Abstract
Topological superconductors have attracted widespreading interests for the bright application perspectives to quantum computing. Cu_{0.3}Bi_{2}Se_{3} is a rare bulk topological superconductor with an oddparity wave function, but the details of the vector order parameter d and its pinning mechanism are still unclear. Here, we succeed 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 the Knight shift, the inplane upper critical field H_{c2} shows a twofold symmetry. However, the angle at which the H_{c2} becomes minimal is different by 90° among them, which indicates that the dvector 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 inplane H_{c2} anisotropy disappears, indicating that the gap symmetry undergoes a transition from nematic to isotropic (possibly chiral) as carrier increases.
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Introduction
Exploring topological materials and their electronic functions are among the frontmost 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 nonAbelian quantum computing^{3,4}. 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 swave superconductor^{5,6,7,8}, or on the surface of ironbased 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 oddparity superconductors with spatial inversion symmetry^{14,15}. For the last case, the criteria for topological superconductivity are effectively two fold. Namely, oddparity of the gap function and an odd number of timereversal invariant momenta in the Brillouin zone^{14}. Experimentally, clear evidence for oddparity superconductivity had not been found until very recently^{16}. Although Cudoped topological insulator Cu_{x}Bi_{2}Se_{3}^{17} had been proposed as a candidate^{14}, experiments had been controversial^{18,19,20}.
The discovery of spontaneous spin rotationsymmetry breaking in the bulk superconducting state of Cu_{0.3}Bi_{2}Se_{3} by nuclear magnetic resonance (NMR) measurements established the spintriplet, oddparity superconducting state^{16}. Since there is only one timereversalsymmetric 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 fullyopened, then Cu_{0.3}Bi_{2}Se_{3} is a class DIII topological superconductor^{22}, where Majorana zeroenergy 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 generallyused term associated with the gap in a spintriplet 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 dvector was found to be parallel to aaxis (the Se–Se bond direction) in Cu_{0.3}Bi_{2}Se_{3}^{16}. This was the first case where the dvector direction was unambiguously determined in any spintriplet superconductor candidate. The emergent twofold 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 twofold symmetry in other physical properties^{24,25,26,27,28}. 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 dvector is oriented to one of aaxes, and why it is robust against heat cycle through the superconducting transition, remain unknown. Note that there are three equivalent aaxis directions. This issue is important as the gap symmetry (“nematicity” indicator) is closely tied to the direction of the dvector^{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^{33}. These are the issues we wish to address in this article.
In this work, we synthesized Cudoped 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 oddparity 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 twofold symmetry in the inplane H_{c2} by ac susceptibility and magnetoresistance measurements, in agreement with previous reports^{16,24,25,26,27,28}. However, the angle at which H_{c2} is a minimum differs by 90°, which means that the direction of the dvector is different for each crystal. In contrast, for x = 0.46 and 0.54, twofold anisotropy disappears, which indicates a nematictoisotropic transition of the gap symmetry as carrier density increases. We discuss possible exotic (chiral) superconducting state for the samples with large x.
Results
Sample characterization
Figure 1 shows the result of dc susceptibility measurements for representative samples. In Fig. 2, we summarize the properties of all the samples we synthesized. The obtained T_{c} and shielding fraction (SF) for most samples are close to the values reported by Kriener et al.^{34}. The SF for x = 0.46 is the highest (56.2%) among those reported so far. In Table 1 we list the properties for the five samples that we will discuss in this paper.
Figure 3a shows the ^{77}Se(I = 1∕2)NMR spectra above T_{c} for five samples with different x. Each spectrum can be fitted by a single Gaussian. The Knight shift is ~0.048% for x = 0.28, 0.36, and 0.37, which is close to the value (K = 0.049%) previously reported for the sample with x = 0.3^{16}. On the other hand, the Knight shift increased substantially in the samples with x = 0.46 and 0.54. Generally, the Knight shift is expressed as,
where K_{orb} is the contribution due to orbital susceptibility and A_{hf} is the hyperfine coupling constant, which are independent of carrier density. The χ_{s} is the spin susceptibility which is proportional to electronic density of states. For this field configuration, K_{orb} = 0.03%^{16}. In Fig. 3b, we plotted the Knight shift as a function of Cu content x. Upon doping, K increases as compared to the parent compound, which implies that the carrier is indeed doped into the sample. The fact that K has a similar value for the samples with x = 0.28–0.37 indicates that the carrier density does not change appreciably in this x region. Such behavior is consistent with other reports by different methods^{33,35}. On the other hand, in the highx region, we discovered for the first time that carriers increase further as x increases beyond 0.37.
