Mechanism for Nodal Topological Superconductivity on PtBi$_2$ Surface

TL;DR

This work addresses intrinsic topological superconductivity on PtBi$_2$ surfaces by proposing a microscopic mechanism where anisotropic electron-phonon coupling on Weyl semimetal surface states combines with statically screened Coulomb repulsion. By solving a full momentum-space gap equation, the authors show that nodal $i\times(p_x+i p_y)$-wave pairing emerges when the surface-state bandwidth is comparable to the maximum phonon energy $\omega_D$, with the nodal structure concentrated at the centers of the Fermi arcs. The study reveals that Coulomb repulsion, especially its momentum dependence from in-plane NN terms, stabilizes nodal pairing and that the nodal state can achieve $T_c$ on the order of several kelvin to tens of kelvin depending on parameter choices; importantly, stronger surface screening is predicted to yield a nodeless gap and higher $T_c$. The results imply PtBi$_2$ is a promising platform for intrinsic topological superconductivity with potential Majorana hinge modes, and that Coulomb engineering could enhance superconductivity in real devices.

Abstract

Experiments show that the Weyl semimetal PtBi$_2$ hosts unconventional superconductivity in its topological surface states. Hence, the material is a candidate for intrinsic topological superconductivity. Measurements indicate nodal gaps in the center of the Fermi arcs. We derive that anisotropic electron-phonon coupling on Weyl semimetal surfaces, combined with statically screened Coulomb repulsion, is a microscopic mechanism for this nodal pairing. The dominant solution of the linearized gap equation shows nodal gaps when the surface state bandwidth is comparable to the maximum phonon energy, as is the case in PtBi$_2$. We further predict that if the screening of Coulomb interaction on the surface is enhanced by Coulomb engineering, the superconducting gap becomes nodeless, and the critical temperature increases.

Figures (11)

