Rafael
Roldán
*^{a},
Luca
Chirolli
^{b},
Elsa
Prada
^{c},
Jose Angel
Silva-Guillén
^{b},
Pablo
San-Jose
^{a} and
Francisco
Guinea
^{bd}
^{a}Instituto de Ciencia de Materiales de Madrid, ICMM-CSIC, Cantoblanco, E-28049 Madrid, Spain. E-mail: rroldan@icmm.csic.es; Fax: +34 913720623; Tel: +34 913349000
^{b}Fundación IMDEA Nanociencia, C/Faraday 9, Campus Cantoblanco, 28049 Madrid, Spain
^{c}Departamento de Física de la Materia Condensada, Condensed Matter Physics Center (IFIMAC) & Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
^{d}Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK

Received
28th March 2017

First published on 22nd June 2017

This tutorial review presents an overview of the basic theoretical aspects of two-dimensional (2D) crystals. We revise essential aspects of graphene and the new families of semiconducting 2D materials, like transition metal dichalcogenides or black phosphorus. Minimal theoretical models for various materials are presented. Some of the exciting new possibilities offered by 2D crystals are discussed, such as manipulation and control of quantum degrees of freedom (spin and pseudospin), confinement of excitons, control of the electronic and optical properties with strain engineering, or unconventional superconducting phases.

In more practical terms, the all-surface nature of 2D crystals exposes them much more directly to influences of all sorts from the environment. As a consequence, their electronic and optical properties can be tuned with a particularly high degree of flexibility. Parameters associated to the electronic structure of crystals, like effective masses, Fermi energy, Fermi velocity or band gap, can be efficiently tuned by controlling the number of layers, by chemical functionalization, by gating or by applying strain to the samples or the substrate on which 2D materials are deposited. As compared to other 2D systems such as conventional thin films, 2D crystals also exhibit a much higher quality and overall coherence, as their strong covalent in-plane bonds allow them to keep disorder under control as their thickness is reduced.

Graphene is the best studied among 2D materials, with recognised hallmark properties that make it particularly attractive, both from a fundamental viewpoint and because of its potential applications.^{2} However, it lacks a band gap, which is necessary to switch between insulating and metallic states, making graphene unsuitable for some electronic devices. Furthermore, a band gap in the visible or infrared range of the spectrum is also required for solar cell and telecommunication applications. Consequently, significant efforts have been devoted to identifying possible 2D semiconducting crystals.^{3} Several classes of layered compounds have received attention recently, including hexagonal boron nitride (h-BN), silicene, MoS_{2}, black phosphorus (BP), etc. Single layers of h-BN are stable insulators with a large gap. Conversely, silicene is highly unstable because single layers can react with air. Among the stable 2D crystals with semiconducting behaviour, some of the best known are transition metal dichalcogenides (TMDs),^{4} which also exhibit strong spin–orbit coupling (SOC) effects. BP is another layered crystal that has been recently synthesized in its single layer form, also known as phosphorene. BP is a stable allotrope of phosphorus and an elemental semiconductor, with a high degree of anisotropy in its electronic and optical properties.^{5} Much less understood are other families of transition metal trichalcogenides like TiS_{3}, single layer Sb (antimonene) or monochalcogenides like GeSe, that have been recently synthesised in their single layer form, joining the growing catalogue of 2D materials. Herein, we review current knowledge on the physical properties of graphene and related 2D crystals. We survey their main electronic and structural features and provide minimal theoretical models that capture their low energy physics. Finally, we discuss some of the novel possibilities afforded by 2D crystals, including spintronics and valleytronics, control of excitons, strain engineering and new aspects of superconductivity.

• Massless carriers: the characteristic linear dispersion relation of electrons in graphene in the vicinity of the Dirac points makes them behave as relativistic quasiparticles with zero effective mass (see Section 3). As a result, the electronic and optical properties of graphene are completely different from those of a standard 2D electron gas with a massive parabolic dispersion relation, as e.g. Si and GaAlAs heterostructures. When graphene is exposed to a strong magnetic field, the massless character of the carriers manifests in an unconventional quantum Hall effect.^{7} Graphene electrons can, moreover, propagate over large (micrometers) distances without scattering due to the chirality of an internal degree of freedom of carriers known as pseudospin.

