Physical properties of YBa
2
Cu
3
O
7d
thin films using microstrip
ring resonators technique
L.S. Lai
a,*, H.K. Zeng
b, J.Y. Juang
a, K.H. Wu
a, T.M. Uen
a, J.Y. Lin
c, Y.S. Gou
a aDepartment of Electrophysics, National Chiao-Tung University, Hsinchu 30050, TaiwanbDepartment of Electronic Engineering, Chung-Yuan Christian University, 200 Chung-Pei Road, Chung-Li 32023, Taiwan c
Institute of Physics, National Chiao-Tung University, Hsinchu 30050, Taiwan Received 3 November 2005; received in revised form 11 April 2006; accepted 21 April 2006
Abstract
Microstrip ring resonators with quality factor (Q) over 104at temperature 5 K, were fabricated using the double-side YBa2Cu3O7d
(YBCO) epitaxial films deposited on LaAlO3(LAO) substrates. By placing a narrow gap in the ring resonator, we observed that the
original fundamental resonating mode (resonance frequency f = 3.61 GHz) splits into a dual-mode with different resonating frequencies (f = 1.80 GHz and f = 5.33 GHz). These two kinds of the resonator allow us to determine the temperature and frequency dependences of the magnetic penetration depth k(T, f) and the surface loss. Several salient features of the above findings related to the nature of low-lying excitations for high-Tcsuperconductivity as a function of oxygen content will be elucidated. In particular, the current models, suggested
by Wen and Lee, will be examined in a quantitative manner. It allows us to give a justification of quasiparticle as Fermi-liquid in the superconducting state. In addition, an equivalent inductance circuit model is suggested to account for the occurrence of the dual-mode resonance.
2006 Published by Elsevier B.V.
PACS: 74.72.Bk; 74.25.Nf; 74.78.Bz
Keywords: Microwave ring resonators; Dual-mode resonance; Magnetic penetration depth
1. Introduction
Microwave ring resonators made of an annular ring with high-Tcsuperconducting (HTS) thin films, combined
with the same superconducting ground plane, are of con-siderable theoretical and practical interests owing to their geometry advantages in greatly resolving the extremely non-uniform current distribution and reducing inductance commonly encountered by the stripline resonators [1–7]. Such resonator always manifests itself with a degenerate fundamental mode. If one put forth a gap structure in the microstrip ring resonator, it is then expected to split
into a mode resonator. The frequencies of the dual-mode are dependent upon the location of the gap with respect to the coupling lines[3]. Such microwave technique provides advantage over the previous one in the measure-ments of the frequency dependences of the penetration depth and its derived and related physical quantity with a unique sample.
Firstly, in an attempt to clarifying the origin of the dual-mode resonance, we manipulated the geometrical configu-ration of the sample by introducing a split gap, which is symmetrical with respect to the coupling lines, into our YBCO ring resonator structure, as shown in Fig. 1. With the insight on the microwave properties of the resonator with and without the gap, a consistent physical property of generating the dual-mode in a unique sample can be elu-cidated in the light of an equivalent inductance-circuit
0921-4534/$ - see front matter 2006 Published by Elsevier B.V. doi:10.1016/j.physc.2006.04.089
*
Corresponding author. Tel.: +886 3 5712121 56111; fax: +886 3 572 5230.
E-mail address:[email protected](L.S. Lai).
model. Using such kinds of samples with a proper control of various oxygen contents[4], a number of unique features of low-lying excitations (quasiparticles) of the high-Tc
underdoped cuprates in the superconducting state can be explored detail.
In this report, the temperature and frequency depen-dences of the penetration depth k(T, f ) and the surface loss will be extracted from the measurements of the surface impedance and the quality factor of the resonator together with the proper geometrical factor, respectively. As com-pared to the previous works, we believe that all these find-ings not only allow us to give some reliable evidences related to the nature of the superconducting carrier density, but also provide a great amount of viable data to shed light on the model construction of the superconductivity mech-anism under the strong impurity scattering.
