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Magnetic susceptibility in the normal phase of Bi2Sr2CaCu2O8+δ single crystals
Physica B xxx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Magnetic susceptibility in the normal phase of Bi2Sr2CaCu2O8+δ single crystals ⁎
Lutiene F. Lopesa, , J. Paola Peñaa, Jacob Schafa, Milton A. Tumeleroa, Valdemar N. Vieirab, Paulo Pureura a b
Instituto de Física, Universidade Federal do Rio Grande do Sul, 90501-970 Porto Alegre, RS, Brazil Instituto de Física e Matemática, Universidade Federal de Pelotas, 96010-900 Pelotas, RS, Brazil
A R T I C L E I N F O
A BS T RAC T
Keywords: Pseudogap Magnetic susceptibility Single crystals Bi-2212
We report on measurements of the c-axis component of the magnetic susceptibility in the normal phase of several single crystal samples of the Bi2Sr2CaCu2O8+δ cuprate superconductor. These crystal were submitted to appropriate heat treatments so that the density of hole carriers could be varied in an extended region of the superconducting dome. In general, the measured susceptibility shows significant temperature dependence, which was attributed to the pseudogap phenomenon. The results were interpreted with basis on a phenomenological model that allows the determination of the pseudogap characteristic temperature T* as a function of the carrier density.
1. Introduction
Whereas some authors argue in terms of a phase boundary that delimitates a pseudogap state characterized by an order parameter distinct from the superconducting gap, others describe this line as a crossover from a normal metallic behavior that becomes gradually modified by some effect leading to a marked decrease in the singleelectron DOS at the Fermi level [2,3,5]. In this context, a debatable issue concerns the termination of the T*(p) line with respect to the superconducting dome. Some tunneling spectroscopy results [4] are in favor to a scenario where T*(p) joints Tc(p) in the overdoped region, the two curves becoming coincident in the point where the superconducting dome vanishes. These results lead to an interpretation of the pseudogap region as a precursor phase of the true superconducting state. That is, in the pseudogap region uncorrelated Cooper pairs would be stabilized without the long range phase coherence that characterizes the genuine superconducting state [2]. On the other hand, experimental techniques involving electrical transport [2], specific heat [6,7], nuclear magnetic resonance [8], neutron scattering [2] and many others are rather consistent with a scenario where T*(p) crosses the superconductor dome around the optimal doping maximum and extends down into the dome. Some authors argue that T*(p) falls to zero in a critical quantum point (QCP) situated at some critical hole concentration located near the optimal doping pc = 0.16. According to this view, the pseudogap is consequence of a phenomenon unrelated to the superconducting ordering and might be either favorable or detrimental to superconductivity in the HTSC systems [2].
The high Tc superconducting cuprates (HTSC) are characterized by temperature versus carrier concentration (T-p) phase diagrams showing universal features [1]. The parent compound for the hole-doped systems exhibits an antiferromagnetic ground state. The Néel ordering temperature falls quickly to zero when the parent compound is slightly hole doped. Increasing the carrier density up to p ≈ 0.05 lead to the stabilization of a superconducting phase at low temperatures [1]. Upon further augmentation of the hole density, the critical temperature Tc increases then goes through a maximum at p ≈ 0.16, which is identified as the optimal doping state, then decreases again describing the characteristic dome-like curve in the T-p diagram. For p < 0.16 the system is considered underdoped and for p > 0.16 is said to be overdoped [2,3]. In the underdoped side of the dome, different experimental techniques furnish solid evidences in favor of the occurrence of the so-called pseudogap phenomenon [2,4]. The pseudogap manifests itself as a depression of the electron density of states (DOS) around the Fermi energy, εF, that occurs below a characteristic temperature, T*, located in the normal state above the superconducting critical temperature. The pseudogap has been identified in large regions of the temperature - carrier concentration phase diagram of most HTSC. Although the large number of studies devoted to the characterization of this phenomenon, its microscopic origin remains unknown. Much controversy also arises about the precise location of the line T*(p) in the T-p phase diagram as well as the physical significance of this line. ⁎
Corresponding author. E-mail address: [email protected] (L.F. Lopes).
http://dx.doi.org/10.1016/j.physb.2017.09.002 Received 1 July 2017; Received in revised form 29 August 2017; Accepted 1 September 2017 0921-4526/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Lopes, L.F., Physica B (2017), http://dx.doi.org/10.1016/j.physb.2017.09.002
Physica B xxx (xxxx) xxx–xxx
L.F. Lopes et al.