• High stiffness and impermeability: graphene is simultaneously flexible and extraordinarily rigid, with the highest elastic constants ever measured in any material. It can be stretched elastically up to ∼20% without rupture. Graphene (as other 2D membranes) presents a negative thermal expansion coefficient: it shrinks with increasing temperature, due to elastic properties that are dominated by out-of-plane flexural phonons.^{8} Despite its one-atom-thickness, graphene is highly impermeable to gases.

Some of these attributes are shared by other 2D crystals that have been exfoliated after graphene.

• Thickness dependence of the nature of the gap: TMDs have an electronic band structure which strongly depends on the number of layers. Single layer TMDs are direct gap semiconductors, with a gap (∼1.9 eV) located at the K and K′ points of the hexagonal BZ. The energy of the band gap lies in the visible range of the spectrum. Multilayer samples are indirect gap semiconductors (of ∼1.3 eV gap) with the maximum of the valence band at Γ and the minimum of the conduction band at an intermediate point between Γ and K.

• Multi-orbital character: TMDs have valence and conduction bands with a rich and complex orbital character. The edge of the valence band is formed by a mixture of d_{x2−y2} and d_{xy} orbitals of the metal M, and p_{x} and p_{y} orbitals of the chalcogen atom X. The edge of the conduction band, on the other hand, is formed by a combination of d_{3z2−r2} of M, plus some minor contribution of chalcogen p_{x} and p_{y} orbitals.

• Strong spin–orbit coupling: TMDs present a strong SOC which, together with the lack of inversion symmetry, leads to a splitting of the valence band of ∼140 meV (for Mo compounds) and ∼400 meV (for W compounds). The conduction band is also split by a few tens of meV.

Other characteristics of TMDs that we will discuss in Section 4 include the coexistence of spin and valley Hall effects, the possibility of tuning the band gap by different means, like strain engineering or application of external electric fields, generation of highly stable excitons (electron–hole pairs bonded by Coulomb interaction), or the emergence of superconductivity in highly doped samples.

While the most commonly studied TMDs in the single-layer form are those with TMs from group VI-B and are semiconducting, those with TMs from groups IV-B and V-B have also been studied. The ones from group V-B with trigonal prismatic coordination are metallic.^{10} The change from semiconductor to metallic phase can be easily explained, since group V-B TMs have one valence electron less than the TMs from group VI-B and, therefore, do not have enough electrons in the unit cell to occupy the valence band like the group VI-B TMDs. The most studied TMDs from group V-B are NbS_{2}, NbSe_{2} and TaS_{2}. These materials present electronic correlation phenomena such as superconductivity and even, in the case of NbSe_{2} and TaS_{2}, a charge density wave phase that can compete with the superconducting state. Those of group IV-B are semimetals and, for example, doped TiSe_{2} exhibits a charge density wave as well as superconductivity.

• Evolution of the band gap with thickness and applied strain: the gap in single-layer as well as in multi-layer samples is direct and located at the Γ point of the BZ. However, while the energy of the gap in bulk BP is ∼0.3 eV, its value increases with decreasing the layer number to ∼1.5 eV for a single layer phosphorene. For a given sample thickness, the band gap is extremely sensitive to external strain.^{12} Therefore, BP provides a high feasibility for photonics and optoelectronics devices that can operate at different frequencies. BP is, in this sense, complementary to some of the most studied 2D crystals, namely graphene and TMDs, that have band gaps ranging from zero in graphene and the ∼2 eV in TMDs (∼2.5 eV if we include correlation effects).

• In-plane anisotropy: the peculiar puckered structure of BP layers leads to highly anisotropic optical and electronic properties in-plane. The non-isotropic band dispersion also yields highly anisotropic excitons and plasmons.^{13}

• Semiconductor-to-semimetal transition: thanks to the strong response of BP to electric and strain fields, it is possible to drive a semiconductor-to-semimetal transition in this material with the appearance of a pair of Dirac-like cones in the spectrum, similarly to graphene.^{14,15} Such transition is accompanied by a change in the topology of the system, due to generation of ±π Berry phases around the Dirac points.

