Abstract
In twodimensional (2D) NbSe_{2} crystal, which lacks inversion symmetry, strong spinorbit 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 NbSe_{2} ICPs across an atomically thin magnetic insulator (MI) Cr_{2}Ge_{2}Te_{6}. 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 momentumconserving tunneling of ICPs across magnetic domains in the barrier. The doubly degenerate ground states in MI JJs provide a twolevel 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 spinorbit coupling, opening up an avenue to designing new superconducting phasecontrolled quantum electronic devices.
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Introduction
In a crystal which lacks inversion symmetry, the relativistic coupling between spins and electron orbits creates a momentumdependent spin splitting, leading to spinpolarization without magnetism^{1,2,3,4}. Twodimensional transition metal dichalcogenides (TMDs), such as 2H phase of NbSe_{2}, MoS_{2}, and TaS_{2}, exhibit unusual superconducting properties stemming from strong spin–orbit coupling (SOC) in combinination with their 2D structure with broken inversion symmetry. Several notable properties include inplane upper critical fields that far exceed the paramagnetic spin limit of Bardeen–Cooper–Schrieffer (BCS) theory^{5,6,7}, coexistance of chargedensitywaves with superconductivity down to monolayer limit^{8}, and higher order paramagneticlimited superconductor–normal metal transitions^{9}. These unusual properties result from an outofplane alignment of electron spins forming ICPs^{5,6,7,8,9}. Early theoretical studies predicted anomalous Josephson coupling between two noncentrosymmeteric superconductors, which can carry a spin current^{10,11}. Josephson coupling between ICPs has been realized in 2D TMD superconductor van dew Waals (vdW) heterostructures, such as NbSe_{2} heterostructures with a stacked interface^{12} or using a graphene layer as a weak link^{13}, and suspended MoS_{2} bilayers with electrical gating^{14}. However, the spindependent coupling in Josephson characteristics originating from ICPs has not been realized in these systems. In this work, we couple the 2D NbSe_{2} to the magnetic insulartor Cr_{2}Ge_{2}Te_{6}, which enables phase modulation of the spin wave functions of ICPs. The atomically sharp vdW interfaces in NbSe_{2}/Cr_{2}Ge_{2}Te_{6}/NbSe_{2} allows momentumconserving tunneling and leads to a doubly degenerate nontrivial JJ phase.
Results
Device characterization
Van der Waals heterostructures are ideal platforms for creating atomically thin Josephson coupled systems. Here, we use few atomic layers of the vdW magnetic insulator (MI) Cr_{2}Ge_{2}Te_{6}^{15} as the magnetic barrier to observe a novel Josephson coupling. NbSe_{2}/Cr_{2}Ge_{2}Te_{6}/NbSe_{2} heterostructures (Fig. 1a, b)^{16} were assembled with a modified drytransfer technique (see Methods for device fabrication). Our heterostructures clearly display Josephson coupling across Cr_{2}Ge_{2}Te_{6} barriers ranging from monolayer (ML) to 6ML. Figure 1c–e shows the current density (J) versus voltage (V) characteristic across the JJs with 1, 2, and 6ML MI barriers. For all devices, we find a clear Josephson supercurrent regime at low bias current, which turns into normal conduction at high bias current. The J–V characteristic is hysteric, indicating a switching current density J_{C} (transition from superconducting to normal state) larger than the retrapping current density J_{R} (transition from normal to superconducting state). J_{C} becomes considerably larger than J_{R} for thinner junctions, which is expected since the junction capacitance is larger for smaller the thickness of Flayer d_{F}. For J > J_{C}, we obtain the normal state resistance