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Calculation of rf magnetic permeability of nearly ferroelectric superconductors in a parallel field
Physica C 453 (2007) 70–75 www.elsevier.com/locate/physc
Calculation of rf magnetic permeability of nearly ferroelectric superconductors in a parallel field Chien-Jang Wu b
a,*
, Hsiu-Li Lee b, Wei-Ching Chuang
c
a Department of Applied Physics, National University of Kaohsiung, Kaohsiung 811, Taiwan, ROC Department of General Education Center, Oriental Institute of Technology, Banciao, Taipei 220, Taiwan, ROC c Department of Electro-Optics Engineering, National Formosa University, Huwei, Yunlin 632, Taiwan, ROC
Received 6 October 2006; received in revised form 4 December 2006; accepted 22 December 2006 Available online 3 January 2007
Abstract ~ ¼ l0 jl00 of nearly ferroelectric superconductors in the Meissner-like response is theoretiThe complex rf magnetic permeability l cally investigated based on the electrodynamics of the nearly ferroelectric superconductors. The complex rf magnetic permeability is calculated and analyzed for a superconducting slab in a parallel applied magnetic field. It is found that the peak of the loss curve (curve of imaginary part, l00 ) occurs at some critical thickness, which is frequency-dependent, decreasing as the frequency is increased. In addition, the magnitude in peak is strongly depressed as the frequency is increased. This critical thickness is also dependent on the damping factor appearing in the dielectric function, increasing with decreasing the damping coefficient. As for the frequency-dependent l00 we find that it increases with increasing the frequency, showing a strong dispersion in the imaginary part of the complex permeability. 2007 Elsevier B.V. All rights reserved. PACS: 74.20.De; 74.25.Nf; 78.20.Bh Keywords: Superconductor; Ferroelectric; Magnetic permeability
1. Introduction The study of the response of a superconductor (SC) to an electromagnetic field has been known as a useful means of exploring the basic physics of superconductivity. The basic issues may include the magnetic penetration depth, the electronic conduction mechanism, and the vortex dynamics in a type-II SC as well. The problem of the electromagnetic response of a SC can be generally classified as two types. One is the propagation-dominated problem in which the wave transmission and reflection coefficients are the main quantities to be interested. The other relating the rf magnetic permeability (or the surface impedance) belongs to the attenuation-dominated problem. Both problems have been extensively studied since the discovery of *
Corresponding author. Tel.: +886 7 5919467; fax: +886 7 5919357. E-mail address: [email protected] (C.-J. Wu).
0921-4534/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.12.014
high-temperature SCs and hence many reports are now available [1–3]. ~ ¼ l0 jl00 is The complex rf magnetic permeability l frequently used to investigate the electromagnetic microwave response of a SC. It conveys information that is of technical use for the applications of SCs. For instance, the imaginary part, l00 , represents the microwave loss and hence determines the performance of a superconductorbased microwave device. The real part, l 0 , is closely related to the resonance frequency when the superconducting sample is used in the resonant circuit. The complex rf magnetic permeability also serves as a fundamental quantity in the study of vortex dynamics for the typical high-temperature type-II superconducting thin-film Y-123 system [4,5]. In the single crystal superconducting platelets it is specifically known as a good and relevant quantity for the purpose of investigating the anisotropic effects in a strongly anisotropic system such as Bi-2212 one [6,7].
C.-J. Wu et al. / Physica C 453 (2007) 70–75
The above-mentioned high-temperature SCs are known as the cuprates. In addition to the cuprates, there exists a special class of non-cuprate SC called the nearly ferroelectric (NFE) SCs. A NFE SC could be a high-temperature one such as the Na-doped WO3 (sodium tungsten bronze NaxWO3 with x 0.05) with a transition temperature Tc 90 K [8]. This non-cuprate superconductor with a high critical temperature was first found by Reich and Tsabba [9] and then was experimentally confirmed by Levi et al. [8]. Such high-Tc nearly ferroelectric superconductors have been theoretically investigated Weger et al. [10] who renormalize the electron–phonon and electron–electron interaction and enhance Tc. They reevaluate the BCS calculation by assuming an electron gas embedded in a dielectric medium with a large dynamic dielectric constant. On the other hand, it could also be a SC having a relatively low Tc as the typical system, n- (or p-) doped SrTiO3 (STO) with Tc 1–3 K [11]. The microscopic theories for the superconductivity in n-STO have been discussed on the strong-coupling basis [12,13]. A NFE SC is a material that a superconducting state and a NFE state coexist in it and the NFE state means a soft-mode ionic system with a fairly large static permittivity. For example, the host STO is a familiar NFE system with a very large permittivity e(0) 104 in the temperature range of the