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Frequency and amplitude responses of ac susceptibility in high temperature superconductors
Physics Letters A 339 (2005) 408–413 www.elsevier.com/locate/pla
Frequency and amplitude responses of ac susceptibility in high temperature superconductors S.L. Liu a,∗ , G.J. Wu a , X.B. Xu a , J. Wu a , H.M. Shao a , Y.M. Cai b , X.C. Jin b a Department of Physics and National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, PR China b Department of Physics, Nanjing University of Technology, Nanjing 210009, PR China
Received 20 January 2005; accepted 10 March 2005 Available online 19 March 2005 Communicated by J. Flouquet
Abstract According to the scaling function of vortex glass phase transition, a general description for the peak temperatures TP of frequency- and amplitude-dependent ac susceptibility has been deduced for high temperature superconductors. The description µ can be written as ln ωωP0 B n hac = A(1 − tP )q . This description is in good agreement of the experiments in our melt-textured YBCO and Pb doped Hg-1234 ceramic superconductors. The values of the exponents are estimated to be q = 3/2 and n = 1. And the comparison of our description with YBCO single crystal also sounds excellent. 2005 Elsevier B.V. All rights reserved. PACS: 74.60.Ge; 74.25.Fy; 74.72.Bk
1. Introduction Ac susceptibility measurement is widely used to study high temperature superconductors (HTSC) since a combination of an external magnetic dc field with a susceptibility method allows one to determine the spatial distribution of the penetrating dc field [1]. If an ac field hac is applied, Fourier transformation of the resulting time dependent magnetization M(t, hac ) yields the real and imaginary parts of the susceptibility * Corresponding author.
E-mail address: [email protected] (S.L. Liu). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.03.028
χ (ω) and χ (ω), respectively. Theoretically, various models, such as flux creep with an activation energy barrier U depending explicitly on current density J as U (J ) = (U0 /µ)[(J /Jc )µ − 1] [2], thermally assisted flux flow (TAFF) resulting from an Ohmic resistive state E = ρJ [3–6], and collective or individual elastic pinning within linear response [7], are introduced to interpret the nature of ac susceptibility. Different sample geometries, e.g., disks, cylinders, rings, and rectangles, are also taken into account [8–10]. It has been well known that the temperature dependent ac susceptibility shows, just below the critical temperature Tc , a sharp decrease in the real part and a peak in the imaginary part, representing energy losses. On
S.L. Liu et al. / Physics Letters A 339 (2005) 408–413
the experimental side, the frequency- and amplitudedependent peak temperature TP , has given several results. Tinkham [11] proposed that the resistance arising from thermally activated flux motion is the same as the case of the thermally activated phase motion in heavily damped Josephson junction, which is worked out by Ambergaokar and Halperin [12]. The resistance can be written as A(1 − t)3/2 −2 ρ = I0 , (1) ρn 2B where ρn is the normal state resistivity, I0 is the modified Bessel function, A is a constant parameter and t is the reduced temperature t = T /Tc . At the peak temperature TP , one has ρ/ρn = ωP /ω0 , with ω0 a characteristic frequency. For the activation energy U0 2kTP , taking into account ρ/ρn = B/Bc2 (T ), one has [13] ωP B (1 − tP ) ≈ , (2) Bc2 (T ) ω0 here tP = TP /Tc , and Bc2 (T ) is the upper critical field. While for the case U0 2kTP , the result is given as [13,14] B ω0 ln , (3) A ωP where A ∼ 4Jc0 /Tc , and Jc0 is the intrinsic Ginzburg– Landau (GL) depairing critical current density. It is well known that the peak temperature is ac amplitude dependent. While in these equations, it seems that the peak temperature is uninfluenced by hac . The ac amplitude dependent peak temperature has been given as [15] α hac ∝ T ∗ − TP , (4) (1 − tP )3/2 =
T∗
where is the peak temperature at the limit of zero amplitude. Likewise, this result cannot give a description of frequency dependent peak temperature. In this work, a general equation of the peak temperature for ac susceptibility will be deduced, and this equation will be applied in YBCO and Pb doped Hg-1234 superconductors.