3. Results and discussion
In this communication, we report on measurements of the c-axis component of the magnetic susceptibility (χc) in the normal phase of single crystal samples of Bi2Sr2CaCu2O8+δ (Bi-2212) with different hole concentrations. The carrier density was varied in such a way that a large extension of the T-p diagram, corresponding to the superconducting dome, could be explored. Results were analyzed with basis on a phenomenological model proposed by Naqib and Cooper [9,10] to describe susceptibility results in YBa2Cu3O7-δ, mostly in the underdoped region.
Typical magnetization versus field measurements in different temperatures are shown in Fig. 1. The c-axis susceptibility is diamagnetic in the studied temperature range (100 K < T < 300 K) for the most underdoped sample, p = 0.092, as shown in panel (a). As the carrier concentration increases, the diamagnetism is weakened. For p = 0.115 the susceptibility is diamagnetic in low temperatures but shows a crossover to a paramagnetic behavior in temperatures above T ≈ 200 K. The susceptibility is paramagnetic for concentrations above p = 0.127 in all temperatures, as exemplified by results in panel (b). In particular, for the optimally doped sample the susceptibility remains practically constant with temperature in the whole studied range between T = 100 K and T = 300 K. The evolution of the magnetic susceptibility with temperature and doping, as shown in Fig. 1, can be associated to an interplay between diamagnetic and paramagnetic susceptibility terms. The diamagnetism is dominant in the most underdoped samples and is mostly due the Langevin contribution form the core atomic electrons. The Landau diamagnetism is estimated to be approximately one order of magnitude smaller than Pauli paramagnetic term and gives a negligible contribution [13,14]. When the doping or temperature increases, the Pauli term eventually overcomes the diamagnetic response. The susceptibility magnitude when plotted as a function of the carrier density p shows a dome-like behavior in all temperatures, as illustrated by the representative results obtained at T = 140 K and shown in Fig. 2. Interesting enough, a peaked maximum is observed when the hole concentration corresponds to that of the optimum doping, pc ≈ 0.16. This feature is also reproduced in all investigated temperatures and might indicate that the electron density of states of Bi-2212 at εF goes through a maximum at this particular carrier concentration. Fig. 3 shows representative susceptibility versus temperature results for Bi-2212 crystals with different hole concentrations. In panels (a) and (b) results are for underdoped samples. In panel (c), measurements for an optimally doped sample are shown, whereas in panel (d) results depicted are for an overdoped crystal. In all cases χc is temperature dependent. For the underdoped and optimally doped crystals, the susceptibility increases with a negative curvature when the temperature is also increased. For the overdoped samples, χc increases linearly with T. This fact obviously makes difficult the description of the susceptibility in the normal phase of Bi-2212 based simply on the additive contributions of the usual Langevin diamagnetic term and the Pauli paramagnetic term. Considering that a similar temperature dependence of the susceptibility in polycrystalline YBa2Cu3O7-δ is an effect of the pseudogap phenomenon, Naqib and Cooper [9,10] proposed a description of the results with basis on an extension of the phenomenological model proposed by them to describe entropy measurements in this same superconducting cuprate [6]. Basically, the model assumes the validity of the Pauli formulation for the paramagnetic susceptibility of an electron gas which is given by
2. Materials and methods Several single crystals of Bi-2212 were grown by the self-flux method. High purity precursors Bi2O3, SrCO3, CaCO3 and CuO were weighed in the ratio 2.4:2:1:2 for Bi, Sr, Ca and Cu respectively. A mixture with total weight 10 g was placed in a conic alumina crucible, then submitted to a heat treatment based in the recipe found in reference [11]. Characterizations with x-ray diffraction and scanning electron microscopy (SEM) were carried out, confirming the good crystalline quality of the obtained crystals. The as-grown crystals have hole density close to the optimum value (maximum Tc). Crystals with different carrier concentrations were obtained by heat treatments aimed to modify the oxygen stoichiometry. Annealing in vacuum were employed to obtain underdoped samples, whereas annealing processes in the presence of oxygen pressure were used to obtain overdoped crystals. Special care was taken to assure the homogeneity of the final oxygen content in the prepared samples. Table 1 lists the measured critical temperatures, estimated hole densities, and parameters as the pressure (P), temperature (T) and time (t) used during the heat treatments employed in the preparation of the ten studied crystals. The magnetic measurements were performed with a Quantum Design XL5-MPMS@SQUID magnetometer. The magnetization versus field of each individual crystal was measured in the range − 50 kOe < H < + 50 kOe in several fixed temperatures. In all cases, the field was applied parallel to the c-axis of the crystals. The dc magnetic susceptibility was obtained by fitting the M versus H results to straight lines and determining their slopes. Due to the smallness of the samples magnetic signal in the normal phase, special care was taken to properly subtract the contribution of the sample holders made of pure Si. For each crystal, a unique and calibrated sample holder was used. In several cases, the magnetization versus field measurements were repeated in order to assure a good estimation of the samples susceptibility χc. The critical temperature for each sample was extracted from zero-field-cooling magnetization versus temperature measurements performed at H = 10 Oe. The intersection of two straight lines extrapolated from the experimental data in the normal and superconductor phases was taken as the location of Tc. From the so-obtained values for Tc, the hole concentration for each sample was estimated using the expression TC = TC, max [1 − 82.6( p − 0.16)2 ] [12].
