dichalcogenides.bib

@article{PhysRevB.94.060501,
  author = {Zhang, Junhua and Aji, Vivek},
  title = {{Topological Yu-Shiba-Rusinov chain in monolayer transition-metal dichalcogenide superconductors}},
  month = aug,
  year = {2016},
  publisher = {American Physical Society},
  journal = {Phys. Rev. B},
  volume = {94},
  issue = {6},
  pages = {060501},
  numpages = {5},
  doi = {10.1103/PhysRevB.94.060501},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.94.060501},
  abstract = {Monolayers of transition-metal dichalcogenides (TMDs) are two-dimensional materials whose low-energy sector consists of two inequivalent valleys. The valence bands have a large spin splitting due to lack of inversion symmetry and strong spin-orbit coupling. Furthermore the spin is polarized up in one valley and down in the other (in directions perpendicular to the two-dimensional crystal). We focus on lightly hole-doped systems where the Fermi surface consists of two disconnected circles with opposite spins. For both proximity induced and intrinsic local attractive interaction induced superconductivity, a fully gapped intervalley pairing state is favored in this system, which is an equal superposition of the singlet and the m = 0 triplet for the lack of centrosymmetry. We show that a ferromagnetically ordered magnetic-adatom chain placed on a monolayer TMD superconductor provides a platform to realize a one-dimensional topological superconducting state characterized by the presence of Majorana zero modes at its ends. We obtain the topological phase diagram and show that the topological superconducting phase is affected not only by the adatom spacing and the direction of the magnetic moment, but also by the orientation of the chain relative to the crystal.}
}

@article{PhysRevB.93.180501,
  author = {Zhou, Benjamin T. and Yuan, Noah F. Q. and Jiang, Hong-Liang and Law, K. T.},
  title = {{Ising superconductivity and Majorana fermions in transition-metal dichalcogenides}},
  year = {2016},
  month = may,
  publisher = {American Physical Society},
  journal = {Phys. Rev. B},
  volume = {93},
  pages = {180501},
  issue = {18},
  numpages = {5},
  doi = {10.1103/PhysRevB.93.180501},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.93.180501},
  abstract = {In monolayer transition-metal dichalcogenides (TMDs), electrons in opposite K valleys are subject to opposite effective Zeeman fields, which are referred to as Ising spin-orbit coupling (SOC) fields. The Ising SOC, originating from in-plane mirror symmetry breaking, pins the electron spins to the out-of-plane directions, and results in Ising superconducting states with strongly enhanced upper critical fields. Here, we show that the Ising SOC generates equal-spin-triplet Cooper pairs with spin polarized in the in-plane directions. Importantly, the spin-triplet Cooper pairs can induce superconducting pairings in a half-metal wire placed on top of the TMD and result in a topological superconductor with Majorana end states. Direct ways to detect equal-spin triplet Cooper pairs and the differences between Ising superconductors and Rashba superconductors are discussed.}
}

@article{1604.02134v2,
  author = {Junhua Zhang and Vivek Aji},
  title = {{Topological Yu-Shiba-Rusinov chain in monolayer transition-metal dichalcogenide superconductors}},
  year = {2016},
  month = apr,
  archiveprefix = {arXiv},
  eprint = {1604.02134v2},
  primaryclass = {cond-mat.supr-con},
  comment = {5 pages + supplemental material, 2 figures},
  x-fetchedfrom = {arXiv.org},
  url = {https://arxiv.org/abs/1604.02134v2},
  abstract = {Monolayers of transition-metal dichalcogenides (TMDs) are two-dimensional materials whose low energy sector consists of two inequivalent valleys. The valence bands have a large spin splitting due to strong spin-orbit coupling. Furthermore the spin is polarized up in one valley and down in the other (in directions perpendicular to the two-dimensional crystal). We focus on lightly hole-doped systems where the Fermi surface consists of two disconnected circles with opposite spins. For both proximity induced and intrinsic local attractive interaction induced superconductivity, a fully gapped intervalley pairing state is favored in this system, which is an equal superposition of the singlet and the m=0 triplet for the lack of centrosymmetry. We show that a ferromagnetically ordered magnetic-adatom chain placed on a monolayer TMD superconductor provides a platform to realize one-dimensional topological superconducting state characterized by the presence of Majorana zero modes at its ends. We obtain the topological phase diagram and show that the topological superconducting phase is affected not only by the adatom spacing and the direction of the magnetic moment, but also by the orientation of the chain relative to the crystal.}
}

