1 Introduction

The 2H-NbSe2, a layered two-dimensional (2D) transition metal dichalcogenide, is a prototypical type-II superconductor that exhibits strong anisotropic responses to external magnetic fields [1, 2], in which the in-plane upper critical field Hc2// is much higher than the out-of-plane one Hc2⊥[3]. When the applied magnetic field is perpendicular to the layers, the field induced Abrikosov flux lattices with six-star shaped vortex cores have been clearly resolved by scanning tunneling microscope (STM) [4]. In an in-plane magnetic field, the vortex cores become stripe-like for STM imaging from the exposed surface [5–7]. When the thickness of NbSe2 is down to the 2D limit with only a few atomic layers, the orbital effect becomes negligible for an in-plane magnetic field because of the reduced interlayer coupling. Consequently, its non-centrosymmetric structure and strong spin-orbital coupling lead to Ising superconductivity [8, 9], in which the Hc2// exceeds the Pauli paramagnetic limit [10]. This exotic phenomenon has revived interest in the electronic states of layered superconductors, particularly under the influence of an in-plane magnetic field.

The electronic states of NbSe2 in the mixed state with out-of-plane magnetic field have been investigated extensively by STM [4, 11, 12]. The bound states inside the vortex cores [13, 14] have been clearly resolved, while the Doppler shift effect induced by the Meissner current outside the vortex cores can hardly be detected [15–17]. For a fully gapped s-wave superconductor, the low energy quasi-particle (qp) excitations under magnetic field mainly come from the vortex bound state, thus the Fermi level density of state (DOS) is expected to be proportional to the field strength [18]. However, it has been reported that the low temperature specific heat coefficient γ in NbSe2 exhibits a sublinear dependence on magnetic field [19–21], just like the behavior for d-wave superconductors [22–24]. This contradiction can be attributed to the anisotropic s-wave superconducting gap structure in NbSe2 [25, 26], which generates more complexities for the superconducting state. Therefore, a systematic measurement of the DOS in the superconducting state of NbSe2 under different magnetic field orientations is highly desired.

Here, we utilize planar tunnel junction technique to investigate the differential conductance (dI/dV) spectroscopy of NbSe2 under out-of-plane and in-plane magnetic fields. The dI/dV value of the tunnel junction is approximately proportional to the spatially averaged electron DOS of the sample. We observe characteristic kink features for weak in-plane magnetic fields, but the overall behaviors are quite similar for different field orientations despite the distinct vortex generation processes and widely different upper critical field values. Especially, the generic square root dependence of the Fermi level DOS on magnetic field indicates that the Doppler shift plays a central role in the low energy excitations of 2H-NbSe2 in the presence of magnetic field.

2 Results

The planar tunnel junction used here is based on the AlOx tunnel barrier grown by atomic layer deposition (ALD). The device fabrication procedures closely follow that of a previous report [27]. The thickness of the NbSe2 flake is about 90 nm, around one fifth of the c-axis penetration depth [28, 29]. Figure 1 provides an illustration of the planar tunnel junction structure, a side view of the device, the tunneling spectroscopy measurement circuit, and an optical image of the NbSe2 device.

Figure 1
figure 1

Illustration of the planar tunnel junction structure and the tunneling spectroscopy measurement circuit. The top and side views of the tunnel junction and the optical image of the NbSe2 device are labelled in the figure. The scale bar in the optical image is 50 μm

Full size image

Figure 2 presents the dI/dV spectroscopy results obtained at \(T= 1.7\) K under varied out-of-plane and in-plane magnetic fields. All spectra exhibit clear superconducting gap and coherence peaks in the absence of a magnetic field. When an out-of-plane magnetic field is applied, as shown in Fig. 2(a), the gap feature and the coherence peaks are rapidly suppressed. It recovers the normal state behavior at 4 T, when the spectrum becomes a nearly bias-independent constant. At low magnetic fields, the coherence peak positions shift only slightly and gradually move towards zero bias near the upper critical field, which is a notable deviation from the behavior observed with increasing temperature, where the peak position energy keeps increasing due to thermal broadening effects [30]. The overall behavior of the spectra under in-plane magnetic fields (Fig. 2(b)) are quite similar, except that the superconducting properties are still present at 12 T, indicating a much higher upper critical field.

Figure 2
figure 2

The planar tunneling spectroscopy of NbSe2 in varied magnetic field. The bias dependence of dI/dV curves in magnetic fields with (a) out-of-plane and (b) in-plane directions at \(T= 1.7\) K. (c) and (d) The normalized field-induced DOSs increase with (c) out-of-plane and (d) in-plane directions. The dashed line in (d) is a parabolic function fit

Full size image

To reveal the main consequence induced by the magnetic field, we normalize the spectrum with the normal state under 5 T out-of-plane field and subtract the spectrum obtained in zero magnetic field. The results are presented in Fig. 2(c) and (d). Specifically, we focus on the spectral range within 0.5 meV and observe a similar trend for both field directions: under low field, the spectral shape is nearly parabolic, which gradually flattens as the magnetic field increases. Notably, the spectrum obtained under 0.5 T out-of-plane field is very similar to that obtained under 2 T in-plane field. At low energy, the result can be fit by a parabolic function, as shown by the dashed line in Fig. 2(d).

