Two-dimensional superconductors with atomic-scale thickness

One of the first experiments on superconducting ultrathin metal films with atomically well-defined thicknesses and high crystallinity was performed using Pb films grown on a clean Si(111)-(7 × 7) surface using the MBE method in UHV [56]. An advantage of using these materials is that Pb and Si do not form an alloy so that their interface is atomically sharp. Furthermore, Pb films can grow on a clean Si surface in a layer-by-layer fashion at low temperatures [57], which makes it possible to fabricate ultrathin films with nearly uniform thicknesses. Thus obtained ultrathin Pb films can be an ideal system for studying 2D superconductivity. In [56], the films were characterized using a spot-profile analysis of low-energy electron diffraction (LEED) and the obtained structural information was utilized for the analysis of the transport data. The transport measurements were performed in situ in UHV to exclude contamination and oxidation of the samples. The T_c of the Pb film was found to remain around 5–7 K down to a thickness of 4ML, which was close to the bulk T_c value of 7.2 K. However, the T_c decreased substantially to 1–2 K in the sub-4ML regime, where structural disorder in the macroscopic scale was unavoidable.

A particular interest in ultrathin metal films with atomically smooth interfaces arises from the fact that conduction electrons are quantized in the out-of-plane direction to form quantum well states (QWS). The quantum confinement occurs when the Bohr–Sommerfeld quantization rule is satisfied [58]:

$$\begin{eqnarray}&&\displaystyle \frac{4\pi d}{\lambda (E)}+{\rm{\Phi }}(E)=2n\pi ,\end{eqnarray} \tag{ 4 }$$

where λ(E) is the electron wavelength at an energy E, d is the film thickness, Φ(E) is the total phase shift at the boundaries, and n is an integer. Since λ(E_F) = 1.06 nm for Pb (E_F: Fermi energy), d must be both of the order of nanometers and atomically uniform for this effect to be observed. The formation of QWS leads to an oscillation in the electron density of states at the Fermi level, ρ(E_F), as a function of d, with the periodicity of λ(E_F)/2. Since the T_c of a BCS-type superconductor is proportional to exp[−1/ρ(E_F)V] [32], it can lead to an oscillation of T_c as a function of d (here, V denotes the effective interaction between electrons). The first clear observation of this phenomenon was reported using ultrathin Pb films on Si(111) surfaces in the thickness regime above 21ML (see figure 4) [19]. The observed periodicity in d was 2ML, approximately equal to λ(E_F)/2 for Pb. The interpretation as a quantum oscillation was supported by the corresponding oscillatory behavior for ρ(E_F), which was determined from the temperature dependence of the out-of-plane upper critical magnetic field H_c2⊥ near T_c. In addition, the electron–phonon coupling constant λ appearing in the Eliashberg–McMillan theory [59] was estimated from the temperature dependence of the quasiparticle lifetime through photoemission spectroscopy. This also showed a similar oscillation as a function of d. In this experiment, the transport measurements were performed ex situ with Au capping layers to protect the superconducting Pb layers from air exposure. The presence of the capping layer may shift the energy positions of the quantum well states due to the change in the boundary condition.

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Another important consequence of the QWS in atomically thin Pb films is the manifestation of 'magic thicknesses' during the growth. When the energy positions of QWS are located far from E_F for a certain thickness, the film becomes stable due to a decrease in electronic energy. This occurs with a periodicity of λ(E_F)/2 and the resulting magic thicknesses are d = 4, 5, 7, 9, ... ML for Pb films [60]. This effect helps to make the film thickness constant at one of these values, and when the nominal coverage of Pb is slightly less than it, flat voids with a lower local magic thickness are created on an otherwise uniform film. The unique morphology leads to a strong pinning of magnetic fluxes at these voids and to the realization of the so-called hard superconductor [32] albeit at atomic-scale thickness. This unexpected robustness of superconductivity was observed for Pb ultrathin films grown on a Si surface by ex situ magnetization measurements using a superconducting quantum interference device (SQUID) [17, 61]. Figures 5(a) and (c) display an STM image of a 7ML thick Pb film with 2ML deep voids and the corresponding d.c. magnetization with a 'hard' hysteresis loop. In contrast, a 7ML thick Pb film with 2ML tall mesas only exhibited a 'soft' hysteresis loop (figures 5(b) and (d)). An analysis of the real and imaginary diamagnetism susceptibility showed that it had a nearly perfect Bean-like critical state corresponding to very hard superconductivity. A critical current density as large as ∼2 × 10⁶ A cm⁻² was obtained at 2 K, which was about 10% of the depairing supercurrent density. It was also shown that the structural stability and the superconducting properties could be tuned by adding Bi into Pb to make ultrathin alloy films [62]. Just below T_c, the temperature dependence of H_c2⊥ was found to markedly deviate from the prediction of the GL theory, i.e. ${H}_{{\rm{c}}2\perp }\propto (1-T/{T}_{{\rm{c}}}).$ This indicated that, in this 2D geometry, the superconducting order was anomalously suppressed by scattering in violation of Anderson's theorem.

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Ultrathin metal films on semiconductor surfaces have been studied not only by the conventional electron transport and magnetic measurements but also by LT-STM/STS. In this case, superconductivity is detected by observing the superconducting energy gap through the measurement of bias-voltage dependent differential conductance (dI/dV). This technique allows direct and real-space investigations of spatially inhomogeneous superconducting properties at atomic scales, which are barely accessible by conventional macroscopic experiments. Since the surfaces of the samples treated here are atomically clean and well-defined, this system is highly suited for STM investigation in UHV. Particularly interesting is the observation of superconducting vortices under application of a magnetic field [63]; the vortex core, where the superconducting energy gap is suppressed due to an excessive supercurrent density, can be imaged as a region with high local density of states (i.e., high dI/dV). Figures 6(a) and (b) show an example of such experiments; 12–14ML thick Pb islands were imaged in (a) the topographic mode and (b) the dI/dV mapping mode with E = E_F [64]. In the dI/dV image, the superconducting regions appeared dark due to the presence of an energy gap while the normal-like vortex cores were bright. The vortices were located at the voids, indicating void-induced trapping as discussed above (see also [65]). Furthermore, hysteretic behaviors of vortex dynamics were observed under varying magnetic fields using a Pb island whose size is about several times the coherence length ξ_GL (~30 nm at 2 K) [66]. The experiment showed that a vortex penetrated into and escaped from the superconducting island at different magnetic fields, revealing the presence of the Bean–Livingston energy barrier at the periphery of the island [67]. In a 2D superconductor with a sufficiently large area, vortices are usually quantized into those with a vorticity of unity and form the Abrikosov lattice, but they can be merged into a giant vortex with a multiple vorticity when squeezed into a narrow region. Such anomalous behaviors were observed for vortices within an ultrathin Pb island using an LT-STM [65, 68].

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When a superconductor is in close contact with a normal metal, the superconducting correlations may propagate into the neighboring region [69] where a superconducting energy gap can be detected by a spectroscopic method. This proximity effect is a spatially inhomogeneous phenomenon and has been successfully studied in real space with an LT-STM using superconducting ultrathin Pb islands on a Si(111) surface [64, 70, 71]. Here, 2D atomic layers of Pb (amorphous or crystalline, depending on the sample preparation) on the Si(111) surface played the role of the normal metal region. In a simple configuration, the length scale with which the superconductivity penetrates into the normal metal region is solely determined by the proximity length ${\xi }_{{\rm{M}}}\equiv \sqrt{\hslash D/2\pi {k}_{{\rm{B}}}T},$ where D is the diffusion constant in the normal metal. However, an LT-STM study revealed the presence of a strong geometric effect on the proximity effect [70]. The normal metal region surrounded by superconductors exhibited an enhanced proximity effect, i.e. the energy gap was found to survive up to distances of several times ξ_M from the boundary. This is due to the quantum interference originating from the multiple Andreev reflection in the confined geometry [72, 73]. At a superconductor–normal metal (S–N) junction, an electron in the N region with an energy −Δ < E < Δ (Δ: energy gap) is reflected as a hole (and vice versa) to allow a Cooper pair injection into the S region, and this Andreev reflection can be multiplied and enhanced in the case of a narrow S–N–S junction. Since the proximity effect is caused by the Andreev reflection, the former is also enhanced by this mechanism. An essentially identical phenomenon was observed when the proximity region was terminated and confined by a surface atomic step [74]. Such an N region in the proximity of an S can host vortices when an out-of-plane magnetic field is applied, which has been successfully imaged using LT-STM [64]. Figures 6(c) and (d) show the superconducting energy gap and its spatial mapping within the proximity N region where a vortex core is visible. The vortex was called a Josephson vortex, since the proximity effect and the multiple Andreev reflection are the origin of the Josephson effect for an S–N–S junction [75]. The locations of these vortices were found to be determined by the spatial distributions of supercurrents within the Pb islands.