We briefly comment on possible mechanisms of Cudoping. In Srdoped system^{35}, Sr goes into multiple sites, namely, the intercalated site inbetween the quintuple layer blocks, interstitial site, or the inplane site substituting for Bi. Cu(Sr) residing on the intercalated or interstitial site contributes electrons, while Cu(Sr) going to the substituting site contributes holes^{17}. The stagelike xdependence of K suggests that Cu doping mechanism in each xrange is not simple, but rather Cu may also go into multiple sites. As post annealing is necessary after electrochemical process in the present case, we speculate that Cu may migrate into a different site after annealing, resulting in the peculiar behavior of K with respect to x. The Cu position is one of the issues that needs to be addressed in future works.
Anisotropy of the upper critical field H _{c2} and its disappearance
Next we present data on the anisotropy of H_{c2}. For x = 0.28, we measured both ac susceptibility (acχ) and electrical resistance by changing the magnetic field for each fielddirection relative to aaxis. The inset to Fig. 4a shows the angle ϕ between the applied magnetic field H and the aaxis. Before Cu doping, the direction of aaxis (Se–Se bond direction) was determined by Laue diffraction, which we assign as ϕ = 0°. The main panel of Fig. 4a shows the magnetic field dependence of the acχ at different ϕ for x = 0.28. As can be seen in the figure, the field dependence of acχ is clearly different for each angle below a certain field (H_{c2}), while above H_{c2} the Hdependence of acχ is the same for all angles. Fig. 4b shows H_{c2} as a function of angle. A clear twofold symmetry is observed.
Such twofold symmetry is also seen in the magnetoresistance measurements, as demonstrated in Fig. 5 where the electrical resistance under a field of 0.9 T and at T = 1.8 K is plotted as a function of angle. We checked the angleresolved resistance at μ_{0}H = 3 T and T = 1.8 K where the sample is in the normal state and found only a random noise with an amplitude no bigger than 0.1 mΩ, thus confirming that such twofold oscillation in the resistance shown in Fig. 5b is caused by the anisotropy of H_{c2}. Fig. 6a shows the resistance as a function of magnetic field at T = 1.8 K for three representative angles. Similar to the results shown in Fig. 4a, a clear angle dependence is found. Fig. 6b shows the angle dependence of \({H}_{{\rm{c}}2}^{\rho }\) obtained from the magnetoresistance data. A twofold symmetry is also clearly seen. This result is in agreement with that seen in the acχ data shown in Fig. 4b, although suchdefined \({H}_{{\rm{c}}2}^{\rho }\) is slightly higher but is not surprising for its definition. An inplane anisotropy of H_{c2} is also resolved from the data of electrical resistance vs temperature under different magnetic fields (see Supplementary Figs. 1, 2).
Quite often, extracting H_{c2} from the magnetic susceptibility has several advantages over that from magnetoresistance measurements. Firstly, the magnetic susceptibility is more sensitive to superconducting volume fraction rather than surface. Secondly, in twodimensional or layered superconductors, H_{c2} determined by resistivity measurements is often inaccurate because of vortex lattice melting^{36}. For example, in highT_{c} cuprates, even in the resistive state, one is still in a regime dominated by Cooper pairings in the presence of vortices^{36}. For these reasons and the technical merit that acχ can be measured to a lower temperature in our case, below we discuss the evolution of gap symmetry based on the acχ data. Data for x = 0.36, 0.37, 0.46, and 0.54 are shown in Fig. 7. For x = 0.36 and 0.37, clear angle dependence was found as in x = 0.28. In contrast, the tendency is completely different for samples with larger x. There is no angle dependence in acχ for the samples with x = 0.46 and 0.54.
The angular dependence of H_{c2} at 1.4 K determined from the acχ for all samples is plotted in Fig. 8 (for more details, see Supplementary Fig. 3). For the samples with smaller x (Fig. 8a, b), H_{c2} shows a large and twofold anisotropy. In contrast, for larger x (Fig. 8c), no anisotropy is observed. The oscillation amplitude of H_{c2} is similar for x = 0.28, 0.36, and 0.37. The magnitude of H_{c2} is also similar between x = 0.28 and 0.37, but is larger for x = 0.36. The origin of this difference is unknown at the moment.
Interestingly, the angle at which H_{c2} becomes minimal is 90° and 150° (perpendicular to Se–Se bond) for x = 0.28 and 0.36, respectively, but is 0° (along Se–Se bond) for x = 0.37. For this crystal structure, the equivalent crystalaxis direction appears every 60°. If two oscillation patterns have a phase difference of 60°, it can be said that they are the same crystallographically. Therefore, the samples with x = 0.28 and 0.36 have the same gap symmetry. However, 90° difference means that the gap symmetry is different. It is noted that the angle at which H_{c2} is minimal corresponds to the direction of dvector^{16}. Therefore, our result indicates that the dvector direction differs for each sample in Cu_{x}Bi_{2}Se_{3} even though the carrier density is the same or very similar. The x = 0.37 sample has the same symmetry as the previous sample used for NMR measurement where the dvector is pinned to Se–Se direction^{16}, while the x = 0.28 and 0.36 samples have the same symmetry as the sample used for specific heat measurements^{24}.
