Abstract
Cooper pairs in noncentrosymmetric superconductors can acquire finite centreofmass momentum in the presence of an external magnetic field. Recent theory predicts that such finitemomentum pairing can lead to an asymmetric critical current, where a dissipationless supercurrent can flow along one direction but not in the opposite one. Here we report the discovery of a giant Josephson diode effect in Josephson junctions formed from a typeII Dirac semimetal, NiTe_{2}. A distinguishing feature is that the asymmetry in the critical current depends sensitively on the magnitude and direction of an applied magnetic field and achieves its maximum value when the magnetic field is perpendicular to the current and is of the order of just 10 mT. Moreover, the asymmetry changes sign several times with an increasing field. These characteristic features are accounted for by a model based on finitemomentum Cooper pairing that largely originates from the Zeeman shift of spinhelical topological surface states. The finite pairing momentum is further established, and its value determined, from the evolution of the interference pattern under an inplane magnetic field. The observed giant magnitude of the asymmetry in critical current and the clear exposition of its underlying mechanism paves the way to build novel superconducting computing devices using the Josephson diode effect.
Main
Semiconductor junctions, which exhibit directiondependent nonreciprocal responses, are essential to modernday electronics^{1,2,3}. On the other hand, a key component of many quantum technologies is the superconducting Josephson junction (JJ) where two superconductors are coupled via a weak link^{4}. JJs can be used for the quantum sensing of small magnetic fields^{5,6}, singlephoton detection^{7,8,9} and quantum computation^{10,11,12}. Despite the longstanding research on superconductivity, the realization of the superconducting analogue of the diode effect, that is, the dissipationless flow of supercurrent along one direction but not the other, has been reported only recently in superconducting thin films^{13} and JJs^{14,15}. However, a clear experimental evidence for a specific mechanism leading to this effect is lacking. Recent theoretical work^{16,17,18} has proposed that in twodimensional (2D) superconductors with strong spin–orbit coupling under an inplane magnetic field, Cooper pairs can acquire a finite momentum and give rise to a diode effect, where the direction of the Cooper pair momentum determines the polarity of the effect. At the same time, although there exist prior theoretical proposals^{19,20,21,22}, none of them have been experimentally realized, and a theoretical description for a fieldinduced diode effect in JJs (the Josephson diode effect (JDE)) that would match experimental observations has not yet been formulated.
In this work, we report the discovery of a giant JDE in a JJ in which a typeII Dirac semimetal NiTe_{2} couples two superconducting electrodes and provide clear evidence of its interrelation with the presence of finitemomentum Cooper pairing. The effect depends sensitively on the presence of a small inplane magnetic field. Nonreciprocity ΔI_{c}—the difference between the critical currents for opposite current directions—is antisymmetric under an applied inplane magnetic field B_{ip}. Further, ΔI_{c} strongly depends on the angle between B_{ip} and the current direction and is the largest when B_{ip} is perpendicular to the current and vanishes when B_{ip} is parallel to it. Moreover, we also observe multiple sign reversals in ΔI_{c} when the magnitude of B_{ip} is varied. Our phenomenological theory shows that the presence of a finite Cooper pair momentum (FCPM) and a nonsinusoidal current–phase relation account for all the salient features of the observed JDE, including the angular, temperature and magneticfield dependences of the observed ΔI_{c}. The distinct evolution of the interference pattern under the inplane magnetic field further establishes the presence of FCPM in this system. We examine the plausibility of the FCPM resulting from the momentum shift of the topological surface states of NiTe_{2} under an inplane magnetic field, by performing angleresolved photoelectron spectroscopy (ARPES) measurements and comparison with density functional theory (DFT) calculations. This paper presents the temperature, field and angle dependences of the JDE, and is the only work to date, to the best of our knowledge, in which the fundamental properties of the effect can be explained within a single model providing clear evidence that the features are consistent with FCPM in the presence of a magnetic field.
