Introduction

The breaking of the out-of-plane or in-plane inversion symmetry in two-dimensional (2D) systems gives rise to Rashba-type or Zeeman-type spin–orbit coupling (SOC), respectively, which plays important roles in spintronics, valleytronics, optoelectronics and superconductivity1,2,3,4,5. Both types of SOCs cause the Fermi surface to be spin-split and the spin-momentum relation to be locked, but the spin polarisation in the momentum space is distinctively different; Rashba-type SOC forces the spins to be polarised in the in-plane direction while Zeeman-type SOC in the out-of-plane direction (Fig. 1a, b)1,2. These unique spin structures have notable implications in terms of superconductivity under strong magnetic fields6,7,8,9,10,11,12,13.

Fig. 1: Schematic illustration of dynamic spin-momentum locking.
figure 1

a Fermi surfaces in the presence of Rashba-type SOC. The spins are polarised in the in-plane directions and locked to the momentum. The split Fermi surfaces are characterised by spin textures with opposite helicities. b Fermi surfaces in the presence of Zeeman-type SOC. The spins are oriented in the out-of-plane directions. The green arrows in (a) and (b) indicate a magnetic field applied in the in-plane direction. c When an electron at the initial state ki is elastically scattered to kf by a non-magnetic scattering centre (depicted by a purple ball), its spin is forced to rotate.

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Suppose the magnetic field is applied to a 2D superconductor precisely in the in-plane direction. Since electron orbitals are barely affected in this configuration, Cooper pairs are destroyed mainly due to the field-induced parallel alignment of the electron spins, which otherwise form an anti-parallel spin-singlet state. This mechanism is called paramagnetic pair breaking, and the upper critical magnetic field Bc2∥ determined by this effect is called the Pauli limit BPauli14,15. In the presence of Zeeman-type SOC, the spins are hardly tilted in the field direction because they are statically locked in the out-of-plane direction. This suppresses the paramagnetic pair breaking effect and substantially enhances Bc2∥ over BPauli8,9. By contrast, the in-plane spin-momentum locking due to Rashba-type SOC can enhance Bc2∥ only by a factor of \(\sqrt{2}\) because of a significant deformation of the Fermi surfaces due to the locking6. Nevertheless, Rashba-type SOC may also strongly enhance Bc2∥ if a dynamic electron scattering process is involved. In this case, because of the spin-momentum locking, the spin is forced to flip at every momentum change accompanied by elastic scattering (Fig. 1c). The mechanism, referred to as dynamic spin-momentum locking here, should cause frequent spin scatterings while preserving the time-reversal symmetry. This enhances Bc2∥ through the suppression of paramagnetic pair-breaking effect even in crystalline systems in an analogous manner as the conventional spin–orbit scattering does in disordered systems. Although such an effect was suggested by Nam et al. for Pb thin films, it was considered dominated by the orbital pair-breaking and hence has remained elusive10. Furthermore, the presence of the Rashba-type SOC itself was an assumption, and the material may include Zeeman-type SOC11. Experimentally resolving this problem requires one to confirm the exclusive presence of the Rashba-type SOC in a relevant system. It is also important to investigate its superconducting properties under a controlled environment to avoid any extrinsic effects.

In the present study, we adopt a crystalline In atomic-layer on a Si(111) surface [Si(111)-(\(\sqrt{7}\times \sqrt{3}\))-In] to fulfil this requirement. Clear Fermi surface splitting and in-plane spin polarisation are demonstrated by angle-resolved photoemission spectroscopy (ARPES) and density functional theory (DFT) calculations, confirming the exclusive presence of the Rashba-type SOC. In situ electron transport measurements under ultrahigh vacuum (UHV) environment reveal that Bc2∥ is anomalously enhanced over the Pauli limit BPauli. The enhancement factor defined by Bc2∥/BPauli reaches ~3 and exceeds the factor of \(\sqrt{2}\) expected for the static locking effect of the Rashba-type SOC. Our quantitative data analysis clarifies that the paramagnetic pair-breaking parameter αP is strongly suppressed by orders of magnitudes from the value estimated for the conventional spin–orbit scattering. The spin scattering times τs directly related to αP are in satisfactory agreement with the electron elastic scattering times τel, proving the idea of spin flipping at every momentum change. These results provide compelling evidence that this 2D superconductor with Rashba-type SOC is protected by dynamic spin-momentum locking.

