Novel phenomena in two-dimensional semiconductors

C H A P T E R

2 Novel phenomena in two-dimensional semiconductors Servet Ozdemir, Yaping Yang, Jun Yin and Artem Mishchenko School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom

2.1 Phenomena and properties on demand in two-dimensional materials and van der Waals heterostructures 2.1.1 Novel two-dimensional materials and heterostructures The tempting idea of combining the best of various ingredients in one ultimate material is a formidable task. Fortunately, the situation seems to be improving recently with the advent of van der Waals technology [1,2], which utilizes layered crystals as starting materials. Layered crystals represent a large class of materials with significant anisotropy in their structure: they have strong in-plane covalent bonds making planar crystal sheets, held together by weak out-of-plane van der Waals forces. By breaking weak interlayer van der Waals forces, individual two-dimensional (2D) layers can be detached from bulk layered crystals; this, for instance, led to the discovery of graphene, exfoliated from graphite using mechanical forces (scotch tape technique). Moreover, individual 2D layers of different layered materials can be assembled into new layered materials (referred to as van der Waals heterostructures), typically via micromechanical manipulation [1,3]. The accessibility of van der Waals technology enabled a breakthrough in the field of 2D materials and heterostructures—hundreds of novel materials have been reported during just the last few years [2]. Thanks to a combination of strong in-plane covalent bonds and weak out-of-plane van der Waals bonds in these heterostructures, the atomically sharp interfaces between the layers are typically free of defects. Surprisingly, the interfaces are also usually free from trapped contaminants, which can be attributed to a huge van der Waals pressure at the interface, which pushes mobile impurities away from the interface— either outside the crystal or in isolated pockets of trapped contaminants, leaving large, clean areas in between [2].

2D Semiconductor Materials and Devices DOI: https://doi.org/10.1016/B978-0-12-816187-6.00002-9

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2. Novel phenomena in two-dimensional semiconductors

In the seminal paper “van der Waals heterostructures,” the first library of 2D materials was compiled [1]. Very rapid progress in the field led to a drastic expansion of the 2D family. For instance, many elemental 2D materials were prepared by either exfoliation or direct growth—in addition to graphene (C), 2D sheets of borophene (B), silicene (Si), phosphorene (P), germanene (Ge), arsenene (As), stannene (Sn), antimonene (Sb), and bismuthylene (Bi) were isolated and characterized [4]. The biggest expansion probably occurred in a 2D metal chalcogenides family. Metals including Ti, Zr, Hf, V, Nb, Ta, Mo, W, Re, Pt, Pd, and Fe, and chalcogens S, Se, and Te were synthesized into binary, ternary, quaternary and quinary stable compounds using standardized chemical vapour deposition (CVD) technique [5]. An important breakthrough in the field was the introduction of 2D metal halides CrCl3, CrBr3, CrI3, and RuCl3, which led to the development of 2D magnetic materials [6
8]. Other magnetic 2D layers include FePS3, Cr2Ge2Te6, VSe2, and MnSe2 materials [9
12]. Novel materials created, using van der Waals technology, made possible the discoveries of a diverse range of exciting new physics within these heterostructures, owing to unique electronic, optical, and mechanical properties of 2D atomic crystals. These discoveries enabled many new functional devices, including tunnel transistors [13
15], light-emitting diodes [16], and photovoltaic sensors [17]. Van der Waals technology has allowed one to precisely control the properties of heterostructures by selection of materials in the stack, adjustment of the built[HYPHEN]in strain, and control of the relative layer orientations. These recent developments are outlined below and also highlighted in a series of review articles which cover many new 2D materials and their heterostructures [2,4,18
21].

2.1.2 Recent developments in the field and the concept of twistronics The electronic properties of 2D materials can be manipulated through controlling the relative twist angle between successive layers in van der Waals heterostructures, which is referred to as “twistronics” [22]. The realization of van der Waals heterostructures with accurate rotational alignment along crystal axes of individual layers is generally achieved through the accurate substrate rotation (Fig. 2.1A). Taking graphene as an example, a controllable twist angle between two graphene layers can be achieved by separating monolayer graphene flake into two sections which are sequentially picked up (“tear-and-stack” method). Between each pick-up step the substrate is rotated by a small angle that can be controlled to B0.1 degrees accuracy. The two graphene sections from the same crystal grain are expected to have rotationally aligned crystal axes, forming the moire´ superlattice [23,26]. Ribeiro-Palau et al. developed an experimental technique which can realize the in situ control over the length of the moire´ superlattice, thus allowing investigation of an arbitrary rotation angle in a single device [27]. In a twisted van der Waals heterostructure a small angle mismatch between two 2D crystals with similar lattice constants creates a moire´ superlattice with the length scale of orders of magnitude larger than that of the underlying atomic lattices. This moire´ superlattice, which forms a spatially periodic lattice potential, modifies the electron spectra of 2D systems [28
30]. Up to now the studies of twisted van der Waals heterostructures mostly concentrate on graphene placed on hexagonal boron nitride (hBN) (Fig. 2.1B).

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FIGURE 2.1 (A) Optical micrograph showing a single graphene flake, subsequently picked up with a twist angle [23]. (B) Schematic cartoon of the device in which two graphene sheets are separated by an hBN tunnel barrier where a small twist angle exists between the graphene layers [13]. (C) Schematic band structure for hBN aligned graphene where a finite bandgap of size 2Δ opens at the charge neutrality point [24]. (D) Longitudinal conductivity as a function of carrier density n and magnetic field B (up) and the Hofstadter butterfly electronic spectrum (down) calculated for the graphene-on-hBN superlattice [25]. hBN, Hexagonal boron nitride.

The graphene subjected simultaneously to both a magnetic field and the spatially periodic lattice potential, exhibits a fractal energy spectrum (known as Hofstadter’s butterfly) (Fig. 2.1D), the band structure with isolated superlattice minibands and an opened bandgap at the charge neutrality (Fig. 2.1C) [25,31,32]. In twisted layers, moire´ pattern also modulates interlayer hybridization. For example, the band structure of twisted bilayer graphene exhibits strong interlayer coupling and hybridization, which depends on the twist angle [33
36]. In a specific range of twist angles, an opened gap will appear at the Γ point of mini Brillouin zone [36
38]. Twisted bilayer graphene also shows novel chiral properties, displaying one of the highest intrinsic ellipticity values (6.5 μm21) ever reported, and a remarkably strong circular dichroism with the peak energy and sign tuned by θ and polarity, due to the large in-plane magnetic moment associated with the interlayer optical transition. At moderate magnetic fields, this twisted bilayer system, with one layer doped into electron band and the other into the hole band by the external electric fields, can host fractional and integer edge states of opposite chiralities [39] (Fig. 2.2A). Such a graphene electron
hole bilayer can be used to realize a helical 1D conductors with fractional quantum statistics. Recent studies of twisted bilayer graphene show that the twist angle of θ  1.1 degrees (the so-called magic angle) provides the right amount of band hybridization for an isolated flat band to form near the Fermi level [23,33,34,42] (Fig. 2.2B). As the Fermi velocity goes

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FIGURE 2.2 (A) Two-probe conductance of the hBN-encapsulated twisted bilayer graphene device with dualgated structure at B 5 4 T, showing broken-symmetry states. A small interlayer displacement field was applied in order to observe all the integer steps. ν tot is the total filling factor [39]. (B) Moire´ bands of twisted bilayer graphene at different twist angle [23]. (C) Four-probe resistance Rxx measured at densities ranging 20.8 3 1012 to 21.8 3 1012 cm22 versus temperature. Two superconducting domes are observed next to the half-filling state, which is labeled “Mott” [40]. (D) Left is the optical micrograph of a MoSe2 and WSe2 heterostructure. Right is the schematic of the band alignment within the MoSe2/WSe2 heterostructure depicting electron hole pairs and an interlayer exciton formation [41].

to zero, the two-particle Coulomb interaction exceeds the kinetic energy of the electrons, giving rise to strongly correlated insulating states at half-band filling and superconductivity (Fig. 2.2C) upon doping slightly away from half-band filling for hole-type carriers [40,43]. Apart from the twist angle, hydrostatic pressure can also be used to precisely tune the interlayer coupling of the heterostructures [44
46]. The increased pressure leads to closer interlayer spacing, and thus increases effective interlayer coupling strength and electronic hybridization. The magic angle is consequently modified, with its 1.1 degrees value at zero-pressure shifting to approximately 2.0 degrees under the compression of 9.2 GPa [44]. Twistronics offers another degree of freedom for dynamically tuning the electron
electron interactions of van der Waals heterostructures. It can be generalized to other 2D systems. The small-angle twisted bilayer
bilayer Bernal-stacked graphene is reported to exhibit a rich phase diagram, with tunable correlated insulating states which are highly

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FIGURE 2.3 (A) Schematic of nanoindentation on suspended graphene membrane [51]. (B) The structure lattice of borophene corresponding to geometries of triangular (up), v1/6 (bottom left), and v1/5 (bottom right) [52]. (C) The puckered structure of phosphorene [53]. (D) Graphene in-plane kirigami springs and graphene spring stretched by about 70% [54]. (E) Tensile stress, σ, as a function of uniaxial strain, ε, along the armchair (left) and zigzag (middle) directions and of biaxial strain (right), respectively, for monolayer MX2 (M 5 Mo, W; X 5 S, Se, Te) TMDCs. Solid and dashed lines are used for WX2 and MoX2, respectively [55]. (F) Volumetric Young’s modulus, and (G), the breaking strength of G and hBN, for various numbers of layers. Young’s modulus and the breaking strength of hBN and graphene monolayer are plotted as a dashed line [56]. G, Graphene; hBN, hexagonal boron nitride; TMDCs, transition metal dichalcogenides.

sensitive to both the twist angle and the electric displacement field [47]. In twisted transition metal dichalcogenide (TMDC) heterostructures the valley-polarized interlayer excitons are observed. At a small twist angle, WX2
MoX2 (where X 5 S, Se) heterostructures have a type II band alignment [48], where the Coulomb-bound electron and hole locate in different monolayers, forming the interlayer excitons, which have shown long lifetimes exceeding a nanosecond due to the spatial separation of electrons and holes which suppresses ultrafast electron
hole recombination and their exchange interaction [41,49] (Fig. 2.2D). The interlayer charge transfer conserves spin-valley polarization, which has only a weak dependence on twist angles in the heterobilayer [49,50].

2.2 Novel nanomechanics phenomena 2.2.1 Mechanical properties of two-dimensional materials In general, 2D materials have extremely high in-plane stiffness and low flexural rigidity due to their strong in-plane covalent bonds. The relatively weak van der Waals forces govern the interlayer interactions of 2D materials, determining their shear, friction, and fracture behavior. The most widely used method for studying the in-plane mechanical properties of 2D materials is nanoindentation via transverse loading on the freestanding films (Fig. 2.3A).

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2.2.1.1 Graphene Graphene is thought of as the strongest material in existence. Extracted by the force
deflection curves from nanoindentation tests, the derived 2D stiffness and strength (E2D, σ2D) of monolayer graphene are 342 and 55 N m21, respectively, corresponding to 1 TPa and 130 GPa for Young’s modulus and intrinsic strength, respectively (assuming a monolayer thickness of  0.334 nm) [57]. The experimental bending rigidity values for mono-, bi-, and trilayer graphene were found to be  7.1,  35.5, and  126 eV, respectively [58], which increases with the increasing thickness. The bending rigidity of monolayer graphene was also found to be size dependent and is considerably higher for large area graphene sheets due to thermal fluctuations [59]. The combination of extremely high stiffness with large flexibility of graphene makes it suitable for kirigami (Fig. 2.3D), which has application opportunities in flexible electronics [54]. The existence of vacancies, grain boundaries, wrinkles, and crumpling can significantly deteriorate the in-plane stiffness and strength of graphene [60
62]. However, a moderate amount and a more ordered arrangement of defects can benefit the stiffness and strength of graphene [63,64]. For example, a small amount of defects (0.2%) can increase the modulus of monolayer graphene up to 550 N m21, while more defects start to decrease the modulus [64]. The strength of graphene degrades with increasing thickness, due to finite shear strength between individual layers [65]. An intriguing finding for few-layer graphene is the room-temperature diamondization of few-layer graphene upon compression [66,67]. The resulted diamond-like layer exhibits a transverse stiffness and hardness comparable to diamond and is a ferromagnetic insulator with different bandgap energies for each spin. Experiments and calculations show that this phase transition is reversible and is not observed for monolayer or graphene films thicker than three layers. 2.2.1.2 Transition metal dichalcogenides 2D semiconducting TMDCs can withstand large deformations, making them attractive for applications in novel strain-engineered and flexible electronic and optoelectronic devices. For TMDCs the overall stress response and strength are determined by the chemical composition, and they show an anisotropic stress
strain response according to the first principles density functional theory (DFT) calculations [55] (Fig. 2.3E). Chalcogenides of tungsten exhibit larger elastic modulus and greater strength than the chalcogenides of molybdenum. For the same transition metal, sulfides are stronger than tellurides. When the strains are more than  4%, the mechanical properties along the armchair directions appear to be consistently stronger than the zigzag direction across all chemical compositions, due to broken hexagonal symmetry. The origin of these mechanical behaviors is attributed to the strain-induced redistribution of the electronic charges between the chalcogen and transition metal atoms. 2.2.1.3 Other two-dimensional materials The hBN, possessing almost identical crystallographic parameters to that of graphene, also has the potential to be extraordinarily resilient and strong. The elastic modulus and fracture strength of monolayer hBN were found to be 0.87 6 0.07 TPa and 71 6 6 GPa,

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respectively [56]. Unlike few-layer graphene, whose strength decreases with increasing thickness, few-layer hBN was as strong as monolayer hBN [56] (Fig. 2.3F and G). This is because hBN is more resistive to interlayer sliding compared to graphene layers due to the more polar bonds. Other 2D materials such as black phosphorous [68,69], borophene [70], silicene, stanene, and germanene [71] which have unique puckering or buckled structures (Fig. 2.3B and C) also exhibit highly anisotropic mechanical properties.