Table 1 Parameters used in oxygenation and deoxygenation processes carried out in order to vary the carrier density, critical temperature and hole density for each studied single crystal. Single crystals
Tc (K)
p (hole/Cu-O2)
Doping State
Heat Treatment
1 2 3 4 5 6 7 8 9 10
56.1 62.8 72.8 79.9 83.0 90.8 91.4 88.3 85.8 82.6
0.092 0.099 0.115 0.121 0.127 0.152 0.160 0.181 0.187 0.194
Underdoped Underdoped Underdoped Underdoped Underdoped Optimal Optimal Overdoped Overdoped Overdoped
P = 5 × 10−2 Torr, T = 350 °C, t = 7 h P = 5 × 10−2 Torr, T = 345 °C, t = 7 h P = 6 × 10−2 Torr, T = 340 °C, t = 7 h P = 5 × 10−2 Torr, T = 320 °C, t = 7 h P = 6 × 10−2 Torr, T = 330 °C, t = 7 h as grown as grown as grown P(O2) = 1 kgf/cm2, T = 450 °C, t = 50 h P(O2) = 1 kgf/cm2, T = 470 °C, t = 50 h
2
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Fig. 1. Magnetization as a function of the applied field for Bi-2212 single crystals with different hole concentrations, (a) p = 0.092 and (b) p = 0.127 in several temperatures between 100 K and 300 K.
⎧ ⎫ 1 χNC = μ0 μB2 N0 ⎨1 − ln [ cosh(D )] ⎬, ⎩ ⎭ D
(3)
where D = Eg /2kB T , and kB is the Boltzmann constant. Fitting experimental data as those exemplified in Fig. 3 to
χC = χdia + χNC ,
where a constant and negative term χdia is added to the paramagnetic contribution given by Eq. (3), in order to account for the observed diamagnetism in the most underdoped crystals. The description of the data for the overdoped crystals with Eq. (4) was not possible, so that the observed temperature dependence in the susceptibility in these samples must be ascribed to a different origin. From the fittings of the experimental data for χc(T) to Eq. (4), values for the parameter D were extracted. Then, as proposed by Naqib and Cooper, the pseudogap temperature could be estimated as T*(p) = Eg(p)/kB. The values found for T* using this procedure are collected in the T-p diagram shown in Fig. 4. In this diagram, the superconducting dome defined by Tc(p) is also shown. The line defined by the values found for T*(p) based on the c-axis susceptibility is in quite good agreement with other experimental determinations for this boundary in Bi-2212 [2]. Indeed, the obtained T*(p) decreases sharply when the hole concentration approaches the optimum hole density pc when this point is approached from below. Interesting enough, a value for T* was obtained inside the superconducting dome, near pc. As indicated by the dashed line in the T-p diagram, this finding is consistent with the crossing of the T*(p) and Tc(p) boundaries near the optimum hole density. Our results thus suggest that the superconducting state and the pseudogap phenomenon have different origins and coexist in a certain region of the phase diagram. In addition, the fact that the results for χc in the overdoped crystals could not be fitted to Eq. (4) leads us to assume the pseudogap is closed in the corresponding region of the T-p diagram. As shown in panel (d) of Fig. 4, the paramagnetic susceptibility remains temperature dependent for the overdoped samples. Tentatively, we attribute this temperature dependence to the close proximity between the Fermi energy and a van Hove singularity in the DOS shown to occur in the slightly overdoped Bi-2212 from tunneling experiments [15].
Fig. 2. Magnitude of the c- axis component of the magnetic susceptibility as a function of the hole concentration for Bi-2212 single crystals, measured at T = 140 K.