Results

Rashba-type SOC revealed by ARPES and DFT

The Si(111)-(\(\sqrt{7}\times \sqrt{3}\))-In (referred to as \(\sqrt{7}\times \sqrt{3}\)-In here) consists of a uniform In bilayer covering the Si(111)-1 × 1 surface with a periodicity of \(\sqrt{7}\times \sqrt{3}\) (Fig. 2a)16,17, and superconductivity occurs below 3 K18,19. The breaking of the out-of-plane inversion symmetry due to the presence of the Si surface leads to the Rashba-type SOC as described below, as reported for other atomic-layer crystals on surfaces20,21,22,23. The material has highly dispersed electronic bands and simple chemical composition without magnetic or heavy elements24, which allows us to neglect complex correlation effects. Despite these ideal features, studying superconducting properties of \(\sqrt{7}\times \sqrt{3}\)-In is challenging because the susceptibility to foreign molecules and surface defects prohibits air exposure and the usage of conventional cryogenic and high-magnetic-field systems25,26. In this study, all experiments, including the transport measurements, were performed in UHV to eliminate the possibility of sample degradation (see “Materials and Methods”).

Fig. 2: Crystal structure and experimentally-observed Fermi surface.
figure 2

a Top view (left) and side view (right) of the crystal structure of \(\sqrt{7}\times \sqrt{3}\)-In. The x, y, and z axes are defined to be \([1\bar{1}0]\), \([11\bar{2}]\), and [111] directions, respectively. b Schematic of the 1st and 2nd Brillouin zones. The dashed box represents the area of the photoelectron intensity map in (c). The characteristic arc feature and butterfly-wing feature of the Fermi surface are depicted. c Photoelectron intensity map at EF. The arrows indicate the portions of the Fermi surface with a clear splitting.

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The details of the electronic structures and the presence of Rashba-type SOC are clarified through ARPES measurements and DFT calculations. Figure 2c shows the photoelectron intensity map at the Fermi energy (EF) measured over the momentum-space region depicted in Fig. 2b. While the result is consistent with the previous studies16,24, it clearly resolves the splitting of the Fermi surfaces for the first time, which is particularly conspicuous on the “arc” and “butterfly-wing” portions (see the pairs of arrows). This finding was fully reproduced by our DFT calculations. The computed Fermi surface structure is essentially identical to the ARPES data (Fig. 3a). While the magnitude of the energy splitting at EF (ΔR) is small along the high-symmetry lines (Y–Γ–X), it is larger at the butterfly-wing along the P–Q line (Fig. 3f). Figure 3c shows that the distribution of ΔR exhibits a peak around 15–20 meV and ranges up to 90 meV. The DFT calculations also confirm that these Fermi surfaces are indeed spin-polarised. As indicated by the arrows in Fig. 3a, the spins are oriented in the in-plane directions, as expected from Rashba-type SOC. The effect of Zeeman-type SOC is negligible, judging from the fact that the out-of-plane components of spins are nearly absent (Fig. 3d). This point will be discussed later in detail. Interestingly, the azimuthal orientation of the spins on the butterfly wing features deviates from the helical spin texture characteristic of the standard Rashba-type SOC. This in-plane spin texture does not affect the conclusion of the present study, and its microscopic origin will be discussed elsewhere27. We also mention the presence of a large anisotropy in the Fermi velocity vF, which was computed as the gradient of band dispersion (Fig. 3b). The histogram in Fig. 3e shows that ∣vF∣ ranges from 2 × 105 to 1.5 × 106 m s−1. The detailed band structure information obtained here will be used later.