2.2.2 Strain engineering The high strength of 2D materials allows them to sustain large strains before fracture. The distortion of the lattice structure induced by strain changes the band structure of 2D materials, thus opening up fertile opportunities for tailoring their electronic, optical, and magnetic properties. 2.2.2.1 Strain-induced modification of electronic bands The breaking of sublattice symmetry happens when the strain is applied to 2D materials, resulting in the change in Raman and photoluminescence (PL). There is a redshift in G and 2D band of monolayer graphene (  14.2 and  27.8 cm21/%, respectively) accompanied by the splitting of G band at large strain [72], as well as the bandgap opening (300 meV/% strain) [73]. The effect of uniaxial tensile strain for MoS2 includes the phonon softening with increased strain and the splitting of E0 Raman mode at high strain, the redshift in PL which indicates the decreased optical bandgap, and the decreased PL intensity indicative of the direct-to-indirect bandgap transition [74,75] (Fig. 2.4B). Strain can also induce the structural phase transition in MoTe2 [79] (Fig. 2.4E). Similar effects also apply for other TMDCs such as NbSe2 [80], and WSe2 [81]. The strain-dependent band structure can be utilized for creating a continuously varying bandgap profile in an initially homogeneous, atomically thin membrane [77,82,83]. By making an “artificial atom” where the MoS2 monolayer has the highest strain at the top of the nanocone, a strain profile is created, which exhibits a novel 2D exciton funnel effect. In the funnel effect the photo-generated excitons drift to areas of higher strain (i.e., lower bandgap) before recombining (Fig. 2.4C). This “artificial atom” proves to be able to absorb a broad range of the solar spectrum and concentrate excitons or charge carriers. 2.2.2.2 The emergence of pseudo-magnetic fields Strained graphene in the form of bubbles (Fig. 2.4A) exhibits strong pseudo-magnetic fields (PMFs) with intensities up to hundreds of tesla [84,85]. An elastic strain not only induces a shift in the Dirac point energy from local changes in electron density but also forms an effective vector potential (an effective gauge field) that arises from changes in the electron-hopping amplitude between carbon atoms. This strain-induced gauge field can give rise to large PMFs with an appropriate geometry of the applied strain. The PMFs in graphene can be measured by analyzing the Landau levels originating from the straininduced PMFs. The PMF magnitude is dependent on the gradient of the strain according to theoretical calculations [86]. This triggers the ideas of creating uniform PMFs in graphene. Uniform

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FIGURE 2.4 (A) Atomic force micrograph of graphene bubbles of different shapes [76]. (B) Photoluminescence spectra of a monolayer MoS2 as it is strained from 0% to 1.5% [74]. (C) Schematic explaining the funnel effect due to the nonhomogeneous strain in wrinkled MoS2 sheets [77]. (D) Subject to uniaxial tension, a suitably patterned graphene ribbon can have a rather uniform and strong pseudo-magnetic field over a large area [78]. (E) Temperature-force phase diagram for semiconducting 2H and metallic 1T0 MoTe2 [79].

distribution of PMFs can be obtained by applying equal-triaxial strain to atomically thin graphene [87], bending graphene into a circular arc [88], or by a simple uniaxial stretching of graphene ribbons with suitably designed nonuniform width [78] (Fig. 2.4D).

2.2.3 Piezoelectricity of two-dimensional materials Piezoelectricity is possible only when the 2D materials meet the two criteria: a noncentrosymmetric crystal structure and a bandgap (being nonmetallic). Piezoelectricity arises from the induced effective piezoelectric charges at the sample edges as a result of the polarization of atoms in the strained crystal, which increase or decrease the Schottky barrier height at the contacts, depending on the direction of the polarization field [89]. The intrinsic piezoelectric 2D materials include hBN, many TMDCs (such as MoS2, MoSe2, and WS2), transition-metal dioxides, II
VI and III
V semiconductors (SCs) (such as AlSb, GaP, and GaAs), and III- and IV-monochalcogenides (such as GeSe and SnS).

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FIGURE 2.5 (A) Monolayer top view geometry of boron nitride (hBN) and c trigonal prismatic molybdenum disulphide (2H-MoS2) where B atoms are red, N atoms are blue, Mo (transition metal) atoms are silver, and S (chalcogen) atoms are yellow. The axes and direction of piezoelectric polarization are labeled, and the primitive hexagonal cell is highlighted in blue [92]. (B) Atomic structure of layered tri-s-triazine (g-C3N4) sheets [93]. (D) Side-view crystal structure of α-In2Se3 in the space group of R3m. The piezoelectric polarization direction is labeled as the arrow shows [94]. (E) Periodic trends for in-plane piezoelectric coefficient in metal dichalcogenides, metal oxides, and group III
V semiconductors as well as for out-of-plane piezoelectric coefficient in group III
V semiconductors [90]. (F) left, the rotation of the crystal with respect to the electric field. Right, the measured piezoelectric coupling strength (square data points) followed the cos 3θ dependence (the red fitting curve) predicted from the crystalline threefold symmetry [90].

The piezoelectric coefficient d of TMDCs obeys the periodic trend since the piezoelectricity of TMDCs originates from the change in the polarization of the atoms due to the applied strain [90,91] (Fig. 2.5E). It is proportional to the ratio of the polarizabilities of the isolated anions and cations. The chalcogenide atoms dominate the piezoelectricity in TMD monolayers, whereas the cation polarization counteracts the piezoelectric effect. Therefore the piezoelectric coefficient of TMDCs can be maximized with a larger chalcogen atom (large polarizability) and a smaller transition-metal atom (small polarizability). The piezoelectric response also shows an angular dependence in agreement with the symmetry of the crystal [90] (Fig. 2.5F). The intrinsic in-plane piezoelectric materials include hBN, TMDC, and graphene nitride (C3N4). hBN, with the boron and nitrogen atoms arranged alternately in the hexagonal vertex sites (Fig. 2.5A), is predicted to have strong in-plane piezoelectricity, with the piezoelectric coefficient ranging from 118 to 139 pC m21 [90,95]. The piezoelectricity of hBN only exists in the odd-layer crystals and vanishes in the even-layer crystals due to the compensation of the piezoelectric contributions of each layer. The piezoelectric coefficient of hBN with an odd number of layers is inversely proportional to the number of layers [96]. Similar layer number dependence has been found in other piezoelectric planar hexagonal materials such as MoS2 [89,90,97].

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The measurement of freestanding monolayer MoS2 shows the in-plane piezoelectric coefficient of 290 6 50 pC m21 [90], which is similar to the theoretical value [92]. The monolayer MoS2 strained by 0.53% has a 5.08% mechanical-to-electrical energy-conversion efficiency [89]. MoS2 monolayer does not have the out-of-plane piezoelectricity because its structure is centrosymmetric if looking into the direction parallel to the atomic plane (Fig. 2.5C). The 2D graphitic form of graphene nitride is g-C3N4, which is nonconducting and possesses intrinsically noncentrosymmetric triangular holes (Fig. 2.5B). Thus g-C3N4 is expected to have in-plane piezoelectricity [93]. Its bulk counterpart also possesses strong in-plane piezoelectricity due to the unusual atom-stacking sequence. The piezoelectricity of g-C3N4 is verified by piezoresponse force microscopy measurements, and the piezoelectric coefficients for single and a stack of layers are calculated to be 0.732 and 0.758 C m22, respectively [93]. Except in-plane 2D piezoelectric materials, there are experimentally reported out-ofplane 2D piezoelectric materials such as alpha phase indium selenide (α-In2Se3) [94], the typical III
VI semiconductor, which has the noncentrosymmetric R3m symmetry (Fig. 2.5D). Graphene is intrinsically a nonpiezoelectric material due to its perfect centrosymmetric hexagonal lattice with one type of atoms. However, when graphene is supported by substrates such as silica, the interfacial chemical interaction between the carbon atoms and oxygen atoms of silica generates nonzero out-of-plane electric dipole moment and polarization, leading to the out-of-plane piezoelectricity [98]. In addition, some special materials such as Janus MoSSe monolayer can exhibit intricate piezoelectricity by the coupling of in-plane and out-of-plane piezoelectricity [99].

2.2.4 Surface and interface properties 2.2.4.1 Friction The friction properties of 2D materials are mainly measured by friction force microscopy (FFM), also known as lateral force microscopy. The lateral torsion deflection of an AFM cantilever is detected at the lattice scale, while a tip is sliding on a sample surface in contact mode. The mechanical properties of friction force, lateral contact stiffness, and shear strength can be quantitatively assessed on the nanoscale by FFM. The friction properties of 2D materials are highly sensitive to the interactions between solid surfaces. The underlying mechanisms that influence the friction includes the wear effect during which energy is dissipated due to shear and removal of material from sliding surfaces; the thermally activated energy barriers to overcome during sliding and the energy dissipation through lattice vibrations; the formation of the chemical bonds between sliding surfaces; the electrostatic charge generation, transfer, and discharge; as well as the surface functionalization and contribution of the surrounding atmosphere during sliding [100]. Here some intriguing findings about friction are briefly introduced as follows. The friction increases with the decrease in the number of layers in many 2D materials such as graphene, MoS2, hBN, and black phosphorus [101
103]. This trend can be explained by the puckering effect [104] (Fig. 2.6A and B), where out-of-plane deformations happen when the tip slides over the material, increasing the contact area between the tip

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FIGURE 2.6 (A) A schematic showing the proposed puckering effect, where adhesion to the sliding atomic force microscopy (AFM) tip creates out-of-plane deformation of a graphene sheet, leading to increased contact area and friction. (B) The variation in friction as a function of the sheet thickness based on the finite element method (FEM) simulation. (C) Friction force images show the changing friction contrast as the sample is rotated counterclockwise from 0 to 184 degrees relative to the horizontal scan direction (red dashed arrow) [105]. (D) Normalized friction force versus rotation angle for each domain [Roman numerals indicate the three friction domains identified in (C)], showing 180 degrees periodicity. The lines show that the variations in friction can be fitted by a simple sine modulus function.

and the surface, thus leading to higher friction. The lower bending rigidity of thinner layers results in higher out-of-plane deformations. Also, thinner layers are more flexible to locally adjust their atomic configuration near the tip. All these factors lead to an increased contact area; thus the friction is strengthened with a decreased number of layers. Anisotropic friction domains are found to exist in exfoliated monolayer graphene [105] (Fig. 2.6C). These friction domains show an anisotropy with a periodicity of 180 degrees, which decreased as the applied load increased (Fig. 2.6D). The anisotropic friction in each domain comes from anisotropic puckering of the graphene, which is formed when the tip pushes the ripple crest forward along the scanning direction. The friction forces are higher when scanning perpendicular than parallel to the wrinkles because of the larger conformability between the tip and the wrinkles in the perpendicular direction [106]. The bulk analogues of lamellar materials, particularly graphite, MoS2, and hBN, are widely used in solid lubricant applications. Reduced to two dimensions, these materials, containing atomically smooth surfaces, perfectly crystalline structure, and high lateral stiffness, show the potential of superlubricity at the nanoscale. Superlubricity is observed

