High-critical-temperature (high-T_c) cuprates exhibit a complex electronic phase diagram where multiple quantum phases, e.g. superconductivity, charge order, and spin order, compete or coexist ^[1]. In such a system, it is important to elucidate the doping level where the phase transition occurs at zero temperature, that is, a quantum critical point (QCP) lying beneath the superconducting dome. For example, doping dependences of T_c and upper critical field of cuprates have been intensively investigated and their unusual behaviors have been discussed in connection with QCPs ^[2,3]. Moreover, recent studies on several quantum critical superconductors including cuprates ^[4.5], heavy fermion superconductors ^[6], and iron-based superconductors ^[7,8], have revealed that the critical current density (J_c) is sharply enhanced around a QCP, demonstrating that J_c can be a powerful tool to investigate a hidden QCP inside the superconducting dome. However, the doping dependence of J_c should be carefully discussed because in general, J_c is significantly affected by the defect structures (pinning centers), which are often extrinsic in origin. In this study, we systematically investigated the doping dependence of J_c using a Bi₂Sr₂CaCu₂O_8+d (Bi2212) single crystal.

Firstly, we demonstrated that a reliable data set can be obtained by repeating the measurement and the control of the doping level on the same crystal, which is expected to reduce the uncertain factors affecting J_c such as crystal quality, chemical compositions, and sample shape. Fig. (a) shows the temperature (T) dependence of J_c at the self field (J_c^sf) for representative doping levels. At any doping level, a sudden change in the T dependence of J_c^sf (J_c^sf – T) was observed (a kink feature indicated by arrows in Fig. (a) and the T is defined as T_0D). As shown in Fig. (a), T_0D is doping dependent, resulting in the crossing behavior of J_c^sf – T curves at different doping levels. As a result, the doping dependence of J_c^sf (J_c^sf – p) significantly depends on T as shown in Fig. (b). At high temperatures above T_0D (50 K (orange circles)), J_c increases with increasing doping, which agrees with those in previous reports ^[4.5], and is usually associated with the increase in the condensation energy and the decrease in the anisotropy. On the other hand, at low temperatures below T_0D (20 K (blue open circles)), the J_c^sf – p curve is characterized by the two peaks located at the underdoped region and the slightly overdoped region, which has never been reported so far.

Then, we focus on the doping dependence of T_0D, which seems to be closely correlated with the change in J_c^sf – p. It should be noted that T_0D has been associated with the change in the dimensionality of vortices. In a strongly anisotropic layered superconductor, it has been discussed that a flux line (a one-dimensional object) becomes a vortex pancake (a zero-dimensional object) when the pinning correlation length (L_c) becomes shorter than the interlayer distance (d) ^[9]. T_0D is considered to be the crossover T where L_c ~ d ^[10,11] and thus inversely correlated with L_c, which can be described as L_c ~ ξ /γ × (J_d /J_c)^1/2 where ξ is the coherence length, γ (> 1) is the anisotropy parameter, and J_d is the depairing current density. Fig. (c) shows the doping dependence of T_0D normalized by T_c (T_0D /T_c). We found that the doping dependence of T_0D /T_c shows a sudden jump at around p₁ ~ 0.13 and p₂ ~ 0.17, where J_c-^sf also shows a sudden change (Fig. (b)). The jump in T_0D indicates a discontinuous change in the fundamental parameters (e.g. ξ, γ, and/or J_d) and/or the pinning landscape (reflected in J_c). Such a discontinuous change suggests the existence of a phase transition, namely, a QCP. When the doping dependences of J_c and T_0D are compared with the phase diagram of cuprates ^[1], the observed p₁ and p₂ seem to correspond to the end points of the charge order phase and the pseudogap, respectively. Thus, it is demonstrated that the QCPs hidden beneath the superconducting dome of Bi2212 system are successfully exposed by the precise J_c measurements, although further studies are necessary to confirm the correspondence between these doping levels (p₁ and p₂) and QCPs.

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[3] M.K.Chan, et. al. PNAS 117, 9782 (2020)
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[5] M.Naamneh, et al. Phys. Rev. B 90, 224501 (2014)
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[11] V.F.Correa, et al. Phys. Rev. B 63, 092502 (2001)

(a) Temperature dependence of J_c^sf of Bi2212 at representative doping levels (p = 0.11-0.20). T_0D is defined as the temperature where a kink appears indicated by arrows.
(b) Doping dependence of J_c^sf at 20 K (blue open circles) and 50 K (orange circles, multiplied by 10). Dashed line indicates T_c dome.
(c) Doping dependence T_0D normalized by T_c.

Keywords: Cuprate High-Tc Superconductor, Bi2212 Single Crystals, Critical Current Density, Electronic Phase Diagram