{'fraction_non_alphanumeric': 0.030398083723052357, 'fraction_numerical': 0.007984487281852401, 'mean_word_length': 4.900067294751009, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Giant magnetoanisotropy in the Josephson effect and switching of staggered order in antiferromagnets

We predict that the amplitude of the Josephson current through an antiferromagnetic weak link changes by several orders of magnitude upon rotation of the N\'eel order parameter characterizing the staggered magnetic order. This occurs due the presence of spin-orbit coupling arising from structural inversion asymmetry which makes the band gap in the antiferromagnet highly sensitive to the staggered order parameter direction. We also demonstrate that when phase-biasing the junction, magnetization dynamics is induced which switches the direction of the N\'eel vector in the antiferromagnet. These results reveal an interesting versatility of antiferromagnetic Josephson junctions as they offer both a large tunability of the supercurrent magnitude via the staggerered magnetization and phase-coherent control over the N\'eel order parameter.


I. INTRODUCTION
In recent years, it has been realized that antiferromagnetic materials can give rise to effects in spin-electronics which are potentially useful technologically as well as interesting from a fundamental physics viewpoint [1]. This includes ultrafast magnetization dynamics on the GHz scale [2] as well as controllable motion of antiferromagnetic domain walls [3]. The use of superconducting materials in spintronics [4] has also been shown to give rise to phenomena such as strongly enhanced spin-accumulation [5,6] and spin Hall effects compared to what is possible in the normal state of metals [7].
Driven by these findings, heterostructures of antiferromagnetic and superconducting materials has very recently also attracted attention [8][9][10][11]13]. In particular the case of uncompensated antiferromagnetic interfaces have been predicted to give rise to interesting spin-dependent effects in superconductors [12], although compensated interfaces have also been shown to offer spin-functionality [10]. Moreover, when spinorbit interactions are present due to e.g. structural inversion asymmetry breaking, additional phenomena emerge such as tunable quantum phase batteries [13].
In a broader perspective, multiple spontaneous symmetry breaking in condensed-matter systems ranks among the most profound emergent phenomena in many-body physics. It is a topic that is not only of interest in terms of studying properties of specific condensed-matter systems, but also because spontaneous breaking of symmetry is responsible for a wide range of physical effects, including the mass differences of elementary particles, phase transitions in condensed matter systems, and emergent phenomena in biology [14].
Here, we investigate the mutual interaction between the supercurrent magnitude and the Néel order parameter in a Josephson junction comprised of two superconductors and an antiferromagnet (Fig. 1). We discover a giant anisotropy in the Josephson effect: by rotating the Néel order parameter, the supercurrent changes by several orders of magnitude. The underlying physics is explained by the momentum-dependence * Corresponding author: jacob.linder@ntnu.no of the band gap in a spin-orbit coupled antiferromagnet. Moreover, we demonstrate that when phase-biasing the junction via e.g. external flux, magnetization dynamics is induced which switches the direction of the Néel vector in the antiferromagnet. This occurs even for extremely small magnetic fluxes which have a negligible Zeeman-coupling to the sublattice magnetization and induce a negligible Meissner response in the superconductors. The results described above show that antiferromagnetic Josephson junctions offer a magnetic transistor-like functionality with strongly tunable supercurrents and that they can be used to both induce magnetization dynamics of the Néel order in addition to detecting its orientation.