2. Theory We start with the vortex glass phase transition theory proposed by Fisher et al. [16–19]. The phase tran-
409
sition of the mixed vortex state is favored by the large anisotropies, high temperatures, and extreme type II character [20]. The vortex solid state is characterized by a nonzero critical current density, while the vortex liquid is dissipative at all currents. The solid-to-liquid phase transition is most likely a first order melting transition in very clean systems, but turns into a second order vortex glass transition for highly disordered systems involving point defects or Bose glass transition in systems with corrected defects like ion-induced columnar defects or twin boundaries [21]. With the assumption of the coherence length ξg (T ) ∼ |T − Tg |−v with v the static critical exponent and the characteristic time scale τg ∝ ξgz , the scaling function of the dc current–voltage (I–V) characteristics for a vortex glass phase transition is scaled with (E/J )ξgz+1 ∝ F± J ξgd−1 , (5) where d is the dimensionality, E is the electrical field, J is the current density, and z is the dynamic critical exponent. Above the phase transition temperature Tg , at low current density limits F+ is constant, i.e., F+ ∼ 1, therefore the electrical field in vortex liquid phase can be written as E ∝ J |T − Tg |v(z+2−d) .
(6)
At the peak temperature of ac susceptibility, the magnetic field penetrates the sample completely, thus J (TP ) = Jc (TP ); and the inducted electrical filed is estimated as E = dB/dt ∼ ωP hac . Consequently, one has ωP hac = Jc |TP − Tg |v(z+2−d) ,
(7)
which agrees with the result calculated by Herzog et al. [8]. The critical current density Jc can be deduced as follows. Usually, the I–V characteristics can be described with the Anderson–Kim model [22], E = LBυ0 exp −U0 (T , B)/kT sinh(J BVc L/kT ), (8) where L is the width of the pinning well, υ0 is the hopping frequency, and Vc is the volume of the flux lattice surrounding that well. Inverting Eq. (6), one result yields Ec kT sinh−1 Jc = exp U0 (T , B)/kT , (9) BVc L LBυ0
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with a typical criterion of Ec = 10−5 V/m, υ0 = 108 Hz, L = 10−8 m, and Vc = 10−24 m3 . To estimate U0 (T , B), Anderson and Kim assumed that U0 (T , B) = (Hc2 /8π)ξ 3 , where Hc is thermodynamic critical field and ξ is the coherence length. Using the GL theory, an explicit expression for the temperature dependent activation energy gives [14] U0 (T , B) A(1 − t)3/2 = . (10) kT B On the other hand, based on the thermal activation equation, the flux flow resistivity is ρ = ρn exp −U0 (T , B, J )/kT , (11) where U0 (T , B, J ) is the activation energy. Solving this formula, one result gives ρn ω0 = kTP ln U0 (T , B, J ) = kT ln (12) . ρ ωP For convenience, we use the experiential relationship of the activation energy of the collective pining, which yields [14,23,24] U0 (T , B, J ) A(1 − t)q −µ (13) J , = kT Bn where µ is a glassy exponent and n is of order of unit. With the combination of these two equations, one has Bn ω0 −1/µ Jc = (14) ln . A(1 − tP )q ωP Consequently, the peak temperature of ac susceptibility can be written as ω0 n µ B hac = A(1 − tP )q , ln (15) ωP where B = Hdc + hac . This is the main equation of this Letter. This result also includes the descriptions of both Eqs. (3) and (4), thus it can be considered as a general description for the peak temperatures of frequency- and amplitude-dependent ac susceptibility for HTSCs.