@article{Saito2016,
  author = {Saito, Yu and Nakamura, Yasuharu and Bahramy, Mohammad Saeed and Kohama, Yoshimitsu and Ye, Jianting and Kasahara, Yuichi and Nakagawa, Yuji and Onga, Masaru and Tokunaga, Masashi and Nojima, Tsutomu and Yanase, Youichi and Iwasa, Yoshihiro},
  title = {{Superconductivity protected by spin-valley locking in ion-gated ${\mathrm{MoS}}_{2}$}},
  year = {2016},
  month = feb,
  publisher = {Nature Publishing Group},
  journal = {Nat Phys},
  volume = {12},
  number = {2},
  pages = {144-149},
  note = {Letter},
  issn = {1745-2473},
  url = {https://dx.doi.org/10.1038/nphys3580},
  abstract = {Symmetry-breaking has been known to play a key role in non-centrosymmetric superconductors with strong spin–orbit interactions (SOIs; refs 1,2,3,4,5,6). The studies, however, have been so far mainly focused on a particular type of SOI, known as the Rashba SOI (ref. 7), whereby the electron spin is locked to its momentum at a right-angle, thereby leading to an in-plane helical spin texture. Here we discuss electric-field-induced superconductivity in molybdenum disulphide (MoS2), which exhibits a fundamentally different type of intrinsic SOI, manifested by an out-of-plane Zeeman-type spin polarization of energy valleys8, 9, 10. We find an upper critical field of approximately 52 T at 1.5 K, which indicates an enhancement of the Pauli limit by a factor of four as compared to that in centrosymmetric conventional superconductors. Using realistic tight-binding calculations, we reveal that this unusual behaviour is due to an inter-valley pairing that is symmetrically protected by Zeeman-type spin–valley locking against external magnetic fields. Our study sheds light on the interplay of inversion asymmetry with SOIs in confined geometries, and its role in superconductivity.}
}

@article{Xi2016,
  author = {Xi, Xiaoxiang and Wang, Zefang and Zhao, Weiwei and Park, Ju-Hyun and Law, Kam Tuen and Berger, Helmuth and Forro, Laszlo and Shan, Jie and Mak, Kin Fai},
  title = {{Ising pairing in superconducting ${\mathrm{NbSe}}_{2}$ atomic layers}},
  year = {2016},
  month = feb,
  publisher = {Nature Publishing Group},
  journal = {Nat Phys},
  volume = {12},
  number = {2},
  pages = {139-143},
  note = {Letter},
  issn = {1745-2473},
  url = {https://dx.doi.org/10.1038/nphys3538},
  abstract = {The properties of two-dimensional transition metal dichalcogenides arising from strong spin–orbit interactions and valley-dependent Berry curvature effects have recently attracted considerable interest1, 2, 3, 4, 5, 6, 7. Although single-particle and excitonic phenomena related to spin–valley coupling have been extensively studied1, 3, 4, 5, 6, the effects of spin–valley coupling on collective quantum phenomena remain less well understood. Here we report the observation of superconducting monolayer NbSe2 with an in-plane upper critical field of more than six times the Pauli paramagnetic limit, by means of magnetotransport measurements. The effect can be interpreted in terms of the competing Zeeman effect and large intrinsic spin–orbit interactions in non-centrosymmetric NbSe2 monolayers, where the electron spin is locked to the out-of-plane direction. Our results provide strong evidence of unconventional Ising pairing protected by spin–momentum locking, and suggest further studies of non-centrosymmetric superconductivity with unique spin and valley degrees of freedom in the two-dimensional limit.}
}

@article{1512.01261v2,
  author = {Evan Sosenko, Junhua Zhang and Vivek Aji},
  title = {{Superconductivity in transition metal dichalcogenides}},
  year = {2015},
  month = dec,
  archiveprefix = {arXiv},
  eprint = {1512.01261v2},
  primaryclass = {cond-mat.supr-con},
  comment = {5 pages, 3 figures},
  x-fetchedfrom = {arXiv.org},
  url = {https://arxiv.org/abs/1512.01261v2},
  abstract = {Strong spin orbit interaction has the potential to engender unconventional superconducting states. Two dimensional dichalcogenides, MX2 (M=Mo, W and X=S,Se,Te), are particularly interesting: the noninteracting electronic states have multiple valleys in the energy dispersion and are topologically nontrivial. We report on the possible superconducting states of hole-doped systems, and analyze to what extent the correlated phase inherits the topological aspects of the parent crystal. We find that local attractive interactions or proximal coupling to s-wave superconductors lead to a pairing which is an equal mixture of spin singlet and m=0 spin triplet. The valley contrasting optical response, where oppositely circularly polarized light couples to different valleys, is present even in the superconducting state but with smaller magnitude. The locking of spin to momentum results in an unusual response to a magnetic field. In the absence of disorder, pair-breaking does not occur for fields in the plane of the crystal. Consequently, the critical magnetic fields are quite large compared to conventional superconductors. These results demonstrate the rich correlated phenomena possible due to the interplay of spin-orbit coupling, interaction, and topology.}
}

