Signature of magnetoelectric coupling driven finite momentum pairing in 3D ising superconductor

Abstract

The finite momentum superconducting paring states (FMPs) represent a forefront of condensed matter physics. Here we report experimental evidence of FMP in a locally noncentrosymmetric bulk superconductor 4H_b-TaS₂. Using hard X-ray diffraction and angle-resolved photoemission spectroscopy, we reveal unusual 2D ferro-rotational charge density wave (CDW) and weak interlayer hopping in 4H_b-TaS₂. The superconducting upper critical field, H_c2, linearly increases via decreasing temperature, and well exceeds the Pauli limit, suggesting the dominant orbital pair-breaking mechanism. Remarkably, we observed evidence of field-induced superconductivity-to-superconductivity transition that breaks continuous rotational symmetry of the s-wave uniform pairing in the Bardeen-Cooper-Schrieffer theory down to the six-fold rotation symmetry. Ginzburg-Landau free energy analysis shows that magnetoelectric coupling, induced by 2D ferro-rotational CDW, stabilizes FMP that provides an explanation of the lowering rotation symmetry. Our results provide a new understanding of unconventional superconducting behaviors of the bulk quantum heterostructure 4H_b-TaS₂.

Introduction

The finite momentum superconducting pairing states (FMPs), where Cooper pairs carry non-zero momentum, are believed to give rise to exotic physical phenomena including the pseudogap phase^1,2,3,4 and Majorana fermions^5,6,7,8. Unlike conventional Bardeen-Cooper-Schrieffer (BCS) superconductors characterized by spatially uniform order parameters, FMPs feature spatially oscillating pairing amplitude, as illustrated in Fig. 1a, b. Since the FMPs and uniform SC are competing orders, the FMPs are thought to arise when uniform SC is suppressed. This condition can be achieved in strongly correlated electronic liquids, such as the cuprate^9,10,11, Fe-based^12,13, heavy fermion^14,15 and kagome superconductors^16,17, where spin, charge, and pairing fields are strongly intertwined. Alternatively, FMPs can be energetically favored in low-dimensional BCS superconductors under external magnetic fields, as originally proposed by Fulde-Ferrell and Larkin-Ovchinnikov (FFLO)^18,19, where the spin Zeeman effect suppresses the uniform SC^{20,21,22,23,24,25}. This Zeeman effect induced FFLO states have been extensively studied in organic superconductor^26,27,28, heavy fermion superconductor²⁹, Fe-based superconductors^30,31, and other van der Waals superconductors^32,33,34.

Fig. 1: FMPs and orbital effect in TMDs with strong Ising SOC.

a Schematic of conventional BCS superconductivity that has spatially uniform order parameter, ${\Delta }_{0}$. Two electrons on the FS with opposite momentum K and -K form a Cooper pair carrying zero total momentum, q = 0. b depicts FMPs featuring finite q pairing, ${\Delta }_{q}$, and spatially oscillating pairing amplitude. c an orbital driven FMP proposed for locally non-centrosymmetric bilayer TMDs. Despite the global inversion symmetry between 1H and 1H’ layer, when $\frac{{t}_{\perp }}{{\beta }_{{SOC}}}\ll \, 1$, the superconducting behavior displays properties of the inversion breaking 1H/1H’ layer, such as the Ising superconductivity. Under external magnetic field shown in (c), the electronic band structure of 1H and 1H’ layers have opposite momentum shifts, ${q}_{x}=\pm \frac{e{B}_{y}{z}_{0}}{2\hslash }$, due to the opposite gauge field effect or magnetoelectric effect. It is important to note that extension of this 2D model to quasi-2D and 3D quantum materials is yet to be established. d illustration of the crystal structure of 4H_b-TaS₂. e The star-of-David superlattice and schematic of flat band and Mottness in the 1T layer. f Resistivity shows three phase transitions below 350 K. The inset of (f) shows a prototypical resistivity curve near the superconducting transition temperature.

While the spin degree of freedom usually plays a key role in microscopic mechanisms of FMPs, the interplay between the Zeeman effect and spin-orbit coupling (SOC) can lead to magnetoelectric (ME) terms (also known as the Lifshitz invariants^35,36,37) that couple magnetic fields to the current operator in the Ginzburg–Landau (GL) formalism, resulting in FMP in non-centrosymmetric superconductors^35,36,37. Particularly, for 2D transition metal dichalcogenides (TMD) (Fig. 1c), the electron spins near the K and K’ valley are pinned to the normal direction of the 2D plane³⁸, which greatly suppresses Zeeman effect and significantly enhances the Pauli limit for in-plane magnetic fields, B_∥^39,40,41,42. Consequently, the orbital degree of freedom becomes critical for the behaviors of SC under B_∥. In bilayer TMD shown in Fig. 1c, the gauge coupling between electrons and B_∥^{43,44,45,46,47} yields relative momentum shift of energy bands between two atomic layers (Fig. 1c), resulting in an orbital FMP in the 2D limit. Recent experiments have revealed that certain bulk TMD superconductors also exhibit a large upper critical field exceeding the Pauli limit^32,48,49,50, suggesting 3D Ising superconductors. This has spurred interests in exploring FMPs in 3D Ising superconductors^47,51,52. If such a pairing state exists, a key question arises: is it still driven by the orbital mechanism, or could other mechanisms be responsible?