It has been generally accepted that metal thin films fabricated on a substrate should lose superconductivity when they approach the atomic-scale limit in thickness. In the case of granular and amorphous films, this phenomenon can be understood in terms of the disorder-driven S–I transition as discussed in section 2. However, crystalline thin films with atomically uniform thicknesses for which the disorder is minimized also tend to lose superconductivity in this limit. For example, ex situ magnetic measurements revealed that the T_c of a Pb ultrathin film decreased as a function of thickness d according to the relation:

$$\begin{eqnarray}&&{T}_{{\rm{c}}}(d)={T}_{{\rm{c}}0}(1-{d}_{{\rm{c}}}/d)\end{eqnarray} \tag{ 5 }$$

where T_c0 is the asymptotic value of T_c(d) for d → ∞ (the bulk limit) and d_c is the critical thickness for the disappearance of superconductivity [17, 61, 62]. Experimentally, d_c was determined to be 0.43–0.615 nm for Pb, which corresponds to 1.5–2ML. Within the framework of the GL theory, this behavior can be naturally explained by the introduction of a surface energy term into the GL free energy [76]. While this term originates from the decrease in density of states near the surface in [76], it could also be attributed to other effects such as surface/interface roughness or disorder, which is hardly avoidable in real experiments (for example, because of a capping layer). Whatever the origin is, superconductivity may persist until the truly monatomic thickness if this effect is negligibly small. In situ UHV-LT-STM experiments on 2D Pb islands on a Si(111) surface showed that this was actually the case [77, 78]. The T_c values, obtained from the analysis of temperature-dependent superconducting energy gap Δ(T), remained between 6.0 K and 6.7 K down to d = 4ML, while T_c = 7.2 K for bulk Pb. T_c was also found to oscillate with a periodicity of 2ML due to the formation of QWS as discussed in section 3.1. Even 2ML thick Pb film, for which only a single QWS channel exists, retained superconductivity with T_c = 3.65 K or 4.9 K depending on the type of Pb–Si interface (i.e. Pb-induced surface reconstructions, see below). It should be noted that a UHV-LT-STM study by another group on the same Pb/Si(111) system found that T_c steadily decreased with decreasing d according to equation (5), where d_c was determined to be 1.88 ML [79]. The reason for this discrepancy is not clear, but may be attributed to a subtle difference in the resulting Pb–Si interface structure originating from the experimental conditions.

The naturally grown ultrathin film of Pb on a clean Si(111) surface retains the atomic structure of a bulk crystal while an amorphous-like wetting layer exists at the interface to the silicon substrate, which is the result of the first stage of the Stranski–Krastanov growth mode [80]. This wetting layer can be transformed into a crystalline atomic layer of Pb through an appropriate annealing process. The structural change is driven by formation of covalent bonding between the metal adatoms and the silicon substrate and leads to a unique lattice periodicity and electronic states distinct from those of bulk Pb. The resulting surface structure is called a Pb-induced (generally, metal-induced) silicon surface reconstruction, and is considered an ultimately thin form of the 2D electron system consisting only of the surface and the interface [81]. It might seem that these surface reconstructions are unlikely to become superconducting due to their ultimate 2D character, but recent experiments have changed this view; UHV-LT-STM observations of the energy gap revealed that three kinds of metal-induced silicon surface reconstructions, Si(111)-SIC–Pb, Si(111)-(√7 × √3)–Pb, and Si(111)-(√7 × √3)–In, exhibited superconductivity with T_c = 1.83 K, 1.52 K, 3.18 K, respectively (see figures 7 and 8(a), (b)) [18]. (Here, SIC stands for 'striped incommensurate' and √7 × √3 for the periodicity against the ideally terminated Si(111) surface.) Compared to T_c = 7.2 K of bulk Pb, those of the two Pb-induced surface reconstructions (Si(111)-SIC–Pb, Si(111)-(√7 × √3)–Pb) are much lower, while that of the In-induced surface (Si(111)-(√7 × √3)–In) is comparable to T_c = 3.4 K of bulk In. For the Si(111)-SIC–Pb surface, the vortices were also observed using STM under an out-of-plane magnetic field. The estimated upper critical field is μ₀H_c2⊥ = 1.45 T, giving the coherence length of ξ_GL(0) = 49 nm. Angle-resolved photoemission spectroscopy (ARPES) measurements on these surface atomic layers revealed highly dispersed metallic band structures, which can be understood based on 2D free-electron-like states [18, 82]. Furthermore, the temperature dependence of the electronic band width showed that the electron–phonon coupling constant λ was ~1 for both Si(111)-SIC–Pb and Si(111)-(√7 × √3)–In, being enhanced from the bulk values. This indicates that the Pb–Si and In–Si interfaces play an important role for the occurrence of the BCS-type superconductivity in this atomically thin limit.

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The above finding was based on the tunneling spectroscopy experiments using STM, but more direct evidence of superconductivity was obtained through in situ electron transport measurements under the UHV environment (see figures 8(c), (d)) [83, 84]. The experiments revealed that the emergence of zero resistance, i.e. the existence of supercurrents through atomic layers, and of the superconducting coherence at macroscopic scales. The observed T_c was 1.1 K for Si(111)-SIC–Pb and 2.4 K or 2.8 K for Si(111)-(√7 × √3)–In. The two different values of T_c for Si(111)-(√7 × √3)–In were attributed to the presence of two types of phases that have different nominal coverages of In for this surface reconstruction [85]. It should be noted that the resistance changes were remarkably sharp just at T_c although the systems were in the atomic-scale limit. The decrease in resistance above T_c was accelerated as the temperature approached T_c, which was well described by the 2D fluctuation theories (see section 2) [84, 86]. Analysis of the magnetic field dependence of T_c gave ξ_GL(0) = 74 nm for Si(111)-SIC–Pb and ξ_GL(0) = 19–29 nm for Si(111)-(√7 × √3)–In [84]. Unexpectedly, the robustness of the superconductivity at this 2D limit was evidenced in terms of a high 2D critical supercurrent density J_2D. For Si(111)-(√7 × √3)–In, I–V characteristics measurements gave J_2D = 1.8 × 10⁻² A cm⁻¹ at 1.8 K (correspondingly, a 3D supercurrent density of 3–6 × 10⁵ A cm⁻²) [83]. The temperature dependence of J_2D was described by the Ambegaokar–Baratoff relation [87], suggesting that J_2D was determined by Josephson junctions that must exist within the current path on the surface. The most likely locations for the junctions are atomic steps since they terminate and separate the surface terraces. This assignment was corroborated by STM observation of Josephson vortices, which in this case were trapped at the atomic steps on the Si(111)-(√7 × √3)–In surface [88]. The Josephson vortices here are anomalously elongated along the steps and superconductivity is significantly recovered within their core regions. These features are distinct from those of Josephson vortices found in the normal region in the proximity of superconducting Pb islands [64]. The presence of Josephson junctions was also indicated for Si(111)-(√7 × √3)–Pb from the observation of suppressed superconductivity near atomic steps [89].

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There are two unique features that should be considered for atomically thin 2D superconductors in general. First, when a magnetic field is applied in the in-plane direction there exists no orbital pair-breaking effect that usually dominates the upper critical magnetic field H_c2. This is because electrons cannot move in the out-of-plane direction (see figure 9(a)). In the simplest case, the in-plane critical magnetic field H_c2// is determined by the Pauli pair-breaking effect, leading to ${\mu }_{0}{H}_{{\rm{c}}2//}={\rm{\Delta }}(0)/\sqrt{2}{\mu }_{{\rm{B}}}(\equiv {\mu }_{0}{H}_{{\rm{P}}})$ at T = 0 where Δ(0) is the BCS energy gap and H_P is called the Pauli paramagnetic limit [32]. The low dimensionality of the system and the dominance of the Pauli effect may also lead to a realization of the Fulde–Ferrell-Larkin–Ovchinnikov (FFLO) state at a high magnetic field around H_P [90, 91]. This unusual superconducting state is the result of competition between an energy gain due to a partial Cooper pair formation and a paramagnetic energy gain due to the remaining normal electrons, both occurring on the Zeeman-split Fermi surfaces. The Cooper pairs have (multiple) non-zero momentum in striking contrast to the BCS state. Consequently, the order parameter ${\rm{\Psi }}(\vec{r})$ becomes spatially modulated with wave vectors of ${\vec{q}}_{m}$ even at equilibrium, and is generally expressed by [92]

$$\begin{eqnarray}&&{\rm{\Psi }}(\vec{r})=\displaystyle \sum _{m=1}^{M}{{\rm{\Psi }}}_{m}\exp ({\rm{i}}{\vec{q}}_{m}\cdot \vec{r}).\end{eqnarray} \tag{ 6 }$$

The second important feature is the manifestation of the Rashba effect due to the space-inversion symmetry breaking of the system and the spin–orbit interaction (SOI) [93]. The Hamiltonian of the Rashba effect is expressed as

$$\begin{eqnarray}&&{ {\mathcal H} }_{{\rm{R}}}(\vec{k})={\alpha }_{{\rm{R}}}(\vec{{\epsilon }}\times \vec{k})\cdot \vec{\sigma },\end{eqnarray} \tag{ 7 }$$

where $\vec{k}=({k}_{x},{k}_{y},0)$ is the kinetic momentum, ${\alpha }_{{\rm{R}}}$ is the strength of the Rashba SOI, $\vec{{\epsilon }}$ is the unit vector perpendicular to the 2D xy plane, and $\vec{\sigma }$ is the Pauli matrix. The Rashba effect leads to spin splitting and spin-momentum locking where electron spins are chirally polarized in the directions perpendicular to both the electron momentum and the electric field (see figure 9(b)). Due to the space-inversion symmetry breaking at surface, this effect has been observed by ARPES for metal-induced surface reconstructions with a strong SOI [94, 95]. In terms of superconductivity,  the Rashba effect may lead to exotic phenomena under high in-plane magnetic fields. As schematically depicted in figure 9(c), the two Rashba-split Fermi surfaces move in the opposite directions by ±q/2, perpendicular to the magnetic field H_// (note that this is a simplified picture and the Fermi surfaces are actually deformed under a high field). This is due to a magnetoelectric effect based on the reconfiguration of momentum-locked spins under a magnetic field [96]. Then the electrons on each Fermi surface can form Cooper pairs with non-zero momentum ±q, leading to ${\rm{\Psi }}(y)\propto \exp ({\rm{i}}qy)+\exp (-{\rm{i}}qy)\propto \,\cos (qy)$ similarly to equation (6) [97] (see figure 9(d); here the x direction is set parallel to H_//). The resulting superconducting state is often called a FFLO-like state, while the original FFLO state is caused purely by the Zeeman effect and does not require the Rashba effect. Another possible consequence of the magnetoelectric effect is induction of unidirectional supercurrents perpendicular to the magnetic field (note that magnetic field usually induces circular supercurrents as a vortex; see also figure 9(d)). Since such stationary supercurrents cannot run in an isolated object, Ψ becomes spatially modulated as ${\rm{\Psi }}(y)\propto \exp ({\rm{i}}q\mbox{'}y)$ to compensate the internal supercurrent (note that q' is different from q in the above equation). This state is called the helical state [98]. The FFLO-like state and the helical state compete with each other [99], but the helical state is more robust against the impurity scattering. These exotic superconducting states have unusually high values of H_c2//, and have been studied within the context of heavy-fermion systems (see section 3.8) [92, 100]. Since metal atomic layers on semiconductor surfaces (metal-induced surface reconstructions) treated here are much simpler systems, they may provide a new platform where complexities such as strong electron-correlation effects can be avoided.