3. Experimental details The melt-textured YBCO samples are prepared by a liquid-phase processing method. Details of the sample preparation have been published elsewhere [25]. The characteristics of large grain YBCO is very close
to single crystal. During the measurements, both the dc and ac fields are parallel to the c axis of the sample. The Pb doped Hg-1234 samples used in this study are prepared from an appropriate mixture of HgO, PbO, Ba, CaO and CuO with predetermined molar ration Hg : Pb : Ba : Ca : Cu = (1 − x) : x : 2 : 3 : 4, where x has values from 0 to 0.5 with an increment of 0.1. Using encapsulation and short-time annealing techniques, we have successfully fabricated a series of as-prepared ceramic superconductors of Hg1–x Pbx Ba2 Ca2 Cu4 Oy . These samples were then post-annealed at 240–280 ◦ C in the flowing O2 gas for several hours. The crystal structures of various phases contained in the specimen are characterized by the powder X-ray diffraction (XRD) technique using CuKα radiation. It is found that our specimen mainly consists of Hg-1234 phase from the XRD results [23]. The size of our samples is of ∼ 1 mm width and of ∼ 8 mm length. The average grain size of our samples is measured by scanning electron microscopy (SEM), and has a value of about 4 µm. The resistivity transition of these specimens is determined by using the conventional four-point probe method. The critical temperature of our specimen for x = 0.31 is measured to be 127 K.
4. Result and discussion The frequency- and amplitude-dependent ac susceptibility measurements are done for several dc fields for both YBCO and Hg-1234 superconductors. A typical result of the temperature dependent 4πχ is shown in Fig. 1 for both various frequencies and various ac amplitudes for melt-textured YBCO. We first devote to the frequency responses. The peaks at Tp , when hac fully penetrates into the sample, shift to higher values with increasing frequencies, which indicates the resistivity at Tp increases with increasing frequency, i.e., ρ/ρn = ωP /ω0 . According to Eq. (15), at a given dc field and a constant ac amplitude, a plot of ln(2πf ) vs. (1 − tP )q should be a straight line. Shown in Fig. 2 are the results of this plot for our Hg-1234 superconductors, where the value of q is chosen to be 3/2, similar with other superconductors, i.e., Rb3 C60 [14]. The results of various current densities are similar and in good agreement of our description. The frequency dependent peak temperatures
S.L. Liu et al. / Physics Letters A 339 (2005) 408–413
Fig. 1. (a) Ac susceptibility for YBCO melt-textured samples at different frequencies: 9.21, 32.1, 91.1, 377, 577, 1771 Hz (from left to right). (b) Ac susceptibility for YBCO melt-textured samples at various ac amplitudes: 16, 2, 0.2, 0.02 G (from left to right).
of melt-textured YBCO are shown in Fig. 3. In the inset of this plot is a same plot of YBCO twined crystal at a dc field of 12 T, taken from Ref. [26]. Now, we turn to discuss the ac amplitude dependent responses. Since the value of q is estimated to be 3/2, the slope of a plot of ln hac vs. ln(1 − tP ) should give the information about q/µ according to our description, if Hdc hac . Shown in Fig. 4 are two results for two different dc fields of our melt-textured YBCO. The estimated values of the slope are 10.1 for Hdc = 0.4 T and 8.6 for Hdc = 0.08 T, respectively. Similar results for various sample shapes of YBCO films are presented in Fig. 5, taken from Ref. [8]. It is interesting that the value of q/µ for our melt-textured sample is much larger that of films. According to the
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Fig. 2. (a) Plot of ln(2πf ) vs. (1 − tP )q for Pb doped Hg-1234 superconductors with various ac amplitudes under dc magnetic field Hdc = 40 mT. (b) Same plot for dc field Hdc = 0.5 T.
Fig. 3. Plot of ln(2πf ) vs. (1 − tP )q for YBCO melt-textured sample. Inset: same plot of YBCO twined crystal (Ref. [26]).
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Fig. 4. Log–log plot of hac vs. (1 − tP ) for melt-textured YBCO samples: left Hdc = 0.4 T, right Hdc = 0.08 T.
Fig. 5. Log–log plot of hac vs. (1 − tP ) for various shapes of YBCO films (Ref. [8]): a—disk with slope 1.7, b—rectangle with slope 1.9, c—25 µm wide ring with slope 1.57, d—50 µm ring with slope 1.15.
collective pinning theory, the current density dependent activation energy can be written as [27] Jc0 µ , U (J ) = U0 (T , B) (16) J where µ = 1/7, 3/2, and 7/9 for a single vortex, small bundles, and large bundles, respectively. Thus, for our melt-textured sample single vortex pinning dominates, while for films bundles pinning is considerable. For the dc magnetic field responses, in the region Bc1 B Bc2 , one has B ≈ Hdc . At constant frequency and ac amplitude, Eq. (15) is simplified as Hdc ∝ (1 − tP )q/n . A typical result is presented in
Fig. 6. Plot of Hdc vs. (1 − tP )3/2 for Pb doped Hg-1234 superconductor with various ac amplitudes.
