An unusual continuous paramagnetic-limited superconducting phase transition in 2D NbSe 2

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1 SUPPLEMENTARY INFORMATION Letters In the format provided by the authors and unedited. An unusual continuous paramagnetic-limited superconducting phase transition in 2D NbSe 2 Egon Sohn 1,2, Xiaoxiang Xi 1,3, Wen-Yu He 4, Shengwei Jiang 1,2, Zefang Wang 1,2, Kaifei Kang 1,2, Ju-Hyun Park 5, Helmuth Berger 6, László Forró 6, Kam Tuen Law 4, Jie Shan 1,2,7 * and Kin Fai Mak 1,2,7 * 1 Department of Physics, The Pennsylvania State University, University Park, PA, USA. 2 Department of Physics and School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA. 3 National Laboratory of Solid State Microstructures, School of Physics, and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, China. 4 Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China. 5 National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL, USA. 6 Institute of Condensed Matter Physics, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. 7 Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA. * jie.shan@cornell. edu; kinfai.mak@cornell.edu Nature Materials Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

2 Supplementary Information An unusual continuous paramagnetic-limited superconducting phase transition in 2D NbSe 2 Egon Sohn, Xiaoxiang Xi, Wen-Yu He, Shengwei Jiang, Zefang Wang, Kaifei Kang, Ju-Hyun Park, Helmuth Berger, László Forró, Kam Tuen Law, Jie Shan*, Kin Fai Mak* *Correspondence to: jie.shan@cornell.edu, kinfai.mak@cornell.edu 1. Differential conductance measurements Differential conductance measurements were performed using a schematic shown in Figure S1a. For instance, to measure junction No. 1 in a four-point geometry, an ac current superimposed on a dc current was sent through contact 1 and 9. The corresponding ac and dc voltage drops between contact 1 and 2 were measured. The differential resistance of junction No. 1 (ratio of the ac current to the ac voltage) was obtained as a function of bias (dc) voltage. Note that although there is a thin layer of AlO x on all of the metal electrodes for our NbSe 2 /AlO x junction devices, the resistance for each junction is about Ω and does not affect the results of our four-point measurements. To confirm this and to show that we indeed measured the junction resistance, we switched to different combinations of connections, e.g. current through 1 and 2 and voltage drop across 1 and 3, and obtained identical results for all combinations (Fig. S1b). a b Figure S1 Differential conductance measurements. a, Schematic for measurements of the differential conductance (I: 1 9, V: 1` 2 ) in a typical junction device. V DC (V AC ), R DC (R AC ) and I DC (I AC ) denote the bias voltage, the bias resistor (1 MΩ) and the bias current for the dc (ac) measurement. b, Raw data of differential conductance spectra for junction No. 1. Each spectrum represents a different combination of connections used to measure the same junction. / (ms) I : 1 9, V : 1 2 I : 1 2, V : 1 3 I : 1 3, V : 1 4 I : 1 4, V : 1 5 I : 1 5, V : 1 6 I : 1 6, V : 1 7 I : 1 7, V : (mv) 2. Alignment of samples parallel to magnetic fields Here we outline the experimental procedure that was used to achieve a parallel alignment of the external magnetic field to the sample plane. First, we made a crude alignment of the samples to the in-plane direction. We then measured the dependence 1

3 of the zero-bias tunnel junction resistance on the applied magnetic field (Fig. S2a). Next, we fixed the magnetic field at the transition edge (shaded red region) and scanned the angle of the sample with respect to the magnetic field (Fig. S2b). The junction resistance reaches its minimum when the sample is perfectly aligned parallel to the magnetic field. The alignment error based on this procedure can be estimated by performing a quadratic fit to the angular dependence of the junction resistance in the vicinity of the minimum. An uncertainty of less than 0.1 degree was estimated. a b ( ) H (T) ( ) Data Fit Angle (degree) Figure S2 Sample alignment with respect to an external magnetic field. a, Junction resistance vs. magnetic field of a trilayer NbSe 2 /AlO x device, aligned roughly parallel to the field, at 0.3 K. The shaded red region corresponds to the transition edge. b, Junction resistance vs. sample angle under a fixed field strength (34 T). The red line is a quadratic fit to the data in the vicinity of the minimum. The uncertainty in the minimum angle position is estimated to be less than 0.1 degree. 3. Optimization of tunnel junction devices 3.1 Tunnel junction with AlO x tunnel barriers In order to achieve a tunnel junction with a reasonably large potential barrier and sharp quasiparticle linewidth (which allows the determination of the quasiparticle gap with high precision under high magnetic fields), we have tried oxidized aluminum (AlO x ) and few-layer MoS 2 as tunnel barriers. Both types of devices yield results supporting the same conclusion. For AlO x barriers, a wide range of AlO x thickness (~ nm) has been explored. While the tunneling resistance generally increased with AlO x thickness (as expected), there was only a narrow range of thickness (~ 0.5 nm) for which the tunnel junctions yielded spectra with well-resolved quasiparticle peaks. The normal-state resistance of these junction devices was in the range of Ω (Fig. S3b) and the typical dimensionless barrier strength Z was ~ 0.5, which did not vary much in this range of AlO x thickness. Such devices possessed stable resistance and did not require large bias currents to acquire reliable spectroscopic information. A small reduction in the AlO x thickness led to junctions that are too transparent, as illustrated in Fig. S3a for the case of ~ nm. Such junctions required a large bias current, which easily exceeds the critical current of the 2D NbSe 2 sample, and reliable spectroscopic information could not be obtained. On the other hand, when the AlO x thickness became a little too thick ( 1 nm), the 2

