Inverse proximity effect in thin-film superconductor/magnet heterostructures with metallic and insulating magnets

TL;DR

This study interrogates the applicability of the effective homogeneous-superconductor model with a Zeeman-like exchange field $h_{ m eff}$ in ballistic thin-film S/magnet heterostructures across ferromagnetic metals/insulators and altermagnets. Using tight-binding Bogoliubov–de Gennes calculations and analytic limits, it shows that S/FI and S/AI heterostructures yield a well-defined, nearly uniform spin splitting of the superconducting spectra and LDOS, validating the $h_{ m eff}$ description (with $d$-wave symmetry in AM cases). In contrast, S/FM and S/AM with metallic magnets produce chaotic, branch-dependent $h_{ m eff}$ and unpredictable LDOS splitting, although robust triplet correlations persist and enable spintronic functionalities such as a spin-valve effect in FM/S/FM trilayers. The work highlights the limits of the effective-model approach for metallic magnets and altermagnets, while pointing to impurity-scattering-induced averaging as a pathway to recover superconducting behavior and informs design principles for superconducting spintronics. Overall, the paper clarifies when simple effective-field pictures apply and when full microscopic treatment of the proximity effect is essential for accurate predictions and device applications.

Abstract

Proximity effect in thin-film superconductor (S)/magnet heterostructures with different types of magnets including ferromagnets, antiferromagnets and altermagnets is widely considered in the framework of an effective model, where the heterostructure is replaced by a homogeneous superconductor in the presence of a homogeneous exchange field of a corresponding type. Here we study the extent to which such a model is actually applicable to ballistic thin-film superconductor/magnetic heterostructures. In particular, a comparative analysis of thin-film superconductor/magnetic metal and superconductor/magnetic insulator heterostructures is performed. Metallic and insulating ferromagnets (FM, FI) and altermagnets (AM, AI) are considered. It is shown that in the S/FI and S/AI heterostructures the the proximity effect creates a well-defined spin splitting of the electronic spectra in the S layer. Thus, they are well described by the effective model. At the same time, the proximity effect in S/FM and S/AM heterostructures also creates a spin splitting of the spectra of the S layer, but it has a chaotic spectral and spatial distribution and unpredictable amplitude and, in general, cannot be detected via the spin splitting of the superconducting density of states. Thus, the effective model is not applicable to such heterostructures. Nevertheless, we demonstrate that they support well-pronounced triplet correlations and, thus, can be used for spintronics applications.

Figures (9)

• Figure 1: (a) Sketch of the S/F bilayer under consideration. Intralayer nearest-neighbor hopping parameter $t$ (the same for the S and F layers) and interlayer hopping parameter $t_{SF}$ are shown. The system is infinite in the $(x,y)$-plane. (b)-(c) Sketches of the superconductor and ferromagnet conduction bands for the case of S/FM bilayer (b) and S/FI bilayer (c). Red and blue colors correspond to spin-up and spin-down density of states, respectively. The bottom of the conduction band in the S(F) layer is determined by the parameter $\mu_{S(F)}$.
• Figure 2: Two discrete spin-split branches of the S layer spectrum $\varepsilon_S^{\uparrow,\downarrow} (\zeta_\parallel, n)$. Spin-splitting of each of the branches, which is equal to $2 h_{\rm{eff}}(n)$ is shown.
• Figure 3: Left column: dependence of $h_{\rm{eff}}$ on the branch number $n$ for different S/F heterostructures. Right column: spatial dependence of $h_{\rm{eff}}$ on the number of the superconducting atomic layer for a corresponding heterostructure. (a),(e) S/FI heterostructure with $\mu_F=-4t$, $h=t$, $t_{SF}=t$, $N_F=70$; (b),(f) S/FM heterostructure with $\mu_F=0.5t$, $N_F=70$; (c),(g) S/FM heterostructure with $\mu_F=0.5t$, $N_F=69$; (d),(h) S/FM heterostructure with $\mu_F=-t$, $N_F=70$. Other parameters are $h=0.3t$, $t_{SF}=0.3t$ for all FM cases. $N_S=100$, $\mu_S=t$ for all panels.
• Figure 4: LDOS calculated according to Eq. (\ref{['eq:LDOS_model']}). Panels (a)-(d) correspond to the S/F systems, for which $h_{\rm eff}(n)$ is represented in Figs. \ref{['fig:h_eff_n_z']}(a)-(d), respectively. The Dynes parameter $\Gamma = 0.025\Delta$. $\Delta=0.002t$.
• Figure 5: Spatial distribution of the main proximity-induced properties across the S layer. Three different S/F bilayers are represented: S/FI bilayer and two S/FM bilayers denoted as S/FM1 and S/FM2. (a) Effective exchange field $h_{\rm eff}$; (b) superconducting order parameter $\Delta$; (c) triplet correlations at first Matsubara frequency $F^t(\omega_1)$. $h_{\rm eff}$ and $\Delta$ are measured in units of the superconducting OP of the isolated S layer $\Delta_0$ taken at the same temperature. $F^t$ is measured in units of the singlet superconducting correlations $F^s(\omega_1)$. The parameters of the considered systems are the following. S/FI: $\mu_F=-10t$, $t_{SF}=t$, $h=4t$. S/FM1: $\mu_F=-t$, $t_{SF}=0.2t$, $h=0.5t$. S/FM2: $h=1.5t$, the other parameters are the same as for S/FM1. For all considered systems $\mu_S=t$, $\Delta_0=0.0195t$, $T=0.051\Delta_0$.