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during the sliding of graphene nanoflakes on a graphene surface at an incommensurate state [107], gold surfaces [108], and diamond-like carbon-coated counterface (with the friction coefficient of  0.004) [109], also between two incommensurately stacked single layers MoS2 (with the friction coefficient being in the range of 0.8 3 1024
2.6 3 1024) [110]. In all these cases the incommensurability (lattice mismatch between two surfaces) plays an important role in achieving superlubricity [111]. In the incommensurate state the lateral corrugation forces between two nonmatching, rigid crystals cancel out systematically, so that the kinetic friction is dramatically reduced at finite sliding speed. In addition, the incommensurability of periodic interfaces cancels the main source of energy dissipation, that is the unavoidable atomic scale stick-slip in the sliding motion. 2.2.4.2 Interfacial interactions For lamellar materials the individual layers are held together by the van der Waals force. It also dominates the interactions between the materials and the supporting substrate. The interaction between the 2D materials and the supporting substrate is mainly assessed by the adhesion energy. For graphene the reported adhesion energy values on different substrates are 0.1
0.45 J m22 on silicon oxide [112,113] up to 6 J m22 on seed copper foils and  3.4 J m22 for graphene cured on epoxy [114]. Values of adhesion energy greater than 1 J m22 means the existence of other interactions apart from van der Waals forces. The capillary and contamination effects may also exist in the interface when the surrounding environment contains specific gas chemistry such as water molecules. The van der Waals interactions play an important role in the interface between different atomic layers in heterostructures of 2D materials. The van der Waals epitaxial growth of 2D materials is the direct usage of these interfacial interactions [115]. It happens between the substrate and the epitaxial layer having a terminated surface without dangling bonds. Another usage, that is by stacking different 2D crystals on top of each other, one can make the van der Waals heterostructures with different materials assembled in a chosen sequence, where van der Waals forces are sufficient to keep the stack together [1]. The van der Waals interactions can also lead to the self-cleansing mechanism [116] and the commensurate
incommensurate surface reconstruction between graphene and hBN [117].

2.3 Valleytronics: exploiting valley degree of freedom Electron, as a fermion, has two intrinsic degrees of freedom, namely, charge and spin. The SC industry is currently based on the control of charge flow but faces lots of limitations that cannot be easily overcome, such as a tremendous energy dissipation especially compared to our brain. Spintronic devices that have the potential to complement or surpass the SC technologies have stimulated extensive collaboration of researchers from various disciplines and gotten notable developments. In contrast, valley, another degree of electron’s freedom, was rarely explored due to the lack of accessibility in traditional threedimensional (3D) materials. The situation changed after the isolation of 2D materials, especially monolayer TMDCs. In this section, we briefly introduce concepts and physics related to valleytronics, as well as some experimental milestones made in the way.

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2.3.1 Valley contrasting physics: the concept of valley electronics and the advantages of two-dimensional materials In momentum space, local minima in conductance band and maxima in the valence band of energy dispersion are referred to as valleys. Controlling the valley degree of freedom could lead to novel electronic applications, that is, valleytronics. Unfortunately, valleys in periodic crystals are energy degenerate, and it is usually impossible to populate or characterize charges in distinct valley states. Only in few systems, including diamond [118], bismuth [119] and aluminum arsenide heterostructures [120], valley polarization was realized but of limited access and at cryogenic temperatures. In general the main challenges of valleytronics are to realize the broken structural inversion symmetry, which is necessary for valley polarization as will be discussed later, as well as coupling between valley index and external fields, such as electric and magnetic fields. Fortunately, 2D materials with honeycomb structures show intrinsic advantages for valleytronics, for the following reasons: (1) the 2D honeycomb structure provides only two valleys, that is, K and K0 , making the problem simpler; (2) inversion symmetry of 2D materials, such as TMDCs, can be easily broken through reducing the thickness to a single layer; (3) for 2D materials which preserve inversion symmetry, such as monolayer graphene and bilayer TMDCs, the inversion symmetry can still be broken through applying electrical displacement or moire´ superlattices [24,121]; and (4) strong spin-orbit coupling (for example, in TMDCs) causes opposite signs of spin splitting at K and K0 valleys, giving rise to coupling between spin and valley pseudospin [122]. This coupling could efficiently enhance the spin and valley polarization lifetime and give more freedom for charge manipulations [123]. Berry curvature (Ω) and orbital magnetic moment (m) are the two physical quantities that can distinguish valley states. Berry curvature, if nonzero, behaves like an effective magnetic field in the momentum space, and electronic equations of motion for Bloch electrons under applied electric (E) and magnetic fields (B) are governed by semiclassical transport equations [123,124], r_ 5

1 @En;k _ 2 k 3 Ωn;k ; ¯hk_ 5 2 eE 2 e_r 3 B ¯h @k

where En;k and Ωn;k are the energy dispersion and Berry curvature of the nth band, k and r are the crystal momentum and position of the electron wave packet. The dot represents the first derivative with respect to time. The term k_ 3 Ωn;k causes an anomalous velocity va perpendicular to E. For systems showing opposite signs of Ω in different valleys, which is the case for graphene with broken inversion symmetry and monolayer TMDCs (Fig. 2.7A), electrons from two valleys will drift to opposite directions perpendicular to E, giving rise to valley Hall effect, Fig. 2.7B. For systems with both time-reversal symmetry (Ωn;k 5 2 Ωn;2k ) and inversion symmetry (Ωn;k 5 Ωn;2k ), the invariance of equations of motion under the symmetry asks for vanishing Berry curvature, forbidding valley-contrasting phenomena. Thus broken inversion symmetry is required for systems for valleytronics applications. Orbital magnetic moment (m) also appears in 2D materials with broken inversion symmetry. It is mainly concentrated in the valleys and gives rise to a valley magnetic moment

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FIGURE 2.7 Valley polarization in systems with broken inversion symmetry. (A) Energy spectrum (ε, dashed curves) of graphene with broken inversion symmetry. Its orbital magnetic moment (m) and Berry curvature (Ω) show a similar dependence on k and have opposite signs in the two valleys (solid red curve) [124]. Many monolayer TMDCs of 2H phase also show contrasting Ω and m between the K and K0 valleys, similar to the one shown here. (B) Valley Hall effect. Electrons with opposite spins drift to opposite directions under an electric field E for valleys K and K0 due to Berry curvature (Ω) induced anomalous velocity (va) perpendicular to E. (C) Valleyselective excitation through circularly polarized light [125]. (D) Splitting of valley degeneracy in magnetic fields by valley Zeeman effect induced energy shift (Δv ).

of opposite signs at K and K0 , respectively (Fig. 2.7A) [124]. By applying a magnetic field perpendicular to the basal plane of the 2D materials, valley Zeeman effect, analogous to spin Zeeman effect, gives rise to an energy difference between the two valleys and breaks the energy degeneracy of two valleys, as illustrated in Fig. 2.7D. Due to its similar behavior as spin, occupation of charges in a specific valley can be easily described by introducing an index, namely, the valley pseudospin. The contrasting m at two valleys also gives rise to valley-dependent optical selection with circularly polarized light in monolayer TMDCs. In these systems, Ω, m, and optical circular dichroism ηk are linked by ηk 5 2





mk z^ Ωk z^ e Δk 52   μBk μBk 2h ¯

where μB 5 eh ¯ =2m , Δ is the bandgap energy. Specifically, right circularly polarized (σ1) light couples to interband transitions in the K valley, and left circularly polarized (σ2) light couples to interband transitions in the K0 valley, Fig. 2.7C. Thus optical injection in a selected valley can be realized through circularly polarized light. In conclusion, contrasting Berry curvature at two valleys gives rise to distinct trajectories of charge from different valleys under electric fields. Opposite signs of orbital magnetic moment at K and K0 endow the systems with valley degeneracy splitting in magnetic fields or via optical circular dichroism. The prerequisite for contrasting Ω and m at two valleys while maintaining time-reversal symmetry is that the material structure should exhibit a lack of inversion symmetry.

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2.3.2 Recent experimental progress in valleytronics: new devices and applications Although theoretical prediction of valley Hall effect in monolayer graphene with broken inversion symmetry was made in 2007 [124], its experimental observation was challenged due to the lack of routes to break the inversion symmetry until graphene/hBN superlattice were introduced [24]. In 2014 using a Hall bar device of graphene/hBN superlattice, a notable nonlocal signal was detected (Fig. 2.8A) far away from the current injection region due to the valley Hall and its inverse effect. The nonlocal voltage can also be observed in bilayer graphene if an out-of-plane electric field is applied to break the inversion symmetry, introducing a bandgap and valley contrasting Berry curvature [121,127]. In TMDCs, such as monolayer MoS2, valley-polarized charges can be optically excited through circularly polarized optical excitation by utilizing the aforementioned valleydependent optical selection. Thus valley Hall effect can be detected by standard Hall measurements [128]. Note that the Berry curvature in the valence band and conduction band are equal but with opposite signs, giving rise to opposite drifting directions of excited electrons and holes from the same valley (Fig. 2.8B). The Hall voltage also changes sign when the helicity of the circularly polarized laser light is flipped. Besides electrical readout, the valley Hall effect has also been directly mapped out through spatially resolved Kerr signal (polarization rotation of injected linearly polarized light) of accumulated valley polarized charges along edges of a gated bilayer MoS2 flake (Fig. 2.8C) [125]. The Kerr signal arises due to the difference in the reflectance for σ1 and σ2 light from charges coming from different valleys. Besides valley Hall effect, valley Zeeman effect has also been experimentally confirmed in monolayer TMDCs [126,129
131]. Splitting of the gap energy at different valleys in a magnetic field was observed through polarization-resolved magnetoluminescence
PL in WSe2, MoSe2 and MoS2. The observed splitting consists of Zeeman shifts from spin, valley orbital angular momenta and atomic orbital angular momenta (Fig. 2.8D). However, the splitting coefficient is limited to 0.1
0.2 meV T21 (Fig. 2.8D), which hinders its applications in valleytronics, especially at high temperature. Theoretical calculations revealed that the interfacial interaction between MoTe2 and ferromagnetic EuO substrate could simultaneously induce a valley splitting of 300 meV, shedding light on valleytronics without external polarization at room temperature [132]. The direct bandgaps of monolayer TMDCs make them ideal materials for optoelectronic applications. Their excitons, that is, bound electron
hole pairs through Coulomb interaction, are of Wannier type: [133,134] wavefunction of the exciton extends over multiple unit cells, electron and hole of bright exciton are confined in the same K and K0 in the momentum space, endowing them with a binary valley pseudospin as well as valley-dependent optical selection. The exceptionally strong Coulomb interaction accompanying the atomically thin geometry gives rise to the extraordinarily large binding energy of exciton at the order of hundreds of millielectronvolts [133,135] and also various strong interaction effects [136
138]. Neutral exciton in combination with an excess electron or hole forms trion with various valley configurations, triggering intriguing physics [139,140]. However, the strong transition dipole moment presented in monolayer TMDCs causes a quite short lifetime at the order of picosecond [141], hindering its applications in valleytronics requiring long-range spatial transport.

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FIGURE 2.8 Experimental progress in valleytronics. (A) Valley Hall effect in graphene/hBN superlattices. Nonlocal signal (red curve) appears at (second) neutrality point, where Berry curvature arises. Right top inset shows the measurement circuit [24]. (B) Valley Hall effect in monolayer MoS2. Valley selectively pumped electrons and holes of opposite spins drift to opposite directions perpendicular to the electrical field through circularly polarized light, giving rise to a net Hall voltage [122]. (C) Spatial maps of Kerr rotation signal of bilayer MoS2 at different doping levels. Note that inversion symmetry is broken when electrostatically doping the flake by a single gate [125]. (D) Splitting of valley degeneracy by valley Zeeman effect. The solid and dashed lines in the left panel show the position of WTe2 bands with and without applied magnetic fields, respectively. Red (blue) color represents spin up (down) electrons. The splitting effect consists of Zeeman shifts from spin (Δs), atomic orbital angular momenta (Δa) and valley orbital angular momenta (Δv). The right panel shows the valley splitting energy as a linear function of the applied magnetic field [126]. (E) Schematic illustration of the formation of spatially separated interlayer excitons in a MoX2/WX2 heterobilayers by the optical pump. (F) Misaligned Brillouin zone of a heterobilayer with a momentum shift (ΔK) due to a twist between two layers in real space. (G) Schematic of a graphene/hBN/graphene tunneling transistor (left). The right map shows conductance dI/dV plot as a function of bias (Vb) and gate voltage (Vg). Solid lines highlight the positions of resonance peaks in the tunneling current [15]. hBN, Hexagonal boron nitride.

