Two-dimensional Superconductors with Atomic-scale Thicknesses

Recent progress in two-dimensional superconductors with atomic-scale thicknesses is reviewed mainly from the experimental point of view. The superconducting systems treated here involve a variety of materials and forms: elemental-metal ultrathin films and atomic layers on semiconductor surfaces; interfaces and superlattices of heterostructures made of cuprates, perovskite oxides, and rare-earth metal heavy-fermion compounds; interfaces of electric-double-layer transistors; graphene and atomic sheets of transition-metal dichalcogenide; iron selenide and organic conductors on oxide and metal surfaces, respectively. Unique phenomena arising from the ultimate two-dimensionality of the system and the physics behind them are discussed.


Introduction
Superconductivity arises from the Cooper pair formation of a huge number of conduction electrons in a metal at sufficiently low temperatures. Since it is a representative order-disorder phase transition, the dimensionality of the system, i.e., one, two, or three dimension (1D, 2D, 3D), can have a crucial influence on its characteristics [1][2][3]. Generally, the lower the dimension, the more difficult for the phase transition to take place, because the interaction between microscopic constituents of the system (in this case, electrons) becomes spatially limited and a less number of partners are available for the interaction with a particular constituent. This means that, even in an ordered phase below the transition temperature (Tc), each subset of the system has a tendency to behave more independently and the order parameter suffers from larger spatial and temporal fluctuations. In the extreme case, the coherence throughout the system is completely lost and the phase transition itself is destroyed. Now let us ask a simple question: Does superconductivity survive in a 2D system, especially when one of the material dimensions is reduced to a truly atomic-scale size? If this is the case, what are the unique characteristics of 2D superconductivity and what kind of new phenomena are expected to occur? These are obviously relevant to the modern state-of-the-art nanotechnology and will be crucial issues when the present superconducting devices are shrunk towards the atomic-scale limit in the future [4][5][6].
There is a long history regarding this problem. First of all, we should note that, for a 2D system, the famous Mermin-Wager theory prohibits the superconducting phase transition that accompanies a symmetry breaking and a long-range correlation of the order parameter [7,8]. This does not mean, however, that 2D superconductivity is unrealistic. The Kosterlitz-Thouless-Berezinskii (KTB) transition, which is compatible with the Mermin-Wager theory, can occur in a 2D system and allows the establishment of a quasi-long-range correlation of the order parameter [9][10][11]. In this case, the zero resistance state is retained for infinitesimally small external perturbation and the Meissner effect can also be well-defined [12,13]. Even without the KTB transition, Cooper pairs can condense at the mean-field level due to the Bardeen-Cooper-Schrieffer (BCS) mechanism (i.e., Tc0 > T KTB, where Tc0 is the Cooper-pair condensation temperature and TKTB is the KTB transition temperature). In a practical sense, the system may be considered superconducting if the correlation of the order parameter is sufficiently developed at low temperatures. Nevertheless, 2D superconductivity is on the verge of transition to a metallic or an insulating state and thus could be fragile. Indeed, introduction of disorder into a 2D superconductor can readily induce a superconductor-insulator (S-I) transition [14], in clear contrast to the 3D counterpart where superconductivity is robust against disorder [15]. Extensive experimental and theoretical efforts have clarified that the S-I transition always occurs when disorder is introduced to such a level that the sheet resistance (2D resistivity) of the sample is on the order of quantum resistance of Coopers, RQ ( h/4e 2 = 6.45 k) [14]. Usually, superconductivity is lost when the thickness of a metal film approaches 1-2 nm and disorder becomes significant [16]. Coming back to the earlier question, the answer should be "yes" in principle, but realization of a 2D superconductor Straightforward way of preparing a superconducting thin film is the vacuum evaporation of an elemental metal such as Pb and Al on an insulating substrate. This can be followed by deposition of a capping layer (e.g. SiOx) for the purpose of avoiding post-oxidation and contamination. If glass or surface-oxidized silicon is used as a substrate, the metal overlayer often grows in the form of a granular film when the thickness is reduced because of a poor wettability of metal on these substrates. This naturally introduces disorder into the metal film, the degree of which can be controlled by the film thickness and is represented typically by the sheet resistance Rsheet [14]. The degree of disorder (and hence Rsheet) can also be controlled by intentional surface oxidation of individual metal granules and by repeating this process several times [33]. Thus prepared granular superconducting film may be modeled as a Josephson-junction coupled island network in the case of strong disorder, where the junctions are of either tunneling or weak-link type. Within individual metal islands, relatively high crystallinity can be obtained. More uniform thin films can be prepared by quench condensation of an evaporated material on a liquid-He cooled substrate, which strongly suppresses diffusion of adsorbed atoms and/or molecules and formation of large clusters [16]. Sputter deposition of certain alloys can also be used for this purpose. To enhance the uniformity of the film, the substrate is often precoated with a buffer layer. The resulting films have amorphous-like microscopic structures with atomic-scale disorder and defects, and are believed to be spatially homogeneous at larger scales [34]. For the quench-condensed films, electron transport measurements should be performed in-situ at the samplepreparation cold stage to avoid morphological changes of the film. This experimental setup is advantageous in terms of precise tuning of the film thickness, which in this case is the dominant parameter representing disorder. A disadvantage is that it is difficult to perform detailed structural analysis of the film. Regarding the superconducting material, elemental metals such as Pb and Bi [34] as well as metal composites and alloys like In2O3 and MoGe have been used [35,36]. Note that thin films of amorphous Bi become superconducting at T ~ 6 K although bulk crystals of Bi do not.
