Chiral antiferromagnetic Josephson junctions as spin-triplet supercurrent spin valves and d.c. SQUIDs

Spin-triplet supercurrent spin valves are of practical importance for the realization of superconducting spintronic logic circuits. In ferromagnetic Josephson junctions, the magnetic-field-controlled non-collinearity between the spin-mixer and spin-rotator magnetizations switches the spin-polarized triplet supercurrents on and off. Here we report an antiferromagnetic equivalent of such spin-triplet supercurrent spin valves in chiral antiferromagnetic Josephson junctions as well as a direct-current superconducting quantum interference device. We employ the topological chiral antiferromagnet Mn3Ge, in which the Berry curvature of the band structure produces fictitious magnetic fields, and the non-collinear atomic-scale spin arrangement accommodates triplet Cooper pairing over long distances (>150 nm). We theoretically verify the observed supercurrent spin-valve behaviours under a small magnetic field of <2 mT for current-biased junctions and the direct-current superconducting quantum interference device functionality. Our calculations reproduce the observed hysteretic field interference of the Josephson critical current and link these to the magnetic-field-modulated antiferromagnetic texture that alters the Berry curvature. Our work employs band topology to control the pairing amplitude of spin-triplet Cooper pairs in a single chiral antiferromagnet.

We consider two different AFMs of topological origin: the conventional itinerant AFM such as IrMn, and the chiral AFM corresponding to our Mn3Ge. The two Hamiltonians are markedly different. Following Krivoruchko S1 , the minimal Hamiltonian in the conventional AFM reads: where is the exchange field characterizing the magnetization of the AFM and † ( ), Ψ ( ⃗) , φ † ( ⃗) and φ ( ⃗) are the field operators for electrons with spin ∈ {↑, ↓} at position ⃗ in the two sublattices of the AFM. In the case of a chiral AFM, the Weyl nodes give rise to a two-orbital structure and a minimal Hamiltonian can be written as S2,S3 : where 1 ,2,3 are Pauli matrices acting in the chirality (orbital) space of the AFM, σ ⃗ ⃗⃗ = (̌1,̌2,̌3) is a spatial vector which encloses the spin Pauli matrices, is the Fermi velocity, is the exchange interaction and δM ⃗⃗⃗⃗⃗⃗ ( ⃗) is the magnetic texture of the AFM. From Eq.(S2), if the uniform component 0 of the magnetization is along the z-axis, the Weyl nodes will be located along the z-axis of the spin states (corresponding to ̌3) at = ± 0 / . We will see below that the spin-momentum locking, the Weyl nodes and the associated magnetic texture deeply modify the superconducting proximity effect with respect to the conventional AFM case.
We now study the physical consequences of the two above Hamiltonians. More precisely, our aim is to study the spatial propagation of superconducting correlations in the presence of the antiferromagnetic order, in the diffusive regime which is relevant for our experiment. To describe the superconducting proximity effect, one can use the matrix Gorkov Green's function ̂ defined as: ̂( ⃗ 1 , ⃗ 2 , 1 , 2 ) = − ( 1 − 2 )〈{Ψ( ⃗ 1 , 1 ), Ψ † ( ⃗ 2 , 2 )}〉 (S3) where 〈 〉 denotes the thermal average, { } is the anticommutator, and Ψ( ⃗, ) = ) . Due to the appearance of the field operators and their conjugates in Ψ( ⃗), the Green's function has a structure not only in the spin and orbital (or chiral) space but also in the Nambu (electron/hole) space, and the matrix Green's function contains both the normal and 'anomalous' we study supercurrents which are equilibrium quantities. Since we consider a stationary problem we make the Fourier time transform ̂( ⃗ 1 , ⃗ 2 , ) = ∫ ( 1 − 2 )̂( ⃗ 1 , ⃗ 2 , 1 , 2 ).