2. Experiment and data analysis
The YBCO thin films used for fabricating the ring reso-nators were deposited epitaxially on both sides of a 0.5 mm thick LaAlO3substrate by pulsed laser deposition[8]. The
substrate temperature was kept at 830C, and the oxygen partial pressure was maintained at 280 mTorr during the deposition. Both films were c-axis oriented, with thickness of 500 nm and Tc= 90 K. One side of the films was then
patterned into the ring shaped microstrip line with or with-out a split gap, as shown schematically in Fig. 1. The line width, split gap and the outer radius of the ring are 0.5 mm, 0.1 mm and 3.625 mm (ro), respectively. The
coupling gap between the microstrip feeding line and the
ring resonator is about 0.4 mm. The main difficulty for double side deposition is the contact between the substrate and the heater, which has been solved by inserting a thin Si-wafer between the substrate and the heater. This effec-tively prevents the backside of the polished substrates as well as the first film during the deposition of the second one from contamination. The fact provides a great advan-tage to give a rather simple treatment in theoretical consid-eration to derive the penetration depth.
To obtain the desired oxygen content of the YBCO films, the sample was put in a YBCO bulk housing and the whole assembly was situated in an oxygen annealing chamber. The oxygen pressure and the corresponding tem-perature were then carefully monitored following the pres-sure-temperature phase diagram established for YBCO system[9]. This process had been confirmed to be capable of obtaining designated oxygen content of YBCO films in a controllable and reversible manner[10].
The microstrip ring resonator was put into an Au-coated aluminum housing with SMA connectors. The package was placed in a vacuum tube and immersed in liquid He. We used a Lake Shore 330 autotune temperature controller to control the temperature of the sample space to better than 0.1 K. The temperature dependence of the res-onance frequency f(T), the resonator bandwidth at3 dB, df(T), and the forward transmission coefficient S21 were
measured by a HP8510C microwave vector network analyzer.
The loaded quality factor QL of the resonator was
derived from the full bandwidth df at half maximum (FWHM) of the transmission curve S21(f ) using QL=
f0/df, where f0is the resonance frequency. Unloaded
qual-ity factor Qu is calculated using Qu= QL/(1 S21). The
dielectric loss of the LaAlO3 substrates is lower than
5· 106 at temperature below Tc of the YBCO, and it
was neglected in our case. Since the losses due to radiation have been minimized by providing effective shielding at half wavelength spacing around the device, they can be neglected as compared to the dielectric loss. Thus the Qu
comes mainly from the surface resistance Rs of the
super-conducting ring and the ground plate. From the unloaded quality factor Qu and the resonance frequency f(T), the
surface resistance Rsand reactance Xsconsisting of surface
impedance Zs(T) = Rs(T) + iXs(T) are calculated with the
aid of the formula[11]:
RsðT Þ ¼ C=QuðT Þ; ð1Þ
XsðT Þ XsðT0Þ ¼ 2C½f ðT Þ f ðT0Þ=f ðT Þ; ð2Þ
where C is the geometry factor of the resonator and T0is
the lowest temperature in the experiment (T0= 5 K).
Actu-ally, C depends [11–13] on the penetration depth k and geometry parameters (such as the microstrip width W, sub-strate thickness h, thin film thickness t) of the superconduc-ting resonator if t/k is not much larger than 1. The value of C can be calculated by the method of incremental fre-quency rules[14–16]via
Fig. 1. An equivalent circuit of the ring shaped microstrip resonator (a) without or (b) with a split gap.
