Introduction

Superconducting states are destroyed when a magnetic field is applied to the superconductor, and the maximum field is the upper critical field \({B}_{\text{c}2}\) in type-II superconductors. Superconductors that yield high \({B}_{\text{c}2}\) have attracted significant attention1,2,3,4,5,6,7,8. \({B}_{\text{c}2}\) is determined by two distinct pair-breaking effects: the paramagnetic pair-breaking effect and orbital pair-breaking effect. The paramagnetic and orbital pair-breaking effects arise from spin polarization which is attributed to the Zeeman effect and vortex formation, respectively. The lower one between the two limits, i.e. between the paramagnetic and orbital limit, is observed as \({B}_{\text{c}2}\) of the examined material. The former effect depends on the pairing symmetry of superconductivity. In spin-singlet superconductors, the limit by the paramagnetic pair-breaking effect is given by the relation \({B}_{\text{P}}=\Delta /\sqrt{g}{\mu }_{\text{B}}\); if we assume the weak-coupling limit \(\Delta =1.76{k}_{\text{B}}{T}_{\text{c}}\) and \(g=2\) (where \(\Delta\), kB, Tc, μB, and \(g\) are the superconducting gap, the Boltzmann constant, the superconducting transition temperature, the Bohr magneton, and \(g\)-factor, respectively), the paramagnetic limit (Pauli limit) is given by the relation \({B}_{\text{P}}=1.86\,{ T}_{\text{c}}\). In contrast, the paramagnetic pair-breaking effect is absent in a specific magnetic field in spin-triplet superconductors. Several uranium-based superconductors such as UCoGe, URhGe, and UTe2 exhibit large \({B}_{\text{c}2}\) values, which remarkably exceed the Pauli limit expected from the aforementioned relation, owing to the spin-triplet superconductivity1,2,3. In non-centrosymmetric heavy fermion superconductors, the paramagnetic pair-breaking effect is suppressed by the unique spin texture of the Rashba-type spin–orbit coupling4,5. The orbital limit is also enhanced in heavy-fermion superconductors because a large effective mass leads to a small coherence length. The orbital pair-breaking effect is also strongly suppressed in two-dimensional (2D) systems and layered superconductors in a magnetic field parallel to the in-plane direction. This can be understood by the out-of-plane coherence length \({\xi }_{\perp }\), which is smaller than the thickness in 2D superconductors, and the large out-of-plane effective mass, which originates from the anisotropic layered structure. In 2D superconductors of transition-metal dichalcogenides, wherein the orbital pair-breaking effect is almost quenched and the paramagnetic pair-breaking effect is predominant, the Zeeman-type spin–orbit coupling originating from the broken in-plane inversion symmetry plays a significant role in superconductivity6,7,8. The Zeeman-type spin–orbit coupling provides an intriguing Ising state which causes the enhancement of the \({B}_{\text{c}2}\).

Recently, theoretical studies have predicted that even if global inversion symmetry is preserved in a material, the breaking of local inversion symmetry leads to interesting physical phenomena such as parity-mixed superconductivity9, stabilised odd-parity superconductivity10, and the possibility of topological crystalline superconductivity with global centrosymmetric systems11. Rashba-type spin–orbit coupling induced by locally non-centrosymmetric systems is expected to suppress the paramagnetic pair-breaking effect and enhance the \({B}_{\text{c}2}\)12. An anomalous \({B}_{\text{c}2}\) was observed in artificial superlattices composed of heavy-fermion superconductor CeCoIn5 and normal metal YbCoIn5, wherein the inversion symmetry is locally broken at the interface between the CeCoIn5 layer, despite the fact that the global inversion symmetry is present13. The Rashba-type spin–orbit coupling can be controlled by tuning the thickness modulation14,15,16. The MoS2 bilayer (2H-MoS2) also has global inversion symmetry, while the inversion symmetry within the individual layers is locally broken. The Zeeman-type spin–orbit interaction and Josephson coupling between the layers can be controlled by tuning the carrier concentration in the individual layers17. Moreover, the complex-strip phase, which has been predicted on the in-plane \({B}_{\text{c}2}\) in the multilayer systems with locally broken inversion symmetry, was confirmed in the bilayer and trilayer of a 2D superconductor18. The transition from the complex-strip phase to the helical phase with layer-dependent Rashba spin–orbit coupling was proposed in its systems18. More recently, a transition between two different superconducting phases and extremely high \({B}_{\text{c}2}\) has been observed in a heavy-fermion superconductor CeRh2As2, which possessed local inversion symmetry breaking at the cerium sites19.