4 junction resistance increased dramatically to ~ 1 MΩ at low temperatures before superconductivity sets in (Fig. S3c). These junctions were typically also unstable with large fluctuating resistance. No reliable tunnel spectra could be acquired either. a b c / ( ) V (mv) I ( A) V (mv) / ( ) V (mv) (k ) T ( ) Figure S3 Tunnel junction characteristics with varying AlO x thickness. Three devices with AlO x tunnel barrier of different thicknesses: nm (a), nm (b) and nm (c). The differential resistance versus bias voltage at the junctions are shown in a and b. The red arrows in a mark the critical current of the sample, and the inset shows the corresponding voltage versus current at the junction. The differential resistance versus bias voltage in b clearly shows the superconducting quasiparticle peaks. The temperature dependent junction resistance is shown in c. There is no indication of superconductivity. The bump in the resistance at K is due to the instability of the junction. 3.2 Tunnel junction with few-layer TMD semiconductor barriers Figure S4 Tunnel junction with a few-layer MoS 2 tunnel barrier. a, Optical image and cross section schematic of a trilayer NbSe 2 device prepared with a 4-layer MoS 2 barrier. The outline of trilayer NbSe 2 and 4-layer MoS 2 is colored in red and blue in the optical image, respectively. b, The differential resistance versus bias voltage at the junction at 2 K. In addition to using AlO x as a tunnel barrier, we have also explored few-layer transition metal dichalcogenide (TMD) semiconductors (e.g. MoS 2 and WSe 2 ) as tunnel barriers. In these devices we first exfoliated 2D NbSe 2 and MoS 2. We then stacked the layers to form a vertical heterostructure of NbSe 2 /MoS 2 on a Si substrate with pre-patterned Au electrodes. Part of NbSe 2 is in direct contact to the Au electrode to form ohmic contacts. Finally, another Au electrode was placed on top of 3

5 the NbSe 2 /MoS 2 heterostructure by e-beam lithography to form a tunnel junction (Fig. S4a). Similar to the AlO x tunnel barrier, only a narrow range of thickness (3-4 layers of MoS 2 ) yielded good tunneling spectra. A representative differential resistance spectrum at 2 K is shown in Fig. S4b. The normal-state resistance (~ 1 kω) is higher than that in AlO x junctions. The superconducting gap can also be clearly seen, and the dimensionless barrier strength is increased to Z ~ 1.5 in these devices. The critical temperature of 2D NbSe 2 in these MoS 2 junction devices was, however, reduced (from 5.8 K to ~ 4.8 K in trilayer NbSe 2, and from 5.2 K to ~ 4.2 K in bilayers), likely caused by the inverse proximity effect from semiconducting MoS 2. The quasiparticle peaks are also broader in these junctions, which somewhat limits the spectral resolution although the dimensionless barrier strength is higher. 4. Raw resistance and differential conductance data In Fig. 1f of the main text, we have shown the temperature dependence of the critical field extracted from four-point measurements of the sample resistance. Figure S5a shows the raw data of the four-point resistance as a function of in-plane field at different temperatures down to 0.3 K. The critical field increases and the transition width narrows as temperature decreases. Figure S5b (identical to Fig. 1f) summarizes the temperature dependence of the field at 10% (blue) and 90% (red) of the normal state resistance. Figure S5 In-plane magnetic field temperature phase diagram. a, Normalized four-point resistance R/R N by the normal-state resistance as a function of in-plane field H at differing temperatures for trilayer NbSe 2. b, Temperature dependence of the critical field H!! defined as the onset of the superconducting transition (90% of the normal-state resistance, red) and of the transition width defined as the shaded region between 90% (red) and 10% (blue) of the normal-state resistance. The critical field and the sample temperature are normalized by the spin paramagnetic limit of the BCS theory H! = 1 T and the critical temperature T! 5.8 K, respectively. 4