C¼ l0x
2=4fDx=Dkg1
; ð3Þ
where Dx/Dk is the derivative of the angular resonance fre-quency (x = 2pf) with respect to the penetration depth k(T). Using Chang’s formula [17] for the inductance and capacitance of a superconducting strip line the angular res-onance frequency is given by
xðT Þ ¼ ðc=e1=2LÞf1 þ kðT Þ=d½2 cothðt=kðT ÞÞ
þ gcschðt=kðT ÞÞg1=2; ð4Þ
where e is the dielectric constant of the substrate, d is the effective substrate thickness, L is the length of the strip line, g is a fringe factor, and c is the velocity of the light in vac-uum. By inserting Dx/Dk obtained from Eq.(4) into Eq. (3), C can be derived as
C¼ ð1=2Þl0xdf1 þ k=d½2 cothðt=kÞ
þ g= sinhðt=kÞg=fcothðt=kÞ þ ðt=kÞ=sinh2ðt=kÞ
þ g½1 þ ðt=kÞ cothðt=kÞ=½2 sinhðt=kÞg; ð5Þ where g is a correction factor accounting for the fringe field effect as the aspect ratio W/d of the stripline near or less than 1. On the other hand, we have the relation between the surface reactance Xs(T) and the penetration depth
k(T) [16,18], i.e., Xs(T) = l0xk(T). Thus, from Eq. (2),
one obtains
l0x½kðT Þ kðT0Þ ¼ 2C½f ðT0Þ f ðT Þ=f ðT Þ: ð6Þ
When t/k(T) 1, the formula(5) can be simplified to:
C¼ ð1=2Þl0xd: ð7Þ
In this case we can get Dk from f(T) curve directly. In gen-eral case t/k(T) 1 is not fulfilled instead, we have to fit the curve, f(T), to give the penetration depth. It is conve-nient to use the normalized form of Eq.(4):
fðT Þ fðT0Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þkðT0Þ d 2 coth t kðT0Þ þ gcsch t kðT0Þ n o r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þkðT Þd 2 coth t kðT Þ þ gcsch t kðT Þ n o r : ð8Þ
Here T0is the lowest temperature in the experiments, which
is around 5 K in our case. To obtain the absolute values of k(T), the modified two-fluid model has to be assumed. Fol-lowing a number of previous studies we will take:
kðT Þ ¼ kð5 KÞ 1 T Tc 2 " #1 2 ; ð9Þ
at 0.3 6 T/Tc60.6. Using (8) and (9), then the best fit to
the experimental curves f(T) gets the k(5 K). For example, the best least-square fit yields d = h/4 and k(5 K) = 150 nm for d = 0.05, as shown in Fig. 2(a)–(c) at frequencies 1.80 GHz, 3.61 GHz, and 5.33 GHz respectively. The ob-tained value of k(5 K) was then inserted into Eq.(8)to ob-tain k(T) for all temperatures. The results for films with various oxygen contents are listed in Table 1. It is noted that, for d = 0.05 (i.e. the optimal doped YBCO), the value
of k(5 K) = 150 nm is very close to that reported for single crystalline YBCO [19,20] and d = h/4 is also consistent with the justification of the geometrical structure of the microwave ring resonator as compared with the strip line. λ λ λ (a) (b) (c)
Fig. 2. The fitting to the normalized temperature dependence of f(T)/ f(5 K) at resonance frequencies: (a) f = 1.80 GHz, (b) f = 3.61 GHz and (c) f = 5.33 GHz in the temperature range 0.3 6 T/Tc60.6 for oxygen
3. Results and discussion
We briefly review the effect of a split gap on a microstrip ring resonator. When the split gap is located at the symmet-rical position of the ring with respect to the coupling lines, the original resonance frequency at 3.61 GHz evidently split into the dual stationary modes at 1.80 GHz and 5.33 GHz, respectively. This result has been studied previ-ously[3]. To understand the origin of this effect, an equiv-alent circuit model has been put forth to describe the stationary behavior of this dual-mode resonance. Since the magnetic field is the dominant relevant parameter inside the superconducting microstrip ring resonator, we will take only the inductive effect into account. By coupling through the input port, the microwave currents flowing inside the microstrip ring resonator can be, to the first approximation, written as L11 dI1 dt ¼ ix0L11I1þ ix0L12I2; L22 dI2 dt ¼ ix0L21I1þ ix0L22I2; ð10Þ
where I1 and I2 are the amplitudes of the current density
wave flowing clockwise and counterclockwise, respectively. L11and L22are self-inductance coefficient (L11= L22in our
case). L12 and L21 are mutual inductance coefficient
(L12= L21, for the symmetrical configuration). The system