NiTe_{2} crystallizes in a CdI_{2}type trigonal crystal structure with the space group P\({\bar 3}\)m1, which is centrosymmetric^{23,24} (Supplementary Section I). This 2D van der Waals material is a typeII Dirac semimetal that hosts several spinhelical topological surface states^{23,24}. As discussed later, these surface states play a key role in the JDE. JJ devices were fabricated on NiTe_{2} flakes that were first mechanically exfoliated from a single crystal (Methods). Figure 1a shows the optical images of several JJ devices formed on a single NiTe_{2} flake, where the edgetoedge separation (d) between the superconducting contacts (formed from 2 nm Ti/30 nm Nb/20 nm Au) in each device is different. A schematic of the JJ device is shown in the absence (Fig. 1b) and presence (Fig. 1c) of a Josephson current, where x is parallel to the current direction and z is the outofplane direction. The temperature dependence of the resistance of the device with d = 350 nm (Fig. 1d) shows two transitions: the first transition (T_{SC}) at T ≈ 5.3 K is related to the superconducting electrodes^{25}. A second transition (T_{J}) takes place at a lower temperature when the device enters the Josephson transport regime such that a supercurrent flows through the NiTe_{2} layer. The dependence of both T_{SC} and T_{J} as a function of the edgetoedge separation d between the electrodes is shown in Fig. 1d, inset. Although T_{SC} is independent of d, T_{J} decreases with increasing d, which corroborates that T_{J} corresponds to the superconducting proximity transition of the JJ device^{26}.
To observe the JDE (Fig. 1b,c), we carried out current versus voltage (I–V) measurements as a function of temperature and magnetic field. Figure 1e shows I–V curves in the presence of an inplane magnetic field B_{y} ≈ 20 mT perpendicular to the direction of the current for the device with d = 350 nm. The device exhibits four different values of critical current with a large hysteresis, indicating that the JJs are in the underdamped regime^{27}. During the negativetopositive current sweep (from −50 to +50 µA), the device shows two critical currents, namely, I_{r–} and I_{c+}, whereas during a positivetonegative current sweep (from +50 to −50 µA), the device exhibits two other critical currents, namely, I_{r+} and I_{c–}. In the rest of the paper, we concern ourselves with the behaviour of the critical currents I_{c–} and I_{c+}, which correspond to the critical values of the supercurrent when the system is still superconducting. For small magnetic fields, we find that the absolute magnitude of I_{c–} is clearly much larger than that of I_{c+} (Fig. 1e) (Supplementary Section II details the zerofield data where I_{c+} = I_{c–}). These different values of I_{c+} and I_{c–} mean that when the absolute value of the applied current lies between I_{c+} and I_{c–}, the system behaves as a superconductor for the current along one direction whereas a normal dissipative metal for the current along the opposite direction. We use this difference to demonstrate a clear rectification effect (Fig. 1f) that occurs for currents larger than I_{c+} but smaller than I_{c–}.
To probe the origin of the JDE, the evolution of ΔI_{c} (ΔI_{c} ≡ I_{c+} – I_{c–}) was examined as a function of the applied inplane magnetic field at various temperatures and angles with respect to the current direction. The dependence of ΔI_{c} on B_{y} (field parallel to the y axis and perpendicular to the current) at different temperatures demonstrates that ΔI_{c} is antisymmetric with respect to B_{y} (Fig. 2a). At 60 mK, ΔI_{c} exhibits the maximum and minimum values at B_{y} = ∓12 mT, respectively (Fig. 2a), and the ratio \(\frac{{{{\Delta }}I_{\mathrm{c}}}}{{ < I_{\mathrm{c}} > }}\) is as large as 60%, where \({ < I_{\mathrm{c}} > } = \frac{{\left( {I_{{\mathrm{c}} + } + \left {I_{{\mathrm{c}}  }} \right} \right)}}{2}\). We find that the magnitude of \(\frac{{{{\Delta }}I_{\mathrm{c}}}}{{ < I_{\mathrm{c}} > }}\) systematically increases as the distance (d) between the superconducting electrodes is decreased: for the device with d = 120 nm, \(\frac{{{{\Delta }}I_{\mathrm{c}}}}{{ < I_{\mathrm{c}} > }}\) is as large as 80% (Supplementary Section III). Such a large magnitude of \(\frac{{{{\Delta }}I_{\mathrm{c}}}}{{ < I_{\mathrm{c}} > }}\) at a low magnetic field (B_{y} ≈ 12 mT) makes this system unique, compared with previous reports where either the magnitude of \(\frac{{{{\Delta }}I_{\mathrm{c}}}}{{ < I_{\mathrm{c}} > }}\) was found to be small or a large magnetic field was required to observe a substantial effect^{13,14,15}. We also observe multiple sign reversals in ΔI_{c} when B_{ip} is increased (Supplementary Section IV), a previously unobserved but interesting dependence that is critical to unravelling the origin of the JDE, as discussed below.
