Unconventional supercurrent phase in Ising superconductor Josephson junction with atomically thin magnetic insulator

In two-dimensional (2D) NbSe2 crystal, which lacks inversion symmetry, strong spin-orbit coupling aligns the spins of Cooper pairs to the orbital valleys, forming Ising Cooper pairs (ICPs). The unusual spin texture of ICPs can be further modulated by introducing magnetic exchange. Here, we report unconventional supercurrent phase in van der Waals heterostructure Josephson junctions (JJs) that couples NbSe2 ICPs across an atomically thin magnetic insulator (MI) Cr2Ge2Te6. By constructing a superconducting quantum interference device (SQUID), we measure the phase of the transferred Cooper pairs in the MI JJ. We demonstrate a doubly degenerate nontrivial JJ phase (ϕ), formed by momentum-conserving tunneling of ICPs across magnetic domains in the barrier. The doubly degenerate ground states in MI JJs provide a two-level quantum system that can be utilized as a new dissipationless component for superconducting quantum devices. Our work boosts the study of various superconducting states with spin-orbit coupling, opening up an avenue to designing new superconducting phase-controlled quantum electronic devices.

2 junction under perpendicular magnetic field. When the field is large, dense vortices in NbSe 2 can form a uniform magnetic structure in Cr2Ge2Te6. Decreasing the strength of the field decreases the density of vortices. The vortex array increases the energy cost to form the stripe pattern, and the magnetic structure with the hexagonal-like pattern becomes more stable (Supplementary Fig. 1c). At small external magnetic fields, the magnetic field is shielded by NbSe 2 (Meissner effect), and the magnetic domain of Cr 2 Ge 2 Te 6 should prefer the stripe pattern ( Supplementary   Fig.1d). At increased fields, vortices are formed in the NbSe2 layer. The energy cost to form bubble-like pattern decreases while there is an energy barrier to switch between the two types of the structures. Finally, increasing the field further leads to the uniform magnetization state again ( Supplementary Fig. 1b).
As shown in Supplementary Fig. 1, one possible scenario to explain our experimental observation (Fig. 2) is to consider the interaction between the magnetic domain in Cr2Ge2Te6 and vortex lattice in NbSe2. Particularly, forming a stripe magnetic domain pattern in the Cr 2 Ge 2 Te 6 layer near the domain reversal can cost extra energy in vortex configuration in the superconductor. Thus, during the transition from the triangular bubble domain to the more stable stripe domain, one expects a reduction of Josephson free energy, resulting in the step in I C (H).
The magnetization of bulk Cr2Ge2Te6 does not show notable hysteresis (Fig. 2c), which is consistent with earlier reports [1,2]. This indicates a small magnetic structure, which is averaged out in the magnetization measurements, may play a role in switching current of Josephson junction. Further, in NbSe 2 , the lower critical field B C1 is found to be 19 mT and 10 mT for the out-of-plane and in-plane fields, respectively [3]. The low field behavior of Figs. 2a and 2b are attributed to this difference.
We also measured the field dependence of the voltage at a high bias (much larger than the critical current).
This measurement did not show notable hysteresis, as shown in Supplementary Fig. 2. The critical current of NbSe2/Cr2Ge2Te6/NbSe2 junction with a 6-ML Cr2Ge2Te6 barrier at the temperature of 0.3 K as a function of magnetic field perpendicular to the 2D layer. Arrows indicate the sweep direction. The shaded areas labeled with b, c and d indicate the field regions corresponding to the configurations of (b) -(d). b-d, Preferential magnetic structure in Cr 2 Ge 2 Te 6 with vortex pattern in NbSe 2 . Green and orange rectangles represent NbSe 2 and Cr 2 Ge 2 Te 6 , respectively. Black rectangles and dots indicate superconducting vortices. The black lines with triangles (arrows) indicate magnetic field lines. The gray arrows in the orange box indicate magnetization direction of Cr2Ge2Te6. Gray filled area in right panels indicate magnetic domain for up-magnetization in Cr 2 Ge 2 Te 6 (completely filled in (b), filled circles in (c), and filled stripes in (d)). As the vortex can penetrate and focus the magnetic field near its core, the part of the Cr 2 Ge 2 Te 6 exposed to the strong fields should be magnetized along that direction. (d) depicts a Meissner phase.
Supplementary Fig.2| The field-dependence of the voltage at high bias for the device NbSe2/Cr2Ge2Te6/NbSe2. The device is the same as the one shown in Fig. 2a. The thickness of the Cr 2 Ge 2 Te 6 is 6ML. The temperature is at 0.3 K. The bias current is 0.172 A and the field direction is out-of-plane.