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Experimentally, there have already been reports on surface 2D superconductivity related to these topics. Notably, the Si(111)-(√3 × √3)–(Tl, Pb) surface reconstruction consisting of 1ML of Tl and 1/3ML of Pb on Si(111) was found to exhibit the Rashba effect based on ARPES measurements (see figure 10(a)) [52]. The same surface also showed a superconducting transition at 2.25 K, opening a route for investigating the exotic superconducting state as stated above. In [89], the STS observation of the superconducting Si(111)-(√7 × √3)–Pb surface and the analysis of its energy gap structure indicated the presence of a large pair-breaking parameter. This was attributed to the scattering of the triplet part of the spin-triplet mixed Cooper pairs, which can be caused by the space-inversion symmetry breaking and the resultant Rashba effect [96]. Furthermore, in situ transport experiments on Pb monatomic layers grown on a vacuum-cleaved GaAs surface showed a surprising robustness against an in-plane magnetic field (see figure 10(b) [101]. The T_c was found to decease only by ~2% under an in-plane field of 15 T. This phenomenon was interpreted as a manifestation of the helical state described above, based on the analysis of elastic and spin–orbit scattering rates deduced from the transport data.

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Finally, 2ML thick Ga atomic layer on GaN(0001) was found to exhibit superconductivity. The measured sheet resistance showed the onset of a transition at ${T}_{{\rm{c}}}^{{\rm{onset}}}=5.4\,{\rm{K}}$ and appearance of the zero resistance at ${T}_{{\rm{c}}}^{{\rm{zero}}}=3.8\,{\rm{K}},$ which were much higher than T_c = 1.08 K for the bulk stable phase of α-Ga [102]. This T_c enhancement was attributed to a possible strong interaction at the interface between the Ga layer and the GaN substrate. It is reminiscent of the high-T_c superconductivity of 1 UC thick FeSe layers on a SrTiO₃ substrate, which will be described in section 3.7.

It is widely recognized that high-T_c cuprate superconductors such as La_2−xSr_xCuO₄, YBa₂Cu₃O₇, and Bi₂Sr₂CaCu₂O_8+x have layered perovskite structures where hole-doped conducting CuO₂ planes are separated from each other by insulating layers [103]. The bulk crystal retains strong 2D characters because the conduction channels of the CuO₂ planes are only weakly coupled (often regarded as Josephson coupled), and the coherence length in the out-of-plane direction (along the c-axis) ξ_GL⊥(0) is comparable to the UC length. This naturally leads to the following questions: what is the minimum number of CuO₂ planes (or unit cells along the c-axis) for the occurrence of superconductivity, and how does the coupling between the CuO₂ planes helps establish high T_c in these systems? Indeed, soon after the finding of cuprate superconductors, the investigation into this problem started [104–108]. Generally, superconductivity was found to survive even when the number of UCs in a cuprate thin film was reduced to one. Nevertheless, the temperatures where the zero resistance sets in $({T}_{{\rm{c}}}^{{\rm{zero}}})$ were substantially suppressed compared to both the bulk T_c values and the temperatures where the resistance starts to decrease $({T}_{{\rm{c}}}^{{\rm{onset}}}).$ For example, a 1 UC thick YBa₂Cu₃O₇ layer sandwiched between 6 UC thick PrBa₂Cu₃O₇ layers showed ${T}_{{\rm{c}}}^{{\rm{zero}}}\sim 30\,{\rm{K}}$ and ${T}_{{\rm{c}}}^{{\rm{onset}}}\sim 80\,{\rm{K}}$ while T_c ≅ 90 K was obtained for a 100 nm thick YBa₂Cu₃O₇ film [107]. Consequently, the superconducting transition was found to be substantially broadened. It was difficult to determine in the early stages of research whether this is the intrinsic effect due to the ultimate 2D character or it is caused by extrinsic factors such as structural and/or stoichiometric disorder at the interface.

Recent remarkable advancements of MBE/PLD growth and surface characterization techniques have now allowed researchers to investigate cuprate superconductors with ultimately thin layers of CuO₂ with unprecedented precision [109, 110]. For example, La_2−xSr_xCuO₄/La₂CuO₄ heterostructures with atomically sharp interfaces were studied through electron transport and magnetic susceptibility measurements [109]. Here neither La_2−xSr_xCuO₄ nor La₂CuO₄ layers are superconducting when isolated, since La_2−xSr_xCuO₄ (x = 0.45) is in the overdoped region of metal and La₂CuO₄ is a Mott insulator. The resulting heterostructure, however, showed superconductivity with T_c ≅ 30 K, which should be attributed to the presence of the La_2−xSr_xCuO₄/La₂CuO₄ interface. When the top La₂CuO₄ insulating layer was transformed into a La₂CuO_4+δ superconducting layer through annealing in ozone atmosphere, T_c reached ≅ 50 K. This value was 25% higher than T_c of a La₂CuO_4+δ single phase. Furthermore, the transitions were sufficiently sharp so that ${T}_{{\rm{c}}}^{{\rm{zero}}}$ was nearly equal to ${T}_{{\rm{c}}}^{{\rm{onset}}}.$ The observed sharp transition strongly suggested the completeness of the interface at the atomic scale. The temperature dependence of critical current density J_c obtained from analysis of diamagnetic response showed that the enhanced superconducting region was spatially limited within 1–2 UC thickness around the interface. A subsequent resonant soft x-ray scanning measurement on the La_2−xSr_xCuO₄/La₂CuO₄ heterostructure (x = 0.36, T_c = 38 K) also revealed that the conducting hole channel was located at the interface over a characteristic distance of ~0.6 nm [111]. This is half the UC length of La₂CuO₄ (=1.3 nm) where only a single CuO₂ plane is included. The existence of the superconductivity within a nominally single CuO₂ plane was further demonstrated using δ-doped heterostructures (see figure 11) [110]. Here one of the CuO₂ planes of La_2−xSr_xCuO₄/La₂CuO₄ (x = 0.45) was selectively doped with Zn to suppress superconductivity, identifying the location where superconductivity occurred. Transport and diamagnetic induction measurements revealed that only the second CuO₂ plane from the interface located on the La₂CuO₄ side was responsible for the phenomenon. The conducting holes were found to arise from charge transfer over the interface due to a chemical potential difference, but its exact location was also influenced by the Sr spatial profile.

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The same group also demonstrated that superconductivity of 1 UC thick La_2−xSr_xCuO₄ (x = 0.06 to 0.2) can be electrically modulated [20]. The samples were fabricated through epitaxial growth on insulating La₂CuO₄/LaSrAlO₄ substrates, and the EDL gating technique based on polymer electrolyte or ionic liquid was applied (see section 3.5) to successfully tune the T_c by up to 30 K. This has become possible due to the fact that the thickness of the conduction layer was only ~1 nm and that a huge local electric field of >10⁹ V m⁻¹ was obtained in this configuration. Using underdoped samples in the same setup, they also investigated the S–I transition by modulating the carrier density to reveal its nature as a quantum phase transition (see figure 12). Accumulated data on the sample sheet resistance R_sheet as a function of temperature T and the number of mobile holes per unit cell, x, were found to follow perfectly the scaling equation function given in equation (2) (see section 2). The analysis gave R_c = 6.45 ± 0.10 kΩ and zν = 1.5 ± 0.1. Remarkably, the critical resistance R_c obtained here is equal to the quantum resistance for Cooper pairs, R_Q = 6.45 kΩ, within an experimental error. This strongly suggests that Cooper pairs exist in the form of localized bosons in the insulating region near the boundary and that the transition is driven by quantum phase fluctuations [42]. The value of zν = 1.5 is clearly different from those of previously investigated systems, such as amorphous MoGe films (zν ≅ 1.3) [35], a LaAlO₃/SrTiO₃ interface (zν ≅ 2/3) (see section 3.4) [112], and amorphous Bi films (zν ≅ 0.7) [46]. This means that they belong to different universality classes and that the observed quantum phase transitions are governed by different physics (for example, zν = 4/3 and 2/3 correspond to the classical percolation and the 3D XY models, respectively). Also, related experiments on hole-doped YBaCuO_7-x in a similar configuration revealed R_c = 6.0 kΩ and zν = 2.2 [113]. Here, the result R_c ≅ R_Q = 6.45 kΩ was also indicative of a phase transition by quantum phase fluctuation of Cooper pairs, but the failure of the scaling analysis at the lowest temperatures suggested a possible presence of an intermediate phase near the transition. For electron-doped Pr_2-xCe_xCuO₄ (x = 0.04, 0.1), R_c = 2.88 kΩ and zν = 2.4 were found [114]. The large deviation of R_c from R_Q indicated that fermionic quasiparticle excitations, arising from the suppression of Cooper pair formation, played an import role in the transition. These results give an important clue for understanding the physics of high-T_c cuprates. It should also be noted that the superfluid density n_s(0) of 2 UC thick Y_1−xCa_xBa₂Cu₃O_7-δ was determined from the a.c. conductivity measured with the two-coil mutual inductance method [115]. The T_c was found to be proportional to n_s(0) near the S–I transition, which was attributed to the quantum fluctuation near a 2D critical point.