superconducting state. The electrodynamics of the NFE SCs has recently reported by Birman and Zimbovskaya [14]. The authors developed a phenomenological theory to model the electromagnetic properties of the NFE SCs. The theory elucidates that the effective wavenumber (or effective penetration length) is strongly dependent on the radiation frequency which in turn leads to two possible responses for a NFE SC material. One is the so-called dielectriclike response in which the electromagnetic wave can propagate in such a material as in a regular dielectric. The other is known as the Meissner-like response where the NFE SC behaves like a regular superconductor. In the Meissner-like response the electromagnetic wave is not allowed to propagate in the material, indicating an attenuation-dominated problem. The dielectriclike response obviously belongs to the propagation-dominated problem and some important results have been predicted, namely the existence of optical anomalies which are the resonance peaks in the transmittance spectrum for a NFE SC film under the normal incidence [14]. These peaks are fairly sharp and a comblike distribution is seen. This comblike spectrum is believed to be experimentally measurable and hence the validity of theory can be tested. It is known that the transmittance is obtained by taking the absolute square of the transmission coefficient and it is a real number. As will be described later, for a NFE SC material the calculated transmission coefficient obviously is related to the dielectric function which may contain the damping factor for the soft-mode oscillator. Also as mentioned in Ref. [14], the effect of damping factor is hardly seen through the transmittance spectrum. Therefore, to
71
explore the effect arising from the damping factor, we shall alternatively capitalize on the complex rf permeability because it can be separately investigated from both the imaginary and real parts of permeability. Another superior feature for choosing the permeability is that the operating frequency for the Meissner-like response lies in the microwave regime which would interest the experimental workers to conduct measurements using the relevant microwave techniques. In addition, the related report of complex rf permeability for this material has been lacking thus far. The complex permeability will be calculated for a slab geometry in a parallel applied rf field. Numerical analysis will be made as a function of the frequency, the slab thickness, and the damping factor as well. 2. Basic equations Following Birman and Zimbovskaya the electromagnetic properties of a NFE SC can be modeled by the coupled field equation derived from the combination of the Maxwell’s equations, the constitutive equations of the medium, the lattice equation, and the London equation of the superconductor, with the result [14] r r E ¼ l0 eðxÞ
o2 l E 0 E; ot2 K
ð1Þ
where K ¼ m =ðN s e2 Þ ¼ l0 k2L where m* is the electron effective mass, Ns is the concentration of the superelectrons, e is the electronic charge, l0 is the permeability of the free space, and kL is the London penetration. In addition, the dielectric function is given by eðxÞ ¼ e0 ðxÞ je00 ðxÞ ¼ e01
x2LO x2 ; x2TO x2 þ jxC
ð2Þ
where xLO and xTO are the longitudinal and transverse soft-mode lattice frequencies, respectively, and e01 ¼ Q0 e1 i x2Li =x2Ti where the primed product over i includes all oscillations except the soft-mode, and the damping factor C indicates the losses in the polaritonic medium. With this complex-valued permittivity the corresponding loss tangent is given by tan d ¼
e00 ðxÞ xC ¼ : e0 ðxÞ x2TO x2
ð3Þ
For a transverse plane-wave solution with a temporal part of exp (jxt), Eq. (1) then becomes l r2 E þ x2 l0 eðxÞ 0 E ¼ 0: ð4Þ K Eq. (4) then defines the wavenumber for a NFE SC material h i1=2 1=2 ~k ¼ x2 l0 eðxÞ l0 ¼ ½x2 l0 eðxÞ k2 ; ð5Þ L K which is obviously a strong function of the frequency. For frequencies such that ~k > 0 one has the dielectriclike response. In this case an electromagnetic wave can propagate
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C.-J. Wu et al. / Physica C 453 (2007) 70–75
in the NFE SC as in a regular dielectric and hence it is referred to as the anomalous frequency region for such a material. For frequencies leading to an imaginary ~k, the wave will be exponentially attenuated and one has the socalled Meissner-like response. The attenuation mode can be best described in terms of the associated complex penetration length given by ~kðxÞ ¼ jk
~1
kL ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k0 ðxÞ jk00 ðxÞ: 1 x2 l0 eðxÞk2L
ð6Þ
xc1;c2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 2 u1 x 2 1 1 xLO 1 x2LO 1 t LO ¼ xTO 1þ 2 4 2; 1þ 2 2 x2TO a 2 xTO a a x2TO ð7Þ