essentially manifests itself as a simple two-state system and any structure perturbation in the perfect ring does may provide a coupling between the two degenerate states. When there is no coupling, as in the case of the perfect ring resonator, the equations become
L11 dI1 dt ¼ ix0L11I1; L22 dI2 dt ¼ ix0L22I2: ð11Þ
Since L11= L22in our case, one obtains in the stationary
case I1¼ Aeix0t;
I2¼ Beix0t;
ð12Þ
with A and B being constants. The solutions indicate the inherent resonance degeneracy of the structure. Eq. (10) though does not specify how coupling between the degen-erate resonant mode can be introduced, it has been demon-strated experimentally that various kinds of the structure perturbation along the periphery of the ring may result in a splitting in the resonance frequencies [7]. In our case, the introduction of a ‘‘split-gap’’, which is symmetrical with respect to the coupling lines, turns out to give rise to an even more pronounced splitting effect. It is conceived that the electromagnetic waves totally reflected at the gap barrier may act as the cause of the mutual inductance. By taking L12= L21=M, the mutual inductance
depend-ing only on the circuit geometry, Eq.(10)can be rewritten as L11 dI1 dt ¼ ix0L11I1 ix0MI2; L22 dI2 dt ¼ ix0L22I2 ix0MI1: ð13Þ
The solutions to Eq.(13)are I1¼ C 2e ix0x0L11M t þD 2e ix0þx0L11M t ; I2¼ C 2e ix0x0L22M t D 2e ix0þx0L22M t ; ð14Þ
where C and D are constants. When either C or D is zero, the system is in stationary resonant mode. Namely, when D = 0, I1and I2are in phase (since L11= L22in our case),
and the system is at a stationary state with a definite fre-quency x0 x0M/L11. On the other hand, when C = 0,
I1and I2are out of phase with another definite frequency
of x0+ x0M/L11. Physically, the scenario is very similar
to other simple two-state systems with coupling, for exam-ple, a coupled two-pendulum problem. In any two-state system the strength of the coupling is, in general, originated from some perturbations in the system. Thus it is also very convenient to obtain the coupling strength from the mea-surements of the resonance frequencies with the aid of the above the circuit model. In our case, the split of the degenerate resonance frequency (x0) to x0/2 and 3x0/2
im-plies an inductance ratio M/L11= 1/2. This result can be
easily justified by the measured resonance frequencies of the ring resonators with and without a split gap.
In order to further confirm the above assertion of regarding the ring resonator as a simple two-state system as well as the split gap induced mutual inductance scenario, numerical simulations using electromagnetic simulator Ensemble of ‘‘ANSOFT’’ corporation to depict the current density wave distributions along the ring were performed. Resonance frequencies and the corresponding standing wave pattern obtained by assuming a perfect conductor ring with the same physical and geometric parameters of the real resonator are depicted inFig. 3(a).Fig. 3(a) shows the fundamental degenerate modes manifested in the full ring structure. The two current density nodes on the opposite sides of the ring resonator with a resonating
Table 1
Some parameters for YBa2Cu3O7dthin films are obtained from
micro-wave ring resonators[30]
7d f (GHz) k(5 K) nm 2D(0)/Tc d[k2(5 K)/ k2(T)]/dT (K1) mF/mD a2 6.95 3.61 150 ± 14 6.1 ± 2.2 1/(250 ± 50) 14 0.43 ± 0.17 1.80 6.8 ± 0.9 1/(212 ± 29) 0.52 ± 0.18 5.33 7.2 ± 1.0 1/(370 ± 100) 0.34 ± 0.16 6.8 3.61 216 ± 16 6.0 ± 0.5 1/(200 ± 26) 10.1 0.35 ± 0.08 1.80 6.5 ± 0.5 1/(189 ± 29) 0.38 ± 0.09 5.33 6.4 ± 0.7 1/(197 ± 17) 0.37 ± 0.07 6.6 3.61 282 ± 20 5.0 ± 0.6 1/(91 ± 10) 8.7 0.53 ± 0.08 1.80 5.9 ± 1.1 1/(119 ± 13) 0.41 ± 0.06 5.33 6.6 ± 0.6 1/(127 ± 10) 0.39 ± 0.05
wavelength kp= 2proare evident. For rings with a split gap
at half-way between the two coupling ports, one obtains two fundamental modes with the resonant frequencies of kp/2 = 2pro(Fig. 3(b)) and 3kp/2 = 2pro(Fig. 3(c)),
respec-tively. In both cases the position of the split gap always coincides with the node of current density waves. In any case, we note that in either mode the phase velocity of the current density wave remains unchanged, indicative of the nondispersive nature of superconductors.