Fig. 7. Scaling plot of E–J characteristic for YBCO melt-textured sample calculated from frequency response. The solid lines are the theoretical results of the scaling function of vortex glass phase transition.
Fig. 6 for our Pb doped Hg-1234 samples. The value of q/n is estimated to be 3/2, indicating n = 1. At the peak temperature TP , the inducted electrical field is estimated to be E = 2πf hac , and the current density can be calculated from Eq. (9). Therefore, the E–J characteristics are achieved from either frequency response or ac amplitude response. Based on the vortex glass phase transition, the E–J characteristics should obey the scaling function. One result of the scaling behavior is shown in Fig. 7. The estimated value of v(z − 1) is found to be 9.3, which is in good agreement with other scaling results [25]. The peak temperatures of ac susceptibility are in the flux flow region, while no point locates on the flux
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creep region, i.e., TP > Tg . This result indicates that at the peak temperatures the resistivity obeys ρ/ρn = |TP − Tg |v(z+2−d) , and F+ is constant in the Ohmic state.
5. Conclusion Based on the vortex glass phase transition theory, a general description of the peak temperature for frequency- and amplitude-dependent ac susceptibility for HTSCs has been deduced from the scaling function. The description is written as ω0 n µ B hac = A(1 − tP )q . ln ωP This description is in good agreement of the experiments in our melt-textured YBCO and Pb doped Hg1234 ceramic superconductors. The values of the exponents are estimated to be q = 3/2 and n = 1. And the comparison of our description with YBCO crystal and films also sounds excellent.
Acknowledgements The authors would like to express their gratitude to the Chinese National Center of Research and Development on superconductivity (grant 95-pre-06-9703) for their financial support. We also thank the anonymous referee for his constructive suggestion.
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Frequency and amplitude responses of ac susceptibility in high temperature superconductors S.L. Liu a,∗ , G.J. Wu a , X.B. Xu a , J. Wu a , H.M. Shao a , Y.M. Cai b , X.C. Jin b a Department of Physics and National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, PR China b Department of Physics, Nanjing University of Technology, Nanjing 210009, PR China
Received 20 January 2005; accepted 10 March 2005 Available online 19 March 2005 Communicated by J. Flouquet
Abstract According to the scaling function of vortex glass phase transition, a general description for the peak temperatures TP of frequency- and amplitude-dependent ac susceptibility has been deduced for high temperature superconductors. The description µ can be written as ln ωωP0 B n hac = A(1 − tP )q . This description is in good agreement of the experiments in our melt-textured YBCO and Pb doped Hg-1234 ceramic superconductors. The values of the exponents are estimated to be q = 3/2 and n = 1. And the comparison of our description with YBCO single crystal also sounds excellent. 2005 Elsevier B.V. All rights reserved. PACS: 74.60.Ge; 74.25.Fy; 74.72.Bk
1. Introduction Ac susceptibility measurement is widely used to study high temperature superconductors (HTSC) since a combination of an external magnetic dc field with a susceptibility method allows one to determine the spatial distribution of the penetrating dc field [1]. If an ac field hac is applied, Fourier transformation of the resulting time dependent magnetization M(t, hac ) yields the real and imaginary parts of the susceptibility * Corresponding author.