where a parameter a2 ¼ e01 x2LO K is introduced. For the case where the damping factor is included, the cutoff frequency can be numerically determined from Eqs. (2) and (5). Using the measured material parameters of n-STO, the shift in the upper cutoff frequency xc2 due to the inclusion of damping factor is negligible, while the shift in the lower one xc1 is also small, say from xc1 = 0.9 to 0.96 · 1011 rad/s [14]. Therefore it suffices to use Eq. (7) to estimate the cutoff frequency even in the presence of the damping factor. With the fact that the calculation of the complex rf magnetic permeability of a superconductor belongs to a typical attenuation-dominated problem, the Meissner-like response will be considered in this paper. Based on the complex penetration length in Eq. (6) one can calculate the complex rf magnetic permeability for some related geometry. Let us consider a superconducting slab to be bounded by the planes, x = ±W/2. The slab is subject to a microwave radiation parallel to the plane boundaries such that one has brf ¼ ^y b0 expðjxtÞ therein. The magnetic induction inside the slab is then expressed as coshðz=~ kÞ : ~ coshðW =2kÞ
ð8Þ
The rf magnetic permeability for a slab in this configuration is defined by Z W =2 1 ~ ¼ l0 jl00 ¼ l by ðzÞdz: ð9Þ Wb0 W =2 Putting Eq. (8) into (9), it is direct to have 2~ k W 0 00 ~ðxÞ ¼ l ðxÞ jl ðxÞ ¼ tanh ; l W 2~ k
and 00 0 0 00 W W k sinh k sin k k 2 2 2 j~ kj j~ kj : l00 ¼ W cosh W k0 þ cos W k00 j~ kj
These two responses are separated by the cutoff frequency at which ~k ¼ 0. If we neglect the damping factor in Eq. (2), then an exact expression for the cutoff frequency can be obtained, namely
by ðzÞ ¼ b0
where the real and imaginary parts are expressed as k0 sinh j~kjW2 k0 þ k00 sin j~kjW2 k00 2 ; ð11Þ l0 ¼ W cosh W k0 þ cos W k00 2 2 ~ ~ jkj jkj
ð10Þ
2
j~ kj
ð12Þ
2
Before presenting the numerical results, we mention that the superconducting cylinder is another geometry that is also frequently studied in the literature. For a cylinder with a radius of q in the parallel field configuration, the complex rf magnetic permeability is given by ~ðxÞ ¼ l0 ðxÞ jl00 ðxÞ ¼ l
2~k I 1 ðq=~kÞ ; q I 0 ðq=~kÞ
ð13Þ
where I0 and I1 are the modified Bessel functions of the first kind with order zero and one, respectively. 3. Numerical results and discussion Let us now apply the above formulas to a typical and prototype NFE SC material. As pointed out in Ref. [14], n-STO is the sole NFE SC material that has all the known material parameters for the numerical calculation. Thus the following numerical analysis will be made on this material. We shall investigate the complex rf permeability as a function of the slab thickness and frequency in the Meissnerlike response. For n-STO the effective mass of the electron is m* 10me with me being the rest mass of electron. The concentration of the superelectrons is NsQ = 9 · 1017 cm3. 0 2 2 Other parameters include e1 = 5.5e0, i xLi =xTi ¼ 4:1, xTO = 1.6 · 1011 rad/s, and xLO = 5.2 · 1012 rad/s. With these parameters, it then follows Eq. (7) that the lower cutoff frequency is calculated to be xc1 = 0.9 · 1011 rad/s, or equivalently, fc1 = 14.3 GHz, being in the microwave region. The graphical solutions of Eq. (5) are shown in Fig. 1, where the solution for frequency near the lower cutoff frequency is plotted in (a), and the one near the upper cutoff frequency is in (b). It is seen that these two cutoff frequencies produce two dielectriclike and two Meissner-like responses in the frequency domain. In what follows we shall consider only the Meissner-like response where the frequencies are lower than xc1. Fig. 2 shows the calculated loss tangent in Eq. (3) at the condition of C2 =x2TO ¼ 0:2. The result indicates that an n-STO behaves as a low-loss material when the relevant damping factor is incorporated in the dielectric function in Eq. (2). The effects of loss on the microwave properties in such a material are thus expected to some extent and will be demonstrated next. The complex rf permeability as a function of the slab thickness is plotted in Fig. 3, in which the imaginary part, l00 , and the real part, l 0 , at four different frequencies 14, 12,
C.-J. Wu et al. / Physica C 453 (2007) 70–75
10 7
250
Meissner-like 10 6
λL = 17.73 μm
Dielectriclike
14 GH z
200
c2
10 5
μ" x 10 3
ε (ω ) ε0
ω 2λ2L
Log Scale
73
150
12 GHz 100
10 GHz 8 GHz
10 4
50
ω = ω TO
ω = ωc1 10 3 0.0
0.5
1.0
1.5
ω / 1011
0
0
5
10
W / λL
2.0
15
20
1.0 100
λL = 17.73 μm
14 GHz
Dielectriclike
Meissner-like 50
0.6
0.4
ω = ω TO
0.2
ω = ωc2
ω = ω LO
-50
To Line ω = ω TO -100
0
20
0.0 0
40
60
80
100
ω / 1011 Fig. 1. The graphical illustration of the solutions of Eq. (5) in the absence of damping, where (a) is plotted around the lower cutoff frequency xc1, whereas (b) around the upper cutoff frequency xc2. Following Eq. (7), the calculated cutoff frequencies are given by xc1 = 0.9 · 1011 rad/s and xc2 = 6.32 · 1012 rad/s, respectively.
0.40
Loss Tangent, tanδ
0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.0
8 GHz
μ'
Linear Scale
ω 2λ2L
12 GHz 10 GHz
ε (ω ) ε0
c2 0
0.8
0.2
0.4
0.6
ω / ω c1
0.8
1.0
Fig. 2. The calculated frequency-dependent loss tangent in the Meissnerlike regime for an n-STO NFE superconductor. The frequency is taken from 1 to 14.2999 GHz, or equivalently, x/xc1 = 0.06981–0.99826. The range of the calculated loss tangent lies in the range of 0.01759–0.36676.
5
10
W / λL
15
20
Fig. 3. The calculated imaginary part, (a) and real part, (b) of rf permeability as a function of the slab thickness at different frequencies. The peak of each curve occurs at the critical thickness Wc/kL = 5.53, 4.82, 4.12, and 3.73 for 14, 12, 10, and 8 GHz, respectively.