The 4H_b-TaS₂ is an intrinsically correlated TMDs that interweaves Ising SC⁴¹ and charge density wave (CDW) induced flat band (Fig. 1e)^53,54,55. The crystal structure of 4H_b-TaS₂ is formed by alternative stacking of the inversion (${{{\mathcal{P}}}}$)-breaking 1H/1H’- TaS₂ and ${{{\mathcal{P}}}}$-preserving 1T-TaS₂ layers (Fig. 1d). 4H_b-TaS₂ belongs to a class of locally non-centrosymmetric superconductors⁵⁶, in which ${{{\mathcal{P}}}}$ connects 1H and 1H’ layers with Ising SC. Figure 1f shows the resistivity of 4H_b-TaS₂. The large jump at 315 K and the quick drop at 25 K correspond to CDWs in the 1T and 1H/1H’ sublayers, respectively⁵⁷. Below T_SC = 3 K, 4H_b-TaS₂ is a clean limit Ising superconductor^58,59 and displays signatures of spontaneous time-reversal symmetry breaking^48,60,61. In this letter, we report the experimental observations of 2D electronic states in the 3D bulk 4H_b-TaS₂ and provide experimental and theoretical evidence of magnetoelectric coupling-driven FMP under magnetic fields.

Results

The 2D electronic structures of the normal state

Figure 2a depicts CDW superlattice peaks in the 1T layers. The 2D star-of-David (SoD) lattice distortions breaks the mirror symmetry, resulting in right-handed (orange) ${Q}^{r}$ and left-handed (cyan) ${Q}^{l}$ superlattice peaks that rotate ±13.9° with respect to the fundamental Bragg peaks (green)⁵⁷. Figure 2b, c show the intensity distributions of CDW superlattice peaks at ${Q}^{r}$ and ${Q}^{l}$ in the HL-diffraction plane, respectively. Surprisingly, while the SoD CDW is ordered in the 2D plane, it is completely disordered in the stacking direction as revealed by the diffraction rods along the L-direction. This observation is in stark contrast to the 3D crystal structure, where the structural Bragg peaks are narrow along the stacking direction (see  Supplementary Materials), and hence establishes an emergent 2D electronic state in 3D systems. The formation of pure 2D CDW in 3D crystal structure, to the best of our knowledge, is very rare even for well-established van der Waals materials, such as the cuprate high T_c-superconductors⁶², 2H/1T TMD^63,64, and AV₃Sb₅ kagome superconductors⁶⁵. Since the CDW is 2D and breaks mirror-symmetry, it falls into the ferro-rotational space group⁶⁶. We thus refer it as ferro-rotational CDW, instead of chiral CDW^67,68 that breaks inversion symmetry in 3D. Figure 2d–f show the angle-resolved photoemission spectroscopy (ARPES) determined electronic structure of 4H_b-TaS₂ at photon energy, hν = 95 eV, and temperature, T = 30 K. Since the 1T and 1H/1H’ layers have different work functions, electrons are transferred from the 1T layers to 1H/1H’ layers. Consequently, the SoD CDW induced flat band crosses the Fermi level and forms windmill-shaped Fermi surfaces (FSs) around the $\bar{\Gamma }$ point in 4H_b-TaS₂. In Fig. 2d, we reveal the windmill-shaped FS⁶⁹ using a photon beam size of ~1 μm. The cyan hexagon indicates the folded Brillouin zone in the left-handed CDW domains.

Fig. 2: The 2D electronic structure in the normal state.

a The CDW in the 1T layer breaks all mirror symmetries of the 2D plane and induces left (cyan) and right (orange) handed CDW superlattice peaks near the fundamental Bragg peaks (green) in 4H_b-TaS₂. b, c Intensity maps of CDW superlattice peaks, ${Q}^{r}$ and ${Q}^{l}$, in the HL-scattering plane. The diffraction rods demonstrate a novel 2D CDW in 3D lattice structure. d ARPES determined FS topology of 4H_b-TaS₂ at 30 K. The black and cyan dashed lines represent the original Brillouin zone and the folded Brillouin zone in the left-handed CDW domain, respectively. The windmill-like FS around the $\bar{\Gamma }$ point arises from the ferro-rotational CDW induced flatband. The dog-bone-shaped Fermi surfaces around $\bar{{{{\rm{K}}}}}$ and $\bar{{{{\rm{M}}}}}$ are from 1H/1H’ layers. The inset shows the enlarged windmill-like FS from ferro-rotational CDW. e ARPES intensity plot along the $\bar{{{{\rm{K}}}}}$-$\bar{\Gamma }$-$\bar{{{{\rm{K}}}}}$ path shows both electronic bands of 1H and 1T layer. The inset shows enlarged band structure from 1T layer. f ARPES intensity plot and tight binding fit of the 1H layer band structure along the $\bar{{{{\rm{K}}}}}$-$\bar{{{{\rm{M}}}}}$-$\bar{\Gamma }$-$\bar{{{{\rm{K}}}}}$ path.

The dog-bone-shaped FSs near the $\bar{{{{\rm{M}}}}}$ and $\bar{{{{\rm{K}}}}}$ points are electronic states from the 1H/1H’-TaS₂⁷⁰. Because of the Ising SOC, two electronic bands with opposite spin are split along the $\bar{\Gamma }$-$\bar{{{{\rm{K}}}}}$ direction, as shown in Fig. 2e. To quantify the size of SOC in the 1H/1H’ layers, we fit the electronic structure using a tight binding model⁷¹. Figure 2f shows the fitted result on top of the ARPES intensity plot along the $\bar{{{{\rm{K}}}}}$-$\bar{{{{\rm{M}}}}}$-$\bar{\Gamma }$-$\bar{{{{\rm{K}}}}}$ path. The extracted ${\beta }_{{soc}}$ is about 220 meV. We find that electronic bands from 1H and 1H’ are degenerated along the $\bar{\Gamma }$-$\bar{M}$ direction, consistent with weak interlayer coupling between adjacent 1H/H’ layers⁵⁷. Furthermore, in agreement with the 2D CDW shown in Fig. 2b, c, we also find that the electronic structure is non-dispersive along the k_z-direction, deriving an upper limit of $\frac{{t}_{\perp }}{{\beta }_{{SOC}}} < 0.02 \, \ll \, 1$ in 4H_b-TaS₂ (see Supplementary Fig. S3).