Supplementary Note 2. Further characteristics of the SQUID with a 1ML Cr 2 Ge 2 Te 6 barrier junction.
In this section, we provide further characteristics of -junction. We studied the device of Fig. 3b after removing a part of the link in the arm in the SQUID connection. This allows us to see two-critical currents without SQUID oscillations. We have confirmed the device shows two different critical currents with one bias polarity, one for each sweep sequence (inset of Supplementary Fig. 3a). Four branches for switching current were characterized by varying current sweep sequences for both positive and negative bias directions ( Supplementary Fig. 3a).
Supplementary Fig. 3b shows the critical current under the in-plane field.
The probability of switching current was also characterized. Supplementary Fig. 4 shows the temperature variation at the field of -4.3 mT, which shows two switching currents ( Supplementary Fig. 3b), indicating suppression of stochastic behavior with raising the temperature.

Supplementary Note 3. Fraunhofer pattern.
We have also studied the response of the in-plane magnetic field for devices besides the 6-ML one shown in Fig.   2. Supplementary Fig. 5 summarizes the response for NbSe 2 /Cr 2 Ge 2 Te 6 /NbSe 2 Josephson junction (JJ) as well as NbSe 2 /NbSe 2 JJ made between two cleaved surfaces. We note the device without Cr 2 Ge 2 Te 6 was fabricated under ambient conditions, and the thickness of the top and bottom flakes are 12.7 nm and 100 nm, respectively. In addition to the clear Fraunhofer pattern in the NbSe2/NbSe2 junction, the devices with Cr2Ge2Te6 show multiple peaks with the main peak around zero field. The second peaks are located at 0.14 T, 0.59 T and 0.79 T (averaged for positive and negative fields) for NbSe 2 /NbSe 2 JJ, NbSe 2 /Cr 2 Ge 2 Te 6 (1ML)/NbSe 2 JJ, and NbSe 2 /Cr 2 Ge 2 Te 6 (6ML)/NbSe 2 JJ, respectively. The critical current in a Fraunhofer pattern is given by

Supplementary Note 4. Model Calculation
In order to corroborate our theoretical arguments in the main paper, we now evaluate the energy-phase relation of a magnetic-insulator Josephson junction based on a concrete microscopic model. We assume translational invariance in the plane perpendicular to the current, which flows along the z direction. Hence our model becomes effectively one-dimensional and the two in-plane momentum components p || enter as parameters of the Hamiltonian.
NbSe2 has an inversion center between two layers that form the unit cell. As a consequence, each layer has strong spin-orbit coupling, which has opposite sign in even and odd layers. This leads to a large spin-orbit splitting  SOC of the Fermi pockets around the the K and K' points within each layer, i.e., a locking of spin and layer degrees of freedom with spins quantized along the z direction [4]. Because of the weak interlayer hopping in NbSe 2 , we can consider the electrons from even and odd layers as forming approximately independent bands. The Josephson effect is dominated by contributions from the first layer on each side of the magnetic barrier so we can focus only on the odd-layer band around the K and K' points. Any coupling from the other spin band at the K and K' points as well as from the pocket around the  point, which does not exhibit spin-layer locking, will yield an additional contribution to Josephson energy with energy minimum at =0, but will not change our general conclusions.
The strong spin-orbit coupling in the odd layers separates the two spins in momentum space around the K and K' points. Because the spin splitting exceeds the out-of-plane hopping strength, the bandwidth along the out-ofplane momentum p z is smaller than the energy separation between different spin bands and we therefore expect the odd-layer Fermi surface sheets for spin up and down to occur at different in-plane momenta. This assumption is corroborated by DFT calculations [5]. Moreover, the spin splitting also exceeds pairing strength,  SOC >> , which means the Cooper pairs have spins quantized along the z axis. As a consequence, we can consider the Hamiltonian H(p || ) at a momentum p || on the Fermi surface as effectively spinless and ignore scattering to the spin which does not cross the Fermi level at that particular in-plane momentum.

