## Abstract

Topological superconductors have attracted wide-spreading interests for the bright application perspectives to quantum computing. Cu_{0.3}Bi_{2}Se_{3} is a rare bulk topological superconductor with an odd-parity wave function, but the details of the vector order parameter d and its pinning mechanism are still unclear. We have succeeded in growing Cu_{x}Bi_{2}Se_{3} single crystals with unprecedented high doping levels. For samples with x = 0.28, 0.36 and 0.37 with similar carrier density as evidenced by Knight shift, the in-plane upper critical field H_{c2} shows a two-fold symmetry. However, the angle at which the H_{c2} becomes minimal is different by 90^{◦} among them, which indicates that the d-vector direction is different for each crystal likely due to a different local environment. The carrier density for x = 0.46 and 0.54 increases substantially compared to x ≤ 0.37. Surprisingly, the in-plane H_{c2} anisotropy disappears, indicating that the gap symmetry undergoes a transition from nematic to isotropic (possibly chiral) as carrier increases. Exploring topological materials and their electronic functions are among the front-most topics of current condensed matter physics. In particular, much attention has been paid in recent years to topological superconductors where Majorana fermions (excitations) are expected to appear on edges or in the vortex cores[1, 2]. Such novel edge states can potentially be applied to fault tolerant non-Abelian quantum computing [4, 5]. So far, great success has been achieved in observing the Majorana bound state on the interface of a ferromagnet or a topological insulator in proximity to an s-wave superconductor[3, 6–8], or on the surface of iron-based superconductors [9]. In contrast, research on bulk topological superconductors progresses much more slowly. Candidates of bulk topological superconductors include superconductors with broken time reversal symmetry [10, 11], superconductors with broken spatial inversion symmetry [12, 13] and odd-parity superconductors with spatial inversion symmetry[14, 15]. For the last case, the criteria for topological superconductivity are effectively two fold. Namely, odd-parity of the gap function and an odd number of time-reversal invariant momenta in the Brillouin zone[14]. Experimentally, clear evidence for odd-parity superconductivity had not been found until very recently [16]. Although Cu-doped topological insulator Cu_{x}Bi_{2}Se_{3}[17] had been proposed as a candidate [14], experiments had been controversial[18–20]. The discovery of spontaneous spin rotation-symmetry breaking in the bulk superconducting state of Cu_{0.3}Bi_{2}Se_{3} by nuclear magnetic resonance (NMR) measurements established the spin-triplet, odd-parity superconducting state [16]. Since there is only one time-reversal-symmetric momentum in the Brillouin zone of Cu_{x}Bi_{2}Se_{3} [21], this material fulfills the twofold criteria and can be classified as a topological superconductor. However, detailed gap function is still unclear. If the gap is fully-opened, then Cu_{0.3}Bi_{2}Se_{3} is a class DIII topological superconductor [22], where Majorana zero-energy modes are expected at edges or vortex cores. If there are nodes in the gap function, the material is nonetheless topological just as the cases of Dirac or Weyl semimetals. A more generally-used term associated with the gap in a spin-triplet superconductor is the vector order parameter d, whose direction is perpendicular to the direction of paired spins and whose magnitude is the gap size. The d-vector was found to be parallel to a-axis (the Se-Se bond direction) in Cu_{0.3}Bi_{2}Se_{3} [16]. This was the first case where the d-vector direction was unambiguously determined in any spin-triplet superconductor candidate. The emergent two-fold symmetry in the Knight shift below T_{c} [16] was interpreted by the concept of nematic order [23], and had triggered many subsequent extensive studies on rotational symmetry breaking by various methods, which also revealed a two-fold symmetry in other physical properties [24–28]. Measurements by transport[29], penetration depth[30], and scanning tunneling microscope (STM) [31, 32] suggesting unconventional superconductivity have also been reported since then. However, why the d-vector is oriented to one of a-axes, and why it is robust against heat cycle through the superconducting transition, remain unknown. Note that there are three equivalent a-axis directions. This issue is important as the gap symmetry (”nematicity” indicator) is closely tied to the direction of the d-vector [23]. From material point of view, it had been unclear whether the carrier density can be controlled and how the physical properties would change with changing carrier density. A previous report showed that the Hall coefficient does not change even though the nominal x increases from 0.15 to 0.45 [34]. These are the issues we wish to address in this article. In this work, we synthesized Cu-doped Bi_{2}Se_{3} single crystals by the electrochemical intercalating method. Through the measurements of the Knight shift, we find unprecedentedly that the carrier concentration further increases with increasing x beyond x=0.37. We study the angle dependence of the upper critical field H_{c2} in different crystals. For an odd-parity gap function, the gap anisotropy will lead to an anisotropic H_{c2}. One then can obtain knowledge about how the superconducting gap evolves with x by measuring the H_{c2} anisotropy. For samples with x=0.28, 0.36 and 0.37 which have the same size of the Knight shift, we find a two-fold symmetry in the in-plane H_{c2} by ac susceptibility and magnetoresistance measurements, in agreement with previous reports [16, 24–28]. However, the angle at which H_{c2} is a minimum differs by 90^{◦}, which means that the direction of the d-vector is different for each crystal. In contrast, for x=0.46 and 0.54, two-fold anisotropy disappears, which indicates a nematic-to-isotropic transition of the gap symmetry as carrier density increases. We discuss possible exotic (chiral) superconducting state for the samples with large x.

Original language | English |
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Journal | Unknown Journal |

Publication status | Published - Dec 19 2019 |

## ASJC Scopus subject areas

- General