II. THEORETICAL FRAMEWORK
The starting point for computing the supercurrent in an antiferromagnetic Josephson junction with spin-orbit coupling arXiv:2108.06342v2 [cond-mat.supr-con] 8 Dec 2022 is the following mean-field Hamiltonian: Here, is the hopping element, is the chemical potential, is the vector connecting sites and , and is the magnetic order parameter on lattice point . The magnetic order parameter on site is obtained from the staggered antiferromagnetic order parameter via = (−1) + with , are labels along the nearest neighbor vector directions and . Moreover, determines the magnitude of the Rashba spin-orbit interaction which is assumed induced from the substrate in both the antiferromagnetic and superconducting region, so thatˆ= −ˆ. Finally, Δ is the superconducting order parameter on site , which is taken as Δ 0 e ±i /2 in the left and right superconducting region. The phase difference drives a supercurrent through the system. Ideally the superconducting order parameter should have been solved self-consistently, however this would have drastically increased the computational complexity. To keep the system solvable would have required greatly reducing the system size, so we have here instead assumed a constant Δ inside the superconductors.
We consider an (110) interface between the antiferromagnet and a superconductor [see Fig. 1(a)], so that all the sites closest to the interface have the same magnetization. For an odd number of chains in the antiferromagnet, there will be a net magnetization proportional to the number of sites on each chain. Such an uncompensated interface with a net magnetization allows for interesting properties not found in compensated interfaces [12,13,15]. Assuming translational invariance in thedirection, the Fourier-transformed Hamiltonian may be written in compact matrix form = 0 + 1 2 † where 0 = − and the basis used is and Δ = Δ with ± ≡ ( +1) ± ( −1) . The indices denote lattice sites along the -direction s.t. even and odd-numbered sites belong to different magnetic sublattices. The matrix is Hermitian and is diagonalized with eigenvalues and eigenvectors [ ↑ , ↓ , ↑ , ↓ ] T . The expectation value for the current flowing across the junction (in the xdirection) is computed using the Heisenberg equation of motion = i/ℏ[ , ] and reads: with similarly defined, and all sums are understood to only go over positive eigenvalues. The current is to be understood as the current passing between lattice site and + 1, and has been normalized to the number of lattice sites in the -direction for convenience. The current may be evaluated at an arbitrary point in the antiferromagnetic region as it is conserved.
To compute the ensuing magnetization dynamics of the staggered order parameter induced by a supercurrent, we use the Landau-Lifshitz-Gilbert equation [16], which in dimensionless form reads for normalized sublattice magnetization , where is the number of lattice sites of sublattice type = { , } in thê -direction, > 0 is the unit-free Gilbert damping and is the dimensionless time coordinate [17]. The effective field entering the LLG equation is obtained from the derivative of the free energy of the system with respect to the sublattice magnetizations and in dimensionless form reads where is the free energy of the Josephson junction and we use the hopping element as the energy scale of the Hubbard model. Hereˆis a unit-vector in the = { , , }-direction, while , Γ and is unit-free parameters denoting the strength of the cubic magnetic anisotropy, hopping parameter and antiferromagnetic exchange coupling respectively. Note that the Helmholtz free energy is used since we will consider a phase-biased Josephson junction rather than a current-biased one.
is obtained from the eigenvalues , of the matrix Hamiltonian through where = 1/ is inverse temperature. As seen, the effective field in Eq. (5) couples the magnetization dynamics equations for the two sublattice magnetizations due to the antiferromagnetic exchange . We assume that the magnetic order parameter in Eq. (1) is proportional to the sublattice magnetization, i.e. = .

A. Giant anisotropy in critical supercurrent
We have computed how the critical supercurrent = |max | changes with the (i) direction of the staggered order parameter and (ii) the magnitude of the spin-orbit coupling in the system. The result is shown in Fig. 2. Due to the possibility of having an anomalous Josephson phase [13], all values of ∈ [− , ) were checked for each parameter set to obtain the critical value of the current. For numerical calculations we assume that = 0, = 0.3 and Δ 0 = 0.1 , with AF = 21 vertical magnetic interlayers and 40 vertical layers in each superconductor. The chosen value of Δ is large compared realistic values, in order for the the superconducting coherence length to be comparable to the junction length of the system. Previous theoretical predictions with similar scaling has been shown to compare well with experiments [18,19]. As the magnitude of the antiferromagnetic gap in ordered-Fe-vacancy ternary iron selenides has been experimentally demonstrated to be of order 100 meV [20] the chosen value of / should be reasonable for a poor conductor. The chosen values of is also expected to be within reach experimentally, it has e.g. been experimentally demonstrated that Ni(111)/graphene bilayers form a gap of around 200 meV [21] due to the Rashba spin-orbit interaction.
Consider first (a) where the staggered order parameter lies either alongˆorˆ. The initial dip upon increasing the spinorbit interaction is a − 0 transition. However, as increases further the critical current is drastically enhanced by several orders of magnitude compared to an antiferromagnetic Josephson junction without spin-orbit coupling = 0. In panel (b), the staggered order parameter is oriented along theˆ-direction. In this case, there is no − 0 transition and the increase in critical current is both delayed to larger values of and is also less dramatic than in panel (a), although still substantial for e.g. = 0.90 . To explain the physical reason this phenomena, we consider the current carried by different transverse modes in the junction, as given by Eq. (3). This is shown in Fig. 3 for several strengths of and orientations of . In the absence of spin-orbit coupling, the current is as expected mostly carried by modes close to normal incidence (small ) as shown in panel (a). The orientation of the staggered order parameter plays no role for = 0. Increasing , the current is initially still carried by small , but is now much larger in the case ˆ becomes sufficiently large in (d), the current is instead carried mostly by modes away from normal incidence when ˆ. To understand this, we finally need to look at the band structure of the spin-orbit coupled antiferromagnet. Assuming translational invariance in all directions, we have computed the band structure in Fig. 4 for = | | = 0.9 and = 0.2 . The first important observation is that the spin-orbit coupling causes the bandgap in the antiferromagnet to narrow for specific values of . In fact, for sufficiently large , the bandgap can close entirely. The key effect which lies behind the giant anisotropy of the critical supercurrent is that the -values at which the bandgap narrows depends on the orientation of , as seen in Fig 4. For ˆ, the gap narrows for small . In contrast, for ˆ, the gap narrows for large . For ˆ, the gap narrows for both small and large .
This effect can be understood from the combination of spinorbit interaction and the antiferromagnetic order. Spin-orbit interaction couples sites on opposite sublattices, where spins prefer anti-parallel alignment along due to the antiferromagnetic order. Normally in a translationally invariant system, spin-orbit interaction prefers the spin to be aligned orthogonally to bothˆand . However for alternating spins, the spin-orbit interaction instead couples to spin-orientations in the plane spanned byˆand . Thus for small the spin-orbit interaction mainly couples to the -and -components of the staggered order parameter, which can cause a narrowing of the gap. For ˆthe spin-orbit coupling at = 0 does not couple to the antiferromagnetic order, and as such cannot narrow the gap.
The critical supercurrent supported by the junction will clearly depend on the magnitude of the band gap in the antiferromagnet, since it becomes exponentially damped with increasing gap. Since we showed above (Fig. 3) that the supercurrent is primarily carried by modes with small , the narrowing gap with increasing for ˆ,ˆthen explains the large increase in critical current seen in Fig. 2(a). In contrast, the narrowing of the gap for large in Fig. 4(c) changes slowly with increasing . This causes the supercurrent to only depend weakly on for ˆuntil a critical value of is reached where the gap is closed for large , in agreement with Fig. 2(b). This closing of the gap in the case ˆfor large values of explains both the -dependence in Fig. 3(d) and why the critical current sharply starts to increase at a critical value of i Fig. 2(b). The main result in the above analysis is thus that for a fixed value of the spin-orbit coupling , the critical current can be tuned several orders in magnitude by rotating the staggered order parameter . We note that a similar effect could also exist in junctions with insulating ferromagnets, the only difference between this model and that of a ferromagnetic junction is the alternating direction of the order parameter . However, in a ferromagnetic junctions much larger values of is required to open a gap in the ferromagnet, which which could change the behaviour of the system. The momentum dependence is also expected to differ as neighboring spins prefer parallel instead of anti-parallel orientations, changing the effect of the spin-orbit interaction.

