Topological electron and phonon flat bands in novel kagome superconductor XPd5 (X=Ca, Sr, Ba)

TL;DR

The paper addresses the coexistence of topological electronic flat bands and topological phonon flat bands in a real kagome metal. It employs first-principles DFT/DFPT calculations and Migdal-Eliashberg theory to map the electronic structure, phonons, surface states, van Hove singularities, and superconducting tendencies in XPd5 (X=Ca,Sr,Ba). Key findings include a topological electronic flat band at the Fermi level with $\mathbb{Z}_2=1$, multiple van Hove singularities arising from Pd $d$ orbitals, and a phonon flat band generated by out-of-plane kagome vibrations that destructively interfere, described by a simple spring-mass model. Electron-phonon coupling yields Tc values of 4.25 K, 2.75 K, and 3.35 K for CaPd5, SrPd5, and BaPd5, respectively, spotlighting XPd5 as a platform to study fermion-boson interplay and novel superconducting states.

Abstract

Fermionic and bosonic localized states induced by geometric frustration in the kagome lattice provide a distinctive research platform for investigating emergent exotic quantum phenomena in strongly correlated systems. Here, we report the discovery of coexisting electronic and phononic flat bands induced by geometric frustration in a novel kagome superconductor XPd5 (X=Ca, Sr, Ba). The electronic flat band is located around the Fermi level and possesses a nontrivial topological invariant with Z2=1. Additionaly, we identify multiple van Hove singularities (vHS) arise from the kagome Pd d orbitals with distinct dispersion and sublattice features, including conventional, higher-order vHS and p-type, m-type vHS. Specifically, our investigation of the vibrational modes of the phononic flat band reveals that its formation originates from destructive interference between adjacent kagome lattice sites with antiphase vibrational modes. A spring-mass model of phonons is established to probe the physical mechanism of the phononic flat bands. Furthermore, the calculations of electron-phonon coupling in the XPd5 reveal superconducting ground states with critical temperatures (Tc) of 4.25 K, 2.75 K, and 3.35 K for CaPd5, SrPd5, and BaPd5, respectively. This work provides a promising platform to explore the Fermion-boson many-body interplay and superconducting states, while simultaneously establishing a novel analytical framework to elucidate the origin of phononic flat bands in quantum materials.

Figures (5)

• Figure 1: Crystal structure of kagome lattice $X$Pd$_5$ in (a) side view and (b) top view. (c) Tight-binding model of a kagome lattice considering the nearest-neighbour electron hopping termyin2022topological. (d) Brillouin zone diagram of $X$Pd$_5$ with labeled high-symmetry points.
• Figure 2: Electronic structure of CaPd$_5$. (a) Band structure without SOC, where the ideal FB, vHS, and DP are labeled. Notably, at the L point near $E_{\mathrm{F}}$, the solid green and blue lines denote p-type vHS, while the solid purple line indicates a m-type higher-order vHS. (b) Band structure with SOC, the irreducible representations of bands near $E_{\mathrm{F}}$ are labeled. (c) The atomic orbital projected band structure and (d) the orbitally resolved band structure are respectively presented. (e) Surface states along the $\bar{\Gamma}-\bar{\mathrm{M}}-\bar{\mathrm{K}}-\bar{\Gamma}$ high-symmetry path on the (001) surface of CaPd$_5$. (f) Calculated Fermi surface at a fixed energy of 0.002 eV above $E_{\mathrm{F}}$. 3D band structures associated with the vHS identified in panel (a): (g) blue vHS, (h) green vHS, and (i) purple vHS. Black lines and curves delineate the constant energy contours at their respective saddle point energies of 0.224 eV, 0.316 eV, and 0.290 eV for the blue, green, and purple vHS, respectively. (j) Top view of the higher-order vHS 3D band structure, with white dashed lines as guides to confirm its higher-order characteristics. (k) Sign structure (blue or red) and spatial orientation of the $B_{3g}$ orbital at kagome lattice sites.
• Figure 3: Kagome flat-band origin of the bosonic mode in CaPd$_5$. (a) Calculated atomic displacements associated with the phonon flat-band in CaPd$_5$, with red arrows indicating the directions of atomic vibrations. (b) Top view of the kagome lattice with three kagome sites labeled as 1, 2, and 3, where $+$ and $-$ represent the vibrational phases of the flat band at neighboring sublattices, respectively. (c) Atomic contribution projection of the kagome phonon flat-band, where orange indicates contributions from Pd atoms on the kagome lattice, and dark blue represents contributions from other atoms. (d) Side view and (e) bottom view of the 3D phonon flat-band. (f) The proportion of $z$-direction vibrations at kagome sites relative to total atomic vibrations. (g) Phase difference of vibrations between sites 2 and 3. (h) Phonon spring-mass model of CaPd$_5$.
• Figure 4: (a) Phonon dispersions weighted by the magnitude of the phonon linewidth for CaPd$_5$. The right panel is the Eliashberg spectral function $\alpha ^2F(\omega )$ (black line), and the integrated strength of EPC $\lambda (\omega )$ (red line). (b) Phonon dispersion weighted by different atomic vibrational modes of CaPd$_5$. The right panel is the total (colored zone) and vibrational mode-resolved (colored lines) phonon density of states. Figs. (c)–(d) illustrate the atomic vibrations corresponding to the principal peaks in the $\alpha ^2F(\omega )$ function shown in Fig. (a). Specifically, Fig. (c) depicts the primary atomic vibrations for peaks i, ii, and iii, while Fig. (d) shows for peak iv.
• Figure 5: Calculated (a) superconducting gap and (b) maximum eigenvalue of CaPd$_5$ as a function of temperature.