Fig. 3: Computed spin-split Fermi surface and Fermi velocity.
figure 3

a Fermi surface obtained from the DFT calculation. The colour indicates the magnitude of band splitting ΔR. Here, ΔR is defined at each point on the Fermi surface as the energy difference from the partner band, as depicted by the double-headed arrows in (f). The small arrows indicate the orientation of the spins. b Fermi velocity ∣vF∣ computed from the band dispersion. c–e Histograms of ΔR (c), the z component of the spin, sz, (d), and ∣vF∣ (e) measured on the Fermi surface. These histograms reflect the weighting factor to the density of states given by dl/∣vF∣, where dl is the line segment on the Fermi surface. c, e share the same colour scales as (a) and (b), respectively. f Energy bands along Y–Γ–X and P–Q. The vertical axis shows the energy measured from the Fermi level. The colour indicates the relative direction of spin with respect to momentum; red and cyan correspond to the clockwise and counterclockwise helicities. For clarity, the states with ΔR < 10 meV are coloured in grey.

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Robust superconductivity in in-plane magnetic fields

Six \(\sqrt{7}\times \sqrt{3}\)-In samples were prepared for electron transport experiments. In addition to three nominally flat Si(111) surfaces (Flat#1/#2/#3), we used three vicinal surfaces (Vicinal#1/#2 with a miscut angle of 0. 5∘ and Vicinal#3 with a miscut angle of 1.1∘) to control the density of scattering sources. These sample surfaces consisted of atomically flat terraces separated by steps, as observed by scanning tunnelling microscopy (STM) (Fig. 4a: Flat#1, Fig. 4b: Vicinal#1). The low-energy electron diffraction (LEED) patterns of Flat#1 and Vicinal#1 confirmed the exclusive presence of \(\sqrt{7}\times \sqrt{3}\) structures with multi- and single-domains, respectively (Insets of Fig. 4a, b). Figure 4c shows the temperature (T) dependence of sheet resistance (Rsheet) recorded at zero magnetic field. The curves of the other four samples are available in Supplementary Fig. 1. All of the samples exhibit sharp superconducting transitions at Tc0, while precursors due to the 2D fluctuation effects are evident at T > Tc028. Here Tc0 is defined as the Bardeen–Cooper–Schrieffer (BCS) mean-field critical temperature, which was determined by fitting to an empirical formula29 (see Supplementary Note 1). The same fitting procedure also gives normal sheet resistance Rn in the absence of the 2D fluctuation effects. The obtained parameters for Tc0 and Rn are presented in Table 1. The small Rn of 36–90 Ω reflects the high crystallinity of the samples. These values are comparable to those reported for transition-metal dichalcogenide samples used in the studies of Zeeman-type SOC8,9.

Fig. 4: Characterisation of samples for electron transport measurements.
figure 4

a, b STM images of Flat#1 and Vicinal#1, respectively. The derivative (gradient) of the topographic data is displayed for highlighting the locations of atomic steps. The insets are the corresponding LEED patterns taken at a 94 eV beam energy. The circles indicate the peaks reflecting the \(\sqrt{7}\times \sqrt{3}\) periodicity. The plus symbol indicates (0, 0). c Resistance curves of Flat#1 and Vicinal#1 measured at zero magnetic field. The solid curves are fitting by an empirical formula (See Supplementary Note 1).