R_{N} = (1/A)dV/dJ, where A is the effective area of the junction. Figure 1f shows the comparison of R_{N}A obtained from devices with three different d_{F}. We find an exponential increase of R_{N}A, fitted well to a exp(d_{F}/t), where the characteristic quasiparticle tunneling length is t ≈1.3 nm and the normalized barrier resistance is a ≈0.34 kΩ μm^{2}. Importantly, our junction resistance is much lower than that of a typical nonvdW ferromagnetic barrier such as EuS (10^{7}−10^{9} Ω μm^{2} for the thickness of 2.5 nm)^{17}. This relatively small junction resistance is consistent with the smaller semiconducting energy gap of Cr_{2}Ge_{2}Te_{6} (~0.4 eV in plane and ~1 eV outofplane)^{18}. Interestingly, we find that while R_{N}A increases exponentially with increasing d_{F}, the critical current density J_{C} decreases more rapidly. Figure 1f shows that the product V_{C} = J_{C}R_{N}A decreases exponentially with increasing d_{F}, following V_{C} = V_{0} exp(−d_{F}/ξ_{F}) with the prefactor V_{0} ≈ 0.8 mV and a characteristic barrier tunneling length in Cr_{2}Ge_{2}Te_{6} ξ_{F} ≈ 1.4 nm. While V_{0} is comparable to V_{C} ~ 0.65 mV in NbSe_{2}/graphene/NbSe_{2} junctions^{13}, the rapid decrease of V_{C} with increasing d_{F} indicates that the JJ coupling becomes weaker with a thicker magnetic barrier, as expected.
Magnetic Josephson junction
To demonstrate the effect of ferromagnetism in Cr_{2}Ge_{2}Te_{6}, we measure the JJ critical current as a function of applied magnetic field. Figure 2a, b shows the inplane and outofplane magneticfielddependent switching current I_{C} = J_{C}A of the NbSe_{2}/Cr_{2}Ge_{2}Te_{6}(6 ML)/NbSe_{2} JJ. We observe hysteretic behavior of I_{C}(H) for both field directions. I_{C}(H) also shows a sudden drop near zero magnetic field. Interestingly, we find that the hysteresis in the magnetic field reaches values of ~±1.5 T, much larger than the saturation field (the field required to reach the saturation magnetization) of our Cr_{2}Ge_{2}Te_{6} bulk crystals (Fig. 2c), and that of reported values in bulk crystals and thin flakes^{15,19,20}. At high bias, which far exceeds the critical current, the voltage across the junction does not show notable hysteresis (Supplementatry Note 1, Supplementary Fig. 2). Furthermore, the magnetization of our bulk Cr_{2}Ge_{2}Te_{6} shows neither a notable hysteresis nor a strong magnetic anisotropy, consistent with earlier reports^{19}. The larger hysteresis field compared to the saturation magnetic field of Cr_{2}Ge_{2}Te_{6} and the strong anisotropy observed in I_{C}(H) thus cannot be simply attributed to the magnetization of Cr_{2}Ge_{2}Te_{6} alone. Rather, the large hysteresis loop for I_{C}(H) can be related to the microscopic magnetic domain structure of Cr_{2}Ge_{2}Te_{6}. Using Lorentz transmission electron microscopy (see Methods for details), we find that thin Cr_{2}Ge_{2}Te_{6} flakes develop two different magnetic domain structures: stripelike (Fig. 2d) and bubblelike (Fig. 2e). The characteristic domain size is ~100 nm, consistent with previously reported multiple domain structures in thicker Cr_{2}Ge_{2}Te_{6} flakes, where the stripephase is more stable than the metastable bubble phase^{21}. We conclude that the interplay between the magnetic domains of Cr_{2}Ge_{2}Te_{6} and the fielddependent Abrikosov vortex lattice in NbSe_{2} can induce a transition between magnetic states and explain the experimental observations, including the sudden drop in I_{C}(H) near zero magnetic field. This critical current drop can be attributed to the differences in the system energy for the two different vortex states, which is interacting underlying magnetic domains (see Supplementary Fig. 1 and Supplementary Note 1 for detail).