6 a / (ms) V (mv) T 16 T 8 T 0 T b / (ms) c I ( A) V (mv) I ( A) V (mv) T 16 T 8 T 0 T / (ms) T 14 T 7 T 0 T 1 Figure S6 Raw differential conductance data. Raw data for the differential conductance as a function of bias voltage (top axis) and current (bottom axis) at 0.3 K for a trilayer NbSe 2 /AlO x junction (a), bilayer NbSe 2 /AlO x junction (b) and trilayer NbSe 2 /MoS 2 junction (c). The arrows indicate the critical currents. The vertical offsets between the curves in a, b and c are 6 ms, 4 ms and 1 ms, respectively I ( A) Figure S6 shows the raw differential conductance as a function of bias voltage and bias current under different in-plane magnetic fields for three junction devices: trilayer NbSe 2 /AlO x junction (a), bilayer NbSe 2 /AlO x junction (b), and trilayer NbSe 2 /MoS 2 junction (c). The critical current, which causes a sudden drop in the differential conductance, is labeled by arrows. Onset of the critical current can be seen for the first two devices within the spectral range. Both have a junction resistance of ~ 170 Ω. The critical current decreases with increasing in-plane magnetic fields. 5

7 However, the important spectral region, where the quasiparticle peaks occur, is always below the critical currents. Our analysis based on the Blonder-Tinkham-Klapwijk (BTK) model 1 has been performed on limited spectral regions that are not influenced by the critical current. The NbSe 2 /MoS 2 junction has a higher junction resistance (~ 1000 Ω) and a larger dimensionless barrier height (Z ~ 1.5). The critical current is beyond the range shown in Fig. S6c. However, the critical temperature of this type of device is lower and the quasiparticle peak is broader compared to devices with AlO x as tunnel barriers. 5. Blonder-Tinkham-Klapwijk (BTK) analysis of tunneling spectra 5.1. The BTK model We extract the superconducting gap 2Δ of 2D NbSe 2 from the differential tunneling conductance spectra of normal metal-insulator-superconductor (NIS) junctions using the Blonder-Tinkham-Klapwijk (BTK) model 1. The tunneling current I NS of the NIS junction as a function of bias voltage V at temperature T can be expressed as!!! I!" (V) de f E ev, T f E, T [1 + A E B(E)], (S1) where E is the electron energy, e is the elementary charge, f(e, T) is the Fermi-Dirac distribution, and A(E) and B(E) are the energy-dependent probability of Andreev reflection and normal reflection at the junction, respectively. While the ordinary reflection of electrons with probability B reduces the tunneling current, Andreev reflection with probability A enhances it by transmitting a Cooper pair over the interface for each incident electron. In the BTK model, the interface between the normal metal and the superconductor is modeled as a delta function with a dimensionless barrier strength Z; the gap size is assumed to be zero on the normal metal side and a constant on the superconductor side, respectively. By matching the boundary conditions, the probabilities A and B can be derived to yield A E B(E) = τ!!!!!!(!)!!(!!!!)!(!)!! where τ! =! is the transparency of the NIS junction, and!!!!!!(!!!!)!(!)!!, (S2) γ E =!!!!!, N!!!! E =!, N!!!!!! E =!!!!!!. (S3) The real part of N! (E) and N! (E) represents the density of states of the quasiparticles and the Cooper pairs, respectively. Note that broadening of the density of states by orbital effects 2 and effects of Zeeman splitting have been neglected because of negligible orbital depairing in 2D NbSe 2 (see Methods) and the presence of strong Ising SOC. The model can be further modified to include the effects of finite quasiparticle lifetime (1/Γ) to account for extra broadening of the differential conductance spectra by replacing E E + iγ. Finally, the normalized differential conductance (G = di NS /dv) can be expressed as!! (V) =!!!!(!")!!!!! de f E ev, T f E, T [1 + A E B E ]τ!, (S4) 6