E-mail address: [email protected] (S.L. Liu). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.03.028
χ (ω) and χ (ω), respectively. Theoretically, various models, such as flux creep with an activation energy barrier U depending explicitly on current density J as U (J ) = (U0 /µ)[(J /Jc )µ − 1] [2], thermally assisted flux flow (TAFF) resulting from an Ohmic resistive state E = ρJ [3–6], and collective or individual elastic pinning within linear response [7], are introduced to interpret the nature of ac susceptibility. Different sample geometries, e.g., disks, cylinders, rings, and rectangles, are also taken into account [8–10]. It has been well known that the temperature dependent ac susceptibility shows, just below the critical temperature Tc , a sharp decrease in the real part and a peak in the imaginary part, representing energy losses. On
S.L. Liu et al. / Physics Letters A 339 (2005) 408–413
the experimental side, the frequency- and amplitudedependent peak temperature TP , has given several results. Tinkham [11] proposed that the resistance arising from thermally activated flux motion is the same as the case of the thermally activated phase motion in heavily damped Josephson junction, which is worked out by Ambergaokar and Halperin [12]. The resistance can be written as A(1 − t)3/2 −2 ρ = I0 , (1) ρn 2B where ρn is the normal state resistivity, I0 is the modified Bessel function, A is a constant parameter and t is the reduced temperature t = T /Tc . At the peak temperature TP , one has ρ/ρn = ωP /ω0 , with ω0 a characteristic frequency. For the activation energy U0 2kTP , taking into account ρ/ρn = B/Bc2 (T ), one has [13] ωP B (1 − tP ) ≈ , (2) Bc2 (T ) ω0 here tP = TP /Tc , and Bc2 (T ) is the upper critical field. While for the case U0 2kTP , the result is given as [13,14] B ω0 ln , (3) A ωP where A ∼ 4Jc0 /Tc , and Jc0 is the intrinsic Ginzburg– Landau (GL) depairing critical current density. It is well known that the peak temperature is ac amplitude dependent. While in these equations, it seems that the peak temperature is uninfluenced by hac . The ac amplitude dependent peak temperature has been given as [15] α hac ∝ T ∗ − TP , (4) (1 − tP )3/2 =
T∗
where is the peak temperature at the limit of zero amplitude. Likewise, this result cannot give a description of frequency dependent peak temperature. In this work, a general equation of the peak temperature for ac susceptibility will be deduced, and this equation will be applied in YBCO and Pb doped Hg-1234 superconductors.
2. Theory We start with the vortex glass phase transition theory proposed by Fisher et al. [16–19]. The phase tran-
409
sition of the mixed vortex state is favored by the large anisotropies, high temperatures, and extreme type II character [20]. The vortex solid state is characterized by a nonzero critical current density, while the vortex liquid is dissipative at all currents. The solid-to-liquid phase transition is most likely a first order melting transition in very clean systems, but turns into a second order vortex glass transition for highly disordered systems involving point defects or Bose glass transition in systems with corrected defects like ion-induced columnar defects or twin boundaries [21]. With the assumption of the coherence length ξg (T ) ∼ |T − Tg |−v with v the static critical exponent and the characteristic time scale τg ∝ ξgz , the scaling function of the dc current–voltage (I–V) characteristics for a vortex glass phase transition is scaled with (E/J )ξgz+1 ∝ F± J ξgd−1 , (5) where d is the dimensionality, E is the electrical field, J is the current density, and z is the dynamic critical exponent. Above the phase transition temperature Tg , at low current density limits F+ is constant, i.e., F+ ∼ 1, therefore the electrical field in vortex liquid phase can be written as E ∝ J |T − Tg |v(z+2−d) .