For larger-scale nanostructures and devices, inaccesible to ab initio approaches, the method of choice is typically tight-binding (TB), which occupies an intermediate level of computational complexity. The TB models for these systems can become moderately complicated, and may include many different crystal fields (onsite energies) and inter-orbital overlaps (hopping amplitudes) beyond nearest neighbors. All these parameters are often tuned to match the electronic structure of ab initio calculations. There are two main ways to do this: on the one hand, one may choose a set of relevant atomic orbitals in the unit cell of the material, fix the range of hoppings to consider and derive a general Slater–Koster model for the system, parametrized by a set of overlap integrals and crystal fields.^{28} These can then be fitted to match the ab initio band structure as closely as possible, potentially taking into account also the orbital character of the bands at different k-points. Such approach produces a sensible Slater–Koster TB model that allows in particular to incorporate the effect of strain in a natural way, by modifying hopping amplitudes t^{αβ}_{ij,0} between orbitals α and β as

(1) |

Fig. 2 MoS_{2} band structure comparison using tight-binding (red) and relativistic density functional theory (black). Panel (a) shows a fit of a Slater–Koster model up to next-nearest-neighbors, adapted with permission from ref. 26. Panel (b) shows a Wannierization calculation with hoppings up to sixth-order neighbors, adapted with permission from ref. 27. |

The final level of the model hierarchy consists of analytical approaches. These are effective models useful to describe in a transparent way the essential physical mechanisms at play in a 2D crystal without aiming for a quantitative description. The relevant model for a given crystal depends on the specific space symmetries of its lattice. The most archetypal of these, relevant for honeycomb lattices (mono- and dichalcogenides, h-BN, graphene) is the massive Dirac equation in two dimensions, in effect a k·p extension of the effective mass approximation of semiconductors. Honeycomb crystals typically possess a band structure with a gap centred at two identical ± valleys located at the K and K′ points in the BZ. Graphene is a special case, with gapless valleys, see red cones in Fig. 3a. The idea is to consider a neighborhood of the band structure around the two valleys, neglecting all but the (spinful) valence and conduction subbands. Expanding these to second order in wavevector = (k_{x}, k_{y}) around K and K′ and imposing invariance under the C_{3} symmetry group of the lattice (120° rotations), we arrive at a valley-degenerate Hamiltonian of the form H = H_{+}⊕H_{−}, where each τ = ± valley is described, in the absence of strains, by

(2) |

The gap is E_{g} = 2m_{0}, v is a velocity and α and β control the valence and conduction effective masses. The H_{τ} matrix above is expressed in the basis of = 0 valence and conduction states, which define a ‘pseudospin’. The = 0 conduction band state (‘pseudo-spin-up’) becomes mixed with the = 0 valence band state (‘pseudo-spin-down’) due to the off-diagonal ħv(τk_{x} ± ik_{y}) terms, which makes pseudospin a non-trivial degree of freedom in any scattering (i.e. -changing) process. The pseudospin should not be confused with the real electron spin s_{z} that enters the SOC term H_{so}, which reads

(3) |

The effect of a strain field ε_{ij} at low energies can also be derived analytically.^{30} Its dominant contribution enters as a valley shift in graphene → − ητ or as a deformation potential H_{τ} → H_{τ} + η′ in other gapped honeycomb crystals. Here, = (ε_{xx} − ε_{yy}, −2ε_{xy}) is a pseudo-gauge vector field, expressed in terms of the strain tensor components, ≈ ε_{xx} + ε_{yy} + (ε_{ij}^{2}) is a scalar and η and η′ are parameters that control the strength of the electromechanical coupling.