The dependence of ΔI_{c} on the direction of the inplane magnetic field with respect to the current is shown for several field strengths (Fig. 2b,c). At smaller fields, ΔI_{c} is the largest when the field is perpendicular to the current (θ = 0°/±180°, where θ is the inplane angle measured with respect to the y axis) and vanishes when the field and current are parallel (θ = ±90°). With regard to the temperature dependence of ΔI_{c}, we find that the magnitude of ΔI_{c} increases monotonically as the temperature is lowered (Fig. 2a). For a quantitative understanding, the temperature dependence of ΔI_{c} for B_{y} = 12 mT (the field at which ΔI_{c} takes the largest value) is shown in Fig. 2d. At temperatures near T_{J}, the variation in ΔI_{c} with temperature can be well fitted by the equation ΔI_{c} = α(T – T_{J})^{2}, where α is a fitting parameter (Fig. 2d, inset).
We propose a possible origin of the JDE as follows. At temperatures close to T_{J} at which superconducting correlations develop in the proximitized region^{26,28,29} and the Josephson effect emerges (Fig. 1d), the free energy F of our system can be expanded in powers of the superconducting order parameters of the two superconducting electrodes, namely, Δ_{1,2}, in the proximitized regions:
where F_{0} is the free energy in the absence of Josephson coupling, and γ_{1} and γ_{2} denote, respectively, the first and secondorder Cooper pair tunnelling processes across the weak link. The presence of higher harmonics account for a nonsinusoidal current–phase relation, as commonly observed in superconductornormal metal–superconductor junctions with high transmission. Importantly, in the absence of timereversal and inversion symmetries, γ_{1,2} are complex numbers, which makes the critical current nonreciprocal, as shown below.
Expressing the order parameters Δ_{1,2} in terms of their amplitude and phase as \({{\varDelta }}_{1,2} = {{\varDelta }}{\mathrm{e}}^{{\mathrm{i}}\varphi _{1,2}}\), the free energy takes the form F = F_{0} – 2γ_{1}Δ^{2}cosφ – γ_{2}Δ^{4}cos(2φ + δ), where φ = φ_{2} – φ_{1} + arg(γ_{1}) is effectively the phase difference between the two superconducting regions. Note that, indeed, when both timereversal and inversion symmetries are broken, the phaseshifted JJ is realized, as observed elsewhere^{30}. Here δ = arg(γ_{2}) – 2arg(γ_{1}) is the phase shift associated with the interference between the firstorder (γ_{1}) and secondorder (γ_{2}) Cooper pair tunnelling processes. The Josephson current–phase relation then includes the second harmonic as
where \({{\varPhi }}_0 = \frac{h}{{2{{e}}}}\) is the superconducting flux quantum, h is Planck’s constant and ℏ = h/2π. When Δ^{4}γ_{2} is small, the critical current of the JJ is reached near a phase difference of φ ≈ ±π/2 and equals
The nonreciprocal part of the critical current is proportional to Δ^{4}.
Since the pairing potential in the proximitized layer behaves as \({{\varDelta }} \propto \sqrt {1  \frac{T}{{T_{\mathrm{J}}}}}\) (refs. ^{26,28,29}), the temperature dependence of ΔI_{c} near T_{J} is then given by
which explains our experimentally measured temperature dependence of ΔI_{c} (Fig. 2d, inset).
Another feature is that ΔI_{c} can change sign as the applied field increases (for example, near B_{ip} ≈ 22 mT (Fig. 2a,c)). Such a sign reversal in ΔI_{c} can be reproduced by including the field dependence of the order parameters \({{\varDelta }}_{1,2} \propto \sqrt {1  \left( {\frac{{\left {{{\bf{B}}}} \right}}{{B_{\mathrm{c}}}}} \right)^2}\) (where B_{c} is the critical field in the proximitized region) and of the phase shift δ due to the Cooper pair momentum.