The effect of the low dimensionality manifests itself as an enhancement of fluctuations as mentioned earlier. This can be seen first of all as a precursor of the superconducting transition. Namely, a 2D superconductor exhibits a sizable decrease in conductivity even above Tc due to the temporal formation of Cooper pairs and the inertia of Cooper pairs after decaying into quasiparticles [37,38].
Within the microscopic theory of superconductivity, the former is expressed by the Aslamazov-Larkin (AL) term and the latter the Maki- Thompson (MK) term. The contribution of the AL term to 2D conductivity, which is added to the normal conductivity, is given by where = 16ℏ = 65.8 kΩ ⁄ is a universal constant. The contribution of the MK term is expressed by a similar form that includes material dependent parameters. These effects should appear commonly in all 2D superconductors.
In a 2D system, conduction electrons can be readily localized due to the quantum interference effect in the presence of disorder, which is known as the Anderson localization [39]. Since the resulting insulating state is incompatible with a superconducting state, a superconductor-insulator (S-I) transition is expected to take place as the degree of disorder in the system is increased. In this viewpoint, the transition is driven by the suppression of amplitude of the superconducting order parameter . Alternatively, superconductivity can also be destroyed when the phase of  fluctuates strongly and its coherence is lost. This leads to a quantum phase transition at zero temperature [14,33,40]. A number of experiments have been performed in this respect since late 1970s, a representative result of which is shown in Fig. 2 [34]. Here, the sheet resistance Rsheet of quench-condensed amorphous Bi films was measured as a function of temperature T while increasing the film thickness.
The data clearly show a S-I transition around Rsheet  RQ  h/4e 2 (= 6.45 k). The presence of this universal constant suggests that the transition is actually a quantum phase transition and can be described by the so-called dirty-boson model at least near the critical point [41,42] [43]. The failure of the dirty-boson picture may be due to extrinsic effects and/or to oversimplification in the theoretical modeling. Otherwise, it suggests that Cooper pairs are at least partially broken at the critical point as a result of the Anderson localization [44]. How satisfactorily this model applies to a real experiment seems to be dependent on the details of the system.
Apart from the particular model, the possibility of a quantum phase transition has also been tested using the finite-size scaling analysis [40,41]. Here the sheet resistance Rsheet of the sample has the following form: Here, x is a control parameter for the phase transition (e.g., disorder and film thickness) and Rc is the critical resistance that separates the superconducting and the insulating phases, corresponding to x = xc. F(u) is a universal function of u that goes to unity for u0. The parameters z and  characterize the critical behaviors near the quantum phase transition, which gives crucial information on the universality class of the system. The experiment cited above concluded that this scaling analysis worked fine, with the extracted parameters of z = 2.4 and 1.2 for the insulating and superconducting sides of the transition, respectively [45]. The success of this analysis was found in many analogous experiments. In addition to disorder, magnetic field induces an S-I transition with Rc ~ RQ, for which the scaling analysis was also successfully applied [35,36]. All these results evidence the existence of quantum phase transitions in these systems. With technological advancements, the study on the S-I transition has seen new developments. For example, carrier density of amorphous Bi layer grown on a thin SrTiO3 substrate was continuously tuned by a gate voltage to investigate this phenomenon quantitatively [46]. High-Tc cuprate superconductors have also become the target of this study.
Remarkably, a S-I transition with Rc = RQ and a nearly perfect scaling behavior were demonstrated using a 1-UC-thick La2-xSrxCuO4 and an ionic liquid EDL gate (see Sec. 3.5) [20].
