Unconventional magnetic properties of cuprates

Unconventional magnetic properties of cuprates

J.L. González; E.S. Caixeiro; E.V.L. de Mello

Departamento de Física, Universidade Federal Fluminense, Niterói, RJ 24210-340, Brazil

ABSTRACT

Recently experiments on high critical temperature superconductors have shown that the doping levels and the superconducting gap are usually not uniform properties but strongly dependent on their positions inside a given sample. We show here that the large diamagnetic signal above the critical temperature T_c and the unusual temperature dependence of the upper critical field H_c2 with the temperature can be explained taking the inomo-geneities and a distribution of different local critical temperatures into account.

There are increasing evidences that high critical temperature superconductors (HTSC) are intrinsic inhomogeneous materials. This is probably the cause of several unconventional behavior. In particular, recent magnetic imaging through a scanning superconducting quantum device (SQUID) microscopy has displayed a static Meissner effect at temperatures as large as three times the T_c of an underdoped LSCO film[1]. Following up SQUID magnetization measurements on powder oriented YBCO and LSCO single crystals[2, 3] have shown a rather high magnetic response which, due to its large signal and structure, cannot be attributed solely to the Ginzburg-Landau (GL) theory of fluctuating superconducting magnetization[4, 8]. On the other hand the H-T phase diagram of the HTSC possess, in certain cases, positive curvature for H_c2(T), with no evidence of saturation at low temperatures [5]. These lack of saturation at low temperatures may minimize the importance of strong fluctuations of the order parameter.

In this paper we develop a unified view for all these anomalous properties. The basic ideas are[,]: the charge distribution inside a HSCT is highly inhomogeneous and may be divided in two types. A hole-poor branch which represents the AF domains and a hole-rich which characterizes the metallic regions. The width of the metallic distribution decreases with the average doping since usually, the samples becomes more homogeneous as the average doping level or average charge density increases. Due to the spatially varying local charge density, it is also expected that the T_c, instead of being a single value as in usual metallic superconductors, becomes locally dependent. Therefore a given HTSC compound with an average charge density n_m possess a distribution of charge density n(r), zero temperature superconducting gap D₀(r) and superconducting critical temperature T_c(r) where the symbol (r) means a point inside the sample. In this scenario we identify the largest T_c(r) with the the pseudogap temperature T^* of the compound[11]. Metallic domains with low (high) doping level have high (low) T_c(r). Upon cooling below T^* the superconducting regions develop at isolated regions as droplets of rain in the air and, eventually they percolates at the superconducting critical temperature T_c of the compound at which superconducting long range order is established.

1 The Model for the Magnetization

In order to estimate the M(B) we follow the ideas and the procedures of the Critical State Model (CSM) to each superconducting droplet. Upon applying an external magnetic field, a critical current (J_c) is established which opposes the field as J_c(B) = a(T)/B according to Ohmer et al[6].

For simplicity we take these superconducting droplets as cylinders of radius R, which is sufficient small in order to have a constant charge density n and consequently the critical temperature T_c(n) is the same within such cylinder region (As the temperature decreases, more droplets appear and the superconducting regions increase by aggregation of droplets of different n). The CSM approach leads to the magnetic field dependence of the magnetization in each small cylinder as[7]:

Since T_c(n) is constant inside a superconducting cylindrical droplet, the critical fields (B_c1 and B_c2) inside the droplets will have their temperature dependence according to the GL theory, e. g., B_c1(T) = B_c1(0)[(1-T/T_c(n)] and B_c2(T) = B_c2(0)[(1-T/T_c(n)]. Taking the dependence of T_c(n) on n as a linear relation, namely T_c(n) = T₀ –b (n – n_c), where n_c = 0.05 is the onset of superconductivity and T₀ is its maximum value, we arrive at the expressions for the critical fields B_c1(T,n) and B_c2(T,n). A similar functional form is supposed for B^* due to the a(T) temperature dependence. Thus,