We start with the case of the AFM. It is important to specify that even though we consider a uniform AFM, the interface between superconductor (SC) and AFM can have a finite (and noncolinear) magnetization, depending on their microscopic details. Such an interface (uncompensated) magnetization could induce the spin-rotation process which in turn leads to the conversion of the singlet superconducting correlations into triplet correlations with either spin 0 or spin 1 S7 . It is thus essential to study the propagation of all types of the superconducting correlations in the AFM. This description can be carried out by generalizing the quasiclassical theory of superconductivity. For this purpose, it is convenient to redefine the Gorkov Green's function as ́( ⃗ 1 , ⃗ 2 , 1 , 2 ) =̂33̂( ⃗ 1 , ⃗ 2 , 1 , 2 ) . This Green's function ́ follows the Gorkov equations: (̂3̂3 + ℏ 2 ∇ ⃗ ⃗⃗ 1 2 2 + −13̂1 +̂)́( ⃗ 1 , ⃗ 2 , ) = 1 ( ⃗ 1 , ⃗ 2 ) (S4) where 1 is the identity in the spin⊗orbital⊗Nambu space. These equations can be obtained by writing down the equations of motion for the field operators due to the Hamiltonian ̂. We have added a self-energy term ̂ which accounts for electronic scattering on the impurities of the material, along the standard approach of the quasiclassical theory of superconductivity.
In the limit where the electronic correlations evolve on a characteristic scale much larger than the Fermi wavelength, one can simplify the Gorkov description by making a quasiclassical approximation. We define the quasiclassical Eilenberger Green's function in the mixed representation: Due to the integration with respect to = 2 /2 , ̂ depends only on the direction ⃗⃗= ⃗/ Equation (S7) is the same as the one found by Krivoruchko S1 but with triplet correlations explicitly taken into account. We now define the isotropic Green's function: is an angular integration on the direction ⃗⃗ . This last Green's function is a relevant quantity when impurity scattering in the material is strong. The impurities yield an isotropization of the electronic correlations S4 , which leads to the Usadel equation: where is the diffusion constant in the material. Equation (S9) resembles that for ferromagnetic metals but with the major difference that the exchange field term in is off-diagonal in orbital and in spin, which leads to substantial differences in the proximity effect.
It is instructive to start by considering the normal state bulk solution of Eq. (S9). In fact, the simplest method is to calculate the bulk value of ́=́0 given by Eqs. (S4) and (S5) without ̂, because in the homogeneous diffusive case ̂=̂= ∮́0 is expected. We have checked that the obtained ̂ is consistent with Eq.(S9) and with the normalization condition ̂2 =1. One has matrix elements: No superconducting correlations are present in ̂ at this stage since it can only occur due to proximity effect, hence terms like † † are zero.
We now consider the spatial evolution of the electronic correlations in the case where the conventional AFM is contacted to the SC. In order to get qualitative insights, it is useful to consider the experimentally relevant limiting case of a weak superconductivity proximity effect S4 . Using a 1D geometry to simplify the discussion, the equations governing the spatial evolution of the triplet and singlet proximity effect, which arise from Eq. (S9), can be approximated as: Eq.(S11), we can replace the normal correlation terms such as † by their bulk value given in (S10). This gives: where the Pauli matrix 3 now acts in the chirality subspace and we have added the self energy ̂′ of scattering impurites. The above equation is similar to that obtained recently for surface states of topological insulators except for the Weyl node structure S5,S6 . The spin-chirality structure as well as its normalization may be found like in these works by looking at the dominant term in the commutator of the right-hand side of Eq. (S15). For well-defined Weyl cones, 0̂33̂3 is a large term S3 which means that the superconducting correlations acquire a spin-chirality structure in the chiral AFM S3 . Similarly to the case of topological insulators S5,S6 , we project the quasiclassical Green's function using the angle -dependent projector 1+⃗⃗⃗.σ ⃗⃗⃗ 2 .