Another important phenomenon that is relevant for 2D superconductors in general is the Kosterlitz-Thouless-Berezinskii (KTB) transition [9,11]. As mentioned earlier, the Mermin-Wagner theory prohibits the emergence of the superconducting phase transition in a 2D system in the strict sense, i.e., the establishment of a long-range correlation at a finite temperature. In a more physical picture, 2D superconductors suffer from phase fluctuations of the order parameter  due to the thermally excited free vortices, even if the amplitude of  is well developed below a Cooper-pair condensation temperature, Tc0. When these vortices are moved by an external current in the transverse direction, the motion gives rise to a voltage drop in the longitudinal direction, resulting in finite energy dissipation. This precludes the realization of the true zero-resistance state [32]. However, a vortex in a 2D superconductor can form a bound state with an antivortex (a vortex with the opposite supercurrent circulation) to form a "neutral" pair. If the vortices interact logarithmically as a function of their separation r like ∝ log ( : interaction energy), all vortices and antivortices form pairs below a certain critical temperature, thus leaving no free vortices. This is called the KTB transition, the transition temperature of which is denoted as TKTB. For T > TKTB (and T < Tc0), the average separation  between free vortices diverges like ∝ | − | − ⁄ as T approaches TKTB, and the zero-bias sheet resistance Rsheet decreases to zero according to a relation ( Since no free vortices exist for T < TKTB, the true zero resistance state can be realized. However, the vortex-antivortex pairs become unbound under a finite external current. This unbinding occurs progressively as the current is increased, leading to a current-voltage (I-V) characteristics of a powerlaw dependence, i.e., ∝ . As T is lowered from above TKTB, exponent a jumps universally from 1 to 3 at T = TKTB and increases further at lower temperatures, which is one of the hallmarks of the KTB transition. The phase transition is unique in the sense that it does not lead to a true long-range order, but rather to a quasi-long-range order where the spatial correlation of the order decays as a function of distance according to a power-law dependence. The general features of the KTB transition have been observed using 2D superconductors with moderately high sheet resistances, which were prepared in the similar method as described above [47][48][49]. Figure 3 shows a representative result of the KTB transition, which was observed for an amorphous homogeneous film made of In/InOx composites with a thickness of 10 nm. The left panel is a logarithmic plot of the sheet resistance as a function of (T-TKTB) -1/2 where the relation of Eq.(3) holds over four orders of magnitude in resistance.
The right panel displays a I-V characteristics in the log-log plot. The exponent a of the relation ∝ was found to cross 3 around TKTB = 1~ 94 K. It should be noted that the occurrence of the KTB transition requires that the logarithmic interaction between vortices be retained up to a sufficiently (ideally, infinitely) long distance. Cutoff of the logarithmic interaction stemming from a finite sample/domain size, a finite perpendicular penetration depth, a residual magnetic field, etc. leads to a non-vanishing zero-bias resistance down to T = 0 [50]. In recent experiments on various 2D superconductors, the KTB transition has been discussed based on the non-linear characteristics ∝ , but this relation was found to apply only at relatively high bias regions [21,[51][52][53][54][55]. Whether its origin is ascribed to the interaction cutoff inevitable in real experiments is not clear in many studies.

Ultrathin metal films and islands on semiconductor surfaces
One of the first experiments on superconducting ultrathin metal films with atomically welldefined thicknesses and high crystallinity was performed using Pb films grown on a clean Si(111)-(77) surface with the molecular beam epitaxy (MBE) method in UHV [56]. An advantage of using these materials is that Pb and Si does 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 to study 2D superconductivity. In Ref. [56], the films were characterized using the 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 Tc of the Pb film was found to remain around 5~7 K down to the thickness of 4 ML, which was close to the bulk Tc value of 7.2 K. However, the Tc decreased substantially to 1~2 K in the sub-4 ML regime, where structural disorder in the macroscopic scale was unavoidable.
A particular interest in ultrathin metal films with the 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]: where (E) is the electron wavelength at an energy E, d is the film thickness, (E) is the total phase shift at the boundaries, and n is an integer. Since (EF) = 1.06 nm for Pb (EF: Fermi energy), d must be both in 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, (EF), as a function of d, with the periodicity of (EF)/2. Since the Tc of a BCS-type superconductor is proportional to [32], it can lead to an oscillation of Tc 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 21 ML (see Fig. 4) [19]. The observed periodicity in d was 2 ML, approximately equal to (EF)/2 for Pb. The interpretation as a quantum oscillation was supported by the corresponding oscillatory behavior for (EF), which was determined from the temperature dependence of out-of-plane upper critical magnetic field Hc2 near Tc. In addition, the electron-phonon coupling constant  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.
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 EF for a certain thickness, the film becomes stable due to a decrease in electronic energy. This occurs with a periodicity of (EF)/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 realization of the so-called hard superconductor [32] albeit the atomic-scale thicknesses.
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]. Bean-like critical state corresponding to a very hard superconductivity. A critical current density as large as ~210 6 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 Tc, the temperature dependence of Hc2 was found to markedly deviate from the prediction of the GL theory, i.e., ⊥ ∝ (1 − ⁄ ).
This indicated that, in this 2D geometry, the superconducting order was anomalously suppressed by scattering in violation of Anderson's theorem.
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 hardly accessible by the 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 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). above (see also Ref. [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 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 a LT-STM [65,68].
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 was successfully studied in real space with a 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 superconductivity penetrate into the normal-metal region is solely determined by the proximity length ≡ ℏ 2 ⁄ , where D is the diffusion constant in the normal metal. However, a 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 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 - < E <  (: 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. The essentially same 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 S can host vortices when out-of-plane magnetic field was applied, which were successfully imaged with a LT-STM [64]. Figure 6(c)(d) shows 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 a 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.