i) Equal spins on both sides of the junction.
We assume for now an orientation of the NbSe 2 such that the Fermi surfaces with the same spin polarization on both sides of the junction are aligned in momentum space. We will later comment on the opposite case. The four Fermi surface sheets can labeled by =1 which equals +1 on the inner (outer) Fermi surface of the K (K') valley and -1 otherwise. The corresponding Hamiltonian in the continuum approximation is given by where  is the chemical potential and the superconducting phase (z)=sign(z)/2 jumps at z=0, the location of the barrier. The narrow magnetic barrier is modeled as a -function and includes potential scattering with strength V and magnetic scattering with strength J, which have both dimensions of velocity. The Hamiltonian has translational invariance in the plane and || / is therefore conserved. The spin-orbit splitting is  SO | || /2 || 8 | such that system behaves as a superconductor for one spin and as an insulator for the other spin.
The scattering matrix of the barrier for states at the Fermi energy is given by where v z is the Fermi velocity in the z direction. At subgap energies, the superconducting leads have zero transmission and the corresponding scattering matrix is given by (iii) Magnetization with arbitrary angle. In the case of a magnetization with both in-plane and out-of-plane components, n = (n x , 0, n z ), the in-plane part, which leads to spin flip scattering, can again be ignored. This case therefore reduces to Eq.(S7) with the replacement J Jn z , i.e., it smoothly interpolates between the cases (i) and (ii). As the magnetization is tilted towards the x -y plane, n z becomes smaller effectively reducing the magnetic scattering amplitude. For sufficiently small out-of-plane magnetization the Josephson energy will always have a global minimum at =0.
The total Josephson energy depends on the spatial distribution of magnetic domains. If we assume that a fraction  of the plane has an out-of-plane magnetization and the rest has an in-plane magnetization, the total ground state energy is (S10) Here we assume for simplicity a magnetization that is either in plane or out of plane, although a more complicated magnetic texture can be easily included. The ground state phase different for  = 0.9 is plotted in Supplementary Fig.   9. There is clearly an extended region of parameter phase with a nonzero phase difference.

ii) Opposite spins on both sides of the junction.
In our model, we have so far assumed that the NbSe2 Fermi surfaces have the same spin in the first layer on both sides of the junction. In the case when the spins are opposite, we can consider the same model with the spin orbit coupling replaced by  SOC   SOC sgn(z). Cooper pair tunneling across the barrier now requires a spin flip, i.e., the magnetization must be in-plane. A similar calculation as above, shows that the ground state energy has a minimum at  = in a sizable fraction of parameter space. Note that in this case we need to choose   0, because in the presence of a hard wall (i.e., =0) the wavefunction would have a node at the -function, which would render spin flips impossible and result in zero Josephson current.
When the magnetization is out of plane, our simple model yields no Josephson current because of spin conservation. In that case, other contributions, e.g., from the  point or due to spin mixing around the K and K' point, 10 would presumably lead to a ground state at zero phase. In conclusion, there is a similar competition between zeroand -junctions that can conceivably result in an overall ground state at a nontrivial phase  0, .