The typical BiCh2-based (Ch: S and Se) system REOBiCh2 (RE: rare earth), which is the target system of this study, has a layered crystal structure composed of REO blocking and BiCh2 conducting layers20,21. The parent phase of REOBiCh2 is a band insulator. A partial F substitution at the O site generates electron carriers, and superconductivity emerges at low temperatures. The superconducting gap structure and pairing mechanism of the system have not been completely clarified. Several theoretical calculations suggest that anisotropic superconductivity, such as extended s-wave, d-wave, and g-wave states, are realised in BiCh2-based superconductors22,23,24,25. However, early-stage experimental results have reported conventional fully gapped s-wave superconductivity; thermal conductivity, specific heat, and magnetic penetration depth measurements support fully gapped s-wave superconductivity26,27,28. In contrast, an anisotropic superconducting gap was observed in a laser angle-resolved photoemission spectroscopy (ARPES) apparatus29. An s-wave superconductor with accidental nodes was proposed based on a comparison with other experimental measurements, although the anisotropy is not contradictory to the d- and g-wave scenarios. The anisotropic s-wave gap structure observed by ARPES is almost consistent with the theoretical calculation, suggesting that the anisotropic gap arises from the Bi p-orbital degrees of freedom only with the use of purely attractive interactions25. Moreover, the absence of an isotope effect was observed, which also implies unconventional superconductivity30. The crystal structure of the BiCh2-based superconductor possesses global inversion symmetry, whereas the inversion symmetry is locally broken in the BiCh2 layer. Theoretical studies predict that hidden spin polarisation by local Rashba-type spin–orbit coupling should exist in the LaOBiS2 system because of the site inversion asymmetry for Bi and S sites31,32. The spin polarisation attributed to local Rashba spin–orbit coupling was observed by spin-ARPES (SARPES) for LaO0.55F0.45BiS233. Furthermore, a high \({B}_{\text{c}2}\) was observed, which implies that the local inversion asymmetry can play a significant role in superconductivity34. However, research focused on local inversion asymmetry has not been extensively developed in BiCh2-based superconductivity. In the BiCh2-based superconductors, we selected LaO0.5F0.5BiS2−xSex (x = 0.22 and 0.69) as the target materials to purely investigate the superconducting properties as the compounds have a bulk superconductivity nature27 and do not contain f-electron elements in the REO blocking layer.

Herein, we show that extremely high in-plane \({B}_{\text{c}2}\) in the LaO0.5F0.5BiS2−xSex (x = 0.22 and 0.69) where the inversion symmetry is locally broken in the BiCh2 layer. The superconducting states were not completely suppressed by applied fields with strengths up to 55 T. The two pair-breaking effects, the paramagnetic pair-breaking effect and the orbital pair-breaking effect, are strongly suppressed by the local inversion symmetry breaking in the BiCh2 layers and the layered structure.

Results and discussion

Figure 1a shows the crystal structure of the target system LaO0.5F0.5BiS2−xSex (x = 0.22 and 0.69) which has a layered structure with tetragonal symmetry (P4/nmm). Although the crystal structure possesses global inversion symmetry (the symbol indicated with P in Fig. 1b shows the global inversion centre), the local inversion symmetry is broken in each BiCh2 layer (dashed rectangles show each BiCh2 layer in Fig. 1b). A partial Se substitution for the in-plane S site (Ch1 site) leads to the enhancement of the bulk nature of superconductivity and a specific heat jump is clearly observed27. Both the in-plane chemical pressure effect and carrier concentration have been revealed to be essential for the emergence of bulk superconductivity in the REOBiCh2 system21. The x value (Se concentration) was estimated using energy-dispersive X-ray (EDX) spectroscopy. The actual atomic ratio is almost consistent with the nominal value. Figure 1c,d show the temperature dependence of the in-plane resistivity ρab at a field strength of 0 T for x = 0.22 and 0.69. The Tc defined as the midpoint of the transition is 3.2 K and 4.1 K for x = 0.22 and 0.69, respectively; the higher Se concentration causes the higher Tc. A weak upward behaviour was observed for x = 0.22 and 0.69, which is similar to previous reports34. The amplitude of the upturn behaviour at low temperatures was suppressed by the Se substitution.