Metal atomic layers on semiconductor surfaces
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 superconductorinsulator (S-I) transition as discussed in Sec. 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 Tc of a Pb ultrathin film decreased as a function of thickness d according to the relation: where Tc0 is the asymptotic value of Tc(d) for d   (the bulk limit) and dc is the critical thickness for the disappearance of superconductivity [17,61,62]. Experimentally, dc was determined to be 0.43-0.615 nm for Pb, which corresponds to 1.5-2 ML. Within the framework of the GL theory, this behavior can be naturally explained by introduction of surface energy term into the GL free energy [76]. While this term originates from the decrease in density of states near the surface in Ref. [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 Tc values, obtained from the analysis of temperature-dependent superconducting energy gap (T), remained between 6.0 K and 6.7 K down to d = 4 ML, while Tc = 7.2 K for bulk Pb . Tc was also found to oscillate with a periodicity of 2 ML due to the formation of QWS as discussed in Sec. 3.1. Even 2-3.65 K or 4.9 K depending on the type of the 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 Tc steadily decreased with decreasing d according to Eq.(5), where dc 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) [85]. It should be noted that the resistance changes were remarkably sharp just at Tc although the systems were in the atomic-scale limit. The decrease in resistance above Tc was accelerated as the temperature approached Tc, which was well described by the 2D fluctuation theories (see Sec.2) [84,86]. Analysis of magnetic field dependence of Tc gave GL(0) = 74 nm for Si (111) A/cm at 1.8 K (correspondingly, 3D supercurrent density of 3-610 5 A/cm 2 ) [83]. The temperature dependence of J2D was described by the Ambegaokar-Baratoff relation [87], suggesting that J2D 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. There are two unique features that should be considered for atomically thin 2D superconductors in general. First, when magnetic field is applied in the in-plane direction, there exists no orbital pairbreaking effect that usually dominates the upper critical magnetic field Hc2. This is because electrons cannot move in the out-of-plane direction (see Fig. 9(a)). In the simplest case, the in-plane critical magnetic field Hc2// is determined by the Pauli pair-breaking effect, leading to is the BCS energy gap and HP 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 HP [90,91]. This unusual superconducting state is the result of the 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 momentums in striking contrast to the BCS state. Consequently, the order parameter ( ⃗ ) becomes spatially modulated with wave vectors of ⃗ even at equilibrium, and is generally expressed by [92] ( ⃗ ) = exp( ⃗ ⋅ ⃗ ) symmetry breaking of the system and the spin-orbit interaction (SOI) [93]. The Hamiltonian of the Rashba effect is expressed as where ⃗ = ( , , 0) is the kinetic momentum, is the strength of the Rashba SOI, ⃗ is the unit vector perpendicular to the 2D xy plane, and is the Pauli matrices. The Rashba effect leads to a spin splitting and the spin-momentum locking where electron spins are chirally polarized in the directions perpendicular to both the electron momentum and the electric field (see Fig. 9(b)). Due to the space-inversion symmetry breaking at surface, this effect has been observed by ARPES for the metal-induced surface reconstructions with a strong SOI [94,95]. In terms of superconductivity, Rashba effect may lead to exotic phenomena under high in-plane magnetic fields. As schematically depicted in Fig. 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 magnetic field [96]. Then the electrons on each  [52]. The same surface also showed a superconducting transition at 2.25 K, opening a route for investigating exotic superconducting state as stated above. In Ref. [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 in-plane magnetic field (see Fig. 10(b) [101]. The Tc was found to decease only by ~2% under in-plane field of 15T. 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.
Finally, 2 ML-thick Ga atomic layer on GaN(0001) was found to exhibit superconductivity. The measured sheet resistance showed an onset of transition at Tc onset = 5.4 K and appearance of the zero resistance at Tc zero = 3.8 K, which were much higher than Tc = 1.08 K for the bulk stable phase of -Ga [102]. This Tc 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-Tc superconductivity of 1-UC-thick FeSe layers on a SrTiO3 substrate, which will be described in Sec. 3.7.

Cuprate: La2-xSrxCuO4/La2CuO4 interface.
It is widely recognized that high-Tc cuprate superconductors such as La2-xSrxCuO4, YBa2Cu3O7, and Bi2Sr2CaCu2O8+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) GL(0) is comparable to the unit-cell (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 the coupling between the CuO2 planes helps establish high Tc in these systems? Indeed, soon after the finding of cuprate superconductors, the investigation into this problem started [104][105][106][107][108]. Generally, superconductivity was found to survive even when the number of unit heterostructures (see Fig. 11) [110]. Here one of the CuO2 planes of La2-xSrxCuO4/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.
The same group also demonstrated that superconductivity of a 1-UC-thick La2-xSrxCuO4 (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 Sec. 3.5) to successfully tune the Tc by up to 30 K. This has become possible due to the facts that the thickness of the conduction layer was only ~1 nm and that a huge local electric field of > 10 9 Vm -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 Fig. 12). Accumulated data on the sample sheet resistance Rsheet 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 Eq.