10, and 8 GHz are shown in (a) and (b), respectively. The damping factor has been considered as given in Fig. 2. Both the thickness-dependent l00 and l 0 for a NFE SC material are similar to those of a high-temperature cuprate [4]. It is known that l00 is an indication of the loss and its loss peak sets a critical thickness, Wc, of the slab. This critical thickness is strongly frequency-dependent and is found to be Wc/kL = 5.53, 4.82, 4.12, and 3.73, for the frequency of 14, 12, 10, and 8 GHz, respectively, where kL = 17.73 lm. The critical thickness, on the order of kL, can be treated as a fundamental length scale characterizing thin-film NFE SC materials. It is also called the impedance of free-space equivalent thickness (IFSET) [15]. In addition, the magnitude in the loss peak is also a strong function of the frequency and is gradually smeared out as the frequency is lowered. As for the real part, l 0 , it is shown to decrease as a function of the slab thickness and its change in magnitude by varying the frequency is not pronounced as l00 . The effect of damping factor is displayed in Fig. 4 where the permeability at a fixed frequency of 14 GHz for
74
C.-J. Wu et al. / Physica C 453 (2007) 70–75
10 0
250
f = 14 GHz 10 -1 10 -2
150
γ = 0.2 γ = 0.1
100
γ = 0.05
W = 5λL
W = 10λL
W = 2λL W = λL
10 -4
γ = 0.025
50 0 0
10 -3
μ"
μ" x 103
200
10 -5 5
10
W / λL
15
20
10 -6 0.0
1.0
0.2
0.4
ω / ωc1
0.6
0.8
1.0
0.6
0.8
1.0
1.0
f = 14 GHz
γ = 0.2
0.8
W = λL 0.8
γ = 0.1 γ = 0.05 γ = 0.025
W = 2λL 0.6
μ'
μ'
0.6
0.4
0.2
0.0 0
5
10
15
20
W / λL Fig. 4. The calculated imaginary part, (a) and real part, (b) of rf permeability as a function of the slab thickness at a fixed frequency of 14 GHz for different damping factor c C2 =x2TO ¼ 0:2, 0.1, 0.05, and 0.025.
different damping factor c C2 =x2TO ¼ 0:2, 0.1, 0.05, and 0.025. It is seen from Fig. 4a that the critical thickness is increased and the loss peak is broadened as the damping is decreased. In addition, the value in the real part, l 0 , is enhanced for a small damping factor. The frequencydependent of rf permeability is plotted in Fig. 5, where four different thicknesses of slab, W = kL, 2kL, 5kL, and 10kL are taken and damping factor, c = 0.2 is considered as well. Fig. 5a illustrates that the loss is increased with increasing the frequency, revealing a strong dispersion in l00 . The real part, however, is not changed as pronouncedly as the imaginary part, as displayed in Fig. 5b. The frequency dependence of the rf permeability in the Meissner-like response is quite different from the transmittance spectrum in the dielectriclike response. In the transmittance spectrum it is found the existence of the anomalous peaks which is believed to be measurable in the NFE SC material, as stated in Ref. [14]. In this work, we have investigated the microwave properties of a NFE SC material through the complex permeability. The complex rf permeability indeed can be measured by related microwave technique. With the
0.4
W = 5λL
0.2
W = 10λL
0.0 0.0
0.2
0.4
ω / ωc1
Fig. 5. The calculated imaginary part, (a) and real part, (b) of rf permeability as a function of the frequency at four different thicknesses of slab, W = kL, 2kL, 5kL, and 10kL and at a fixed damping factor, c = 0.2.
measured results, a test of the electromagnetic theory of such a material can be achieved. 4. Summary The complex rf magnetic permeability for a NFE SC material in the Meissner-like response has been theoretically studied based on the electrodynamics of a nearly ferroelectric superconductor. We have investigated the permeability for a superconducting slab in a parallel field configuration. It is shown that the imaginary part attains maximum at a critical thickness of slab. This critical thickness can be defined as a length scale which is of use in characterizing the NFE SC films. We have found that the critical thickness is strongly dependent on the frequency as well as the damping factor. The effects of damping factor on the rf permeability are also specifically illustrated. The study of complex rf permeability provides a possible alternative for experimentally exploring the microwave properties of NFE SCs. It could be further beneficial to the understanding of electrodynamical model of these materials.
C.-J. Wu et al. / Physica C 453 (2007) 70–75
Acknowledgement C.-J. Wu acknowledges the financial support from the National Science Council of the Republic of China under Grant No. NSC-95-2112-M-390-003-MY2. References [1] M. Golosovsky, M. Tsidlekht, D. Davidov, Supercond. Sci. Technol. 9 (1996) 1. [2] V.M. Pan, D.A. Luzhbin, A.A. Kalenyuk, A.L. Kasatkin, V.A. Komashko, A.V. Velichko, M. Lancaster, Low Temp. Phys. 31 (2005) 254, and references therein. [3] M.R. Trunin, J. Supercond. 11 (1998) 381.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
75
M.W. Coffey, J.R. Clem, Phys. Rev. B 45 (1992) 9872. L.-W. Chen, M.C. Marchetti, Phys. Rev. B 50 (1994) 6382. C.E. Gough, N.J. Exon, Phys. Rev. B 50 (1994) 488. C.J. Wu, T.Y. Tseng, Physica C 259 (1996) 61. Y. Levi, O. Millo, A. Sharoni, Y. Tsabba, G. Leitus, S. Reich, Europhys. Lett. 51 (2000) 564. S. Reich, Y. Tsabba, Eur. Phys. J. B 9 (1999) 1. D. Fay, M. Weger, Phys. Rev. B 62 (2000) 15208. C.S. Koonce, M.L. Cohen, J.E. Schooley, W.R. Hosler, L.F. Pfeiffer, Phys. Rev. 163 (1967) 380. Y. Takada, J. Phys. Soc. Jpn. 49 (1980) 1267. A. Baratoff, G. Binnig, Physica B 108 (1981) 1335. J.L. Birman, N.A. Zimbovskaya, Phys. Rev. B 64 (2001) 144506. H. Contopanagos, E. Yablonovitch, J. Opt. Soc. Am. A 16 (1999) 2294.
