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Thickness-dependent effective surface resistances of nearly ferroelectric superconductors
Physics Letters A 364 (2007) 163–166 www.elsevier.com/locate/pla
Thickness-dependent effective surface resistances of nearly ferroelectric superconductors Chien-Jang Wu ∗ Department of Applied Physics, National University of Kaohsiung, Kaohsiung 811, Taiwan, ROC Received 9 October 2006; accepted 26 November 2006 Available online 8 January 2007 Communicated by R. Wu
Abstract The effective surface resistance of nearly ferroelectric superconducting film in the dielectriclike response is theoretically investigated based on the electrodynamics of the nearly ferroelectric superconductors. We calculate the intrinsic film surface resistance for isolated thin film and the effective surface resistance for a superconductor/dielectric layered structure. It is found that the thickness-dependent surface resistance has two different behaviors separated by a critical film thickness being equal to the London penetration length. That is, a nonresonant dependence is seen when the film thickness is less than the London penetration length, and an anomalously resonant behavior is found when the film thickness is larger than the London penetration length. The nonresonant dependence is similar to that of a cuprate superconductor and it further is characterized by some other critical thicknesses. As for the anomalous resonant region it is seen only in a nearly ferroelectric superconductor. © 2007 Elsevier B.V. All rights reserved. PACS: 74.20.De; 74.25.Nf; 78.20.Bh Keywords: Surface impedance; Ferroelectric; Superconductor; Meissner state
1. Introduction The surface impedance Zs = Rs + j Xs of a superconductor is being known as a popular quantity in the study of basic physics of superconductivity. Surface impedance is also of technical use in the applicational aspect. For instance, the real part, Rs , of the surface impedance called the surface resistance is related to the power dissipation in a superconductor and therefore determines the performance of a superconductorbased microwave device. A superconducting microwave passive device is commonly made by a thin film on a dielectric substrate, forming a layered structure. In this case the surface impedance is generally referred to as the effective surface impedance, Zs,eff = Rs,eff + j Xs,eff , that can be determined based on the electrodynamics of superconductor and on the substrate material as well. In addition, Zs,eff is also strongly dependent on * Tel.: +886 7 5919467, fax: +886 7 5919357.
E-mail address: [email protected] (C.-J. Wu). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.11.101
the thickness of superconducting film. The study of thicknessdependent Zs,eff is thus indispensable. To date there have been many reports on the thicknessdependent microwave surface resistances of superconductors [1–6]. As seen in the literature, theoretical and experimental works on the microwave surface resistances are largely focused on the high-temperature cuprate superconductors, especially on the YBa2 Cu3 O7−x system and its related derivatives. In addition to the familiar cuprate superconductors, there exists a special class of noncuprate superconductor called the nearly ferroelectric superconductor (NFE SC). NFE SC is a material such that the superconducting state and the NFE state coexist in it. The NFE state means a soft-mode ionic system with a fairly large static permittivity. Two typical materials of such kind are now available. One is the Na-doped WO3 (sodium tungsten bronze Nax WO3 with x ∼ 0.05) which has a relatively high transition temperature Tc ∼ 90 K [7]. The other having a relatively low Tc ∼ 1–3 K is n- (or p-) doped SrTiO3 (STO) [8]. The host STO is a familiar NFE system with a very large relative permittivity εr ∼ 104 in the temperature range of the super-
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C.-J. Wu / Physics Letters A 364 (2007) 163–166
conducting state. The study of the thickness-dependent surface resistance for such a superconducting material is however rarely seen thus far. The purpose of this Letter is to investigate theoretically the effective microwave surface resistance as a function of the thickness of the NFE SC film. It will be analyzed within the framework of electromagnetic model of the NFE SC materials reported by Birman and Zimbovskaya [9]. Based on this phenomenological model the wave propagation characteristics can be described by an effective wavenumber (or effective penetration length) that is a strong function of the radiation frequency. In some frequency regime NFE SC behaves like a regular dielectric and electromagnetic wave can propagate in it. In this case it is called the dielectriclike response. In some other regime it may act as a regular superconductor in the Meissner state. In the Meissner state the electromagnetic wave is not allowed to propagate in superconductor, only exponentially decaying. In this work we shall calculate the effective surface impedance under the dielectriclike response. The effective surface resistance will be examined as a function of the film thickness.