Fig. 4(a) and (b) show the systematic changes of thejS21j
obtained at 5 K for both resonance frequencies (1.80 GHz and 5.33 GHz) when the oxygen content of the same YBCO ring resonator with a split gap was varied intention-ally from d = 0.05 to d = 0.4. The removal of oxygen in YBCO is known to take place primarily at the Cu–O chain, leading to a reduction of superfluid density and longer London penetration depths (kL). As a result, the shifts to
lower resonance frequencies observed in both modes with decreasing oxygen content are due primarily to the increas-ing kL. The other peculiar feature to be noted is the
magni-tude ofjS21j appears to decrease with decreasing oxygen in
YBCO for the lower frequency mode, while the opposite trend is evident for the higher frequency mode. The fact how reducing oxygen would enhance jS21j of one mode
while suppress the other is not clarified yet.
In Fig. 5, the temperature dependence of the unloaded quality factor, Qufor fully oxygenated YBCO ring
resona-tor displays essentially the same behavior for the cases of the frequencies at 1.80 GHz and 3.61 GHz regardless of
Fig. 4. (a) The frequency dependence of the forward transmission coefficient jS21j of the split gap resonator obtained at 5 K on the one
mode (f = 1.80 GHz) for d = 0.05, 0.2 and 0.4, respectively. The result of another mode (f = 5.33 GHz) is shown in (b).
Fig. 3. (a) The fundamental degenerate modes (kp= 2pro) manifested in
the perfect ring structure. For rings with a split gap at half-way between the two coupling ports, one obtains two fundamental modes with the resonant frequencies of (b) kp/2 = 2proand (c) 3kp/2 = 2pro, respectively.
the existence of the split gap. This implies that Qu is not
affected even when the geometrical symmetry of the ring resonator is broken. Moreover, from the measurement of Qu and the calculation of the geometrical factor C, we
can obtain the surface resistance Rs, as shown in Fig. 6.
Fig. 6also shows that the ratio of Rs values for 3.61 and
1.80 GHz is about 4, while the ratio between 5.33 and
1.80 GHz rises by a factor of 9. This is consistent with the fact that Rsshould be proportional to x
2
in the London limit. Herein the values of penetration depth k are indepen-dent of frequencies.