Calculation of rf magnetic permeability of nearly ferroelectric superconductors in a parallel field Chien-Jang Wu b
a,*
, Hsiu-Li Lee b, Wei-Ching Chuang
c
a Department of Applied Physics, National University of Kaohsiung, Kaohsiung 811, Taiwan, ROC Department of General Education Center, Oriental Institute of Technology, Banciao, Taipei 220, Taiwan, ROC c Department of Electro-Optics Engineering, National Formosa University, Huwei, Yunlin 632, Taiwan, ROC
Received 6 October 2006; received in revised form 4 December 2006; accepted 22 December 2006 Available online 3 January 2007
Abstract ~ ¼ l0 jl00 of nearly ferroelectric superconductors in the Meissner-like response is theoretiThe complex rf magnetic permeability l cally investigated based on the electrodynamics of the nearly ferroelectric superconductors. The complex rf magnetic permeability is calculated and analyzed for a superconducting slab in a parallel applied magnetic field. It is found that the peak of the loss curve (curve of imaginary part, l00 ) occurs at some critical thickness, which is frequency-dependent, decreasing as the frequency is increased. In addition, the magnitude in peak is strongly depressed as the frequency is increased. This critical thickness is also dependent on the damping factor appearing in the dielectric function, increasing with decreasing the damping coefficient. As for the frequency-dependent l00 we find that it increases with increasing the frequency, showing a strong dispersion in the imaginary part of the complex permeability. 2007 Elsevier B.V. All rights reserved. PACS: 74.20.De; 74.25.Nf; 78.20.Bh Keywords: Superconductor; Ferroelectric; Magnetic permeability
1. Introduction The study of the response of a superconductor (SC) to an electromagnetic field has been known as a useful means of exploring the basic physics of superconductivity. The basic issues may include the magnetic penetration depth, the electronic conduction mechanism, and the vortex dynamics in a type-II SC as well. The problem of the electromagnetic response of a SC can be generally classified as two types. One is the propagation-dominated problem in which the wave transmission and reflection coefficients are the main quantities to be interested. The other relating the rf magnetic permeability (or the surface impedance) belongs to the attenuation-dominated problem. Both problems have been extensively studied since the discovery of *
Corresponding author. Tel.: +886 7 5919467; fax: +886 7 5919357. E-mail address: [email protected] (C.-J. Wu).
0921-4534/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.12.014
high-temperature SCs and hence many reports are now available [1–3]. ~ ¼ l0 jl00 is The complex rf magnetic permeability l frequently used to investigate the electromagnetic microwave response of a SC. It conveys information that is of technical use for the applications of SCs. For instance, the imaginary part, l00 , represents the microwave loss and hence determines the performance of a superconductorbased microwave device. The real part, l 0 , is closely related to the resonance frequency when the superconducting sample is used in the resonant circuit. The complex rf magnetic permeability also serves as a fundamental quantity in the study of vortex dynamics for the typical high-temperature type-II superconducting thin-film Y-123 system [4,5]. In the single crystal superconducting platelets it is specifically known as a good and relevant quantity for the purpose of investigating the anisotropic effects in a strongly anisotropic system such as Bi-2212 one [6,7].