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Table 1 List of parameters obtained for the six samples.
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We now focus on the effects of strong magnetic fields on superconductivity of \(\sqrt{7}\times \sqrt{3}\)-In. Figure 5a shows the temperature dependence of sheet resistance Rsheet of Vicinal#1 measured under magnetic fields, which were applied precisely in the in-plane direction. The data of the other samples are presented in Supplementary Figure 2. While slight shifts and broadenings of the resistive transition were detected, superconductivity persisted even at the maximum magnetic field of B = 5 T. Figure 5c shows the magnetic field dependence of Tc of all six samples, where Tc is determined from T at which Rsheet decreases to half of Rn. The data show that the lowering of Tc as a function of B is quadratic and reaches 23% of Tc0 at 8.25 T for Flat#3. By contrast, for out-of-plane magnetic fields, the superconducting transition was rapidly suppressed and disappeared above B = 0.5 T (Fig. 5b). The lowering of Tc as a function of B is linear (Fig. 5d). Our detailed analysis for out-of-plane upper critical field Bc2⊥ shows that the observed rapid quenching of superconductivity is due to penetration of vortices, i.e. to orbital pair-breaking effect (see Supplementary Fig. 3 and Supplementary Note 2). The robust superconductivity against the in-plane fields, in contrast, indicates that the pair-breaking is not caused by the orbital effect but rather by the paramagnetic effect as expected. For the present superconductor with Tc0 = 2.97–3.14 K, the Pauli limit BPauli is equal to 5.5–5.8 T from the relation BPauli = 1.86 (T K−1)Tc014,15. Since the observed Bc2∥ apparently exceeds this limit as T → 0, the paramagnetic pair breaking effect must be substantially suppressed.

Fig. 5: Superconducting properties in in-plane and out-of-plane magnetic fields.
figure 5

a Resistance curves of Vicinal#1 in in-plane magnetic fields of B = 0, 1, 2, 3, 4 and 5 T. b Resistance curves of Vicinal#1 in out-of-plane magnetic fields of B = 0, 0.02, 0.1, 0.15, 0.2, 0.25, 0.3 and 0.5 T. The insets of (a) and (b) show the directions of the magnetic fields with respect to the samples. The samples were patterned in a shape suitable for four-terminal measurements, as represented by areas coloured in light blue. I+(−) and V+(−) are the current and voltage terminals, respectively. c Field dependence of Tc in in-plane magnetic fields. d Field dependence of Tc in out-of-plane magnetic fields. The dotted curves in (c) and (d) are the quadratic and linear functions fitted to the data. e Comparison of Bc2∥ and Bc2⊥. For clarity, Bc2⊥ is scaled by a factor of 10. The dotted curves are the universal function Eq. (1) plotted with parameters determined from the fitting analyses. The dashed horizontal line indicates the enhancement factor \(\sqrt{2}\) for static locking effect of Rashba-type SOC. The relatively large variation in the fitting curves for Bc2∥ originates mainly from the angular error of the sample orientation denoted by θe in the main text.

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Paramagnetically limited upper critical field

It is widely known that spin scattering is induced occasionally at an elastic electron scattering event by the atomistic SOC. In the presence of this conventional spin–orbit scattering, Cooper pairs are no longer exact spin-singlet states. It induces a finite spin susceptibility in the system and lowers the Zeeman energy gain acquired by breaking a Cooper pair under a magnetic field, thus suppressing the paramagnetic pair-breaking effects30. Here we assume this mechanism and, without taking account of the Rashba-type SOC, analyse the magnetic field effects on superconductivity in terms of pair-breaking parameters. The dependence of Tc on magnetic field B can be described using a universal function given by

$${\mathrm{ln}}\,\left(\frac{{T}_{{\rm{c}}}}{{T}_{{\rm{c}}0}}\right)=\psi \left(\frac{1}{2}\right)-\psi \left(\frac{1}{2}+\frac{\alpha ({\bf{B}})}{2\pi {k}_{{\rm{B}}}{T}_{{\rm{c}}}}\right),$$
(1)

where ψ is the digamma function, and α(B) denotes field-dependent pair-breaking parameter31. α(B) is the sum of three contributions: αO⊥ and αO∥ representing the orbital effects due to out-of-plane (B⊥) and in-plane (B∥) fields and αP the paramagnetic effect due to the total field ∣B∣ in the presence of frequent spin scatterings. It is given by the equation

$$\alpha ({\bf{B}})={\alpha }_{{\rm{O}}\perp }+{\alpha }_{{\rm{O}}\parallel }+{\alpha }_{{\rm{P}}}$$
(2)
$$={c}_{{\rm{O}}\perp }{B}_{\perp }+{c}_{{\rm{O}}\parallel }{B}_{\parallel }^{2}+{c}_{{\rm{P}}}| {\bf{B}}{| }^{2},$$
(3)

where cO⊥, cO∥ and cP are coefficients for individual contributions32. This form of the pair-breaking parameter is closely related to the Klemm-Luther-Beasley (KLB) model proposed for 2D superconductors with conventional spin–orbit scattering33,34.

