Abstract
Spinmomentum locking is essential to the spinsplit Fermi surfaces of inversionsymmetry broken materials, which are caused by either Rashbatype or Zeemantype spinorbit coupling (SOC). While the effect of Zeemantype SOC on superconductivity has experimentally been shown recently, that of Rashbatype SOC remains elusive. Here we report on convincing evidence for the critical role of the spinmomentum locking on crystalline atomiclayer superconductors on surfaces, for which the presence of the Rashbatype SOC is demonstrated. Insitu electron transport measurements reveal that inplane upper critical magnetic field is anomalously enhanced, reaching approximately three times the Pauli limit at T = 0. Our quantitative analysis clarifies that dynamic spinmomentum locking, a mechanism where spin is forced to flip at every elastic electron scattering, suppresses the Cooper pairbreaking parameter by orders of magnitude and thereby protects superconductivity. The present result provides a new insight into how superconductivity can survive the detrimental effects of strong magnetic fields and exchange interactions.
Introduction
The breaking of the outofplane or inplane inversion symmetry in twodimensional (2D) systems gives rise to Rashbatype or Zeemantype spin–orbit coupling (SOC), respectively, which plays important roles in spintronics, valleytronics, optoelectronics and superconductivity^{1,2,3,4,5}. Both types of SOCs cause the Fermi surface to be spinsplit and the spinmomentum relation to be locked, but the spin polarisation in the momentum space is distinctively different; Rashbatype SOC forces the spins to be polarised in the inplane direction while Zeemantype SOC in the outofplane direction (Fig. 1a, b)^{1,2}. These unique spin structures have notable implications in terms of superconductivity under strong magnetic fields^{6,7,8,9,10,11,12,13}.
Suppose the magnetic field is applied to a 2D superconductor precisely in the inplane direction. Since electron orbitals are barely affected in this configuration, Cooper pairs are destroyed mainly due to the fieldinduced parallel alignment of the electron spins, which otherwise form an antiparallel spinsinglet state. This mechanism is called paramagnetic pair breaking, and the upper critical magnetic field B_{c2∥} determined by this effect is called the Pauli limit B_{Pauli}^{14,15}. In the presence of Zeemantype SOC, the spins are hardly tilted in the field direction because they are statically locked in the outofplane direction. This suppresses the paramagnetic pair breaking effect and substantially enhances B_{c2∥} over B_{Pauli}^{8,9}. By contrast, the inplane spinmomentum locking due to Rashbatype SOC can enhance B_{c2∥} only by a factor of \(\sqrt{2}\) because of a significant deformation of the Fermi surfaces due to the locking^{6}. Nevertheless, Rashbatype SOC may also strongly enhance B_{c2∥} if a dynamic electron scattering process is involved. In this case, because of the spinmomentum locking, the spin is forced to flip at every momentum change accompanied by elastic scattering (Fig. 1c). The mechanism, referred to as dynamic spinmomentum locking here, should cause frequent spin scatterings while preserving the timereversal symmetry. This enhances B_{c2∥} through the suppression of paramagnetic pairbreaking 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 pairbreaking and hence has remained elusive^{10}. Furthermore, the presence of the Rashbatype SOC itself was an assumption, and the material may include Zeemantype SOC^{11}. Experimentally resolving this problem requires one to confirm the exclusive presence of the Rashbatype 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 atomiclayer on a Si(111) surface [Si(111)(\(\sqrt{7}\times \sqrt{3}\))In] to fulfil this requirement. Clear Fermi surface splitting and inplane spin polarisation are demonstrated by angleresolved photoemission spectroscopy (ARPES) and density functional theory (DFT) calculations, confirming the exclusive presence of the Rashbatype SOC. In situ electron transport measurements under ultrahigh vacuum (UHV) environment reveal that B_{c2∥} is anomalously enhanced over the Pauli limit B_{Pauli}. The enhancement factor defined by B_{c2∥}/B_{Pauli} reaches ~3 and exceeds the factor of \(\sqrt{2}\) expected for the static locking effect of the Rashbatype SOC. Our quantitative data analysis clarifies that the paramagnetic pairbreaking parameter α_{P} is strongly suppressed by orders of magnitudes from the value estimated for the conventional spin–orbit scattering. The spin scattering times τ_{s} directly related to α_{P} are in satisfactory agreement with the electron elastic scattering times τ_{el}, proving the idea of spin flipping at every momentum change. These results provide compelling evidence that this 2D superconductor with Rashbatype SOC is protected by dynamic spinmomentum locking.