To gain a more intuitive understanding of the influence of magnetic field on low-energy excitations, we measure the zero-bias differential conductance (\(G_{0}\)), which reflects the Fermi level DOS, with continuous magnetic field changes in both out-of-plane and in-plane directions, as shown in Fig. 3(a) and (b). We take experiments at several specific temperature points below \(T_{\mathrm{c}}= 6.8\) K and at \(T= 7\) K in the normal state. As shown in Fig. 3(a) with an out-of-plane magnetic field, the \(G_{0}\)–H curves exhibit a similar line shape at different temperatures, indicating that the qp excitations induced by vortices have very weak temperature dependence. In contrast, the \(G_{0}\)–H curves under in-plane magnetic field display two apparent kinks below 2 T in Fig. 3(b), and the kink positions remain the same at different temperatures.

Figure 3
figure 3

The zero bias dI/dV (\(G_{0}\)) of NbSe2 in magnetic field with (a) out-of-plane and (b) in-plane directions at several temperatures. The derivative of the \(G_{0}\)–H curves with (c) out-of-plane and (d) in-plane directions. Each spectrum is shifted vertically by −0.15 in (c) and −0.08 in (d) for clarity

Full size image

The derivative of the \(G_{0}\)–H curves provide more remarkable characteristics of the superconducting transition in both field directions, as shown in Fig. 3(c) and (d). In Fig. 3(c), the superconducting transition in the out-of-plane magnetic field occurs as a step in the derivative curve, which is a much sharper feature than the \(G_{0}\)–H curves. We can define the upper critical field as the field scale when the curve begins to remain at a constant value. The superconducting transition under in-plane magnetic field in Fig. 3(d) shows an overall similar behavior, and the kink features are more apparent at the fields of 0.5 T and 1.2 T.

Figure 4 provides an analysis of the variations of the spectrum with temperature and magnetic field. To illustrate the change of Fermi level DOS induced by the magnetic field, in Fig. 4(a), we normalize the zero bias dI/dV as following form: GFI = [\(G_{0} - G_{0}\)(0T)]/[\(G_{\mathrm{N}} - G_{0}\)(0T)], where the subscript of GFI stands for field-induced, \(G_{0}\)(0T) corresponds to the thermal excitation contribution at the given temperature, and \(G_{\mathrm{N}}\) corresponds to the normal state DOS. In Fig. 3(c), the upper critical field can be determined at every temperature. We plot the relationship between GFI and H/\(H_{\mathrm{c}}\) below 5 K, and the curves roughly collapse onto each other. Therefore, we conclude that the field-induced DOS increase at the Fermi level shares the same mechanism at different temperatures. For the in-plane magnetic field case, the upper critical fields exceed the equipment limit at 1.7 K and 3 K. We can extrapolate the values from the ratio between Hc// and Hc⊥ at 4 K and 5 K, and assume that ratio is also valid at lower temperature. Then we can obtain the Hc// values at 1.7 K and 3 K to be 19.2 T and 15.4 T, respectively. In Fig. 4(b), the curves also collapse onto each other for the in-plane field, but the kink position is shifted by the normalization. Then we put the normalized curves below 4 K in Fig. 4(c), which share the same line shape except for the kinks under in-plane field, similar to the square root of magnetic field relation as shown by the dashed line. That means the orbital effect of magnetic field has the same influence on the low energy qp excitations with different temperatures and field directions, despite the sharp contrast on the vortex formations and upper critical field values between out-of-plane and in-plane field directions.

Figure 4
figure 4

The consistent behavior of spectroscopy induced by magnetic field with different temperatures and field directions. The normalized field-induced zero bias dI/dV (GFI) \(-- H\)/\(H_{\mathrm{c}}\) curves of NbSe2 in magnetic field with (a) out-of-plane and (b) in-plane directions at different temperatures. (c) GFI \(--H\)/\(H_{\mathrm{c}}\) curve roughly collapse onto each other with out-of-plane (OP) and in-plane (IP) directions. The dashed line is the square root of magnetic field. (d) Comparison of three pairs of spectra at 1.7 K with similar ratios between out-of-plane and in-plane magnetic fields

Full size image

Figure 4(d) further shows that the energy spectrum at 1.7 K for both field directions have similar overall behavior. For example, the pairs of spectra for out-of-plane (OP) 0.5 T and in-plane (IP) 2 T, OP 1 T and IP 4 T, and OP 2 T and IP 10 T roughly coincide with each other. Considering that the ratio of the upper critical field between out-of-plane and in-plane direction is about 0.21 at 1.7 K, these results demonstrate the consistent behavior of spatially-averaged DOS between two forms of vortices under different magnetic field orientations.

