Commun. Theor. Phys. (Beijing, China) 49 (2008) pp. 1631–1634 c Chinese Physical Society Vol. 49, No. 6, June 15, 2008 An Extended Model for Coexistence of Superconductivity and Paramagnetism in High-Tc Superconductors F. Inanir Department of Physics, Rize University, Rize, 53100, Turkey (Received November 7, 2007) Abstract In the present work, the total magnetization in superconducting state is separated into critical state and paramagnetic components in terms of an H(x)-dependent magnetic flux density. Utilizing this model, we reproduce successfully M -H curves measured by Sandu et al. [Phys. Rev. B 74 (2006) 184511] and Sandu et al. [J. Supercond. Incorp. Novel Magn. 17 (2004) 701] for different forms of Jc . PACS numbers: 74.72.-h, 74.25.Ha, 74.25.Qt Key words: superconductivity, paramagnetism, high-Tc superconductor, critical-state model 1 Introduction 2 Result and Discussion Important characteristics of hard type-II superconducting materials such as critical fields, Meissner expulsion, trapped flux, hysteresis losses, interplay of superconductivity, and magnetism, etc., can be determined from their magnetization curves. The critical state model (CSM) has extensively been employed to analyze magnetization data on these materials. It gives the relationship between the measured irreversible magnetization Mirr and the bulk supercurrent density J introduced by Bean.[1,2] This model is based on two simple assumptions: (i) the supercurrent density is defined by a critical current density Jc , and (ii) any changes in the flux distribution are introduced at the sample surface. The Bean model takes Jc to be independent of local flux density B. Later, various forms of Jc , which is dependent on B have been proposed by others.[3−10] Many workers have reported that the magnetic moment of the various type-II superconductors especially containing paramagnetic ions changes its sign at some value of the increasing magnetic field after zero-field cool (ZFC) processes.[11−27] This unusual behavior has generally been attributed to the coexistence of superconductivity and paramagnetism. Common Bean’s CSM[1,2] is insufficient to explain these observations and to reproduce experimental M -H curves with this unusual magnetic crossing. To the best of our knowledge, some theoretical calculations have been presented in literature to reproduce this abnormality based on CSM. For instance, Qin et al. and Hari Babu et al. employed the critical state model in combination with the standard theory of paramagnetism with the extended Brillouin function and Fisher et al. have developed two velocity hydrodynamic model of the CSM. In this paper, we extend our approach to calculate the families of magnetization curves by considering the influence of volume currents on local paramagnetism based on critical state model and compare our values with the available magnetization data in the literature. The experimental results have been justified quite well in the frame work of our model by taking into account proper field dependence of Jc (B). 2.1 Outline of the Model We have considered a type-II superconducting slab with thickness 2D (−D < x < D), which is infinite in the y- and z-directions, initially cooled to low temperature in the absence of a magnetic field. The thickness of the slab is assumed to be much larger than London penetration depth λ. The influence of the demagnetization effect and surface barrier effects are neglected in the present treatment since surface barrier effect does not essentially cause an explicit deformation to the M (H) curves.[29−31] The magnetic field is applied parallel to the wide faces of the slab. The magnetic field penetrates into the superconductor and induces shielding current in the regions of flux penetration. The flux penetrated regions carry a shielding current density Jc determined by Ampére’s law, ∂B = ±µJc . (1) ∂x Here, µ = µ0 (1 + χ) is permeability of material, where µ0 is permeability of vacuum and χ is magnetic susceptibility. χ > 0 corresponds to paramagnetism and χ < 0 for diamagnetism. Jc is a function of B(x) = µ0 (1 + χ)H(x) and thus, paramagnetism and supercurrents can interact with each other. The sign ± is determined by the slope of the flux density in the specimen, i.e., the direction to which the critical current flows. The boundary condition for Eq. (1) is B(x = 0) = B(x = ±D) = µHa . From the symmetry, it is sufficient to consider only the half-part of the slab. Following critical state approach,[1,2] magnetization M for an idealized slab arising from the flux produced by the external magnetic field is calculated by Z D 1 M= B(x) dx − Ba , (2) µ0 D 0 where D is the thickness of the slab, B(x) is the local flux density permeating the slab transverse to the x axis and Ba is the applied field with Ba = µ0 Ha . Substituting the definition B(x) = µ0 (1 + χ)H(x) into Eq. (2), we have Z Z D 1 D 1 M= H(x) dx − Ha + χ H(x) dx . (3) D 0 D 0 RD The part (1/D) 0 H(x) dx − Ha in Eq. (3) defines the superconducting