Emergent quasi-2D superconductivity in 4H _b-TaS₂

The observations of 2D electronic states with $\frac{{t}_{\perp }}{{\beta }_{{SOC}}}\, \ll \, 1$ indicate that the spin degree of freedom is pinned along the c-axis due to the large Ising SOC and hence explain the enhanced upper critical field beyond the standard Pauli limit under B_∥ in the 3D bulk 4H_b-TaS₂⁵⁶. Figure 3a, b show the magnetic field dependence of current density (J) vs voltage (V) maps, $V\left({B}_{\perp },J\right)$ and $V\left({B}_{\parallel },J\right)$ at T = 30 mK, where B_⊥ is the magnetic field perpendicular to the 2D plane. The SC, depicted as the dark blue region, is vulnerable to ${B}_{\perp }$ as expected for clean limit BCS superconductors, but robust under B_∥ as expected for Ising SC. Indeed, the zero-temperature upper critical field, H_c2, is about 4 times of the standard Pauli limit under B_∥ according to B_P = 1.86T_SC ~ 4.2 T (Fig. 3c). This is because the Zeeman term of in-plane magnetic field can only couple a spin-polarized band at K/K’ valley in the 1H layer to the opposite spin polarized band at the same valley in the 1H’ layer due to the Ising SOC (Fig. 1c). Consequently, the corresponding in-plane Zeeman spin splitting is reduced by a factor of $\frac{{t}_{\perp }}{{\beta }_{{SOC}}} < 0.02$ ¹¹. Thus, the in-plane upper critical field is mainly determined by the orbital pair-breaking mechanism for the 3D bulk 4H_b-TaS₂.

Fig. 3: Polar magnetic field induced 3D SC to 2D SC transition in 4H_b-TaS₂.

a, b Color maps of $V\left({B}_{\perp },J\right)$ and $V\left({B}_{\parallel },J\right)$ at T = 30 mK, respectively. Dark blue region represents the superconducting phase. Inset in (b) shows the experimental setup. The θ represents the polar angle, where $\theta=0$ and 90° are corresponding to ${B}_{\parallel }$ and ${B}_{\perp }$, respectively. c, Phase separation lines between the superconducting state and normal state at various temperatures. The text denotes the measurement temperature in units of Kelvin. Inset: Typical I-V curves data at 30 mK under B_⊥ = 0, 0.4 and 2T. d The polar angle dependence of ${H}_{c2}\left(\theta \right)$. Here ${H}_{c2}$ is defined as the magnetic field where resistivity reaches 50% of its normal state value. Black and red curves are fittings with the anisotropic Ginzburg-Landau model and 2D Tinkham model, respectively. A SC-to-SC transition around ${\theta }_{c} \sim 1^\circ$ is observed. e, f Schematics of A_brikosov vortex lattice under ${B}_{\perp }$ and Josephson vortex under ${B}_{\parallel }$ in 4H_b-TaS₂. Due to the Ising superconductivity in the ${{{\mathcal{P}}}}$-breaking 1H-TaS₂, Josephson flux is confined in the ${{{\mathcal{P}}}}$-preserving 1T-TaS₂.

Figure 3d shows the polar angle, $\theta$, dependence of ${H}_{c2}(\theta )$ at 30 mK. The experimental geometry is shown in the inset of Fig. 3b. Tilting magnetic field away from θ = 0°, ${H}_{c2}(\theta )$ displays humps at ${\theta }_{c} \sim \pm 1^\circ$, suggesting a SC-to-SC transition. Fitting of the experimental curve shows that ${H}_{c2}(\theta > {\theta }_{c})$ is described by the anisotropic 3D GL model, ${({H}_{c2}\left(\theta \right)\cos \theta /{H}_{c2}^{\parallel })}^{2}+{({H}_{c2}\left(\theta \right)\sin \theta /{H}_{c2}^{\perp })}^{2}=1$, whereas ${H}_{c2}(\theta < {\theta }_{c})$ is captured by the 2D Tinkham model, ${({H}_{c2}\left(\theta \right)\cos \theta /{H}_{c2}^{\parallel })}^{2}+\left|{H}_{c2}\left(\theta \right)\sin \theta /{H}_{c2}^{\perp }\right|=1$ ⁷². We thus conclude that the H_c2(${\theta }_{c}$) anomaly corresponds to a transition from 3D BCS state, where magnetic flux pass through the Abrikosov vortex, as depicted in Fig. 3e, to an unconventional quasi-2D SC, where the Josephson vortex is confined in the inversion-symmetric 1T-TaS₂ layers, as depicted in Fig. 3f. These observations suggest that the superconducting phase of 4H_b-TaS₂ under large B_∥ is an effective 2D bilayer system, where the 1H and 1H’ layers are weakly coupled via Josephson tunneling assisted by the electronic states at 1T layers.