Graphene's low energy spectrum is thus composed of two isotropic massless Dirac cones (the α, β corrections in eqn (2) can be neglected for most purposes, and spin–orbit is also negligible due to the low atomic mass of Carbon). The massless Dirac spectrum exhibits pseudospin chirality and a unique scale invariance ( → Λ and ε → ε/Λ), responsible for many of graphene's remarkable electronic properties. Amongst these are an absence of Anderson localization (provided disorder is smooth on the scale of the lattice spacing), the associated Klein tunneling phenomenon, or a logarithmic Fermi velocity renormalization from interactions.^{2} One of the most remarkable consequences of graphene's scale invariance is the way strain couples to electrons. As mentioned above, a given (possibly position dependent) strain tensor ε_{ij} applied to the system enters the effective massless Dirac model as an effective pseudogauge field that shifts the position of the two Dirac cones in opposite directions, see green cones in Fig. 3a. If a sufficiently strong uniaxial strain is applied along the armchair direction, it can in principle have the dramatic effect of fusing the two Dirac cones at the Γ point of the BZ. This possibility requires unrealistic deformations beyond graphene's point of rupture, but signals a possibility that is actually materialized in a different 2D crystal: phosphorene.

This process can actually be reversed using strain. If we apply uniaxial compression to phosphorene along the armchair direction, |t_{1}| is increased, while t_{2} slightly decreases, so that the quasiparticle gap (E_{g} ≈ 1.8 eV using DFT-GW at equilibrium) decreases (increases) by a huge ∼6% per 1% of uniaxial compression (expansion). For a sufficient compression, we may reach t_{2} ≤ −2t_{1}, which corresponds to a gapless spectrum. This situation may be achieved in practice for multilayers,^{25} which then transition into a semimetallic phase with shifted Dirac cones.

The large atomic mass of the metallic species endows these materials with a very sizeable spin–orbit coupling. This leads to a significant splitting of the valence bands (the splitting in the conduction band is quite smaller). The corresponding spin structure around the gap is sketched in Fig. 3b. The opposite spin-polarization of degenerate states in opposite valleys opens unique opportunities for spintronics in these materials, as discussed below.

Fig. 4 (a) Schematic representation of the single-particle energy bands, with an exciton state at energy E_{b} below the gap E_{g}. (b) Schematic representation of the particle-hole excitation spectrum. Excitons within the light cone can decay radiatively and are thus “bright”, see ref. 33. (c) Excitons in phosphorene from ref. 34. Top: Schematic plot showing the measured ground-state exciton energy (red) and the energy corresponding to the quasiparticle band edge (blue). Bottom: Calculated optical absorption from first-principles of monolayer black phosphorus with e–h interactions (excitonic absorption, red curve) and without e–h interactions (quasiparticle absorption, blue curve). (d) Photoluminescence spectra (red, green, blue and purple curves) and differential reflectance spectra (grey curves) from ref. 35 for a variety of TMDs exhibiting a number of different excitonic peaks. Adapted from ref. 33 with permission of the American Physical Society [panel (a)], from ref. 34 and 35 with permission from Nature Publishing Group [panels (b) and (c)]. |

The typical quantities that describe an exciton are its binding energy, E_{b} ≡ E_{g} − E_{ex} and its size, parametrized by the exciton radius, a. In a few materials where the Coulomb interaction is very strong, as is the case of fullerenes, small excitons of a size comparable to the lattice constant may form, which are known as Frenkel excitons. These have binding energies of the order of 1 eV. In the most common cases, the semiconductor dielectric constant is large and electric field screening tends to reduce the Coulomb interaction, producing bigger exciton radii, smaller binding energies of the order of 10–100 meV, and longer exciton lifetimes. These are called Wannier–Mott excitons.

To find the exciton solution exactly is certainly complicated, but can in principle be done by solving the so-called Bethe–Salpeter equation for excitons. In many cases, though, we can perform a series of simplifications valid for most common semiconductors. First, we consider that the electron that is excited into the conduction band can be simply described as a quasiparticle with mass m_{e} as obtained by the conduction band minimum. Its interaction with the rest of the electrons of the valence band is replaced by its interaction with a hole with opposite charge and mass m_{h} given by the valence band structure at its maximum. This is known as the effective mass approximation. Second, we consider that different e–h pairs are so far apart (in space or in time), that they can be considered independent. These two assumptions allow us to neglect exchange and correlation contributions to the Coulomb interaction. The two-particle problem is then solved by going to the center of mass and relative coordinate reference system.