(2) (see Sec. 2). The analysis gave Rc = 6.450.10 k and z = 1.50.1. Remarkably, the critical resistance Rc obtained here is equal to the quantum resistance for Cooper pair, RQ = 6.45 k, 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 fluctuation [42]. The value of z = 1.5 is clearly different from those of previously investigated systems, such as amorphous MoGe films (z  1.3) [35], [112], and amorphous Bi films (z  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, z = 4/3 and 2/3 correspond to the classical percolation and the 3D XY models, respectively). Also, related experiments on hole-doped YBaCuO7x in the similar configuration revealed Rc = 6.0 k and z = 2.2 [113]. Here, the result Rc  RQ = 6. should also be noted that the superfluid density ns(0) of 2-UC-thick Y1-xCaxBa2Cu3O7- was determined from the ac conductivity measured with the two-coil mutual inductance method [115]. The Tc was found to be proportional to ns(0) near the S-I transition, which was attributed to the quantum fluctuation near a 2D critical point.
Preparation of cuprate thin layers with an atomic-scale precision based on MBE or PLD methods as shown above requires a highly advanced instrumentation. However, the recent progress in mechanical exfoliation of atomic sheets, which was first demonstrated for graphene [25,116], has now allowed preparation of atomic sheets of Bi2Sr2CaCu2O8+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 Rsheet of the sample increased from a few  to 5 k as the sheet thickness was reduced from 270 to 0.5 UC, sharp superconducting transitions were consistently observed, with nearly constant Tc of ~82 K. This indicates that the interlayer coupling does not play an important role for the superconductivity of Bi2Sr2CaCu2O8+x. Superconductivity of exfoliated atomic sheets of other layered materials will be described in Sec.3.6.

LaAlO3/SrTiO3 interface
As seen in Sec. 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 interface of oxide heterostructure 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 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]. generally has a high carrier density (~10 13 cm -2 ) and a low sheet resistance (~10 3 ) 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 (~10 4 cmV -1 s -1 ), indicating the electron scatterings due to impurities and defects are 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].
Remarkably, electron transport measurements have revealed that the LaAlO3/SrTiO3 interface become superconducting at low temperatures (see Fig. 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 Tc  200 mK, 2D critical supercurrent density Jc,2D = 98 A/cm, out-of-plane critical magnetic field 0Hc2 = 65 mT. The observed Tc 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 GL(0) ~70 nm from the temperature dependence of Hc2. Other 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 Tc, 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 Ref. [21], the residual ohmic regime at small currents observed even below TKTB 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 GL(0) ~70 nm. A subsequent experiment on the same system revealed a strong anisotropy of critical magnetic field (Hc2///Hc2  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. In contrast, it could spread up to as large as ~500 m when a different annealing process was taken due to the 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 (r ~300 at RT 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 TKTB reached 310 mK [112].
Clear S-I transitions were also demonstrated through electrostatic gate control. The critical sheet resistance for the transition was Rc  4.5 k, which was close to but lower than the quantum resistance of Cooper pairs, RQ = 6.45 k [14]. The transition temperature identified with TKTB was found to scale with the carrier density n2D according to a relation Intriguingly, the LaAlO3/SrTiO3 interface exhibits not only superconductivity but also ferromagnetism, as in a theoretical prediction of ferromagnetic alignment of local spins at Ti 3+ sites of the interface TiO layer [128]. Experimentally, the existence of 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 Rsheet was found as an evidence for the formation of ferromagnetic domains, and a logarithmic temperature dependence of Rsheet 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 Tc, which was electrically tuned by a back gate, was found to have hysteresis within the range of -30 mT < 0H < -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 measurement 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 Fig. 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 -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 Sec. 3.2). The spin-orbit interaction (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.

Electric-field induced interface superconductivity
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 electric-double-layer (EDL) as a gate electrode (also see Sec. 3.3) [23,24]. The conventional FET device uses a gate insulator made of 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 n2D ~110 13 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 gate voltage [22,[135][136][137]. The resulting EDL works as a subnanometer-gap capacitor and can induce carriers with an areal density up to ~110 15 cm -2 at the interface (more precisely, at the subsurface) of the target material.
In the early stage of the EDL device, a polymer electrolyte consisting of KClO4 and polyethylene oxide was used to fabricate a n-type FET with a 2D channel at the SrTiO3 interface [23]. This device successfully induced superconductivity at Tc ~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 n2D = 110 13~1 10 14 cm -2 was confirmed through Hall resistance measurement. This corresponds to a 3D carrier density as high as 310 18~1 10 20 cm -3 , which is roughly equal to that of superconducting Nb-doped SrTiO3 bulk samples with Tc = 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 that is on the verge of 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 even an 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][140][141]. The experiment on thin layers of ZrNCl, obtained by mechanical micro-cleavage, realized for the first time superconducting transitions with Tc = 12 ~ 15.2 K under the gate voltages of 4~5 V using a EDL-FET device (see Fig. 15) [24]. The measured net carrier density amounted up to 1.710 14 cm -2 . This seminal work was followed by a more elaborate experiment that revealed many intriguing phenomena (see Fig. 16) [142]. Generally, quasi-2D bulk superconductors made of weakly coupled conducting layers have an angular dependence of the upper critical magnetic field Hc2(), where  is the angle of magnetic field relative to the in-plane direction (note that, in Fig.   15(a),  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]: This form applies when the coherence length in the out-of-plane direction,  GL, is larger than the interlayer distance. In contrast, for an isolated 2D superconductor with a thickness much smaller than the magnetic penetration depth, Hc2() obeys the relation: which is called the 2D Tinkham model [143]. The clear difference in the angle dependence between the two model can be found around  = 0; for the former, the change in Hc2() is smooth (i.e., | ⁄ | = = 0) while, for the latter, it exhibits a cusp-like shape (i.e., | ⁄ | = > 0). In Ref.