χ = μ0 μB2
∫0
∞
⎛ ∂f ⎞ ⎜ − ⎟ N (ε ) dε, ⎝ ∂ε ⎠
(1)
where f is the Fermi-Dirac distribution function, N(ε) is the total density of states, μ0 is the vacuum permeability and μB is the Bohr magneton. The usual approximation consists to consider (−∂f /∂ε ) ≅ δ (ε − εF ) , resulting in the Pauli susceptibility, which is temperature independent Naqib and Cooper [9,10] proposed a DOS of the type
N (ε ) = N0 for ε − εF > Eg
(2.a)
N (ε ) = N0 ε − εF for ε − εF < Eg
(2.b)
(4)
where Eg is an energy scale related to the characteristic temperature T* that is considered to denote the opening of the pseudogap. When substituting expressions (2.a) and (2.b) in Eq. (1), these authors recalculate the susceptibility using the exact expression for (−∂f /∂ε ). They obtained the paramagnetic susceptibility given by 3
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Fig. 3. Representative c-axis susceptibility results as functions of the temperature for Bi-2212 single crystals with different hole concentration. Panels (a) and (b) show data for underdoped samples. Data for optimally doped and overdoped crystals are shown in panels (c) and (d), respectively. The full lines are fittings of the experimental data to Eq. (4) that includes the paramagnetic term given by Eq. (3).
phenomenon, which is more prominent in the underdoped regime. Applying to our data a phenomenological model for the electron paramagnetic susceptibility, proposed by Naqib and Cooper [9,10], we were able to obtain the boundary T*(p) that denotes the opening of the pseudogap in the T-p phase diagram for Bi-2212. This result is in good agreement with other estimations for this line. Our results indicate that the pseudogap line enters into the superconducting dome, suggesting that the pseudogap phenomenon and the superconducting state are different in origin. It is somewhat surprising that the c-axis component of the susceptibility in Bi-2212 is sensible to the pseudogap phenomenon. The relevance of the V-shaped DOS, as proposed in the Naqib-Cooper model, for the description of our χc(T) results might be an indication that the Bi-2212 system behaves as a zero gap semiconductor when probed along the c-axis. Acknowledgments This work was supported by Brazilian agency “Conselho Nacional de Pesquisas Científicas e Tecnológicas” (CNPq) (165419/2014-8).
Fig. 4. Boundaries Tc(p) and T*(p) = Eg(p)/kB for Bi-2212 single crystals. The values for Eg(p)/kB were obtained from fittings of the experimental data for χc(T) to Eq. (4) (see text). The experimental Tc(p) defining the superconducting dome are also shown.
References 4. Conclusions
[1] J.L. Tallon, C. Bernhard, H. Shaked, R.L. Hitterman, J.D. Jorgensen, Generic superconducting phase behavior in high-Tc cuprates: Tc variation with hole concentration in YBa2Cu3O7-δ, Phys. Rev. B 51 (1995) 12911. [2] J.L. Tallon, J.W. Loram, The doping dependence of T* – what is the real high-Tc phase diagram?, Physica C 359 (2001) 53–68. [3] M. Hashimoto, I.M. Vishik, R. He, T.P. Devereaux, Z. Shen, Energy gaps in hightransition-temperature cuprate superconductors, Rev. Nat. Phys. 10 (2014) 483–495. [4] C. Renner, B. Revaz, J. –Y. Genoud, K. Kadowaki, Q. Fisher, Pseudogap precursor of the superconducting gap in under-and overdoped Bi2Sr2CaCu2O8+δ, Phys. Rev. Lett. 80 (1998) 149.
We study experimentally the c-axis component of the magnetic susceptibility in the normal phase of Bi-2212 in a large region of the temperature – hole density phase diagram. The superconducting dome in the underdoped and overdoped regimes could be accessed. Usually, the unconventional dependence of the normal phase susceptibility with temperature and hole doping observed in the HTSC [13,14], as also observed in our Bi-2212 samples, has been attributed to the pseudogap 4
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susceptibility of Y1−xCaxBa2Cu3O7-δ, Physica C 460–462 (2007) 750–752. [11] W. Wenbin, L. Fanqing, J. Yunbo, Z. Guien, Q. Yitai, Q. Qiaonan, Z. Yuheng, Growth of Bi2Sr2CaCu2O8 single crystals from Bi-rich melts, Physica C 213 (1993) 133–138. [12] Z. Konstantinovic, Z.Z. Li, H. Raffy, Temperature dependence of the Hall effect in single-layer and bilayer Bi2Sr2Can-1CunOy thin films at various oxygen contents, Phys. Rev. B. 62 (2000) R11989–R11992. [13] D.B. Mitzi, L.W. Lombardo, A. Kapitulnik, S.S. Laderman, R.D. Jacowitz, Growth and properties of oxygen- and ion-doped Bi2Sr2CaCu2O8+δ single crystals, Phys. Rev. B 41 (1990) 6564–6574. [14] C. Allgeier, J.S. Schilling, Magnetic susceptibility in the normal state: a tool to optimize Tc within a given superconducting oxide system, Phys. Rev. B 48 (1993) 9747–9753. [15] A. Matsuda, S. Sugita, T. Fujii, T. Watanabe, Study of pseudogap by STM and other probes, J. Phys. Chem. Solids 62 (2001) 65–68.