Stacking 2D materials into heterostructures provides another platform for exploring valleytronics. When stacking two TMDC monolayers, for example, monolayer MoX2 and monolayer WX2, the alignment of their band structures results in an atomically sharp p
n junction in the out-of-plane direction without a depletion region. It is predicted that

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type II SC junctions form at the heterointerface, facilitating ultrafast electron-hole separation [142,143]. Under light irradiation, electrons can be pumped from one layer to another layer, forming spatially separated interlayer excitons, as illustrated in Fig. 2.8E. A lattice mismatch and twist between the two layers further give rise to an indirect momentum correlation between electrons and holes. The spatial separation in combination with the momentum mismatch efficiently increases the lifetime of interlayer exciton by orders of magnitude as compared to intralayer exciton [41]. Circularly polarized light pumping could create interlayer excitons of long-lived valley polarization up to 40 ns in WSe2
MoSe2 heterobilayers [49,131]. With such a long lifetime, the exciton can expand over several micrometers. Another advantage of 2D heterostructures is that the valley configuration between two layers can be facilely tuned through their relative alignment. Their relative rotation angle in momentum space (Brillouin zone) is exactly the same as their twist angle in real space, as shown in Fig. 2.8F. Control of alignment in momentum space has the potential to enable new valley functionalities, such as interlayer valley Hall effect in heterobilayers [144]. In the case of two graphene layers separated by a thin hBN flake, the tunneling current between the two graphene electrodes shows resonance peaks if two graphenes are carefully aligned because the alignment provides conservation of momentum [15,145] (Fig. 2.8G). Negative differential conductance associated with the resonance conductance has the potential for applications in high-frequency devices [15]. Despite the short time since the first experimental investigation of valley Hall effect in graphene, exponential growth in the publications have been reported in this emerging area. Here, we briefly introduced only a tiny part of them. For more information, please refer to recent reviews [146
150].

2.4 Two-dimensional superconductivity 2.4.1 Introduction to two-dimensional superconductivity Superconductivity—a zero resistance state accompanied by perfect diamagnetism— originates from pairing of electrons into a Cooper pair state, which is a purely quantum phenomenon. Zero resistance at low temperatures in bulk metals was discovered by Heike Kamerlingh Onnes in 1911. Perfect expulsion of a magnetic field (perfect diamagnetism or Meissner effect) was demonstrated by Walther Meissner and Robert Ochsenfeld in 1933. For bulk superconductors a beginning of the theory of superconductivity can be attributed to Leon Cooper when he has shown that two electrons near the Fermi surface with mutually opposite momenta, and spins can have a bound state with negative coupling energy [151]. Works of John Bardeen, Leon Cooper, John Schrieffer, and Nikolay Bogolyubov developed this principle into a systematic theory of superconductivity (Bardeen
Cooper
Schrieffer (BCS) theory). In BCS theory the ground state of interacting electrons has lower energy as compared to the Fermi energy of electrons in the normal state. This ground state of interacting fermions is separated from the excited states by an energy gap, Δ, of the order of the coupling energy of a single Cooper pair. Lev Gor’kov realized that the transition to a superconducting state can be considered as the

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Bose
Einstein condensation of Cooper pairs and developed a Green’s function theory of this instability [152]. Vitaly Ginzburg applied a general theory of phase transitions developed by Lev Landau, to superconductors, for an introductory overview of the theory see Ref. [153]. A superconductor represents a separate thermodynamic phase of matter, and a second-order phase transition between high-temperature normal and low-temperature superconducting phases can be described in a framework of this phenomenological Ginzburg
Landau theory [154]. Central to Ginzburg
Landau theory is the notion of an order parameter Ψ(r) (Cooper pair wavefunction): Ψðr Þ 5 Δðr Þe2iϕ

(2.1)

where Δ(r) is the superconducting gap which depends on the coordinates r, and φ is the phase. |Ψ(r)|2 is the density of Cooper pairs. Interestingly, Ginzburg
Landau theory is handy for the description of thin superconducting films in the clean (disorder-free) limit, where, as first proposed by Pierre-Gilles de Gennes, the boundary condition at the surface of a thin superconducting film affects the superconducting gap [155]. Simonin performed a careful analysis of Ginzburg
Landau theory for thin films, and using the approach, developed by Gor’kov, derived the following expression for the boundary condition for the superconducting gap [156]:



rΔðr Þ s^ jS 5 2 CΔS

(2.2)

where sˆ is the normal to the surface unit vector, ΔS is the superconducting gap at the surface of the film, C  a=N0 Vξ 2 ð0Þ, a is the Thomas
Fermi screening length, N0 is the density of states of normal electrons at the Fermi surface, V is the pairing potential, and ξ(0) is the superconducting coherence length at zero temperature. While the effect of the boundary condition in Eq. (2.2) on Δ can be ignored in thick films and bulk superconductors, it dominates in the 2D limit, as can be seen from strong dependence of superconducting transition temperature TC on the film thickness d:
 dC TC ðdÞ 5 TC;bulk 1 2 (2.3) d where dC 5 2Cξ2 ð0Þ  2a=N0 V is the critical thickness at zero temperature.

2.4.2 Recent experimental progress in highly crystalline two-dimensional superconductors 2D superconductivity was studied for decades both experimentally and theoretically because 2D superconductors are associated with a broad range of interesting physics, including famous Berezinskii
Kosterlitz
Thouless (BKT) transitions and debated quantum phase transitions. In 1938 the superconductivity was reported in thin films of lead and tin (down to 5 nm thick) [157]. Both much lower critical currents and much higher critical magnetic fields, than in the bulk samples, were found. Relatively recent work (2009) demonstrated superconductivity in the extreme 2D limit on epitaxial two-atomthick lead film [158]. Transition temperature (Tc) had an interesting thickness dependence: for films above four atomic layers thick, Tc was  6.2K independent of thickness,

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four-layer thick lead shows enhanced Tc  6.7K, while for lead bilayer, Tc was significantly smaller, 3.6K
4.9K, depending on sample preparation (lead trilayer is thermodynamically unstable which precluded Tc measurements for this thickness). A dramatic drop in Tc for bilayer lead can be attributed to the straining effect of a substrate, leading to the crystallographic reconstruction of atomically thin lead film. In the case of lead bilayer grown on Si (1 1 1) 7 3 7 surface, two types of reconstruction were observed: [158] 1 3 1 with moire´ superlattice of  3 nm and O3 3 O3 with moire´ of  4.4 nm. Although electronic structures of these two types of reconstruction are very similar, the superconducting gaps and critical temperatures differ drastically: Tc for a 1 3 1 type is  4.9K, while O3 3 O3 has significantly lower Tc  3.6K. Since 2015, in addition to research on amorphous films and epitaxial layers, studies of 2D highly crystalline superconductors emerged, thanks to rapid progress in van der Waals technology [2]. Some of this progress is summarized in a 2017 review article [159], where 2D superconductors were grouped into five broad classes: (1) deposited and typically disordered films of Ga, Sn, Bi, Pb, Al, In, YBa2Cu3O72x, InOx, MoGe, (2) epitaxial layers of Pb or In, (3) interfacial superconductors—LaAlO3/SrTtO3, La22xSrxCuO4\La2CuO4, FeSe/ SrTtO3, (4) exfoliated 2D materials—NbSe2, Bi2Sr2CaCuO81x, TaS2, and (5) gate-induced superconductors—SrTtO3, ZrNCl, MoS2. Exfoliated superconductors (class iv) were inspired by the isolation of graphene and enabled by the van der Waals technology. The first 2D superconductor produced by exfoliation was a single layer of Bi2Sr2CaCuO81x (BSCCO), which was protected from the environment by graphene capping layer [160]. Tc of BSCCO single layer (0.5 unit cell thick) was reported  82K (cf. bulk BSCCO of Bi-2212 variant studied in which work has Tc  96K) [160]. Another layered superconductor, NbSe2, was also extensively studied. First work on superconductivity in ultrathin NbSe2 crystals dates back to 1972; curiously, thin crystals were prepared using van der Waals technology into heterostructures of NbSe2/mica, and samples down to a trilayer thickness were obtained [161]. More recently, with the advances of van der Waals technology, NbSe2 superconductivity was studied in a 2D limit [162
167]. Earlier studies of 2D NbSe2 flakes were inconsistent mainly due to oxidation of the crystals in an ambient environment. Introduction of van der Waals technology in a controlled environment together with exploiting encapsulation of air- and moisturesensitive crystals by chemically inert hBN and graphene films enabled fabrication of high-quality devices [162,163,167], Fig. 2.9A. Improved electronic quality of NbSe2 crystals led to the discovery of an interesting quantum metal phase adjacent to the superconducting state [163], Fig. 2.9B. Phase transition between the superconducting phase and the quantum metal phase can be induced by applying a small perpendicular magnetic field. Measured resistance in the quantum metal phase followed power law dependence on the magnetic field, RB(H 2 Hc0)α(T), where critical exponent 2ν 5 α  3 at T 5 0K. The observed power-law scaling was attributed to the crossover from fermionic tunneling of vortex quasiparticles to Bose-metal behavior in the limit of vanishing disorder. Eq. (2.3) from Ginzburg
Landau theory suggests the suppression of superconducting transition temperature with the film thickness. This effect was thoroughly studied by Khestanova et al. via performing measurements of the superconducting energy gap and

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FIGURE 2.9 Superconductivity of NbSe2 in the 2D limit. (A) Schematic of environmentally controlled device fabrication. hBN/G on a polymer stamp (PDMS) is used to encapsulate NbSe2 in an inert atmosphere. The heterostructure is lithographically patterned, and the edge of graphite is metallized with Cr/Pd/Au. (B) Emergence of the quantum metal. Full H
T phase diagram of the bilayer NbSe2 device. The red circles are Hc2(T). The purple squares, dividing the Bose metal from the TAFF regime, mark the transition from activated behavior RBexp(U (H)/T) to temperature-independent resistance R 5 R(H). The blue triangles denote the boundary of the superconducting phase Hc0(T). (C) top: sketch of a tunneling device; bottom: optical image of layer 2 and 3 NbSe2 devices. The red and black dashes outline NbSe2 and thin hBN crystals providing the tunneling barrier, respectively. Scale bar, 10 μm. (D) Tunneling conductance dI/dV as a function of bias voltage, Vb, for crystals with different N (symbols). Black solid curves are fits to Dynes formula. The spectra, labeled by N, are shifted for clarity. (E) Thickness dependence of the energy gap and TC. Main panel: symbols show Δ0 from tunneling spectra (red symbols) and TC from transport measurements (blue symbols) as functions of N. The dashed lines best fits to TC(1/N) and Δ0(1/N). Error bars for TC correspond to the width of the SC transition and for Δ0 to the accuracy of the fit. Top inset: Δ0 versus N. Bottom inset: dependence of the anisotropy parameter A on the electron mean-free path, l. 2D, Twodimensional; hBN, boron nitride; G, graphite; TAFF, thermally assisted flux flow. Source: From (A and B) A.W. Tsen, et al., Nature of the quantum metal in a two-dimensional crystalline superconductor, Nat. Phys. 12 (2015) 208-212, doi:10.1038/nphys3579 and (C
E) E., Khestanova, et al., Unusual suppression of the superconducting energy gap and critical temperature in atomically thin NbSe2, Nano Lett. 18 (2018) 2623
2629, doi:10.1021/acs.nanolett.8b00443.

the critical temperature in high-quality crystals of NbSe2 [167]. Schematic and micrograph of typical tunneling devices are shown in Fig. 2.9C; measured tunneling spectra and theoretical fits (using Eqs. 2.4
2.5) are presented in Fig. 2.9D. Differential tunneling conductance spectra have the typical shape of normal metal-superconductor tunneling junctions—clearly visible superconducting energy gap, Δ, (zero conductance) sandwiched by sharp quasiparticle peaks (due to a high density of states at the gap edges). To extract