In the present study, the addition of the αO∥ term allows us to account for the orbital effect within the superconducting layer under the in-plane magnetic field, which is not included in the KLB model. This effect played a crucial role in few-layer Pb films studied previously10. For the in-plane configuration, B⊥ ≃ θe∣B∣ and B∥ ≃ ∣B∣, where θe is the angular error. All coefficients were determined by fitting Eq. (1) to the experimental data in Fig. 5c, d, and the results are listed in Table 1 (for details, see “Materials and Methods”). From the values of cO⊥, cO∥, cP, we conclude αO⊥, αO∥ ≪ αP in the in-plane configuration, meaning that the pair breaking is dominated by the paramagnetic effect. This is distinct from the finding by Nam et al. that the orbital effect is the primary pair breaking mechanism for 5–13 Pb monolayers on the Si(111) surface10. Figure 5e plots Bc2∥/BPauli and Bc2⊥/BPauli as a function of Tc/Tc0, along with their extrapolations down to T = 0 calculated with the universal function of Eq. (1). Bc2∥/BPauli is found to reach ~ 3 at T = 0. We note that this enhancement factor exceeds the value of \(\sqrt{2}\), which is expected for the static effect of Rashba-type spin momentum locking. This claim is directly evidenced by the maximum value of Bc2∥/BPauli = 1.43 obtained for Flat#3.

Spin flipping rate enhanced by dynamic spin-momentum locking

The strong enhancement of Bc2∥ observed above is actually not attributed to the atomistic SOC, but to the Rashba-type SOC as explained in the following. We first estimate elastic scattering time τel from the normal-state sheet resistance Rn. The calculation was carried out by explicitly considering the anisotropy of Fermi velocity vF computed above (Fig. 3c, d) and by employing the Boltzmann theory under relaxation approximation35. The sheet conductance is given by

$${\sigma }_{\mu \mu }={\tau }_{{\rm{el}}}{I}_{\mu \mu }$$
(4)

with

$${I}_{\mu \mu }\equiv \frac{{e}^{2}}{{(2\pi )}^{2}\hslash }{\int}_{\!\!\!\!{\rm{FS}}}dk\frac{{v}_{{\rm{F}}\mu }{({\bf{k}})}^{2}}{| {{\bf{v}}}_{{\rm{F}}}({\bf{k}})| },$$
(5)

where vFμ (μ = x or y) is the μ component of vF. The integral was taken over all the spin-split Fermi surfaces, yielding Ixx = 3.9 × 10−4 Ω−1 fs−1 and Iyy = 4.5 × 10−4 Ω−1 fs−1. τel was evaluated from \({R}_{{\rm{n}}}^{-1}=({\sigma }_{xx}+{\sigma }_{yy})/2\) for multi-domain flat samples (Flat#1/#2/#3) and from \({R}_{{\rm{n}}}^{-1}={\sigma }_{xx}\) for single-domain vicinal samples (Vicinal#1/#2#3). This gives τel = 71.7, 53.3, 44.9 fs for Flat#1/#2/#3 and τel = 28.6, 39.2, 31.4 fs for Vicinal#1/#2/#3, respectively (Table 1). We then estimate the spin scattering time τs from the coefficient cP for paramagnetic pair breaking effect. τs is calculated with an equation

$${c}_{{\rm{P}}}=\frac{3{\tau }_{{\rm{s}}}{\mu }_{{\rm{B}}}^{2}}{2\hslash },$$
(6)