Rotational invariance breakdown and FMP

To determine the nature of the 2D SC under magnetic field, we determine the magnetoresistance, ${R}({B}_{\parallel },\varphi )$ of 4H_b-TaS₂, where $\varphi$ is azimuthal angle defined in the inset of Fig. 3b. Figure 4a shows the polar plot of ${R}({B}_{\parallel },\varphi )$ at T = 1.9 K. We find that under small ${B}_{\parallel }(\varphi )$, the ${R}({B}_{\parallel },\varphi )$ is twofold symmetric. This observation is consistent with a spatially isotropic superconductivity, where the continuous rotational symmetry ${C}_{\infty }$ is reduced to ${C}_{2}$ in the presence of external currents. Indeed, we have confirmed that the ${C}_{2}$ axis is always pinned to the current direction in all measurements with different samples and geometries (see  Supplementary Materials). Remarkably, increasing ${B}_{\parallel }(\varphi )$ at a fixed temperature to a critical ${B}_{c}$, we find a ${C}_{2}$ to ${C}_{6}$ transition. Figure 4b compares the two-fold symmetric ${R}({B}_{\parallel }=1T,\varphi )$ and the sixfold symmetric ${R}({B}_{\parallel }=4T,\varphi )$. The ${R}({B}_{\parallel }=1T,\varphi )$ exhibits a maxima and minima at B⊥I and B//I configurations, respectively. The ${R}({B}_{\parallel }=4T,\varphi )$ shows minima when magnetic field directions are parallel or anti-parallel to the crystalline direction at φ = −60°, 0° and 60°. These observations establish that the rotational invariance of the isotropic s-wave SC is broken down to six-fold rotation ${C}_{6}$ that is determined by the crystal directions under high ${B}_{\parallel }(\varphi )$. Increasing temperature above T_SC, as shown in Fig. 4d, restores the ${C}_{2}$ symmetry as expected for the isotropic Fermi liquid normal state. The recovery of ${C}_{2}$ symmetry in the normal state confirms that the ${C}_{6}$ symmetric SC under magnetic field is an intrinsic 2D superconducting phase that is different from the spatially isotropic SC. Similar to finite-q electronic orders, such as the spin and charge density waves, the FMPs will also break the continuous rotational symmetry as a consequence of directional translational symmetry breaking. Therefore, the observations of SC-SC transition and rotational invariance breakdown provide experimental evidence of an FMP in 4H_b-TaS₂. Indeed, a similar rotational symmetry breaking of SC pairing has been reported in multilayer⁴⁵ and bulk 2H-TMD⁴⁷ and has been interpreted as the signature of orbital FMPs.

Fig. 4: Signature of magnetoelectric coupling driven FMP in 4H_b-TaS₂.

a In-plane magnetic field dependent magnetoresistance ${R}({B}_{\parallel },\varphi )$ from B = 0 to 6T with field step 0.5T at T = 1.9 K. The colorbar labels the magnetic field strength. b Representative normalized magnetoresistance ${R}({B}_{\parallel }=1T,\varphi )$ and ${R}({B}_{\parallel }=4T,\varphi )$ showing emergence of C₆ symmetry at high field. c Intensity plot of ${R}({B}_{\parallel },\varphi )$ at T = 1.9 K shows a first order-like C₂ to C₆ transition at B_c. d Intensity plot of the temperature dependent ${R}({B}_{\parallel }=2T,\varphi )$. The C₆ changes back to C₂ above the superconducting transition temperature T_c(2T) ~ 2.5 K. e The ${B}_{\parallel }$-T phase diagram of 4H_b-TaS₂. Experimentally determined phase boundaries between the uniform SC, FMP, and the paramagnetic normal state are marked as black and red dots. The error bars for the C₂ to C₆ transition (black) represent standard deviations of fitting procedure (see “Methods”).

Discussion

To understand the FMPs in 3D, we first consider a Lawrence-Doniach type of model^73,74 with the standard superconducting GL free energy for each 1H/1H’-TaS₂ layer. The adjacent superconducting layers are coupled by a weak Josephson tunneling and the orbital effect of in-plane magnetic fields is incorporated via replacing the spatial derivative by covariant derivative, namely the gauge coupling (see “Methods”). This model has been previously applied to 2D bilayer and multilayer Ising superconductor films, exhibiting the FMP phase^43,44,74. However, our numerical simulations for this model in the 3D limit do not show FMP (See “Methods” and Supplementary Materials). Thus, the orbital mechanism alone fails to explain experimental observations of rotational symmetry breaking of SC pairing, and additional mechanism is required. We note that the observations of 2D ferro-rotational CDW and forbidden Bragg peaks imply local mirror symmetry breaking in the superconducting 1H or 1H’ layers of 4H_b-TaS₂. This local symmetry breaking is important as it allows ME coupling terms in the GL free energy and give rise to the FMP in the intermediate magnetic field range (See “Methods” and Supplementary Materials Fig. 12). Importantly, the higher-order terms of ME coupling naturally lead to the breaking of rotational invariance of SC pairing down to six-fold rotation ${C}_{6}$, in agreement with the experimental observations (See Supplementary Materials Fig. 13).

Figure 4e summarizes the experimentally determined B_∥-T phase diagram. The extracted phase boundaries between BCS, FMPs, and Fermi liquid normal states are marked by black and red dots. When the B_∥ is lower than B_c, the system is a uniform Josephson coupled Ising superconductor. When B_∥ is greater than B_c, the FMPs state is formed in 1H and 1H’ layers. The proximity-induced SC in 1T layer⁶¹ is completely suppressed by external magnetic fields due to the absence of Ising-type SOC and the formation of Josephson vortices. Physically, the first order phase boundary separating the uniform SC and FMPs is determined by competition between interlayer Josephson coupling and the energy gain of FMP under in-plane magnetic fields. Due to the dominating orbital effect, the ${H}_{c2}$ separating the normal metal and FMP is linear in T, which is also captured by our GL calculations (Supplementary Materials). The experimental and numerical results thus support magnetoelectric coupling driven FMP and call for diffraction measurement of FMP superlattice peak to directly prove the translational symmetry breaking.