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Δ for devices of different thickness, the tunneling spectra were fitted using standard BCS expression [168]: ð GNN 1N @fðE 1 eVb Þ GNS 5 dI=dV 5 dE (2.4) NS ðE; Γ; ΔÞ @ðeVb Þ NN ð0Þ 2N where GNN is the conductance in the normal state, NN(0) and NS(E,Γ ,Δ) are the density of states in the normal and superconducting phase, respectively; f is the Fermi
Dirac distribution. Γ is the quasiparticle lifetime broadening. The superconducting density of states is given by the Dynes formula [169,170] 9 8 > > = < E 2 iΓ (2.5) NS ðE; Γ; ΔÞ 5 Re qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ; : ðE2iΓ Þ2 2 Δ2 > Fig. 2.9E summarizes the main result of [167]—the suppression of Tc closely follows the decrease in Δ, in excellent agreement with the Ginzburg
Landau theory applied to thin films. This ruled out the contribution of disorder and BKT transition to the observed suppression of Tc in high-quality NbSe2 crystals. Instead, the suppression of Tc originates from the boundary condition (Eq. 2.2) which leads to 1/d correction of Δ, negligible in bulk crystals but becoming dominant in thin films. The predicted 1/d behavior of Δ and Tc (Eq. 2.3) agrees well with the experiment, Fig. 2.9E. The difference in the 1/d slopes for Δ and Tc (together with noticeable deviation for NbSe2 bilayer) was attributed to changes in the anisotropy of Δ in the momentum space—from a highly anisotropic gap of bulk NbSe2 to isotropic bilayer NbSe2 (bottom inset in Fig. 2.9E). Another interesting phenomenon observed in 2D NbSe2 is related to competing between the Zeeman effect and large intrinsic spin-orbit interactions in noncentrosymmetric NbSe2 monolayers, where the electron spin is locked to the out-of-plane direction [164]. In conventional superconductors, superconductivity can be quenched by an external magnetic field via the vortex formation (orbital effects) or spin alignment (Zeeman effect). For the in-plane magnetic field the orbital effects are absent, and the spin-momentum locking (Ising pairing) is expected to significantly enhance the Pauli paramagnetic limit (direct spin-flip). This effect was indeed observed by Xi et al. as they discovered very high critical in-plane magnetic fields induced by the spin-momentum locking in the superconducting monolayer of NbSe2 [164], Fig. 2.10. In contrast to NbSe2, where Tc is suppressed in the monolayer limit, TaS2 shows strongly enhanced superconductivity in the 2D limit [171]. One of the possible explanations of the observed unusual behavior can be related to the suppression of the commensurate charge density wave order, which directly competes with the superconducting order. Alternatively, the enhanced effective coupling in a 2D limit due to the presence of van Hove singularity near the Fermi level can also account for increased Tc in monolayer TaS2. Dramatic enhancement of Tc in a monolayer limit was also observed in one-unit-cell FeSe films [172]. The potential mechanisms of this unusual enhancement are discussed in a topical review [173]. An in-depth analysis of both TaS2 and FeSe, as well as other superconducting systems is presented in a recent work by Talantsev et al. [174]. Their central

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FIGURE 2.10 Spin-momentum and spin-layer locking in NbSe2. (A and B) Illustration of spin-momentum locking in monolayer (A) and spin-layer locking in bilayer/bulk (B) NbSe2. Top: monolayer NbSe2 consists of a layer of Nb atoms (blue balls) sandwiched between two layers of Se atoms (yellow balls) in the trigonal prismatic structure. Bulk 2H-NbSe2 is made of monolayers stacked in the ABAB sequence. Bottom: Brillouin zone in the in-plane direction and Fermi surface near the Γ , K and K0 point. The Fermi surface is spin split in the monolayer and is spin degenerate in the bilayer/bulk. (C) H
T superconducting phase diagram for atomically thin NbSe2. The critical field Hc2/HP as a function of transition temperature TC/TC0 for NbSe2 samples of differing thickness under both out-of-plane H\ (open symbols) and in-plane HO (filled symbols) magnetic fields. Filled circles and triangles represent data acquired from different devices. The dashed line corresponds to the Pauli paramagnetic limit. The solid blue line is the best fit to the solution of the pair breaking equation. The solid green line is a linear fit. The inset shows the zero-bias peak area and width as a function of HO at 0.36K obtained from differential conductance measurements in a monolayer device. The result shows that superconductivity in monolayer NbSe2 survives under HO 5 31.5 T. Source: From X. Xi, et al., Ising pairing in superconducting NbSe2 atomic layers, Nat. Phys. 12 (2016) 139
143, doi:10.1038/nphys3538.

finding is that the enhancement of Tc arises from the opening of a second, larger, superconducting gap, while the smaller “bulk” superconducting gap remains almost unchanged. According to Talantsev et al. [174], the electron
phonon interactions remain unchanged, and the enhancement effect cannot be associated with the presence of a van Hove singularity or with the effect of fluctuations. Thus further experimental and theoretical work is required to clarify the enhancement mechanism.

2.4.3 Gate-induced superconductivity in two-dimensional materials Electrostatic gating provides an efficient way to control the carrier density in 2D systems without introducing disorder. Since conventional oxide dielectrics do not allow to induce high carrier densities due to the dielectric breakdown, an alternative solution is usually employed—an electric-double-layer electrochemical gating [159]. At the electrode surfaces, mobile cations and anions of liquid or solid electrolytes form a bilayer of positively and negatively charged layers with only a nanometer separation between them. This nanometer gap allows to produce large electric fields ( . 109 V m21) leading to extremely

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FIGURE 2.11 2D superconductivity in MoS2 and ZrNCl crystals. (A) Schematic of an ionic liquid gated MoS2 field effect transistor device. (B) Superconductivity transitions at different thicknesses. (C) Layer dependence of critical magnetic field. (D) Ball-and-stick model of a ZrNCl single-layer crystal. The monolayer is 0.92 nm thick. (E) Sheet resistance, Rsheet, of a ZrNCl-EDLT as a function of magnetic field at field steps of 0.1
0.3, 0.2
0.9, 0.3
1.8, 0.5
5.0, and 1
9 T. (F) Phase diagram of a ZrNCl electric double-layer transistor obtained based on experimental data. 2D, Two-dimensional. Source: Obtained from (A
C) D. Costanzo, S. Jo, H. Berger, A.F. Morpurgo, Gate-induced superconductivity in atomically thin MoS2 crystals, Nat. Nanotechnol. 11 (216) 339
344, doi:10.1038/nnano.2015.314 and (D
F) Y. Saito, T. Nojima, Y. Iwasa, Quantum phase transitions in highly crystalline two-dimensional superconductors, Nat. Commun. 9 (2018), ARTN 778, doi:10.1038/s41467-018-03275-z [178].

high charge carrier doping, reaching over 1014 cm22. Doping 2D SCs, such as MoS2 to a high carrier density induces a superconducting state with Tc . 10K [175
177]. In experiments with electric double layer transistor (EDLT) devices an exfoliated flake of 2D material is contacted using conventional microfabrication techniques, and then a droplet of ionic liquid or polymer electrolyte is applied onto the surface of the flake also covering a side gate electrode. A voltage applied between the side electrode and the flake drives anions (cations) onto the flake surface under positive (negative) bias, inducing large carrier density in a 2D material. The charge carrier density induced by the liquid gate can be tuned only above certain temperatures, at which the ions preserve enough mobility to enable gating. For typical ionic liquids these temperatures are usually in the range of 200K
250K. Thus measurements of Tc
n2D phase diagram require multiple thermo-cycling of EDLT device. These measurements enable a detailed study of the superconducting phase induced by the electric field effect, avoiding phase separation. Fig. 2.11A shows schematic of an ionic liquid gated MoS2 device and Fig. 2.11B and C shows the layer-dependent superconducting transition temperature and magnetic field,

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2. Novel phenomena in two-dimensional semiconductors

respectively [176]. Recently, the same group led by Alberto Morpurgo, combining two powerful techniques, EDLT and tunneling spectroscopy, explored gate-induced superconductivity in MoS2, and directly measured density of states and the superconducting gap in this material [177]. They concluded that low-energy density of states depends linearly on energy which suggests the absence of a full gap, questioning the current understanding of gate-induced superconductivity in MoS2, and asking for further research beyond conventional phonon-driven BCS mechanism. Another layered SC which shows insulator-to-superconductor transition under electrochemical gating is ZrNCl [179]; Fig. 2.11D displays a monolayer of ZrNCl. Fig. 2.11E highlights magnetic field dependence of superconducting transition temperature and a quantum phase diagram of a ZrNCl system is shown in Fig. 2.11F. Temperature dependence of the resistivity drastically depends on the applied electrochemical gate voltage VG. Above VG 5 6 V, ZrNCl becomes superconducting. 2D superconductivity in ZrNCl could not described by dirty-boson model due to the appearance of a metallic intermediate state. This metallic state was found by the application of a weak magnetic field [179]. Instead, the 2D superconductivity of ZrNCl was dominated by the BKT transition [179]. Electric-field-induced superconductivity was also observed in a topologically nontrivial material WTe2—a monolayer topological insulator [180]. This is an important breakthrough since materials combining nontrivial topology and superconductivity are promising for future topological superconducting devices and applications for quantum information schemes based on topological protection.

2.5 Two-dimensional ferromagnetism and other magnetic phenomena 2.5.1 Introduction to two-dimensional ferromagnetism Magnetism is a fascinating phenomenon dating back to ancient times. One of the first known practical uses of magnetism took place using magnetic needles in compasses, which was documented as early as the 11th century. The understanding of the relationship between electricity and magnetism began emerging in the early 1800s with the socalled Oersted’s experiment where compasses nearby a current carrying wire were influenced by the generated magnetic field. Following on from there, a series of pioneering experiments was done including the ones by Michael Faraday all in the 19th century, culminating with the renowned “Maxwell’s equations” which unified the fields of electricity, magnetism, and optics. In the early 20th century, Pierre-Ernest Weiss came up with domain theory, where he suggested parallel magnetic moments of many atoms explain magnetism. Weiss’ theory
suggested magnetism arises because of an effective Weiss molecular magnetic field which is proportional to the magnetization of the material, that is, Be 5 αMs where Ms is the saturation magnetization and α is the mean field constant. Effective field, Be , is the field at which domain walls seize to become magnetic. Weiss’ phenomenological

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approach was complemented with the development of quantum theory in the early 1900s where an exchange interaction between fermions was predicted due to the Pauli Exclusion Principle independently by Heisenberg and Dirac in 1926. Taking into account the exchange interaction between the nearest neighbor atoms i and j in a solid (as atoms themselves also possess an overall spin), an indirect exchange is obtained with the electrons in the solid (i.e., with their spin degree of freedom) leading to the Heisenberg Hamiltonian describing the magnetic properties of a solid as XX H52 Jij Si Sj i

j6¼i



where Si and Sj correspond to spins of nearest neighbor atoms, and Jij corresponds to exchange interaction between the spins. Heisenberg Hamiltonian predicts three possible ground states of a solid which are 1. Jij . 0 ferromagnetic 2. Jij , 0 antiferromagnetic 3. Jij 5 0 noninteracting spins The merger of the Weiss’ theory (effective magnetic field) and Heisenberg Hamiltonian gave rise to what is known as the Ising model which was found to be an excellent model in describing magnetic properties of the bulk solids. In 1966 Wagner and Mermin predicted an interesting phenomenon, which was the absence of the validity of an isotropic Heisenberg Hamiltonian (i.e., isotropic magnetism) in a 2D system [181]. For decades, 2D magnetism was studied using thin films and quantum wells [182,183]. Thus the intersection of 2D matter and magnetism was a very stimulating dream since the advent of graphene [184]. Various methods were used in order to induce magnetic properties in graphene such as defect engineering [185]. Finally, during the last couple of years, intrinsic magnetism was observed on mechanically exfoliated 2D crystals which created a playground for studies of magnetism in truly 2D systems [6,10,186
191]. This means that on top of the Ising model of magnetism mentioned earlier, alternative models of magnetism, such as XXZ and XY, can finally be tested experimentally [192
194] as well as Kitaev magnetism and quantum spin liquids [195]. Many van der Waals systems have been predicted to exhibit magnetism in a 2D limit including the transition metal trihalide systems [196
199]. Out of these materials, Cr2Ge2Te6 and CrI3 were the first 2D magnets confirmed through magneto-optic Kerr effect (MOKE) experiments [6,10]. CrI3 was found to show out of plane easy axis magnetism, and it was interpreted within more classical Ising model, whereas the observed magnetic behavior of Cr2Ge2Te6 was interpreted with a more complicated Heisenberg Hamiltonian taking into account the anisotropy required by Mermin
Wagner theorem [181]. Following on CrBr3 was demonstrated to be a 2D insulating magnet with a magnon excitation gap [186]. Having more structural similarity to Cr2Ge2Te3, Fe3GeTe2 was demonstrated to be undergoing room temperature ferromagnetism while doped through a powerful method of ionic liquid gating [190].