[142], Hc2() for the field-induced superconductivity of ZrNCl exhibited a huge anisotropy between the in-plane ( = 0) and out-of-plane ( = 90) directions. Furthermore, a cusp-like angle dependence of Hc2() was found around  = 0, which was well described by Eq. (9) of the 2D Tinkham model. 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 ~10 14 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 cm 2 /Vs 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-Tc superconductors [103]. Namely, Tc plotted as a function of 2D carrier density n2D was found to have a maximum value of 10.8 K at n2D = 1.210 14 cm -2 . Further increase in n2D resulted in a lower Tc. The Tc = 10.8 K is ~40% higher than the maximum Tc 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 Fig. 17) [54,147]. MoS2 as a bulk crystal has the global inversion symmetry due to its D6h symmetry, and its superconducting state is considered conventional. However, monolayer of MoS2 lacks the in-plane inversion symmetry because it has half the unit cell of a bulk crystal and the symmetry is lowered to D3h [148]. Together with a large spinorbit interaction (SOI) originating from the Mo d-orbitals, this causes an effective Zeeman field in the order of ~100 T and a 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 the spin-valley locking [149]. The Hamiltonian for this effect is expressed as where ⃗ = ( , , 0) is the kinetic momentum in the K and K' valleys, is the momentum of the  [27] are about 4-5 times larger than the Pauli paramagnetic limit HP.
These Hc2// values are also an order of magnitude larger than those of alkali-doped bulk MoS2 crystal, for which Hc2// is dominated by the orbital pair-breaking effect caused by the interlayer coupling. It should be noted that a huge Hc2// value surpassing the Pauli limit was also indicated for the 2D superconductor with Rashba effect as discussed in Sec.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.

Atomic sheets: graphene and transition-metal dichalcogenide atomic layers
In 2004, graphene was mechanically exfoliated from a piece of graphite to fabricate an highperformance 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. Until today, however, superconductivity was not 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 Tc = 0.14 K [153], CaC6 with Tc = 11.5 K, and YbC6 with Tc = 6.5 K [154,155]. The metal layers play important roles in terms of the BCS-type superconductivity because they donate electrons into the *bands of graphite and modify the electron-phonon interaction. They also form electronic bands by themselves, which affect 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 Tc = 4.5 K was found based on magnetic susceptibility measurement [156]. Similarly prepared Li-intercalated few-layer graphene was found to have Tc = 7.4 K from magnetization measurement, although transport measurement 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 10 ML, with the maximum Tc 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 fourpoint-probe technique under magnetic fields, which revealed Tc onset  4 K and Tc zero  2 K (see Fig.   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 the similar method indicated the emergence of superconductivity at Tc ~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 electron-phonon coupling constant  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 Tc 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 , in line with the experiment. The same theory also predicted Tc = 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 measurement. 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 Tc of Ca-intercalated graphene laminates was found to be ~6 K at maximum, but it was strongly dependent on the sample condition. None of K, Cs, or Li induced superconductivity in the temperature range of T > 1.8 K.
The mechanical exfoliation of graphene from graphite has motivated creation of atomic sheets from layered materials using the same method. Particularly, the studies on atomic sheets of transitionmetal 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 force, 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 Tc of 7.2 K in the bulk form. It also has a phase transition accompanying charge density waves (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 Tc  2 K. Similarly prepared monolayer NbSe2 showed a superconducting transition at 3 K, while a strong enhancement of the CDW transition up to Tc = 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 Tc onset = 1.9 K and Tc zero = 0.46 K. The suppression of Tc 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 Tc enhancement observed in Ref. [168].
Exactly like MoS2, 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 the spin-valley locking as observed for the field-induced superconductivity in the MoSe2 EDL transistors [54,147] (see Sec. 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 Tc = 3.0 K [27].
Magnetotransport measurements showed that superconductivity was remarkably robust against inplane magnetic field, with an estimated upper critical field at zero temperature 0Hc2//(0) ~ 35 T. This value is more than six times the Pauli paramagnetic limit 0HP. Analysis of the temperature dependence of Hc2// gave a spin splitting energy 2SO ~76 meV, equivalent to a spin-orbit field 0HSO ~660 T (HSO  SO/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 NbSe2 bilayer with a 1-UC-thickness (Tc = 5.26K) showed that its true superconducting state was easily destroyed by a small out-of-plane magnetic field of 0.175T (see Fig. 19) [169]. At higher fields, the presence of 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.

One-unit-cell thick FeSe layer on SrTrO3
The tetragonal phase -FeSe with PbO-type structure belongs to the iron-based high-Tc superconductor family (see Fig. 20(a)) and exhibits superconductivity with Tc ~ 8 K at ambient pressure and with Tc ~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 makes FeSe a promising material for the study of superconductivity in the 2D limit as well.