In a Josephson coupling, as a phase difference φ develops between two superconductors, a DC Josephson supercurrent I_{S} = I_{C} sin(φ) flows through the junction. At equilibrium, the vanishing supercurrent at the minimum energy imposes the condition that φ can only be 0 or π. For conventional superconductors with spinsinglet pairing, the spatially symmetric Cooper pair wavefunction enforces φ = 0 as the ground state. When the superconducting (S) electrodes are separated by a ferromagnetic barrier (F), Cooper pairs can acquire an additional phase when tunneling through the magnetic barrier, yielding a spatial oscillation of the superconducting order parameter in the barrier^{22,23,24}. Tuning the thickness of the Flayer, d_{F}, can reverse the sign of the superconducting order parameter across the barrier owing to an exchangeenergy driven phase shift^{23,25,26,27}, resulting in a πphase JJ.
SQUID and ϕ phase
To probe possibly anomalous Josephson phase, we have realized SQUIDs consisting of one MI JJ (NbSe_{2}/Cr_{2}Ge_{2}Te_{6}/NbSe_{2}) and one reference JJ (NbSe_{2}/NbSe_{2}). After the assembly, we create the device by etching away the unnecessary areas (Fig. 3a, note the edges of the NbSe_{2} flakes were aligned parallel to each other (see Methods for details)). The wider MI JJ allows us to balance the critical currents for each JJ for a maximal SQUID critical current \({I}_{{{{{{\rm{SQUID}}}}}}}^{{{{{{\rm{C}}}}}}}(\varPhi )\) as a function of magnetic flux \(\varPhi\) threaded through the SQUID loop. The critical current measured in the SQUID (Fig. 3b) exhibits oscillations with the periodicity \({\varPhi }_{0}=h/2e\). However, we observe an irregular SQUID response in the field range between −1.2 and −2.2 mT, with a telegraphlike signal oscillating between two metastable critical current branches (Fig. 3b, c). This bistability is possibly related to the sudden change in critical current seen in Fig. 2a caused by the change of magnetic structure in the junction. This bistable switching state is an indirect indication of a doubly degenerate ground state of the system (see Supplementary Note 2 for details). Nevertheless, the regular oscillation around zero magnetic field allows us to extract the phase of the MI JJ (NbSe_{2}/Cr_{2}Ge_{2}Te_{6}/NbSe_{2}). In a previous study of SQUIDs with ferromagnetic metallic spin valves^{28}, a controllable switching between 0 and π Josephson junctions has been demonstrated. A SQUID that combines 0/0 or π/π JJs shows a maximal \({I}_{{{{{{\rm{SQUID}}}}}}}^{{{{{{\rm{C}}}}}}}(0)\) (defined as a 0phase JJ), whereas a SQUID combining 0/π JJs shows a minimal \({I}_{{{{{{\rm{SQUID}}}}}}}^{{{{{{\rm{C}}}}}}}\left(0\right)\) (defined as a πphase JJ).
For our SQUID with the MI JJ, we use two schemes to measure the two different switching currents (Fig. 3d): a switching current I_{C−} obtained by sweeping from large negative bias to positive bias and another one, I_{C0}, obtained by sweeping from large positive bias to zero and then back to positive bias. Generally, we find I_{C−} >I_{C0}. More importantly, the phases of their oscillations are different, as shown in Fig. 3e. To obtain the absolute phase of \({I}_{{{{{{\rm{SQUID}}}}}}}^{{{{{{\rm{C}}}}}}}(\varPhi )\), we have carefully calibrated our electromagnet for zero magnetic field using several onchip Al SQUIDs with different sizes (see Supplementary Fig. 7 for details). Strikingly, we find that none of the switching schemes provide 0 or π phase but ϕ_{C−} = 259° and ϕ_{C0} = 59° as shown in Fig. 3e.