No multiple domains
As for the disappearance of oscillation in H_{c2} for x = 0.46 and 0.54, a possibility of multiple domains each of which is a nematic phase with dvector pointing to a different direction, can be ruled out. In that case, the NMR spectrum for T < T_{c} would be broadened compared to that at T > T_{c}. This is because the spectrum coming from the domains with H∥ dvector will shift to a lower frequency^{16}, but that from domains with H ⊥ dvector will not. However, our result (Fig. 9a–c) shows no such broadening which is not compatible with the multipledomains scenario. In Fig. 9c, we show a simulation for the spectrum in case of multiple domains, assuming that various domains are distributed randomly as to wipe out oscillations in the angle dependence of H_{c2}. The simulation also assumed that, for H∥ dvector, the spin Knight shift (K_{s}) below T_{c} follows the same temperature dependence as found in Cu_{0.3}Bi_{2}Se_{3}^{16}; K_{s} = 0.046% and the reduction of K_{s} at T = 1.5K, ΔK_{s} = 0.043%, were used in the simulation.
Discussion
Fu pointed out that the spin rotationsymmetry breaking (the inplane Knight shift nematicity) can be understood if the pairing function is a doublet representation^{23}, Δ_{4x} or Δ_{4y}, both of which being pwave. For Δ_{4x} state, the dvector is along the principle crystal axis while it is orthogonal to the principle crystal axis for Δ_{4y} state. In the Δ_{4x} state, there are two point nodes in the superconducting gap, which are in the direction perpendicular to the Se–Se bond. In the Δ_{4y} state, there is no node but a minimum in the superconducting gap, and the dvector is oriented perpendicular to the gapminimum direction. Applying Fu’s theory to our results, the x = 0.37 and the previous x = 0.3^{16} samples would correspond to Δ_{4x}, and the samples with x = 0.28 and 0.36 would correspond to Δ_{4y} state.
As mentioned in ref. ^{23}, the theory does not take crystalline anisotropy into consideration so that Δ_{4x} and Δ_{4y} are degenerate. In real crystals, however, crystalline distortions due to dopants exist, and it was suggested that the dopant position is important for superconductivity^{35,37}. Hexagonal distortion was indeed reported in Sr_{x}Bi_{2}Se_{3}^{38}. The same can be expected in Cu_{x}Bi_{2}Se_{3}^{39}. Moreover, strains induced by quenching can vary from one sample to another. The chemical processes to obtain a sample with high SF are complex which includes a quenching process. Only those quenched from a narrow temperature (~560 °C) show high SF, which suggests that it is important to seize a metallurgically metastable phase to obtain the superconductivity. However, the quenching process is less controllable so that strain induced during this process is random among samples. We believe that dvector pointing to different directions in samples with the same carrier density is due to a different local structural environment such as strain caused by quenching, dopantinduced crystal distortion, etc.
The most intriguing and surprising finding of this work is that the oscillation in H_{c2} disappears for the samples with x = 0.46 and 0.54, as seen in Fig. 8b. Judging from the NMR spectrum width which is almost the same for x = 0.36, 0.48, and 0.54 in particular, we conclude that the sample homogeneity is very similar among them. Also, a multipledomains scenario can be ruled out as discussed in the previous section. We interpret the surprising evolution of H_{c2} as due to an emergent fullyopened isotropic gap for large x. It was previously pointed out that the oddparity superconductivity with twocomponent E_{u} representation admits two possible phases, nematic and chiral^{40,41}. Under some circumstances, a chiral, fullygapped, state becomes more stable as compared to a nematic state. One of the crucial parameters important for selecting between the two states is the Fermi surface shape. It has been theoretically shown by several groups that the chiral state can be stabilized when the Fermi surface becomes more twodimensional^{42,43,44}. Experimentally, photoemission and quantum oscillation measurements have suggested that the Fermi surface becomes more twodimensional as carrier density increases^{45,46}. Indeed, the Knight shift result indicates that carriers increased in the samples of x = 0.46 and 0.54 as compared to x ≤ 0.37. Therefore, the dramatic change in the angle dependence of the H_{c2} can be naturally explained as due to a gap symmetry change from nematic to chiral. The lack of change of the Knight shift below T_{c} for all angles (Fig. 9) as contrary to what found in the nematic phase^{16} is consistent with existing theory that the dvector for the chiral phase would be pointing to the caxis^{42}. Thus, our finding suggests that superconductors with strong spinorbital interaction^{14} is a land much more fertile than we thought.