8 where G! and G! are the differential conductance of the superconducting and the normal state, respectively. We have performed the linear least-squares fit using Eqn. S4 to our experimental differential tunneling conductance spectra. In this process, the temperature was fixed to the value from experiment; the dimensionless barrier strength Z, the superconducting gap Δ, and lifetime broadening Γ were used as free fitting parameters. An integration range of ±10Δ was used Analysis of the temperature-dependent tunneling spectra under zero field a / K 5.1 K 4.2 K 3.2 K 2.0 K 0.9 K 0.3 K b (mev) c Z 1.2 BCS model BTK fit gap (K) d (mv) e (K) K / K 3.8 K 3.0 K 1.6 K 0.9 K (mev) f BCS model BTK fit gap (K) K Z (mv) (K) 7

9 Figure S7 Temperature-dependent tunneling spectra under zero magnetic field. Zero-field differential conductance spectra (symbols) and BTK fits (solid red lines) at varying temperatures for a trilayer (a) and bilayer (d) NbSe 2 /AlO x device. b, e, Superconducting gap; c, f, Dimensionless barrier height from the BTK analysis. The predicted temperature dependence from the BCS model is shown as solid grey lines in b and e. The vertical dashed line denotes the critical temperature. In Fig. S7a and 7d we show the tunneling spectra of the trilayer and bilayer NbSe 2 /AlO x tunnel junction devices together with the BTK fits at different temperatures under zero magnetic field (the trilayer data are shown in Fig. 1 of the main text). Fig. S7b, c, e, f summarize the temperature dependence of the superconducting gap and the dimensionless barrier height. Note that since the peak width here is largely dominated by thermal broadening, we have ignored the quasiparticle broadening in the BTK fits and only included the Fermi-Dirac distribution. The superconducting gap closes continuously at the critical temperature, reflecting a second-order phase transition. This behavior agrees very well with the BCS model 2 (solid grey lines in Fig. S7b and 7e). Our tunneling experiment is therefore able to capture the continuous closure of the superconducting gap and a second-order phase transition even though the quasiparticle peaks are broadened near the critical point Analysis of the field dependent tunneling spectra Figure S8 shows the measured tunneling spectra together with the BTK fits for the three devices shown in Fig. S6. All fitting parameters (the gap Δ, the dimensionless barrier height Z, and the peak width Γ) are shown in Fig. S9. In all cases, the superconducting gap decreases continuously with magnetic field; the barrier height remains largely constant; and the peak width increases monotonically to ~ 0.3 mev at magnetic fields near the upper critical field. For the trilayer NbSe 2 /AlO x device, the peak width seems to decrease at the highest field > 30 T. Such a behavior is not evident in the other two devices. This behavior, as well as the overall increased broadening at high fields, remains not understood, and further investigations are required. a b c / T 7 T 14 T 21 T 28 T 37 T (mv) / (mv) Figure S8 Magnetic field-dependent tunneling spectra at base temperature. Normalized 4 T 10 T 16 T 22 T 28 T 37 T / T 7 T 12 T 16 T 20 T 26 T (mv) 8

10 differential conductance spectra at 0.3 K under different in-plane magnetic fields for the trilayer NbSe 2 /AlO x junction (a), bilayer NbSe 2 /AlO x junction (b) and trilayer NbSe 2 /MoS 2 junction (c) as shown in Fig. S6. Measured spectra (black symbols) are plotted together with the modified BTK fits (red lines). All spectra are vertically shifted for clarity. Separation of the two short blue lines corresponds to 2Δ. A continuous closure of the superconducting gap is evident. a b c 1.2 (mev) BTK fit Peak position (T) (T) Figure S9 Summary of the BTK analysis of tunneling measurements. Results from the modified BTK fits (red symbols) for all junction devices shown in Fig. S7. The results in (a-c), (d-f) and (g-i) correspond to the device in Fig. S7a, b and c, respectively. The blue symbols in (a,d) are extracted from directly reading the peak separations in the differential conductance spectra. The dashed blue lines represent the upper critical field and the dotted blue lines in (a-f) indicate the maximum magnetic field below which the quasi-particle peaks are clearly separated. Z d e f (mev) g h i (mev) BTK fit Peak position (T) 0.1 BTK fit (T) Z Z (T) (T) (mev) (mev) (mev) (T) (T) (T) Near the upper critical field, the error bars of all fitting parameters become larger, reflecting the reduced precision. As in all other tunneling measurements 3, 4, this is unavoidable due to the weaker quasiparticle peak features and the smaller peak separations near the critical point. However, the quasiparticle peaks are well resolved (i.e. peak separation larger than the peak width) for a large range of magnetic field, up 9