We would like to turn the attention to examine some fundamental property of k(T) using the present resonating structures. This is one of the most intriguing issues on high-Tc superconductivities. In particular, the temperature
dependence and the slope of the dependence are the impor-tant relevant parameters revealing of the order-parameter symmetry as well as providing rich information about the nature of low-lying excitations. As shown inFigs. 7 and 8, Dk(T) as a function of oxygen content derived from the low frequency mode (1.80 GHz) in the split YBCO ring resonator and 3.61 GHz in the perfect one, respectively, display the same linear-temperature dependent features. However, the linear-temperature dependent range of Dk(T) is different for the two resonance frequencies. For d= 0.05, Dk(T) displays a linear behavior up to tempera-ture T = 35 K for the perfect ring resonator, as shown in Fig. 8. But for the ring resonator with a split gap, Dk(T) displays a linear behavior at T < 25 K for d = 0.05. Fur-thermore, for d = 0.2 and 0.4 [21], similar peculiarities are observed, except that the respective corresponding tem-peratures are lower. All the above-mentioned features of the resonator with a split gap at 5.33 GHz were similar to those at 1.80 GHz but were different comparing with the perfect ring resonator. The effect of introducing a geo-metrical gap in suppressing the temperature dependence of
Fig. 5. The temperature dependence of the quality factor for the ring resonators with and without a split gap.
Fig. 6. The temperature dependence of surface resistance Rs(d = 0.05) of
the YBCO ring resonator with and without a split gap measured for f = 1.80, 3.61, and 5.33 GHz, respectively.
Fig. 7. The temperature dependence of the Dk(T) as a function of oxygen content derived from the low frequency mode (1.80 GHz) in split gap ring resonator. The inset shows the normalized temperature dependence of the Dk(T) for d = 0.05, 0.2 and 0.4, respectively.
Dk(T) in the low temperature regime was observed here, and this effect was very similar to the study done by Bonn et al.[22] who observed that the properties of Dk(T) were very sensitive to defects. The interesting problem is, how-ever, not clarify yet.
At the low temperature regime (T < Tc/3), moreover, we
can estimate the ratio of the maximum energy gap D(0) and the critical temperature Tc, from Scalapino’s[23]
theoreti-cal formula, Dk/T = k(0)ln 2/D(0) for the d-wave pairing, where we assume k(0) k(5 K). The value of 2D(0)/Tc is
approximated to a constant value (6.0 ± 0.6), which is independent of oxygen concentrations. The value of k(5 K) depends on the hole concentration greatly. And all k(5 K) are independent of the frequency. Finally, k(T) behaves as a universal manner in terms of the normalized temperatures, (T/Tc). The fact is thus strongly evident that
a unique nature of the high-Tcsuperconducting mechanism
is shown in the CuO2plane.
Fig. 9shows the normalized temperature dependence of the k2(5K)/k2(T) at 1.80 GHz, which denotes the change of the superfluid density by decreasing temperature. It is clearly evident that it is almost independent of the doping concentrations in terms of the normalized temperature (T/Tc< 0.3). In the low temperature regime, the linear T
dependence is also observed, as shown in Fig. 10, while the slope increases as d increasing. The independent prop-erty of the doping concentration reflects a fact that the Cu– O chain in the YBCO material is not a crucial factor in the consideration of superconductor properties. In fact the lin-ear temperature dependence will provide a valuable data to estimate the Fermi-liquid correction factor, a2, together with the measurement of the thermal conductivity. It allows us to test the theoretical model suggested by Lee and Wen [24,25]in a quantitative way. The slope of
nor-malized superfluid density k2(0)/k2(T), following their model, is related to the Fermi-liquid correction factor, a2, and the thermal conductivity [26], j0/T, as
dk2ðT Þ
dT ¼ 2:93 10
13j0
T a
2: ð15Þ
For the optimally doped sample, the j0/T value of
0.14 mW/K2cm can be taken from the thermal conductiv-ity measurements of YBCO single crystal [26]for the thin films. Hence the Fermi-liquid correction factor a2 could be calculated as
Fig. 10. The k2(5 K)/k2(T) vs. temperature T in the lower temperature
region for the low frequency mode (1.80 GHz) in split gap ring resonator with various oxygen contents.
Fig. 8. The temperature dependence of the Dk(T) as a function of oxygen content in the perfect ring resonator. The inset shows the normalized temperature dependence of the Dk(T) for d = 0.05, 0.2 and 0.4, respectively.