The presence of these two nontrivial phases (i.e., not simple multiples of 180°) is reminiscent of two switching current states in a metallic ferromagnetic (F) ϕJJ^{29,30}. It is reported that an arbitrary ϕphase between 0 and π can be realized by engineering the combination of 0 and πJJs^{29}. One example is a long channel metallic FJJ (typically 100 μm)^{30}, where the doubly degenerate ground states are realized by laterally connecting 0junction and πjunction. Here, the πJJ requires the thickness of the Flayer to be comparable to the wavelength of the orderparameter oscillation, implying d_{F} ~10 nm, set by the exchange energy^{24}. In addition, the widths of the 0JJ and πJJ perpendicular to the supercurrent flow direction is restricted to be much longer than the Josephson length, \(\normalsize {\lambda }_{{{\rm{J}}}}=\sqrt{{{\Phi }}_{0}/2\pi {\mu }_{0}{J}_{{{{{{\rm{C}}}}}}}{\lambda }_{{{{{{\rm{C}}}}}}}}\), where λ_{C} is the magnetic penetration length, because the formation of a ϕFJJ in the 0π JJ arrays needs a Josephson vortex pinned at the 0π junction. Such values of d_{F} and \({\lambda }_{{{{{{\rm{J}}}}}}}\) are incompatible with our atomically thin Cr_{2}Ge_{2}Te_{6} based JJ. Specifically, our Cr_{2}Ge_{2}Te_{6} barrier is an atomically thin insulator (d_{F} = 1 nm), which is too thin to exhibit spatial orderparameter oscillations. Furthermore, the lateral size of our JJ, L < 5 μm, is much smaller than \({\lambda }_{{{{{{\rm{J}}}}}}}\) ~10 μm, estimated using our experimentally obtained J_{C} and λ_{C} ≈ 0.1 μm for NbSe_{2} reported previously^{31}. Therefore, the observed ϕJJ formation in our atomically thin MIJJ, manifested by the appearance of doubly degenerate nontrivial phase shifts, requires an alternative mechanism.
Interplay between Ising superconductivity and ferromagnetism
The singlecrystallinity of our vdW heterostructure combined with the strong SOC in the quasi2D superconductor (S) constituent provides two new characteristics for Josephson coupling that are absent in conventional metallic FJJs. First, in contrast to the FJJs constructed by sputtered heterostructures^{32}, momentumconserving tunneling in crystalline vdW heterostructures is allowed between the closely aligned Fermi surfaces of two Slayers in vdW JJ, as the top and bottom S layers in our JJ are aligned along the same crystallographic axis (< 2°–5° misalignment; see “Methods”). Second, the strong SOC in NbSe_{2} fixes the spin quantization axis of the Cooper pairs^{6,33} normal to the substrate, denoted by ↑ and ↓. In NbSe_{2}, due to weak interlayer tunneling and strong inversion symmetry breaking within each layer, two spin components remain localized predominantly in even or odd layers with a sizable spin splitting Δ_{SOC} \(\simeq\)100 meV^{7}. This spinlayer locking results in unconventional Ising Cooper pairing (K↑–K′↓ or K↓–K′↑) inside each layer where K and K′ denote the electronic band near K and K′ points.
The Josephson phase between ICPs on the surfaces of NbSe_{2} across the MI barrier can be sensitively modified by the magnetization direction. For outofplane magnetization, the spin of the ICPs is aligned parallel or antiparallel with the spin of the MI. Similar to a previous theoretical study of JJs with magnetic impurities^{34}, the wave function of ICPs tunneling across the ferromagnetic junction can acquire an additional minus sign with respect to the nonmagnetic junction for sufficiently strong magnetic scattering (Fig. 4a, see “Methods” for details). Importantly, this sign flip of the Josephson coupled ICPs sets the phase of the JJ ground state to be φ = π. As the magnetization of the MIlayer is tilted away from the tunneling direction, the spin of the ICPs can flip during the tunneling process (see “Methods” and Supplementary Note 4 for details). As a result, the ground state of the Josephson junction is at φ = 0 (Fig. 4b). Unlike metallic FJJs, our heterostructure allows for both 0 and πJJs by adjusting the direction of the magnetization in the MI.