(6)
At the peak temperature of ac susceptibility, the magnetic field penetrates the sample completely, thus J (TP ) = Jc (TP ); and the inducted electrical filed is estimated as E = dB/dt ∼ ωP hac . Consequently, one has ωP hac = Jc |TP − Tg |v(z+2−d) ,
(7)
which agrees with the result calculated by Herzog et al. [8]. The critical current density Jc can be deduced as follows. Usually, the I–V characteristics can be described with the Anderson–Kim model [22], E = LBυ0 exp −U0 (T , B)/kT sinh(J BVc L/kT ), (8) where L is the width of the pinning well, υ0 is the hopping frequency, and Vc is the volume of the flux lattice surrounding that well. Inverting Eq. (6), one result yields Ec kT sinh−1 Jc = exp U0 (T , B)/kT , (9) BVc L LBυ0
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with a typical criterion of Ec = 10−5 V/m, υ0 = 108 Hz, L = 10−8 m, and Vc = 10−24 m3 . To estimate U0 (T , B), Anderson and Kim assumed that U0 (T , B) = (Hc2 /8π)ξ 3 , where Hc is thermodynamic critical field and ξ is the coherence length. Using the GL theory, an explicit expression for the temperature dependent activation energy gives [14] U0 (T , B) A(1 − t)3/2 = . (10) kT B On the other hand, based on the thermal activation equation, the flux flow resistivity is ρ = ρn exp −U0 (T , B, J )/kT , (11) where U0 (T , B, J ) is the activation energy. Solving this formula, one result gives ρn ω0 = kTP ln U0 (T , B, J ) = kT ln (12) . ρ ωP For convenience, we use the experiential relationship of the activation energy of the collective pining, which yields [14,23,24] U0 (T , B, J ) A(1 − t)q −µ (13) J , = kT Bn where µ is a glassy exponent and n is of order of unit. With the combination of these two equations, one has Bn ω0 −1/µ Jc = (14) ln . A(1 − tP )q ωP Consequently, the peak temperature of ac susceptibility can be written as ω0 n µ B hac = A(1 − tP )q , ln (15) ωP where B = Hdc + hac . This is the main equation of this Letter. This result also includes the descriptions of both Eqs. (3) and (4), thus it can be considered as a general description for the peak temperatures of frequency- and amplitude-dependent ac susceptibility for HTSCs.
3. Experimental details The melt-textured YBCO samples are prepared by a liquid-phase processing method. Details of the sample preparation have been published elsewhere [25]. The characteristics of large grain YBCO is very close
to single crystal. During the measurements, both the dc and ac fields are parallel to the c axis of the sample. The Pb doped Hg-1234 samples used in this study are prepared from an appropriate mixture of HgO, PbO, Ba, CaO and CuO with predetermined molar ration Hg : Pb : Ba : Ca : Cu = (1 − x) : x : 2 : 3 : 4, where x has values from 0 to 0.5 with an increment of 0.1. Using encapsulation and short-time annealing techniques, we have successfully fabricated a series of as-prepared ceramic superconductors of Hg1–x Pbx Ba2 Ca2 Cu4 Oy . These samples were then post-annealed at 240–280 ◦ C in the flowing O2 gas for several hours. The crystal structures of various phases contained in the specimen are characterized by the powder X-ray diffraction (XRD) technique using CuKα radiation. It is found that our specimen mainly consists of Hg-1234 phase from the XRD results [23]. The size of our samples is of ∼ 1 mm width and of ∼ 8 mm length. The average grain size of our samples is measured by scanning electron microscopy (SEM), and has a value of about 4 µm. The resistivity transition of these specimens is determined by using the conventional four-point probe method. The critical temperature of our specimen for x = 0.31 is measured to be 127 K.
4. Result and discussion The frequency- and amplitude-dependent ac susceptibility measurements are done for several dc fields for both YBCO and Hg-1234 superconductors. A typical result of the temperature dependent 4πχ is shown in Fig. 1 for both various frequencies and various ac amplitudes for melt-textured YBCO. We first devote to the frequency responses. The peaks at Tp , when hac fully penetrates into the sample, shift to higher values with increasing frequencies, which indicates the resistivity at Tp increases with increasing frequency, i.e., ρ/ρn = ωP /ω0 . According to Eq. (15), at a given dc field and a constant ac amplitude, a plot of ln(2πf ) vs. (1 − tP )q should be a straight line. Shown in Fig. 2 are the results of this plot for our Hg-1234 superconductors, where the value of q is chosen to be 3/2, similar with other superconductors, i.e., Rb3 C60 [14]. The results of various current densities are similar and in good agreement of our description. The frequency dependent peak temperatures
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Fig. 1. (a) Ac susceptibility for YBCO melt-textured samples at different frequencies: 9.21, 32.1, 91.1, 377, 577, 1771 Hz (from left to right). (b) Ac susceptibility for YBCO melt-textured samples at various ac amplitudes: 16, 2, 0.2, 0.02 G (from left to right).