Figure 1
figure 1

Crystal structure of LaO0.5F0.5BiS2−xSex and electrical resistivity in the absence of a magnetic field for LaO0.5F0.5BiS2−xSex. (a) Crystal structure of LaO0.5F0.5BiS2−xSex with a space group of P4/nmm (No. 129). The crystal structure image is prepared by VESTA35. (b) Two BiCh2 layers and LaO blocking layer. The inversion symmetry is locally broken in the each BiCh2 layer. The symbol P in the LaO blocking layer denotes the global inversion centre for the LaO0.5F0.5BiS2−xSex system. The thicknesses of the BiCh2 and the blocking layers were evaluated as the length between the centres of the two atomics by structural analysis of powder X-ray diffraction. (c,d) Temperature dependence of the resistivity in the absence of a magnetic field for x = 0.22 (c) and 0.69 (d). The insets show the enlarged resistivity curves near the superconducting transition. The superconducting transition temperature, Tc is defined as the midpoint of the resistive transition and was observed at Tc = 3.2 K and 4.1 K.

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Figure 2a,b respectively show the temperature dependence of the in-plane resistivity ρab for x = 0.22 at various fields parallel to the ab-plane and c-axis. The applied electric current was perpendicular to the magnetic field in \(B\parallel c\) and parallel in \(B\parallel ab\). In the case of \(B\parallel c\), the superconducting states are immediately suppressed by the applied field. In contrast, in the case of \(B\parallel ab\), the superconductivity is robust against the applied field. Figure 2c,d exhibit the field dependence of the ρab for x = 0.22 from 3.0 to 2.0 K with \(B\parallel ab\) and \(B\parallel c\). The ρab(B) data show a trend similar to that of ρab(T). Figure 2e–h show the x = 0.69 data. The behaviours of both ρab(T) and ρab(B) are similar to that of x = 0.22. For x = 0.69 with \(B\parallel ab\), zero resistivity is retained up to 9 T for temperatures below 2.6 K. The temperature and field dependencies of ρab show a large anisotropic factor γ for both x = 0.22 and 0.69, thus indicating that the superconductors have strong anisotropic (two-dimensional) superconducting characteristics. \({B}_{\text{c}2}\) was estimated from the midpoint of the resistive transition of the ρab(B) data (see Supplementary Fig. 2); a similar criterion has been used in several studies6,7,13,14,34. Moreover, we plotted the \({B}_{\text{c}2}\) defined as the beginning of the resistive increase from zero resistivity of the ρab(B) data and from the ρab(T) data, respectively, in Supplementary Figs. 4 and 5.

Figure 2
figure 2

In-plane resistivity ρab for x = 0.22 and 0.69 in the presence of a magnetic field. (a,b) Temperature dependence of the in-plane resistivity ρab in the presence of a magnetic field for x = 0.22 in \(B\parallel ab\) (a) and \(B\parallel c\) (b). The arrows denote the direction from low to high fields. (c,d) Field dependence of the ρab at different temperatures for x = 0.22 in \(B\parallel ab\) (c) and \(B\parallel c\) (d). The arrows denote the direction from low to high temperatures. (e–h) ρab(T) in \(B\parallel ab\) (e) and \(B\parallel c\) (f), and ρab(B) in \(B\parallel ab\) (g) and \(B\parallel c\) (h) for x = 0.69.