A parallel arrangement of 0 and πJJ can lead to a degenerate ϕJJ. Our MIJJ offers such lateral arrays, created by magnetic domain structures in the MI, similar to what is shown in Fig. 2d,e. Here, the domains with outofplane magnetization separated by boundaries with titled spins could lead to a coexistence of 0 and π junction segments. Using a simple model based on a short junction with finite transparency D and a fraction of the plane λ that favors a π junction, we can estimate the Josephson energy E_{J} of the junction, which provides two ϕ values for the degenerate ground states (see Methods section for more details). As an example, Fig. 4d shows E_{J}(ϕ) computed using this model with λ = 0.53 and D = 0.75, resulting in the appearance of two ground states at different nontrivial Josephson phases ϕ_{1} \(\simeq\)100° and ϕ_{2} \(\simeq\)260°. These two states can be obtained by sweeping back from positive or negative bias as the switching currents can be simply controlled by choosing two metastable states in the bistable potential. The telegramlike signal in Fig. 3b is found in the negative magnetic field region. A careful examination of this two state switching behavior implies that two metastable SQUID oscillations with different phases involve (Fig. 3c), suggesting that the bistability can also be controlled by applied magnetic fields. We also note that the experimental value of ϕ_{1} deviates substantially from the theoretical value obtained above. This can be attributed to the experimental anisotropy and inhomogeneity of the devices as observed in the inplane field Fraunhofer pattern (see Supplementary Note 3), which is not considered in our theoretical analysis above.
Discussion
Experimentally, the presence of two minimal phase angles in E_{J}(ϕ) can be directly revealed by measuring the JJ switching current distributions. This distribution is sensitively determined by the escape rate τ ^{−1} from a tilted washboard potential that is created by biasing E_{J}(ϕ) with current I (insets of Fig. 4f). Figure 4e shows the switching current distribution measured using the two abovementioned sweep schemes, as a function of current I(t) increasing monotonically with time t. The switching current distribution shows not only different values of the critical current but also a much wider distribution for the ϕ_{c0} state than for the ϕ_{c−} state (the bottom panel of Fig. 4e). Figure 4f shows the escape rate for both ground states, which is calculated using the normalized distribution function P(I_{C}) and the Fulton and Dunkleberger formula \(\tau =(1\int _{0}^{I}P\left(u\right){du})/[P\left(I\right)\frac{{dI}}{{dt}}]\)^{35}. Generally, we find the escape rate for the ϕ_{c0} state to be larger than for the ϕ_{C−} state, suggesting the ϕ_{C−} state is more stable than the ϕ_{C0} state under a bias current. Assuming the escape process is dominated by thermal activation, it gives \({\tau }^{1} \sim {e}^{\varDelta /{kT}}\), where Δ is the barrier height in the tilted washboard potential and k is the Boltzmann constant. Since the experimentally estimated \({\tau }^{1}\) is smaller for ϕ_{C−} than that for ϕ_{C0}, we infer that Δ_{1} < Δ_{2} (Fig. 4d), where Δ_{1} is responsible for the switching of ϕ_{C0} and \({\varDelta }_{2}\) for the switching of ϕ_{C−}, in agreement with the model presented in Fig. 4d. For an applied bias current I smaller than the switching current of the ϕ_{C−} state, retrapping is allowed (inset of Fig. 4f), which consistently explains the slower increase of \({\tau }^{1}\) of the ϕ_{C0} state before the switching of the ϕ_{C−} state at higher current.
To date, a metallic ferromagnetic barrier requires d_{F} ≥ 5 nm and a macroscopic lateral junction size, which is not in general suitable for use in dissipationless and compact quantum device components. JJs using magnetic semiconducting GdN barriers in the spin filter device geometry^{32} exhibit an unconventional second harmonic currentphase relation and switching characteristics^{36,37}. In contrast, we have demonstrated a Josephson phase engineering in dissipationless magnetic JJs. ϕphase JJs can serve as useful components for various superconducting quantum electronic devices, such as phase batteries that can be used to bias both classical and quantum circuits, superconductingmagnet hybrid memories and JJbased quantum ratchets^{28,38,39,40}. The spin sensitivity of an Ising Josephson junction together with atomically thin magnetic tunneling barriers provides a route to the fabrication of novel superconducting and spintronic devices.