Finally, we discuss other scenarios for SC-SC transition and the rational invariance breakdown. The multi-component SC has proposed for 4H_b-TaS₂^75,76. Since SC transition temperatures of 4H_b-TaS₂ and monolayer 1H-TaS₂ are close to each other⁷⁷, the same s-wave Ising SC pairing mechanism is expected in both systems. Nevertheless, as we showed in Fig. 2, the ferro-rotational CDW induced flat band in the 1T layer is crossing the Fermi level, which may host a uniform chiral SC in the 1T layer via the superconducting proximity effect. Furthermore, a recent study of the chiral molecules intercalated TaS₂ have revealed evidence of unconventional superconductivity⁷⁸. In our case, the ferro-rotational structure of the 1T layer may play a similar role. However, the rotational-invariance breakdown at high B_∥ is unlikely related to SC in the 1T layer due to the absence of Ising SOC and small superfluid density in the 1T-layer. The Rashba effect has also been proposed to induce FMP states^20,21. However, as shown in Fig. 2, the ARPES determined band structure do not show Rashba band splitting, suggesting a minor Rashba effect in 4H_b-TaS₂. We also note that a recent theory paper⁵¹ suggested that the first order transition line due to Josephson vortex lattice melting can exist in layered Ising superconductors, which may provide an explanation of the SC-SC transition, but it remains unclear how the rotational symmetry could be lowered in the vortex lattice melting phase. Furthermore, since the rotational symmetry breakdown is observed under magnetic field, defects and change transfer are unlikely the driving factors. Lastly, we note that recent transport study of 1% Se-doped 4H_b-TaS₂ thin flakes do not observe six-fold symmetry⁷⁵. This suggests that the superconducting state of 4H_b-TaS₂ is sensitive to doping level and sample thickness. Importantly, the CDW in 1H layer is completely suppressed by just 1% Se-doping⁵⁷, highlighting the intriguing interplay between CDW, superconductivity, and structural dimensionality.

In summary, using ARPES and x-ray diffraction, we established a 2D electronic band structure and a 2D ferro-rotational CDW. The CDW together with lattice distortions break mirror symmetry in the bulk 4Hb-TaS₂. The 2D electronic states collaborate with a large SOC giving rise to 2D Ising superconductivity that is essential for the emergence of magnetoelectric coupling-driven FMP under an external magnetic field. Our discovery opens new avenue to realize unconventional SC and highlight the bulk TMDs quantum heterostructures as an ideal platform for novel quantum states.

Methods

Sample preparation and characterizations

High-quality single crystals of 4H_b-TaS₂ were grown by using the chemical vapor transport method⁵⁹. A stoichiometric mix of Ta and S powders with additional 0.15 g of I₂ was sealed under high vacuum in silicon quartz tubes. These tubes heated for 15 days in a two-zone furnace, where the temperature of source and growth zones were fixed at 820 °C and 750 °C, respectively. Electrical transport characterizations^58,59 found a normal state electron mean free around 941~1570 Å, that is significantly larger than the in-plane superconducting coherence length around 173 Å, suggesting 4H_b-TaS₂ is a clean limit superconductor.

X-ray scattering measurements

The single crystal elastic X-ray diffraction was performed at the 4-ID-D beamline of the Advanced Photon Source (APS), Argonne National Laboratory (ANL), and the integrated in situ and resonant hard X-ray studies (4-ID) beam line of National Synchrotron Light Source II (NSLS-II). The photon energy, which is selected by a cryogenically cooled Si(111) double-crystal monochromator, is 9.88 keV.

The 4-ID-D, APS

The X-rays higher harmonics were suppressed using a Si mirror and by detuning the Si (111) monochromator. Diffraction was measured using a vertical scattering plane geometry and horizontally polarized (σ) X-rays. The incident intensity was monitored by a N₂ filled ion chamber, while diffraction was collected using a Si-drift energy dispersive detector with approximately 200 eV energy resolution. The sample temperature was controlled using a He closed cycle cryostat and oriented such that X-rays scattered from the (001) surface.

The 4-ID, NSLS2

The sample is mounted in a closed-cycle displex cryostat in a vertical scattering geometry. The incident X-rays were horizontally polarized, and the diffraction was measured using a silicon drift detector.

ARPES measurements

The ARPES experiments are performed at beamline 21-ID-1 of NSLS-II at BNL. The 4H_b-TaS₂ samples are cleaved in situ in a vacuum better than 3 × 10⁻¹¹ Torr. The measurements are taken with synchrotron light source and a Scienta-Omicron DA30 electron analyzer with a beam size ~ 1 μm. The total energy resolution of ARPES measurement is approximately 15 meV. The sample stage is maintained at 30 K throughout the experiments.

Transport measurements

Transport measurements were performed on Triton Cryofree Dilution Refrigerator (Oxford instruments) and Physical Property Measurement System (Quantum Design). Polar angle (θ) dependent resistivity data were measured on a piezo rotator with built-in resistive gauge. We use a lock-in amplifier (SR860, Stanford Research) for θ readings, with typical angular resolution at about 0.03 degrees. The resistivity measurement current is supplied by K6221 source meter (Keithley), and the voltage readings were monitored by K2182 nanovolt meter (Keithley). Azimuthal angle (φ) dependent resistivity data were measured on the rotator option from Quantum design. The typical angular resolution is near 1 degree, and one must pay special attention to the backlash problem of the factory default setup. The resistivity data is measured by the resistivity option from Quantum Design.

