Crossed surface flat bands in three-dimensional superconducting altermagnets

Abstract

Superconducting altermagnets have proven to be a promising ground for emergent phenomena but their study has involved two dimensional systems. In this work, we investigate three-dimensional $d$- and $g$-wave altermagnets with chiral $d$-wave superconductivity and show the formation of crossed surface flat bands due to the underlying symmetries. We find that these crossed flat bands appear at zero energy in the surface along $z$ due to the superconducting nodal lines in the $xy$-plane, while the number of corners is determined by the crystal symmetry of altermagnets. We also show that the superconducting nodal lines give rise to Bogoliubov-Fermi surfaces, which then affect the appearance of zero-energy arcs in the surface along $x$. Moreover, we demonstrate that the crossed surface flat bands, surface arcs, and Bogoliubov-Fermi surfaces give rise to distinct signals in charge conductance, hence offering a solid way for their detection and paving the way for realizing higher dimensional topological phases using altermagnets.

Figures (8)

• Figure 1: (a,b) Sketches of the studied junctions along $z$- (a) and $x$-directions (b), formed by a 3D superconducting altermagnet (light blue) and a normal metal (light red). (c-f) Bulk and surface properties of $d_{xy}$- and $g_{xy(x^2-y^2)}$-wave altermagnets with chiral $d$-wave superconductivity. (c,d) Bulk Fermi volumes for $d_{xy}$- (c) and $g_{xy(x^2-y^2)}$-wave (d) altermagnets, where the cyan and magenta colors indicate up and down spins. The line nodes of the Fermi volumes give rise to Bogoliubov-Fermi surfaces (gray) under chiral $d$-wave superconductivity. (e) A projection of the Fermi volumes on the [100] surface along $x$ leads to surface arc states (orange), whose ends are marked by the point nodes. (f,g) When projecting the Fermi volumes on the [001] surface along $z$, crossed flat bands emerge (green) with their corners defined by the nodes of the 2D spin-polarized altermagnetic Fermi surfaces, indicated by cyan and magenta ellipses for down and up spins. The crossed flat bands appear at zero energy, which is ensured by the nodal lines of chiral $d$-wave of superconductivity.
• Figure 2: (a,c) Zero-bias conductance along the $z$-direction [001] as a function of momenta $k_{x,y}$ for a superconducting AM with $d_{xy}$-wave (a) and $g_{xy(x^2-y^2)}$-wave (c) altermagnetism. (b,d) Normalized total conductance along the $z$-direction as a function of $eV$ for distinct values of the altermagnetic strength $t_{\alpha}$. $\sigma_\mathrm{N}$ indicates the conductance in the normal state at $eV=0$. The insets in (b,d) indicate the normal state Fermi surfaces projected onto the [001] surface for up (magenta) and down spins (cyan). Parameters: $t_{\alpha}=\Delta$ (a,c), $\mu=-4.5t$, $\Delta=0.01t$, $U_\mathrm{b}=5t$, and $\delta=0.01\Delta$.
• Figure 3: (a,c) Zero-bias conductance along the $x$-direction [100] as a function of $k_{y,z}$ for a superconducting AM with $d_{xy}$-wave (a) and $g_{xy(x^2-y^2)}$-wave (c) altermagnetism. (b,d) Normalized total conductance along the $x$-direction [100] as a function of $eV$ for distinct values of the altermagnetic strength $t_{\alpha}$. $\sigma_\mathrm{N}$ indicates the conductance in the normal state at $eV=0$. The insets in (b,d) indicate the normal state Fermi surfaces projected onto the [100] surface for up (magenta) and down spins (cyan). Parameters: $t_{\alpha}=\Delta$ (a,c), $\mu=-4.5t$, $\Delta=0.01t$, $U_\mathrm{b}=5t$, $\delta=0.01\Delta$.
• Figure 4: (a) Zero-bias conductance along the $z$-direction [001] for a superconducting AM with $d_{x^2-y^2}$-wave altermagnetism as a function of $k_{x,y}$. (b) Normalized total conductance along the $z$-direction as a function of $eV$ for distinct values of the altermagnetic strength $t_{d2}$; $\sigma_\mathrm{N}$ indicates the conductance in the normal state at $eV=0$. The inset in (b) shows the normal state Fermi surfaces projected onto the [001] surface for up (magenta) and down spins (cyan). Parameters: $t_{d2}=\Delta$ (a), $\mu=-4.5t$, $\Delta=0.01t$, $U_\mathrm{b}=5t$, $\delta=0.01\Delta$.
• Figure 5: (a-c) Projected zero-energy DOS on the [001] surface as a function of $k_{x,y}$ for a superconducting AM with $d_{xy}$- (a), $d_{x^2-y^2}$- (b), and $g_{xy(x^2-y^2)}$-wave altermagnetism. Parameters: $t_{\alpha}=\Delta$, $\mu=-4.5t$, $\Delta=0.01t$, $U_\mathrm{b}=5t$, $\delta=0.01\Delta$.