Results
Rashbatype 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 K^{18,19}. The breaking of the outofplane inversion symmetry due to the presence of the Si surface leads to the Rashbatype SOC as described below, as reported for other atomiclayer crystals on surfaces^{20,21,22,23}. The material has highly dispersed electronic bands and simple chemical composition without magnetic or heavy elements^{24}, 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 highmagneticfield systems^{25,26}. In this study, all experiments, including the transport measurements, were performed in UHV to eliminate the possibility of sample degradation (see “Materials and Methods”).
The details of the electronic structures and the presence of Rashbatype SOC are clarified through ARPES measurements and DFT calculations. Figure 2c shows the photoelectron intensity map at the Fermi energy (E_{F}) measured over the momentumspace region depicted in Fig. 2b. While the result is consistent with the previous studies^{16,24}, it clearly resolves the splitting of the Fermi surfaces for the first time, which is particularly conspicuous on the “arc” and “butterflywing” 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 E_{F} (Δ_{R}) is small along the highsymmetry lines (Y–Γ–X), it is larger at the butterflywing along the P–Q line (Fig. 3f). Figure 3c shows that the distribution of Δ_{R} exhibits a peak around 15–20 meV and ranges up to 90 meV. The DFT calculations also confirm that these Fermi surfaces are indeed spinpolarised. As indicated by the arrows in Fig. 3a, the spins are oriented in the inplane directions, as expected from Rashbatype SOC. The effect of Zeemantype SOC is negligible, judging from the fact that the outofplane 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 Rashbatype SOC. This inplane spin texture does not affect the conclusion of the present study, and its microscopic origin will be discussed elsewhere^{27}. We also mention the presence of a large anisotropy in the Fermi velocity v_{F}, which was computed as the gradient of band dispersion (Fig. 3b). The histogram in Fig. 3e shows that ∣v_{F}∣ ranges from 2 × 10^{5} to 1.5 × 10^{6} m s^{−1}. The detailed band structure information obtained here will be used later.
Robust superconductivity in inplane 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 lowenergy electron diffraction (LEED) patterns of Flat#1 and Vicinal#1 confirmed the exclusive presence of \(\sqrt{7}\times \sqrt{3}\) structures with multi and singledomains, respectively (Insets of Fig. 4a, b). Figure 4c shows the temperature (T) dependence of sheet resistance (R_{sheet}) 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 T_{c0}, while precursors due to the 2D fluctuation effects are evident at T > T_{c0}^{28}. Here T_{c0} is defined as the Bardeen–Cooper–Schrieffer (BCS) meanfield critical temperature, which was determined by fitting to an empirical formula^{29} (see Supplementary Note 1). The same fitting procedure also gives normal sheet resistance R_{n} in the absence of the 2D fluctuation effects. The obtained parameters for T_{c0} and R_{n} are presented in Table 1. The small R_{n} of 36–90 Ω reflects the high crystallinity of the samples. These values are comparable to those reported for transitionmetal dichalcogenide samples used in the studies of Zeemantype SOC^{8,9}.