3 Discussion

STM experiments on NbSe2 have demonstrated that vortex cores have six-star shapes and are arranged in a triangular lattice under out-of-plane magnetic fields [11, 31], while they have stripe shapes and are parallel-aligned under in-plane magnetic fields [5]. The bound state is observed as a broad peak centered at the Fermi level in the vortex core and splits into two peaks on the positive and negative sides of the Fermi level. At positions far away from the vortex core, the tunneling spectrum is similar to that under zero magnetic field. Our planar tunneling spectroscopy measures the spatial average of DOS in three regions: the vortex core region with bound states, the screening current region where the Doppler shift effect dominates, and the region far from the vortex cores that is not affected by magnetic field. The STM result shows that the third part disappears under a magnetic field as low as 0.1 T, when the inter-vortex distance is reduced to the level of penetration depth [16]. A theory work [32] calculates the spatially averaged spectrum in the vortex core, yielding a linear term α|E| in low energy DOS that fits well with the STM data below 0.4 T. That theory also suggests a quadratic term attributed to other effects besides bound states, which was recently confirmed to be due to the contribution of screening current in other planar tunnel junction experiment [33]. In our experiment, the low energy parabolic excitation in Fig. 2(d) suggests that the Doppler shift effect induced by screening current plays a more important role than the bound states inside the vortex core. As the magnetic field increases, the number of vortex cores increases accordingly, causing a broadening of the low energy spectrum. Under higher magnetic fields, the contribution of vortex core bound states may deviate from the energy linear term due to the overlap between vortex cores [34].

The \(G_{0}\)–H curves shown in Fig. 3 roughly follow the square root of magnetic field, consistent with the specific heat results [19, 21]. In high-\(T_{\mathrm{c}}\) cuprates, such relationship was attributed to the Doppler shift effect of nodal superconducting gaps [35]. Similarly, the anisotropic s-wave gap of NbSe2 will also have a ‘node’ when some gapped regions in k-space are partially filled, due to the applied magnetic field. Previous planar tunnel junction measurements [33, 36] revealed that a two-gap feature can be resolved at the temperature of 70 mK. The smaller gap with a value of 0.6 meV has already closed under an out-of-plane field of 60 mT [33], satisfying the gap closing relationship \(\xi \sim \frac{\hslash v_{\mathrm{F}}}{\pi \Delta} \sim \sqrt{\phi _{0} /\pi H_{\bot}} \). Therefore, our results are consistent with the picture that under a magnetic field larger than 60 mT, NbSe2 behaves like a superconductor with a gap node.

When magnetic field is applied in-plane, the sudden jump in \(G_{0}\) at the kink position corresponds to the nucleation of stripe vortex cores at the surface [33]. These cores lack the necessary energy to penetrate into the bulk. Furthermore, the kink position remains fixed as the temperature rises, suggesting that the vortex physics is not strongly dependent on temperature. These kink features reveal the characteristic vortex formation process for NbSe2 in an in-plane magnetic field. Except for these kink features, the overall profile of the spectrum strongly resembles those under an out-of-plane magnetic field, which indicates that the Doppler shift caused by the screening current also makes the dominant contribution to the Fermi level DOS even for in-plane magnetic field. This is further confirmed by the square root dependence of the \(G_{\mathrm{FI}}\text{--}H/H_{\mathrm{c}}\) curves shown by the dashed line in Fig. 4(c), which represents the Doppler shift contribution of a node gap as discussed above. At field below 0.1 \(H_{\mathrm{c}}\), the curves are below the square root and behave more like an anisotropic gap [18]. At higher fields, the curves are higher than the square root of magnetic field, which indicates that the increase of bound states in the vortex core must be taken into account.

In conclusion, our planar tunneling spectroscopy measurements on 2H-NbSe2 with both out-of-plane and in-plane magnetic fields reveal overall similar behaviors despite the differences in vortex generation processes and upper critical field values. The DOS at the Fermi level induced by the magnetic field follows the \(\sqrt{H}\) relation due to the Doppler shift contribution of a nodal gap, and shows weak dependence on temperature and field direction. These findings provide a comprehensive picture regarding the effect of magnetic fields on the superconducting properties of 2H-NbSe2.

4 Materials and methods

The 2H-NbSe2 single crystals, grown by the chemical vapor transport (CVT) method, are purchased from PrMat company. Transport measurement gives a Residual Resistance Ratio (RRR) of approximately 10 and superconducting transition at \(T_{\mathrm{c}}= 6.8\) K.

The fabrication process of the tunnel junction is almost the same as that in a previous work [27]. The AlOx tunnel barrier is grown on a pre-patterned Cr/Au (3 nm/30 nm) electrode by atomic layer deposition (ALD) with ozone and trimethylaluminum (TMA) as precursors. The ALD growth cycle is repeated for 4 times, followed by several hours of post-oxidation. Four more Cr/Au (3 nm/30 nm) electrodes are then patterned onto the wafer as contact electrodes. Finally, NbSe2 flakes are exfoliated and transferred onto the substrate, covering the tunnel barrier and at least two of the contact electrodes.

The differential conductance (dI/dV) of the tunnel junction is measured using the standard lock-in method. The measurements of dI/dV spectra are performed with AC modulation voltage d\(V = 0.14\) mV and frequency \(f = 37\) Hz. The zero bias dI/dV–H curves are measured with the same frequency and a slightly larger d\(V = 0.21\) mV to obtain a higher Signal-to-Noise ratio.