Josephson diode effect from Cooper pair momentum in a topological semimetal

Cooper pairs in non-centrosymmetric superconductors can acquire finite centre-of-mass momentum in the presence of an external magnetic field. Recent theory predicts that such finite-momentum 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 type-II Dirac semimetal, NiTe2. 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 finite-momentum Cooper pairing that largely originates from the Zeeman shift of spin-helical topological surface states. The finite pairing momentum is further established, and its value determined, from the evolution of the interference pattern under an in-plane 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.


I.
Crystal structure, atomic force microscopy and other electrical characterization of NiTe2 II.
Characterization of Josephson junction devices at zero field III.
The dependence of ∆ on superconducting electrode separation and JDE effect in thick films IV.
Multiple sign reversals in ∆ as a function of in-plane magnetic field V.
Determination of and for the theoretical modeling VI.
Tilting correction and slope determination from the interference pattern to estimate finite momentum Cooper pairing VII. Calculation of the evolution of the Fraunhofer pattern VIII. Effect of the finite film thickness of NiTe2 on the evolution of the Fraunhofer pattern IX.
ARPES data and Fermi velocity of two surface states X.
Photon energy dependent ARPES measurements for the identification of the surface states

I. Crystal structure, atomic force microscopy and other electrical characterization of NiTe2
NiTe2 is a layered van der Waals materials which can be easily exfoliated. NiTe2 crystallizes in a trigonal crystal structure with the space group # $ %& . The crystal structure is comprised of NiTe2 tri-layers stacked along the c axis, each tri-layer composed of a Ni sheet sandwiched between two Te sheets. The crystal structure of NiTe2 along two crystallographic directions is shown in Fig. S1a   Hexagonal crystal structure of NiTe2 viewed along two crystallographic directions. Ni and Te atoms are shown in green and violet. c, AFM image of a NiTe2 flake with an estimated thickness of ~22 nm. This flake is used for the results that are presented in the main text. d, and e, Distance between the superconducting electrodes for 2 devices measured from an AFM image.
Transport properties of the exfoliated NiTe2 flake are shown in Fig. S2. The temperature dependence of the sample resistance shows that the sample is metallic. The optical image of the flake with contacts is shown in the inset in Fig. S2a. The magnetoresistance was measured using magnetic fields of up to 14 T that were applied along various directions (Fig. S2b). A large magnetoresistance of ~ 80% is observed at 2 K when the applied field is perpendicular to the plane of the crystal. These results are consistent with previous reports 1 .

Non-reciprocity in a Josephson junction comprised of 60 nm thick NiTe2:
The non-reciprocal effect in a Josephson junction formed from a 60 nm thick NiTe2 flake is shown in Fig. S5. Fig. S5 (a) shows an optical micrograph of the junctions formed on a single NiTe2 flake. This 60 nm thick sample shows a JDE effect as can be observed from the I-V curve in Fig. S5 (b) that was acquired for an in-plane magnetic field 0 = 40 .   shows that the sign of ∆ ( reverses multiple times, which should take place whenever the phase shift equals an integer multiple of (i.e. n ).

Dependence of sign reversal of ∆ on at various temperatures
The color plot in Fig. S7 shows the variation of ∆ ( on 0 at various temperatures. The magnitude of ∆ ( oscillates as 0 is varied and decreases monotonically as the temperature is increased. This variation is explained in detail in the main manuscript.

V. Determination of and for the theoretical model
When the magnetic field is applied along the -direction, the dependence of the nonreciprocal part of the critical current is given by: Thus, the sign reversal is achieved whenever 0 = 5 . Therefore, we estimate 5 ≈ 22 mT from the position of the first sign change in the experiment.
In the main text of the paper, we concentrate on small values of magnetic fields and use the conventional dependence of the order parameters on the magnetic field ∆ &,7 ∝ E1 − ?
in order to find the qualitative behavior of the JDE effect with the magnitude of the in-plane field.
To find ( , we compare the ratio of the magnitudes of the first and second maxima of the field dependence of ∆ ( to the same ratio extracted from the experiments. This yields ( ≈ 45 mT and the corresponding dependence of ∆ ( on the magnitude of the magnetic field in the ydirection is shown in Fig. S8.