• Figure 1: Illustration of gap symmetries on a circular Fermi surface around the $\boldsymbol{\Gamma}$ point. Even-parity $i$-wave pairing multiplied by $p_x+i p_y$-wave momentum dependence from spin-orbit coupling yields an odd-parity gap function. The angular extent of the Fermi arcs is indicated in gray for the absolute value of the gap $|\Delta|$. The absolute value of the gap reproduces the experimentally observed gap profile in the Fermi arcs of PtBi$_2$Changdar2025iwave.
• Figure 2: (a) Electron bands in slab geometry shown along a path between high symmetry points in the first Brillouin zone. The inset illustrates the surface states close to the Fermi level. The bands are colored by the location of their eigenstate $W_{\boldsymbol{k}n}$ along the $z$-direction, as indicated in (b). In panel (c), the colored dashed lines show the three bulk acoustic phonon modes, while the black lines illustrate the phonon spectrum in slab geometry. The parameters are $t_o/t = 0.9$, $\beta/t = -1.1$, $\mu/t = -0.01, \mu_o/t = 0.05, \alpha/t = -0.025$, $\gamma/t = -0.025$, $\gamma_1 = -(0.01t)^2$, $\gamma_3 = 0.43 \gamma_1$, $\gamma_4 = 0.9\gamma_1$, $\gamma_6 = 1.5 \gamma_1$, and $L = 10$.
• Figure 3: (a) The dimensionless coupling $\lambda$ as a function of $U$ and $V_N$, scaled by $\lambda_0$ which is $\lambda$ at $U=V_N = 0$. Colors indicate the symmetry of the superconducting gap with the largest critical temperature. The black line shows $V_N = U$ and the region below that line is most realistic. Nodal $i\times (p_x+ip_y)$ pairing dominates the phase space if the axes are extended. Stars indicate the value of $U$ and $V_N$ used for the rows in (b), where the real, imaginary, and absolute value of the gap is shown on the bottom surface Fermi arc. The gaps are shown in units of their own largest absolute value. The parameters are $J = 0.2U$, $V_S = 0.3V_N$, $V_L = 0.5V_N$, $\chi_0 = 10$, $\chi_1 = 22$, $Mt = 24350$, $L=20$, $N_{\text{samp}} = 150$, and otherwise the same as Fig. \ref{['fig:bands']}.
• Figure S1: (a) The surface band $\epsilon_{\boldsymbol{k}}$ in the 1BZ with high-symmetry points marked. The values are shown at the 6498 points used in the adaptive quadrature and the size of each marker is scaled by the weight of the point. We found the adaptive quadrature by integrating $\chi_{\boldsymbol{k}}$ at $T=t/500$ with a tolerance of $10^{-2}$. In the white regions there is no surface state. The red points between $\boldsymbol{\Gamma}$ and $\boldsymbol{M}$ show where $|\epsilon_{\boldsymbol{k}}| < \omega_D$. (b) Real part, (c) imaginary part, and (d) absolute value of the gap, all scaled by the largest absolute value $\Delta_{\text{max}}$. (e) Gap on the FS shown as a function of the angle $\theta$ that $\boldsymbol{k}_F$ makes with the $k_x$-axis. Note that a bit more than one complete revolution is shown. The red regions show where points correspond to the red crosses in (d). The black regions show directions where there is no FS and so the gap is simply linearly interpolated here and has no meaning. (f) [(g)] shows the gap along the pink [orange] crosses shown in (d). In (e), (f), and (g) the real part of the gap is shown in blue, the imaginary part in orange, and the absolute value in green, as indicated in (f). Panel (h) is a zoomed in version of (d) close to one Fermi arc, denoted by red crosses. The parameters are $t_o/t = 1.5$, $\beta/t = -1.5$, $\mu/t = -0.05, \mu_o/t = 0.2, \alpha/t = -0.18$, $\gamma/t = -0.2$, $\gamma_1 = -(0.005t)^2$, $\gamma_3 = 0.45\gamma_1$, $\gamma_4 = \gamma_1$, $\gamma_6 = 1.5 \gamma_1$, $Mt = 4.87\times 10^{4}$, $\chi_0 = \chi_1 = 8$, and $L=20$. For Coulomb we set $U = 2t$, $J=0.2U$, $V_N = 0.4t$, $V_S = 0.3V_N$, and $V_L = 0.5V_N$. If $t = 1$ eV, we get $T_c \approx 4.72$ K.
• Figure S2: Solution of the coupled FS average gap equation in Eq. \ref{['eq:D12gapeq']}, with $c_{12} = c_2 = 1$. (a) Real, imaginary and absolute values of $\Delta_{\boldsymbol{k}}^{(1)}$ and $\Delta_{\boldsymbol{k}}^{(2)}$ shown as functions of the angle $\theta$ that $\boldsymbol{k}$ makes with the $k_x$ axis. (b) Real part (solid lines) and imaginary part (dashed lines) of the phonon mediated interaction $\bar{V}_{\boldsymbol{k} \boldsymbol{k}'}^{\text{ph}}$ as a function of the angle $\theta'$ that $\boldsymbol{k}'$ makes with the $k_x$ axis. The color denotes at what angle $\theta$ that $\boldsymbol{k}$ is fixed. All momenta are on the FS. For the blue curves, $\boldsymbol{k}$ is in the center of the rightmost Fermi arc. For the orange curves, $\boldsymbol{k}$ is fixed at the midpoint between the center of the Fermi arc and its endpoint. (c) $\bar{V}_{\boldsymbol{k} \boldsymbol{k}'}^{\text{rep}} = \bar{V}_{\boldsymbol{k} \boldsymbol{k}'}^{C}$ and (d) $\bar{V}_{\boldsymbol{k} \boldsymbol{k}'}^{\text{attr}} = \bar{V}_{\boldsymbol{k} \boldsymbol{k}'}^{\text{ph}} + \bar{V}_{\boldsymbol{k} \boldsymbol{k}'}^{C}$ plotted in the same way. The parameters are $W = 0.35t$, $\omega_D = 0.0205t$, $N_{\text{samp}} = 138$, and otherwise the same as in Fig. \ref{['fig:fullgapPRB']}.