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 R_{sheet} of Vicinal#1 measured under magnetic fields, which were applied precisely in the inplane 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 T_{c} of all six samples, where T_{c} is determined from T at which R_{sheet} decreases to half of R_{n}. The data show that the lowering of T_{c} as a function of B is quadratic and reaches 23% of T_{c0} at 8.25 T for Flat#3. By contrast, for outofplane magnetic fields, the superconducting transition was rapidly suppressed and disappeared above B = 0.5 T (Fig. 5b). The lowering of T_{c} as a function of B is linear (Fig. 5d). Our detailed analysis for outofplane upper critical field B_{c2⊥} shows that the observed rapid quenching of superconductivity is due to penetration of vortices, i.e. to orbital pairbreaking effect (see Supplementary Fig. 3 and Supplementary Note 2). The robust superconductivity against the inplane fields, in contrast, indicates that the pairbreaking is not caused by the orbital effect but rather by the paramagnetic effect as expected. For the present superconductor with T_{c0} = 2.97–3.14 K, the Pauli limit B_{Pauli} is equal to 5.5–5.8 T from the relation B_{Pauli} = 1.86 (T K^{−1})T_{c0}^{14,15}. Since the observed B_{c2∥} apparently exceeds this limit as T → 0, the paramagnetic pair breaking effect must be substantially suppressed.
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 spinsinglet 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 pairbreaking effects^{30}. Here we assume this mechanism and, without taking account of the Rashbatype SOC, analyse the magnetic field effects on superconductivity in terms of pairbreaking parameters. The dependence of T_{c} on magnetic field B can be described using a universal function given by
where ψ is the digamma function, and α(B) denotes fielddependent pairbreaking parameter^{31}. α(B) is the sum of three contributions: α_{O⊥} and α_{O∥} representing the orbital effects due to outofplane (B_{⊥}) and inplane (B_{∥}) fields and α_{P} the paramagnetic effect due to the total field ∣B∣ in the presence of frequent spin scatterings. It is given by the equation
where c_{O⊥}, c_{O∥} and c_{P} are coefficients for individual contributions^{32}. This form of the pairbreaking parameter is closely related to the KlemmLutherBeasley (KLB) model proposed for 2D superconductors with conventional spin–orbit scattering^{33,34}.
In the present study, the addition of the α_{O∥} term allows us to account for the orbital effect within the superconducting layer under the inplane magnetic field, which is not included in the KLB model. This effect played a crucial role in fewlayer Pb films studied previously^{10}. For the inplane configuration, B_{⊥} ≃ θ_{e}∣B∣ and B_{∥} ≃ ∣B∣, where θ_{e} 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 c_{O⊥}, c_{O∥}, c_{P}, we conclude α_{O⊥}, α_{O∥} ≪ α_{P} in the inplane 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) surface^{10}. Figure 5e plots B_{c2∥}/B_{Pauli} and B_{c2⊥}/B_{Pauli} as a function of T_{c}/T_{c0}, along with their extrapolations down to T = 0 calculated with the universal function of Eq. (1). B_{c2∥}/B_{Pauli} 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 Rashbatype spin momentum locking. This claim is directly evidenced by the maximum value of B_{c2∥}/B_{Pauli} = 1.43 obtained for Flat#3.