The x-ray diffraction measurement of CDWs in the 1H and 1T layer⁵⁷ showed a CDW correlation length ~170 Å in the ab-plane. Therefore, the samples (~2 mm) used in the transport measurements must be structurally twined. For this reason, the C2-symmetric magnetoresistance at low magnetic field is likely an extrinsic effect. Indeed, as we showed in the Supplementary Fig. 8, the C2 axis always aligns with the direction of the electrodes.

Fitting method for the phase diagram

The experimental data show both six-fold and two-fold components. We use function, ${\cos }^{2}(\varphi )$, to fit the twofold background. After subtraction, we define the amplitude of six-fold component ${R}_{{six}}$ as the average of resistance value at $\varphi=0,\pm 60$ deg. At a fixed temperature, the ${R}_{{six}}$ displays a linear dependence on the magnetic field ${B}_{\parallel } > {B}_{c}$. Extrapolating the linear line to ${R}_{{six}}=0$ gives the phase boundary between the uniform SC and FMP states. The error bars of the phase boundary represent the standard deviations of the linear fittings.

Ginzburg-Landau Theory of bulk layered Ising superconductors

To model the bulk 4H_b-TaS₂, we consider the Lawrence-Doniach type of model^44,73,74 in the context of GL theory with the form ${{{\mathcal{F}}}}={{{{\mathcal{F}}}}}_{0}+{{{{\mathcal{F}}}}}_{J}+{{{{\mathcal{F}}}}}^{\left(1\right)}_{{ME}}$, where

$${{{{\mathcal{F}}}}}_{0}={\sum}_{l{{{\rm{\eta }}}}}\int {d}^{2}r\left({{{\rm{\alpha }}}}{\left|{\psi }_{l,{{{\rm{\eta }}}}}\right|}^{2}+\frac{{{{\rm{\beta }}}}}{2}{\left|{\psi }_{l,{{{\rm{\eta }}}}}\right|}^{4}+\frac{{\hslash }^{2}}{2{m}^{*}}{\left|{\vec{D}}_{l\eta }{\psi }_{l,{{{\rm{\eta }}}}}\right|}^{2}\right)$$

$${{{{\mathcal{F}}}}}_{J}=-{J}_{0}{\sum}_{l}\int {d}^{2}r\left({{\psi }_{l,+}{\psi }_{l, -}^{*}}+{{\psi }_{l+1, -}{\psi }_{l,+}^{*}}+h.c.\right)$$

$${{{{\mathcal{F}}}}}^{\left(1\right)}_{{ME}}={\sum}_{l{{{\rm{\eta }}}}}{{{{\rm{\eta }}}}{{{\rm{\gamma }}}}}_{0}\int {d}^{2}r\left({B}_{y}{j}_{l,{{{\rm{\eta }}}}}^{x}-{B}_{x}{j}_{l,{{{\rm{\eta }}}}}^{y}\right).$$

Here we consider the TaS₂ layers stacked along the z direction and ${{{\boldsymbol{r}}}}=(x,y)$ for the in-plane directions. ${\psi }_{l,{{{\rm{\eta }}}}}$ is the superconducting order parameter with $\eta=\pm$ the layer index for 1H/1H’ layer in the unit cell $l$. ${J}_{0}$ is the Josephson interlayer coupling parameter, ${\gamma }_{0}$ is the strength of the ME effect, $\alpha,\beta$ are the coefficients of the second and fourth order terms of the order parameter. We choose $\alpha={\alpha }_{0}(T-{T}_{0})$ with the parameter ${\alpha }_{0}$ and ${T}_{0}$. The covariant derivative ${\vec{D}}_{l,\eta }=-{{{\rm{i}}}}\vec{\nabla }+\frac{2{{{\rm{e}}}}}{\hslash }{\vec{A}}_{l,\eta }$ and the current operator is given by ${j}_{l,{{{\rm{\eta }}}}}^{i}={\psi }_{l,{{{\rm{\eta }}}}}^{*}{D}_{l,{{{\rm{\eta }}}}}^{x}{\psi }_{l,{{{\rm{\eta }}}}}+{\psi }_{l,{{{\rm{\eta }}}}}{\left({D}_{l,{{{\rm{\eta }}}}}^{x}{\psi }_{l,{{{\rm{\eta }}}}}\right)}^{*}$. ${{{{\mathcal{F}}}}}_{0}$ models the superconducting state in 1H/1H’-TaS₂ layers, ${{{{\mathcal{F}}}}}_{J}$ describes the inter-layer Josephson coupling, and ${{{{\mathcal{F}}}}}_{{ME}}$ is the ME coupling term^40,41,42 due to the local inversion symmetry breaking (see  Supplementary Materials). Here we assume the inter-layer Josephson coupling is weak, and the parameter ${J}_{0}$ can be positive or negative, depending on the inter-layer electron hopping form. Particularly, it was suggested⁷⁹ that the negative Josephson coupling ${J}_{0} < 0$, together with dislocation, can provide an explanation of the $\pi$ phase shift of the Little-Parks experiments in the 4H_b-TaS₂⁷⁶. The full form of the ME coupling term ${{{{\mathcal{F}}}}}_{{ME}}$ can be constructed from the crystal symmetry in the Supplementary Materials and the form of ${{{{\mathcal{F}}}}}_{{ME}}$ shown above is a simplified version and used for our numerical calculations. Here we consider the ${C}_{3v}$ point group symmetry for the 1H/1H’-TaS₂ layers, in which the mirror symmetry along the z direction is broken as supported by the x-ray diffraction (see Supplementary Materials). We also note that the in-plane magnetic field $\vec{B}=\left({B}_{x},{B}_{y}\right)$ appears in the covariant derivative ${\vec{D}}_{l,\eta }$ and the ME coupling term ${{{{\mathcal{F}}}}}_{{ME}}$. Both ${\vec{D}}_{l,\eta }$ and ${{{{\mathcal{F}}}}}_{{ME}}$ are connected to the orbital effect of magnetic fields. The Zeeman effect of magnetic fields is dropped in the above GL free energy. This is because the Ising SOC suppresses the Zeeman effect and significantly enhances the Pauli limit^{38,39,40,41,42}. Thus, the above GL free energy ${{{\mathcal{F}}}}$ is suitable for the description of the bulk layered Ising superconducting materials.