Methods
Crystal synthesis
NbSe_{2} crystals were grown from precleaned elemental starting materials in an evacuated quartz glass tube, in a 700 to 650 °C temperature gradient, by iodine vapor transport. Cr_{2}Ge_{2}Te_{6} was grown out of a ternary melt that was rich in the Ge–Te eutectic. High purity elements we placed into fritted alumina crucibles^{41} in a ratio of Cr_{5}Ge_{17}Te_{78}, sealed in an amorphous silica ampoule under roughly 1/4 atmosphere of high purity Ar. The ampoule was heated over 5 h to 900 °C, held at 900 °C for an additional 5 h, and then cooled to 500 °C over 99 h. At 500 °C, the excess liquid was separated from the Cr_{2}Ge_{2}Te_{6} crystals with the aid of a centrifuge^{41}. The single crystals grew as plates with basal plane dimensions of up to a cm and had mirrored surfaces perpendicular to the hexagonal caxis (inset to Supplementary Fig. 6). Low field magnetization on a bulk sample is shown in Supplementary Fig. 6 and is consistent with a ferromagnetic transition near 65 K.
Device fabrication
NbSe_{2} and Cr_{2}Ge_{2}Te_{6} crystals of the desired thickness were mechanically exfoliated onto a pdoped silicon chip terminated with 285 nm SiO_{2}. The following fabrication procedure was employed unless explicitly noted otherwise. Exfoliated crystals were identified by optical contrast (some of them were separately characterized with atomic force microscope) in an argonfilled glove box. The thickness of NbSe_{2} flakes ranges from 8 to 16 nm (on average 12 nm) except for the top flake of the 2ML junction (100 nm) and the NbSe_{2}/NbSe_{2} junction in Supplementary Note 2. The NbSe_{2}/Cr_{2}Ge_{2}Te_{6}/NbSe_{2} heterostructure was prepared by polymerbased dry transfer technique, inside of the Ar glove box, with maximum process temperature of typically between 60 and 80 °C so that degradation of the flakes is prevented^{42}. The surface of the stack was examined by atomic force microscopy and/or scanning electron microscopy to identify the clean and atomically flat parts of the junction. Unnecessary parts were removed by reactive ion etching with fluorine gas using an electron (e)beam (Elionix ELSF125) patterned mask. For the SQUID device with fieldcalibration sensors, a doublelayer resist was patterned by ebeam lithography followed by oblique deposition of aluminum. To form Al/Al_{2}O_{3}/Al junctions, an insitu oxidization process (1 mTorr, 10 min) was utilized between the ebeam evaporation of Aluminum. After the liftoff of the ebeam pattern, Ti/Au contacts were patterned by ebeam lithography using a polymethyl methacrylate mask followed by ebeam evaporation to contact both NbSe_{2} and Al. Before this evaporation, the surface was insitu cleaned by ionmilling.
Preparation of the states for different switching current
For the measurement in Fig. 3e, the switching current branch was prepared as follows: first, the device was measured by specific bias sweeps (from positive bias to negative bias, then returning to zero bias) at a magnetic field of 8.2 mT. Then we swept the magnetic field within the range between −0.4 and 0.4 mT to determine the switching current. The other switching current branch was characterized in the same manner, but the initial state was prepared by sweeping from positive bias to zero bias (at the same field of 8.2 mT).
