Direct Visualization of the Nematic Superconductivity in ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$

${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ hosts both topological surface states and bulk superconductivity. It has been identified recently as a topological superconductor (TSC) with an extraordinary nematic, i.e., $C_2$-symmetric superconducting state and odd-parity pairing. Here, using scanning tunneling microscopy, we directly examine the response of the superconductivity of ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ to magnetic field. Under out-of-plane fields ($B_{\perp }$), we discover elongated magnetic vortices hosting zero-bias conductance peaks consistent with the Majorana bound states expected in a TSC. Under in-plane fields ($B_{//}$), the average superconducting gap exhibits twofold symmetry with field orientation; the long $C_2$ symmetry axes are pinned to the dihedral mirror planes under $B_{//}=0.5\mathrm{T}$ but rotate slightly under $B_{//}=1.0\mathrm{T}$. Moreover, a nodeless ${\mathrm{\Delta }}_{4x}$ gap structure is semiquantitatively determined for the first time. Our data paint a microscopic picture of the nematic superconductivity in ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ and pose strong constraints on theory.

Popular Summary

Topological superconductors (TSCs) are exotic materials with the potential for use in quantum computing. To make these materials, normal superconductors are a good starting point. The superconductor ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ has recently been identified as a candidate TSC, however, theory and experiment do not agree on various properties and behaviors. To reach a comprehensive microscopic understanding of this compound and to reconcile previous contradicting reports, we investigate the superconducting properties of ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ with a scanning tunneling microscope at ultralow temperatures under various magnetic fields.

Under out-of-plane fields, we discover elongated magnetic vortices hosting a state at zero energy that might be its own antiparticle, a manifestation of the Majorana fermion expected in a TSC. Under in-plane fields, the superconducting gap, which reflects how electrons form Cooper pairs in a superconductor, exhibits an unexpected twofold symmetry on the threefold symmetric lattice of ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$. This form of the superconducting gap, along with the absence of states on the step edges of the material, is not compatible with the existing theories of either 2D or 3D TSCs.

Our insight into the microscopic behavior of superconductivity in ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ will not only facilitate the understanding of this remarkable material, but will also have profound implications for topological superconductivity and quantum computation, which is underdeveloped compared to other topological materials.

Figure 1

The three types of ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ surfaces and corresponding $dI/dV$ spectra. (a),(b) Typical topographic images of the NSC region: (a) $V_b=3\mathrm{V}$, $I=20\mathrm{pA}$; (b) $V_b=1\mathrm{V}$, $I=20\mathrm{pA}$. Two kinds of typical defects are marked by green circles. The inset of (b) shows an atomically resolved image ($V_b=30\mathrm{mV}$, $I=70\mathrm{pA}$). (c),(d) Typical $dI/dV$ spectra of the NSC region: (c) $V_b=2\mathrm{mV}$, $I=150\mathrm{pA}$, $\mathrm{\Delta }V=30\mu \mathrm{V}$; (d) $V_b=1\mathrm{V}$, $I=100\mathrm{pA}$, $\mathrm{\Delta }V=10\mathrm{mV}$. (e) Terrace heights in the NSC region along linecut 1 in (a). (f),(g) Typical topographic images of the SC-I region: (f) $V_b=3\mathrm{V}$, $I=10\mathrm{pA}$; (g) $V_b=3\mathrm{V}$, $I=10\mathrm{pA}$. (h),(i) Typical $dI/dV$ spectra of the SC-I region: (h) $V_b=2\mathrm{mV}$, $I=100\mathrm{pA}$, $\mathrm{\Delta }V=50\mu \mathrm{V}$; (i) $V_b=1\mathrm{V}$, $I=100\mathrm{pA}$, $\mathrm{\Delta }V=10\mathrm{mV}$. The orange lines in (d) and (i) show the linear dispersion of the DOS; the energies of Dirac points (DP) are indicated. (j) Terrace heights in the SC-I region along linecut 2 in (f). (k) Superconducting spectra collected across a step edge along linecut 3 in (f). (l),(m) Typical topographic images of the SC-II region: (l) differential image, $V_b=3\mathrm{V}$, $I=20\mathrm{pA}$; (m) $V_b=2\mathrm{V}$, $I=20\mathrm{pA}$. The inset of (m) shows an atomically resolved image ($V_b=5\mathrm{mV}$, $I=100\mathrm{pA}$). (n),(o) Typical $dI/dV$ spectra of the SC-II region: (n) $V_b=2\mathrm{mV}$, $I=200\mathrm{pA}$, $\mathrm{\Delta }V=30\mu \mathrm{V}$; (o) $V_b=1\mathrm{V}$, $I=150\mathrm{pA}$, $\mathrm{\Delta }V=10\mathrm{mV}$.