B. Staggered magnetization dynamics in phase-biased junctions
The above analysis showed that the supercurrent is highly sensitive to the orientation of the antiferromagnetic order parameter . This suggests the existence of the reciprocal effect: a rotation of the sublattice magnetization caused by a phasedifference across the junction. To determine if this is possible, we start by analyzing the free energy of the system and how it depends on the orientation of the sublattice magnetizations.
The total free energy of the junction including an staggered order parameter anisotropy term anis = − AF ( 2 − 2 )/2 is shown in Fig. 5. The system minimizes the free energy for a specific orientation of the order parameter based on the non-trivial competition between the superconducting proximity effect, band-structur effects, and magnetic anisotropies in the system. The key observation is seen in the inset of Fig.  5: the orientation of the staggered order which minimizes depends on the phase difference across the junction. For = 0, the preferred orientation is ˆ, whereas = favors ˆ. The phase of the Josephson junction is controllable by integrating the junction into a superconducting ring threaded by a weak magnetic flux.
We have solved the LLG equations Eqs. (4) and (5) numerically with zero phase difference = 0 as initial condition. As expected from the free energy analysis above, the system attempts to orient alongˆ. However, when switching to , ˆis energetically preferred, causing a reorientation of the sublattice magnetization. The anomalous phase difference in the junction plays no particular role in this dynamic behavior, since the chosen parameters ensure that such an anomalous phase is essentially absent for all orientations of . Switching the phase back to = 0, switches back to ˆ. It is interesting to note that the speed at which the transitionsˆ→â ndˆ→ −ˆfor the staggered order take place are different. This is due to the free energy landscape in the − plane, which has a steepness that is not identical for = 0 and = , as seen in Fig. 5.

IV. CONCLUSION
Our results enable both Néel order parameter control via the superconducting phase and a magnetic transistor-like functionality for a supercurrent with a critical current varying by several orders of magnitude when rotating the Néel order parameter. These findings reveal an interesting versatility of antiferromagnetic Josephson junctions resulting from the interplay of staggered and phase-coherent order which can be tested experimentally.

ACKNOWLEDGMENTS
We thank M. D. Hansen and I. Bobkova for useful discussions. We acknowledge funding from the Research Council of Norway via its Centres of Excellence funding scheme, project number 262633. Appendix A: Effect of self-consistency in Δ Selfconsistency in the order parameters [22] was not considered in the main text. This is a common approximation due to the large increase in computational resources required to study a system selfconsistency. However, to make sure that our results do not change significantly when doing the computations selfconsistency, we show in Fig. 7 a comparison between selfconsistent and non-selfconsistent calculations for the superconducting order parameter. Due to the timeconsumption required for the selfconsistent solution, a smaller number of values have been considered compared to the non-selfconsistent solution. As seen, the two procedures match qualitatively and are comparable quantitatively. Moreover, since the superconducting gap is the smallest energy scale in the problem, it is expected to be the most strongly affected by selfconsistency. Therefore, we have not considered the effect of selfconsistency on the antiferromagnetic order parameter, since we do not expect any substantial change in the results.