11 to ~ 85% of the upper critical field for our devices. In fact, we can directly read off the peak separations from the tunneling spectra and estimate the superconducting gap, (blue symbols in Fig. S9). The results agree well with those from the BTK fits. For the remaining ~ 15% range of magnetic fields near the upper critical field, we rely on the BTK fits for an estimate of the gap values. Given the continuous decrease of the superconducting gaps in all three different devices (including different sample and tunnel barrier thickness, as well as different tunnel barrier materials), the evidence for a continuous paramagnetic-limited phase transition is strong Simulation of field-dependent tunneling spectra for a first-order and second-order phase transition by the BTK model Figure S10 Simulated tunneling spectra for a first- and second-order phase transition. a - c, First-order transition. d - f, Second-order transition in the paramagnetic limited simulated by the modified BTK model. a, d, Input parameters (Δ, Z, Γ) for the BTK model. The grey dashed lines indicate the upper critical field. b, e, Simulated differential conductance spectra from 0 T to 40 T in a step of 5 T with vertical offsets of 0.5 for clarity. The blue short lines indicate the position of the superconducting gap. c, f, Contour plot of normalized differential conductance as a function of bias voltage (horizontal axis) and magnetic field (vertical axis). The dashed arrows show the evolution of the quasiparticle peak position. The red dashed lines indicate the upper critical field. We have simulated the tunneling spectra using the BTK model at different in-plane magnetic fields for a first- and second-order paramagnetic limited transition (Fig. S10). The input parameters (Fig. S10a and S10d for the first- and second-order transition, respectively) have been chosen to be close to the experimental values. For the first-order transition, a constant gap below the upper critical field was used, in 10

12 accordance with the first-order paramagnetic-limited transition in aluminum thin films 5 (Fig. 2e of the main text). The magnetic field dependence of the gap for the second-order transition was obtained by fitting a polynomial function to the experimental dependence in Fig. S9a. (Note that the precise field dependence of the gap does not alter our conclusion.) Furthermore, a dimensionless barrier height Z = 0.5 and a linear increase with magnetic field in the quasiparticle width Γ were assumed. Clearly, the simulated spectra (Fig. S10b and 10e) and the contour plots (Fig. S10c and 10f) are very different between the two cases. Only the simulated spectra for the second-order transition match our experimental results. The simulated spectra for the first-order transition cannot reproduce the observed continuous closure of superconducting gaps (Fig. S8, Fig. S9 and Fig. 2 of the main text). 5.5 Tunneling spectroscopy of trilayer NbSe 2 at elevated temperatures and analysis In the main text we have shown the tunneling spectra and the fitting result for trilayer NbSe 2 at varying temperatures in the absence of field (Fig. 1d) and under varying in-plane fields at 0.3 K (Fig. 2b). In Fig. S11 we show the raw experimental data for the normalized differential conductance G! /G! and fits to the BTK model under varying in-plane fields at K, 2.3 K, 3.4 K, and 3.8 K. The extracted gap size as a function of in-plane field at these temperatures has been included in Fig. 2c. 11

13 Figure S11 Tunneling spectra at elevated temperatures. Contour plot of the normalized differential conductance G! /G! for trilayer NbSe 2 as a function of bias voltage V and in-plane magnetic field at K (a), 2.3 K (b), 3.4 K (c), and 3.8 K (d). Selected line cuts of the contour plots (symbols) are shown on the right, vertically displaced for clarity. The red solid lines represent the BTK fits to experimental data. 12

14 6. Calculation of in-plane spin susceptibility and superconducting gap 6.1. Tight binding model for trilayer NbSe 2 Trilayer NbSe 2, mechanically exfoliated from bulk 2H-NbSe 2 single crystals, is stacked in the ABA sequence. The Hamiltonian for trilayer NbSe 2 can be constructed from that of monolayer NbSe 2 as follows H ( k) = H 0 ( k) H int 0 H 0 ( k) H int. (S5) 0 H int H 0 ( k) H int Here H 0 is the Hamiltonian for monolayer NbSe 2, and H int represents the interlayer coupling matrix. The tight binding Hamiltonian H 0 for monolayer NbSe 2 is constructed from the d z 2, d xy and d x 2 y 2 orbitals of the Nb atoms 6, 7. In the basis of the electron annihilation operators [c,c,c k,dz 2 k,d xy k,d,c x 2 y 2 k,d,c,c z 2 k,d xy k,d x 2 y 2 ], it can be written as H 0 ( k) = H TNN ( k) σ λl σ, (S6) z z with H TNN ( k) = V 0 V 1 V 2 * V 1 * V 2 V 11 V 12 * V 12 V 22 and L z = i 0 i 0. (S7) Here σ! and σ! denote the 2x2 identity matrix and the z-component of the Pauli matrix, respectively; λ is the spin-orbit coupling constant; and L! is the matrix representation of the z-component angular momentum operator in the basis of d z 2, d xy and d x 2 y 2 orbitals. Since all the matrix elements of L! and L! are zero under this basis, the atomic spin-orbit coupling is expressed in the form of Lz σ z. Since the Hamiltonian H! (k) is block diagonalized in the spin up and spin down sector, one can show that the time-reversal symmetry is satisfied: H ( k) = H ( ) ( ) ( ) H = H / TNN ± λl z k with 1 k k. In order to explicitly see the antisymmetric spin-orbit 2 H / coupling at the Fermi energy, we can diagonalize ( ) k to obtain the energy 13

