Quasiparticle Interference of Spin-Triplet Superconductors: Application to UTe$_2$

Abstract

Quasiparticle interference (QPI) obtained from scanning tunneling microscopy (STM) is a powerful method to help extract the pairing symmetry of unconventional superconductors. We examine the general properties of QPI on surfaces of spin-triplet superconductors, where the properties of the $\vec d$-vector order parameter and topological surface bound states offer important differences from QPI on spin-singlet superconducting materials. We then apply the theory to a model specific to UTe$_2$, and compare the resulting QPI with recent STM measurements. We conclude that the two candidate Cooper pair instabilities $B_{2u}$ and $B_{3u}$ exhibit distinct features in the QPI intensity to discriminate these using the experimental data. Characteristic features of the emergent topological surface states protected by chiral symmetry in general, and by mirror symmetries in the case of UTe$_2$, provide further unique signatures to help pinpointing the pairing symmetry channel in this material.

Figures (3)

• Figure 1: QPI in triplet superconductors (a) Fermi surface and $\vec{d}$ vector of a $p_x$ SC with $\vec{d}=p k_x\vec{e}_x$ in 2D. (b) Two contours of constant energy of the quasiparticle dispersion $E_{\mathbf k}$. For small energies, a "banana" close to the nodal point at $k_x=0$ occurs such that scattering processes with relative angle 0 between the $\vec{d}$-vectors (labeled by ${\bf q}_0$) and relative angle $\pi$ (labeled by ${\bf q}_\pi$) are difficult to resolve experimentally. At larger energies $\mathbf q_0$ and $\mathbf q_\pi$ can be more easily resolved and contain information about the relative direction of the $\vec{d}$-vector. (c) 3D analogon for a $B_{3u}$ state on a spherical Fermi surface, compare Fig. S3 in the SM. With the vector $\vec{d}_{B_{3u}}=(p_1k_xk_yk_z, p_2k_z, p_3k_y)$, there are point nodes at the $k_x$ axis such that at small energy the contours of constant energy form small "lentils" centered around the $k_x$ axis (red surface). Scattering at this energy is dominated by processes from the edges (blue lines) similar to the scattering processes of the tips of the "bananas" in two dimensions. (d) The $\vec{d}$-vector along these lines winds around such that there are scattering processes with all relative angles between the $\vec{d}$-vectors. Three example vectors with relative angle of the $\vec{d}$-vector of $0$, $\pi/2$ and $\pi$ are shown. At low energies, the "lentils" (and therefore the blue circles of large DOS) are small and $\mathbf q_0$ and $\mathbf q_\pi$ may not be resolvable experimentally, similar to the 2D case.
• Figure 2: (a) Fermi surface at $k_z=0$ with red (blue) bulk bands dominated by U (Te) orbital content. (b) (0-11) spectral function in the normal state, arising mainly from the U bands. The locations of symmetry-imposed and additional nodes for both B$_{2u}$ and B$_{3u}$ have been indicated in both panels, along with the TRIM points relevant for the TSS. The orange lines in (b) indicate the surface Brillouin zone.
• Figure 3: QPI signal $\delta\rho(\mathbf{q}^{\parallel},\omega)$ and spectral functions $A_{s/b}(\mathbf{q}^\parallel,\omega)=-\frac{1}{\pi}\mathrm{Im}G_{s/b}(\mathbf{q}^{\parallel},\omega)$ on the (0-11) surface of UTe$_2$ in the B$_{2u}$ and B$_{3u}$ phases. (a-b) The bulk U (red) and Te (blue) spectral functions along with the surface states (green). (c,d) [e,f] QPI versus surface momentum at $\omega=0.05\Delta_0$ including the surface-projected bulk [surface] states. Characteristic scattering vectors $\mathbf q_i$ as seen by the STM experiment in Ref. SeamusQPI are indicated by the white arrows. (g,h) [(i,j)] Momentum cuts of the respective panels (c,d) [e,f] at $q_{c^*}=0, \pi$. In panels (g-j) we have also indicated the locations of $\mathbf q_i$. As seen, only the B$_{3u}$ pairing symmetry features significant enhancements at $\mathbf q_1$ and $\mathbf q_5$ compared to the normal state QPI response. In panels (c-f) the plots are in units of $V_0/{\mathrm{(eV)}^2}$ while in panels (g-j) the density modulations have been normalized such that $\sum_{\mathbf{q}^\parallel}\delta\rho(\mathbf{q^\parallel},\omega)=1$.