The wave function of the resulting bound state is said to be hydrogenic because it is similar to the one of the hydrogen atom. However, the Coulomb interaction in a typical 3D semiconductor V_{C}(r) = − e^{2}/(4πε_{0}εr) is screened by all other electrons in the valence band. This effect is captured by a dielectric constant ε ≫ 1. The effective masses m_{e} and m_{h} are, moreover, much smaller than the free electron mass m. This results in an exciton binding energy E_{b} much smaller than the Rydberg energy, and an exciton radius a much bigger than the hydrogen atom,^{36}

(4) |

Now, in a 2D crystal, the electron and hole are confined to move in lower dimensions, thus increasing their attraction. Constrained to 2D space, the same Coulomb interaction V_{C}(r) translates into a bigger E^{2D}_{b} = 4E^{3D}_{b} and a smaller a^{2D} = a^{3D}/2. This is a rather rough simplification of the problem, however, that neglects the different screening between the 2D material and its surrounding environment. Keldysh showed that in thin films of thickness d, the actual interaction V_{K}(r) between charges at distances bigger than d is influenced by the dielectric constant of the medium surrounding the film, ε_{1} and ε_{2} (typically the substrate and vacuum). Solving the electrostatic problem with the use of image charges,^{37} he obtained V_{K}(r) = R_{∞}(πa_{0}/r_{0})[Y_{0}(r/r_{0}) − H_{0}(r/r_{0})], where = (ε_{1} + ε_{2})/2, r_{0} = dε/(ε_{1} + ε_{2}) is an effective screening length and Y_{0} and H_{0} are second-kind Bessel and Struve functions, respectively. This potential presents a logarithmic divergence for r → 0 (like the potential of a charged string) and reduces to the unscreened Coulomb potential at large distances,

(5) |

For excitons larger than r_{0}, the binding energy and the exciton radius have the same form as E_{b}^{2D} and a^{2D}, but with ε replaced by (note that = 1 for the 2D dielectric in vacuum). In the opposite limit of small excitons, the e–h pair experiences mostly the logarithmic part of the Keldysh interaction, and^{39}

(6) |

The excitons influence the optical properties of semiconductors and can be detected in experiments. One such property is the optical absorption, which at low energies is the conversion of a photon into an exciton. Due to conservation of energy and momentum, this process occurs at points in momentum space where the photon light cone overlaps with the exciton dispersion surface, see Fig. 4b. The existence of excitons lowers the threshold of photon absorption, which shows peaks at the energies of different internal states of the exciton, see Fig. 4c for the case of phosphorene. Said absorption may be measured through the differential reflectance ΔR/R,^{35} which thus exhibits signatures from different internal exciton states that may decay radiatively or non-radiatively, see Fig. 4d. It is also possible to measure the opposite phenomenon, the material's photoluminescence (PL). In PL, the electrons in the sample are excited electrically or optically and, after some energy loss (relaxation), they recombine and return to the ground state radiatively, i.e. by emitting light. Fig. 4d shows different PL spectra for a series of TMDs.

Generation of pseudo-magnetic fields.
As discussed in Section 3, when graphene is subjected to strain fields, the modification of hopping amplitudes between carbon atoms gives rise to effective gauge fields, whose effect is similar to that of a magnetic field applied perpendicular to the graphene plane, albeit opposite in different valleys.^{42} Therefore, strain engineering can be used to discretize the graphene band structure into a set of pseudo-Landau levels, corresponding to counterpropagating cyclotron orbits in opposite valleys. This effect has been observed experimentally and pseudo-magnetic fields exceeding 300 Tesla have been reported in trigonally distorted graphene nanobubbles.^{43} MoS_{2} and related TMDs have, like graphene, an hexagonal lattice structure. However, the simple pseudo-magnetic field picture of strained graphene does not carry over to deformed monolayer TMDs. Due to the complex orbital contributions leading to the formation of the valence and conduction bands, we have seen in Section 3 that the effect of strains in the low energy model is dominated by different terms than in the massless Dirac Hamiltonian relevant for graphene. As a consequence, not only one gauge field but several pseudo-vector potentials and scalar fields appear in strained single layer TMDs.^{30} The inclusion of strain-displacement relations from valence force-field models leads to additional correction to the strain-induced pseudo-magnetic fields.^{44}