15 eigenstates / φ 0k. The effective spin-orbit magnetic field at the Fermi energy then eff reads ( k ) 1 2 i j HSOC = φ0k λl z σ z φ0k = βkσ z. Hence, in the basis of / φ 0k, one recovers the antisymmetric spin-orbit coupling explicitly, i.e. β = β k k. Up to the third-nearest-neighbor hopping, the matrix elements V 0, V 1, V 2, V 11, V 12, V 22 can be expressed as V 0 = ε 1 + 2t 0 ( 2cosα cosβ + cos2α ) + 2r 0 ( 2cos3α cosβ + cos2β ), (S8) +2u 0 2cos2α cos2β + cos4α ( ) V 1 = 2 3t 2 sinα sinβ + 2( r 1 + r 2 )sin3α sinβ 2 3u 2 sin2α sin2β +i2t 1 sinα 2cosα + cosβ ( )sin3α cos β, (S9) ( ) + i2 r 1 r 2 ( ) +i2u 1 sin2α 2cos2α + cos2β V 2 = 2t 2 ( cos2α cosα cosβ ) 2 3 r + r 1 2 ( )( cos3α cosβ cos2β ) +2u 2 ( cos4α cos2α cos2β ) + i2 3t 1 cosα sinβ, (S10) +i 2 ( 3 r r 1 2 )sinβ ( cos3α + 2cosβ ) + i2 3u 1 cos2α sin2β V 11 = ε 2 + ( t t 22 )cosα cosβ + 2t 11 cos2α + 4r 11 cos3α cosβ +2( r r 12 )cos2β + ( u u 22 )cos2α cos2β + 2u 11 cos4α, (S11) V 12 = 3( t 22 t 11 )sinα sinβ + 4r 12 sin3α sinβ + 3( u 22 u 11 )sin2α sin2β, (S12) +i4t 12 sinα ( cosα cosβ ) + i4u 12 sin2α ( cos2α cos2β ) V 22 = ε 2 + ( 3t 11 + t 22 )cosα cosβ + 2t 22 cos2α + 2r 11 ( 2cos3α cosβ + cos2β ) r 4cos3α cosβ cos2β 12 ( ) + ( 3u 11 + u 22 )cos2α cos2β + 2u 22 cos4α. (S13) Here α,β ( ) = 1 2 k x a, 3 2 k a y is defined through the lattice constant a and the crystal momentum (k!, k! ); and ε!, t!, t!", r!, r!", u! and u!" are the various intralayer coupling parameters. The intra-orbital tunneling is considered for the interlayer coupling and H int reads as 14

16 t 01 H int = t 02 t 02, (S14) where t!! are the interlayer hopping parameters. The intralayer hopping parameters and interlayer coupling parameters were obtained by fitting the first principles band structure calculated using the ABINIT package 8. The values are summarized in Table S1. TABLE S1: Fitting parameters (in units of ev) for the Hamiltonian of Eqn. S5. ε 1 ε 2 t 0 t 1 t 2 t 11 t 12 t 22 r 0 r 1 t t 02 r 2 r 11 r 12 u 0 u 1 u 2 u 11 u 12 u 22 λ With the tight binding model above, the band structure for trilayer NbSe 2 is obtained and shown in Fig. S12a. Figure S12b shows the Fermi pockets. Interestingly, due to the exchange symmetry between the top and bottom layer, the trilayer NbSe 2 Hamiltonian can be simplified into that of a monolayer and an effective bilayer as follows U H ( k)u = H 0 ( k) H 0 k 0 2H int ( ) 2H int ( k) H 0, with U = (S15) Therefore, the energy spectrum for trilayer NbSe 2 is composed of two spin-split bands from the effective monolayer and two doubly degenerate bands from the effective bilayer system. Note that the interlayer interaction is strong (weak) for the Γ (K/K ) pocket, and the spin-orbit splitting is large (small) for the K/K (Γ) pocket. Layer stacking thus significantly changes the nature of the Γ pocket although it only has a weak effect on the K/K pocket. a b 0.5 M E(eV) μ Γ K 0.5 Γ K M Γ 15