Lorentz transmission electron microscopy
Cr_{2}Ge_{2}Te_{6} flakes were obtained by mechanical exfoliation and transferred onto 50nmthick SiN membranes with holes supported by an Si frame in an Argonfilled glove box, by a similar procedure to the device fabrication. The samples were mounted on the liquidHeliumcooling holder (ULTDT, Gatan) and inserted to the 300kV fieldemission TEM (HF3300S, Hitachi HighTech) specially designed for eliminating the magnetic field in the sample area. The Lorentz micrographs were taken using a defocusing condition in the electron optical system and the sample tilting condition (45°–30° tilt measured from the normal position for the optical axis)^{43}. The thin flake (approximately 18nmthick) over the hole showed the disappearance of the stripe pattern near the holder temperature of 62 K, consistent with Curie temperature of Cr_{2}Ge_{2}Te_{6}.
Theoretical model
The spectrum of monolayer NbSe_{2} has Fermi pockets around the Γ point and around the K and K′ points, and the bands around the K and K′ points have sizable spin splitting Δ_{SOC} \(\simeq\)100 meV due to inversionsymmetry breaking spin–orbit interaction, which results in Ising superconductivity^{7}. In contrast, bulk NbSe_{2} has an inversion center between the layers, such that the SOC is opposite in even and odd layers destroying the spinmomentum locking. Because of weak interlayer hopping, however, bulk NbSe_{2} can be thought of a stack of weakly coupled Ising superconductors with opposite spin polarization in even and odd layers. Tunneling in a Josephson junction predominantly originates from the layer adjacent to the junction and as a result the supercurrent will be carried by ICPs. There can be nonvanishing contribution from the Γ point in the Josephson coupling, which modifies the relative amplitudes of π and 0couplings of the ICP (see Supplementary Note 4).
In our phenomenological model, we neglect any coupling between the layers. An effective lowenergy Hamiltonian for the K and K′ valleys in a single layer can be written in Bogoliubovde Gennes form in the Nambuspinor basis \(\psi =({\psi }_{\uparrow },{\psi }_{\downarrow },{\psi }_{\downarrow }^{{{\dagger}} },{\psi }_{\uparrow }^{{{\dagger}} })\) as
where σ, τ, and λ are Pauli matrices acting in spin, particlehole and valley space, respectively. The SOC is opposite in the two valleys thus preserving timereversal symmetry. The superconducting phase ϕ/2 has opposite signs in the two leads, such that ϕ is the phase difference across the junction. For concreteness, we here assume that the SOC has the same sign both sides of the junctions. In the case of opposite signs, a similar argument for a ϕ junction can be made. We moreover assume the pairing strength Δ to be small compared to the spin splitting 2v p_{SOC} and hence the Cooper pairs consist of two electrons with opposite spins aligned with the z direction. The magnetic layer is approximated by a single insulating band for each spin, whose energy bands are flat in twodimensional momentum space. In the Nambu spinor basis \(({d}_{\uparrow },{d}_{\downarrow },{d}_{\downarrow }^{{{\dagger}} },{d}_{\uparrow }^{{{\dagger}} })\), the Hamiltonian reads
where V and J denote the potential and exchange energy and n is a unit vector describing the direction of the magnetization. An extension to more complicated band structures is possible but will not qualitatively change our conclusions. The superconductors and the magnet are coupled by the hopping Hamiltonian
where t is positive. We now calculate the spectrum of Andreev bound states in the junction. For offresonant tunneling, t ≪ V, J, we can obtain an effective hopping between the left and right superconductor from secondorder perturbation theory
where the effective hopping strength is
If the junction is nonmagnetic, J = 0, we obtain \(\widetilde{t}\) = t^{2}/V and the Andreev spectrum simply is that of a narrow Josephson junction in a BCS superconductor.