Figure 2

Vortex state in the SC-I region of ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$. (a) Vortex mapping in the SC-I region ($225\times 225{\mathrm{nm}}^2$) at $V_b=0\mathrm{mV}$ under $B_{\perp }=0.2\mathrm{T}$. The profile of one vortex core is indicated by the green ellipse. (b) Exponential fits to line profiles of a single vortex along its long and short axes. (c) Field dependence of the coherence lengths along the long and short axes of the vortex. (d),(e) Evolution of the $dI/dV$ spectra ($V_b=2\mathrm{mV}$, $I=100\mathrm{pA}$, $\mathrm{\Delta }V=50\mu \mathrm{V}$) taken along the two orthogonal linecuts in (a) as indicated. The green numbers indicate the distances of the spectra from the vortex core center. (f) $dI/dV$ spectra as a function of field strength, comparing the vortex core center (red curve) with the bulk tens of nanometers away.

Figure 3

The average superconducting gap as a function of $\theta$ in the SC-I region. (a) Typical $dI/dV$ spectra measured under $B_{//}=0.5\mathrm{T}$ with different field orientations. The definition of $\theta$ is indicated in the inset, where the purple spheres represent Se atoms. The definitions of ${\mathrm{\Delta }}_{\mathrm{CP}}^{*}$ and ${\mathrm{\Delta }}_{\mathrm{LE}}^{*}$ are indicated by the double-arrowed lines. (b) Angular dependence of ${\mathrm{\Delta }}_{\mathrm{CP}}^{*}$ and ${\mathrm{\Delta }}_{\mathrm{LE}}^{*}$ under $B_{//}=0.5\mathrm{T}$. The sinusoidal fit highlights the clear twofold symmetry. (c),(d) Same as (a) and (b) except that $B_{//}$ is increased to 1.0 T. (e) In-plane magnetic field dependence of ${\mathrm{\Delta }}_{\mathrm{LE}}^{*}$ for $\theta =85°$, 40°, and 0°. (f) Field dependence of the oscillation amplitudes of ${\mathrm{\Delta }}_{\mathrm{CP}}^{*}$ and ${\mathrm{\Delta }}_{\mathrm{LE}}^{*}$. The oscillation amplitude is defined here as the ratio of the peak-to-peak amplitude of the oscillation to the maximal gap size at that field.

Figure 4

Nematicity of the superconducting gap in different domains. (a) Comparison of the profile of elliptical vortex and twofold symmetry of ${\mathrm{\Delta }}_{\mathrm{LE}}^{*}$ under $B_{//}=0.5\mathrm{T}$ in the SC-I region, with respect to the Se lattice (the purple spheres represent Se atoms, gray solid lines and dash-dot lines indicate the $\mathrm{Se}─\mathrm{Se}$ bond directions and the dihedral mirror planes, respectively). (b) Twofold symmetry of ${\mathrm{\Delta }}_{\mathrm{LE}}^{*}$ under $B_{//}=1.0\mathrm{T}$ in the SC-I region. (c),(d) Twofold symmetry of ${\mathrm{\Delta }}_{\mathrm{LE}}^{*}$ in the SC-II region observed in domain $A$ under $B_{//}=0.5\mathrm{T}$ and domain $B$ under $B_{//}=1.0\mathrm{T}$, respectively. Red squares are the experimental data, while the magenta symbols in (d) denote the symmetric data obtained by shifting the experimental data by 180°. Red dashed lines show the long axis of ${\mathrm{\Delta }}_{\mathrm{LE}}^{*}$ through its two gap maxima. Polar coordinates are plotted out by gray dashed circles on a scale that differs among the panels; the sinusoidal fit functions describing the twofold symmetry of ${\mathrm{\Delta }}_{\mathrm{LE}}^{*}$ are also listed to show the oscillation amplitudes and the directions of the $C_2$ axes. The rotation of the long $C_2$ axis of ${\mathrm{\Delta }}_{\mathrm{LE}}^{*}$ under $B_{//}=1.0\mathrm{T}$ is marked in (b) and (d). The angular distribution of ${\mathrm{\Delta }}_k$ in various superconducting regions is sketched by thick blue curves, whose anisotropies are extracted from the fitted gap anisotropy in Figs. 1 and 1, and Fig. S8 in Ref. [20]. The fitting functions for ${\mathrm{\Delta }}_k$ are also listed. (e) Schematics of the in-plane-field angular-dependent quasiparticle excitation (red dots) in the low-temperature and low-field condition [39].