17 Figure S12 Band structure and Fermi surface of trilayer NbSe 2 from the tight binding model. a, Band structure of trilayer NbSe 2 in the normal state. The dash-dotted line represents the Fermi level (μ = 0). b, Energy contour at the Fermi level. The orange bands are doubly degenerate and occupied by electronic states from the effective bilayer. The blue and red bands represent the spin-split bands from the effective monolayer In-plane spin susceptibility The spin susceptibility in the superconducting state can be calculated by means of Green s function. For trilayer NbSe 2 the Bogliubov-de Gennes Hamiltonian in the Nambu spinor representation becomes H BdG ( k) = ( ) µ iδi 3 σ y I 3 ( ) + µ H k iδi 3 σ y I 3 H * k, (S16) where I 3 is the 3 3 identity matrix and µ is the chemical potential. In the k space the Matsubara Green s function is defined as ( ) F ( k,ω n ) ( k,ω n ) G T ( k, ω n ) G k,ω n F = 1 iω n H BdG ( k). (S17) The spin susceptibility tensor χ ij s in the superconducting state reads 9, 10 χ ij s = µ B 2 k B T k ω n { ( ) s i F ( k,ω n )s T j F ( k,ω n )} tr s i G( k,ω n )s j G k,ω n, (S18) where s! is the matrix representation for spins along the i-direction, G( k,ω n ) and F ( k,ω n ) are the normal and anomalous Green s function, respectively. In each dω Fermi pocket we can replace the sum over k by N 0 2π dξ, where N 0 is the density of states at the Fermi level. Taking both the effective monolayer and s bilayer bands into account, we obtain the spin susceptibility χ xx k s χ xx = χ n 1 Δ 2 k B Tπ ω n 1 ω 2 n + Δ 2 2 ( + β k ) + t 2 + ω 2 n + Δ 2 ω 2 n + Δ 2 t 2 + β 2 k + ω 2 n + Δ 2 ( )( ω 2 n + Δ ) k (S19) 1 where β φ k = 0k λl z σ z φ0k is the effective spin-orbit coupling at the Fermi level, 2 as defined above in Sect. 6.1, and t is the effective interlayer coupling, which is 16

18 obtained by the energy difference between the two layer-split bands in the effective bilayer Hamiltonian (Eqn. S15) Superconducting gap as a function of in-plane field In the presence of an in-plane magnetic field, the Bogliubov-de Gennes Hamiltonian for trilayer NbSe 2 is modified as H BdG k ( ) = H BdG ( k) + τ z I gµ B! H σ!, (S20) where I 9 is the 9 9 identity matrix. In the isotropic s-wave pairing channel, the free energy density at temperature T = 1 k B β (k! denoting the Boltzmann constant) reads ( ) F = Δ2 U 1 1 A 2β ln 1+ e βe k,n. (S21) k,n Here U is the intra-orbital attractive interaction between electrons in each layer and E k,n is the eigenvalue adopted from the Bogliubov-de Gennes Hamiltonian H BdG ( k). In the BCS regime, we take Δ = 1.76k B T c as the pairing gap at zero temperature to determine U through the self-consistent gap equation Δ = U c A k,,o,l c k,o,l k,,o,l at zero magnetic field, where c k, /,o,l is the electron annihilation operator for orbital o in layer l, and A is the sample area. After finding the interaction U at zero magnetic field, the dependence of Δ on in-plane magnetic field H at different temperatures can be determined by minimizing the free energy density F. The calculated free energies of the normal (blue) and the superconducting (orange) states as a function of in-plane magnetic field in the zero-temperature limit are shown in Fig. S13. The cases without including the SOC (i.e. a BCS superconductor) and with the SOC in 2D NbSe 2 are shown in Fig. S13a and S13b, respectively. The results confirm our physical picture shown in Fig. 1a of the main text. 17