Spin flipping rate enhanced by dynamic spinmomentum locking
The strong enhancement of B_{c2∥} observed above is actually not attributed to the atomistic SOC, but to the Rashbatype SOC as explained in the following. We first estimate elastic scattering time τ_{el} from the normalstate sheet resistance R_{n}. The calculation was carried out by explicitly considering the anisotropy of Fermi velocity v_{F} computed above (Fig. 3c, d) and by employing the Boltzmann theory under relaxation approximation^{35}. The sheet conductance is given by
with
where v_{Fμ} (μ = x or y) is the μ component of v_{F}. The integral was taken over all the spinsplit Fermi surfaces, yielding I_{xx} = 3.9 × 10^{−4} Ω^{−1} fs^{−1} and I_{yy} = 4.5 × 10^{−4} Ω^{−1} fs^{−1}. τ_{el} was evaluated from \({R}_{{\rm{n}}}^{1}=({\sigma }_{xx}+{\sigma }_{yy})/2\) for multidomain flat samples (Flat#1/#2/#3) and from \({R}_{{\rm{n}}}^{1}={\sigma }_{xx}\) for singledomain vicinal samples (Vicinal#1/#2#3). This gives τ_{el} = 71.7, 53.3, 44.9 fs for Flat#1/#2/#3 and τ_{el} = 28.6, 39.2, 31.4 fs for Vicinal#1/#2/#3, respectively (Table 1). We then estimate the spin scattering time τ_{s} from the coefficient c_{P} for paramagnetic pair breaking effect. τ_{s} is calculated with an equation
where μ_{B} is the Bohr magneton and ℏ the reduced Plank constant^{36}. This gives τ_{s} = 86 ± 12, 52 ± 14, 33 ± 18 fs for Flat#1/#2/#3, and τ_{s} = 69 ± 12, 70 ± 12, 57 ± 12 fs for Vicinal#1/#2/#3 (see Table 1). These results lead to τ_{el}/τ_{s} ≃ 0.5 − 1. Nevertheless, if only the conventional spin–orbit scattering is considered, τ_{s} should be much larger than the τ_{el}. In this case, the ratio τ_{el}/τ_{s} should be on the order of (Zα)^{4}, where Z is the atomic number and α is the fine structure constant^{30}. For In (Z = 49), τ_{el}/τ_{s} ~ 1/60. An experimental study reported an even smaller τ_{el}/τ_{s} of about 10^{−3} for thin In films^{37}. Therefore, the spin–orbit scattering that occurs in the absence of the Rashbatype SOC cannot account for our result. In contrast, if the Rashbatype SOC is considered, it can be reasonably explained based on the concept of dynamic spinmomentum locking; namely, every elastic scattering should contribute a spin flipping and τ_{el}/τ_{s} approaches unity. The decrease in τ_{s} together with Eq. (3) and Eq. (6) means the paramagnetic pair breaking parameter α_{P} is suppressed by orders of magnitude from the value expected for the conventional spin–orbit scattering.
Remarkably, for the flat samples, τ_{s} falls equal to τ_{el} within the experimental error. By contrast, τ_{s} is larger than τ_{el} by a factor of two for the vicinal samples. This can be reasonably explained by an energy broadening caused by electron elastic scattering, ℏ/τ_{el}. For vicinal samples, ℏ/τ_{el} = 16–24 meV is comparable to the peak energy in the distribution of Δ_{R} (see Fig. 3b). This energy broadening degrades the spin polarisation at a large portion of the Fermi surface and partially unlocks the spinmomentum relation, resulting in a recovery of spin scattering time τ_{s}. For flat samples, ℏ/τ_{el} = 9–14 meV < Δ_{R}, 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 Rashbatype SOC.
Finally, we note that the static spinmomentum locking due to the Rashbatype SOC can enhance the inplane critical field B_{c2∥} 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 τ_{s} (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 τ_{s} obtained above is doubled, leading to τ_{el}/τ_{s} = 0.25−0.5. These values are still much higher than 1/601/1000 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 Rashbatype superconductors with nonmagnetic impurities^{10,38,39,40}. These studies predict that upper critical field increases with the decrease in elastic scattering time τ_{el}. In 2D, the enhancement factor corresponds to a pairbreaking parameter \(\alpha =(2{\mu }_{{\rm{B}}}^{2}{\tau }_{{\rm{el}}}/\hslash ){B}^{2}\) in the limit of strong SOC (ℏ/τ_{el} ≪ Δ_{R})^{10}. This expression is equivalent to Eq. (6) if τ_{s} is replaced by (4/3)τ_{el}. The agreement allows us to interpret the above theoretical result in terms of dynamic spinmomentum locking. Theories also claim that the ground state of a 2D superconductor with Rashbatype SOC has a helical state with a spatially modulated order parameter^{38,39,40}. The formation of the helical state may increase B_{c2∥}, and a previous study on a quenchcondensed monolayer Pb film attributed their observation of giant B_{c2∥} to this effect^{7}. However, the enhancement factor is only in the order of \(\sim {({{{\Delta }}}_{{\rm{R}}}/{E}_{{\rm{F}}})}^{2}\) and is usually negligible because Δ_{R} ≪ E_{F}^{40}. Therefore, the observed large B_{c2∥} 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 Zeemantype SOC, which is suggested from the nonzero outofplane 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 inplane directions for the most of energy regions (Supplementary Figure 6). The spins tend to tilt toward the outofplane direction below 30 meV, but the offangle is about 45^{∘} at most. Namely, there is no region where the Zeemantype SOC is dominant. This nondominant Zeeman SOC confined to the small area of the Fermi surface can barely enhance B_{c2∥} because the enhancement factor is determined by an average over the whole Fermi surface^{41}. If the dynamics of spins is considered, the effect of the Zeemantype SOC can be suppressed even more. Thus, we conclude that the Zeemantype 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 Rashbatype 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 pairbreaking effect^{32}, the dynamic spinmomentum 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 superconductivity^{42,43,44}. The new insight into the spinmomentum locking obtained in the present study will form the basis of such an unexplored realm of research.