C.-J. Wu et al. / Physica C 453 (2007) 70–75
The above-mentioned high-temperature SCs are known as the cuprates. In addition to the cuprates, there exists a special class of non-cuprate SC called the nearly ferroelectric (NFE) SCs. A NFE SC could be a high-temperature one such as the Na-doped WO3 (sodium tungsten bronze NaxWO3 with x 0.05) with a transition temperature Tc 90 K [8]. This non-cuprate superconductor with a high critical temperature was first found by Reich and Tsabba [9] and then was experimentally confirmed by Levi et al. [8]. Such high-Tc nearly ferroelectric superconductors have been theoretically investigated Weger et al. [10] who renormalize the electron–phonon and electron–electron interaction and enhance Tc. They reevaluate the BCS calculation by assuming an electron gas embedded in a dielectric medium with a large dynamic dielectric constant. On the other hand, it could also be a SC having a relatively low Tc as the typical system, n- (or p-) doped SrTiO3 (STO) with Tc 1–3 K [11]. The microscopic theories for the superconductivity in n-STO have been discussed on the strong-coupling basis [12,13]. A NFE SC is a material that a superconducting state and a NFE state coexist in it and the NFE state means a soft-mode ionic system with a fairly large static permittivity. For example, the host STO is a familiar NFE system with a very large permittivity e(0) 104 in the temperature range of the superconducting state. The electrodynamics of the NFE SCs has recently reported by Birman and Zimbovskaya [14]. The authors developed a phenomenological theory to model the electromagnetic properties of the NFE SCs. The theory elucidates that the effective wavenumber (or effective penetration length) is strongly dependent on the radiation frequency which in turn leads to two possible responses for a NFE SC material. One is the so-called dielectriclike response in which the electromagnetic wave can propagate in such a material as in a regular dielectric. The other is known as the Meissner-like response where the NFE SC behaves like a regular superconductor. In the Meissner-like response the electromagnetic wave is not allowed to propagate in the material, indicating an attenuation-dominated problem. The dielectriclike response obviously belongs to the propagation-dominated problem and some important results have been predicted, namely the existence of optical anomalies which are the resonance peaks in the transmittance spectrum for a NFE SC film under the normal incidence [14]. These peaks are fairly sharp and a comblike distribution is seen. This comblike spectrum is believed to be experimentally measurable and hence the validity of theory can be tested. It is known that the transmittance is obtained by taking the absolute square of the transmission coefficient and it is a real number. As will be described later, for a NFE SC material the calculated transmission coefficient obviously is related to the dielectric function which may contain the damping factor for the soft-mode oscillator. Also as mentioned in Ref. [14], the effect of damping factor is hardly seen through the transmittance spectrum. Therefore, to
71
explore the effect arising from the damping factor, we shall alternatively capitalize on the complex rf permeability because it can be separately investigated from both the imaginary and real parts of permeability. Another superior feature for choosing the permeability is that the operating frequency for the Meissner-like response lies in the microwave regime which would interest the experimental workers to conduct measurements using the relevant microwave techniques. In addition, the related report of complex rf permeability for this material has been lacking thus far. The complex permeability will be calculated for a slab geometry in a parallel applied rf field. Numerical analysis will be made as a function of the frequency, the slab thickness, and the damping factor as well. 2. Basic equations Following Birman and Zimbovskaya the electromagnetic properties of a NFE SC can be modeled by the coupled field equation derived from the combination of the Maxwell’s equations, the constitutive equations of the medium, the lattice equation, and the London equation of the superconductor, with the result [14] r r E ¼ l0 eðxÞ
o2 l E 0 E; ot2 K
ð1Þ
where K ¼ m =ðN s e2 Þ ¼ l0 k2L where m* is the electron effective mass, Ns is the concentration of the superelectrons, e is the electronic charge, l0 is the permeability of the free space, and kL is the London penetration. In addition, the dielectric function is given by eðxÞ ¼ e0 ðxÞ je00 ðxÞ ¼ e01
x2LO x2 ; x2TO x2 þ jxC
ð2Þ
where xLO and xTO are the longitudinal and transverse soft-mode lattice frequencies, respectively, and e01 ¼ Q0 e1 i x2Li =x2Ti where the primed product over i includes all oscillations except the soft-mode, and the damping factor C indicates the losses in the polaritonic medium. With this complex-valued permittivity the corresponding loss tangent is given by tan d ¼
e00 ðxÞ xC ¼ : e0 ðxÞ x2TO x2
ð3Þ
For a transverse plane-wave solution with a temporal part of exp (jxt), Eq. (1) then becomes l r2 E þ x2 l0 eðxÞ 0 E ¼ 0: ð4Þ K Eq. (4) then defines the wavenumber for a NFE SC material h i1=2 1=2 ~k ¼ x2 l0 eðxÞ l0 ¼ ½x2 l0 eðxÞ k2 ; ð5Þ L K which is obviously a strong function of the frequency. For frequencies such that ~k > 0 one has the dielectriclike response. In this case an electromagnetic wave can propagate
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in the NFE SC as in a regular dielectric and hence it is referred to as the anomalous frequency region for such a material. For frequencies leading to an imaginary ~k, the wave will be exponentially attenuated and one has the socalled Meissner-like response. The attenuation mode can be best described in terms of the associated complex penetration length given by ~kðxÞ ¼ jk