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Preparation of cuprate thin layers with an atomic-scale precision based on MBE or PLD methods as shown above requires highly advanced instrumentation. However, recent progress in mechanical exfoliation of atomic sheets, which was first demonstrated for graphene [25, 116], has now allowed preparation of atomic sheets of Bi₂Sr₂CaCu₂O_8+x in a simple and cost-effective way [117]. The prepared atomic sheets with thicknesses down to half UC (including two CuO₂ planes) were protected by graphene sheets from degradation and transport measurements were performed. While the sheet resistance R_sheet of the sample increased from a few Ω to 5 kΩ as the sheet thickness was reduced from 270 to 0.5 UC, sharp superconducting transitions were consistently observed, with nearly constant T_c of ~82 K. This indicates that the interlayer coupling does not play an important role for the superconductivity of Bi₂Sr₂CaCu₂O_8+x. Superconductivity of exfoliated atomic sheets of other layered materials will be described in section 3.6.

As seen in section 3.3, recent technological advancements have allowed researchers to grow complex oxides in a layer-by-layer fashion and to fabricate oxide heterostructures with atomically sharp and well-defined interfaces. Since many oxide materials have unique properties originating from strong electron correlations and orbital/spin degrees of freedom, the interfaces of oxide heterostructures have a huge potential for realizing exotic and functional artificial 2D materials [118]. One of the most famous examples is the interface between two perovskite transition-metal oxides, LaAlO₃ and SrTiO₃, where LaAlO₃ layers are epitaxially grown on a TiO₂-teminated (100) surface of SrTiO₃ due to a small lattice mismatch [119, 120]. Although LaAlO₃ and SrTiO₃ are both wide-bandgap insulators, their interface is known to possess a 2D metallic conduction channel with a high carrier mobility. This has been widely ascribed to a mechanism called 'polar catastrophe', which arises from the discontinuity of ionic characters of oxide layers [121]. While the layers consisting of SrTiO₃ are formally charge-neutral [(SrO)⁰–(TiO₂)⁰], those of LaAlO₃ are ionic [(LaO)⁺–(AlO₂)⁻] (see figure 13(a)). Within the LaO–AlO₂ stacking layers, an electrostatic voltage is accumulated and becomes linearly divergent as the number of layers increases, which is apparently unrealistic. This difficulty can be solved by the polar catastrophe where half of the charge of the top-surface LaO layer is transferred to the TiO₂ layer of the interface. Consequently, in the simplest case, the TiO₂ sheets of the LaAlO₃–SrTiO₃ interface are doped by 1/2 electron per 2D unit cell to have a carrier density of n_2D = 3.4 × 10¹⁴ cm⁻². The actual interface is much more complex, due to effects such as structural and electronic reconstructions, extrinsic electron doping from oxygen defects, and stoichiometric inhomogeneity. Nevertheless, it generally has a high carrier density (~10¹³ cm⁻²) and a low sheet resistance (~10³ Ω) even when extrinsic electron doping from oxygen defects is minimized by choosing the appropriate growth condition. [119]. The interface also has a high carrier mobility (~10⁴ cmV⁻¹s⁻¹), indicating the electron scattering due to impurities and defects is weak. Generally speaking, LaAlO₃ growth at a lower oxygen pressure leads to a higher density of oxygen-deficiency defects in the SrTiO₃ substrate and the electron conduction becomes more bulk-like due to the doping [122].

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Remarkably, electron transport measurements have revealed that the LaAlO₃/SrTiO₃ interface becomes superconducting at low temperatures (see figures 13(b)–(e)) [21]. For a representative result using a sample with a 8 UC thick LaAlO₃ layer, important physical quantities related to superconductivity were given as follows: transition temperature T_c ≅ 200 mK, 2D critical supercurrent density J_c,2D = 98 μA cm⁻¹, out-of-plane critical magnetic field μ₀H_c2⊥ = 65 mT. The observed T_c was within the range of those previously reported for oxygen-defect-doped bulk SrTiO₃ [123]. The GL coherence length at zero temperature was estimated to be ξ_GL(0) ~70 nm from the temperature dependence of H_c2⊥. Other parameters regarding the electron transport were as follows: sheet resistance R_sheet ≅ 300 Ω, Hall carrier density ≅4 × 10¹³ cm⁻², carrier mobility ≅350 cm V⁻¹s⁻¹ (all at T = 4.2 K). The significant contribution of the interface conduction channel to the transport phenomena and the 2D character of superconductivity were discussed from the viewpoint of the KTB transition. T_KTB = 188 mK was obtained from the analysis of the temperature dependence of the I–V characteristics, where the exponent a of the relation $V\propto {I}^{a}$ was found to cross the universal constant of three. In addition, the zero-bias resistance was found to follow equation (3), giving T_KTB = 190 mK, consistent with the above value. It should be noted that the KTB transition considered here is not of the conventional type where vortex–antivortex bound pairs are formed from thermally excited free vortices in a 2D superconductor; rather it is analogous to the dislocation-induced melting of a 2D crystal treated in the original paper by Kosterlitz and Thouless [9]. Just below T_c, a high density of vortices and antivortices can form an ionic-like crystal if the energy required for thermal excitation of a vortex is sufficiently small. In this case, the KTB transition is characterized by the freezing of vortex-lattice defect motion, which leads to the occurrence of true zero resistance in the limit of an infinitely small bias current [124, 125]. In [21], the residual ohmic regime at small currents observed even below T_KTB was attributed to the finite sample size effect [50].

The KTB transition observed here indicates that superconductivity in this system is of 2D character, thus limiting the maximum thickness of the conducting layer sufficiently below the coherence length of ξ_GL(0) ~70 nm. A subsequent experiment on the same system revealed the strong anisotropy of  a critical magnetic field (H_c2///H_c2⊥ ≅ 20) and showed that the LaAlO₃/SrTiO₃ superconductivity had indeed a strong 2D character [126]. The thickness of the conduction layer was estimated to be 11 ± 3 nm from the analysis of temperature dependence of the critical magnetic fields. More direct information on the conducting layer thickness was obtained with a microscopic imaging technique based on atomic force microscopy (AFM) [122]. The local resistance was mapped over the cross-section cut of a LaAlO₃/SrTiO₃ heterostructure using an AFM tip to investigate the extent of the conducting region near the interface. For samples prepared by annealing around 750 °C at an oxygen pressure of 300 mbar after the LaAlO₃ growth, the spatial extension of the conducting layer was found to be smaller than 7 nm. By contrast, it could spread up to as large as ~500 μm when a different annealing process was taken due to oxygen-deficiency doping. Furthermore, hard x-ray photoelectron spectroscopy taken on similarly prepared samples revealed that doped electrons are located at Ti sites of the TiO₂ layers within a distance of 4 nm from the interface [127].

Thanks to an extremely high dielectric constant of SrTiO₃ (ε_r ~ 300 at room temperature and ∼20 000 at LT) and a carrier density much lower than those of usual metals, the 2D conduction can be electrically tuned using a back gate on the substrate [120]. Superconductivity in this system was electrically enhanced based on this method where the maximum KTB transition temperature T_KTB reached 310 mK [112]. Clear S–I transitions were also demonstrated through electrostatic gate control. The critical sheet resistance for the transition was R_c ≅ 4.5 kΩ, which was close to, but lower than, the quantum resistance of Cooper pairs, R_Q = 6.45 kΩ [14]. The transition temperature identified with T_KTB was found to scale with the carrier density n_2D according to a relation ${T}_{{\rm{KTB}}}\propto \,{| {n}_{2{\rm{D}}}-{n}_{2{\rm{D}},{\rm{c}}}| }^{z\nu },$ where n_2D,c is the critical carrier density for the S–I transition and $z\nu$ is a parameter related to the critical behavior near the transition (see equation (2) of section 2). The successful application of the scaling theory and the value of $z\nu =2/3$ indicate that the transition is driven by the quantum fluctuations of Cooper pairs in a clean 2D system. Furthermore, a large negative magnetoresistance observed for the insulating phase in the same experiment was attributed to the weak localization of electrons.

Intriguingly, the LaAlO₃/SrTiO₃ interface exhibits not only superconductivity but also ferromagnetism, as in a theoretical prediction of ferromagnetic alignment of local spins at Ti³⁺ sites of the interface TiO layer [128]. Experimentally, the existence of a ferromagnetic phase was first reported using samples prepared in high oxygen pressures (i.e. with small oxygen-deficiency doping) nearly simultaneously as superconductivity was discovered for the same system [129]. A clear magnetic-field hysteresis of sample sheet resistance R_sheet was found as an evidence for the formation of ferromagnetic domains, and a logarithmic temperature dependence of R_sheet and a large negative magnetoresistance observed at low temperatures were attributed to the presence of local magnetic moments. Subsequent studies have revealed that superconductivity and ferromagnetism can coexist within the same sample. Magnetic-field dependence of superconducting transition temperature T_c, which was electrically tuned by a back gate, was found to have hysteresis within the range −30 mT < μ₀H < −30 mT [130]. This clearly indicated that two conduction channels, corresponding to superconducting and ferromagnetic regions, existed within the interface. More direct evidence for this unexpected phenomenon was soon reported by two independent groups. A combined experiment based on high-resolution magnetic torque magnetometry and electron transport measurements revealed the presence of magnetic ordering in a superconducting LaAlO₃/SrTiO₃ interface [131]. Also reported was the real-space 2D mappings of magnetization and diamagnetic susceptibility using a SQUID, demonstrating the existence of nanoscale phase separation of ferromagnetic and superconducting regions (see figure 14) [132]. Ferromagnetic domains were observed as separate dipoles and persisted over the whole temperature range of the experiment, while inhomogeneous superconducting domains appeared below 100 mK. A control experiment using Nb δ-doped samples showed that magnetism in this system could only arise from the interface. It should be noted that the space-inversion symmetry is broken at the interface, which is important for the emergence of the Rashba effect (see section 3.2). The SOI needed for the Rashba effect was also detected and was successfully controlled by a gate voltage [133]. Together with the electric-field tunability and the coexistence with ferromagnetism, the LaAlO₃/SrTiO₃ system offers a platform for exploring exotic superconducting phenomena.