@article{1510.06289v2,
  author = {Tong Zhou and Hong-Liang Jiang and Noah F.Q. Yuan and K. T. Law},
  title = {{Ising Superconductivity and Majorana Fermions in Transition Metal Dichalcogenides}},
  year = {2015},
  month = oct,
  archiveprefix = {arXiv},
  eprint = {1510.06289v2},
  primaryclass = {cond-mat.supr-con},
  comment = {3 figures. Comments are welcome},
  x-fetchedfrom = {arXiv.org},
  url = {https://arxiv.org/abs/1510.06289v2},
  abstract = {In monolayer transition metal dichalcogenides (TMDs), electrons in opposite $K$ valleys are subject to opposite effective Zeeman fields, which are referred to as Ising spin-orbit coupling (SOC) fields. The Ising SOC, originated from in-plane mirror symmetry breaking pins the electron spins in out-of-plane directions, and results in the newly discovered Ising superconducting states with strongly enhanced upper critical fields. In this work, we show that the Ising SOC generates equal-spin triplet Cooper pairs with spin polarization in the in-plane directions. Importantly, the spin-triplet Cooper pairs can induce superconducting pairings in a half-metal wire placed on top of the TMD and result in a topological superconductor with Majorana end states. Direct ways to detect equal-spin triplet Cooper pairs and the differences between Ising superconductors and Rashba superconductors are discussed.}
}

@article {Lu1353,
  author = {Lu, J. M. and Zheliuk, O. and Leermakers, I. and Yuan, N. F. Q. and Zeitler, U. and Law, K. T. and Ye, J. T.},
  title = {{Evidence for two-dimensional Ising superconductivity in gated ${\mathrm{MoS}}_{2}$}},
  year = {2015},
  month = dec,
  publisher = {American Association for the Advancement of Science},
  journal = {Science},
  volume = {350},
  number = {6266},
  pages = {1353--1357},
  issn = {0036-8075},
  doi = {10.1126/science.aab2277},
  url = {http://science.sciencemag.org/content/350/6266/1353},
  abstract = {In Cooper pairs{\textemdash}pairs of electrons responsible for the exotic properties of superconductors{\textemdash}the two electrons{\textquoteright} spins typically point in opposite directions. A strong-enough external magnetic field will destroy superconductivity by making the spins point in the same direction. Lu et al. observed a two-dimensional superconducting state in the material MoS2 that was surprisingly immune to a magnetic field applied in the plane of the sample (see the Perspective by Suderow). The band structure of MoS2 and its spin-orbit coupling conspired to create an effective magnetic field that reinforced the electron pairing, with spins aligned perpendicular to the sample.Science, this issue p. 1353; see also p. 1316The Zeeman effect, which is usually detrimental to superconductivity, can be strongly protective when an effective Zeeman field from intrinsic spin-orbit coupling locks the spins of Cooper pairs in a direction orthogonal to an external magnetic field. We performed magnetotransport experiments with ionic-gated molybdenum disulfide transistors, in which gating prepared individual superconducting states with different carrier dopings, and measured an in-plane critical field Bc2 far beyond the Pauli paramagnetic limit, consistent with Zeeman-protected superconductivity. The gating-enhanced Bc2 is more than an order of magnitude larger than it is in the bulk superconducting phases, where the effective Zeeman field is weakened by interlayer coupling. Our study provides experimental evidence of an Ising superconductor, in which spins of the pairing electrons are strongly pinned by an effective Zeeman field.}
}