The inplane magnetic field B_{ip} can induce an FCPM in the junction. We discuss the possible origins of the finitemomentum pairing: the screening current^{31} and/or the Zeeman effect on topological surface states^{32,33,34}. We discuss both these possible origins in detail later in the text, but note that they have the same symmetry. We also note that the phenomenological theory presented here applies to a wider class of JJs made with strongly spin–orbitcoupled materials, where the Zeeman effect can lead to FCPM^{33}.
In the presence of momentum shift q_{x} under B_{y}, the proximitized region effectively turns into a finitemomentum superconductor. The presence of Cooper pair momentum results in a phase shift accumulated during the Cooper pair propagation across the junction: δ ≈ 2q_{x}d. At small values of the field, q_{x} must be linear in B_{y} such that
where B_{d} is a property of the junction geometry and material that can, in principle, be determined based on the specific microscopic origin of the fieldinduced Cooper pair momentum. As a result, ΔI_{c} will have the following field dependence:
Depending on the ratio \(\frac{{B_d}}{{B_{\mathrm{c}}}}\), different scenarios can be realized from this equation; for \(\frac{1}{{2(n + 1)}} < \frac{{B_d}}{{B_{\mathrm{c}}}} < \frac{1}{n}\), there are n sign reversals in ΔI_{c} when the magnetic field is applied in the y direction.
Figure 2e,f shows the dependences of ΔI_{c} on the magnitude of B_{y} and the direction of the inplane magnetic field as obtained from our phenomenological model (equation (7)), where we used B_{c} = 45 mT and B_{d} ≈ 22 mT (Supplementary Section V). To explain the angular direction dependence, we take into account that the nonreciprocal part of the current is proportional only to the x component of momentum shift q_{x} that is proportional to the B_{y} = B_{ip}cosθ component of the inplane magnetic field. In Fig. 2f, it is evident that in each domain \( \frac{\uppi }{2} + \uppi n < \theta < \frac{\uppi }{2} + \uppi n\), the sign reversal of ΔI_{c} occurs where the condition \(\sin \left( {\uppi \frac{{B_{{\mathrm{ip}}}\cos \;\theta }}{{B_d}}} \right) = 0\) is fulfilled, which is evident in Fig. 2c. Thus, our model successfully captures the main features of the JDE, as seen in our experimental data.
To confirm the emergence of a FCPM under an inplane magnetic field, we examine the evolution of the interference pattern (\(\frac{{{\mathrm{d}}V}}{{{\mathrm{d}}I}}\) versus B_{z}) with the field B_{x} parallel to the current direction (Fig. 3b). This interference pattern has a similar resemblance with the Fraunhofer pattern such that a higher I_{c} in the latter translates to a lower \(\frac{{{\mathrm{d}}V}}{{{\mathrm{d}}I}}\) in the former^{33,34}. For this inplane field orientation (B_{x}), the Cooper pairs acquire finite momentum 2q_{y} along the y direction (Fig. 3a), which should not generate a JDE but is expected to change the interference pattern. When a Cooper pair tunnels from position (x = 0, y_{1}) in the left superconductor with order parameter \({{\varDelta }}_1\left( {y_1} \right) = {{\varDelta }}{\mathrm{e}}^{2{\mathrm{i}}q_yy_1}\) to the superconductor on the right at (x = d_{eff}, y_{2}) with \({{\varDelta }}_2\left( {y_2} \right) = {{\varDelta }}{\mathrm{e}}^{2{\mathrm{i}}q_yy_2 + {\mathrm{i}}\varphi _0}\) (where d_{eff} = d + 2λ is the effective length of the junction and λ = 140 nm is the London penetration depth in Nb (ref. ^{35})), the contribution of this trajectory to the current involves a phase factor proportional to the Cooper pair momentum^{33,34}:
in addition to the usual phase factor \(\frac{{2\uppi B_zd_{{\mathrm{eff}}}\left( {y_1 + y_2} \right)}}{{{{\varPhi }}_0}}\) due to the magnetic flux associated with B_{z}. The interference pattern is a result of interference from all such trajectories (Fig. 3b,c).