where μB is the Bohr magneton and ℏ the reduced Plank constant36. This gives τs = 86 ± 12, 52 ± 14, 33 ± 18 fs for Flat#1/#2/#3, and τs = 69 ± 12, 70 ± 12, 57 ± 12 fs for Vicinal#1/#2/#3 (see Table 1). These results lead to τel/τs ≃ 0.5 − 1. Nevertheless, if only the conventional spin–orbit scattering is considered, τs should be much larger than the τel. In this case, the ratio τel/τs should be on the order of (Zα)4, where Z is the atomic number and α is the fine structure constant30. For In (Z = 49), τel/τs ~ 1/60. An experimental study reported an even smaller τel/τs of about 10−3 for thin In films37. Therefore, the spin–orbit scattering that occurs in the absence of the Rashba-type SOC cannot account for our result. In contrast, if the Rashba-type SOC is considered, it can be reasonably explained based on the concept of dynamic spin-momentum locking; namely, every elastic scattering should contribute a spin flipping and τel/τs approaches unity. The decrease in τs together with Eq. (3) and Eq. (6) means the paramagnetic pair breaking parameter αP is suppressed by orders of magnitude from the value expected for the conventional spin–orbit scattering.

Remarkably, for the flat samples, τs falls equal to τel within the experimental error. By contrast, τs is larger than τel by a factor of two for the vicinal samples. This can be reasonably explained by an energy broadening caused by electron elastic scattering, ℏ/τel. For vicinal samples, ℏ/τel = 16–24 meV is comparable to the peak energy in the distribution of ΔR (see Fig. 3b). This energy broadening degrades the spin polarisation at a large portion of the Fermi surface and partially unlocks the spin-momentum relation, resulting in a recovery of spin scattering time τs. For flat samples, ℏ/τel = 9–14 meV < ΔR, meaning that the spin texture of the energy bands remains intact for the whole Fermi surface. This argument further supports our conclusion on the critical role of the dynamic effect of the Rashba-type SOC.

Finally, we note that the static spin-momentum locking due to the Rashba-type SOC can enhance the in-plane critical field Bc2∥ by a factor of \(\sqrt{2}\) from the Pauli limit. This effect is likely to be weakened by electron scattering and mixing between different spin states, but here we estimate the upper limit of error in spin scattering time τs (for a detailed discussion, see Supplementary Note 3). When it is taken into account as an effective magnetic field \({B}_{{\rm{eff}}}=(1/\sqrt{2})B\), the value of τs obtained above is doubled, leading to τel/τs = 0.25−0.5. These values are still much higher than 1/60-1/1000 expected from the atomistic spin–orbit scattering mechanism. Therefore, the result is not attributable only to the conventional mechanism, and our conclusion remains the same.

Discussion

Here we discuss the consistency with the theoretical studies of Rashba-type superconductors with non-magnetic impurities10,38,39,40. These studies predict that upper critical field increases with the decrease in elastic scattering time τel. In 2D, the enhancement factor corresponds to a pair-breaking parameter \(\alpha =(2{\mu }_{{\rm{B}}}^{2}{\tau }_{{\rm{el}}}/\hslash ){B}^{2}\) in the limit of strong SOC (ℏ/τel ≪ ΔR)10. This expression is equivalent to Eq. (6) if τs is replaced by (4/3)τel. The agreement allows us to interpret the above theoretical result in terms of dynamic spin-momentum locking. Theories also claim that the ground state of a 2D superconductor with Rashba-type SOC has a helical state with a spatially modulated order parameter38,39,40. The formation of the helical state may increase Bc2∥, and a previous study on a quench-condensed monolayer Pb film attributed their observation of giant Bc2∥ to this effect7. However, the enhancement factor is only in the order of \(\sim {({{{\Delta }}}_{{\rm{R}}}/{E}_{{\rm{F}}})}^{2}\) and is usually negligible because ΔR ≪ EF40. Therefore, the observed large Bc2∥ in the present and previous studies are not attributable to the formation of the helical state.