where the transparency is D = π^{2}\({\widetilde{t}}^{2}\)ν^{2}/(1 + π^{2}\({\widetilde{t}}^{2}\)ν^{2}) with the ν normal density of states in the superconductors. This is a regular Josephson junction whose ground state is at ϕ = 0. For a purely magnetic junction with n along the z axis, we instead obtain an effective hopping parameter
The hopping has a different sign for the two spin components and, hence, a Cooper pair tunneling across the junction acquires an additional minus sign with respect to a nonmagnetic junction. We can show this explicitly by doing a gauge transformation \({\psi }_{L,\downarrow }\to {\psi }_{L,\downarrow }\) while leaving all other fermions invariant. In this new gauge the hopping is nonmagnetic \(\widetilde{t}\to\)t^{2}/J and the pairing term in the left superconductor changes sign Δ_{L} = \(\langle{\psi }_{L,\uparrow },{\psi }_{L,\downarrow }\rangle \to\)−Δ_{L} while the remaining terms are unchanged. This shows that we obtain the same spectrum as before but with a π phase shift
Hence the ground state of the Josephson junction is at ϕ = π. In fact, the system always forms a π junction when J > V as was first noted in ref. ^{44}.
Now we consider a junction with magnetization along the x direction so that scattering in the junction can result in spin flips. Due to the strong SOC, however, the band structure in the superconductor is helical, meaning that at any particular inplane momentum there is only one spin component at the Fermi level. Thus, if a spin flip occurs in the barrier the other spin component has a large momentum mismatch when entering the superconductor. The latter therefore acts as a hard wall for flipped spins as long as the superconductormagnet interface is sufficiently clean, such that scattering approximately conserves the inplane momentum. Andreev reflection can therefore only happen after an even number of spin flips in the barrier, which means the supercurrent is an even function of J in this case and all spin dependence drops out. This implies in particular that hopping has the same sign for electrons with different spins and hence the junction always has a ground state at zero.
Now let us assume that the magnet is inhomogeneous and there are regions with magnetization along z and x. This means critical current changes sign as a function of the inplane position. When the length scale of the spatial variations is smaller than the Josephson screening length the critical current is simply the spatial average of the current. As a simple model, we assume a fraction λ of the plane favors a π junction described by Eq. (8). The remaining fraction (1 − λ) is instead described by Eq. (6). Note that the latter also includes a conventional Josephson current due to electron near the Γ point. In Fig. 4d we plot the spectrum of the Josephson junction when the transparency is D = 0.75 in both regions and λ = 0.53. See the Supplementary Note 4 for the microscopic description of the theoretical model.
Data availability
The datasets generated during and/or analyzed in the current study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank K.F. Mak, J. Shen, and Y. Otani for a fruitful discussion. The major part of the experiment performed by H.I. was supported by ARO (W911NF1710574) and measurement performed by K.F.H. was supported by NSF (QIITAQS MPS 1936263). The sample fabrication was supported by DOE QPress (DESC0019300). P.K. acknowledges support from the DoD Vannevar Bush Faculty Fellowship N000141812877. H.I. acknowledges JSPS Overseas Research Fellowship and the Nakajima Foundation for support. L.T.N. and R.J.C. acknowledge the US Department of Energy, Division of Basic Energy Sciences, grant DEFG02 98ER45706 for supporting the growth of the NbSe_{2} crystals. Work done at Ames Lab (P.C.C. and N.H.J.) was supported by the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Sciences and Engineering. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DEAC0207CH11358. N.H.J. was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4411.
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H.I. fabricated the sample and analyzed the data. H.I. and K.F.H. performed the measurements. K.H. and D.S. performed TEM experiments. H.I. and P.K. conceived the experiment. F.P. developed the theoretical description. Y.J.S. provided the polymer and optimized transfer process. N.H.J. and P.C.C. grew and characterized single crystals of Cr_{2}Ge_{2}Te_{6}. L.T.N. and R.J.C. grew the NbSe_{2} crystals. Ö.G. contributed to the interpretation of the results. H.I. and P.K. wrote the paper with input from all other authors.
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Idzuchi, H., Pientka, F., Huang, KF. et al. Unconventional supercurrent phase in Ising superconductor Josephson junction with atomically thin magnetic insulator. Nat Commun 12, 5332 (2021). https://doi.org/10.1038/s41467021256081
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DOI: https://doi.org/10.1038/s41467021256081
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