43 (5) (1995) 1053. J.G. Ma, I. Wolff, IEEE Trans. MTT 44 (4) (1996) 537. C.P. Poole, Handbook of Superconductivity, Academic Press, 2000. A.A. Gallitto, I. Ciccarello, M. Guccione, M.L. Vigni, D.P. Adorno, Phys. Rev. B 56 (1997) 5140. J. Kang, R. Han, X. Liu, Y. Wang, IEEE Trans. Appl. Supercond. 7 (1) (1997) 27. D.A. Bonn, R. Liang, T.M. Riseman, D.J. Baar, D.C. Morgan, K. Zhang, P. Dosanjh, T.L. Duty, A. Mac Farlane, G.D. Morris, J.H. Brewer, W.N. Hardy, Phys. Rev. B 47 (17) (1993) 11314. A. Heinrich, M. Kuhn, B. Schey, W. Beigel, B. Stritzker, Physica C 405 (2004). P. Lahl, R. Wordenweber, Appl. Phys. Lett. 81 (3) (2002) 505. A. Trotel, M. Pyee, B. Lavigne, D. Chambonnet, P. Lederer, Appl. Phys. Lett. 68 (18) (1996) 2559. J. Wosik, L.M. Xie, M. Strikovski, P. Przyslupski, M. Kamel, V.V. Srinivasu, S.A. Long, Jour. Appl. Phys. 91 (8) (2002) 5384. C. Song, T.W. Heitmann, M.P. DeFeo, K. Yu, R. McDermott, M. Neeley, J.M. Martinis, B.L.T. Plourde, Phys. Rev. B 79 (2009) 174512. M. Golosovsky, M. Tsindlekht, D. Davidov, Supercond. Sci. Technol. 9 (1996) 1. N. Belk, D.E. Oates, D.A. Feld, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 53 (6) (1996) 3459. N. Belk, D.E. Oates, D.A. Feld, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 56 (18) (1997) 11966. J.R. Powell, A. Porch, R.G. Humphreys, F. Wellhofer, M.J. Lancaster, C.E. Gough, Phys. Rev. B 57 (9) (1998) 5474. M.I. Tsindlekht, E.B. Sonin, M.A. Golosovsky, D. Davidov, X. Castel, M. GuillouxViry, A. Perrin, Phys. Rev. B 61 (2) (2000) 1596. Y. Tsuchiya, K. Iwaya, K. Kinoshita, T. Hanaguri, H. Kitano, A. Maeda, K. Shibata, T. Nishizaki, N. Kobayashi, Phys. Rev. B 63 (2001) 184517. T. Banerjee, V.C. Bagwe, J. John, S.P. Pai, R. Pinto, D. Kanjilal, Phys. Rev. B 69 (2004) 104533. D.A. Luzhbin, J. Phys.: Conf. Ser. 43 (2006) 243. A.G. Zaitsev, R. Schneider, R. Hott, Th. Schwarz, J. Geerk, Phys. Rev. B 75 (2007) 212505. M.S. Grbic, D. Janjusevic, M. Pozek, A. Dulcic, T. Wagner, Physica C 460 (2007) 1293. N. Pompeo, E. Silva, Phys. Rev. B 78 (2008) 094503. Y. Mawatari, J.R. Clem, Phys. Rev. B 74 (2006) 144523. Y. Mawatari, J.R. Clem, Phys. Rev. B 74 (2006) 214505. K.H. Kuit, J.R. Kirtley, W. Veur, C.G. Molenaar, F.J.G. Roesthuis, A.G.P. Troeman, J.R. Clem, H. Hilgenkamp, H. Rogalla, J. Flokstra, Phys. Rev. B 77 (2008) 134504. P.J. Curran, J.R. Clem, S.J. Bending, Y. Tsuchiya, T. Tamegai, Phys. Rev. B 82 (2010) 134501. M.W. Coffey, J.R. Clem, Phys. Rev. Lett. 67 (1991) 386. M.W. Coffey, J.R. Clem, Phys. Rev. B 45 (1992) 9872. C.J. Wu, J. Appl. Phys. 96 (2004) 3348. M.W. Coffey, Low Temp. Phys. 108 (1997) 331. T. Itoh, R. Mittra, IEEE Trans. MTT 21 (1973) 496. J.M. Pond, C.M. Krowne, W.L. Carter, IEEE Trans. MTT 37 (1989) 181. K.K. Mei, G.C. Liang, IEEE Trans. MTT 39 (1991) 1545.