is introduced. where a parameter In order to calculate the intrinsic film surface impedance, Zs,int (ω, L), for a superconducting film occupying space, 0 z L, one must first determine the surface impedance, Zs , of a bulk material. The definition of Zs is given by
2. Calculations of surface impedances of a NFE SC film
Zs = Rs + j Xs =
The calculation of the surface impedance of a superconductor is, in principle, dependent on the electrodynamical model of superconductor. For a NFE SC material the electromagnetic properties 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. This governing field equation can be expressed as [9]
Eq. (5) is generally defined in a superconductor that occupies the half space, z 0. This impedance is generally referred to as the intrinsic bulk surface impedance. Simply calculation leads to ωμ0 . Z s = Rs + j X s = (6) k For the dielectriclike response we see that the surface impedance is purely resistive, namely ωμ0 = ωμ0 λ∗L , Z s = Rs = (7) k where we have introduced the effective penetration length λ∗L = k −1 . Then we can obtain Zs,int (ω, L) by using the impedance transform method, namely
∇ × ∇ × E = −μ0 ε(ω)
∂2 μ0 E− E, 2 Λ ∂t
(1)
where Λ = m∗ /(Ns e2 ) = μ0 λ2L where m∗ is the electron effective mass, Ns is the concentration of the superelectrons, e is the electron charge, μ0 is the permeability of the free space, and λL is the London penetration. In addition, the dielectric function is given by ε(ω) = ε∞
2 − ω2 ωLO 2 ωTO
− ω2
,
(2)
where ωLO and ωTO are the longitudinal and transverse soft2 ωLi = ε mode lattice frequencies, respectively, and ε∞ ∞ i 2 ωTi
where the primed product over i includes all oscillations except the soft-mode. Let us consider a plane-wave solution proportional to exp(−j kz + j ωt), in which k = β − j α with β being the phase constant and α being the attenuation constant as well. Then Eq. (1) enables us to find the wavenumber in a NFE SC material given by 1/2 . k = ω2 μ0 ε(ω) − λ−2 (3) L With Eq. (2) for the dielectric function, it is seen that the wavenumber is strongly dependent on the frequency. For frequencies such that k > 0, the electromagnetic wave will be
allowed to propagate in a NFE SC material as in a regular dielectric, leading to the so-called dielectriclike response. It is also referred to as the anomalous frequency region because of the salient dielectric properties in a superconductor. For some frequencies leading to a purely imaginary k, the electromagnetic wave can only be exponentially attenuated as a regular superconductor. In this case the response to an electromagnetic field is called the Meissner-like response. At frequency such that k = 0 we have the cutoff frequency which can be determined from Eqs. (2) and (3) and is given by 2 2 2 1 ωLO ωLO 1 1 1 ωLO 1+ 1 ωc1,c2 = ωTO −4 2, 1 + ± 2 2 2 ωTO 2 ωTO a2 a 2 ωTO a
(4) a2
ω2 Λ = ε∞ LO
Ex (0) . Hy (0)
(5)
Zs,int (ω, L) = Rs,int + j Xs,int = Zs
Z0 + Zs tanh(j L/λ∗L ) , (8) Zs + Z0 tanh(j L/λ∗L )
√ where Z0 = μ0 /ε0 = 377 is the intrinsic impedance of free space. In the layered structure where superconducting film (0 z L) deposited on a dielectric substrate occupying the space of L z L + d. For an electromagnetic wave normally incident on the plane boundary, z = 0, we have the forward effective surface impedance, Zs,eff,f , which can be obtained by the successive impedance transform, namely Zs,eff,f = Rs,eff,f + j Xs,eff,f = Zs
Zs,L + Zs tanh(j L/λ∗L ) , Zs + Zs,L tanh(j L/λ∗L )
(9)
where Zs,L = Zd
Z0 + Zd tanh(j d/λd ) , Zd + Z0 tanh(j d/λd )
(10)
where the substrate wavelength is defined by λd = 1/kd = √ c/(ω εd ), where c is the speed of light in free space and
C.-J. Wu / Physics Letters A 364 (2007) 163–166
165
εd is the relative permittivity of the dielectric substrate, and √ Zd = Z0 / εd is the wave impedance of the substrate. For the case of backward incidence where the electromagnetic wave normally impinges on the plane boundary, z = L + d, we have the backward effective surface impedance, Zs,eff,b , which is also obtainable from the impedance transform, namely Zs,eff,b = Rs,eff,b + j Xs,eff,b Zs,int + Zd tanh(j d/λd ) = Zd , Zd + Zs,int tanh(j d/λd )
(11)
