Magnetic field filtering of the boundary supercurrent in unconventional metal NiTe2-based Josephson junctions

Topological materials with boundary (surface/edge/hinge) states have attracted tremendous research interest. Additionally, unconventional (obstructed atomic) materials have recently drawn lots of attention owing to their obstructed boundary states. Experimentally, Josephson junctions (JJs) constructed on materials with boundary states produce the peculiar boundary supercurrent, which was utilized as a powerful diagnostic approach. Here, we report the observations of boundary supercurrent in NiTe2-based JJs. Particularly, applying an in-plane magnetic field along the Josephson current can rapidly suppress the bulk supercurrent and retain the nearly pure boundary supercurrent, namely the magnetic field filtering of supercurrent. Further systematic comparative analysis and theoretical calculations demonstrate the existence of unconventional nature and obstructed hinge states in NiTe2, which could produce hinge supercurrent that accounts for the observation. Our results reveal the probable hinge states in unconventional metal NiTe2, and demonstrate in-plane magnetic field as an efficient method to filter out the bulk contributions and thereby to highlight the hinge states hidden in topological/unconventional materials.

Topological materials with boundary (surface/edge/hinge) states have attracted tremendous research interest.Besides, unconventional (obstructed atomic) materials have recently drawn lots of attention owing to their obstructed boundary states.Experimentally, Josephson junctions (JJs) constructed on materials with boundary states produce the peculiar boundary supercurrent, which was utilized as a powerful diagnostic approach.Here, we report the observations of conspicuous hinge supercurrent in NiTe2-based JJs.Particularly, applying an inplane magnetic field along the Josephson current could rapidly suppress the bulk supercurrent and retain the nearly pure hinge supercurrent, namely the magnetic field filtering of supercurrent.Further systematic comparative analysis and theoretical calculations demonstrate the existence of unconventional nature and obstructed hinge states in NiTe2.Our results revealed the unique hinge states in unconventional metal NiTe2, and demonstrated in-plane magnetic field as an efficient method to filter out the futile bulk contributions and thereby to highlight the hinge states hidden in topological/unconventional materials.
On the other hand, a new category of unconventional materials has been proposed with the obstructed atomic nature, where the electrons are located away from the nuclei in crystals [22][23][24][25][26][27][28][29] .As a result of the mismatch between average electronic centers and atomic positions, the obstructed states emerge on the boundary, whose bulk band gaps could be much larger than those of topological materials.In general, both the topological and obstructed boundary states can be used for constructing topological superconductivity and Majorana zero modes with the assistance of superconducting proximity effect (SPE) [30][31][32][33] .However, in principle it is a challenge to distinguish the boundary states hidden in a semimetal/metal from bulk states, because both of them are metallic.But for their JJs, the supercurrent on bulk states is expected to suffer larger decoherence and dephasing effects than the boundary states, and therefore, the JJs would exhibit a particular behaviour based on the boundary supercurrent channels [12][13][14][15][16] .
In this work, we report the observations of the hinge supercurrent in unconventional metal NiTe2-based JJs.Particularly, an in-plane magnetic field (only few tens of millitesla) applied parallel to the Josephson current could filter out the bulk supercurrent and retain the robust hinge supercurrent.Based on a further comparison with a JJ which did not include the hinges of the sample, the effect of an in-plane magnetic field perpendicular/parallel to the Josephson current, and theoretical calculations, these observations could be attributed to obstructed hinge states in the unconventional metal NiTe2.Especially, the magnetic field filtering of the supercurrent functions as a compelling route to acquire the nearly pure boundary supercurrent in topological/unconventional materials-based JJs.