For the in-plane magnetic field $\vec{B}=({B}_{x},{B}_{y})$, we can choose the vector potential as ${\vec{A}}_{l\eta }=\left(2l+\frac{{{{\rm{\eta }}}}}{2}\right){d}_{0}(\hat{z}\times \vec{B})$, where ${d}_{0}$ is the distance between two adjacent 1H and 1H’ layers, so the lattice constant along the z direction is $2{d}_{0}$. As the vector potential ${\vec{A}}_{l\eta }$ relies on the unit cell $l$, the translation symmetry along the z direction is broken. We perform a gauge transformation

$${\tilde{\psi }}_{l,\eta }\left(\vec{r}\right)={e}^{-i{\vec{{{{\rm{q}}}}}}_{{{{\rm{l}}}}{{{\rm{\eta }}}}}\cdot r}{\psi }_{l,{{{\rm{\eta }}}}}\left(\vec{r}\right),$$

With ${\vec{q}}_{l,\eta }=\frac{2{{{\rm{e}}}}}{\hslash }\left(2{{{\rm{l}}}}+\frac{{{{\rm{\eta }}}}}{2}\right){{{{\rm{d}}}}}_{0}(\hat{{{{\rm{z}}}}}\times \vec{{{{\rm{B}}}}})$, to restore the z-directional translation invariance with the price that the in-plane continuous translation symmetry is broken down to the discrete translation by forming the magnetic unit cell. We then carry out the Fourier transformation for ${\widetilde{\psi }}_{l,\eta }\left(\vec{r}\right)$ into the momentum space and obtain

$${\widetilde{\psi }}_{l,\eta }\left(\vec{r}\right)=\frac{1}{\sqrt{{N}_{z}S}}{\sum}_{n,\vec{k}}{e}^{{il}\frac{2{{{\rm{\pi }}}}n}{{N}_{z}}}{e}^{i\vec{k}\cdot \vec{r}}{{{{\rm{\psi }}}}}_{n{{{\rm{\eta }}}}}\left(\vec{k}\right)$$

where ${N}_{z}$ is the total number of unit cells and S defines the area of each layer. After the Fourier transformation, the GL free energy in the momentum space takes the form

$${{{\mathcal{F}}}}= {\sum}_{n,\vec{k},{{{\rm{\eta }}}}}\left({{{\rm{\alpha }}}}+\frac{{\hslash }^{2}{k}^{2}}{2{m}^{*}}+2{{{\rm{\eta }}}}{{{{\rm{\gamma }}}}}_{0}\left({k}_{x}{B}_{y}-{k}_{y}{B}_{x}\right)\right){\left|{\psi }_{n,{{{\rm{\eta }}}}}\left(\vec{k}\right)\right|}^{2}\\ -{J}_{0}{\sum}_{{nk}}{\psi }_{n+}\left(\vec{k}\right){\psi }_{n-}^{*}\left(\vec{k}+\delta \vec{q}\right)+{e}^{-2\pi n/{N}_{z}}{\psi }_{n+}\left(\vec{k}\right){\psi }_{n-}^{*}\left(\vec{k}-\delta \vec{q}\right)\\ +c.c+\frac{\beta }{2S}{\sum}_{n,{n}^{{\prime} },m}{\sum}_{{k}_{1},{k}_{2},p}{\psi }_{n\eta }^{*}\left({k}_{1}\right){\psi }_{m\eta }^{*}\left({k}_{2}\right){\psi }_{{n}^{{\prime} }\eta }\left({k}_{1}-p\right){\psi }_{n+m-{n}^{{\prime} }\eta }\left({k}_{2}+p\right).$$

The GL equation can be derived by taking the functional derivative $\frac{\partial {{{\mathcal{F}}}}}{\partial {\psi }_{l\eta }^{*}\left(k\right)}=0$ and has the form

$$ -\left\{\frac{{\hslash }^{2}{k}^{2}}{2{m}^{*}}+2{{{\rm{\eta }}}}{{{{\rm{\gamma }}}}}_{0}\left({k}_{x}{B}_{y}-{k}_{y}{B}_{x}\right)\right\}{{{{\rm{\psi }}}}}_{n{{{\rm{\eta }}}}}\left(\vec{k}\right)+{J}_{0}{{{{\rm{\delta }}}}}_{{{{\rm{\eta }}}}+}\left[{{{{\rm{\psi }}}}}_{n-}\left(\vec{k}+{{{\rm{\delta }}}}\vec{q}\right)\right.\\ \left.+{e}^{i2\pi n/{N}_{z}}{{{{\rm{\psi }}}}}_{n-}\left(\vec{k}-{{{\rm{\delta }}}}\vec{q}\right)\right]+{J}_{0}{\delta }_{\eta -}\left[{\psi }_{n+}\left(\vec{k}-\delta \vec{q}\right)+{e}^{-i2\pi n/{N}_{z}}{\psi }_{n+}\left(\vec{k}+\delta \vec{q}\right)\right]\\ ={\alpha }_{0}\left(T-{T}_{0}\right){{{{\rm{\psi }}}}}_{n{{{\rm{\eta }}}}}\left(\vec{k}\right).$$