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In the preceding two sections, we have seen that the interface between two non-superconducting (metallic or insulating) materials may spontaneously host a conduction channel with 2D superconductivity. Realization of such an interface superconductivity in an artificial and controlled way, preferably in a field-effect transistor (FET) configuration, should provide great opportunities for basic physics, materials science, and device applications [134]. In the case of the SrTiO₃ substrate, the back-gate control was possible due to its high dielectric constant [112], but the real power of this approach was demonstrated based on an electrochemical method, i.e., using an EDL as a gate electrode (also see section 3.3) [23, 24]. The conventional FET device uses a gate insulator made of a solid-state dielectric material such as SiO₂ and high-k complex oxides, but the accumulation of carriers at the interface is limited up to the level of n_2D ∼ 1 × 10¹³ cm⁻² because of current leakage and dielectric breakdown of the insulator. In an EDL transistor, however, the solid-state gate insulator is replaced by a liquid electrolyte, which allows free cations or anions to be assembled at the liquid–substrate interface under the application of a gate voltage [22, 135–137]. The resulting EDL works as a subnanometer-gap capacitor and can induce carriers with an areal density up to ∼1 × 10¹⁵ cm⁻² at the interface (more precisely, at the subsurface) of the target material.

In the early stages of the EDL device, a polymer electrolyte consisting of KClO₄ and polyethylene oxide was used to fabricate an n-type FET with a 2D channel at the SrTiO₃ interface [23]. This device successfully induced superconductivity at T_c ∼ 0.4 K under the application of the gate voltage above 2.5 V. This is the first report on electric-field-induced superconductivity in an insulator without chemical doping. A substantially high density of electron carriers with n_2D = 1 × 10¹³ ∼ 1 × 10¹⁴ cm⁻² was confirmed through Hall resistance measurements. This corresponds to a 3D carrier density as high as 3 × 10¹⁸ cm⁻³–1 × 10²⁰ cm⁻³, which is roughly equal to that of superconducting Nb-doped SrTiO₃ bulk samples with T_c = 0.4 K. The effective thickness of the conduction channel was estimated to be 5–15 nm based on the calculated spatial distribution of carriers. This relatively large value is due to the large dielectric constant of SrTiO₃, which is on the verge of a transition to ferroelectricity. The analysis showed that several 2D quantized subbands were involved within the conduction channel; in this sense, the system is not considered a superconductor in the 2D limit.

Substitution of the polymer electrolyte with ionic liquid, which is an organic salt in the liquid state even at room temperature, enables an even stronger gating function and accumulation of a higher density of carriers at the interface [138]. This has allowed investigation of field-induced superconductivity using various kinds of insulators including ZrNCl, KTaO₃, and 1T-TiSe₂. Some of the insulators also exhibited many-body quantum states featuring, e.g., magnetic ordering and charge density waves (CDW) [24, 139–141]. The experiment on thin layers of ZrNCl, obtained by mechanical micro-cleavage, realized for the first time superconducting transitions with T_c = 12 ∼ 15.2 K under the gate voltages of 4–5 V using a EDL-FET device (see figure 15) [24]. The measured net carrier density amounted to 1.7 × 10¹⁴ cm⁻². This seminal work was followed by a more elaborate experiment that revealed many intriguing phenomena (see figure 16) [142]. Generally, quasi-2D bulk superconductors made of weakly coupled conducting layers have an angular dependence on the upper critical magnetic field H_c2(θ), where θ is the angle of magnetic field relative to the in-plane direction (note that, in figure 15(a), θ is measured relative to the out-of-plane direction). Within the 3D anisotropic GL model, the anisotropy can be expressed by the following equation [32]:

$$\begin{eqnarray}&&{\left[\displaystyle \frac{{H}_{{\rm{c}}2}(\theta )\cos \theta }{{H}_{{\rm{c}}2}(0^\circ )}\right]}^{2}+{\left[\displaystyle \frac{{H}_{{\rm{c}}2}(\theta )\sin \theta }{{H}_{{\rm{c}}2}(90^\circ )}\right]}^{2}=1.\end{eqnarray} \tag{ 8 }$$

This form applies when the coherence length in the out-of-plane direction, ξ_GL⊥, is larger than the interlayer distance. By contrast, for an isolated 2D superconductor with a thickness much smaller than the magnetic penetration depth, H_c2(θ) obeys the relation:

$$\begin{eqnarray}&&{\left[\displaystyle \frac{{H}_{{\rm{c}}2}(\theta )\cos \theta }{{H}_{{\rm{c}}2}(0^\circ )}\right]}^{2}+\left[\displaystyle \frac{{H}_{{\rm{c}}2}(\theta )\sin \,\theta }{{H}_{{\rm{c}}2}(90^\circ )}\right]=1,\end{eqnarray} \tag{ 9 }$$

which is called the 2D Tinkham model [143]. The clear difference in the angle dependence between the two model can be found around θ = 0°; for the former, the change in H_c2(θ) is smooth (i.e. ${| \partial {H}_{{\rm{c}}2}/\partial \theta | }_{\theta ={0}^{^\circ }}=0)$, while for the latter it exhibits a cusp-like shape (i.e. ${| \partial {H}_{{\rm{c}}2}/\partial \theta | }_{\theta ={0}^{^\circ }}\gt 0).$ In [142], H_c2(θ) for the field-induced superconductivity of ZrNCl exhibited a huge anisotropy between the in-plane (θ = 0°) and out-of-plane (θ = 90°) directions. Furthermore, a cusp-like angle dependence of H_c2(θ) was found around θ = 0°, which was well described by equation (9) of the 2D Tinkham model. From the analysis of the angle dependence, the thickness of the superconducting region was determined to be 1.8 nm. This value corresponds to the (ZrNCl)₂ bilayer thickness and is smaller than the 1 UC thickness of ZrNCl (=2.76 nm). Thus it is considered a superconductor in the 2D limit. Correspondingly, the 2D signatures of the KTB transition and the superconducting fluctuation were found from the analysis of temperature dependence of the zero-bias resistance (see section 2). This atomically thin 2D superconductor is relatively clean; the electron mean free path was estimated to be 18–35 nm from the normal sheet resistance, and this value is comparable to Pippard's coherence length ξ_Pippard = 43.4 nm (${\xi }_{{\rm{Pippard}}}\equiv \hslash {v}_{{\rm{F}}}/{\rm{\Delta }}(0),$ ${v}_{{\rm{F}}}:$ Fermi velocity, ${\rm{\Delta }}(0):$ BCS energy gap at T = 0). This ideal 2D character and high crystallinity led to an almost free vortex motion under magnetic fields of μ₀H_c2⊥ > 1.3 T. For 0.05 T < μ₀H_c2⊥ < 1.3 T, a quantum creep motion of the vortex lattice was indicated, and the detected finite residual resistances at the lowest temperatures demonstrated the presence of metallic states in this regime [144]. True superconducting states with undetectable residual resistance were obtained only for μ₀H_c2⊥ < 0.05 T. The result is contradictory to the theoretical prediction of the dirty-boson model where there should be no metallic state between the superconducting and the insulating states [41]. Accordingly, the magnetic-field dependence of the R_sheet–T relations did not follow the scaling equation of equation (2), thus clarifying the limitation of the conventional theories of the magnetic-field-induced S–I transition when applied to a clean system.

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2H-type MoS₂, a band insulator and a representative transition metal dichalcogenide, was also turned into a superconductor using an EDL transistor device with ionic liquid [145, 146]. The induced carriers with an areal density of ∼10¹⁴ cm⁻² were estimated to concentrate within a thickness of ∼0.6 nm, which corresponds to a monolayer of MoS₂ (half the unit cell of a MoS₂ layered crystal). In addition, a high Hall mobility of ∼240 cm² V⁻¹ s⁻¹ was obtained at 20 K. Most intriguingly, the superconducting state had a dome-like phase diagram [146] that was reminiscent of those of cuprate high-T_c superconductors [103]. Namely, T_c plotted as a function of 2D carrier density n_2D was found to have a maximum value of 10.8 K at n_2D = 1.2 × 10¹⁴ cm⁻². Further increase in n_2D resulted in a lower T_c. The T_c = 10.8 K is ∼40% higher than the maximum T_c of alkali-doped bulk MoS₂ crystal.