where Zs,int is given by Eq. (8). 3. Numerical results and discussion In what follows we shall numerically investigate a typical and prototype NFE SC material such as the n-STO which is the sole NFE SC material having all the known material parameters for the numerical calculation [9]. The material parameters include effective mass of the electron, m∗ ≈ 10me with me being the rest mass of electron and concentration of 9 × 1017 cm−3 . Other parameters are the superelectrons, 2 Ns = 2 = 4.1, ω 11 rad/s, and /ωTi ε∞ = 5.5ε0 , i ωLi TO = 1.6 × 10 12 ωLO = 5.2 × 10 rad/s. With these parameters, the London penetration length is calculated to be λL = 17.73 µm and it follows Eq. (4) that the lower and upper cutoff frequencies are calculated to be ωc1 = 0.9 × 1011 rad/s (fc1 = 14.3 GHz) and ωc1 = 1.6 × 1011 rad/s (fc1 = 25.4 GHz), respectively, which are all in the microwave region. The dielectriclike response thus falls in the frequency range of fc1 < f < fc2 . In Fig. 1(a) we plot the intrinsic film surface resistance and forward effective surface resistance as a function of the thickness of the superconducting film at a fixed frequency of 20 GHz. In calculation of forward effective surface resistance we have taken LaAlO3 (εd = 24) as the substrate and its thickness is taken to be d = 0.5 mm. The intrinsic bulk impedance at this frequency is Zs = Rs = ωμ0 λ∗L = 1.797 . The overall behaviors between Rs,int and Rs,eff,f are similar and they can be divided into two different regions separated by a critical thickness of L = λL . For L < λL we have a nonresonant region and it is seen that both Rs,int and Rs,eff,f decrease with increasing the film thickness. Some features are of note in this region. Firstly, two critical points A: (0.044, 1) and B: (0.059, 1) indicate that the film can be, in effect, regarded as a bulk material because Rs,eff,f = Rs,int = Rs . Point C: (0.027, 2.663) characterizes another critical thickness for which the influence of dielectric substrate can be neglected due to the fact that Rs,eff,f = Rs,int . Secondly, in the limit of L λL , Rs,int = 209.751Rs = 377 equal to the wave impedance of free space and Rs,eff,f = 11.326Rs = 20.357 which is contributed only by the dielectric substrate. For L > λL we have a resonant behavior and it is replotted in Fig. 1(b) for convenience of illustration. For an isolated film at the resonant point, Rs,int = 377 , indicating that the film is totally transparent to the electromagnetic wave which in turn shows that the film effectively acts as a free-space film. As for the forward effective surface resistance at the resonant point we have Rs,eff,f ∼ = 113 , smaller
Fig. 1. (a) Intrinsic film surface resistance and forward effective surface resistance as a function of the film thickness at 20 GHz. The resonant part in (a) is replotted in (b). The black curve is for Rs,int while the grey (red in the web version of this Letter) one is for Rs,eff,f .
than Rs,int . The existence of substrate has lowered down the peak height of the surface resistance. It is further seen that the resonant peaks occur around the even multiples of λL for both Rs,int and Rs,eff,f . These peak positions can be well described as follows. Substituting Eq. (7) into Eq. (8) an analytic expression for Rs,int can be found, namely Rs,int =
1 Rs2 . Z0 sin2 (L/λ∗ ) + Rs2 cos2 (L/λ∗ ) 2 L L
(12)
Z0
It is seen from Eq. (12) that the resonance occurs at the condition of sin(L/λ∗L ) = 0 → L/λ∗L = N π , N = 1, 2, 3, . . . . With λ∗L = 0.642λL (at 20 GHz), it is direct to have L/λL = 2.0169 (N = 1), 4.0338 (N = 2), 6.0507 (N = 3), 8.0676 (N = 4), which exactly are the four peak positions as shown in Fig. 1(b). It should also be noted that they are regularly spaced in the thickness-domain. In Fig. 2 we plot the forward effective surface resistance at three different frequencies, 15, 20, and 25 GHz. The critical thickness for which Rs,eff,f = Rs occurs at point B : (0.012, 1), B: (0.059, 1), and B : (0.272, 1) for 25, 20, and 15 GHz, respectively. It shows that this critical thickness increases with decreasing frequency. In addition, the number of resonant peaks
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C.-J. Wu / Physics Letters A 364 (2007) 163–166
tive surface resistance is greater than forward one by about 12.4 times. It can be reasoned that at resonance the forward resistance is low because the superconducting film is itself regarded as a low resistance compared to the substrate in the backward case. Other than the resonant points, the difference between the effective backward and forward surface resistances is not substantial. 4. Summary
Fig. 2. Calculated forward effective surface resistance versus the film thickness at 15, 20, and 25 GHz.
