Proximity-induced superconductivity and emerging topological phases in altermagnet-based heterostructures

TL;DR

This paper develops a microscopic framework for superconducting proximity effects in a 2D altermagnet layered on a 3D $s$-wave superconductor. By integrating out the SC degrees of freedom, the authors derive an effective self-energy and Green's function for the AM, revealing proximity-induced even-parity singlet and odd-frequency triplet correlations, with RSOC generating additional in-plane odd-parity triplets. The work demonstrates the emergence of weak and strong topological superconducting phases, evidenced by edge Majorana modes and topological invariants, and validates the approach with exact diagonalization. The findings position AM–SC heterostructures as versatile platforms to realize and tune topological superconductivity in two dimensions, with experimental control via interfacial coupling and RSOC, and point to candidate materials such as RuO$_2$, MnTe, and layered altermagnet–superconductor hybrids.

Abstract

We present a theoretical framework for investigating superconducting proximity effect in altermagnet (AM)-superconductor (SC) heterostructures. In general, AMs, characterized by vanishing net magnetization but spin-split electronic spectra, provide a promising platform for realizing unconventional magnetic phases. We consider a two-dimensional $d$-wave AM proximity coupled to a three dimensional ordinary $s$-wave SC. By integrating out the superconducting degrees of freedom, we derive an effective Hamiltonian that describes the proximity-induced modifications in the AM layer in the form of a self-energy. We then derive an effective Green's function to obtain the proximity-induced pairing amplitudes in the AM layer and classify the induced pairing amplitudes according to their parity, frequency, and spin. We find the presence of even-parity singlet and triplet pairing amplitudes in the AM layer. To achieve the odd-parity triplet components, important to realize topological superconductivity, we introduce a layer of Rashba spin-orbit coupling (RSOC) in the heterostructure. We analyse the band topology of this proximity-induced AM-RSOC layer and demonstrate the emergence of both weak and strong topological superconducting phases with edge-localized modes, characterized by winding number and Chern number. These findings highlight the role of AM-SC hybrid setup as a versatile platform for realizing odd-parity triplet pairings and engineering topological superconductivity in two-dimension.

Figures (10)

• Figure 1: Schematic illustration of our heterostructure comprised of a 2D $d$-wave AM layer placed on top of a three-dimensional bulk $s$-wave SC. Cooper-pair tunneling across the interface is described by the effective coupling strength $\lambda_{s}$.The two lobes of the spin-up and spin-down Fermi surfaces of the $d$-wave AM are schematically shown, and in this work we focus on the case where they are rotated by $90^{\circ}$ relative to each other.
• Figure 2: The total DOS $D(\omega)$ of the AM layer is shown for two cases: superconductivity is induced via the proximity effect [panels (a) and (b)] and superconducting pairing incorporated directly into the bulk Hamiltonian [panel (c)]. Panel (a) presents the results for $J_{a}=0$ with different values of $\lambda_{s}$, while panel (b) depicts the case for $\lambda_{s}=0.5\Delta$ with varying $J_{a}$. On the other hand, panel (c) illustrates the bulk SC case for different values of $J_{a}$. The other model parameters used are $\mu=0$ and $\Delta=0.3t$.
• Figure 3: In panels (a) and (b), we display the real and imaginary parts of $\psi(\omega, \mathbf{k})$ and $d_z(\omega, \mathbf{k})$, respectively, as functions of $\omega$ for fixed momentum values $(k_x, k_y) = (1, 0.3)$ (in units of the inverse lattice constant). Panel (c) showcases the absolute value of the triplet amplitude $|d_z|$ in the $k_x$–$k_y$ plane at $\omega = 0.5\Delta$, where the white diagonal lines indicate nodes at which $|d_z|$ vanishes below a numerical tolerance of $10^{-18}$. Panel (d) depicts the ratio of the integrated squared magnitudes of OTE to ESE amplitudes in the $J_a$–$\omega$ plane. Panels (a)–(c) are obtained for $J_a = 0.1\Delta$. The other model parameters are chosen as $\mu = 0$, $\Delta = 0.3t$, and $\lambda_s = 0.5\Delta$.
• Figure 4: Momentum-resolved pairing amplitudes are depicted in the $k_{x}$-$k_{y}$ plane choosing $\omega = 0.5\Delta$. Panels (a) and (b) show the real and imaginary parts of the singlet component $\psi(\mathbf{k})$. On the other hand, panels (c)–(h) display the real and imaginary parts of the triplet components $d_x(\mathbf{k})$, $d_y(\mathbf{k})$, and $d_z(\mathbf{k})$. The finite in-plane triplet components $d_x$, $d_y$ arise in the presence of RSOC [$\alpha=0.3\Delta$]. The other model parameters used are $J_a=0.1\Delta$, $\mu=0 t$, $\Delta=0.3 t$, $\lambda_s=0.5 \Delta$.
• Figure 5: Panels (a)–(d) demonstrate the frequency dependence of the pairing amplitudes at an arbitrary fixed momentum point $(k_x, k_y) = (1, 0.4)$ (in units of the inverse lattice constant) in the Brillouin zone. Panel (a) displays the real and imaginary parts of the singlet component, panels (b) and (c) present the in-plane triplet components $d_x$, $d_y$, and panel (d) illustrates the transverse triplet component $d_z$. In panel (e), we exhibit the absolute magnitudes of the triplet components as a function of the Rashba SOC strength $\alpha$ (the inset depicts the singlet component), while panel (f) presents the absolute magnitudes of both singlet and triplet components as a function of the AM-SC coupling strength $\lambda_s$, plotted on a logarithmic scale. The other model parameters are chosen as $J_a=0.1\Delta$, $\mu=0 t$, $\Delta=0.3 t$, $\lambda_s=0.5 \Delta$.