Another issue to be discussed is the possible effect of a finite Zeeman-type SOC, which is suggested from the non-zero out-of-plane spin polarisations shown in Fig. 3d. From the spin polarisation direction calculated as a function of energy splitting, one sees that the spins align in the in-plane directions for the most of energy regions (Supplementary Figure 6). The spins tend to tilt toward the out-of-plane direction below 30 meV, but the off-angle is about 45∘ at most. Namely, there is no region where the Zeeman-type SOC is dominant. This non-dominant Zeeman SOC confined to the small area of the Fermi surface can barely enhance Bc2∥ because the enhancement factor is determined by an average over the whole Fermi surface41. If the dynamics of spins is considered, the effect of the Zeeman-type SOC can be suppressed even more. Thus, we conclude that the Zeeman-type SOC plays only a minor role in the present system. For more discussions, see Supplementary Note 4.

The present result has significant implications in terms of robustness of a superconductor with the Rashba-type SOC in general under a strong magnetic field as well as in the proximity of a ferromagnet. The presence of a strong exchange interaction at the interface with a ferromagnet usually leads to the destruction of superconductivity to the depth of the coherence length. However, since the destruction of superconductivity by exchange interaction is caused by the same paramagnetic pair-breaking effect32, the dynamic spin-momentum locking revealed here may help superconductivity to persist even in this situation. This makes realistic the coexistence of a 2D superconductor and a ferromagnet at atomic scales, which has been proposed to realise emergent phenomena such as chiral topological superconductivity42,43,44. The new insight into the spin-momentum locking obtained in the present study will form the basis of such an unexplored realm of research.

Methods

ARPES

The high-resolution ARPES experiment was conducted in a UHV environment with a base pressure better than 1 × 10−8 Pa. The substrate was cut from an n-type (resistivity ρ < 0.001 Ω cm) vicinal Si(111) wafer with a miscut angle of 0.5∘ in the \([\bar{1}\bar{1}2]\) direction. The \(\sqrt{7}\times \sqrt{3}\)-In surface was prepared by thermal deposition of In onto a clean Si(111)-7 × 7 surface followed by annealing at 600 K for 2 min. The sample quality was confirmed from the sharp spots with low background intensity in the LEED patterns. The photoelectrons were excited by a vacuum-ultraviolet laser (hν = 6.994 eV) and were collected by a hemispherical photoelectron analyser45. The sample temperature was maintained at 35 K during the ARPES measurement. The energy and momentum resolutions were 3 meV and 1.4 × 10−3 Å−1, respectively.

DFT

The DFT calculations were performed using the Quantum ESPRESSO package46. We employed the augmented plane wave method and used the local density approximation (LDA) for the exchange correlation. The crystal structure of \(\sqrt{7}\times \sqrt{3}\)-In was modelled by a repeated slab consisting of an In bilayer, six Si bilayers, a H layer for termination, and a vacuum region of thickness 3 nm. We used a cutoff energy of 680 eV for the wave functions and a 6 × 8 × 1k-point mesh for the Brillouin zone. The geometry optimisation was performed without the SOC until all components of all forces became less than 2.6 × 10−3 eV ⋅ Å−1. Based on the optimised structure, we performed band calculations that included the SOC. To check the reproducibility of our result, we carried out the same calculation from scratch using another DFT package OpenMX47,48. The result by OpenMX is essentially the same as the one by Quantum ESPRESSO (See Supplementary Figs. 5, 6, as well as Supplementary Note 5).

