[Paper Review] Finite-momentum Cooper pairing in proximitized altermagnets

Song-Bo Zhang, Lun‐Hui Hu+1 more|arXiv (Cornell University)|Feb 25, 2023

Physics of Superconductivity and MagnetismPhysics and Astronomy53 references8 citations

TL;DR

The paper shows that in altermagnetic metals proximitized by s-wave superconductors, Cooper pairs acquire finite momentum with strong angular anisotropy, leading to oscillatory order parameters and 0-π transitions in Josephson junctions without net magnetization.

ABSTRACT

Finite-momentum Cooper pairing is an unconventional form of superconductivity that is widely believed to require finite magnetization. Altermagnetism is an emerging magnetic phase with highly anisotropic spin-splitting of specific symmetries, but zero net magnetization. Here, we study Cooper pairing in metallic altermagnets connected to conventional $s$-wave superconductors. Remarkably, we find that the Cooper pairs induced in the altermagnets acquire a finite centre-of-mass momentum, despite the extit{zero} net magnetization in the system. This anomalous Cooper-pair momentum strongly depends on the propagation direction and exhibits unusual symmetric patterns. Furthermore, it yields several unique features: (i) highly orientation-dependent oscillations in the order parameter, (ii) controllable 0-$π$ transitions in the Josephson supercurrent, (iii) large-oblique-angle Cooper-pair transfer trajectories in junctions parallel with the direction where spin splitting vanishes, and (iv) distinct Fraunhofer patterns in junctions oriented along different directions. Finally, we discuss the implementation of our predictions in candidate materials such as RuO$_{2}$ and KRu$_{4}$O$_{8}$.

Motivation & Objective

• Motivate studying superconductivity in altermagnets with zero net magnetization.
• Characterize how proximity-induced Cooper pairs acquire finite momentum in an altermagnet.
• Understand the angular dependence and its impact on order parameter oscillations and Josephson currents.
• Predict experimental signatures such as 0-π transitions and dominant oblique-angle Cooper-pair transport.

Proposed method

• Model the altermagnet with a 2D d-wave-like spin-splitting Hamiltonian H(k)=t(kx^2+ky^2)+Jkxky sz.
• Compute the Cooper-pair propagator (Cooperon bubble) D(r2;r1) and identify the finite momentum q via q=√(2μ)(J+−J−).
• Derive the proximity-induced order parameter from D via Ψ(r′)=λ∫Dx′1 D(r′;x′1,0).
• Analyze the large-width junction limit to obtain an oscillatory, decaying Ψ(r′) with distance y′, Ψ∝cos(√μ Fθ′ y′+π/4)/(y′)^{3/2}μ^{1/4}.
• Extract the dominant propagation angle θm and its dependence on junction orientation φ.
• Compute the Josephson critical current Ic from the δφ-dependent free energy via Ic=(4e/ħ)∫∫ D(x2′,L;x1′,0) dx1′ dx2′.

Experimental results

Research questions

• RQ1 Do altermagnetic systems proximitized by s-wave superconductors host finite-momentum Cooper pairs despite zero net magnetization?
• RQ2 How does the anisotropic spin splitting in altermagnets affect the momentum q and its angular dependence?
• RQ3 What are the observable consequences for the order parameter and Josephson currents in AM-SC and AM-AM junctions as a function of doping μ, junction length L, and orientation φ?
• RQ4 Can RuO2 or similar altermagnets exhibit 0-π transitions without net magnetization, and what are the dominant Cooper-pair trajectories in different junction geometries?

Key findings

• Cooper pairs in the altermagnet acquire a finite momentum q that depends strongly on propagation angle θ and vanishes along crystalline axes.
• The induced order parameter Ψ decays as (y′)^{-3/2} and oscillates with period Pθ′∝1/|Fθ′|, with orientation-dependent periodicity.
• Josephson junctions show 0-π transitions as μ, L, or φ are varied, with Ic given by an expression resembling the order-parameter oscillation term.
• The dominant Cooper-pair transfer direction θm shifts with φ, leading to large oblique-angle transport when the junction is parallel to crystalline axes and junction-direction transport when diagonal (φ≈π/4).
• The results contrast with ferromagnets/antiferromagnets by producing finite-momentum pairing without net magnetization and by symmetry-driven angular patterns.
• RuO2 is identified as a realistic candidate material with estimated oscillation periods on the order of 16–20 nm for typical parameters.