Here we have linearized the GL equation by dropping the fourth-order terms, which is a valid approximation for the superconducting states close to the critical temperature so that the SC order parameter is a small number. The above GL equation has a similar form as the Bloch equation for electrons in a periodic potential, and the momentum shift $\delta \vec{q}=\frac{2{{{\rm{e}}}}}{\hslash }{{{{\rm{d}}}}}_{0}(\hat{{{{\rm{z}}}}}\times \vec{{{{\rm{B}}}}})$ plays the role of reciprocal lattice vector. This GL equation can be regarded as an eigen-equation problem, which can be solved numerically (See Supplementary Materials), and the eigen-values give ${\alpha }_{0}(T-{T}_{0})$, which is a function of momentum $\vec{k}=({k}_{x},{k}_{y})$ and ${k}_{z}=\frac{2\pi n}{{N}_{z}}$. The largest eigen-value for $T(\vec{k},n)$ gives rise to the SC critical temperature T_c in the channel $(\vec{k},n)$. We maximize $T(\vec{k},n)$ with respect to both $\vec{k}$ and $n$ to get the true T_c for the system and if the corresponding optimal momentum, denoted as ${\vec{k}}_{{{{\boldsymbol{0}}}}}$, is non-zero, we obtain the superconducting state with FMP. Our numerical results suggest that the pairing momentum ${\vec{k}}_{{{{\boldsymbol{0}}}}}$ is always zero if the ME coupling parameter ${{{{\rm{\gamma }}}}}_{0}$ vanishes, thus excluding the orbital mechanism alone for FMP. In contrast, when the ME coupling term ${{{{\rm{\gamma }}}}}_{0}$ is non-zero, the FMPs can exist in the intermediate strength range of in-plane magnetic fields. To see how ME coupling term ${{{{\rm{\gamma }}}}}_{0}$ can induce FMP, we may consider the limit with zero inter-layer Josephson coupling ${J}_{0}=0$, which essentially corresponds to the 2D limit, the critical temperature for the momentum channel $\vec{k}$ is then $T\left(\vec{k}\right)={T}_{0}-\frac{1}{{\alpha }_{0}}\left(\frac{{\hslash }^{2}{k}^{2}}{2{m}^{*}}+2{{{\rm{\eta }}}}{{{{\rm{\gamma }}}}}_{0}\left({k}_{x}{B}_{y}-{k}_{y}{B}_{x}\right)\right)={T}_{0}-\frac{{\hslash }^{2}}{2{m}^{*}}{\left(\vec{k}+\eta \gamma \frac{2{m}^{*}}{{\hslash }^{2}}\left(\vec{B}\times \hat{z}\right)\right)}^{2}+\frac{2{m}^{*}{\gamma }^{2}}{{\hslash }^{2}}{B}^{2}$. Thus, in this limit the critical temperature is given by ${T}_{c}={T}_{0}+\frac{2{m}^{*}{\gamma }^{2}}{{\hslash }^{2}}{B}^{2}$ for a finite momentum ${\vec{k}}_{0}=-\eta {\gamma }_{0}\frac{2{m}^{*}}{{\hslash }^{2}}\left(\vec{B}\times \hat{z}\right)$, from which one can clearly see a non-zero ${\gamma }_{0}$ is substantial for FMP. More detailed behaviors of the superconducting states with FMPs for the general choice of parameters are discussed in the Supplementary Materials.

We also notice that the crystal symmetry also allows for higher-order ME coupling terms in the free energy

$${{{{\mathcal{F}}}}}_{{ME}}^{\left(2\right)}= {\sum}_{n,\vec{k}}{\Psi }_{{{{\rm{n}}}}}^{{*} }(\vec{k})\left.\left\{{\gamma }_{1}{Im}\left[{k}_{+}{B}_{+}^{2}\right]{\tau }_{y}+{\gamma }_{2}{Re}\left[{k}_{+}^{2}{B}_{+}^{4}\right]{\tau }_{0}\right.\right.\\ \left.\left.+{\gamma }_{3}{Re}\left[{k}_{+}^{4}{B}_{+}^{2}\right]{\tau }_{0}+{\gamma }_{4}{Im}\left[{p}_{+}^{3}{B}_{+}^{3}\right]{\tau }_{z}\right.\right\}{\Psi}_{n}(\vec{k})$$

where ${\Psi }_{n}(\vec{k})={\left\{{\psi }_{n+}\left(\vec{k}\right),{\psi }_{n-}\left(\vec{k}\right)\right\}}^{T}$ is the eigenstate for a bilayer unit cell. $\tau_{0}$ and $\tau_{x,y,z}$ are identity and Pauli matrices on the layer basis $(\eta=\pm )$ in each unit cell. We treat this term as a perturbation and its correction to the critical temperature is derived as

$$\Delta {T}_{c}\left(B,\phi \right)={\Psi }_{0}^{{{\dagger}} }\left({k}_{0}\right){{{{\mathcal{F}}}}}_{{ME}}^{\left(2\right)}{\Psi }_{0}({k}_{0})$$

where $\vec{B}=B\left(\cos \phi,\sin \phi \right)$ and the ${k}_{0}$ is the optimal momentum found from the lowest order G–L equation, which can be solved numerically for the critical temperature, and ${\Psi }_{0}({k}_{0})$ is the eigenstate corresponding to this solution. Our numerical results in Fig. S13 of  Supplementary Materials show the breaking of full rotation symmetry to six-fold rotation symmetry of T_c due to the higher-order ME coupling term ${{{{\mathcal{F}}}}}_{{ME}}^{\left(2\right)}$.