Due to the additional phase Δφ from the inplanefieldinduced Cooper pair momentum, the interference pattern evolves as the inplane field is increased, splitting into two branches. The Cooper pair momentum 2q_{y} can be extracted from the slope of the side branches^{33,34} (Fig. 3b, solid lines). For the d = 350 nm device, we estimate the average slope to be \(\frac{{B_x}}{{B_z}}\) ≈ 13 (Supplementary Section VI details the estimation process). In Fig. 3c, we show the calculated critical Josephson current, obtained by summing over quasiclassical trajectories (Supplementary Sections VII and VIII), which has a qualitatively similar behaviour to the differential resistance (Fig. 3b), with the same period of oscillations (ẟB_{z} ≈ 0.8 mT) and the slope of the side branches. The slope of the side branches can be expressed as
From this and the value of the slope \(\frac{{B_x}}{{B_z}}\) extracted from the experiment, we find that at B_{x} = 12 mT, the Cooper pair momentum is 2q_{y} ≈ 1.6 × 10^{6} m^{–1}.
Let us compare the estimate of the Cooper pair momentum based on the evolution of the interference pattern with the results of the JDE measurements mentioned above. The maximum JDE is achieved when the phase shift ẟ = 2q_{x}d equals approximately 0.5π, which corresponds to the Cooper pair momentum 2q_{x} = \(\frac{\delta }{d}\) ≈ 4.5 × 10^{6} m^{−1} at B_{x} = 12 mT. Although the JDE and interference peak splitting are measured under inplane magnetic fields along two orthogonal directions, the fieldinduced Cooper pair momenta 2q_{x} and 2q_{y} estimated from these measurements are of the same order of magnitude, further strengthening our conclusion that both effects arise from the FCPM.
It is remarkable that a JJ device comprising a centrosymmetric material such as NiTe_{2} exhibits a large JDE effect arising from finitemomentum pairing. There exist several possible mechanisms that could be responsible for FCPM in our experiment. One candidate is the current screening effect. This mechanism, which lies in the emergence of the finitemomentum Cooper pairing due to Meissner screening, has already been demonstrated to be able to give rise to FCPM^{31} and has been predicted to be responsible for a robust and large JDE in short junctions^{36}. We estimate the value of finitemomentum Cooper pairing arising from screening in Nb contacts. The thickness of Nb leads (30 nm) is much smaller than the London penetration depth, and therefore, we estimate \(q_x \approx \frac{{{e}}}{\hbar }B_y\frac{{h_{{\mathrm{Nb}}}}}{2}.\) At B_{y} = 20 mT, this corresponds to 2q_{x} ≈ 10^{6} m^{–1}. Since this is smaller than the observed value of Cooper pairing, it alone cannot account for the observed effect.
Another plausible origin of the FCPM is related to the Zeeman shift of the proximitized topological surface states of NiTe_{2}. Previous studies^{23,24} have reported the presence of spinpolarized surface states in NiTe_{2} that cross the Fermi level. Here we have used ARPES to examine the detailed fermiology of these surfaces states and estimate their Fermi energy and velocity (Supplementary Sections IX and X). Figure 4a displays a closeup of the experimental Fermi surface, showing both sharp surfacestate bands and diffuse bulk bands that are broadened due to the limited penetration depth of the photoelectrons. The full Fermi surface in the surface Brillouin zone of NiTe_{2} is shown in Fig. 4a (inset). A schematic representing the surface electron and hole pockets (indicated by green and orange lines, respectively) is shown in Fig. 4b,c, where the arrows indicate the spin texture. The electronlike and holelike character of the Fermi surface pockets can be examined from the linecuts (Fig. 4d,e). The energy–momentum dispersion of two topological surface states along the \({{{\bar{\mathrm \Gamma }}}}  \bar M\) direction are shown in Fig. 4e, which is qualitatively reproduced by our ab initio calculations (Fig. 4f) that also indicate their spin polarization. Note that the calculation slightly underestimates the energy at which the lowerlying surface state merges with the valenceband bulk continuum in the experiment. The origin of these surface states is the band inversion of the valence and conduction bands above the Fermi level^{24,37}, which leads to the formation of a spinhelical Dirac surface state that connects the conduction and valence bands (Fig. 4g), similar to the surface state in a topological insulator. The upper branch of this Dirac surface state forms an electron pocket with a relatively small Fermi energy (E_{F} ≈ 15 meV) and Fermi velocity v_{F} ≈ 0.4 × 10^{5} m s^{–1}, as well as a hole pocket with a larger Fermi velocity of 3.3 × 10^{5} m s^{–1}.