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 Tc ranges from 23.5 K to 109 K depending on the type of measurement and the experimental environment. The first indication of a high Tc of the monolayer FeSe was given by UHV-LT-STM experiments with a similar setup described in Sec. 3.1 and 3.2 [28], where degradation of FeSe layers by air exposure was avoided. Atomically well-ordered FeSe monolayer was fabricated on Nb-doped SrTiO3 substrates by optimizing the growth condition ( Fig. 20(b)). The STS measurements revealed the presence of a large gap structure with an energy gap   20 meV at 4.2 K (Fig. 20(c)), which persisted at least up to 43 K. Assuming that the relation between the energy gap  and Tc of bulk FeSe (2/kBTc  5.5) still holds, it indicates that Tc would be as high as 80 K.
This value could break the Tc record of the iron-based superconductors, 56 K, found for Sr1−xSmxFFeAs [175]. STM measurements under an out-of-plane magnetic field of 11 T revealed formation of vortices as expected ( Fig. 20(d)), from which the GL coherence length 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, Tc was found to decrease with decreasing thickness in accordance with Eq. More direct evidence for high-Tc 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 Tc, with Tc onset = 40.2 K and Tc zero = 23.5 K. The latter value is close to Tc = 21 K determined from the diamagnetic response. In accordance with the high Tc, a large out-of-plane upper critical field 0Hc2 > 52 T was indicated at 1.4 K from the magnetotransport measurements. A large critical current density Jc = 1.710 6 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 Tc of 109 K was locally detected from the emergence of the zero resistance state (see Fig. 20(e)). Accordingly, 0Hc2(0)  116 T, was deduced from magnetic field dependence of Tc. Although the samples suffered from spatial inhomogeneity and hence the high-Tc superconductivity was not found all over the surface, the result suggests that macroscopically uniform 2D superconductors with Tc > 100 K may be available if the sample fabrication process is optimized. In terms of spectroscopic measurements, ARPES is a powerful surface-sensitive tool to reveal the electronic band structure and the superconducting energy gap that should appear below Tc. All ARPES experiments reported the opening of an energy gap that amounts to 15-20 meV in the gap size at the lowest temperatures [178][179][180][181]. Tc 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 (Tc onset = 40.2K) [53] but is lower than the maximum value in the in-situ local transport study (Tc = 109K) [29]. This discrepancy may be explained considering the fact that ARPES is performed in-situ but at a macroscopic scale.
The emergence of high-Tc superconductivity driven by the atomic-scale thinning is highly unusual because Tc becomes generally suppressed with decreasing layer thickness as in Eq. (5). This certainly calls for investigations 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 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-Tc superconductivity [180]. However, it is becoming clearer from the following experiments that the 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  point and an electron-like pocket near the M point in the Brillouin zone, making FeSe a multiband superconductor.
In 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][179][180][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 Tc [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 Fig. 21) [178]. When the doping was tuned to the optimum level, a superconducting energy gap that corresponds to Tc ~ 50 K was observed for the trilayer, revealing that the monolayer was not the prerequisite for the emergence of a high Tc. The Tc enhancement in a FeSe multilayer due to the 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, Tc onset ~40 K was observed even for 10-UC-thickness FeSe multilayers when electrons were doped via gate voltage. Furthermore, the essentially 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 Tc enhancement is to work as a source of carrier doping. Nevertheless, in Ref. [178], the maximum available Tc estimated from the energy gap is dependent on the number of FeSe layers and indeed the monolayer had the highest Tc. 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 Tc up to ~50 K is possible only from substrate-induced electron doping, the presence of the FeSe-SrTiO3 interface itself contributes to the additional Tc 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-Tc superconductivity in this system.

Monolayer organic conductor and heavy-fermion superlattice
In this final subsection, we will treat 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 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 worthwhile to be introduced here.
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 whether 1-UC-thick layer (monolayer) of (BETS)2GaCl4 can become superconducting or not, as in the case of cuprates discussed in Sec. 3.3. Such an experiment was performed for monolayers of (BETS)2GaCl4 grown on a Ag(111) surface using a UHV-LT-STM (see Fig. 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   12 meV at 6 K that was attributed to the emergence of superconductivity. The Tc 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 Tc of ~ 8 K. The ratio of the energy gap to Tc, 2/kBTc, 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 dxy 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 unit. 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 insulating 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   12 meV at 4.6 K as seen in the molecular chains of Ref. [30].
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 conductors. Use of a closely related organic conductor -(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 energy gap can be safely attributed to the emergence of superconductivity.
As for rare-earth based heavy-fermion compounds, there have been no report on a 2D superconductor with an atomic-scale thickness, but superlattices made of 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 meff. In terms of superconductivity, this has a significant influence on Hc2, because Hc2 is usually determined by the orbital pair-breaking effect and its value, Hc2 orb , is enhanced due to the relation Hc2 orb  meff 2 . Indeed, 0Hc2 orb exceeds 10 T in a bulk CeCoIn5, which is well above the Pauli limit 0HP = 2/gB (: superconducting energy gap, g: g- factor; see Sec.3.2). Therefore, Hc2 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 Sec. 3.2) [92,100].