In summary, we have investigated the thickness-dependent effective microwave surface resistance for a NFE SC film. It can be divided into two different regions by the critical thickness of superconducting film being equal to the London penetration length, λL . For film thickness less than λL one has a nonresonant region where the resistance decreases with increasing the thickness. In this region we also find other critical thickness such that the film can behave effectively like a bulk material. The other anomalous resonance region is found when the film thickness is larger than λL . The positions of the resonance peaks are located around the even multiple of λL . The resonant part is believed to be seen only in such a superconductor and is expected to be measurable. With the measured results it is beneficial to testing the electrodynamics of the NFE SC materials. Acknowledgements The author 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
Fig. 3. Calculated forward and backward effective surface resistances as a function of the film thickness at 20 GHz.
is strongly increased at a higher frequency such as 25 GHz. Only a peak is seen at 15 GHz. Fig. 3 shows the calculated forward and backward effective surface resistances as a function of film thickness at 20 GHz. A severe change between Rs,eff,b and Rs,eff,f can be seen only at the resonant points. At resonant peak, Rs,eff,b = 1404 , and Rs,eff,f = 113 . It is seen that at resonance the backward effec-
[1] X.Q. Jia, L. Kang, J. Chen, W.W. Xu, P.H. Wu, Physica C 49 (2006) 29. [2] A.G. Zaitsev, R. Schneider, J. Greek, G. Linker, F. Ratzel, R. Smithey, Appl. Phys. Lett. 75 (1999) 4165. [3] C.-J. Wu, T.-Y. Tseng, IEEE Trans. Magnetics 33 (1997) 2348. [4] R. Pinto, A.G. Chourey, P.R. Apte, J. Appl. Phys. 64 (1994) 2166. [5] N. Klein, H. Chaloupka, G. Muller, S. Orbach, H. Piel, B. Roas, L. Schultz, U. Klein, M. Peiniger, J. Appl. Phys. 67 (1990) 6950. [6] P. Hartemann, IEEE Trans. Appl. Supercond. 2 (1992) 228. [7] Y. Levi, O. Millo, A. Sharoni, Y. Tsabba, G. Leitus, S. Reich, Europhys. Lett. 51 (2000) 564. [8] C.S. Koonce, M.L. Cohen, J.E. Schooley, W.R. Hosler, L.F. Pfeiffer, Phys. Rev. 163 (1967) 380. [9] J.L. Birman, N.A. Zimbovskaya, Phys. Rev. B 64 (2001) 144506.
Thickness-dependent effective surface resistances of nearly ferroelectric superconductors Chien-Jang Wu ∗ Department of Applied Physics, National University of Kaohsiung, Kaohsiung 811, Taiwan, ROC Received 9 October 2006; accepted 26 November 2006 Available online 8 January 2007 Communicated by R. Wu
Abstract The effective surface resistance of nearly ferroelectric superconducting film in the dielectriclike response is theoretically investigated based on the electrodynamics of the nearly ferroelectric superconductors. We calculate the intrinsic film surface resistance for isolated thin film and the effective surface resistance for a superconductor/dielectric layered structure. It is found that the thickness-dependent surface resistance has two different behaviors separated by a critical film thickness being equal to the London penetration length. That is, a nonresonant dependence is seen when the film thickness is less than the London penetration length, and an anomalously resonant behavior is found when the film thickness is larger than the London penetration length. The nonresonant dependence is similar to that of a cuprate superconductor and it further is characterized by some other critical thicknesses. As for the anomalous resonant region it is seen only in a nearly ferroelectric superconductor. © 2007 Elsevier B.V. All rights reserved. PACS: 74.20.De; 74.25.Nf; 78.20.Bh Keywords: Surface impedance; Ferroelectric; Superconductor; Meissner state
1. Introduction The surface impedance Zs = Rs + j Xs of a superconductor is being known as a popular quantity in the study of basic physics of superconductivity. Surface impedance is also of technical use in the applicational aspect. For instance, the real part, Rs , of the surface impedance called the surface resistance is related to the power dissipation in a superconductor and therefore determines the performance of a superconductorbased microwave device. A superconducting microwave passive device is commonly made by a thin film on a dielectric substrate, forming a layered structure. In this case the surface impedance is generally referred to as the effective surface impedance, Zs,eff = Rs,eff + j Xs,eff , that can be determined based on the electrodynamics of superconductor and on the substrate material as well. In addition, Zs,eff is also strongly dependent on * Tel.: +886 7 5919467, fax: +886 7 5919357.