Direct-to-indirect gap and semiconducting-to-metallic transitions.
The outstanding stretchability of 2D semiconductors can be used to tune the size and the nature of their band gap. While single layer TMDs are direct gap semiconductors, uniaxial strain can produce a shift of the band edges and drive a transition to an indirect gap. The opposite trend takes place in group IV-B monochalcogenides, which are indirect gap semiconductors that can become direct gap crystals under tensile strain. Controlling the nature of the gap with strain (straintronics) may become a powerful strategy for photonics applications. One may envision combining dark and bright regions for photoexcitation within the same sample by tailoring substrate-induced tensions. Furthermore, by applying higher amounts of strain, but still below the fracture limit, it is possible to drive a semiconducting-to-metallic transition. Such a huge modification of the gap in 2D semiconductors (from ∼1.9 eV to 0 eV for the case of MoS_{2}) has to be compared with the poor tunability of 3D semiconductors like silicon of only ∼0.25 eV under ∼1.5% of biaxial strain.

Strain induced funnel of excitons.
Strain engineering can be used for the creation of a broad-band optical funnel for excitons in semiconducting 2D crystals.^{45} By continuously changing the strain across a sheet of MoS_{2} or black phosphorus, for example, a continuous variation of the optical band gap is produced, allowing for the capture of photons with different energies. Furthermore, it is possible to tune the strain profile so as to drive the photogenerated excitons towards the regions of minimal gap, creating a funnel for excitons.^{46} The funnel effect in BP is much stronger than in MoS_{2} and of opposite sign.^{33} While excitons in MoS_{2} are driven isotropically towards regions of maximum tension, excitons in BP move away from tensile regions. This difference stems from the fact that MoS_{2} and BP respond differently to the application of strain: the gap in MoS_{2} (BP) is reduced (increased) with tensile strain. Furthermore, the exciton drift in MoS_{2} is isotropic, while the funnelling in BP is highly anisotropic, with much larger drift lengths along one crystallographic (armchair) direction (see Fig. 5). The inverse funnel effect can be beneficial for manipulation and harvesting of light, and in particular for the design of more efficient solar cells.

Fig. 5 Strained induced funnel effect of excitons in 2D crystals. An indenter creates an inhomogeneous profile of strain in single layer MoS_{2} (a) and phosphorene (b). The strain gradient modulates the gap of the two crystals as sketched in the bottom panel. Photogenerated excitons (green arrows) are pushed isotropically towards the indenter center in MoS_{2} (funnel effect), while they are pushed anisotropically away it in black phosphorus (inverse funnel effect). Adapted from ref. 33 with permission of the American Physical Society. |

Piezoelectricity.
A large number of 2D crystals lack inversion symmetry. This gives them piezoelectric properties, that is the potential ability to convert mechanical to electric energy. Stretching or compressing a piezoelectric crystal generates an electrical voltage. Conversely, an applied voltage produces expansion or contraction of the crystal. MoS_{2}, which is centrosymmetric in its 3D bulk configuration, has been shown to become piezoelectric when it is reduced to its monolayer form.^{47} Recently, single layer monochalcogenides have been predicted to show an anomalously strong piezoelectric response, with piezoelectric coefficients that can be two order of magnitude larger than those in MoS_{2}. Furthermore, due to their anisotropic puckered lattice, the piezoelectric properties are strongly angle-dependent.^{48} Considering the rapid advances in nanofabrication techniques, their amazing elastic properties and the possibility to withstand large strains, 2D piezoelectric crystals are viewed as promising platforms for applications in nano-sensors or portable electronic devices for energy harvesting.