The crystal of CeCoIn5 has the 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 Ref. [31], CeCoIn5(n)/YbCoIn5(m) superlattices (n, m: number of UC 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, Tc 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 Fig. 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 Tc 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 finding here was that Hc2, measured in both outof-plane and in-plane directions, did not decrease from the bulk values in proportion to the reduction in Tc. Since Hc2 can be identified with the Pauli limit HP here and HP is proportional to the superconducting energy gap , the ratio / kBTc must be enhanced. The estimated ratio of / kBTc > 10 is much larger than the BCS value of 3.54, suggesting the realization of an extremely strongcoupling superconductor.
Furthermore, anomalous behaviors were observed in the angular dependence of Hc2(), where  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 Hc2() obeys the 2D Tinkham model (see Eq. (9)) as far as the orbital pair-breaking effect is dominant.
However, when Hc2() 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 Eq. (8)). In this case, the anisotropy of superconductivity reflects that of g-factor in the Zeeman term. In Ref. [195], for n = 4,5 (n: number of UC for the CeCoIn5 layers), the angular dependence of Hc2() was expressed by the 3D anisotropic GL model in a wide temperature range (see the lower panel of Fig. 23(b); the variation around  = 0 is smooth). This was ascribed to the strong Pauli paramagnetic effect in CeCoIn5. For n =3, however, the angular dependence of Hc2() at T = 0.8 K (Tc = 1.04 K) was described by the 2D Tinkham model (see the upper panel of Fig. 23(b); the variation around  = 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 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 (m  m') [196].

Summary
In the present review, we have seen that the existence of 2D superconductors with atomic-scale thicknesses have already been established in a wide range of fields and that they are now under extensive investigations from different viewpoints. The relevant superconducting materials encompass almost all categories to think of, from simple elemental metals on semiconductor surfaces, graphene, transition-metal chalcogenides, and FeSe to more complex compounds and molecules such as cuprates, perovskite oxides, rare-earth based heavy-fermion systems, and organic conductors. This means that the degree of electron correlation, which is one of the key factors in the modern solid-state physics and materials science, ranges from the weak to the strong limit. In terms of experimental technique, the studies of 2D superconductors feature a large variety as well. For example, LT-STM/STS and ARPES are powerful tools to directly probe the superconducting energy gaps of atomically thin superconductors exposed to the surface. Their usage is not limited to the studies treated here, but they surely have great advantages because of their surface sensitivity. Highly advanced MBE and PLD techniques allow researchers to grow different kinds of heterostructures in a layer-by-layer fashion and to fabricate atomically well-defined interfaces and superlattices with ideal 2D characters. Two recent technical breakthroughs, the EDL-FET device and the mechanical exfoliation of atomic sheets, have also greatly activated and broadened the present topics. Undoubtedly, the state-of-the-art nanotechnology including the above examples has been essential for the studies on superconductors in the 2D limit and will surely continue to drive their developments in the coming future.
New phenomena and physics revealed in these systems are so diverse and significant that they are also worthwhile to repeat here. Contrary to the general belief that superconductivity is fragile and must be suppressed as the material thickness approaches the atomic-scale limit, many 2D systems have been found to exhibit robust superconductivity at low temperatures as far as the structural and compositional quality of the sample is sufficiently high. This promises practical applications of these materials and devices in future. In some cases, the observed Tc is comparable to or even higher than that of the bulk counterpart, and particularly for FeSe monolayers grown on SrTiO3 substrates, Tc is surprisingly high. Its origin is now under hot debate, but it is clear that the interface (or charge transfer through the interface) rather than the atomic-scale thickness itself plays the essential role. This may be called an interface-induced high-Tc superconductor, but the ultimately small layer thickness helps Although the studies on 2D superconductors with atomic-scale thicknesses have already been developed to a very high level, further important progress is sure to come in the next decade. This will be brought not only by focusing on individual topics described in Sec. 3, but also by crossing their boundaries, i.e., by promoting the overlap among the categories shown in Fig. 1. Along the horizontal axis, different materials may be combined to fabricate 2D hybrid systems, for example, using organic molecules and metal atomic layers on semiconductor surfaces. In the vertical direction, different experimental techniques may be combined. For example, surface atom layers treated only in UHV so far may be investigated using an EDL-FET device after an appropriate protection process. The 2D superconductivity at the buried interface may be exposed to the surface for the purpose of LT-STM and ARPES studies. This certainly calls for an interdisciplinary approach involving material science, surface science, and device physics. In terms of new phenomena and physics, finding the topological superconductivity will be one direction to take, since combination of the conventional (BCS-type) superconductivity and the Rashba effect is one of the most promising methods to realize it [197]. The two dimensionality is also a key factor in this respect, in view of observing and manipulating the in place of topological insulator-superconductor heterostructures, for which unusual vortex core states were observed by LT-STM [200,201].