19 0 Condensation energy 0 Condensation energy F" BCS superconductor Normal state Ising superconductor Normal state a" b" H/H P$ H/H P$ Figure S13 Magnetic-field dependence of free energy in trilayer NbSe 2. Calculated free energy density as a function of in-plane magnetic field in the zero-temperature limit for the normal state (blue line), without SOC (solid orange line in a and dashed orange line in b) and with SOC for trilayer NbSe 2 (solid orange line in b). The transition is first order in (a) and second order in (b). The quasiparticle band dispersions under zero and 1.5H! in-plane magnetic fields are shown in Fig. S14. One can see that the quasiparticle gap in different regions of the Brillouin zone evolves differently under the magnetic field. This is caused by the details of the band structure discussed in Sect. 6.1, including multi-fermi surface 11-14, anisotropic SOC 11 and interlayer interactions. These details give rise to the fine structures in the calculated gap as a function of magnetic field at 0.1 T! (Fig. 3b of the main text). Figure S14 Quasiparticle band dispersions of superconducting trilayer NbSe 2. a, Zero field. b, 1.5H! in-plane magnetic field. By minimizing the free energy density F, we further calculate the superconductor-normal metal phase diagram in the parameter space of magnetic field! H and temperature T (Fig. S15). It considers the case of both in-plane and out-of-plane magnetic field and of Ising SOC (out-of-plane) and Rashba SOC 18

20 (in-plane). The continuous phase transition occurs when the SOC field is perpendicular to the external magnetic field (Fig. S15a, d), whereas the phase transition becomes first order when the two fields are parallel to each other (Fig. S15b, c). However, we note that the phase transition under an out-of-plane field is often limited by the orbital effect and the paramagnetic-limited transition studied here cannot be observed experimentally. Figure S15 Calculated magnetic field temperature phase diagrams for a Rashba- and Ising-type superconductor. Superconducting gap size as a function of magnetic field and temperature for systems with Ising SOC under an in-plane magnetic field (a), Rashba SOC under an in-plane field (b), Ising SOC under an out-of-plane field (c), and Rashba SOC under an out-of-plane field (d). The gap size is normalized to the zero-temperature zero-field value!. The SOC-induced spin splitting at the Fermi level is set to be 6.2 mev in the calculation Tunneling spectroscopy and spin susceptibility for bilayer NbSe 2 Similar to trilayer NbSe 2, a continuous second-order paramagnetic-limited transition has been observed experimentally in bilayer NbSe 2 (Fig. S8 and S9). This can also be understood as the result of large spin susceptibility in the superconducting state (χ! ) at low temperatures. Although bilayer NbSe 2 has global inversion symmetry in the crystal structure (with the inversion center in between the layers), the constituent non-centrosymmetric monolayers are weakly coupled 15. Ising spins in each monolayer are, therefore, largely preserved. This is consistent with the lack of any observable Zeeman splitting in the quasiparticle peaks. The Ising spins also give rise to the large spin susceptibility. Figure S16 shows the theoretical result for the 19

21 in-plane spin susceptibility of the superconducting state normalized by the normal state value χ! /χ! as a function of temperature (red line). It is close to unity even at low temperatures. As a comparison, we have also included the result for which the Ising spin-orbit coupling (SOC) has been turned off in the band structure (blue line). In this case, χ! /χ! is nearly zero and the behavior is identical to that of a BCS superconductor. Figure S16 Temperature-dependent in-plane magnetic susceptibility of bilayer NbSe 2. Temperature dependence of the in-plane spin susceptibility of bilayer NbSe 2 with (red line) and without (blue line) Ising SOC, calculated using the same band parameters as for trilayer NbSe 2. We also briefly comment on the behavior of χ! /χ! in bilayer as a function of interlayer coupling. In the limit of zero interlayer coupling, the temperature dependence of χ! /χ! becomes identical to that for monolayer NbSe 2. In the opposite limit (i.e. strong interlayer coupling), the temperature dependence of χ! /χ! becomes identical to that for a BCS superconductor because the structure is globally inversion symmetric and the electron spins become Heisenberg-like. Realistic bilayer NbSe 2 has an interlayer coupling much weaker than the intralayer SOC, and is close to the limit of zero interlayer coupling. References: 1. Blonder, G.E., Tinkham, M. & Klapwijk, T.M. Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion. Physical Review B 25, (1982). 2. Tinkham, M. Introduction to superconductivity (McGraw-Hill Book Co., New York, 2004). 3. Daghero, D. & Gonnelli, R.S. Probing multiband superconductivity by point-contact spectroscopy. Superconductor Science and Technology 23, (2010). 4. Wolf, E.L. Principles of electron tunneling spectroscopy (Oxford University Press, New York, 1985). 5. Meservey, R. & Tedrow, P.M. Spin-polarized electron-tunneling. Physics 20

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