The data shown in Fig. 4d–g were used for deducing the spin structure of the surface states shown in Fig. 4b,c, which is in agreement with the experimental reports^{23,24,37}. The spin texture of the surface states (Fig. 4c) clearly shows one (inner) helical hole state and one (outer) helical electron state. The helicity of the hole and electron states is opposite. In the presence of an inplane magnetic field, the Fermi surfaces of these states shift in the same direction; furthermore, in the proximity of the superconductor, finitemomentum Cooper pairing is realized. The momentum shift of these states in a magnetic field can be substantial: the surface states of topological insulators can have large gfactors, for example, a gfactor of ~60 has been found in a previous report^{38}. However, because of the complexity of the surfacestate structure, we leave quantifying the FCPM to a further study. However, the presented analysis suggests that there will be a finite, and likely substantial, contribution of the spin–momentumlocked surface states of NiTe_{2} to the finitemomentum Cooper pairing.
In summary, we have shown that a JJ device involving a typeII Dirac semimetal NiTe_{2} exhibits a large nonreciprocal critical current in the presence of a small inplane magnetic field oriented perpendicular to the supercurrent. The behaviour of the critical current together with the evolution of the interference pattern under the application of an inplane magnetic field provides compelling evidence for finitemomentum Cooper pairing. Whether or not this mechanism is generic to the family of Dirac semimetals remains an open question.
Methods
Exfoliation of NiTe_{2} flakes
Thin NiTe_{2} flakes were exfoliated from a highquality NiTe_{2} single crystal (from HQ Graphene) using a standard Scotchtape exfoliation technique and placed on a Si(100) substrate with a 280nmthick SiO_{2} on top. Although NiTe_{2} is known to be very stable under ambient conditions^{39}, the exfoliation was carried out in a glove box under a nitrogen atmosphere, with water and oxygen levels each below 1 ppm. The thinnest flakes were identified from their optical contrast in an optical microscope and subsequently used to prepare the devices.
Device fabrication and electrical measurements
All the devices used in our experiments were fabricated using an electronbeamlithographybased method. Before exposure, each substrate was spin coated with an ARP 679.03 resist at 4,000 rpm for 60 s followed by annealing at 150 °C for 60 s. After electronbeam exposure and subsequent development, contact electrodes were formed from sputterdeposited trilayers of 2 nm Ti/30 nm Nb/20 nm Au for the JJ devices. The contacts formed from 2 nm Ti/80 nm Au were deposited for the Hall bar devices.
Electrical transport measurements were performed in a Bluefors LD400 dilution refrigerator with a base temperature of 20 mK and equipped with highfrequency electronic filters (QDevil ApS). Angledependent magneticfield measurements were carried out using a 2D superconducting vector magnet integrated within the Bluefors system. A small consistent offset in the magnetic field, of the order of ~1.5 mT, was observed as the magnetic field was swept due to likely flux trapping within the superconducting magnet coils. Direct current (d.c.) voltage characteristics of the JJ devices were measured using a Keithley 6221 current source and a Keithley 2182A nanovoltmeter. Differential resistance measurements were performed using a Zurich Instruments MFLI lockin amplifier with a multidemodulator option using a standard lowfrequency (3–28 Hz) lockin technique. The critical currents for the Fraunhofer pattern were measured in combination with a Keithley 2636B voltage source to sweep the d.c. bias.
ARPES experiments
To explore the momentumresolved electronic structure, ARPES measurements were carried out on a NiTe_{2} single crystal cleaved along the [0001] direction. All the experiments were carried out at the ULTRA endstation of the SIS beamline at the Swiss Light Source, Switzerland, with a Scienta Omicron DA30L analyser. Each crystal was cleaved in situ under ultrahigh vacuum at 20 K to ensure no contamination of the surface. The base pressure of the system was better than 1 × 10^{−10} mbar.
