Rotational Symmetry Breaking in a Trigonal Superconductor Nb-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$

The search for unconventional superconductivity has been focused on materials with strong spin-orbit coupling and unique crystal lattices. Doped bismuth selenide (${\mathrm{Bi}}_2{\mathrm{Se}}_3$) is a strong candidate, given the topological insulator nature of the parent compound and its triangular lattice. The coupling between the physical properties in the superconducting state and its underlying crystal symmetry is a crucial test for unconventional superconductivity. In this paper, we report direct evidence that the superconducting magnetic response couples strongly to the underlying trigonal crystal symmetry in the recently discovered superconductor with trigonal crystal structure, niobium (Nb)-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. As a result, the in-plane magnetic torque signal vanishes every 60°. More importantly, the superconducting hysteresis loop amplitude is enhanced along one preferred direction, spontaneously breaking the rotational symmetry. This observation indicates the presence of nematic order in the superconducting ground state of Nb-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$.

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Computers perform calculations by manipulating ones and zeros, but these devices are reaching physical limitations on how densely information can be stored. Quantum computers, in contrast, can take advantage of bizarre behaviors of atoms—such as the ability to spin in two directions at once—to pack information more densely and perform calculations more quickly than any traditional computer. But these quantum states are fragile and easily destroyed by heat or other particles. This is the advantage of a new class of materials called topological superconductors (TSCs). Their strange “quasiparticles,” which have properties of partial electrons, are robust against environmental interference. Confirming the existence of TSCs has been hampered, however, because their predicted properties are difficult to detect. Using a novel technique called torque magnetometry, we have detected one of these properties in a TSC candidate, a crystal of Nb-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$.

TSCs are predicted to exhibit rotational symmetry breaking—a magnet will tug on the crystal more strongly in two directions, for example, despite the crystal’s hexagonal arrangement of atoms. We rotated ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ in a magnetic field and observed its response. The crystal is glued to a diving board, and when the magnet is applied, the crystal is either attracted or repulsed, thus pushing or pulling on the board. Under normal conditions, the response is the same in six directions. But when cooled to a superconducting state, the crystal responds strongly in only two directions.

Our results not only reveal broken rotational symmetry in Nb-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$—a telltale sign of topological superconductivity—but also demonstrate the power of torque magnetometry. This technique could be used to detect rotational symmetry breaking in other materials and identify other topological superconductors.

Figure 1

Experimental setup, sample orientation, and torque curves of ${\mathrm{Nb}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$. (a) Schematic sketch of torque magnetometry under in-plane field rotation. The magnetic field is applied in plane. The azimuthal angle $\varphi$ is defined as the angle between the external magnetic field and the cantilever arm ($x$ axis). Torque $\overrightarrow{\tau }={\mu }_{0}V\overrightarrow{M}\times \overrightarrow{H}$ is tracked by measuring the capacitance between the cantilever and the gold film beneath it. (b) Crystal structure of ${\mathrm{Nb}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ viewed down the crystalline $\widehat{c}$ axis. The dashed lines are the mirror planes of the crystal, and the blue arrows are the crystal axes. These crystal axes are shown to be the directions where the torque loop vanishes. (c) Meissner effect from the sample. The volume magnetic susceptibility reaches close to $-1$, indicating a fully superconducting volume. (d) Selected torque curves at 0.3 K with external magnetic fields between $-1$ and 1 T. The magnitude of the hysteresis loop is at a maximum at around 120° and is nearly zero at 30° and 90°.

Figure 2

The angular dependence of susceptibility of ${\mathrm{Nb}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ in a normal state and superconducting state. (a) Examples of the hysteretic $M\text{−}H$ curves demonstrate the definition of $M$, $\mathrm{\Delta }M$, and $\chi$. Note that $M_{\pm }={\tau }_{\pm }/{\mu }_{0}H$, $M=(M_{+}+M_{-})/2$, $\mathrm{\Delta }M=(M_{+}-M_{-})/2$, and $\chi =dM/{\mu }_{0}dH$, where ${\tau }_{+}$ is the torque signal from the up-sweep of the magnetic field, and ${\tau }_{-}$ is the torque signal from the down sweep of the magnetic field. (b) Angular dependence of effective susceptibility $\chi$ in a normal state at 0.3 K. A magnetic field of 0.8 T was applied to suppress the superconductivity. The angle $\varphi$ is defined as the tilt angle between the positive $x$ axis and the magnetic field direction. Reversing the sign of the magnetic field is equivalent to rotating the cantilever 180°. We thus used the negative field values to complete the 360° angular dependence (open circles). It is clearly shown that $\sin 6\varphi$ is dominant. This indicates that the crystal has hexagonal symmetry. (c) The FFT plot of the data shown in (b). Here, $A_{6\varphi }$ is dominant, while $A_{2\varphi }$ and $A_{4\varphi }$ are very small. (d) Angular dependence of effective susceptibility $\chi$ in a superconducting state at 0.3 K. The crystal symmetry is lower than the normal state, resulting in a lower $n$ periodic function of $\varphi$. (e) FFT plot of the data shown in (d). Note that $A_{2\varphi }$ and $A_{4\varphi }$ are dominant, while $A_{6\varphi }$ is very small. The contrast between the pattern in the normal state and the superconducting state demonstrates the breaking of the rotational symmetry in the superconducting state.

Figure 3

The angular dependence of spontaneous effective magnetization. (a) The angular dependence of spontaneous effective magnetization $\mathrm{\Delta }M-\varphi$. The data were taken at 0.3 K at a few selected $H$ fields. Data taken from the positive field sweep are plotted as filled symbols, and data from the negative field sweep are plotted as open symbols. The solid lines show $f(\varphi )=2A_{2\varphi }\sin (\varphi -30°)\cos 3\varphi$. (b) FFT of $\mathrm{\Delta }M-\varphi$ at ${\mu }_{0}H=0.05\mathrm{T}$. The first peak, $A_{2\varphi }$, is the amplitude of the nematic order term. The second peak, $A_{4\varphi }$, represents the $\sin 4\varphi$ term originating from the product fit function $f(\varphi )=2A_{2\varphi }\sin (\varphi -30°)\cos 3\varphi$. (c) The magnetic field dependence of the FFT amplitudes $A_{2\varphi }$ and $A_{4\varphi }$ in the superconducting state of Nb-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The FFT amplitude is plotted in logarithmic scale for clarity. Above 0.6 T, the superconducting hysteresis loop quickly vanishes as $H$ approaches the upper critical field.