When a given sample is submitted to an applied external magnetic field B, the superconducting droplets with carrier concentration n for which the applied field is higher than their second critical field B_c2(T,n) = B_c2(0)[(1-T/(T₀ – b*(n – n_c)], do not contribute to the sample magnetization. This condition is verified for droplets for which n > n_max, where n_max(B_c2) = n_c+T₀/b–(T/b)/[1 – B/B_c2(0)] is obtained inverting Eq.5. Since T_c(n) is a decreasing function of n, only the droplets with n bigger than n_max do not contribute to the sample's magnetization because their superconductivity is destroyed by the field B. Thus we expect that

Where P(n) is the distribution function for the charge level inside a given HTCS inferred in Ref.[10]. However, in the context of the CSM, depending on the intensity of the applied field, there are different possibilities in which each domain contributes to M(B). In the low field regime the superconducting clusters will contribute to the magnetization of the sample in three forms: there are some clusters, which are not penetrated by the field B, that is B < B_c1(T,n) and they contribute to the magnetization with perfect diamagnetism (Eq.1). These clusters have their carrier concentration in the interval, n < nc +T₀/b–(T/b)/[1–B/B_c1(0)]. The second group of clusters have their B_c1(T,n) lower than the applied field but B is also lower than B^*(n,T). This group is partially penetrated by the field and they contribute to M(B) according Eq.2. These domains have their carrier concentration in the interval n_c+T₀/b–(T/b)/[1–B/B_c1(0)] < n < n_c + T₀/b – (T/b)/[1 – B/B^*(0)]. Lastly, there are some superconducting granules for which the applied field is higher than B^*(T,n) but also lower than B_c2(T,n). These domains contribute to the magnetization according Eq.3. Therefore, for a sufficient low applied field, the general expression for the M(B) is given by:

The above theory was developed to model the measured magnetization curves of the La_1–xSr_xCuO₄ family of compounds. In Fig. 1 we plot the results of our model with the parameter which corresponds to a n_m = 0.1. The qualitative features of the measurements are entirely reproduced and are simply explained by our model; at low fields the perfect diamagnetism is expected for droplets for which the fields are lower than their B_c1. We expect B_c1 to be weak because the superconducting regions formed above T_c are small and isolated and the penetration depth l is large for HTCS. By the same token, the droplets penetrating field B^* should not be very strong what decreases rapidly the overall diamagnetic signal for field much weaker than B_c2. As the applied field increases, the magnetic response dies off and is reduced to the fluctuations. This is the reason why M(B) has a minimum at very low applied fields. In order to obtain a reliable value of M(B) and to compare with the experimental results, we have incorporated the fluctuation magnetizations induced by the superconducting order parameter, an effect which should be always present, regardless whether the superconductor is more or less inhomogeneous. As noted in reference[7], for superconducting droplets with a homogeneous order parameter and with dimensions d approximately equal to the coherent length x(T), the Ginzburg-Landau model provides an exact solution for M_fluct(B). Here we use a simplified "zero-dimensional" for superconducting clusters of radius d smaller or near the coherence length x(T), namely[2]:

where k_B is the Boltzmann constant, F₀ is the quantum flux and d » x µ (T – T_c)/T_c. This last expression yields a linear M_fluct(B) dependence for low fields and it has been incorporated in our calculations. The specific results for M_fluct are shown in the inset of Fig.1.

We can see that the up-turn field is near B_up = 0.001T which agrees with the experimental values[3]. It is worthwhile to mention that previously estimation for the up-turn field considering only the Lawrence-Doniach fluctuations[4] in a layered superconductor[2] yields expected values near B_up = 10T. These figures bring out the importance of the CSM applied to the superconducting islands above T_c to explain the experimental results.