Electron transport

For transport experiments, six samples were grown on substrates cut from Si(111) wafers (3 mm × 8 mm × 0.38 mm) with miscut angles of 0∘ (Flat#1, Flat#2, and Flat#3), 0.5∘ (Vicinal#1 and Vicinal#2), and 1.1∘ (Vicinal#3) in the \([\bar{1}\bar{1}2]\) direction. The non-doped wafers (ρ ≳ 1000 Ω cm) were chosen so that the electron conduction in bulk can be ignored at low temperatures. The \(\sqrt{7}\times \sqrt{3}\)-In surface was prepared under the UHV condition (base pressure 1 × 10−8 Pa) by depositing In onto a clean Si(111)-7 × 7 surface followed by annealing at 600 K for 10 s. The samples were then characterised by LEED and STM. The current path was defined by Ar+ sputtering using a shadow mask technique19,29. Electric contact was made at room temperature by mildly pressing four Au-plated spring probes. The samples were then cooled down to ~ 0.9 K or to ~ 0.4 K by pumping condensed 4He or 3He with a charcoal sorption pump. The magnetic fields were applied with a superconducting solenoid magnet. The maximum field was 5 T in the experiment of Flat#1/#2 and Vicinal#1/#2/#3 and was 8.25 T in the experiment of Flat#3. The sample was rotated in-situ to tune the angle of the magnetic fields with respect to the sample. The parallel field alignment was judged within an accuracy better than 0.1∘ from the minimum of sample resistance measured at a constant temperature near the Tc. The sample temperature was measured with a Cernox thermometer calibrated in magnetic fields. The DC resistance of the samples was measured using a nanovoltmeter (Keithley 2182A) with a bias current of 1 μA generated by a source meter (Keithley 2401).

Fitting analysis of pair-breaking effects

In the following, we denote ∣B∣ as B for simplicity. The pair-breaking parameters for orbital effects are given by

$${\alpha }_{{\rm{O}}\parallel }=\frac{D{e}^{2}{\delta }^{2}{B}_{\parallel }^{2}}{6\hslash }\equiv {c}_{{\rm{O}}\parallel }{B}_{\parallel }^{2}$$
(7)

for the in-plane component, and

$${\alpha }_{{\rm{O}}\perp }=De{B}_{\perp }\equiv {c}_{{\rm{O}}\perp }{B}_{\perp }$$
(8)

for the out-of-plane component of the magnetic fields32. Here, D is the diffusion coefficient, and δ represents the thickness of the superconducting layer. We assume δ = 4.5 Å, which is twice the height difference between the upper and lower atoms in the In bilayer (Fig. 1a). Note that cO∥ is related to cO⊥ as follows:

$${c}_{{\rm{O}}\parallel }=\frac{\pi {\delta }^{2}}{6{{{\Phi }}}_{0}}{c}_{{\rm{O}}\perp }.$$
(9)

The pair-breaking parameter for the paramagnetic effect in the presence of spin–orbit scattering is given by

$${\alpha }_{{\rm{P}}}=\frac{3{\mu }_{{\rm{B}}}^{2}{B}^{2}{\tau }_{{\rm{so}}}}{2\hslash }\equiv {c}_{{\rm{P}}}{B}^{2}.$$
(10)

The total pair-breaking parameter is the sum of all contributions and is given by Eq. (3). Near Tc0, the universal function Eq. (1) has an approximate form given by,

$${T}_{{\rm{c}}}={T}_{{\rm{c}}0}-\frac{\pi \alpha }{4{k}_{{\rm{B}}}}.$$
(11)

For the in-plane field, we take B⊥ ≃ θeB and B∥ ≃ B in Eq. (3). The explicit form of the fitting function becomes,

$${T}_{{\rm{c}}}={T}_{{\rm{c}}0}-\frac{\pi }{4{k}_{{\rm{B}}}}{c}_{{\rm{O}}\perp }{\theta }_{{\rm{e}}}B-\frac{\pi }{4{k}_{{\rm{B}}}}({c}_{{\rm{O}}\parallel }+{c}_{{\rm{P}}}){B}^{2}.$$
(12)

For the out-of-plane field, we take B⊥ ≃ B and assumed that \({c}_{{\rm{O}}\parallel }{B}_{\parallel }^{2},{c}_{{\rm{P}}}{B}_{\parallel }^{2}\ll {c}_{{\rm{O}}\perp }{B}_{\perp }\). The fitting function becomes,

$${T}_{{\rm{c}}}={T}_{{\rm{c}}0}-\frac{\pi }{4{k}_{{\rm{B}}}}{c}_{{\rm{O}}\perp }B.$$
(13)

These functions, (12) and (13), are fitted to the Tc curves in Fig. 5b, d, respectively. cO∥ and cP can be separated using Eq. (9). We confirmed that the estimated θe is within the accuracy in the sample angle control (see Supplementary Fig. 4).