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To investigate the in-plane \({B}_{\text{c}2}\) variation at higher fields, we measured the field dependence of ρab for x = 0.22 and 0.69 based on the use of pulsed high fields up to 55 T. Figure 3a,b show the ρab(B) data acquired by the pulsed fields parallel to the ab plane from 4.2 to ~ 0.47 K for x = 0.22 and 0.69, respectively. Notably, the superconducting states are not completely destroyed up to 55 T at ~ 0.47 K for both x = 0.22 and 0.69. Moreover, for x = 0.69, the superconducting states survive at the highest field of 55 T even at 1.39 K. The gradient of the ρab(B) curves from zero resistivity to the normal state became smaller in the low-temperature and high-field regions for both samples. We defined the normal state resistivity ρn as the black dashed lines in Fig. 3a,b because it is not largely changed by the scanning field. Thus, the strength of \({B}_{\text{c}2}\) was estimated from the midpoint of zero resistivity and ρn (see Supplementary Fig. 2). We also plotted the \({B}_{\text{c}2}\) defined as the beginning of the resistive increase from zero resistivity of the ρab(B) curves by the pulsed field (see Supplementary Figs. 3, 4). We summarise the \({B}_{\text{c}2}\) obtained by the static fields (Fig. 2) and pulsed fields (Fig. 3) for x = 0.22 and 0.69 in Fig. 4a,b, respectively. The pulsed-field data are not contradictory to the static-field data because we used the same single crystal samples. Note that there is uncertainty in the field directions in the pulsed-field measurements to deviate slightly from the ab plane because we could not use a rotator system in the setup; therefore, we added the error bar to the blue circles in Fig. 4. The upturn of \({B}_{\text{c}2}^{\parallel }\) at the low-temperature region indicates that \({B}_{\text{c}2}^{\parallel }(0)\) is higher than our minimum temperature value. The orbital limit \({B}_{\text{orb}}(0)\) is estimated to be 9.1 and 0.33 T within the in-plane and out-of-plane direction for x = 0.22 and 17.2, and 0.58 T within the in-plane and out-of-plane direction for x = 0.69 from the initial slope of \({B}_{\text{c}2}\) at Tc based on the relation \({B}_{\text{orb}}\left(0\right)=0.69{T}_{\text{c}}{\left(-d{B}_{\text{c}2}/dT\right)}_{{T}_{\text{c}}}\) in the dirty limit36. We describe the Werthamer–Helfand–Hohenberg (WHH) curves (dashed lines in Fig. 4a,b). The observed \({B}_{\text{c}2}^{\parallel }\) clearly exceeds the in-plane orbital limit \({B}_{\text{orb}}^{\parallel }(0)\) and the WHH curves are not suitable for \({B}_{\text{c}2}^{\parallel }\) at x = 0.22 and 0.69. Moreover, the out-of-plane orbital limit \({B}_{\text{c}2}^{\perp }\) also deviates from the WHH curves (see the inset of Fig. 4), which is consistent with a previous report27,34. For another pair-breaking effect, the Pauli limit is estimated from the relation \({B}_{\text{P}}=1.86\,{T}_{\text{c}}\). Pauli limit is estimated to be 5.9 and 7.6 T for x = 0.22 and 0.69, respectively; these values (black diamonds in Fig. 4) are much lower than the observed \({B}_{c2}^{\parallel }\). We performed linear fitting (solid lines in the inset of Fig. 4) to evaluate the \({B}_{\text{c}2}^{\perp }(0)\) because the WHH curves are not applicable to the \({B}_{\text{c}2}^{\perp }(T)\) for both samples. The \({B}_{\text{c}2}^{\perp }(0)\) values were estimated to be 0.57 T and 0.98 T for both x = 0.22 and 0.69, respectively. The in-plane coherence length \({\xi }_{\parallel }\) was evaluated from the relation \({B}_{\text{c}2}^{\perp }={\Phi }_{0}/2\pi {\xi }_{\parallel }^{2}\) based on \({B}_{\text{c}2}^{\perp }(0)\) obtained from the linear fitting. This led to \({\xi }_{\parallel }\sim 24\) and \(\sim 18\) nm for x = 0.22 and 0.69, respectively. These values are comparable to those reported previously for x = 027,34. To estimate the out-of-plane coherence length \({\xi }_{\perp }\), we used the \({B}^{*}\) defined as the intersection of the extrapolation of the ρab(B) curves at the minimum temperature of ~ 0.47 K and ρn (see the insets of Fig. 3). Specifically, \({B}^{*}\) is expected to be the field which completely destroys the superconducting states at ~ 0.47 K. By using the values of \({\xi }_{\parallel }\) and \({B}^{*}\) at the minimum temperatures, we evaluated the out-of-plane coherence length \({\xi }_{\perp }\) at ~ 0.47 K from the relation \({B}_{\text{c}2}^{\parallel }={\Phi }_{0}/2\pi {\xi }_{\parallel }{\xi }_{\perp }\). This resulted in \({\xi }_{\perp }\sim 0.22\) and \(\sim 0.23\) nm for x = 0.22 and 0.69, respectively.