The fact that electron carriers are confined within a MoS₂ monolayer has a remarkable consequence, as explained in the following (see figure 17) [54, 147]. MoS₂ as a bulk crystal has global inversion symmetry due to its D_6h symmetry, and its superconducting state is considered conventional. However, the monolayer of MoS₂ lacks in-plane inversion symmetry because it has half the unit cell of a bulk crystal and the symmetry is lowered to D_3h [148]. Together with a large SOI originating from the Mo d-orbitals, this causes an effective Zeeman field of the order of ∼100 T and spin polarization in the out-of-plane direction. In the momentum space, carriers induced in the MoSe₂ monolayer are located at electron pockets around the K and K' points in the Brillouin zone. The sign of the Zeeman splitting is inverted between the K and K' points due to the time-reversal relation, which is called spin-valley locking [149]. The Hamiltonian for this effect is expressed as

$$\begin{eqnarray}&&{ {\mathcal H} }_{{\rm{sv}}}(\vec{k}+{\epsilon }\vec{K})={\epsilon }{\beta }_{{\rm{SO}}}{\sigma }_{{\rm{z}}}\end{eqnarray} \tag{ 10 }$$

where $\vec{k}=({k}_{x},{k}_{y},0)$ is the kinetic momentum in the K and K' valleys, $\vec{K}\,$ is the momentum of the K valley, ${\epsilon }=\pm 1$ is the valley index, ${\beta }_{{\rm{SO}}}$ is the strength of the intrinsic SOI, and ${\sigma }_{{\rm{z}}}$ is the z-component of the Pauli matrices. When Cooper pairs are formed by two electrons with opposite spins (up and down) and momenta (at K and K'), the only allowed spin directions are along the out-of-plane direction. The resultant state, called Ising superconductivity, is robust against the in-plane magnetic field because the spin polarization direction is perpendicular to the field. Although the spin-valley locking is analogous to the in-plane spin-momentum locking that originates from the Rashba effect (see equation (7) in section 3.2) [52, 101], they are different in terms of the locking direction. Experimentally, huge in-plane critical magnetic fields H_c2// were observed for MoS₂ EDL transistors and were attributed to the spin-valley locking phenomena. The observed μ₀H_c2// = 52 T at 1.5 K (for T_c = 9.7 K) [147] and μ₀H_c2// > 20 T at 1.46 K (for T_c = 2.37 K) [27] are about four to five times larger than the Pauli paramagnetic limit H_P. These H_c2// values are also an order of magnitude larger than those of alkali-doped bulk MoS₂ crystal, for which H_c2// is dominated by the orbital pair-breaking effect caused by the interlayer coupling. It should be noted that a huge H_c2// value surpassing the Pauli limit was also indicated for the 2D superconductor with the Rashba effect, as discussed in section 3.2, but its origin is different. In the system with in-plane inversion symmetry breaking like the MoS₂ monolayer, the presence of the Rashba effect tends to weaken the spin-momentum locking because their spin polarization directions are orthogonal.

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In 2004, graphene was mechanically exfoliated from a piece of graphite to fabricate an high-performance ambipolar FET device for the first time [25]. This invention was followed by the discovery of the half-integer quantum hall effect at low temperatures that signified the presence of massless Dirac fermions [150, 151], and quickly by an explosive number of graphene-related work. Finding superconductivity in graphene has been a long-standing goal in this field and should also greatly contribute to the study on 2D superconductivity. Up to now, however, superconductivity has not been reported in pristine graphene except for graphene-based junctions where Josephson supercurrents were detected to run [152]. This is presumably because of a small density of states at the Fermi level near the Dirac point. Nevertheless, it is well known that graphite intercalation compounds with foreign alkaline or alkaline-earth metal layers exhibit superconductivity, examples of which include KC₈ with T_c = 0.14 K [153], CaC₆ with T_c = 11.5 K, and YbC₆ with T_c = 6.5 K [154, 155]. The metal layers play important roles in terms of the BCS-type superconductivity because they donate electrons into the π*-bands of graphite and modify the electron–phonon interaction. They also form electronic bands by themselves, which affect the total properties of graphite. This raises a hope that intercalated or chemically doped graphene may become superconducting.

Recently, metal-doped few-layer graphene was reported to exhibit superconductivity in a variety of experiments. For example, a wet chemistry method allowed preparation of K-doped few-layer graphene from graphite flakes in dimethoxyethane solution, and superconductivity with T_c = 4.5 K was found based on magnetic susceptibility measurements [156]. Similarly prepared Li-intercalated few-layer graphene was found to have T_c = 7.4 K from magnetization measurements, although transport measurements did not show the evidence of resistance decrease around that temperature [157]. In terms of controlling the quality of graphene and the number of its layers, it is preferable to grow graphene in a layer-by-layer fashion on a semiconducting substrate such as SiC under the UHV environment. Thus prepared multi-layer graphene was Ca-intercalated ex situ by following the standard intercalation method used for bulk compounds [158]. Both magnetic and transport measurements revealed superconducting transitions down to a thickness of 10ML, with the maximum T_c of 7 K. However, since the intercalated dopants of alkaline or alkaline-earth metal are very reactive in air, the above experiments may have been influenced by sample degradation due to air exposure, particularly when the number of layers is very small. To avoid such an undesirable effect, Ca-intercalated bilayer graphene (C₆CaC₆) was fabricated by growing graphene on a 6H-SiC(0001) substrate and subsequently by depositing Ca in UHV [159]. The samples were characterized by STM and ARPES to reveal their atomic-scale structures and free-electron-like interlayer electronic bands. Furthermore, transport measurements were taken on similarly prepared C₆CaC₆ samples in situ using a micro four-point-probe technique under magnetic fields, which revealed ${T}_{{\rm{c}}}^{{\rm{onset}}}\cong 4\,{\rm{K}}$ and ${T}_{{\rm{c}}}^{{\rm{zero}}}\cong 2\,{\rm{K}}$ (see figure 18) [160]. Analysis by reflection high-energy electron diffraction (RHEED) showed that intercalated Ca atoms formed a (√3 × √3)-R30° structure against the C(1 × 1) surface of the host graphene and that this structural ordering was crucial to have a clear superconducting transition. However, they did not observe a superconducting transition for Li-intercalated bilayer graphene (C₆LiC₆). This is puzzling because an ARPES study on Li-intercalated monolayer graphene (LiC₆) prepared using a similar method indicated the emergence of superconductivity at T_c ∼ 5.9 K through observation of an energy gap opening at the Fermi level [161]. In this experiment, the occurrence of superconductivity was ascribed to the enhancement of the electron–phonon coupling constant λ to 0.58, which was estimated from the ARPES measurement of the electronic bands. Indeed, first-principle calculations on LiC₆ assuming the (√3 × √3)-R30° structure of Li adatoms on a monolayer graphene predicted that T_c would be strongly enhanced to 8.1 K from the bulk value of 0.9 K [162]. This enhancement was attributed to an increase in the electron–phonon coupling constant λ, in line with the experiment. The same theory also predicted T_c = 1.4 K for Ca-intercalated monolayer graphene (CaC₆), which is suppressed significantly from the bulk value of 11.5 K. Nevertheless, direct evidence of superconductivity of doped monolayer graphene is so far missing from the viewpoint of electron transport or magnetization measurements. Coming back to the bulk material, superconductivity of intercalated graphene laminates were recently studied [163]. This layered material is similar to bulk graphite, but the coupling between the individual layers is weaker due to the presence of rotational disorder. Thus its electronic structure should be close to that of isolated monolayer graphene. The T_c of Ca-intercalated graphene laminates was found to be ∼6 K at maximum, but it was strongly dependent on the sample condition. K, Cs, nor Li induced superconductivity in the temperature range of T > 1.8 K.

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The mechanical exfoliation of graphene from graphite has motivated the creation of atomic sheets from layered materials using the same method. In particular, studies on atomic sheets of transition metal dichalcogenides are now very active, in view of valleytronics (electronics combined with the valley degree of freedom in certain semiconductors) and optoelectronics applications [164]. Since the layers of transition metal dichalcogenides are only weakly bonded through van der Waals forces, the technique of mechanical exfoliation is readily applicable. In terms of studies on superconductivity in the 2D limit, NbSe₂ is a promising material because it has a relatively high T_c of 7.2 K in the bulk form. It also has a phase transition accompanying CDW at 33 K, so the competition or coexistence of the two ordered states in the 2D limit would also be interesting to investigate. An early experiment in 1972 already reported on the exfoliation of atomic sheets of NbSe₂ from a bulk crystal and the electron transport data indicative of the superconducting transition at a few UC thickness [165]. However, recent studies reported the absence of superconductivity in this regime [166], suggesting that the sample degradation due to air exposure and the lithography process was a serious problem. This was solved by performing the whole sample treatment in a controlled inert atmosphere and by encapsulating the target atomic sheet using a graphene or BN atomic sheet [167]. With this technique, monolayer (half UC thick) NbSe₂ was found to exhibit superconductivity with T_c ≅ 2 K. Similarly prepared monolayer NbSe₂ showed a superconducting transition at 3 K, while a strong enhancement of the CDW transition up to T_c = 145 K was detected from the Raman spectroscopy measurement [168]. High-quality monolayers of NbSe₂ were also prepared by MBE growth on epitaxial bilayer graphene on a 6H-SiC(0001) substrate under UHV environment [26]. After addition and removal of a Se capping layer, electron transport measurements were taken to reveal a superconducting transition with ${T}_{{\rm{c}}}^{{\rm{onset}}}=1.9\,{\rm{K}}$ and ${T}_{{\rm{c}}}^{{\rm{zero}}}=0.46\,{\rm{K}}.$ The suppression of T_c compared to that of bulk NbSe₂ was partly attributed to the reduction of density of states at the Fermi level, which was caused by changes in electronic band structures in the monolayer of NbSe₂. STM/STS observations also clarified the occurrence of the CDW phase in the same sample. The 3 × 3 CDW superlattice was found to set in around 25 K and to develop fully at 5 K, while there was no signature of the CDW phase at 45 K. This is contradictory to the T_c enhancement observed in [168].