NiTe2 crystallizes in the CdI2-type trigonal structure with a P3 m1 space group (number 164), as schematically illustrated by the left inset of Fig. 1a.The NiTe2 layers individually stack along the c-axis (C3 rotation axis) via van der Waals (vdW) force.It was reported as a type-II Dirac material by ab initio calculations and angle-resolved photoemission spectroscopy measurements 34,35 .Exfoliated NiTe2 nanoplates with a thickness more than 30 nm were used in this work.Figure 1a shows the temperature dependence of the resistance (R) for a NiTe2 nanoplate from room temperature to 1.55 K with a typical metallic behaviour.The magnetic field dependence of R suggests a nearly non-saturating linear or sublinear magnetoresistance as shown by the right inset of Fig. 1a, which is similar to the bulk materials 34,36 .
We fabricated JJs on NiTe2 nanoplates with superconducting electrodes NbTiN as shown in Fig. 1b.JJs of this type have been successfully implemented on many topological materials to explore the boundary states 13,14,[37][38][39][40] .The current-voltage (I-V) characteristics of device D1 is shown in Fig. 1c, indicating a Josephson critical supercurrent (Ic) of ~1 A.When a magnetic field is applied perpendicular to the junction (Bz), the superconducting interference pattern (SIP) could be obtained as illustrated in Fig. 2a.The SIP is characterized by the periodic oscillations of Ic as marked by the whitish envelope which separates the superconducting and normal states.
We note that the Ic decays very slowly with increasing | |, which is in stark contrast to the standard one-slit Fraunhofer-like pattern with the form |(Φ/Φ )/ (Φ/Φ )| in conventional JJs, denoted by the red line in Fig. 2a, where Φ =   is the magnetic flux,  and  are the effective length and width of the junction, respectively, and Φ = h/2e is the flux quantum (h is the Planck constant, e is the elementary charge) 11 .A similar phenomenon has been reported on various materials which was attributed to the large boundary (edge or hinge) supercurrent density in the JJs [13][14][15][16][17][18][19][20][21] .As for JJs, the supercurrent density Js as a function of position y, Js(y), can be extracted from the Bz dependence of Ic, Ic(Bz), through the Fourier transform (Dynes-Fulton approach) 41 .Figure 2b depicts the supercurrent density profile Js(y) extracted from the Ic(Bz) curves, retrieved from Fig. 2a accordingly (see Supplementary Section I).The center of the junction corresponds to the position y = 0.
Note that large supercurrent densities appear around y = ±0.7 μm, which locate at the hinges or side surfaces of the sample and give rise to the boundary supercurrent.
Therefore, the SIP on D1 is constituted by the bulk and hinge/side-surface supercurrent.
In the following, boundary refers to the hinges or side surfaces of the NiTe2 nanoplate.
We next investigate the effect of an in-plane magnetic field on the SIP.We further compared the effect between Bx and By on D4-1 and realized that By has a negligible filter effect on the bulk supercurrent.Without the in-plane magnetic field, D4-1 exhibits a SIP with a bulk-dominant supercurrent, as plotted in Fig. 4a.Applying Bx = 0.04 T is successful in presenting the SQUID-like pattern as shown in Fig. 4b, as expected.However, the SQUID-like pattern is always absent for By = 0.04 T, 0.06 T and 0.2 T, as shown in Figs.4c, 4d and 4e, respectively (The different critical magnetic field of the bulk supercurrent between Bx and By is discussed in Supplementary Section III).
Regarding to the origin of the observed boundary supercurrent, it is commonly attributed to the proximity-induced superconductivity on hinge/side-surface channels 13- 21 .However, the bending of the magnetic field lines around the edges of the electrodes was also proposed 42 .In order to further clarify the essential role of the sample hinges/side surfaces, we fabricated a JJ whose junction region did not include the hinges/side surfaces of the sample, as shown by the left inset of Fig. 5a (device D2-2; the upper left junction of D2 shown in the inset of Fig. 3a).The SIP only presents a central lobe (Fig. 5a), which corresponds to the supercurrent density decaying from the center to the edges of the JJ region, as depicted in the right inset of Fig. 5a.Consistently, the SIP does not show any SQUID-like signal even if applying a magnetic field Bx = 0.05 T, as displayed in Fig. 5b (The width of central lobe is smaller than Fig. 5a, primarily due to the large suppression of supercurrent at Bx = 0.05 T, which can also be seen in Fig. 4e at By = 0.2 T).It would be a critical evidence to pin down the role of the sample hinges/side surfaces for the boundary supercurrent in our JJs.