[5] B. Keimer, S.A. Kivelson, M.R. Norman, S. Uchida, J. Zaanen, From quantum matter to high-temperature superconductivity in cooper oxides, Rev. Nat. 518 (2015) 179–186. [6] J.W. Loram, J. Luo, J.R. Cooper, W.Y. Liang, J.L. Tallon, Evidence on the pseudogap and condensate from electronic specific heat, J. Phys. Chem. Solids 62 (2001) 59–64. [7] W. Loram, K.A. Mirza, J.R. Cooper, W.Y. Liang, Electronic specific heat of YBa2Cu3O7-δ, Phys. Rev. Lett. 71 (1993) 1740–1743. [8] W.W. Warrem Jr., R.E. Walstedt, G.F. Brennert, R.J. Cava, R. Tycko, R.F. Bell, G. Dabbagh, Cu spin dynamics and superconducting precursor effects in planes above Tc in YBa2Cu3O7-δ, Phys. Rev. Lett. 62 (1989) 1193–1196. [9] S.H. Naqib, R.S. Islam, Extraction of the pseudogap energy scale from the static magnetic susceptibility of single and double CuO2 plane high-Tc cuprates, Supercond. Sci. Technol. 21 (2008) 105017–105021. [10] S.H. Naqib, J.R. Cooper, Effect of the pseudogap on the uniform magnetic
5
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Magnetic susceptibility in the normal phase of Bi2Sr2CaCu2O8+δ single crystals ⁎
Lutiene F. Lopesa, , J. Paola Peñaa, Jacob Schafa, Milton A. Tumeleroa, Valdemar N. Vieirab, Paulo Pureura a b
Instituto de Física, Universidade Federal do Rio Grande do Sul, 90501-970 Porto Alegre, RS, Brazil Instituto de Física e Matemática, Universidade Federal de Pelotas, 96010-900 Pelotas, RS, Brazil
A R T I C L E I N F O
A BS T RAC T
Keywords: Pseudogap Magnetic susceptibility Single crystals Bi-2212
We report on measurements of the c-axis component of the magnetic susceptibility in the normal phase of several single crystal samples of the Bi2Sr2CaCu2O8+δ cuprate superconductor. These crystal were submitted to appropriate heat treatments so that the density of hole carriers could be varied in an extended region of the superconducting dome. In general, the measured susceptibility shows significant temperature dependence, which was attributed to the pseudogap phenomenon. The results were interpreted with basis on a phenomenological model that allows the determination of the pseudogap characteristic temperature T* as a function of the carrier density.
1. Introduction
Whereas some authors argue in terms of a phase boundary that delimitates a pseudogap state characterized by an order parameter distinct from the superconducting gap, others describe this line as a crossover from a normal metallic behavior that becomes gradually modified by some effect leading to a marked decrease in the singleelectron DOS at the Fermi level [2,3,5]. In this context, a debatable issue concerns the termination of the T*(p) line with respect to the superconducting dome. Some tunneling spectroscopy results [4] are in favor to a scenario where T*(p) joints Tc(p) in the overdoped region, the two curves becoming coincident in the point where the superconducting dome vanishes. These results lead to an interpretation of the pseudogap region as a precursor phase of the true superconducting state. That is, in the pseudogap region uncorrelated Cooper pairs would be stabilized without the long range phase coherence that characterizes the genuine superconducting state [2]. On the other hand, experimental techniques involving electrical transport [2], specific heat [6,7], nuclear magnetic resonance [8], neutron scattering [2] and many others are rather consistent with a scenario where T*(p) crosses the superconductor dome around the optimal doping maximum and extends down into the dome. Some authors argue that T*(p) falls to zero in a critical quantum point (QCP) situated at some critical hole concentration located near the optimal doping pc = 0.16. According to this view, the pseudogap is consequence of a phenomenon unrelated to the superconducting ordering and might be either favorable or detrimental to superconductivity in the HTSC systems [2].