Fig. 6 (a) Experimental variation of the resistance vs. temperature for different number of layers of NbSe_{2}. Shown in the inset is the dependence of T_{c} with the number of layers. Adapted with permission from ref. 49. (b) Experimental tunnelling differential conductance dI/dV of a superconducting monolayer of Pb for different representative tip positions (red and blue solid lines refer to regions of high and low coherence peak, respectively, and the black solid line is the BCS best fit). The dashed orange line corresponds to the theoretical zero-temperature dI/dV spectra of a BCS superconductor, proportional to the density of states (DoS). The DoS is zero within the gap Δ and exhibits pronounced coherence peaks at ±Δ. Adapted with permission from ref. 50. |

The celebrated Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity (SC) explains the phenomenon as a many body phase-coherent state, where electrons with opposite spin and momentum pair up across the Fermi surface via a phonon-mediated attractive interaction. This pairing leads to the formation of new elementary entities, the so-called Cooper pairs, that Bose-condense to form a new ground state with a finite excitation gap Δ that reflects the binding energy of Cooper pairs. The BCS theory was a tremendous success, and was able to explain and predict all the essential properties of many 3D superconductors.

Thermal fluctuations break Cooper pairs apart. This makes the gap temperature-dependent, with Δ(T = 0) = Δ_{0} and Δ(T_{c}) = 0. For T < T_{c}, the coherence length ξ(T) ∝ 1/Δ(T) measures the size of the Cooper pairs. At T = T_{c}, ξ → ∞ and the Cooper pairs unbind. According to Anderson theorem, T_{c} for s-wave SC is not affected by weak conventional disorder due to time-reversal (TR) invariance. A magnetic field breaks TR in a way that makes states of opposite spin and momentum on the Fermi surface no longer degenerate. For low fields the non-dissipative Cooper pair motion screens the external field completely. The magnetic flux is thus completely expelled from the superconductor, which behaves as a perfect diamagnet. When the magnetic field is increased beyond a critical value, H_{cr}, SC is destroyed. The value of H_{cr} corresponds to a magnetic length for Cooper pairs equal to their size ξ, H_{cr} = Φ_{0}/(2πξ^{2}), with Φ_{0} = h/2e the superconducting flux quantum.

When lowering the dimensionality of the system, these well-defined results must be revised. The density of states (DoS) at the Fermi level decreases, and so does the density of electrons available for pairing, and consequently T_{c}. In a 2D material with N layers, the BCS theory predicts that T_{c}(N) =T_{c}exp(−1/(Uρ_{0}N)) (Cooper law), with ρ_{0} the single layer DoS at the Fermi level, and U the pairing interaction strength. For a quasi-2D slab of thickness d ≪ ξ, the in-plane critical field changes to H_{cr,‖} ∝ Φ_{0}/(2πξd), so that the system may support much higher in-plane fields than the 3D case. In the extreme 2D limit d → 0, H_{cr,‖} formally diverges. In this case, a second critical field appears, that is related to the pair-breaking action of the applied field via Zeeman spin polarization. Thus, the critical field in a strictly 2D BCS superconductor is the Pauli paramagnetic limit, , with μ_{B} the Bohr magneton. Above the Pauli field H_{p}, the Zeeman splitting of the Cooper pairs compensates the energy gained from creating the BCS condensate, and 2D SC is suppressed.

The Bose condensation of Cooper pairs into the same state implies that the number of pairs (and hence of electrons) ceases to be well-defined. It follows that the BCS ground state is a superposition of states with different number of particles. The quantum uncertainty on the number of particles fixes the phase of the BCS ground state. It is thus said that SC spontaneously breaks the global gauge symmetry of the system, and develops a complex order parameter Δ(r) = Δ_{0}e^{iϕ(r)}. In this language, the spectral gap is given by the average |〈Δ〉|. Thermal fluctuations of ϕ are responsible for the suppression of the gap with temperature. In a 3D superconductor below T_{c}, the phase ϕ(r) fluctuates around a well-defined average that is fixed across the entire sample at equilibrium, so that the gap is finite and SC remains stable against fluctuations.