Calculation of evolution of interference pattern in an inplane magnetic field
We calculate the Josephson current by considering the quasiclassical trajectories of Cooper pairs across the junction^{34} (Supplementary Fig. 12):
where
is the total phase difference for a trajectory that starts at (0, y_{1}) and ends at (d_{eff}, y_{2}), d_{eff} = d + 2λ and λ = 140 nm is the London penetration depth^{35}. Further, Δφ_{0} is the phase difference between the order parameters in the two superconducting leads in the absence of an applied field. We have neglected the effect of the finite thickness of NiTe_{2}.
To calculate the evolution of the interference pattern, we compute the Josephson current using the equation above and maximize it by varying Δφ_{0}, which allows one to find I_{c}. To compare the theoretical and experimental predictions, we further adjust the value of the effective junction separation d_{eff} because of flux focusing^{34}, which is carried out by calculating the Fraunhofer pattern at a zero inplane magnetic field (giving a slightly different but still qualitatively similar dependence to \(\frac{{\sin \left( \frac{{\uppi {{\varPhi }}}}{{{{\varPhi }}_0}}\right) }}{\left({{\frac{{\uppi {{\varPhi }}}}{{{{\varPhi }}_0 }}}}\right)}\)) and by fitting the position of the first minimum to the experimental value. We find the linear dependence of the parameter q_{y}(B_{x}) and the vertical scale of the theoretical plot by using the average slope of the side branches \(\frac{{B_x}}{{B_z}}\) from the experiment and matching it to the slope of the calculated pattern^{34}: \(\frac{{2q_y}}{{B_z}} \approx \frac{{\uppi d_{{\mathrm{eff}}}}}{{{{\varPhi }}_0}}\).
DFT calculation of NiTe_{2} energy spectrum
For the Dirac semimetal NiTe_{2}, the lattice parameters are a = b = 3.857 Å, c = 5.262 Å, α = β = 90° and γ = 120°. We first perform DFT calculations with the fullpotential localorbital program^{40}. The bandstructure calculation is done with a fine kpoint mesh including up to 50 points. Since we are using atomic basis sets, a highsymmetry tightbinding Hamiltonian is constructed using the automatic Wannier projection flow^{41}. The Bloch states are projected onto a small local basis set (34 bands for 3 atoms) around the Fermi level, which consists of 3d, 4s and 4p orbitals for Ni and 5s and 5p orbitals for Te. The projected tightbinding Hamiltonian perfectly reproduces the DFT band structures even up to 5 eV away from the Fermi level. The spinprojected surface states are calculated in a slab geometry with 10 to 20 layers from a Wannier tightbinding model.
Data availability
Source data are provided with this paper. All other data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank T. Kontos and S.H. Yang for valuable discussions. S.S.P.P. acknowledges the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—project no. 443406107, Priority Programme (SPP) 2244. The work at the Massachusetts Institute of Technology was supported by a Simons Investigator Award from the Simons Foundation. L.F. was partly supported by the David and Lucile Packard foundation. J.A.K. acknowledges support by the Swiss National Science Foundation (SNF grant no. P500PT_203159). We acknowledge the Paul Scherrer Institut for provision of synchrotron radiation beam time at the ULTRA end station of the SIS Beamline at the Swiss Light Source and we thank N. Plumb, M. Shi, M. Radovic, A. Pfister, L. Nue and H. Li for their help with the ARPES measurements. We thank Saumya Mukherjee for sharing with us his raw ARPES data from NiTe_{2}.
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S.S.S.P. and B.P. conceived the project. A.C. and B.P. performed the exfoliation and fabricated the JJ and Hall bar devices. B.P. and P.K.S. perfomed all the electrical measurements. A.C., A.K.G. and A.K.P. characterized the flakes. M. Davydova, Y.Z., N.Y. and L.F. performed the phenomenological, theoretical and DFT calculations. J.A.K., M. Date, S.J., B.P. and N.B.M.S. collected the ARPES data. B.P. and N.B.M.S. performed the ARPES analysis. S.S.S.P., B.P., N.B.M.S., M.D. and L.F. wrote the manuscript with help from all the coauthors.
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Supplementary Sections I–X and Figs. 1–16.
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Pal, B., Chakraborty, A., Sivakumar, P.K. et al. Josephson diode effect from Cooper pair momentum in a topological semimetal. Nat. Phys. 18, 1228–1233 (2022). https://doi.org/10.1038/s41567022016995
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DOI: https://doi.org/10.1038/s41567022016995
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