The high Tc superconducting cuprates (HTSC) are characterized by temperature versus carrier concentration (T-p) phase diagrams showing universal features [1]. The parent compound for the hole-doped systems exhibits an antiferromagnetic ground state. The Néel ordering temperature falls quickly to zero when the parent compound is slightly hole doped. Increasing the carrier density up to p ≈ 0.05 lead to the stabilization of a superconducting phase at low temperatures [1]. Upon further augmentation of the hole density, the critical temperature Tc increases then goes through a maximum at p ≈ 0.16, which is identified as the optimal doping state, then decreases again describing the characteristic dome-like curve in the T-p diagram. For p < 0.16 the system is considered underdoped and for p > 0.16 is said to be overdoped [2,3]. In the underdoped side of the dome, different experimental techniques furnish solid evidences in favor of the occurrence of the so-called pseudogap phenomenon [2,4]. The pseudogap manifests itself as a depression of the electron density of states (DOS) around the Fermi energy, εF, that occurs below a characteristic temperature, T*, located in the normal state above the superconducting critical temperature. The pseudogap has been identified in large regions of the temperature - carrier concentration phase diagram of most HTSC. Although the large number of studies devoted to the characterization of this phenomenon, its microscopic origin remains unknown. Much controversy also arises about the precise location of the line T*(p) in the T-p phase diagram as well as the physical significance of this line. ⁎
Corresponding author. E-mail address: [email protected] (L.F. Lopes).
http://dx.doi.org/10.1016/j.physb.2017.09.002 Received 1 July 2017; Received in revised form 29 August 2017; Accepted 1 September 2017 0921-4526/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Lopes, L.F., Physica B (2017), http://dx.doi.org/10.1016/j.physb.2017.09.002
Physica B xxx (xxxx) xxx–xxx
L.F. Lopes et al.
3. Results and discussion
In this communication, we report on measurements of the c-axis component of the magnetic susceptibility (χc) in the normal phase of single crystal samples of Bi2Sr2CaCu2O8+δ (Bi-2212) with different hole concentrations. The carrier density was varied in such a way that a large extension of the T-p diagram, corresponding to the superconducting dome, could be explored. Results were analyzed with basis on a phenomenological model proposed by Naqib and Cooper [9,10] to describe susceptibility results in YBa2Cu3O7-δ, mostly in the underdoped region.
Typical magnetization versus field measurements in different temperatures are shown in Fig. 1. The c-axis susceptibility is diamagnetic in the studied temperature range (100 K < T < 300 K) for the most underdoped sample, p = 0.092, as shown in panel (a). As the carrier concentration increases, the diamagnetism is weakened. For p = 0.115 the susceptibility is diamagnetic in low temperatures but shows a crossover to a paramagnetic behavior in temperatures above T ≈ 200 K. The susceptibility is paramagnetic for concentrations above p = 0.127 in all temperatures, as exemplified by results in panel (b). In particular, for the optimally doped sample the susceptibility remains practically constant with temperature in the whole studied range between T = 100 K and T = 300 K. The evolution of the magnetic susceptibility with temperature and doping, as shown in Fig. 1, can be associated to an interplay between diamagnetic and paramagnetic susceptibility terms. The diamagnetism is dominant in the most underdoped samples and is mostly due the Langevin contribution form the core atomic electrons. The Landau diamagnetism is estimated to be approximately one order of magnitude smaller than Pauli paramagnetic term and gives a negligible contribution [13,14]. When the doping or temperature increases, the Pauli term eventually overcomes the diamagnetic response. The susceptibility magnitude when plotted as a function of the carrier density p shows a dome-like behavior in all temperatures, as illustrated by the representative results obtained at T = 140 K and shown in Fig. 2. Interesting enough, a peaked maximum is observed when the hole concentration corresponds to that of the optimum doping, pc ≈ 0.16. This feature is also reproduced in all investigated temperatures and might indicate that the electron density of states of Bi-2212 at εF goes through a maximum at this particular carrier concentration. Fig. 3 shows representative susceptibility versus temperature results for Bi-2212 crystals with different hole concentrations. In panels (a) and (b) results