In summary, single crystal samples of Cu_{x}Bi_{2}Se_{3} with unprecedented high doping levels were newly synthesized and investigated. By NMR measurements, we found for the first time that the carrier density increased further by Cu doping beyond x = 0.37. By magnetic susceptibility measurements, we found that the inplane H_{c2} shows a clear twofold oscillation for the samples with x = 0.28, 0.36, and 0.37 which have similar carrier density as evidenced by the Knight shift. However, the angle at which H_{c2} becomes minimal is different by 90° between different samples. This indicates that the direction of the dvector is different from crystal to crystal due to a different local structure caused by strain during the quenching process, dopantinduced crystal distortion, etc. In the samples with x = 0.46 and 0.54, the twofold oscillation is completely suppressed, which indicates a gap symmetry change from nematic to isotropic as carrier density increases. These findings enriched the contents of topological superconductivity in doped Bi_{2}Se_{3}, and we hope that our work will stimulate further studies on possibly even more exotic superconducting state (possible chiral state) in the highdoped region of Cu_{x}Bi_{2}Se_{3} as well as on bulk topological superconductors in general.
Methods
Single crystal growth and characterization
Single crystals of Cu_{x}Bi_{2}Se_{3} were prepared by intercalating Cu into Bi_{2}Se_{3} by the electrochemical doping method described in ref. ^{34}. First, single crystals of Bi_{2}Se_{3} were grown by melting stoichiometric mixtures of elemental Bi (99.9999%) and Se (99.999%) at 850 °C for 48 h in sealed evacuated quartz tubes. After melting, the sample was slowly cooled down to 550 °C over 48 h and kept at the same temperature for 24 h. Those meltgrown Bi_{2}Se_{3} single crystals were cleaved into smaller rectangular pieces of about 14 mg. They were wound by bare copper wire (dia. 0.05 mm), and used as a working electrode. A Cu wire with diameter of 0.5 mm was used both as the counter (CE) and the reference electrode (RE). We applied a current of 10 μA in a saturated solution of CuI powder (99.99%) in acetonitrile (CH_{3}CN). The obtained crystals samples were then annealed at 560 °C for 1 h in sealed evacuated quartz tubes, and quenched into water. After quenching, the samples were covered with epoxy (STYCAST 1266) to avoid deterioration. The Cu concentration x was determined from the mass increment of the samples. To check the superconducting properties, dc susceptibility measurements were performed using a superconducting quantum interference device (SQUID) with the vibrating sample magnetometer (VSM).
NMR measurements
The ^{77}SeNMR spectra were obtained by the fast Fourier transformation of the spinecho at a fixed magnetic field (1.5 T or 0.7 T). The Knight shift K was calculated using nuclear gyrometric ratio γ_{N} = 8.118 MHz/T for ^{77}Se.
Angleresolved H _{c2} measurements
H_{c2} was determined from ac susceptibility by measuring the inductance of insitu NMR coil. Angledependent measurements were performed by using a piezodriven rotator (Attocube ANR51) equipped with Hall sensors to determine the angle between magnetic field and crystal axis. The acχ vs. H data in the normal state were fitted by a linear function (a constant line). H_{c2} was defined as a point off the straight line. A typical example is shown in Supplementary Fig. 3.
Magnetoresistance measurements
The angledependent electrical resistance was measured by the standard fourelectrode method in a physical properties measurement system (PPMS, Quantum Design) with a mechanical rotating probe. The building of electrodes were carried out in a glove box filled with highpurity Ar gas to prevent sample from degradation. The electrodes were made such that the current direction is along the aaxis. The excitation currents are 0.1–1 mA to make a compromise of the Joule heating and the measurement accuracy.
Data availability
The data that support the findings of this study are available on reasonable request.
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Acknowledgements
We thank Y. Inada for help in Laue diffraction measurements and S. Kawasaki for help in some of the H_{c2} measurements, Markus Kriener and T. Mizushima for useful discussions. This work was supported in part by the JSPS/MEXT Grants (Nos. JP15H05852, JP16H04016, and JP17K14340) and MOST grant No. 2017YFA0302904.
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G.q.Z. planned and supervised the project. T.Kawai, Y.K., and T.Kambe synthesized the single crystals. T.Kawai, Y.K., Y.H., and K.M. performed magnetic susceptibility, Y.H. and K.M. conducted NMR, and C.G.W. performed magnetoresistance measurements. G.q.Z. wrote the manuscript with inputs from K.M. All authors discussed the results and interpretation.
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Kawai, T., Wang, C.G., Kandori, Y. et al. Direction and symmetry transition of the vector order parameter in topological superconductors Cu_{x}Bi_{2}Se_{3}. Nat Commun 11, 235 (2020). https://doi.org/10.1038/s4146701914126w
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DOI: https://doi.org/10.1038/s4146701914126w
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