Next, we will demonstrate that the boundary supercurrent comes from the sample the Fraunhofer-like decay and the probable orbital effect is expected for By if assuming a side-surface supercurrent.Therefore, it points to the hinge supercurrent.
We next investigate the origin of the hinge states.Comparing with the hinge supercurrent originating from the higher-order topology in Cd3As2 and WTe2-based JJs [13][14][15][16] , it is intriguing to scrutinize the topological hinge states in NiTe2.However, the type-II Dirac point in NiTe2 is embedded in the bulk bands, and there is no clue yet that it could present topological hinge states.
Instead, our detailed calculations show that NiTe2 has the unconventional nature of charge mismatch, which gives rise to the obstructed hinge states.Meanwhile, the locked spin of the hinge states could explain the observed magnetic field filtering effect of the hinge supercurrent (as shown later).We calculated the NiTe2 rod and obtained the projected spectrum on the hinge atoms shown in the inset of Fig. 6f.Since it is a vdW layered compound, we tried to investigate the monolayer for convenience.We slightly enlarged the interlayer Te-Te distance (only modifying Te-pz dispersion) and computed the orbital-resolved band structures (Figs.6a, b and c) and Wannier charge centers (Fig. 6d).The results show that the Te-px, py and Ni-d orbitals have a strong hybridization.
The Te 2-valence state usually means that the Te-p orbitals are fully occupied.
Surprisingly, there is a large weight of Te-px/py orbitals in the conduction bands, which contradicts with the Te 2-valence state.On the other hand, using z-directed 1D Wilson ).Thus, we demonstrated that the hinge states are regarded as the remanence of the obstructed states of the unconventional metal NiTe2 26,27 .Intriguingly, such unconventional materials were also found to be suitable to construct Josephson diode 25,26,43 .
Due to the existence of time reversal symmetry and mirror symmetry (mx), the electron spins of the hinge states are locked to be in the plane perpendicular to the hinges (along the x direction), which could explain the filter effect of the supercurrent under Bx.The 1D hinge channel undergoes less scattering than the bulk 13 due to the locked spin, and thereby produces a more robust supercurrent under the in-plane magnetic field Bx.
However, By could act the Lorentz force on the hinge channel and weaken the hinge supercurrent.Therefore, the magnetic field filtering of supercurrent is absent for By.
In conclusion, we uncovered the hinge supercurrent in NiTe2-based JJs.Our observations combined with the theoretical calculations revealed the unconventional nature and hinge states in NiTe2.In particular, we demonstrated the in-plane magnetic field filtering as a route of vital importance to eliminate unserviceable contributions from bulk states in topological/unconventional materials with hinge states.
were set to  20 Å.The Brillouin zone was sampled by Γ-centered Monkhorst-Pack method in the self-consistent process, with a 9×9×6 k-mesh for NiTe2 bulk and a 10×1×1 k-mesh for the NiTe2 bulk and monolayer with open boundary conditions.The insets are corresponding supercurrent density profiles Js(y).
, which produces the supercurrent density profile as shown in Fig. S1d (the same as Fig. 2b).Section II.Magnitude of Bx to kill the bulk supercurrent for different devices.
We note that the Bx for killing the bulk supercurrent on D1 is much larger than D2, D3 and D4.We think it is caused by the detailed fabrication process (e.g., the quality of the superconductor films), because D2, D3 and D4 are fabricated simultaneously.However, D1 was fabricated and measured around six months earlier than them.In Fig. S2, we present another device D5, which was fabricated together with D1, and it also shows a larger Bx for killing the bulk supercurrent.
Section III.Anisotropy of bulk supercurrent between Bx and By.
As shown by Fig. 4 in the main text, the critical field of bulk supercurrent along Bx is lower than By.We think it is a common phenomenon that has been reported on other Josephson junctions with a similar electrode configuration 1 .We attribute it to the vimineous shape of the electrodes that exhibit anisotropic demagnetization N 2 .It is natural to assume the proximity-induced superconducting region beneath the electrodes to be a stripe shape.The effective demagnetization can be approximated as 3 :