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2.5.2 Emerging two-dimensional magnetic materials and phenomena 2.5.2.1 Magneto-optic Kerr effect experiments of Cr2Ge2Te3 and CrI3 Two first simultaneous systematic reports of an intrinsic (anti)ferromagnetic behavior of a van der Waals system in 2D limit came through with MOKE results on Cr2Ge2Te3 and CrI3 systems [6,10]. MOKE spectroscopy allows one to have an insight into magnetic properties as the optical birefringence (polarization dependent refractive index) allows one to distinguish magnetic properties. MOKE gives insights into magnetic properties through an experimentally obtainable quantity, “complex Kerr angle” as shown in the following equation: φK 5 θK 1 iεK The real part of the complex Kerr angle is referred to as the Kerr rotation angle, and it is the quantity that relates to the magnetic properties of the dielectric medium being investigated. The latter quantity is the Kerr ellipticity. The quantities described are obtained through a 3 3 3 dielectric tensor describing the reflection dynamics on the magnetic surface. One can qualitatively imagine the 45 degrees (w.r.t. electric field) linearly polarized light used in the experiments as a combination of right- and left-handed circularly polarized light of the same amplitude. As a result of dielectric properties arising from intrinsic magnetism of the crystals, the two components making up the linearly polarized light interact differently with the medium mathematically giving rise to the “complex Kerr angle” is defined above. The work carried out by Gong et al. was interpreted within the Heisenberg Hamiltonian with an anisotropy term where a magnon excitation gap was found as a result [10], as shown in Fig. 2.12A. Fig. 2.12B shows the identification of a number of layers through the use of the optical contrast. A most striking observation was that the transition temperature was found to go down with a decreasing number of layers (from B70K in bulk to B30K for bilayer). Unlike Cr2Ge2Te6, the overall (anti)ferromagnetic behavior of the CrI3 system was interpreted within the Ising model, the magnetic configuration of a monolayer system is shown in Fig. 2.13A. The spins of the Cr atoms arrange themselves in an aligned orientation leading to an easy axis pointing out of the plane in monolayer limit [6]. Kerr rotation angle obtained at various temperatures for zero magnetic field (red) and 0.15 T (blue) is shown in Fig. 2.13B, suggesting a Curie temperature below 60K. The hysteresis curves obtained for mono- (Fig. 2.13C), bi- (Fig. 2.13D), and trilayer (Fig. 2.13E) system exhibit ferromagnetic, antiferromagnetic, and ferromagnetic behavior, respectively. The spin ordering leading to the antiferromagnetic behavior is illustrated in Fig. 2.13D. The Kerr results shown were reproduced by Wang et al. on the same material, CrI3 considering thicker films [189]. 2.5.2.2 Tunneling magnetoresistance on mono-, bi-, tri-, tetra-layer CrI3 systems Following from the magneto-optic confirmation of the intrinsic ferromagnetism on transition metal trihalide family member CrI3 electron tunneling experiments were reported on the same material by three groups simultaneously. Fig. 2.14 shows the tunneling magnetoresistance data obtained by Klein et al., on bi- and tetra-layer CrI3 systems, and associated physical interpretation is depicted on the schematics [187].

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FIGURE 2.12 (A) Schematic illustrating the crystal structure of a CrI3 layer. (B) Optical micrograph of a mechanically exfoliated CrI3 flake. (C) Number of layers of flake identified through optical contrast [10].

The confirmation of the antiferromagnetic behavior on even-layer atomically thin CrI3 systems through electron tunneling can be seen in Fig. 2.14A and D for bi- and tetra-layer devices utilizing graphite electrodes where a typical magneto-conductance attributed to a nonspin polarized magnetic system is observable. Fig. 2.14B shows expected tunnel barrier for spin-up and spin-down electrons for an antiferromagnetic bilayer system, and Fig. 2.14D shows the tunnel barrier once the system becomes spin-polarized tunnel barrier with the application of a perpendicular magnetic field. Fig. 2.14E and F is analogous (tunnel barriers) for a tetra-layer system. The maximum tunneling magnetoresistance reported in that work was 95%. The other study by Song et al. [188] is summarized in Fig. 2.15. Devices used in that work utilized graphene electrodes in order to perform electron tunneling study. The maximum reported magnetoresistance was 19,000%, four orders of magnitude higher than the previous one. Fig. 2.15A depicts the magnetic ordering of a bilayer CrI3 system illustrating how spin ordering is manipulated with respect to out-of-plane and in-plane magnetic field above a critical value. Fig. 2.15B shows the schematic of a typical device and the influence

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FIGURE 2.13 (A) Schematic of magnetization of a single layer CrI3 crystal. (B) Kerr rotation angle as a function of temperature at zero (blue) and 0.15 T (red) magnetic field. (C
E) Magnetic field sweep dependence of the Kerr rotation angle (a.k.a. magnetization) in single, bi- and trilayer CrI3 systems, respectively [6].

of both out-of-plane and in-plane magnetic field on the tunneling current is shown in Fig. 2.15C. It can be seen that both out-of-plane and in-plane magnetic field increase the conductance of the magnetic tunnel barrier. Tunneling experiments with CrI3 devices were also done by Wang et al. where they investigated thickness range of 5.5
14 nm, and the critical switching magnetic field was found to be bigger than in the other studies, up to 2 T, as shown in Fig. 2.16 [189]. As it can be seen in Fig. 2.16A, there are three critical fields (namely, J1, J2, and J3) at which potential magnetic switching occurs which are at around 6 0.5, 6 1, and 6 2 T. As the temperature is increased, magnetic switchings that take place at J2 and J3 become less clear and completely disappearing at 50K, below the bulk Curie temperature of 61K. 2.5.2.3 Magnon assisted tunneling in atomically thin CrBr3 tunnel barriers Magnons are bosonic excitations that arise due to the inherent spin ordering of magnetic matter. The simplest way of viewing magnons would be through an analogy to phonons. Phonons arise due to quantized vibrations of lattice sites. Similarly, magnons arise due to quantized in-phase precession of spin moments on the lattice sites (atoms), in other words, due to the quantization of spin waves. Magnons were considered in the work

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FIGURE 2.14 (A) Bilayer CrI3 tunneling magneto-conductance showing antiferromagnetic hysteresis with respect to magnetic field sweeps. (B and C) The tunnel barriers of a bilayer CrI3 system at antiferromagnetic and ferromagnetic (spin polarized) states, respectively. (D) Tetra-layer CrI3 tunneling magneto-conductance showing antiferromagnetic hysteresis. (E and F) Tunnel barriers of a tetra-layer system in the ground antiferromagnetic state and spin polarized state, respectively [187].

reporting MOKE results on Cr2Ge2Te6 [10], and one of the electrons tunneling works on CrI3 (Klein et al.) [187]. However, a more systematic study of the magnons was carried by Ghazaryan et al. in a team led by Kostya Novoselov on a new material that is also a transition metal trihalide, CrBr3 [186]. Motivated by their earlier work which reported that gate-independent features in differential conductance (corresponding to inelastic tunneling events) of graphene/hBN/graphene tunneling devices were typically attributed to phonons [200]; they were able to show that magnons dominate the tunneling in the case of replacement of hBN with a magnetic material CrBr3. The tunneling features and their associated back-gate dependence are shown in Fig. 2.17A. The diagonal features typically attributed to the Dirac point in hBN

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FIGURE 2.15 (A) Spin ordering of a bilayer CrI3 in the absence of magnetic field, above a critical perpendicular field and above a critical parallel field respectively. (B) A bilayer CrI3 tunneling device sandwiched between two graphene electrodes. (C) The influence of spin switching by perpendicular and parallel magnetic field to the tunneling characteristic of the bilayer CrI3 device [188].

FIGURE 2.16 Thickness-independent magnetoresistance in exfoliated CrI3 crystals at temperatures between 10K and 65K. It can be seen that negative magnetoresistance is still present at 65K with the absence of any jumps that could be attributed to magnetic properties. Black and red colors denote trace and retrace, respectively, at a DC bias of 500 mV [189].

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FIGURE 2.17 (A) Back-gate dependence of differential conductance in a six-layer CrBr3 system sandwiched between two graphene electrodes. The right axis corresponds to the differential conductance curve superimposed to the map. (B) Back-gate dependence of the CrCl3 tunneling device at a perpendicular magnetic field of 12.5 T. The diagonal features correspond to Landau quantization of the lower graphene electrode. (C) In-plane magnetic field dependence of the differential conductance curve [186].

tunneling devices are absent and gate-independent features are present, especially at positive gate voltages. In order to clarify that the observed features are indeed coming from magnon-assisted tunneling, magnetic field dependence was studied, as shown in Fig. 2.17B (out-of-plane) and Fig. 2.17C (in-plane). Perpendicular field of 12.5 T, as shown in Fig. 2.17B, leads to diagonal features (marked by black arrows). The observed diagonal features arise in one diagonal direction only (i.e., with only a positive slope) and can be attributed to Landau quantization of the bottom graphene electrode. To confirm that the inelastic tunneling events arise as a result of magnons, bias dependence was studied for in-plane magnetic field up to 12 T. The features at lower bias range were dispersing with respect to in-plane magnetic field and, as a result, they are attributed to magnon-assisted tunneling. The temperature dependence of the tunnelling conductance as a function of bias for a tetra-layer CrBr3 system at different magnetic fields (B|| 5 0, B|| 5 10, B|| 5 15 T) is shown in Fig. 2.18. The temperature dependence maps manifest the antiferromagnetic transition with a trend of decrease in conductance being marked at the transition temperature, TC. The transition temperature is increased from  40K to  75K by the presence of in-plane magnetic field approaching to the one for bulk crystals. 2.5.2.4 Room temperature ferromagnetism in Fe3GeTe2 A metallic 2D magnet with ferromagnetism persisting up to room temperature (310K) was reported with mechanical exfoliation of Fe3GeTe2 down to monolayer thickness [190]. Similarly to CrI3, magnetization was found to be out of plane due to magnetocrystalline anisotropy; however, unlike CrI3, Fe3GeTe2 crystals are found to be metallic. A team led by Yuanbo Zhang investigated electronic transport characteristics of the few-layer systems using the powerful method of ionic liquid gating [190]. Through the observation of Hall-resistance hysteresis, that is, an anomalous Hall effect, they found that room temperature ferromagnetism of bulk crystals is suppressed down to 205K when the sample thickness is decreased down to atomic thicknesses. Using an ionic liquid gate as

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FIGURE 2.18 Temperature dependence of differential conductance at varying DC bias across a four-layer CrBr3 tunneling device at in-plane magnetic fields of (A) 0, (B) 10, and (C) 15 T [186].

FIGURE 2.19 (A) Schematic of an ionic liquid gated Hall bar device of Fe3GeTe2. (B) Hysteresis in Hall resistance persisting up to 310K in the presence of an ionic gate voltage applied. (C) Temperature dependence of zero magnetic field Hall resistance [190].

shown in Fig. 2.19A, they were able to increase the temperatures at which anomalous Hall effect was observed up to above the room temperature (310K) as shown in Fig. 2.19B. Fig. 2.19C shows zero magnetic field Hall resistance observed at various temperatures and a linear extrapolation down to 0 Ω suggests a ferromagnetic transition exceeding 310K in the presence of an ionic liquid gate voltage of 2.1 V. 2.5.2.5 Hall magnetometry of CrBr3 Following up on the earlier tunneling work on insulating transition metal trihalide CrBr3, the group led by Andre Geim carried out magnetometry measurements on CrBr3 flakes that are in proximity to ballistic graphene Hall bars [193]. It has been shown that the measured Hall resistance of a ballistic system in proximity to a magnetic material is influenced by the magnetic flux of the material, allowing quantitative analysis of magnetization. Using this technique, varying thickness of individual 2D atomically thin CrBr3 magnets were investigated. Fig. 2.20A shows saturation magnetization of a monolayer CrBr3

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FIGURE 2.20 (A) Ferromagnetic hysteresis of a monolayer CrBr3 flake measured through ballistic Hall magnetometry at various temperatures up to 35K above the Curie temperature. (B) Saturation magnetization at various temperatures as well as fits according to various theoretical models including XXZ. (C) Curie temperature at varying number of layers [193].

magnet with respect to temperature. Magnetization becomes absent once the temperature is above the Curie temperature, that is, 30K. Curie temperatures determined with respect to the number of layers are shown in Fig. 2.20C where decreasing critical temperature with respect to decreasing number of layers can be seen. The most striking finding in the paper, however, is shown in Fig. 2.20B. Motivated by the different interpretations of the magnetic behavior of Cr2Ge2Te6 and CrI3, Xu et al. carried out calculations looking at the magnetism in these systems from an alternative viewpoint, and speculate that it may well be an XXY interaction leading to the observed 2D magnetism accompanied by additional Kitaev physics [192]. Fig. 2.20B is a plot of the temperature dependence of magnetization, and it can be seen for the case of sister compound of CrI3, XXZ theory describes the data better than the 2D Ising model used to explain the findings in the earlier work. The findings show that both further experiments and theory are needed to understand transition metal trihalides as well as other atomically thin magnets. 2.5.2.6 In-plane easy axis transition metal trihalide CrCl3 The reports of the first in-plane 2D magnet took place with data coming from a CrCl3 electron tunneling device as shown in Fig. 2.21A [194,201]. It was found that a bilayer CrCl3 flake sandwiched between two graphite electrodes was sensitive to both in-plane and out-of-plane magnetic field, as shown in Fig. 2.21B. Moreover, it was found to exhibit

