Inherent momentum-dependent gap structure of altermagnetic superconductors

TL;DR

Altermagnets exhibit momentum-dependent spin splitting without net magnetization, raising questions about superconductivity arising from these metals. The authors develop minimal two-sublattice altermagnetic models and solve self-consistently with Bogoliubov–de Gennes theory for onsite and extended-range attractive interactions. They show that onsite attraction induces a highly anisotropic gap with nodes on the Brillouin-zone edges due to a blocking effect from sublattice-to-band weights, while extended-range interactions allow both spin-singlet and equal-spin-pairing triplet states, with ESP favored when spin-splitting is large and generally non-unitary. The ESP states display a nontrivial d-vector structure and can emerge via two phase transitions as temperature decreases. The work provides a concrete framework to understand and identify altermagnetic superconductors in real materials, with implications for materials like strained RuO2 and related compounds.

Abstract

Altermagnetic metals break time-reversal symmetry and feature spin-split Fermi surfaces generated by compensated Néel-ordered collinear magnetic moments. Being metallic, such altermagnets may undergo a further instability at low temperatures to a superconducting state, and it is an interesting open question what are the salient features of such altermagnetic superconductors? We address this question on the basis of realistic microscopic models that capture the altermagnetic sublattice degrees of freedom. We find that the sublattice structure can strongly affect the superconducting gap structure in altermagnetic superconductors. In particular, it imposes nodes in the gap on the Brillouin zone edges for superconductors stabilized by momentum-independent bare attraction channels. We contrast this to the case of superconductivity generated by extended range interactions where pairing is allowed on the Brillouin zone edges and both spin-singlet and equal-spin-pairing triplet states can be stabilized. Equal-spin-pairing triplet superconductivity is generically favored in the limit of large altermagnetic spin-splitting of the bands compared to the superconducting gap scale, and features characteristic non-unitary properties due to the altermagnetic order.

Figures (5)

• Figure 1: (a) Sublattice structure of the 2D lattice model with orange sites defining the $A$-sublattice and blue sites the $B$-sublattice. The different hoppings entering the model are indicated and labeled by different colors. (b) The electronic band structure along the high-symmetry path $\Gamma$-$X$-$M$-$\Gamma$ with sublattice coloring and three different Fermi levels indicated by the dashed horizontal lines. All energies are calculated in the energy scale $t$. Panels (c)-(h) show the Fermi surfaces for the three Fermi levels in (b). Panels (c)-(e) are colored according to spin and (f)-(h) according to sublattice weight.
• Figure 2: The superconducting order parameter stabilized by onsite bare attraction. (a) The structure of the combined form factor $l_{\uparrow,\mathbf{k}}l_{\downarrow,\mathbf{-k}}+m_{\uparrow,\mathbf{k}}m_{\downarrow,\mathbf{-k}}$. (b)-(d) display the superconducting gap structure on the Fermi surface for the chemical potentials shown in Fig. \ref{['fig:model']}(b). The onsite interaction used for the results shown in (b)-(d) is $V=1.5t$. $\Delta_\mathrm{max}$ for (b) $0.17t$, (c) $0.24t$ and (d) $0.18t$. The superconducting order parameter is color-coded according to its maximum value. The coordinates to the path used in Fig. \ref{['fig:coeff_ll_mm_spin_weights']}(c)-(d) is marked in (b).
• Figure 3: In the upper panel, Fourier coefficients $c^i_{\nu,\eta}$ from the decoupling of (a) $l_{\uparrow,\mathbf{k}}l_{\downarrow,\mathbf{-k}}$ and (b) $m_{\uparrow,\mathbf{k}}m_{\downarrow,\mathbf{-k}}$. The color scale is chosen such that the structure of the neighboring sites are clearly visible, as the onsite value $c^l_{00} = c^m_{00} = 0.414$ dominates. The lower panels show the sublattice weights on the $d_{\alpha,\mathbf{k},\sigma}$ band with color-scheme red (blue) for spin-up (down) and solid (dashed) for sublattice A (B). (c) is along the path $\Gamma-X-M-\Gamma$ and (d) along the path $\Gamma-Y-M^\prime-\Gamma$ shown in Fig. \ref{['fig:gap_amplitude']}(b)
• Figure 4: The superconducting order parameter stabilized by NN bare attraction in the regime of weak to intermediate altermagnetic spin splitting. The superconducting order parameter in the Brillouin zone for chemical potentials of (a) $0.3t$, (b) $0.6t$, and (c) $1.0t$. Panels (d)–(f) show the corresponding values of the superconducting order parameter projected onto the Fermi surface. The NN-interaction used and the self-consistent order parameters are for (a) $V^\mathrm{NN}=0.5t$ and $\Delta_{\mathrm{max}}=0.13t$, for (b) $V^\mathrm{NN}=0.6t$ and $\Delta_\mathrm{max}=0.23t$ and for (c) $V^\mathrm{NN}=0.9t$ and $\Delta_\mathrm{max}=0.17t$.
• Figure 5: The superconducting order parameter stabilized by NNN bare attraction. Each horizontal row displays the momentum dependence of the ESP triplet states. All four states are degenerate. In the panels in (a) we show the real and imaginary parts of $\Delta^{\uparrow\uparrow}_{\mathbf{k}}$ and $\Delta^{\downarrow\downarrow}_{\mathbf{k}}$ where an overall phase from the random complex-valued initial guess has been removed, whereas panels in (b) display the associated d-vector components. Panel (c) reveal the non-unitary nature of the ESP states. Finally, panel (d) show that all four states exhibit identical $|\bf d|$. Since $|\bf d|$ and $\mathbf{d}\times\mathbf{d}^*$ are identical for all four states, they exhibit the same spectroscopic gap and ground state energy. For all four self-consistent runs the parameters are $V^\mathrm{NNN}=1t,N_z = 0.2t, \mu=0.6t$.