of melt-textured YBCO are shown in Fig. 3. In the inset of this plot is a same plot of YBCO twined crystal at a dc field of 12 T, taken from Ref. [26]. Now, we turn to discuss the ac amplitude dependent responses. Since the value of q is estimated to be 3/2, the slope of a plot of ln hac vs. ln(1 − tP ) should give the information about q/µ according to our description, if Hdc hac . Shown in Fig. 4 are two results for two different dc fields of our melt-textured YBCO. The estimated values of the slope are 10.1 for Hdc = 0.4 T and 8.6 for Hdc = 0.08 T, respectively. Similar results for various sample shapes of YBCO films are presented in Fig. 5, taken from Ref. [8]. It is interesting that the value of q/µ for our melt-textured sample is much larger that of films. According to the
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Fig. 2. (a) Plot of ln(2πf ) vs. (1 − tP )q for Pb doped Hg-1234 superconductors with various ac amplitudes under dc magnetic field Hdc = 40 mT. (b) Same plot for dc field Hdc = 0.5 T.
Fig. 3. Plot of ln(2πf ) vs. (1 − tP )q for YBCO melt-textured sample. Inset: same plot of YBCO twined crystal (Ref. [26]).
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Fig. 4. Log–log plot of hac vs. (1 − tP ) for melt-textured YBCO samples: left Hdc = 0.4 T, right Hdc = 0.08 T.
Fig. 5. Log–log plot of hac vs. (1 − tP ) for various shapes of YBCO films (Ref. [8]): a—disk with slope 1.7, b—rectangle with slope 1.9, c—25 µm wide ring with slope 1.57, d—50 µm ring with slope 1.15.
collective pinning theory, the current density dependent activation energy can be written as [27] Jc0 µ , U (J ) = U0 (T , B) (16) J where µ = 1/7, 3/2, and 7/9 for a single vortex, small bundles, and large bundles, respectively. Thus, for our melt-textured sample single vortex pinning dominates, while for films bundles pinning is considerable. For the dc magnetic field responses, in the region Bc1 B Bc2 , one has B ≈ Hdc . At constant frequency and ac amplitude, Eq. (15) is simplified as Hdc ∝ (1 − tP )q/n . A typical result is presented in
Fig. 6. Plot of Hdc vs. (1 − tP )3/2 for Pb doped Hg-1234 superconductor with various ac amplitudes.
Fig. 7. Scaling plot of E–J characteristic for YBCO melt-textured sample calculated from frequency response. The solid lines are the theoretical results of the scaling function of vortex glass phase transition.
Fig. 6 for our Pb doped Hg-1234 samples. The value of q/n is estimated to be 3/2, indicating n = 1. At the peak temperature TP , the inducted electrical field is estimated to be E = 2πf hac , and the current density can be calculated from Eq. (9). Therefore, the E–J characteristics are achieved from either frequency response or ac amplitude response. Based on the vortex glass phase transition, the E–J characteristics should obey the scaling function. One result of the scaling behavior is shown in Fig. 7. The estimated value of v(z − 1) is found to be 9.3, which is in good agreement with other scaling results [25]. The peak temperatures of ac susceptibility are in the flux flow region, while no point locates on the flux
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creep region, i.e., TP > Tg . This result indicates that at the peak temperatures the resistivity obeys ρ/ρn = |TP − Tg |v(z+2−d) , and F+ is constant in the Ohmic state.
5. Conclusion Based on the vortex glass phase transition theory, a general description of the peak temperature for frequency- and amplitude-dependent ac susceptibility for HTSCs has been deduced from the scaling function. The description is written as ω0 n µ B hac = A(1 − tP )q . ln ωP This description is in good agreement of the experiments in our melt-textured YBCO and Pb doped Hg1234 ceramic superconductors. The values of the exponents are estimated to be q = 3/2 and n = 1. And the comparison of our description with YBCO crystal and films also sounds excellent.
Acknowledgements The authors would like to express their gratitude to the Chinese National Center of Research and Development on superconductivity (grant 95-pre-06-9703) for their financial support. We also thank the anonymous referee for his constructive suggestion.
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