VI. Tilting correction and slope determination from the interference pattern to estimate finite momentum Cooper pairing
Previous reports 4,5 show that the dependence of the differential resistance 5-vs ; ), we apply an AC excitation (without any dc current) and measure the differential resistance 5: 5-as a function of ; and < . We note that a lower differential resistance corresponds to a higher critical current in the conventional Fraunhofer pattern 4,5 . 5-on ; and < . As discussed in the previous section, an application of < in the direction of the supercurrent flow causes the interference pattern to shift in a certain direction. To remove this shift/tilt, we estimate the slope of the tilt to be ~ 1/65 and the resulted 5: 5-map after slope correction is shown in Fig. S9b. The tilting is performed so that the central lobe is oriented vertically to allow for the calculation of the angles of the side branches.  (Fig. S10a). For several specific values of ; (Fig. S10a), the dependence of 5:

5-on
< is plotted, as shown for ; = -13.46 mT and -9.83 mT in Fig. S10b and c, respectively. Using a polynomial fit, we calculate the value of < which corresponds to the minimum value  5-on ; and < (this figure is the same as Fig. 3b). b, and c, Dependence of 5: 5-on < for two specific values of ; . A polynomial is used to fit the data (red solid line) and to calculate < that corresponds to the minimum value of

VII. Calculation of the evolution of the Fraunhofer pattern
We calculate the Josephson current by summing quasi-classical trajectories of Cooper pairs across the junction (Fig. S12) 4 : where, is the total phase difference for a trajectory that starts at (0, & ) and ends at ( EFF , 7 ) , and EFF = + 2λ, λ = 140 nm is the London penetration depth 6 . ∆φ A is the phase difference between the order parameters in the two superconducting leads in the absence of applied field, and we have neglected the effect of the finite thickness of NiTe2 (see the next section for a justification). In order to calculate the evolution of the Fraunhofer pattern, we compute the Josephson current using eq. (S2) and maximize it by varying ∆ A , which allows us to find ( . To compare theoretical and experimental predictions, we further adjust the value of the effective junction separation EFF because of flux focusing 4 , which is carried out by calculating the Fraunhofer pattern at zero in-plane magnetic field using eq. (S2) (which gives a slightly different, but still qualitatively similar dependence to and is shown in Fig. S13 below) and fit the position of the first minimum to the experimental value from the experiments shown in Fig. S8. As we see in Fig. S13, the central peak disappears, and two side branches emerge as the value of the magnetic field is increased. We find the linear dependence of the parameter ( < ) in eq. (S3) and the vertical scale of the theoretical plot by using the average slope of the side branches 4 $ 4 % from experiment and matching it to the slope that the calculated pattern has 4 :

Fig. S13| Theoretical calculation of Fraunhofer pattern evolution with magnetic field.
Fraunhofer patterns at several values of in-plane magnetic field as computed from maximizing eq. (S2).

VIII. Effect of the finite film thickness of NiTe2 on the evolution of the Fraunhofer pattern
When the thickness is finite, there will be another contribution to the phase in eq. (S3) from the flux piercing the device through the y-z plane.
Let us compare the maximum contribution of this phase in comparison to the other two in (S3).
For example, let us compare ∆ J ( < ) to the phase due to the Cooper pair momentum at typical values of fields for the Fraunhofer map: < ≈ 100 mT and ; ≈ 10 mT: As discussed above, we can estimate the Cooper pair momentum from 2 0 ≈ which yields: We have used a typical thickness of 20 nm and EFF ~1 m. Therefore, we can neglect the contribution from the finite thickness effect in comparison to the phase due to the finite Cooper pair momentum.
Additionally, we estimate the flux piercing through the cross-section of the device perpendicular to the y-axis (x-axis is parallel to the current), corresponding to maximum dimension 20nm x 350nm, is 5 ( 1(0 ( *0 ) ) ( @ (see ref. [5]), and the average flux is this case is similarly small.

IX. ARPES data and Fermi velocity of two surface states
Previous reports 7,8 show that NiTe2 hosts spin-split topological surface states (SS) along with a bulk type-II Dirac semi-metallic ground state. Our theoretical calculations suggest that both the low Fermi velocity and small Fermi energy of one of these SSs play key roles in creating the JDE (see main text). In order to extract these material parameters, angle-resolved photoelectron spectroscopy (ARPES) experiments were carried out on a NiTe2 single-crystal cleaved along the [0001] direction. The energy-momentum dispersion along the G W → W direction is given in Fig. S14a, where two topological surface states (SS1 and SS2) are indicated with arrows. These experimental findings can be well reproduced by our ab-initio calculations (Fig. S14b), which also indicate the spin-polarization of SS1 and SS2.

X. Photon energy dependent ARPES measurements for the identification of the surface states:
We identify the existence of the topological surface states by carrying out photon energy dependent ARPES measurements, as shown in Fig. S16. One can see that the bands identified as e-SS and h-SS in Fig. S16 have negligible out-of-plane dispersion (photon energy dependent dispersion), which is the hallmark of a state confined to the 2D surface of the crystal. This is in line with previous investigations in the literature 7,8 .