~1
kL ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k0 ðxÞ jk00 ðxÞ: 1 x2 l0 eðxÞk2L
ð6Þ
xc1;c2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 2 u1 x 2 1 1 xLO 1 x2LO 1 t LO ¼ xTO 1þ 2 4 2; 1þ 2 2 x2TO a 2 xTO a a x2TO ð7Þ
where a parameter a2 ¼ e01 x2LO K is introduced. For the case where the damping factor is included, the cutoff frequency can be numerically determined from Eqs. (2) and (5). Using the measured material parameters of n-STO, the shift in the upper cutoff frequency xc2 due to the inclusion of damping factor is negligible, while the shift in the lower one xc1 is also small, say from xc1 = 0.9 to 0.96 · 1011 rad/s [14]. Therefore it suffices to use Eq. (7) to estimate the cutoff frequency even in the presence of the damping factor. With the fact that the calculation of the complex rf magnetic permeability of a superconductor belongs to a typical attenuation-dominated problem, the Meissner-like response will be considered in this paper. Based on the complex penetration length in Eq. (6) one can calculate the complex rf magnetic permeability for some related geometry. Let us consider a superconducting slab to be bounded by the planes, x = ±W/2. The slab is subject to a microwave radiation parallel to the plane boundaries such that one has brf ¼ ^y b0 expðjxtÞ therein. The magnetic induction inside the slab is then expressed as coshðz=~ kÞ : ~ coshðW =2kÞ
ð8Þ
The rf magnetic permeability for a slab in this configuration is defined by Z W =2 1 ~ ¼ l0 jl00 ¼ l by ðzÞdz: ð9Þ Wb0 W =2 Putting Eq. (8) into (9), it is direct to have 2~ k W 0 00 ~ðxÞ ¼ l ðxÞ jl ðxÞ ¼ tanh ; l W 2~ k
and 00 0 0 00 W W k sinh k sin k k 2 2 2 j~ kj j~ kj : l00 ¼ W cosh W k0 þ cos W k00 j~ kj
These two responses are separated by the cutoff frequency at which ~k ¼ 0. If we neglect the damping factor in Eq. (2), then an exact expression for the cutoff frequency can be obtained, namely
by ðzÞ ¼ b0
where the real and imaginary parts are expressed as k0 sinh j~kjW2 k0 þ k00 sin j~kjW2 k00 2 ; ð11Þ l0 ¼ W cosh W k0 þ cos W k00 2 2 ~ ~ jkj jkj
ð10Þ
2
j~ kj
ð12Þ
2
Before presenting the numerical results, we mention that the superconducting cylinder is another geometry that is also frequently studied in the literature. For a cylinder with a radius of q in the parallel field configuration, the complex rf magnetic permeability is given by ~ðxÞ ¼ l0 ðxÞ jl00 ðxÞ ¼ l
2~k I 1 ðq=~kÞ ; q I 0 ðq=~kÞ
ð13Þ
where I0 and I1 are the modified Bessel functions of the first kind with order zero and one, respectively. 3. Numerical results and discussion Let us now apply the above formulas to a typical and prototype NFE SC material. As pointed out in Ref. [14], n-STO is the sole NFE SC material that has all the known material parameters for the numerical calculation. Thus the following numerical analysis will be made on this material. We shall investigate the complex rf permeability as a function of the slab thickness and frequency in the Meissnerlike response. For n-STO the effective mass of the electron is m* 10me with me being the rest mass of electron. The concentration of the superelectrons is NsQ = 9 · 1017 cm3. 0 2 2 Other parameters include e1 = 5.5e0, i xLi =xTi ¼ 4:1, xTO = 1.6 · 1011 rad/s, and xLO = 5.2 · 1012 rad/s. With these parameters, it then follows Eq. (7) that the lower cutoff frequency is calculated to be xc1 = 0.9 · 1011 rad/s, or equivalently, fc1 = 14.3 GHz, being in the microwave region. The graphical solutions of Eq. (5) are shown in Fig. 1, where the solution for frequency near the lower cutoff frequency is plotted in (a), and the one near the upper cutoff frequency is in (b). It is seen that these two cutoff frequencies produce two dielectriclike and two Meissner-like responses in the frequency domain. In what follows we shall consider only the Meissner-like response where the frequencies are lower than xc1. Fig. 2 shows the calculated loss tangent in Eq. (3) at the condition of C2 =x2TO ¼ 0:2. The result indicates that an n-STO behaves as a low-loss material when the relevant damping factor is incorporated in the dielectric function in Eq. (2). The effects of loss on the microwave properties in such a material are thus expected to some extent and will be demonstrated next. The complex rf permeability as a function of the slab thickness is plotted in Fig. 3, in which the imaginary part, l00 , and the real part, l 0 , at four different frequencies 14, 12,
C.-J. Wu et al. / Physica C 453 (2007) 70–75
10 7
250
Meissner-like 10 6
λL = 17.73 μm
Dielectriclike
14 GH z
200
c2
10 5
μ" x 10 3
ε (ω ) ε0
ω 2λ2L
Log Scale
73
150
12 GHz 100
10 GHz 8 GHz
10 4
50
ω = ω TO
ω = ωc1 10 3 0.0
0.5
1.0
1.5
ω / 1011
0
0
5
10
W / λL
2.0
15
20
1.0 100
λL = 17.73 μm
14 GHz
Dielectriclike
Meissner-like 50
0.6
0.4
ω = ω TO
0.2
ω = ωc2
ω = ω LO
-50
To Line ω = ω TO -100
0
20
0.0 0
40
60
80
100
ω / 1011 Fig. 1. The graphical illustration of the solutions of Eq. (5) in the absence of damping, where (a) is plotted around the lower cutoff frequency xc1, whereas (b) around the upper cutoff frequency xc2. Following Eq. (7), the calculated cutoff frequencies are given by xc1 = 0.9 · 1011 rad/s and xc2 = 6.32 · 1012 rad/s, respectively.
0.40
Loss Tangent, tanδ
0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.0
8 GHz
μ'
Linear Scale
ω 2λ2L
12 GHz 10 GHz
ε (ω ) ε0
c2 0
0.8
0.2
0.4
0.6
ω / ω c1
0.8
1.0
Fig. 2. The calculated frequency-dependent loss tangent in the Meissnerlike regime for an n-STO NFE superconductor. The frequency is taken from 1 to 14.2999 GHz, or equivalently, x/xc1 = 0.06981–0.99826. The range of the calculated loss tangent lies in the range of 0.01759–0.36676.