critical state magnetization. The second integral on the right-hand side of Eq. (3) is the measure of a paramagnetic or strong diamagnetic contribution to the superconducting magnetization. In this picture, total magnetization contains both irreversible critical state 1632 F. Inanir magnetization Mcrt and normal state-like magnetization (paramagnetism and diamagnetism) Mnorm . Vol. 49 The coexistence of superconductivity and paramagnetism of single crystal Y0.47 Pr0.53 Ba2 Cu3 07−δ specimen was investigated by Sandu et al. Figure 1 displays the magnetization versus applied magnetic field up to 1 kOe at 2 K reproduced from Sandu et al. The most prominent feature of the data of Sandu et al. is the migration of the magnetization curves from the negative (diamagnetic) part to the positive (paramagnetic) part at certain value of the increasing magnetic fields. Our primary interest in this analysis is to reproduce the change of this sign as a function of applied field. The hysteretic loop is typical for the superconducting systems coexisting with local paramagnetic moments. They have stated that total magnetization loop is the superposition of superconducting magnetization and paramagnetic moment due to Pr ions. Experimental (circle) and the best fitted curve (full line) are displayed in Fig. 2. The experimental data in Fig. 2 were carefully extracted from Ref. . The hysteresis loop shows a little indication of reversible component in the magnetization. The experimental data could be fitted directly by the model. Evidently, the model reproduces successfully the entire hysteretic behaviour and the crossing from diamagnetism to paramagnetism. The experimental values can be generated theoretically quite well by employing exponential critical state model,[8,32] |B| Jc (B) = Jc0 exp − , (4) B0 which leads to the description of many observations of hysteretic magnetization curves in high Tc superconductor. Substituting Eq. (4) into Eq. (1), for the given boundary condition, we derive the flux profile equation as follows: x i 1 h , (5) H± (x) = ln exp(p Ha ) ∓ p 1 − p D where p is called the pinning parameter which determines the order of field dependence of the critical current, B = µ H, and Ha denotes the magnetic flux density at the specimen surface. The parameters Jc0 and H0 are related to Hp and p by Hp = (1/p) ln(1 + p) and p = Jc0 D/H0 , in which Jc0 and H0 are temperature-dependent. Here, Hp is the full penetration field, i.e. the field required to penetrate the sample up to the center. Chen et al. have given expressions for M -H curves of type II superconductors based on exponential model. The geometry for our calculation is considered as an infinite slab and therefore the correlation of the demagnetization field was omitted. Theoretical contribution of χhH(x)i-dependent paramagnetic magnetization to the critical state magnetization can be visually determined by comparing each pair of adjacent curves displayed in Fig. 3. Fig. 2 Observed magnetization data extracted from Ref.  (open circle) together with a fitted model behaviour (full line). The fitting curve was calculated employing Eq. (3) and exponential CSM (Eqs. (4) and (5)). The parameters are: p = Jc0 D/H0 = 0.2, χ = 0.075. Fig. 3 Effect of paramagnetic magnetization on superconducting one. The upper magnetization curve (black line) of Fig. 2 and the curve (gray line) calculated by the same input as the former except that χ = 0, hence the paramagnetic magnetization component of Eq. (3) is absent. 2.2 Comparisons with Experimental Results (i) Coexistence of superconductivity and paramagnetism Fig. 1 Magnetization M versus applied field H curve in a single crystal Y0.47 Pr0.53 Ba2 Cu3 O7−δ reproduced from Sandu et al. The measurement was carried out at 2 K in increasing and decreasing magnetic field up to 1 kOe, perpendicular to the ab plane of the specimen. (ii) Coexistence of superconductivity and paramagnetism with Meissner diamagnetism: observations of Sandu et al. No. 6 An Extended Model for Coexistence of Superconductivity and Paramagnetism in High-Tc Superconductors 1633 Figure 4 reproduces the measurements of M -H loop at 65 K (Tc = 90 K on a polycrystalline Eu0.7 Sm0.3 Ba2 Cu3 O7−δ reported by Sandu et al. (Fig. 1 of Ref. ). The irreversible magnetization chances sign (negative to positive) at the certain value of the increasing magnetic field after ZFC and recovers the negative values for the reverse cycle. We have confined ourselves to generate theoretically the anomalous sign alternations in the experiment employing our approach. From the behavior of the curve it is deduced that total magnetization comprises two contributions to the superconducting magnetization. One of them is paramagnetic magnetization which arises from paramagnetic Eu ions, that is responsible for the sign alternation (negative to positive) in the ascending branch. Another one is the reversible or equilibrium magnetization which causes the second sign change in the descending branch (positive to negative), some downward displacement and some reversible part in curves (see the inset Fig. 4). Fig. 4 Magnetization curve measured by Sandu et al. on a Eu0.7 Sm0.3 Ba2 Cu3 O7−δ at 65 K. The dotted line represents equilibrium magnetization Meq calculated by Eq. (3) in that reference. This displays their observations of sign alternations in magnetic moment in forward and reverse field cycle. Inset: M -H loop at 80 K (lower curve), and at 86 K (upper curve). Fig. 5 Corresponding curve of Fig. 4 calculated via Eqs. (3), (6), and (7), introducing contribution of Meissner current. The parameters are: n = 0.4, χ = 0.000 95, Hc1 = 0.5 H∗ , H0 = H∗ , s = 0.2. We have adopted the generalized Kim’s critical state model[3,4] to fit experimental data. In this model, Jc (B) is given by Jc (B) = Jc0 /[(1 + B/B0 )n ] , (6) where B0 and n are phenomenological parameters and Jc0 is the critical current density at zero magnetic field. Chen and Goldfarb calculated the magnetization curves of an infinitely long orthorhombic specimen with finite rectangular cross section for n = 1. Figure 5 shows the corresponding curve combining Eq. (5) and the Kim’s critical state model (Eq. (6)) including Meissner current, IM = Mrev = (µ0 Ha − µ Hs ). Here, IM denotes the diamagnetic Meissner surface current flowing perpendicularly to Ha per unit dimension along Ha , and Mrev denotes the reversible Abrikosov diamagnetism. Again, the magnetic inductions are normalized with respect to H∗ = Jc0 D and the calculated magnetization is normalized accordingly. Over the range 0 ≤ Ha ≤ Hc1 , IM = µ0 Ha , and Hs = 0 for all previous histories of µ0 Ha , hence for all configurations of H(x). For the case Hc1 < Ha < Hc2 , we employ the simple expression,[34−36] s+1 IM = Hc1 /Has , (7) where s is Meissner exponent, Hc1 is the lower critical fields and both are regarded as temperature-dependent parameters. The influence of paramagnetic and Meissner diamagnetism contribution to the shape of CSM magnetization of the specimen can be visually assessed comparing each pair of adjacent curves displayed in Fig. 6. In these calculations, the adjustable parameters are n, s, and Hc1 . The parameters χ and Hc1 /H∗ are adopted in order to produce the observed magnetic behaviors of the specimen. The parameter n for bulk current and exponent s for Meissner current are assessed by various trials to obtain the best fit of the calculated and experimental curves. We find that the calculated magnetizations are nearly equal to the experimental values. But the pattern of the calculated M -H loop exhibits some little deviation from the experimental data at the lower values of the reverse field cycle. We adopted Kim-type dependence of the critical current density to fit experimental data since it provides smooth descending function with a wide change in range. However, both Kim’s and exponential models are not perfect to fit experimental data. Therefore, more complex Jc (B) functions are required to improve the results. In addition, we note that it is complicate to analyze the magnetic behaviour of the granular superconductor samples since there are many other factors effecting material data, such as size, shape and orientation of the grains, the current circulating inside and between the grains, flux pinning properties of grain and matrix etc. Overall, it is remarkable that theoretical curves employing a simple expression for the dependence of the critical current density Jc (B) on the magnetic field and temperature reproduce striking features of the experimental data pretty well as seen from Figs. 4 and 5. 1634 F. Inanir Vol. 49 Our treatment especially chances the results at low fields when it is compared to the previous works that the paramagnetism is directly calculated from the applied field (see Refs.  and ). At the initial increase of field, a low-field diamagnetic peak appears in all measurements (see Figs. 1 and 4), which is generally typical of magnetization of type-II superconductors. This low field peak can be reproduced quite well by our treatment, besides migration from negative magnetic moment to positive magnetic moment can be generated. The influence of the surface barrier effects are neglected in the present treatment, since surface barrier effect does not cause an explicit deformation to the M (H) curves, but merely expands them up and down. Fig. 6 Influence of Meissner surface current (b) and paramagnetic magnetization (c) to the calculated M -H curves. The parameters are (a) χ = 0, Hc1 = 0 (b) χ = 0, Hc1 = 0.5 H∗ , and (c) χ = 0.000 95, Hc1 = 0.5 H∗ . Other parameters are the same as adopted in Fig. 5. 3 Conclusions The sign change of magnetic moment in the M -H loop of several high-Tc superconductors owing to the coexistence of superconductivity and paramagnetism could be described quite well by critical state combined with a normal state χH(x)-dependent paramagnetic contribution to the magnetization. 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