𝑙 ∥ 𝑙
where the symbol  ∥ represents the length along the magnetic field direction,  is the length perpendicular to the magnetic field.Suppose  ∥ and  are comparable to the size of the electrodes on the sample (2.1 μm × 0.5 μm), and N is estimated to be around 0.84 for Bx and 0.24 for By, which produce a large anisotropy.
Section IV.Analysis of the critical side-surface supercurrent (side-surface-  ) under By.
As we mentioned in the main text, if the boundary supercurrent originates from the side surfaces (rather than the hinges), we can calculate its value at a certain By through the Fraunhofer-like decay curve.To do so, we need to supress the contribution of the bulk supercurrent, which can be achieved by applying a small Bz to render the Fraunhoferlike 1/| | decay of the bulk supercurrent itself.Therefore, we inspect the side lobes of SIP in Fig. 2a in the main text, where the bulk supercurrent has been supressed due to the Fraunhofer diffraction in Bz.Of course, the larger the serial number |n| of the side lobes, the heavier the suppression of the bulk supercurrent.Nevertheless, we assign the height of the side lobes (excluding the central lobe) of the SIP at By = 0 T in Fig. 2a as the critical supercurrent of the side-surface, side-surface- .In the main text, we assume that the effective junction length of the side surface is comparable to the separation of the electrodes, i.e., ~300 nm.In fact, it could be underestimated if considering the flux focusing effect of the electrodes.We thus test several conditions here, as shown in Fig. S3, and side-surface- is always smaller than the height of the first lobe ~0.21 , where  is the side-surface- at By = 0 T.However, the experimental boundary- at By = 0.2 T is much larger than 0.21 as shown in Fig. 5c in the main text, in contrast to the side-surface scenario.
Section V. SIP for D1.
Fig.2d, with the increase of Bx the contribution from the bulk supercurrent decreases significantly and finally a SQUID-like pattern emerges with the boundary-dominant supercurrent.Therefore, the in-plane magnetic field Bx could filter out the bulk supercurrent, i.e., the magnetic field filtering of supercurrent is observed in our experiments.
Fig.2afor device D1 assuming a side-surface supercurrent, as shown by the pink triangles in Fig.5c(detailed analysis is shown in Supplementary Section IV).However, loop technique, Wannier charge centers (WCC) are obtained (Fig. 6d), and the two average charge centers in the red box are quite away from the Te atoms (the dashed lines), indicating the unconventional nature of NiTe2 monolayer.Then, when we performed the calculation in an open boundary condition, the obstructed states were obtained on the edge (Fig. 6e) (red and blue bands indicate the different spin channels due to spin-orbit interactions).To investigate the side surface and hinge states of the bulk NiTe2, we have performed a rod calculation with open boundary conditions in both b and c directions.The results in Fig.6f show the hinge states clearly (projected onto the hinge atoms in the red box), while the side-surface states are much weaker than the hinge states due to the interlayer hybridization (see Fig.S4 in the Supplementary Section VI