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FIGURE 2.21 (A) A bilayer CrCl3 system sandwiched between two graphite electrodes forming a tunneling device encapsulated by hBN. (B) In-plane (orange) and out-of-plane (green) magnetic field dependence of tunneling current at an optimized bias. (C) In-plane magnetic field hysteresis of a tunneling device. (D) Experimentally measured phase diagram of CrCl3 [190].

antiferromagnetic hysteresis within the application of in-plane magnetic field only, thus confirming in-plane field ordering as shown in Fig. 2.21C. Fig. 2.21D shows the experimentally measured phase diagram of a bilayer CrCl3 tunneling device. The state device switches to (from an antiferromagnetic state) with the application of both in-plane and out-of-plane magnetic field is a spin-polarized state. At temperatures higher than the Neel temperature (17K), CrCl3 exhibits conventional paramagnetism. A remarkable feature of CrCl3 as a layered magnetic system is that its in-plane magnetization was explained in the context of XY Hamiltonian or 2D XY interaction. XY interactions have close links with topological 2D systems and in principle could allow studies of BKT transition [7,194]. Demonstration of the possibility of in-plane magnetization in 2D limit is also a breakthrough. Studies of more exotic layered material systems such as α-RuCl3 hinted to exotic Kitaev antiferromagnetism and Majorana fermions through neutron scattering and heat capacity experiments [8,202,203]. For further information on magnetic 2D materials and heterostructures, we recommend recent review articles [7,204].

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2.6 Novel topological states Topological effects in 2D semiconducting systems have their origins from the honeycomb lattice crystal structure arising from the presence of A and B sublattices [124]. The massless Dirac Hamiltonian describing monolayer graphene could be extended to a massive Dirac Hamiltonian for monolayer TMDCs [122]. The mass arises due to the presence of a bandgap because of inversion asymmetry due to the presence of different atoms in A and B sublattices. Two orbital wave-functions describing the sublattices give rise to pseudospin degree of freedom with Berry phase and associated Berry curvature. The absence of inversion or time reversal symmetry leads to the presence of a finite Berry curvature that can be classified topologically in terms of Chern numbers leading to potential zero magnetic field quantum Hall states [205,206]. However, it has been theoretically shown that one of these quantum phases, quantum spin Hall state, exists in the presence of both inversion and time-reversal symmetry [207]. This state is linked to spinrotational symmetry through Kramers degeneracy theorem which states that eigenstates of a system with half-integer total spins are at least doubly degenerate. As a result, this state is described by another quantity, Z2 index and its value is dictated by the presence of either odd or even number of Kramer’s degenerate pairs [208]. Generalization of Z2 index to three dimensions has led to the predictions of topological insulators giving rise to low energy metallic surface states in the insulating bandgap, where the number of Z2 indices describing a 3D system was found to be four, unlike the single Z2 index describing a 2D system [209]. Shortly after Weyl and Dirac semimetal systems were identified in gapless SCs as 3D analogues of graphene possessing low energy linear dispersion relation in 3D momentum space [210]. A remarkable connection between 3D topological matter and 2D SCs is that a reasonable number of these systems are layered van der Waals crystals that can be exfoliated, showing 2D SCs properties in their monolayer form [211,212]. In this section, topological concepts including Berry phase, Berry curvature, and Chern numbers will be introduced on the simplest two band Hamiltonian describing monolayer honeycomb lattice systems. Spontaneous zero magnetic field quantum Hall states associated with the absence of respective symmetries will then be explained. Lastly, developments will be highlighted on systems where the topological effects were found to be generalized onto three dimensions in topological insulators as well as Weyl, Dirac, and nodal line semimetal systems.

2.6.1 Massless Dirac Hamiltonian for graphene Effective tight binding Hamiltonian from two atom basis of a graphene lattice approximated near the Brillouin zone corners K and K0 points is given by



¯ rÞ H^ 5 6 ¯h vF σ ð 2ih

(2.6)

where ¯h is reduced Planck constant, vF is the Fermi velocity, σ is Pauli spin matrices σx and σy. The solution of the Hamiltonian at K and K0 point is obtained through the eigenvectors

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1 tΨ 1 i 5 pffiffiffi 2





 1 1 1 ik  r ik  r e ; tΨ 2 i 5 pffiffiffi 2iφ e 6 e1iφ 2 7e

(2.7)

with φ being polar angle of momentum in x
y plane. A remarkable property of the eigenvectors is that they resemble two-component spinors describing a system consisting of either spin-up or spin-down states. However, unlike spinor functions, the eigenvectors obtained above are a result of A and B sublattices making the honeycomb lattice structure of graphene, behaving as an analogue of spin and leading to the pseudospin degree of freedom. The solution of the Hamiltonian in Eq. (2.6) using the eigenvectors in Eq. (2.7) leads to the low energy limit linear dispersion relation of graphene describing massless Dirac fermions [213,214] εðkÞ 5 6 ¯hvF jkj

(2.8)

where 6 refers to the conduction and valance bands.

2.6.2 Chirality and Berry phase Using the eigenvectors defined in Eq. (2.7), effective massless Dirac Hamiltonian defined in Eq. (2.6) could be rewritten around K point as



H^ 5 6 ¯hvF σ Ψ

(2.9)

where σ consists of Pauli spin matrices σx and σy and Ψ is the eigenvector. The scalar product in Eq. (2.9) implies pseudospin degree of freedom originating from eigenvectors describing A and B sublattices is linked to the direction of momentum leading to the chirality of the massless Dirac electrons. This can also be pictured by another examination of eigenfunctions shown in Eq. (2.7). It can be seen that the pseudospins are linked to a momentum through the phase component, which is a polar angle of the momentum in the x
y plane. Furthermore, solving the eigenvector in Eq. (2.6) for a closed circular loop in momentum space, corresponding to a change in the value of φ of 2π, a phase change equal to π is obtained as shown in the following equation: ð 2π d π5i (2.10) dφΨ  Ψ dφ 0 The integral in Eq. 2.(10) is known as the Berry phase integral [215], and the quantity of π obtained above is the experimentally verified characteristic Berry phase for graphene [216,217].

2.6.3 Inversion asymmetry, Berry curvature, and Chern numbers The extension of 2 3 2 massless Dirac Hamiltonian to 2D SC TMDC systems takes place by the inclusion of inversion asymmetry (for the case of 1H TMDC structures). The massive Dirac Hamiltonian for gapped low energy bands becomes [124]   Δ H^ 5 6 ¯hvF k cosðφÞσx 1 sinðφÞσy 1 σz 2

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(2.11)

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where Δ is the magnitude of the gap, cosðφÞ and sinðφÞ denote the x and y components of momentum. Low energy dispersion relation is obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εðkÞ 5 6 Δ2 1 ¯h2 v2F k2 (2.12) Defining eigenvectors Ψ(k) at K and K0 valleys for the modified Hamiltonian in Eq. (2.11) yields Berry curvature of 2D system lacking inversion symmetry [24,124]   h¯ 2 v2F Δ ΩðkÞ 5 rk 3 AðkÞ 5 rk 3 i ΨðkÞjrk jΨðkÞ 5 2εðkÞ3

(2.13)

where A(k) is referred to as the Berry potential. The physical significance of both Berry curvature and Berry potential is that they are analogous to the magnetic field and vector potential in momentum space. It must be noted that Berry curvature also exists for systems lacking time-reversal symmetry. For systems with time-reversal symmetry, Bloch functions dictate Berry curvature to be an odd function of k, whereas for the case of inversion symmetry dictates that Berry curvature is an even function of k, as is the case in Eq. (2.13). These two cases mean that Berry curvature of a system could only be finite in the presence of either of the two symmetries [211]. Integration of Berry curvature over any closed surface yields I ΩðkÞ dS 5 2πm (2.14)



where m is an integer and is referred to as the Chern number. Presence of nonzero Chern number in a system means that the system is topologically nontrivial. One of the best examples of the manifestation of Chern numbers are the quantization factors of integer quantum Hall effect observed through electronic transport experiments.

2.6.4 Two-dimensional semiconductors as topological insulators Since the explosion of research on graphene and 2D materials, a variety of 2D SCs have been predicted to possess physics of topological origin [218
224] and partially verified experimentally as 2D topologically insulating systems [225]. Different symmetry properties exhibited by the exhaustive range of potential 2D materials allow one to study a variety of interesting physics of topological origins. Here we will highlight the three different spontaneous topological phases, namely, quantum anomalous Hall insulator arising in the absence of time-reversal symmetry, quantum valley Hall insulator arising in the absence of inversion symmetry, and quantum spin Hall insulator arising from the absence of spin-rotational symmetry. 2.6.4.1 Quantum anomalous Hall insulator Shortly after the discovery of quantum Hall effect by von Klitzing et al. [226] in 1981, its topological nature [205] was demonstrated in 1982. Thouless et al. showed that intrinsic Hall conductivity in the quantum Hall effect is described by

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σH 5

ð 2e2 d2 k f ðkÞΩ ðkÞ h ð2πÞ2

where f(k) is the Fermi
Dirac distribution and factor of 2 takes care of the spin degeneracy. The formula above and its mathematical analogues are known as the Thouless
Kohmoto
Nightingale
den Nijs invariant. Following on from initial efforts, in 1988 Haldane (constructing his model on then hypothetical material graphene) showed that in 2D systems lacking time-reversal symmetry (e.g., due to spontaneous magnetic ordering), a quantized Hall conductance could be observed in the absence of any magnetic field, as a result of presence of a spontaneous Berry curvature, thus predicting the quantum anomalous Hall effect [227]. This prediction led to Haldane being awarded the 2016 Nobel Prize in physics alongside Thouless and Kosterilitz. Experimental realization of quantum anomalous Hall effect has so far been limited to more traditional heterostructures, thin films of magnetically doped Bi2Sb2Te3 where quantization of Hall signal in the absence of magnetic field was indeed experimentally observed [228]. Fig. 2.22 shows quantized anomalous Hall resistance for a V-doped (Bi, Se)2Te3 film [230]. As demonstrated theoretically by Haldane, 2D materials and specifically 2D SCs lacking time-reversal symmetry are all candidates for experimental realization of a quantum anomalous Hall insulator. Explicit theoretical predictions of quantum anomalous Hall Effect in 2D systems include graphene [231], silicene [224], and 2D sheets formed from self-assembled organic molecules such as triphenyl-manganese [222]. 2.6.4.2 Quantum spin Hall insulator Inspiration for the theoretical prediction of quantum spin Hall insulator came following the isolation of graphene and demonstration of its topological properties [216,217]. In 2005 Kane and Mele found that inclusion of a gap, induced by spin-orbit coupling, leads to a

FIGURE 2.22

(A) Schematic of a magnetic topological insulator thin film device lacking time reversal symmetry due to its intrinsic ferromagnetism [229]. (B) Quantized anomalous Hall resistance with a conductance value of e2/h at charge neutrality point of a (Bi,Se)2Te3 film at zero magnetic field [230].