Such a spin structure implies that there are triplet superconducting correlations. In order to derive the Usadel equations, we now expand in spherical harmonics the Eilenberger Green's function using the expression: is very similar to the form found for topological insulator surface states. The right hand side does not involve any characteristic energy as a consequence of spin-momentum locking. This implies that the magnetic texture plays the role of a vector potential which appears in (S18).
The expression of the average current flowing through the junction can be expressed as: Above, we use Usadel Green's functions which depend on the Matsubara frequencies = (2 + 1) instead of the real energy because it will simplify the calculation of the current (one can formally relate ̂ and ̂ using − → ). The value of I depends both on the position of the Weyl nodes (located at ⃗⃗ = (0,0, ± 0 ℏ )) and on the effective axial magnetic field ⃗⃗ = ∇ ⃗ ⃗⃗ × δM ⃗⃗⃗⃗⃗⃗ ( ⃗⃗ ). It is instructive to solve the above problem for δM ⃗⃗⃗⃗⃗⃗ ( ⃗⃗ ) = 0, to put forward the effect of the Weyl nodes, which are directly related to the Berry curvature, and to get a simple idea of the functional form of the supercurrent in Mn3Ge. We work in the weak proximity effect (like in the previous case) for each chirality sector. We thus take as before the bulk value for the normal part of ̂, and we note: We get the following equations for the anomalous parts of the Green's functions: where is the chirality quantum number. We have assumed here that M ⃗⃗⃗⃗ ( ⃗⃗ ) = M 0 ⃗ and ℱ ± ( ⃗⃗ ) = ℱ ± ( ). We also use the following boundary conditions: Bovenzi et al. S3 Deriving the full spin/chiral dependent boundary conditions is beyond the scope of this section S8 and will be the subject of a subsequent theoretical work S10 . Nevertheless, given the structure of the problem which is diagonal in the chiral index, it is reasonable, in the regime we consider, to use the above ansatz that the barrier has one chirality independent term and one term proportional to 3. The solutions of the differential equations (S20a), (S20b) and (S20c) are functions of the type: with = ±1 . Note that ℱ + ( ) and ℱ − ( ) have both spatially increasing and decreasing components, as expected for a finite size layer. Letting be the length of the junction along the Weyl node vectors and defining = √2| |/ℏ , we get the following expression for the current: where ∆ is the superconducting gap in the superconducting electrodes. One can draw important conclusions at this stage and also test experimentally formula (S22) using the Fraunhoffer pattern. In the case where = 0, we can further simplify the above formula to: = sin( )cos (  . This allows us to write how the magnetic texture modifies Eq. (S22): This yields the following expression for the critical current, found as the maximum of I as a function of : Equation (S29) shows that the antiferromagnetic-spin texture shifts the Fraunhofer pattern by a chirality dependent phase ℏ . Since this phase term is proportional to , the current is a hysteretic function of the applied magnetic field with the characteristic mirror symmetric behavior with respect to zero magnetic field. The resulting Fraunhofer patterns are shown in Fig. 1d (on the top of the experimental data, main text) and Fig. S1 as well. They account well for the peculiar hysteretic shifts in Fig. 1d of the main text. The hysteresis of the Fraunhofer pattern stems both from the fact that there is a chirality dependent phase shift arising from the antiferromagnetic-spin texture and that the interface parameter is non-zero.
Important aspects of Eq. (S22) and its generalization in the textured case under a finite OOP magnetic field [Eq. (S28)] are therefore experimentally tested in Fig. 1d (of the main text). In addition, from the above theory, it follows that, in general (as in the topological insulator case S5,S6 ), the CPR of a chiral antiferromagnet Josephson junction has both the critical current We further note that the almost absence of the supercurrent SV signature in the ≈ 80 JJ albeit its larger critical supercurrent being by a factor of 7 (~ 0.7 mA, Extended Data Figure   5) than the ≈ 199 JJ (~ 0.1 mA, Figs. 1d and 2d) allows one to completely exclude any feasibility of the self-field-driven hysteretic effect S14 .