Methods
ARPES
The highresolution ARPES experiment was conducted in a UHV environment with a base pressure better than 1 × 10^{−8} Pa. The substrate was cut from an ntype (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 vacuumultraviolet laser (hν = 6.994 eV) and were collected by a hemispherical photoelectron analyser^{45}. 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 package^{46}. 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 × 1kpoint 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 OpenMX^{47,48}. 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 nondoped 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 technique^{19,29}. Electric contact was made at room temperature by mildly pressing four Auplated spring probes. The samples were then cooled down to ~ 0.9 K or to ~ 0.4 K by pumping condensed ^{4}He or ^{3}He 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 insitu 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 T_{c}. 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 pairbreaking effects
In the following, we denote ∣B∣ as B for simplicity. The pairbreaking parameters for orbital effects are given by
for the inplane component, and
for the outofplane component of the magnetic fields^{32}. 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 c_{O∥} is related to c_{O⊥} as follows:
The pairbreaking parameter for the paramagnetic effect in the presence of spin–orbit scattering is given by
The total pairbreaking parameter is the sum of all contributions and is given by Eq. (3). Near T_{c0}, the universal function Eq. (1) has an approximate form given by,
For the inplane field, we take B_{⊥} ≃ θ_{e}B and B_{∥} ≃ B in Eq. (3). The explicit form of the fitting function becomes,
For the outofplane 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,
These functions, (12) and (13), are fitted to the T_{c} curves in Fig. 5b, d, respectively. c_{O∥} and c_{P} can be separated using Eq. (9). We confirmed that the estimated θ_{e} is within the accuracy in the sample angle control (see Supplementary Fig. 4).
Data availability
The data that support the finding of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank Y. Higashi, S. Ichinokura and T. Shishidou for helpful discussions. We also thank K. Kuroda and A. Harasawa for their technical supports during the ARPES experiments. This work was supported financially by JSPS KAKENHI (Grant Numbers 18H01876, 16K17727, 25247053, 19H02592, 19H00651, 18K03484, 17H05211, and 17H05461), by Advanced Technology Institute (ATI) Research Grants 2017, and by World Premier International Research Center (WPI) Initiative on Materials Nanoarchitectonics, MEXT, Japan.
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S.Y. and T.U. conceived the experiment and wrote the manuscript. S.Y, K. Yokota, and T.U. carried out the electron transport experiments. S.Y. analysed the transport data and performed the DFT calculations. T.K., Y.N., K. Yaji, and K.S. measured the ARPES data with supports from F.K. and S.S. All the authors discussed the results and contributed to finalising the manuscript.
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Yoshizawa, S., Kobayashi, T., Nakata, Y. et al. Atomiclayer Rashbatype superconductor protected by dynamic spinmomentum locking. Nat Commun 12, 1462 (2021). https://doi.org/10.1038/s41467021216421
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DOI: https://doi.org/10.1038/s41467021216421
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