Figure 3
figure 3

Field dependence of the resistivity by a pulsed magnetic field. (a,b) Field dependence of the in-plane resistivity in \(B\parallel ab\) for x = 0.22 (a) and 0.69 (b). The ρn (black dashed lines) denotes normal resistivity in the pulsed-field measurement. The insets show magnified views of the ρab(B) curves at ~ 0.47 K near ρn. B* is defined as the intersection of the extrapolation of the ρab(B) curves and the ρn.

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Figure 4
figure 4

Huge upper critical field for locally non-centrosymmetric superconductor LaO0.5F0.5BiS2−xSex (x = 0.22 and 0.69). (a,b) Upper critical field as a function of temperature for x = 0.22 (a) and 0.69 (b). Red and blue circles show the \({B}_{\text{c}2}^{\parallel }\left(T\right)\) by static field and pulsed field, respectively. We added the error bars to the blue circles as it is possible for the applied field to slightly deviate from the ab plane owing to the lack of rotator measurements. Additionally, there is uncertainty related to the stability of temperature. Red squares show the \({B}_{\text{c}2}^{\perp }(T)\) by the static field. The black dashed curves show the Werthamer–Helfand–Hohenberg (WHH) fits. Black diamonds denote the Pauli limit. The insets show magnified views of the low-field region for \({B}_{\text{c}2}^{\perp }(T)\). Black solid lines show linear fits.

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The angular (θ) dependence of the \({B}_{\text{c}2}\) at 2.5 K for x = 0.22 and 3.5 K for x = 0.69 are displayed in Fig. 5a,b (θ represents the angle between the c-axis and the direction of the applied magnetic field). \({B}_{\text{c}2}(\theta )\) was also estimated from the middle point of the resistive transition. Generally, \({B}_{\text{c}2}(\theta )\) for layered superconductors is described by the anisotropic three-dimensional (3D) Ginzburg–Landau (GL) model or 2D Tinkham’s formula. The anisotropic 3D GL model is given by the relation \({\left({B}_{\text{c}2}\left(\theta \right)\text{cos}\theta /{B}_{\text{c}2}^{\perp }\right)}^{2}+{\left({B}_{\text{c}2}\left(\theta \right)\text{sin}\theta /{B}_{\text{c}2}^{\parallel }\right)}^{2}=1\) and explains \({B}_{\text{c}2}(\theta )\) anisotropic 3D systems. In contrast, the 2D Tinkham’s formula is given by the \(|\left.{B}_{\text{c}2}\left(\theta \right)\text{cos}\theta /{B}_{\text{c}2}^{\perp }\right|+{\left({B}_{\text{c}2}\left(\theta \right)\text{sin}\theta /{B}_{\text{c}2}^{\parallel }\right)}^{2}=1\) and describes \({B}_{\text{c}2}(\theta )\) in the case of the 2D systems6. The 2D Tinkham’s formula exhibits cusp-like behaviour around the magnetic field parallel to the in-plane. The observed \({B}_{\text{c}2}(\theta )\) is not contradictory to the anisotropic 3D GL model (solid line in Fig. 5). The broadening of \({B}_{\text{c}2}(\theta )\) at x = 0.69 can be attributed to the flux flow and/or tiny single crystals with an inclination of a few degrees. The 2D Tinkham’s formula (dashed line in Fig. 5) is not favourable for the broad \({B}_{\text{c}2}(\theta )\) behaviour. Furthermore, we performed the \({B}_{\text{c}2}(\theta )\) measurements at different temperatures (Supplementary Fig. 1), which is consistent with Fig. 5.