E-mail address: [email protected] (C.-J. Wu). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.11.101
the thickness of superconducting film. The study of thicknessdependent Zs,eff is thus indispensable. To date there have been many reports on the thicknessdependent microwave surface resistances of superconductors [1–6]. As seen in the literature, theoretical and experimental works on the microwave surface resistances are largely focused on the high-temperature cuprate superconductors, especially on the YBa2 Cu3 O7−x system and its related derivatives. In addition to the familiar cuprate superconductors, there exists a special class of noncuprate superconductor called the nearly ferroelectric superconductor (NFE SC). NFE SC is a material such that the superconducting state and the NFE state coexist in it. The NFE state means a soft-mode ionic system with a fairly large static permittivity. Two typical materials of such kind are now available. One is the Na-doped WO3 (sodium tungsten bronze Nax WO3 with x ∼ 0.05) which has a relatively high transition temperature Tc ∼ 90 K [7]. The other having a relatively low Tc ∼ 1–3 K is n- (or p-) doped SrTiO3 (STO) [8]. The host STO is a familiar NFE system with a very large relative permittivity εr ∼ 104 in the temperature range of the super-
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C.-J. Wu / Physics Letters A 364 (2007) 163–166
conducting state. The study of the thickness-dependent surface resistance for such a superconducting material is however rarely seen thus far. The purpose of this Letter is to investigate theoretically the effective microwave surface resistance as a function of the thickness of the NFE SC film. It will be analyzed within the framework of electromagnetic model of the NFE SC materials reported by Birman and Zimbovskaya [9]. Based on this phenomenological model the wave propagation characteristics can be described by an effective wavenumber (or effective penetration length) that is a strong function of the radiation frequency. In some frequency regime NFE SC behaves like a regular dielectric and electromagnetic wave can propagate in it. In this case it is called the dielectriclike response. In some other regime it may act as a regular superconductor in the Meissner state. In the Meissner state the electromagnetic wave is not allowed to propagate in superconductor, only exponentially decaying. In this work we shall calculate the effective surface impedance under the dielectriclike response. The effective surface resistance will be examined as a function of the film thickness.
is introduced. where a parameter In order to calculate the intrinsic film surface impedance, Zs,int (ω, L), for a superconducting film occupying space, 0 z L, one must first determine the surface impedance, Zs , of a bulk material. The definition of Zs is given by
2. Calculations of surface impedances of a NFE SC film
Zs = Rs + j Xs =
The calculation of the surface impedance of a superconductor is, in principle, dependent on the electrodynamical model of superconductor. For a NFE SC material the electromagnetic properties 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. This governing field equation can be expressed as [9]
Eq. (5) is generally defined in a superconductor that occupies the half space, z 0. This impedance is generally referred to as the intrinsic bulk surface impedance. Simply calculation leads to ωμ0 . Z s = Rs + j X s = (6) k For the dielectriclike response we see that the surface impedance is purely resistive, namely ωμ0 = ωμ0 λ∗L , Z s = Rs = (7) k where we have introduced the effective penetration length λ∗L = k −1 . Then we can obtain Zs,int (ω, L) by using the impedance transform method, namely
∇ × ∇ × E = −μ0 ε(ω)
∂2 μ0 E− E, 2 Λ ∂t
(1)
where Λ = m∗ /(Ns e2 ) = μ0 λ2L where m∗ is the electron effective mass, Ns is the concentration of the superelectrons, e is the electron charge, μ0 is the permeability of the free space, and λL is the London penetration. In addition, the dielectric function is given by ε(ω) = ε∞
2 − ω2 ωLO 2 ωTO
− ω2
,
(2)
where ωLO and ωTO are the longitudinal and transverse soft2 ωLi = ε mode lattice frequencies, respectively, and ε∞ ∞ i 2 ωTi