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gapped phase preserving both time-reversal and inversion symmetry as the spindependent term describing the interaction is even under parity (inversion) and odd under time reversal [207]. A crucial property of the spin-orbit interaction is that it was found to give rise to a gap of opposite signs at K and K0 points, respectively, with low energy bands describing the edge states connecting the conduction and valance bands of K and K0 valley pairs forming a Kramer’s doublet at their intersection point. The spin-orbit coupling term could be viewed as a replacement of the term Δ2 σz in Eq. (2.11) by ðΔ=2Þσz τ z sz where τ z takes care of valley degeneracy and sz takes care of the spin degeneracy. Considered separately for sz 5 6 1, the effective Hamiltonians for each spin violate time-reversal symmetry and giving rise to a Berry curvature of opposite values for each spin and associated quantized Hall conductivity [207]. However, aside from the quantized spin-filtered edge currents and the quantum spin Hall phase is characterized by a mathematical quantity, namely, Z2 index [208] and a possible point of viewing it is as [232] # " I ð 1 2 Z2 5 dk AðkÞ 2 d kΩz ðkÞ modð2Þ (2.15) 2π



dHBZ

HBZ

where the limits of the integral are related to the half Brillouin zone defined so that only one of k or 2 k is included at a time. The factor mod(2) makes sure the value of the Z2 index obtained from the equation above is either 0 or 1, meaning that the system is topologically nontrivial in the case of it being equal to 1. Despite the first theoretical prediction for graphene, the first experimental realization of quantum spin Hall effect happened in another candidate system, HgTe quantum wells [233]. Prediction on graphene, even though provided the necessary theoretical grounds, was experimentally unrealistic given the low spin-orbit coupling of graphene. The realistic 2D material candidates were the TMDCs which possess heavier atoms resulting in larger intrinsic spin-orbit coupling. Indeed, monolayer TMDCs in 1T0 form [MX2, M 5 (W,Mo), X 5 (Te, Se, S)] were predicted to possess a Z2 index equal to 1 making them candidates for the observation of quantum spin Hall phase [218]. These predictions were indeed confirmed for the case of WTe2 where edge currents were observed [212] initially, and later quantization was found to persist at temperatures up to 100K [225]. 2.6.4.3 Quantum valley Hall Quantum valley Hall effect, as the name implies, is the effect where zero magnetic field quantized Hall conductivity arises in systems lacking inversion symmetry [124]. A quantum valley Hall state exists due to the presence of finite Berry curvature as in the case of quantum anomalous Hall state; however, a crucial difference is that this state exists due to lack of inversion symmetry, and due to the presence of time-reversal symmetry. Berry curvature is an odd function of k, having the values of Ω ðkÞ and 2Ω ð 2 kÞ at K and K0 valleys, respectively. This results in nonzero but opposite sign Chern numbers at each valley and a quantization arising from it as shown in Eq. (2.15). Application of a net electric field leads to the presence of current; in the absence of scattering, this is described by the semiclassical equation of motion

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FIGURE 2.23

Valley Hall currents in a bilayer graphene system due to finite Berry curvature in the absence of inversion symmetry. Source: From Y. Shimazaki, et al., Generation and detection of pure valley current by electrically induced Berry curvature in bilayer graphene, Nat. Phys. 11 (2015) 1032, doi:10.1038/nphys3551.

¯hk_ 5 eE

(2.16)

where E is the electric field. The resulting velocity of the electrons in the presence of Berry curvature is given by vðkÞ 5

1 @εðkÞ _ 1 k 3 ΩðkÞ ¯h @k

(2.17)

The first term in the equation originates from the standard definition of Fermi velocity as the group velocity of the dispersion relation. It can be seen that this term indeed resembles the Lorentz force in the presence of a magnetic field, vðkÞ 3 B, and is therefore responsible for the zero magnetic field Hall conductivity, often referred to as valley Hall effect, which has been demonstrated through both electrical transport and optical characterization, Fig. 2.23. Quantum analogue of the valley Hall effect arises in the absence of intervalley scattering, where the Hall conductivity, as a result of valley Hall currents, is expected to approach a quantized value equal to 6 e2 =h [124,206]. Quantum valley Hall (QVH) insulator has been predicted in inversion symmetry broken graphene layers, where A/B sublattice symmetry is induced through the use of a substrate such as the alignment with underlying hBN crystals. It has also been suggested to be present in AB-stacked bilayer graphene systems where the inversion symmetry is tuned through the use of a net electric field as well as AB
BA stacking domain walls. However, the experimental realization of quantization of a 2D nature in these systems has been hindered by the presence of intervalley scattering [24,121,127]. However, imperfect quantization of valley Hall currents has been demonstrated in AB
BA stacked domain walls [234] (Fig. 2.24), and, more recently, quantization of nonlocal signals has been claimed on a ballistic graphene/hBN superlattice system and attributed to quantized valley Hall currents [235]. In 2D semiconducting systems, QVH effect theoretically has been predicted in silicene [221] as well as BiX/SiX monolayers (X 5 F, Cl, and Br) [223]. However, the stability of these systems is currently a remaining issue prohibiting detailed experimental investigations.

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FIGURE 2.24 (A) Conductance at AB
BA stacking domains of bilayer graphene due 1D valley Hall currents for varying sizes of domain walls, approaching quantized limit for smaller domains. (B) Schematic illustration of the opposite direction of currents for each valley at each domain leading to quantization in a ballistic limit. Source: Adapted from L. Ju, et al., Topological valley transport at bilayer graphene domain walls. Nature 520 (2015) 650
655, doi:10.1038/nature14364.

2.6.5 Topological insulators First hints of generalization of topological effects discussed in previous sections to three dimensions happened with a theoretical consideration of systems with a relatively narrow bandgap arising from spin-orbit coupling effects. Extending the Z2 index to these 3D systems, it was found that due to the presence of spin-orbit coupling effects, eight timereversal invariant points arise in the Brillouin zone giving rise to four invariants, describing a 3D topological insulator system [209], namely, ν 0 and ν 1 ; ν 2 ; ν 3 . The first invariant, ν 0 , is robust against the presence of disorder, and it is the parameter that distinguishes weak and strong topological insulators [209,236]. Weak topological insulators, where ν 0 invariant is 0, are the simplest types of 3D systems which could be constructed through stacking of multiple 2D quantum spin Hall insulator systems, where the invariants ν 1 ν 2 ν 3 would be acting as Miller indices and edge states would be turning into anisotropic surface states. A fundamental difference with the 2D quantum spin Hall state and the weak topological insulator is that surface states are no longer protected by time reversal symmetry and the system is not robust against the presence of disorder. For strong topological insulators, ν 0 invariant is equal to one. The ν 0 invariant determines whether there is odd (1) or even (0) number of Kramers degenerate points, giving rise to low energy surface states for the case of it being equal to one. The most remarkable feature of these low energy states is that each Kramers degenerate point is a Dirac point. For the simplest case, where the number of degenerate pairs is one, a single Dirac point is present, and it is described with the identical massless Dirac Hamiltonian to graphene highlighted in Section 6.1 with the difference being the absence of K and K0 valleys. As a result, rather than the pseudospin degree of freedom for the case of graphene and 2D semiconducting systems, in topological insulators, chirality originates from linking of the spin degree of freedom with momentum, and the Berry phase of π originates from rotation of spins in momentum space [236].

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FIGURE 2.25 (A) The low energy surface bands of a Bi2Se3 system intersecting at a Kramers point and undergoing linear dispersion leading to associated massless helical Dirac fermions (B). Source: From D. Hsieh, et al., A tunable topological insulator in the spin helical Dirac transport regime, Nature 460 (2009) 1101
1105, doi:10.1038/nature08234.

Shortly after the theoretical predictions, experimental breakthroughs came through the angle resolved photoemission spectroscopy (ARPES) experiments carried out by the Zahid Hasan group in 2008 and 2009 where they were initially able to identify Kramers points and gapless surface bands in Bi12xSbx systems [237], and later the predicted spinmomentum locked Dirac cones in Bi2Se3 systems [238], Fig. 2.25. Due to the large bandgap it possesses (0.3 eV), as well as the single Dirac cone verified by ARPES experiments, Bi2Se3 in the years to follow has become the reference material, especially in electrical transport experiments. A fundamental property of Bi2Se3 is that it is a layered van der Waals crystal, and it can be studied through mechanical exfoliation similarly to 2D SC systems. As a result of these studies, it has been found that there is indeed a crossover to the 2D limit in exfoliated Bi2Se3 crystals where the surface states become gapped with Rashba spin-orbit splitting [211].

2.6.6 Weyl and Dirac semimetals The extension of Dirac systems to 3D with the prediction and discovery of topological insulators showed the significance of systems exhibiting strong spin-orbit coupling effects [239]. A further extension to relativistic condensed matter physics systems came with the discovery of low energy Weyl nodes in systems exhibiting large spin-orbit coupling in the absence of either inversion or time reversal symmetries [240]. It was found that when a topological insulator undergoes a continuous transition into a topologically trivial gapless metal, at a certain point, in the presence of inversion and time-reversal symmetries, a fourfold degenerate Dirac cone is obtained in 3D momentum space when crossings occur between twofold degenerate bands, giving rise to a Dirac semimetal [210,237]. In the absence of either inversion or time-reversal symmetry, two twofold degenerate 3D Dirac cones separated in momentum space are obtained, giving rise to a Weyl semimetal [240]. The name originates from Hermann Weyl who in 1929 proposed the existence of a solution to the Dirac equation in 3D, predicting right- and left-handed chiral massless fermions [241]. Neutrinos were prime candidates as Weyl fermions; however, this was ruled out with the discovery of their intrinsic mass. As a result, attention was shifted to condensed matter systems with the observation of Weyl nodes in gapless semimetals with high spinorbit coupling such as TaAs [242]. Indeed, many of the predicted phenomena for Weyl

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FIGURE 2.26 (A) Schematic illustrating linearly dispersing 3D Dirac cones and the associated Fermi arcs of a Weyl semimetal. (B) In plane current and magnetic field angle dependence of negative magnetoresistance in exfoliated WTe2 films. Source: From (A) B.A. Bernevig, It’s been a Weyl coming, Nat. Phys. 11 (2015) 698
699 [243] and (B) Y.J. Wang, et al., Gate-tunable negative longitudinal magnetoresistance in the predicted type-II Weyl semimetal WTe2, Nat. Commun. 7 (2016), doi:ARTN314210.1038/ncomms13142 [244].

fermions, such as chiral anomaly, were confirmed in these systems through observed negative longitudinal magnetoresistance in electronic transport experiments (Fig. 2.26B) [245,246]. As highlighted earlier, the consequence of the Weyl equation is that Weyl fermions arise as right- and left-handed pairs. In Weyl semimetals, the Weyl node pairs act as sources and sinks of Berry curvature in momentum space. Analogous to graphene, Weyl fermions also possess a pseudospin degree of freedom, where the pseudospins point from one Weyl node to the other, coupling to the direction of momentum and giving rise to topological surface Fermi arcs originating from the Berry curvature monopole, Fig. 2.26A. Presence of Fermi arcs was confirmed through ARPES investigations in materials such as TaAs and TaP [242,247]. Fermi arcs have also been observed in topological Dirac semimetals that possess fourfold degenerate Dirac point pairs that are separated in kz momentum space respecting rotation symmetry around the z-axis in material such as Na3Bi [242,248].

2.6.7 Nodal line semimetals Nodal line semimetals are zero gap SCs where band touching points of conduction and valance bands occur along lines or loops in 3D Brillouin zone instead of points as in the case of Weyl and Dirac semimetals [210], Fig. 2.27A. It has been shown that these nodal lines are symmetry-protected, and being analogous to other 3D topological matter, they are also characterized by Z invariants as well as Berry phase [210]. Nodal lines have been

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FIGURE 2.27 (A) 3D plot illustrating the two merging Dirac cones forming the nodal line of a line node semimetal [210]. (B) Butterfly positive magnetoresistance of a nodal line semimetal ZrSiSe for varying angles between in-plane current and magnetic field [249].

theoretically shown to be originating in different systems such as ZrSiS, TlTaSe2, 3D graphene networks, Ca3P2, Cu3N, BaSn2 and BaVS3 from a variety of symmetry considerations including mirror reflection, inversion, time-reversal, and spin and screw rotations [19,250
253], see Fig. 2.27B for a ZrSiSe nodal line semimetal. Various topological consequences exist for nodal line semimetals dependent on the symmetry properties of the system, which include low energy flat bands as well as drumhead such as surface states [254,255]. However, the complicated electronic structure has hindered detailed experimental investigations into these systems. Nevertheless, ARPES and limited electrical transport characterization of these materials have recently been obtained by verifying the predicted topological bulk and surface features [19,256,257]. A fundamental property of nodal line semimetals, linking them to 2D SCs is that a variety of them are layered crystals suggesting possibility of their exfoliation down to monolayer limit and becoming 2D topological insulators such as ZrSiS, ZrSiSe, and ZrSiTe for which devices obtained through exfoliation were used in transport experiments showing a combination of 2D and 3D effects [257].

2.7 Summary and perspectives 15 years the field of 2D materials and 2D SCs particularly remains at the forefront of mesoscopic and nanoscale physics displaying a variety of stimulating phenomena ranging from superconductivity and magnetism to topological states. In particular the field of twistronics has recently enabled observation of flat-band superconductivity and correlated insulating states, while a variety of new exotic physics is still expected. Superconductivity is also being studied extensively in materials such as NbSe2, MoS2, and ZrNCl. Nanomechanics saw developments such as the emergence of gigantic PMFs in strained bubbles and the observation of 2D piezoelectricity. Another degree of freedom, the presence of multiple valleys in the electronic band structure, arising from hexagonal crystal

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