Fig. 1 .
Fig. 1.Characterization of the NiTe2-based JJ. a, Temperature dependence of resistance R of an exfoliated NiTe2 nanoplate.The left inset is a schematic illustration of the atomic structure of NiTe2 crystal.The right inset is magnetic field dependence of R at 1.55 K.The black dashed line represents a linear fit to the data.b, False-colour scanning electron microscopic image of a typical NiTe2-based Josephson junction D1, where the purple colour represents superconducting NbTiN electrodes with a width t ~ 500 nm.The separation L between electrodes is ~300 nm.The width W of the NiTe2 nanoplate (red colour) between the two electrodes is ~1.5 μm.c, The I-V characteristic curve showing the Josephson supercurrent at 70 mK.

Fig. 3 .
Fig. 3. In-plane magnetic field Bx filtered boundary supercurrent on junctions D2-1 and D3-1.a, c, SIP for D2-1 and D3-1 at 10 mK without the in-plane magnetic field, respectively.The red line represents the standard Fraunhofer-like curve.The left insets are optical images for D2-1 and D3-1, indicated by red frames, and with the sample width of 1.9 μm and 1.1 μm, respectively.The right insets depict supercurrent density profiles Js(y).b, d, SIP for D2-1 and D3-1 at 10 mK under Bx = 0.04 T, respectively.

Fig. 4 .
Fig. 4. Comparison of the effect of Bx and By on SIP for D4-1.a, SIP for D4-1 at 10 mK without the in-plane magnetic field.The red line represents the standard Fraunhofer-like curve.The inset displays the optical image for D4-1, indicated by the red frame, and with the sample width of 2.1 μm.b, SIP for D4-1 at 10 mK under Bx = 0.04 T. c, d, e, SIP for D4-1 at 10 mK under By = 0.04 T, 0.06 T and 0.2 T, respectively.

Fig. 5 .
Fig. 5. Evidence for hinge supercurrent.a, SIP for D2-2 at 10 mK without the inplane magnetic field.The red line represents the standard Fraunhofer-like curve.The left inset is the schematic illustration of D2-2.The right inset shows the supercurrent density profile Js(y).b, SIP for D2-2 at 10 mK under Bx = 0.05 T. c, The black balls denote the extracted boundary- from each center of the side lobes in Fig. 2a with Bx = By = 0 T. The red cycles and blue squares denote the extracted boundary- from each center of the side lobes in Fig. 2d and Fig. S4 with Bx = 0.2 T and By = 0.2 T, respectively.The pink triangles represent the calculated boundary- for each center of side lobes when By = 0.2 T, if assuming the existence of side-surface supercurrent.n denotes the serial number of the side lobes.

Fig. 6 .
Fig. 6.Calculation of the obstructed hinge states.a, b, c, The orbital-resolved band structure for NiTe2 (with dz = 6.84 in the inset), which shows the strong hybridization between the Te-px, py orbitals and Ni-d orbitals.d, The z-directed Wannier charge centers for the occupied nine bands.The dashed lines (0.19c) indicate the locations of the Te atoms.Two Wannier charge centers in the red box are quite away from the Te atoms, with the average of 0.116c.It indicates that the NiTe2 layer is unconventional with mismatched electronic charge centers.e, The obtained obstructed states of the NiTe2.The hinge states are highlighted in the red and blue lines.f, The hinge spectrum of the NiTe2 bulk with open boundary conditions in both b and c directions.The inset shows the projected atoms on the hinge.

Fig
Fig. S1.a, SIP for D1, the same as Fig. 2a in the main text.The red curve illustrates Ic(Bz) taken following the whitish envelope for an example.b, The even part of  () recovered from the red curve in a. c, The odd part of  () recovered from the red curve in a. d, Supercurrent density profile for Ic(Bz) in a.

Fig
Fig. S2.a, The optical image for D5.b, SIP for D5 under Bx = 0.1 T.

Figure
Figure S4 shows the SIP for D1 at By = 0.2 T, which was used to extract the boundary- at By = 0.2 T for the side lobes presented in Fig. 5c in the main text.

Fig. S5 .
Fig. S5.The bands of a, hinge state and b, side-surface state.Compared with the sidesurface atoms' projection (projection #B), the hinge atoms' projection (projection #A) contributes more around EF, which indicates that the band contribution near the Fermi level is dominated by the hinge state rather than side-surface state.