where the primed product over i includes all oscillations except the soft-mode. Let us consider a plane-wave solution proportional to exp(−j kz + j ωt), in which k = β − j α with β being the phase constant and α being the attenuation constant as well. Then Eq. (1) enables us to find the wavenumber in a NFE SC material given by 1/2 . k = ω2 μ0 ε(ω) − λ−2 (3) L With Eq. (2) for the dielectric function, it is seen that the wavenumber is strongly dependent on the frequency. For frequencies such that k > 0, the electromagnetic wave will be
allowed to propagate in a NFE SC material as in a regular dielectric, leading to the so-called dielectriclike response. It is also referred to as the anomalous frequency region because of the salient dielectric properties in a superconductor. For some frequencies leading to a purely imaginary k, the electromagnetic wave can only be exponentially attenuated as a regular superconductor. In this case the response to an electromagnetic field is called the Meissner-like response. At frequency such that k = 0 we have the cutoff frequency which can be determined from Eqs. (2) and (3) and is given by 2 2 2 1 ωLO ωLO 1 1 1 ωLO 1+ 1 ωc1,c2 = ωTO −4 2, 1 + ± 2 2 2 ωTO 2 ωTO a2 a 2 ωTO a
(4) a2
ω2 Λ = ε∞ LO
Ex (0) . Hy (0)
(5)
Zs,int (ω, L) = Rs,int + j Xs,int = Zs
Z0 + Zs tanh(j L/λ∗L ) , (8) Zs + Z0 tanh(j L/λ∗L )
√ where Z0 = μ0 /ε0 = 377 is the intrinsic impedance of free space. In the layered structure where superconducting film (0 z L) deposited on a dielectric substrate occupying the space of L z L + d. For an electromagnetic wave normally incident on the plane boundary, z = 0, we have the forward effective surface impedance, Zs,eff,f , which can be obtained by the successive impedance transform, namely Zs,eff,f = Rs,eff,f + j Xs,eff,f = Zs
Zs,L + Zs tanh(j L/λ∗L ) , Zs + Zs,L tanh(j L/λ∗L )
(9)
where Zs,L = Zd
Z0 + Zd tanh(j d/λd ) , Zd + Z0 tanh(j d/λd )
(10)
where the substrate wavelength is defined by λd = 1/kd = √ c/(ω εd ), where c is the speed of light in free space and
C.-J. Wu / Physics Letters A 364 (2007) 163–166
165
εd is the relative permittivity of the dielectric substrate, and √ Zd = Z0 / εd is the wave impedance of the substrate. For the case of backward incidence where the electromagnetic wave normally impinges on the plane boundary, z = L + d, we have the backward effective surface impedance, Zs,eff,b , which is also obtainable from the impedance transform, namely Zs,eff,b = Rs,eff,b + j Xs,eff,b Zs,int + Zd tanh(j d/λd ) = Zd , Zd + Zs,int tanh(j d/λd )
(11)
where Zs,int is given by Eq. (8). 3. Numerical results and discussion In what follows we shall numerically investigate a typical and prototype NFE SC material such as the n-STO which is the sole NFE SC material having all the known material parameters for the numerical calculation [9]. The material parameters include effective mass of the electron, m∗ ≈ 10me with me being the rest mass of electron and concentration of 9 × 1017 cm−3 . Other parameters are the superelectrons, 2 Ns = 2 = 4.1, ω 11 rad/s, and /ωTi ε∞ = 5.5ε0 , i ωLi TO = 1.6 × 10 12 ωLO = 5.2 × 10 rad/s. With these parameters, the London penetration length is calculated to be λL = 17.73 µm and it follows Eq. (4) that the lower and upper cutoff frequencies are calculated to be ωc1 = 0.9 × 1011 rad/s (fc1 = 14.3 GHz) and ωc1 = 1.6 × 1011 rad/s (fc1 = 25.4 GHz), respectively, which are all in the microwave region. The dielectriclike response thus falls in the frequency range of fc1 < f < fc2 . In Fig. 1(a) we plot the intrinsic film surface resistance and forward effective surface resistance as a function of the thickness of the superconducting film at a fixed frequency of 20 GHz. In calculation of forward effective surface resistance we have taken LaAlO3 (εd = 24) as the substrate and its thickness is taken to be d = 0.5 mm. The intrinsic bulk impedance at this frequency is Zs = Rs = ωμ0 λ∗L = 1.797 . The overall behaviors between Rs,int and Rs,eff,f are similar and they can be divided into two different regions separated by a critical thickness of L = λL . For L < λL we have a nonresonant region and it is seen that both Rs,int and Rs,eff,f decrease with increasing the film thickness. Some features are of note in this region. Firstly, two critical points A: (0.044, 1) and B: (0.059, 1) indicate that the film can be, in effect, regarded as a bulk material because Rs,eff,f = Rs,int = Rs . Point C: (0.027, 2.663) characterizes another critical