Figure 5
figure 5

Three-dimensional (3D) nature of the upper critical field. (a,b) Angular θ dependence of the upper critical field at 2.5 K for x = 0.22 (a) and at 3.5 K for x = 0.69 (b). θ represents the angle between the c-axis and the direction of the applied magnetic field. The insets show magnified views of the region around θ = 90°. The solid curves denote the anisotropic 3D Ginzburg–Landau (GL) model. The dashed curves show the two-dimensional (2D) Tinkham’s formula.

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The paramagnetic pair-breaking effect and the orbital pair-breaking effect should be suppressed because the observed \({B}_{\text{c}2}^{\parallel }\) clearly exceeds both the Pauli and the orbital limits. For the paramagnetic pair-breaking effect, the strong coupling nature leads to the enhancement of the Pauli limit. While we assumed the validity of the weak-coupling limit in the above explanation, some experimental results in BiCh2-based superconductors indicate that the strong coupling limit applies27,28. However, even if we use the reported value of \(\Delta =2.25\, {k}_{\text{B}}\,{T}_{\text{c}}\) for the single crystal of LaO0.5F0.5BiSSe (x = 1.0)27, the Pauli limit is much smaller than the observed \({B}_{\text{c}2}^{\parallel }\). A previous study proposed that the behaviour of in-plane \({B}_{\text{c}2}\) for LaO0.5F0.5BiS2 (x = 0) can be explained by the two-gap nature34. By contrast, specific heat measurements suggested a single gap for the LaO0.5F0.5BiSSe (x = 1.0) single-crystal sample27. We consider that the single-gap scenario is likely valid for the present samples because the superconducting properties, including the bulk nature of superconductivity of our single crystals, are similar to those for x = 1.0. Therefore, we suggest that the breaking of the local inversion symmetry in the BiCh2 layer leads to a large \({B}_{\text{c}2}^{\parallel }\). The Rashba-type spin–orbit coupling due to the lack of local inversion symmetry enhances the Pauli limit because the spin direction is locked onto the ab plane and the spin texture protects the Cooper pairs from de-pairing against the applied fields12,14,34,37. The local Rashba-type spin texture near the Fermi energy was observed by SARPES for LaO0.55F0.45BiS233. Moreover, the spin-singlet and spin-triplet states can be mixed in a material by breaking the local inversion symmetry9. If the spin-triplet component is the predominant component in the superconductivity in the present phases, the paramagnetic pair-breaking effect may be absent in this system. In any case, the paramagnetic pair-breaking effect was largely suppressed by the local inversion symmetry breaking in the BiCh2 layer.

The orbital limit should be enhanced by a layered structure. The Rashba-type spin–orbit coupling generally weakens the interlayer coupling and enhances the 2D nature of the superconductivity12. This situation allowed us to establish the Josephson vortex state in \(B\parallel ab\). In these conditions, we can deduce that the Josephson vortices penetrate in the LaO blocking layer when the out-of-plane coherence length \({\xi }_{\perp }\) is smaller than the thickness of the blocking layer, in which the vortices may induce the orbital pair-breaking effect. However, \({B}_{\text{c}2}(\theta )\) in Fig. 5 indicates that the 2D superconductivity is not very strong because the anisotropic 3D GL model is well fitted to the \({B}_{\text{c}2}(\theta )\) data. Therefore, it is reasonable to expect that the crossover of the conventional Abrikosov and Josephson vortex states was realised in this system. This description is almost consistent with the upturn behaviour of \({B}_{\text{c}2}(T)\) at Tc17. To confirm whether the angular-dependent \({B}_{\text{c}2}\) is changed by temperature, we may need to investigate \({B}_{\text{c}2}(\theta )\) at the lower-temperature region by using dilution systems in future work. Evaluation of the out-of-plane coherence length \({\xi }_{\perp }\sim 0.22\) (x = 0.22) and \(\sim 0.23\) nm (x = 0.69) at the minimum temperature indicated that the outcomes are comparable to the LaO blocking layer thicknesses \(\sim 0.26\) nm in the cases of both samples. The blocking layer thicknesses were estimated from powder X-ray diffraction at 298 K. We expect that \({B}_{\text{c}2}\) could be determined by the orbital pair-breaking effect from the Josephson vortices if we could confirm the change from the anisotropic 3D GL model to 2D Tinkham’s formula in the lower temperature region.