Exactly like MoS₂, the NbSe₂ monolayer is of half UC thickness and lacks the in-plane inversion symmetry. Together with a strong SOI of transition metal Nb, this can lead to spin-valley locking, as observed for the field-induced superconductivity in the MoSe₂ EDL transistors [54, 147] (see section 3.5). This expectation was recently demonstrated by using a mechanically exfoliated NbSe₂ monolayer that was encapsulated by thin layers of BN, which exhibited superconductivity with T_c = 3.0 K [27]. Magnetotransport measurements showed that superconductivity was remarkably robust against an in-plane magnetic field, with an estimated upper critical field at zero temperature μ₀H_c2//(0) ∼ 35 T. This value is more than six times the Pauli paramagnetic limit μ₀H_P. Analysis of the temperature dependence of H_c2// gave a spin splitting energy 2Δ_SO ∼ 76 meV, equivalent to a spin–orbit field μ₀H_SO ∼ 660 T (H_SO ≡ Δ_SO/μ_B). These results indicate that the NbSe₂ monolayer is an Ising superconductor where the spins of Cooper pairs are aligned in the out-of-plane direction due to the spin-valley locking. Furthermore, magnetotransport measurement on a NbSe₂ bilayer with a 1 UC thickness (T_c = 5.26 K) showed that its true superconducting state was easily destroyed by a small out-of-plane magnetic field of 0.175 T (see figure 19) [169]. At higher fields, the presence of the so-called Bose metal phase was indicated, in which uncondensed Cooper pairs and vortices are responsible for the non-zero resistances [170, 171]. In view of the emergence of a metallic ground state under application of small out-of-plane magnetic fields, the obtained H_⊥–T phase diagram was similar to that of field-induced superconductivity of a ZrNCl-EDL transistor [142]. The absence of the magnetic-field-induced S–I transition, unlike the conventional theories on 2D superconductors, should be attributed to the extremely small disorder and weak vortex pinning in these systems.

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The tetragonal phase α-FeSe with PbO-type structure belongs to the iron-based high-T_c superconductor family (see figure 20(a)) and exhibits superconductivity with T_c ∼ 8 K at ambient pressure and with T_c ∼ 37 K at a high pressure of 8.9 GP [172, 173]. FeSe has attracted extensive interest because of its compositional and structural simplicity and of a variety of unique physical properties attributable to its strong electron correlation [174]. The bulk crystal of FeSe consists of covalent-bonded 2D layers that are weakly coupled via van der Waals interaction. Together with its simple chemical form, this also makes FeSe a promising material for the study of superconductivity in the 2D limit.

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Surprisingly, 1 UC thick FeSe layers (monolayers) epitaxially grown on a SrTiO₃(001) substrate using the MBE technique were found to exhibit superconductivity at much higher temperatures than for bulk crystals. The reported T_c ranges from 23.5 K to 109 K depending on the type of measurement and the experimental environment. The first indication of a high T_c of the monolayer FeSe was given by UHV-LT-STM experiments with a similar setup to that described in sections 3.1 and 3.2 [28], where degradation of FeSe layers by air exposure was avoided. An atomically well-ordered FeSe monolayer was fabricated on Nb-doped SrTiO₃ substrates by optimizing the growth condition (figure 20(b)). STS measurements revealed the presence of a large gap structure with an energy gap Δ ≅ 20 meV at 4.2 K (figure 20(c)), which persisted at least up to 43 K. Assuming that the relation between the energy gap Δ and T_c of bulk FeSe (2Δ/k_BT_c ≅ 5.5) still holds, it indicates that T_c could be as high as 80 K. This value could break the T_c record of iron-based superconductors, 56 K, found for Sr_1−xSm_xFFeAs [175]. STM measurements under an out-of-plane magnetic field of 11 T revealed formation of vortices as expected (figure 20(d)), from which the GL coherence length ξ_GL was estimated to be a few nm. The result is in striking contrast to a related work on FeSe layers grown on bilayer graphene that was prepared on a SiC(0001) substrate [176]. In this case, the superconducting energy gap was not detected for 1 UC thick FeSe layers down to T = 2.2 K. In addition, T_c was found to decrease with decreasing thickness in accordance with equation (5), from which d_c = 0.7 nm was deduced as the critical thickness for the disappearance of superconductivity. This form of thickness dependence of T_c was also observed for ultrathin Pb films (see section 3.2) [17, 79], suggesting the same physical mechanism behind it. In this case, the FeSe layers should have only weak van der Waals interaction with the bilayer graphene underneath, judging from the observation of frequent displacements and rotations of FeSe islands during STM imaging. By contrast, for FeSe/SrTrO₃ the overlayer should have a strong interaction with the substrate considering the heteroepitaxial growth under a tensile strain. Therefore, the interaction at the interface, whatever it is, should be the origin of the high T_c of monolayer FeSe.

More direct evidence for high-T_c superconductivity of the 1 UC thick FeSe layer was provided by ex situ electron transport and diamagnetic response measurements using insulating SrTrO₃ substrates. The samples were capped with protection layers made of FeTe and amorphous-Si [53, 177]. The transport measurements indeed revealed a high T_c, with ${T}_{{\rm{c}}}^{{\rm{onset}}}=40.2\,{\rm{K}}$ and ${T}_{{\rm{c}}}^{{\rm{zero}}}=23.5\,{\rm{K}}.$ The latter value is close to T_c = 21 K determined from the diamagnetic response. In accordance with the high T_c, a large out-of-plane upper critical field μ₀H_c2⊥ > 52 T was indicated at 1.4 K from the magnetotransport measurements. A large critical current density J_c = 1.7 × 10⁶ A cm⁻² was also observed at 2 K, which is two orders of magnitude larger than that of bulk FeSe. Nevertheless, the protection layers used in these experiments may have caused structural and chemical disorder in the FeSe layer and have adversely affected the observed superconducting properties. To avoid this problem, in situ electron transport measurements were performed using a micro four-point-probe technique in UHV without any protection layers [29]. A surprisingly high T_c of 109 K was locally detected from the emergence of the zero-resistance state (see figure 20(e)). Accordingly, μ₀H_c2⊥(0) ≅ 116 T was deduced from magnetic field dependence of T_c. Although the samples suffered from spatial inhomogeneity and hence the high-T_c superconductivity was not found all over the surface, the result suggests that macroscopically uniform 2D superconductors with T_c > 100 K may be available if the sample fabrication process is optimized. In terms of spectroscopic measurements, ARPES is a powerful surface-sensitive tool for revealing electronic band structure and the superconducting energy gap that should appear below T_c. All ARPES experiments reported an opening of an energy gap that amounted to 15–20 meV in gap size at the lowest temperatures [178–181]. T_c determined from the temperature dependence of the energy gap ranges from 55 K to 65 K. These values are higher than that reported from the ex situ macroscopic transport measurement $({T}_{{\rm{c}}}^{{\rm{onset}}}=40.2\,{\rm{K}})$ [53] but is lower than the maximum value in the in situ local transport study (T_c = 109 K) [29]. This discrepancy may be explained considering the fact that ARPES is performed in situ but at a macroscopic scale.

The emergence of high-T_c superconductivity driven by atomic-scale thinning is highly unusual because T_c generally becomes suppressed with decreasing layer thickness, as in equation (5). This certainly calls for investigation into its mechanism and the physics behind it, which are currently under intense debate. As mentioned above, the interface between the FeSe monolayer and the SrTiO₃ substrate must play a crucial role. There are several possibilities for the main mechanism, such as strain effects due to a lattice mismatch, enhancement of electron–phonon coupling, polaronic effects associated with the high dielectric constant of the SrTiO₃ substrate, and carrier doping from the interface [28]. Some of the ARPES experiments indicated the presence of optical phonon modes at the interface [182] and of spin density waves (SDW) in the FeSe film, both of which may contribute to the high-T_c superconductivity [180]. However, it is becoming clearer from the following experiments that electron doping from the SrTiO₃ substrate plays a central role.

The Fermi surface of a bulk FeSe crystal consists of a hole-like pocket near the Γ point and an electron-like pocket near the M point in the Brillouin zone, making FeSe a multiband superconductor. By contrast, the FeSe monolayer on a SrTiO₃ substrate has only an electron-like pocket near the M point, featuring a complete change in the topology of the Fermi surface [178–181]. The disappearance of the hole-like Fermi surface is naturally attributed to electron doping from the SrTiO₃ substrate with oxygen vacancies, which was confirmed by step-by-step annealing taken after the growth of a FeSe monolayer [179]. On this Fermi surface, a nearly isotropic (s-wave-like) superconducting energy gap was detected below T_c [181]. The band structure change due to the electron doping was also demonstrated by intentional chemical doping on the surface with K atoms, not only for a monolayer but also for a trilayer of FeSe (see figure 21) [178]. When the doping was tuned to the optimum level, a superconducting energy gap that corresponds to T_c ∼ 50 K was observed for the trilayer, revealing that the monolayer was not the prerequisite for the emergence of a high T_c. The T_c enhancement in a FeSe multi-layer due to surface K-doping was also observed by STS when bilayer graphene on SiC(0001) was used as a substrate [183]. The idea was further supported by an experiment using an EDL transistor device [184]. Here, in addition to the field-induced carrier doping, electrochemical etching of FeSe layers by ionic liquid was performed through the gate operation by adjusting the working temperature and voltage. This allowed the layers to be thinned down to atomic-scale thicknesses. After the thinning process was terminated, ${T}_{{\rm{c}}}^{{\rm{onset}}}\sim \; 40\; {\rm K}$ was observed even for 10 UC thick FeSe multi-layers when electrons were doped via gate voltage. Furthermore, essentially the same result was obtained when the substrate material was changed from SrTiO₃ to MgO. These results suggest that the main role of the SrTiO₃ substrate in T_c enhancement is to work as a source of carrier doping. Nevertheless, in [178], the maximum available T_c estimated from the energy gap is dependent on the number of FeSe layers and indeed the monolayer had the highest T_c. Very recently, the same trend was also observed in STS studies on the thickness-dependent energy gap of FeSe on SrTiO₃ [185, 186]. Correspondingly, the contribution of the intrinsic interface effect rather than doping was also noticed in related experiments [186, 187]. Thus it can be concluded that, although the emergence of a high T_c up to ∼50 K is possible only from substrate-induced electron doping, the presence of the FeSe–SrTiO₃ interface itself contributes to the additional T_c enhancement up to 65–100 K, e.g., through electron–phonon coupling and polaronic effects. More experimental and theoretical investigations are needed to clarify the origin of the high-T_c superconductivity in this system.