2 The Model for H_c2

In the case of an external magnetic field parallel to the c-direction, i.e. perpendicular to the CuO₂ planes (ab-direction), the Ginzburg-Landau (GL) upper critical field may be given by[7]

where F₀ = hc/ 2e is the flux quantum and x_ab(T) is the GL temperature dependent coherence length in the ab plane. In terms of the GL parameters the coher ence length is given

where (0)=²/2m_abaT_c is the extrapolated coherence length, m_ab is the part of the mass tensor for the ab plane and a is a constant[12]. Therefore,

For the LSCO series a coherence length of x_ab(0)=30Å was adopted.

Now, assuming that each isolated or connected superconducting region displays a local which is given by the above linearized GL equation, the total H_c2 is the sum of these contributions. Since a given local superconducting region ''i'' has a local temperature T_c(i) and probability P_i, it will contribute to the upper critical field with a local linear upper critical field (T) near T_c(i). Therefore, the total contribution of the local superconducting regions to the upper critical field is the sum of all the (T)'s. Thus, applying Eq.(12), the H_c2 for an entire sample is

where N the number of superconducting regions, or superconducting islands each with its local T_c(i) < T_c and W = P_i is the sum of all the P_i's. As we have already mentioned, at temperatures above T_c there are isolated superconducting regions, while below T_c these regions percolate and the system may hold a dissipationless current. Since H_c2 is experimentally measured at T < T_c(H=0), it is the field which destroys the superconducting clusters with T < T_c(i) < T_c, leading the system without percolation. The first superconducting regions which are destroyed by the external field are the weakest ones, that is, those which have critical temperatures T_c(i)'s lower than the critical temperature T_c(H=0).

The mechanism is the following: at a temperature T < T_c most of the system is superconducting and a small applied field destroys first the superconducting regions at lower T_c(i)'s, without loss of long range order. Increasing the applied field causes more regions to become normal and eventually when the regions with T_c(i) » T_c turn to the normal phase, the system is about to have a nonvanishing resistivity. This value of the applied field is taken as the H_c2 in our theory, and it is the physical meaning of Eq.(13). Thus, at a given temperature T, we sum the superconducting regions with < T_c(i) < T_c, with their respective probabilities.

The experimental upper critical field H_c2 of the HTSC may be obtained from the resistivity measurements as it is the field relative to a fraction of the ''normal-state'' resistivity [5]. By definition, H_onset from the resistive measurements is defined as the magnetic field at which the resistivity r first is detected to deviate from the zero in the r vs H curves, and this is the assumption used in Eq.(13) and, therefore, it is our definition of H_c2(T). Below we plot H_c2(T) with the measured H_onset for the cases of n=0.15 (Fig. 2a) and n=0.17 (Fig. 2b) of the LSCO series. For the case of n=0.08 (Fig. 2c) we compared our results with H90 since this field vanishes at T_c » 24K, which is the value of T_c obtained from the phase diagram of Ref [10], while H_onset vanishes at T » 12K.

In Fig. 3 one can see the results for n=0.20 compared with H^* from the Nernst-signal measurements of Ref. [13, 14]. By the same token, for the Nernst signal[14] measurements of Ref. [14], H^* may be considered the upper critical field since it represents an intrinsic field which controls the onset of the flux-flow dissipation and vanishes at a temperature close to T_c. Therefore, H^* may be compared with H_onset. Also, in Fig. 2a we plot the results of a linear normalized charge distribution together with the bimodal distribution of Ref. [10]. As one can see, both distributions yield very similar results, which shows that the calculations do not depend on the details of the charge distribution.

3 Conclusion

We have been able to reproduce the qualitative features of two nonconventional behavior of HTSC: the unusual diamagnetic signal above T_c and the H_c2 dependence with the temperature. The basic hypothesis is the non-uniform distribution of charge which was introduced before and which was used to interpret the HTSC phase diagram. Our results demonstrated that the measured normal state magnetization curves, the B_c2 fields, Nernst signal and the STM magnetic imaging results may be interpreted through the formation of static superconducting islands at temperatures above the sample's T_c and below T^*.

Received on 23 May, 2003

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Publication Dates

• Publication in this collection
  25 Nov 2005
• Date of issue
  Dec 2003

History

• Received
  23 May 2003