Fig. 9. The k2(5 K)/k2(T) vs. reduced temperature T/T
c for the low
frequency mode (1.80 GHz) in split gap ring resonator with various oxygen contents.
a2¼ 2:44 10 12
k2ð5 KÞ
d½k2ð5 KÞ=k2ðT Þ
dT : ð16Þ
For the underdoped samples, more elaborate consider-ation is necessary to elucidate the accurate value of a2. In the cuprates, j0=T ðk2B=3hÞðn=dÞðvF=vDÞ where n/d is the
stacking density of CuO2planes, vF is the Fermi velocity,
and vDis the energy dispersion along the Fermi surface at
nodes[27,28]. Assuming the tendency of vF/vDwith respect
to p of single crystal data in Ref. [29] is qualitatively and quantitatively applicable to our samples, the values of a2 for various oxygen contents are now accessible. The values of a2 for various oxygen contents [30] and some other important findings are listed as follows and inTable 1. It is found that in these oxygen contents (d = 0.05–0.4) stud-ied in this paper, a2 is less than 1. Therefore, the Fermi liquid corrections for quasiparticles in the underdoped cup-rates are not negligible. The values of a2are between 0.4 and 0.6, and are almost identical to that in Ref. [29]. At last, the values of a2for the cuprate samples studied in this work are nearly independent of frequencies.
4. Summary
Microstrip ring resonators made of double-side YBCO films have been demonstrated to have great potential for science and engineering applications. From the frequency dependence of the forward transmission coefficient jS21j,
it was found that the resonating frequency split into a dual-mode by introducing a geometric gap. The equivalent mutual-inductance-circuit model was proposed to delineate the occurrence of the dual stationary modes. Taking advantage of the characteristic features of the device, some intriguing results of the temperature and frequency depen-dences of the physical properties were obtained in the sam-ples with various oxygen contents. Some salient results are listed below:
1. For fully oxygenated case (d = 0.05), the resonator exhibits a quality factor Q > 104 around 15 K, and Dk(T) = k(T) k(5 K) displays a linear behavior at T < 35 K for the perfect ring resonator (f = 3.61 GHz).
But for the ring resonator with a split gap
(f = 1.80 GHz and f = 5.33 GHz), the linear-T depen-dent range of Dk(T) becomes smaller (T < 25 K). With increasing d (e.g. d = 0.2, 0.4), although Dk is still linear in temperature, the slope changes with increasing oxy-gen deficiency.
2. Following the model suggested by Scalapino, the energy gap could be derived and the value of the ratio of energy gap to the critical temperature, we found 2D(0)/kBTc,
around 6.0 ± 0.6, which is independent of the doping concentrations.
3. The function of k(T) versus the normalized temperature T/Tcexhibits a universal form. It implies a unique
high-Tc mechanism in the underdoped cuprate, which is
occurred in the CuO2planes only. This fact is consistent
with the observations of lSR data by Tallon et al.[19]. The values of k(5 K) with various oxygen contents are 150 ± 14 (d = 0.05), 216 ± 16 (d = 0.2) and 282 ± 20 nm (d = 0.4), respectively and found to be indepen-dent of frequencies.
4. The Fermi-liquid correction factor a2predicted by Lee and Wen’s model can be obtained with the values between 0.4 and 0.6 from the optimum (d = 0.05) to the underdoped (d = 0.4) cuprates. The result gives a justification of the quasiparticle as Fermi-liquid in the superconducting state.
Acknowledgement
This work was supported by the National Science Coun-cil of Taiwan, ROC under grant: NSC 89-2112-M-009-028. References
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[30] In Physica C 364–365 (2001) 408 we have reported a value of a2= 0.95–1.07 for d = 0.4, which appeared to be in contradiction to our assertion of noting that a2is always smaller than 1. We note here that the discrepancy was mainly due to the limited available data onvF
v2
at that time. The experimental data reported by Sutherland et al.[29]
showed that vF
v2 is strongly doping dependent and decreases with
decreasing oxygen contents. With the correction made by using more accuratevF
v2values at each d, the values of a
2are indeed all well below 1