thickness for which the influence of dielectric substrate can be neglected due to the fact that Rs,eff,f = Rs,int . Secondly, in the limit of L λL , Rs,int = 209.751Rs = 377 equal to the wave impedance of free space and Rs,eff,f = 11.326Rs = 20.357 which is contributed only by the dielectric substrate. For L > λL we have a resonant behavior and it is replotted in Fig. 1(b) for convenience of illustration. For an isolated film at the resonant point, Rs,int = 377 , indicating that the film is totally transparent to the electromagnetic wave which in turn shows that the film effectively acts as a free-space film. As for the forward effective surface resistance at the resonant point we have Rs,eff,f ∼ = 113 , smaller
Fig. 1. (a) Intrinsic film surface resistance and forward effective surface resistance as a function of the film thickness at 20 GHz. The resonant part in (a) is replotted in (b). The black curve is for Rs,int while the grey (red in the web version of this Letter) one is for Rs,eff,f .
than Rs,int . The existence of substrate has lowered down the peak height of the surface resistance. It is further seen that the resonant peaks occur around the even multiples of λL for both Rs,int and Rs,eff,f . These peak positions can be well described as follows. Substituting Eq. (7) into Eq. (8) an analytic expression for Rs,int can be found, namely Rs,int =
1 Rs2 . Z0 sin2 (L/λ∗ ) + Rs2 cos2 (L/λ∗ ) 2 L L
(12)
Z0
It is seen from Eq. (12) that the resonance occurs at the condition of sin(L/λ∗L ) = 0 → L/λ∗L = N π , N = 1, 2, 3, . . . . With λ∗L = 0.642λL (at 20 GHz), it is direct to have L/λL = 2.0169 (N = 1), 4.0338 (N = 2), 6.0507 (N = 3), 8.0676 (N = 4), which exactly are the four peak positions as shown in Fig. 1(b). It should also be noted that they are regularly spaced in the thickness-domain. In Fig. 2 we plot the forward effective surface resistance at three different frequencies, 15, 20, and 25 GHz. The critical thickness for which Rs,eff,f = Rs occurs at point B : (0.012, 1), B: (0.059, 1), and B : (0.272, 1) for 25, 20, and 15 GHz, respectively. It shows that this critical thickness increases with decreasing frequency. In addition, the number of resonant peaks
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C.-J. Wu / Physics Letters A 364 (2007) 163–166
tive surface resistance is greater than forward one by about 12.4 times. It can be reasoned that at resonance the forward resistance is low because the superconducting film is itself regarded as a low resistance compared to the substrate in the backward case. Other than the resonant points, the difference between the effective backward and forward surface resistances is not substantial. 4. Summary
Fig. 2. Calculated forward effective surface resistance versus the film thickness at 15, 20, and 25 GHz.
In summary, we have investigated the thickness-dependent effective microwave surface resistance for a NFE SC film. It can be divided into two different regions by the critical thickness of superconducting film being equal to the London penetration length, λL . For film thickness less than λL one has a nonresonant region where the resistance decreases with increasing the thickness. In this region we also find other critical thickness such that the film can behave effectively like a bulk material. The other anomalous resonance region is found when the film thickness is larger than λL . The positions of the resonance peaks are located around the even multiple of λL . The resonant part is believed to be seen only in such a superconductor and is expected to be measurable. With the measured results it is beneficial to testing the electrodynamics of the NFE SC materials. Acknowledgements The author 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
Fig. 3. Calculated forward and backward effective surface resistances as a function of the film thickness at 20 GHz.
is strongly increased at a higher frequency such as 25 GHz. Only a peak is seen at 15 GHz. Fig. 3 shows the calculated forward and backward effective surface resistances as a function of film thickness at 20 GHz. A severe change between Rs,eff,b and Rs,eff,f can be seen only at the resonant points. At resonant peak, Rs,eff,b = 1404 , and Rs,eff,f = 113 . It is seen that at resonance the backward effec-
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