Herein, we discuss the intriguing complex-strip phase, which is a unique property of multilayer systems with local inversion symmetry breaking18,38. This phase is induced by the vortices penetrating the blocking layer and is thus regarded as the Josephson vortex state18. Although our system behaviours are not expected to be similar to the simple Josephson vortex states as discussed above and the upturn of \({B}_{\text{c}2}(T)\) at Tc does not seem suitable for the complex-strip phase, the behaviours of the \({B}_{\text{c}2}(T)\) obtained by the pulsed field (high-field regions) are similar to the transition of the complex-stripe phase to the helical phase18. If the mixture of spin-singlet and spin-triplet states is significant in our system, it may be difficult for the complex-stripe phase to emerge in this system because the complex-stripe phase is in spin-singlet superconductivity without a mixture of spin-triplet superconductivity38.

Finally, we briefly discuss the pairing symmetry of the superconductivity of the BiCh2-based superconductor. Many experiments are consistent with fully gapped s-wave superconductivity26,27,28, while the ARPES study found an anisotropic superconducting gap29. The breaking of local inversion symmetry may be a key factor in the solution of the puzzle because spin-singlet and spin-triplet superconductivity can be mixed in the local inversion symmetry breaking. We expect that additional work on the local inversion asymmetry in BiCh2-based superconductors will reveal the pairing symmetry of superconductivity.

In conclusion, we have observed extremely high in-plane \({B}_{\text{c}2}\) for the centrosymmetric superconductor LaO0.5F0.5BiS2−xSex with local inversion symmetry breaking in the BiCh2 layer. The superconducting states were not completely destroyed, even at the field strength of 55 T. The paramagnetic pair-breaking effect should be suppressed by the local Rashba-type spin–orbit coupling which arises from the lack of local inversion symmetry, and was observed by SARPES33. The orbital limit should be enhanced by the layered structure and strong local Rashba spin–orbit coupling. Local inversion symmetry breaking may be a clue for the solution of the pairing symmetry of superconductivity in the BiCh2-based superconductor family. Our results pave the way for the exploration of superconductors which have a high \({B}_{\text{c}2}\), and also lead to the in-depth understanding of the relationship between superconductivity and local inversion symmetry breaking.

Methods

Single crystal growth

Single crystal samples of LaO0.5F0.5BiS2−xSex (x = 0.25 and 0.75) were grown by a high-temperature flux method in a quartz tube in vacuum39. First, polycrystalline samples were prepared by a conventional solid-state reaction method to obtain a nominal composition. The obtained polycrystalline (0.4 g) samples were mixed with CsCl/KCl flux (5.0 g) at a molar ratio of 5:3, ground, and then sealed in a quartz tube in vacuum. The quartz tube was heated at 1223 K for 10 h, cooled to 873 K at a rate of − 2 K /h, and furnace-cooled to 298 K. The quartz tube was opened in air, and the flux was dissolved in a quartz tube using pure water. The real atomic ratio of the single-crystal samples for the resistivity measurement was estimated by EDX analysis, and its value was consistent with the nominal composition. The sizes of the plate-like single crystals for resistivity measurement are 1.53 mm × 2.05 mm ×  ~ 0.03 mm and 0.6 mm × 0.89 mm ×  ~ 0.04 mm for x = 0.22 and 0.69, respectively.

Transport measurements

Resistivity measurements for both static and pulsed fields were performed using the conventional four-probe method. The resistivity measurements in a static field up to 9 T and 2 K were performed using a physical property measurement system (PPMS) with a horizontal rotator probe. Pulsed magnetic field measurements were performed at the Institute for Solid State Physics (ISSP) at the University of Tokyo.