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In this final subsection, we will discuss two important superconducting materials: organic conductors and rare-earth based heavy-fermion compounds. These two systems are known to exhibit varieties of exotic phenomena originating from strong electron correlations and/or intrinsic low dimensionality of the crystal. Examples include CDW/SDW formation, non-BCS-type superconductivity such as d-wave Cooper pairing, the Kondo lattice, and the coexistence and competition of superconductivity and magnetism [188, 189]. The studies on 2D superconductors with atomic-scale thicknesses made of these materials have been relatively limited so far, but the reported results are definitely worth  introducing here.

λ-(BETS)₂GaCl₄ (BETS=bis(ethylenedithio)tetraselenafulvalene) belongs to the family of charge-transfer complex type organic conductors and exhibits superconductivity at 4.7 K [190, 191]. Within a bulk crystal, 2D conduction layers consisting of hole-doped BETS molecules are stacked in the perpendicular direction while separated by GaCl₄ insulating layers, forming a highly anisotropic quasi-2D electron system. This native 2D character of the organic crystal poses a question of whether a 1 UC thick layer (monolayer) of (BETS)₂GaCl₄ can become superconducting or not, as in the case of the cuprates discussed in section 3.3. Such an experiment was performed for monolayers of (BETS)₂GaCl₄ grown on a Ag(111) surface using UHV-LT-STM (see figure 22) [30]. The organic molecule was deposited through thermal evaporation on a clean Ag substrate cooled down to ∼120 K and was subsequently imaged with STM. The arrangement of BETS and GaCl₄ molecular units on the Ag substrate was found to be identical to that of a bulk crystal, confirming the 1 UC limit of the 2D organic crystal. STS measurements revealed that the monolayer of (BETS)₂GaCl₄ indeed exhibited an energy gap Δ ≅ 12 meV at 6 K that was attributed to the emergence of superconductivity. The T_c was estimated to be ∼10 K from the disappearance of the gap. It should be noted that this value is comparable to, or higher than, the bulk T_c of ∼8 K. The ratio of the energy gap to T_c, 2Δ/k_BT_c, was ∼13.6, which is much larger than the BCS value of 3.52. Furthermore, the spectral shape of the observed energy gap was best fitted by assuming a d_xy symmetry of Cooper pairs. These results showed that the superconductivity detected here is of non-BCS-type, as found for many organic superconductors [192]. Surprisingly, the energy gap structure at the Fermi level persisted even in isolated molecular chains and the gap size remained nearly constant until the number of (BETS)₂GaCl₄ molecule pairs were reduced to 15. A small gap-like structure was noticed even for just four pairs of molecule units. This means that superconductivity can survive nearly down to the zero-dimensional limit. A subsequent experiment on the same system revealed that the obtained molecular structure was sensitive to the substrate temperature during the deposition [193]. Kagome-like lattices were formed when the substrate temperature was kept at 125 K, which were found to be insulated from STS measurements. For room temperature deposition, BETS molecules grew on a closely packed GaCl₄ layer on a Ag(111) surface. Despite the structural difference, this phase also exhibited a gap structure with Δ ≅ 12 meV at 4.6 K, as seen in the molecular chains of  [30].

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There are many open questions for the monolayer of (BETS)₂GaCl₄ and its superconductivity. For example, precise determinations of molecular arrangement and of its electronic states are still far from complete. Particularly, charge transfers between the BETS/GaCl₄ molecules and the Ag substrate are very likely to take place as a result of their chemical potential difference. This should have a strong influence on the system, since the charge transfer within the molecular unit (2BETS + GaCl₄) is the driving force for the emergence of superconductivity in this type of complex organic conductor. Use of a closely related organic conductor λ-(BETS)₂FeCl₄ would also be very interesting because it is one of the few examples of magnetic-field-induced superconductors [194]. Since there has been no direct evidence of superconductivity by electron transport or magnetic measurements, the finding calls for further investigations on whether the signatures of the energy gap can be safely attributed to the emergence of superconductivity.

As for rare-earth based heavy-fermion compounds, there have been no reports on a 2D superconductor with an atomic-scale thickness, but superlattices made of a few UC thick layers of heavy-fermion superconductor CeCoIn₅ and normal metal YbCoIn₅ were successfully created [31, 195, 196]. 4f electrons in CeCoIn₅ are localized at the Ce atomic sites at high temperatures, but as the temperature is decreased, they form a narrow conduction band together with sp itinerant electrons as a consequence of the Kondo effect. The resulting electronic states are highly unusual because of strong electron correlations and a very large effective mass m_eff. In terms of superconductivity, this has a significant influence on H_c2, because H_c2 is usually determined by the orbital pair-breaking effect and its value, ${{H}_{{\rm{c}}2}}^{{\rm{orb}}},$ is enhanced due to the relation ${{H}_{{\rm{c}}2}}^{{\rm{orb}}}$ ∝ ${{m}_{{\rm{eff}}}}^{2}.$ Indeed, ${\mu }_{0}{{H}_{{\rm{c}}2}}^{{\rm{orb}}}$ exceeds 10 T in a bulk CeCoIn₅, which is well above the Pauli limit μ₀H_P = √2Δ/gμ_B (Δ: superconducting energy gap, g: g-factor; see section 3.2). Therefore, H_c2 is dominated by the Pauli paramagnetic effect, not by the orbital effect as in the usual superconductors. This offers a unique opportunity for studying exotic phenomena such as the FFLO states and the helical states induced by the Rashba effect (see section 3.2) [92, 100]. The crystal of CeCoIn₅ has space-inversion symmetry, but introduction of a superlattice with atomically thin layers can artificially break this symmetry, consequently inducing the Rashba effect in a controllable way [195, 196].

In [31], CeCoIn₅(n)/YbCoIn₅(m) superlattices (n, m: number of UCs in each layer) were epitaxially grown by the MBE technique and their superconducting properties were revealed by electron transport measurements for the first time. As n was decreased while m = 5 was fixed, T_c was found to decrease from the bulk value of 2.3 K, but a clear superconducting transition at 1.0 K was still observed for n = 3 (see figure 23(a)). For n = 2 and 1, the zero-resistance state was not observed but magnetotransport measurements indicated the onset of superconductivity. Although suppression of superconductivity may be expected because of the presence of the neighboring normal metal layers of YbCoIn₅, the observed decrease in T_c should be mostly intrinsic. This is because the large Fermi velocity mismatch between the two layers suppresses the proximity effect and confines the Cooper pairs within the CeCoIn₅ layers. One of the intriguing findings here was that H_c2, measured in both the out-of-plane and in-plane directions, did not decrease from the bulk values in proportion to the reduction in T_c. Since H_c2 can be identified with the Pauli limit H_P here and H_P is proportional to the superconducting energy gap Δ, the ratio Δ/k_BT_c must be enhanced. The estimated ratio of Δ/k_BT_c > 10 is much larger than the BCS value of 3.54, suggesting the realization of an extremely strong-coupling superconductor.

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Furthermore, anomalous behaviors were observed in the angular dependence of H_c2(θ), where θ is the angle of magnetic field relative to the in-plane direction [195]. When the conducting layers consisting of a quasi-2D superconductors are sufficiently decoupled, the angle dependence of H_c2(θ) obeys the 2D Tinkham model (see equation (9)) as long as the orbital pair-breaking effect is dominant. However, when H_c2(θ) is dominated by the Pauli paramagnetic pair-breaking effect, the angle dependence is expressed by the 3D anisotropic GL model regardless of the interlayer coupling strength (see equation (8)). In this case, the anisotropy of superconductivity reflects that of the g-factor in the Zeeman term. In [195], for n = 4, 5 (n: number of UCs for the CeCoIn₅ layers), the angular dependence of H_c2(θ) was expressed by the 3D anisotropic GL model in a wide temperature range (see the lower panel of figure 23(b); the variation around θ = 0 is smooth). This was ascribed to the strong Pauli paramagnetic effect in CeCoIn₅. For n =3, however, the angular dependence of H_c2(θ) at T = 0.8 K (T_c = 1.04 K) was described by the 2D Tinkham model (see the upper panel of figure 23(b); the variation around θ = 0 is cusp-like). This means that the orbital pair-breaking effect is stronger than the Pauli effect in this regime, considering the fact that the CeCoIn₅ layers are well decoupled from each other. Therefore, it was concluded that a decrease in the layer thickness makes the Pauli effect weaker compared to the orbital effect. This phenomenon can be explained by the fact that the thinning of the CeCoIn₅ layers leads to increasingly important roles of the CeCoIn₅–YbCoIn₅ interface and of the space-inversion symmetry breaking. Namely, the resultant manifestation of the Rashba effect significantly weakens the Pauli effect, because the spins are already polarized due the Rashba effect. Nevertheless, the Pauli effect was found to dominate again at lower temperatures (T ≪ T_c), which was interpreted as the effect of the FFLO phase appearing in CeCoIn₅. The influence of the Rashba effect on H_c2(θ) was confirmed in a subsequent experiment, where additional space-inversion symmetry breaking was introduced by an asymmetric stacking sequence of CeCoIn₅(n)/YbCoIn₅(m)/CeCoIn₅(n)/YbCoIn₅(m') (m ≠ m') [196].