Anomalous phase shift and superconducting diode effect in Josephson junctions via thin films of rare-earth intermetallic magnets

TL;DR

This work combines density functional theory and Bogoliubov-de Gennes calculations to predict a material-specific φ0-S/F/S Josephson junction using an ultrathin GdIr$_2$Si$_2$ interlayer. A two-band tight-binding model fitted to DFT captures strong Rashba spin-orbit coupling and exchange splitting, enabling accurate current-phase relation predictions. The CPR exhibits a sizable anomalous phase $\varphi_0$ and a zero-field Josephson diode effect with efficiency up to about 0.3, both showing strong anisotropy with the in-plane magnetization orientation $m_y$. The study highlights the Ln$T_2X_2$ family as a tunable platform for controllable $\varphi_0$-junctions with potential for superconducting memory, logic, and diode-based devices, and points to gating and magnetization dynamics as avenues for further control.

Abstract

The superconductor/ferromagnet/superconductor (S/F/S) Josephson junctions (JJs) with an anomalous ground state phase shift $\varphi_0 \neq 0,π$ ($\varphi_0$-S/F/S JJs) enable the implementation of the zero-field Josephson diode effect with the possibility to control the diode efficiency and polarity. It is just as important that in this case $\varphi_0$ provides a coupling between the superconducting phase and the magnetization of the interlayer. Such $\varphi_0$-S/F/S JJs can be used for superconducting memory and logic circuit applications. Here we present the results of theoretical calculation of the current-phase relationship (CPR), exhibiting the Josephson diode effect and $\varphi_0\neq 0,π$, for a JJ through a specific magnetic material. As the interlayer of the JJ we consider an ultra-thin film of intermetallic lanthanide ($Ln$)-based compound $\mathrm{GdIr_2Si_2}$. Using a combination of density functional theory (DFT) methods and Bogoliubov-de Gennes equations, we study the electronic structure and magnetic properties of the film, construct the effective tight-binding Hamiltonian, perfectly describing its electronic properties, and calculate CPR. The CPRs demonstrate a pronounced $\varphi_0$ of the order of unity and a pronounced Josephson diode effect with the diode efficiency $ \lesssim 0.3$. Moreover, the efficiency can be controlled via rotation of in-plane magnetization in the interlayer. The prospects for utilizing alternative magnetic $Ln$-based materials of the $LnT_2X_2$ family ($T$ is a transition metal and $X$ is a $p$-element from groups III-V) for the implementation in $\varphi_0$-S/F/S JJs are also discussed.

Figures (6)

• Figure 1: (a) Translationally invariant along the $(x,y)$-plane minimal cell of the $\mathrm{GdIr_2Si_2}$ symmetric slab of the $I4/mmm$-phase with Si-termination, which is used as an interlayer of the JJ. (b) Sketch of the planar JJ via the $\mathrm{GdIr_2Si_2}$ interlayer. The length of the magnetic weak link is $L$, the length of the magnet covered by the superconductor is $L_s$, the thickness of the superconducting leads is $h_S$.
• Figure 2: DFT-calculated electron spectra along the $\bar{\rm X}_3 \bar{\rm M} \bar{\rm X}_1$ direction [see inset in (a)] and corresponding atomic structure for the $\mathrm{GdIr_2Si_2}$ slab (shown on the right). (a) P-phase (Ir-termination), (b) P-phase (Si-termination), (c) I-phase (Si-termination). In all cases, magnetic moments of gadolinium atoms are ordered ferromagnetically in each of the layers with magnetization lying in-plane (along the $x$-axis), while the interlayer ordering is antiferromagnetic. Spin expectation value $S_y$ is shown by color and thickness. The spin quantization axis is chosen perpendicular to the magnetization direction to clearly resolve the spin-orbit splitting of the bands without contamination from the Zeeman contribution.
• Figure 3: DFT-calculated electronic spectra for the I-phase $\mathrm{GdIr_2Si_2}$ film, used in calculations of the Josephson characteristics [for crystal structure see Fig. \ref{['fig:different_phases']}(c)]. The magnetization is aligned with the $x$-axis.
• Figure 4: (a) TBH electronic spectra $\varepsilon_n^{\rm{TBH}}(\bm p)$ (lines) and DFT electronic spectra $\varepsilon_n^{\rm{DFT}}(\bm p)$ (dots) calculated along $\bar{\rm X}_3 \bar{\rm M} \bar{\rm X}_1$ momentum direction exploiting the hopping amplitudes presented in Tab. \ref{['tab:hopping_par']}. The $x$ spin component of the $\varepsilon_n^{\rm{TBH}}(\bm p)$ states is shown by color. The magnetization is aligned with the $x$-axis, $\bm m = \bm e_x$. (b) The same for $\bar{\Gamma}_3 \bar{\rm M} \bar{\Gamma}_1$ momentum direction. (c) Sketch of all 12 hops (numbered 0--11) used for construction of the tight-binding Hamiltonian.
• Figure 5: Josephson characteristics of S/${\rm GdIr_2Si_2}$/S JJ with $L=30 a = 12.1$ nm. (a) CPR at $m_y = 0.9$. Inset: Josephson diode efficiency as a function of $m_y$. (b) Positive and negative critical Josephson currents as functions of $m_y$. (c) Anomalous ground state phase $\varphi_0$ as a function of $m_y$.