5
10
W / λL
15
20
Fig. 3. The calculated imaginary part, (a) and real part, (b) of rf permeability as a function of the slab thickness at different frequencies. The peak of each curve occurs at the critical thickness Wc/kL = 5.53, 4.82, 4.12, and 3.73 for 14, 12, 10, and 8 GHz, respectively.
10, and 8 GHz are shown in (a) and (b), respectively. The damping factor has been considered as given in Fig. 2. Both the thickness-dependent l00 and l 0 for a NFE SC material are similar to those of a high-temperature cuprate [4]. It is known that l00 is an indication of the loss and its loss peak sets a critical thickness, Wc, of the slab. This critical thickness is strongly frequency-dependent and is found to be Wc/kL = 5.53, 4.82, 4.12, and 3.73, for the frequency of 14, 12, 10, and 8 GHz, respectively, where kL = 17.73 lm. The critical thickness, on the order of kL, can be treated as a fundamental length scale characterizing thin-film NFE SC materials. It is also called the impedance of free-space equivalent thickness (IFSET) [15]. In addition, the magnitude in the loss peak is also a strong function of the frequency and is gradually smeared out as the frequency is lowered. As for the real part, l 0 , it is shown to decrease as a function of the slab thickness and its change in magnitude by varying the frequency is not pronounced as l00 . The effect of damping factor is displayed in Fig. 4 where the permeability at a fixed frequency of 14 GHz for
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C.-J. Wu et al. / Physica C 453 (2007) 70–75
10 0
250
f = 14 GHz 10 -1 10 -2
150
γ = 0.2 γ = 0.1
100
γ = 0.05
W = 5λL
W = 10λL
W = 2λL W = λL
10 -4
γ = 0.025
50 0 0
10 -3
μ"
μ" x 103
200
10 -5 5
10
W / λL
15
20
10 -6 0.0
1.0
0.2
0.4
ω / ωc1
0.6
0.8
1.0
0.6
0.8
1.0
1.0
f = 14 GHz
γ = 0.2
0.8
W = λL 0.8
γ = 0.1 γ = 0.05 γ = 0.025
W = 2λL 0.6
μ'
μ'
0.6
0.4
0.2
0.0 0
5
10
15
20
W / λL Fig. 4. The calculated imaginary part, (a) and real part, (b) of rf permeability as a function of the slab thickness at a fixed frequency of 14 GHz for different damping factor c C2 =x2TO ¼ 0:2, 0.1, 0.05, and 0.025.
different damping factor c C2 =x2TO ¼ 0:2, 0.1, 0.05, and 0.025. It is seen from Fig. 4a that the critical thickness is increased and the loss peak is broadened as the damping is decreased. In addition, the value in the real part, l 0 , is enhanced for a small damping factor. The frequencydependent of rf permeability is plotted in Fig. 5, where four different thicknesses of slab, W = kL, 2kL, 5kL, and 10kL are taken and damping factor, c = 0.2 is considered as well. Fig. 5a illustrates that the loss is increased with increasing the frequency, revealing a strong dispersion in l00 . The real part, however, is not changed as pronouncedly as the imaginary part, as displayed in Fig. 5b. The frequency dependence of the rf permeability in the Meissner-like response is quite different from the transmittance spectrum in the dielectriclike response. In the transmittance spectrum it is found the existence of the anomalous peaks which is believed to be measurable in the NFE SC material, as stated in Ref. [14]. In this work, we have investigated the microwave properties of a NFE SC material through the complex permeability. The complex rf permeability indeed can be measured by related microwave technique. With the
0.4
W = 5λL
0.2
W = 10λL
0.0 0.0
0.2
0.4
ω / ωc1
Fig. 5. The calculated imaginary part, (a) and real part, (b) of rf permeability as a function of the frequency at four different thicknesses of slab, W = kL, 2kL, 5kL, and 10kL and at a fixed damping factor, c = 0.2.
measured results, a test of the electromagnetic theory of such a material can be achieved. 4. Summary The complex rf magnetic permeability for a NFE SC material in the Meissner-like response has been theoretically studied based on the electrodynamics of a nearly ferroelectric superconductor. We have investigated the permeability for a superconducting slab in a parallel field configuration. It is shown that the imaginary part attains maximum at a critical thickness of slab. This critical thickness can be defined as a length scale which is of use in characterizing the NFE SC films. We have found that the critical thickness is strongly dependent on the frequency as well as the damping factor. The effects of damping factor on the rf permeability are also specifically illustrated. The study of complex rf permeability provides a possible alternative for experimentally exploring the microwave properties of NFE SCs. It could be further beneficial to the understanding of electrodynamical model of these materials.
C.-J. Wu et al. / Physica C 453 (2007) 70–75
Acknowledgement C.-J. Wu acknowledges the financial support from the National Science Council of the Republic of China under Grant No. NSC-95-2112-M-390-003-MY2. References [1] M. Golosovsky, M. Tsidlekht, D. Davidov, Supercond. Sci. Technol. 9 (1996) 1. [2] V.M. Pan, D.A. Luzhbin, A.A. Kalenyuk, A.L. Kasatkin, V.A. Komashko, A.V. Velichko, M. Lancaster, Low Temp. Phys. 31 (2005) 254, and references therein. [3] M.R. Trunin, J. Supercond. 11 (1998) 381.
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