In 2D systems a dramatic change takes place. Thermal fluctuations in this case have a stronger effect, and destroy the long-range rigidity of the phase across the system. The non-local correlation function 〈Δ*(r)Δ(r′)〉 = Δ_{0}^{2}〈e^{−iϕ(r)}e^{iϕ(r′)}〉, which remains finite at long distances |r′ − r| ≫ ξ in 3D, is in contrast suppressed in 2D, reflecting a lack of long-range superconducting order,

(7) |

The fundamental arguments sketched above suggest that 2D systems should be host to particularly non-trivial superconducting phases. Superconducting thin films have been the subject of intense investigation during the last decades of the past century and most of their basic superconducting properties have been unveiled. It has been found that, as their thickness is reduced, thin films usually exhibit disordered structures, mostly amorphous and granular, that do not favor SC, or that show unusual behaviours, such as strong spatial fluctuations of the quasiparticle peaks (see Fig. 6b). In contrast, 2D crystals fabricated using methods such as exfoliation are very pure and clean down to atomic scales. They thus allow us to pursue 2D superconductivity into new territory. The newly emerging field of 2D superconductors^{51,52} is now actively exploring the fundamentally new physics in these systems and their possible applications. We now present an overview of some of the recent experimental achievements in this domain with simple theoretical explanations.

T
_{c}
vs. thickness.
Besides an overall decrease of T_{c} with sample thickness, experiments in thin slabs of Pb^{52} have revealed oscillations of T_{c} with the number of layers. This phenomenon is explained as follows. As the thickness of a film is reduced to the nanometer scale, the film surface and interface confine the motion of the electrons, leading to the formation of discrete electronic states known as quantum well states. This quantum size effect changes the overall electronic structure of the film and determines oscillations of T_{c}. A much more dramatic deviation from Cooper law has been reported in a TMD layered material, TaS_{2}, for which T_{c} shows an enhancement upon reducing the number of layers,^{53} contrary to the behavior of NbSe_{2}. These findings point to an interplay between dimensionality, strong Coulomb repulsion and existence of van Hove singularities in the DoS (i.e. logarithmic divergences in the DoS associated to saddle points in the band structure of 2D crystals).

Layered SC.
Despite many theoretical predictions, superconductivity in graphene has long remained elusive. At the Dirac point the DoS is zero, so that SC is not expected at charge neutrality. The high Fermi velocity at the Fermi level also requires strong doping in order to achieve a sizeable DoS and electron density. Superconductivity has been recently reported in graphene decorated with alkali metals,^{54,55} with the dopants intercalated between effectively decoupled graphene layers. The dopants increase the electronic concentration and enhance the electron–phonon coupling. The T-dependent H_{cr,‖}(T) exhibits a positive curvature that is consistent with the behavior of superconductors made of weakly coupled superconducting layers. This system realises layered SC, in which the vortex lines move between the layers, allowing the individual layers to remain superconducting at much higher fields.

Ising SC.
We have seen that, for 2D crystals, the in-plane critical field is given by the Pauli paramagnetic limit H_{p}. In MoS_{2} and NbSe_{2},^{51} it has been shown experimentally that H_{cr,||} greatly exceeds the expected value from the Pauli limit. This behaviour has been ascribed to the strong spin–orbit coupling that arises in monolayer TMDs, see Section 2, which produces a strong spin-valley locking. The Fermi surface of this materials is formed by pockets around the K and K′ points (in the case of MoS_{2} this is achieved upon doping). As discussed in Section 3, the SOC produces an effective Zeeman field H_{SO} with opposite sign at the K and K′ pockets (see Fig. 7). The SOC spin-splits the Fermi pockets and polarizes the spin along the out-of-plane direction. Singlet superconductivity pairs electron of opposite spin across the Fermi surface, and Cooper pairs may form either as |K,↑;K′,↓〉 − |K′,↓;K,↑〉 or as |K,↓;K′,↑〉 − |K′,↑;K,↓〉, resulting in a so-called Ising pairing, as schematically depicted in Fig. 7. An external in-plane field H_{‖} tends to tilt the spin towards the plane, in competition with the out-of-plane effective SOC field H_{SO}, that tends to keep the spin aligned to the out-of-plane direction. This allows greater fields to be applied before destroying SC. The in-plane critical field may be estimated by noting that the in-plane component of the spin magnetic moment is reduced to ∼H_{‖}/H_{SO}. Pair breaking occurs when the modified Zeeman energy ∼(H_{‖}^{2}/H_{SO})μ_{B} (known as van Vleck paramagnetism) overcomes the superconducting gap. For strong SOC, an Ising superconductor therefore exhibits an enhanced upper critical field .

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