are for underdoped samples. In panel (c), measurements for an optimally doped sample are shown, whereas in panel (d) results depicted are for an overdoped crystal. In all cases χc is temperature dependent. For the underdoped and optimally doped crystals, the susceptibility increases with a negative curvature when the temperature is also increased. For the overdoped samples, χc increases linearly with T. This fact obviously makes difficult the description of the susceptibility in the normal phase of Bi-2212 based simply on the additive contributions of the usual Langevin diamagnetic term and the Pauli paramagnetic term. Considering that a similar temperature dependence of the susceptibility in polycrystalline YBa2Cu3O7-δ is an effect of the pseudogap phenomenon, Naqib and Cooper [9,10] proposed a description of the results with basis on an extension of the phenomenological model proposed by them to describe entropy measurements in this same superconducting cuprate [6]. Basically, the model assumes the validity of the Pauli formulation for the paramagnetic susceptibility of an electron gas which is given by
2. Materials and methods Several single crystals of Bi-2212 were grown by the self-flux method. High purity precursors Bi2O3, SrCO3, CaCO3 and CuO were weighed in the ratio 2.4:2:1:2 for Bi, Sr, Ca and Cu respectively. A mixture with total weight 10 g was placed in a conic alumina crucible, then submitted to a heat treatment based in the recipe found in reference [11]. Characterizations with x-ray diffraction and scanning electron microscopy (SEM) were carried out, confirming the good crystalline quality of the obtained crystals. The as-grown crystals have hole density close to the optimum value (maximum Tc). Crystals with different carrier concentrations were obtained by heat treatments aimed to modify the oxygen stoichiometry. Annealing in vacuum were employed to obtain underdoped samples, whereas annealing processes in the presence of oxygen pressure were used to obtain overdoped crystals. Special care was taken to assure the homogeneity of the final oxygen content in the prepared samples. Table 1 lists the measured critical temperatures, estimated hole densities, and parameters as the pressure (P), temperature (T) and time (t) used during the heat treatments employed in the preparation of the ten studied crystals. The magnetic measurements were performed with a Quantum Design XL5-MPMS@SQUID magnetometer. The magnetization versus field of each individual crystal was measured in the range − 50 kOe < H < + 50 kOe in several fixed temperatures. In all cases, the field was applied parallel to the c-axis of the crystals. The dc magnetic susceptibility was obtained by fitting the M versus H results to straight lines and determining their slopes. Due to the smallness of the samples magnetic signal in the normal phase, special care was taken to properly subtract the contribution of the sample holders made of pure Si. For each crystal, a unique and calibrated sample holder was used. In several cases, the magnetization versus field measurements were repeated in order to assure a good estimation of the samples susceptibility χc. The critical temperature for each sample was extracted from zero-field-cooling magnetization versus temperature measurements performed at H = 10 Oe. The intersection of two straight lines extrapolated from the experimental data in the normal and superconductor phases was taken as the location of Tc. From the so-obtained values for Tc, the hole concentration for each sample was estimated using the expression TC = TC, max [1 − 82.6( p − 0.16)2 ] [12].
Table 1 Parameters used in oxygenation and deoxygenation processes carried out in order to vary the carrier density, critical temperature and hole density for each studied single crystal. Single crystals
Tc (K)
p (hole/Cu-O2)
Doping State
Heat Treatment
1 2 3 4 5 6 7 8 9 10
56.1 62.8 72.8 79.9 83.0 90.8 91.4 88.3 85.8 82.6
0.092 0.099 0.115 0.121 0.127 0.152 0.160 0.181 0.187 0.194
Underdoped Underdoped Underdoped Underdoped Underdoped Optimal Optimal Overdoped Overdoped Overdoped
P = 5 × 10−2 Torr, T = 350 °C, t = 7 h P = 5 × 10−2 Torr, T = 345 °C, t = 7 h P = 6 × 10−2 Torr, T = 340 °C, t = 7 h P = 5 × 10−2 Torr, T = 320 °C, t = 7 h P = 6 × 10−2 Torr, T = 330 °C, t = 7 h as grown as grown as grown P(O2) = 1 kgf/cm2, T = 450 °C, t = 50 h P(O2) = 1 kgf/cm2, T = 470 °C, t = 50 h
2
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Fig. 1. Magnetization as a function of the applied field for Bi-2212 single crystals with different hole concentrations, (a) p = 0.092 and (b) p = 0.127 in several temperatures between 100 K and 300 K.
⎧ ⎫ 1 χNC = μ0 μB2 N0 ⎨1 − ln [ cosh(D )] ⎬, ⎩ ⎭ D
(3)
where D = Eg /2kB T , and kB is the Boltzmann constant. Fitting experimental data as those exemplified in Fig. 3 to
χC = χdia + χNC ,
where a constant and negative term χdia is added to the paramagnetic contribution given by Eq. (3), in order to account for the observed diamagnetism in the most underdoped crystals. The description of the data for the overdoped crystals with Eq. (4) was not possible, so that the observed temperature dependence in the susceptibility in these samples must be ascribed to a different origin. From the fittings of the experimental data for χc(T) to Eq. (4), values for the parameter D were extracted. Then, as proposed by Naqib and Cooper, the pseudogap temperature could be estimated as T*(p) = Eg(p)/kB. The values found for T* using this procedure are collected in the T-p diagram shown in Fig. 4. In this diagram, the superconducting dome defined by Tc(p) is also shown. The line defined by the values found for T*(p) based on the c-axis susceptibility is in quite good agreement with other experimental determinations for this boundary in Bi-2212 [2]. Indeed, the obtained T*(p) decreases sharply when the hole concentration approaches the optimum hole density pc when this point is approached from below. Interesting enough, a value for T* was obtained inside the superconducting dome, near pc. As indicated by the dashed line in the T-p diagram, this finding is consistent with the crossing of the T*(p) and Tc(p) boundaries near the optimum hole density. Our results thus suggest that the superconducting state and the pseudogap phenomenon have different origins and coexist in a certain region of the phase diagram. In addition, the fact that the results for χc in the overdoped crystals could not be fitted to Eq. (4) leads us to assume the pseudogap is closed in the corresponding region of the T-p diagram. As shown in panel (d) of Fig. 4, the paramagnetic susceptibility remains temperature dependent for the overdoped samples. Tentatively, we attribute this temperature dependence to the close proximity between the Fermi energy and a van Hove singularity in the DOS shown to occur in the slightly overdoped Bi-2212 from tunneling experiments [15].
Fig. 2. Magnitude of the c- axis component of the magnetic susceptibility as a function of the hole concentration for Bi-2212 single crystals, measured at T = 140 K.
χ = μ0 μB2
∫0
∞
⎛ ∂f ⎞ ⎜ − ⎟ N (ε ) dε, ⎝ ∂ε ⎠
(1)
where f is the Fermi-Dirac distribution function, N(ε) is the total density of states, μ0 is the vacuum permeability and μB is the Bohr magneton. The usual approximation consists to consider (−∂f /∂ε ) ≅ δ (ε − εF ) , resulting in the Pauli susceptibility, which is temperature independent Naqib and Cooper [9,10] proposed a DOS of the type
N (ε ) = N0 for ε − εF > Eg
(2.a)
N (ε ) = N0 ε − εF for ε − εF < Eg
(2.b)
(4)
where Eg is an energy scale related to the characteristic temperature T* that is considered to denote the opening of the pseudogap. When substituting expressions (2.a) and (2.b) in Eq. (1), these authors recalculate the susceptibility using the exact expression for (−∂f /∂ε ). They obtained the paramagnetic susceptibility given by 3
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Fig. 3. Representative c-axis susceptibility results as functions of the temperature for Bi-2212 single crystals with different hole concentration. Panels (a) and (b) show data for underdoped samples. Data for optimally doped and overdoped crystals are shown in panels (c) and (d), respectively. The full lines are fittings of the experimental data to Eq. (4) that includes the paramagnetic term given by Eq. (3).
phenomenon, which is more prominent in the underdoped regime. Applying to our data a phenomenological model for the electron paramagnetic susceptibility, proposed by Naqib and Cooper [9,10], we were able to obtain the boundary T*(p) that denotes the opening of the pseudogap in the T-p phase diagram for Bi-2212. This result is in good agreement with other estimations for this line. Our results indicate that the pseudogap line enters into the superconducting dome, suggesting that the pseudogap phenomenon and the superconducting state are different in origin. It is somewhat surprising that the c-axis component of the susceptibility in Bi-2212 is sensible to the pseudogap phenomenon. The relevance of the V-shaped DOS, as proposed in the Naqib-Cooper model, for the description of our χc(T) results might be an indication that the Bi-2212 system behaves as a zero gap semiconductor when probed along the c-axis. Acknowledgments This work was supported by Brazilian agency “Conselho Nacional de Pesquisas Científicas e Tecnológicas” (CNPq) (165419/2014-8).
Fig. 4. Boundaries Tc(p) and T*(p) = Eg(p)/kB for Bi-2212 single crystals. The values for Eg(p)/kB were obtained from fittings of the experimental data for χc(T) to Eq. (4) (see text). The experimental Tc(p) defining the superconducting dome are also shown.
References 4. Conclusions
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We study experimentally the c-axis component of the magnetic susceptibility in the normal phase of Bi-2212 in a large region of the temperature – hole density phase diagram. The superconducting dome in the underdoped and overdoped regimes could be accessed. Usually, the unconventional dependence of the normal phase susceptibility with temperature and hole doping observed in the HTSC [13,14], as also observed in our Bi-2212 samples, has been attributed to the pseudogap 4
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