@article{1508.03068,
  author = {Jieun Lee and Kin Fai Mak and Jie Shan},
  title = {{Electrical control of the valley Hall effect in bilayer ${\mathrm{MoS}}_{2}$ transistors}},
  year = {2015},
  month = aug,
  archiveprefix = {arXiv},
  eprint = {1508.03068},
  primaryclass = {cond-mat.mes-hall},
  comment = {10 pages, 4 figures},
  x-fetchedfrom = {arXiv.org},
  url = {https://arxiv.org/abs/1508.03068},
  abstract = {The valley degree of freedom of electrons in solids has been proposed as a new type of information carriers beyond the electronic charge and spin. Recent experimental demonstrations of the optical orientation of the valley polarization and generation of the valley current through the valley Hall effect in monolayer MoS2 have shown the potential of two-dimensional semiconductor transition metal dichalcogenides for valley based electronic and optoelectronic applications. The valley Hall conductivity in monolayer MoS2, a non-centrosymmetric crystal, however, cannot be easily tuned, presenting a challenge for valley-based applications. Here we report the control of the valley Hall effect in bilayer MoS2 transistors through a gate. The inversion symmetry present in bilayer MoS2 was broken by the gate applied electric field perpendicular to the plane. The valley polarization near the edges of the device channels induced by the longitudinal electrical current was imaged by use of Kerr rotation microscopy. The polarization is out-of-plane, has opposite sign for the two edges, and is strongly dependent on the gate voltage. The observation is consistent with the symmetry dependent Berry curvature and valley Hall conductivity in bilayer MoS2. Our results are another step towards information processing based on the valley degree of freedom.}
}

@article{PhysRevB.91.094510,
  author = {Das, Tanmoy and Dolui, Kapildeb},
  title = {{Superconducting dome in ${\mathrm{MoS}}_{2}$ and ${\mathrm{TiSe}}_{2}$ generated by quasiparticle-phonon coupling}},
  year = {2015},
  month = mar,
  journal = {Phys. Rev. B},
  volume = {91},
  pages = {094510},
  issue = {9},
  numpages = {9},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevB.91.094510},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.91.094510},
  abstract = {We use a first-principles based self-consistent momentum-resolved density fluctuation (MRDF) model to compute the combined effects of electron-electron and electron-phonon interactions to describe the superconducting dome in the correlated MoS2 thin flake and TiSe2. We find that without including the electron-electron interaction, the electron-phonon coupling and the superconducting transition temperature (Tc) are overestimated in both these materials. However, once the full angular and dynamical fluctuations of the spin and charge density induced quasiparticle self-energy effects are included, the electron-phonon coupling and Tc are reduced to the experimental value. With doping, both electronic correlation and electron-phonon coupling grows, and above some doping value, the former becomes so large that it starts to reduce the quasiparticle-phonon coupling constant and Tc, creating a superconducting dome, in agreement with experiments.}
}

@article{doi:10.1146/annurev-conmatphys-020911-125138,
  author = {William Witczak-Krempa and Gang Chen and Yong Baek Kim and Leon Balents},
  title = {{Correlated Quantum Phenomena in the Strong Spin-Orbit Regime}},
  year = {2014},
  month = dec,
  journal = {Annual Review of Condensed Matter Physics},
  volume = {5},
  number = {1},
  pages = {57-82},
  doi = {10.1146/annurev-conmatphys-020911-125138},
  url = {https://dx.doi.org/10.1146/annurev-conmatphys-020911-125138},
  abstract = {We discuss phenomena arising from the combined influence of electron correlation and spin-orbit coupling (SOC), with an emphasis on emergent quantum phases and transitions in heavy transition metal compounds with 4d and 5d elements. A common theme is the influence of spin-orbital entanglement produced by SOC, which influences the electronic and magnetic structure. In the weak-to-intermediate correlation regime, we show how nontrivial band-like topology leads to a plethora of phases related to topological insulators (TIs). We expound these ideas using the example of pyrochlore iridates, showing how many novel phases, such as the Weyl semimetal, axion insulator, topological Mott insulator, and TIs, may arise in this context. In the strong correlation regime, we argue that spin-orbital entanglement fully or partially removes orbital degeneracy, reducing or avoiding the normally ubiquitous Jahn-Teller effect. As we illustrate for the honeycomb-lattice iridates and double perovskites, this leads to enhanced quantum fluctuations of the spin-orbital entangled states and the chance to promote exotic spin liquid and multipolar ordered ground states. Connections to experiments, materials, and future directions are discussed.}
}

@article{PhysRevLett.113.097001,
  author = {Yuan, Noah F. Q. and Mak, Kin Fai and Law, K. T.},
  title = {{Possible Topological Superconducting Phases of ${\mathrm{MoS}}_{2}$}},
  month = aug,
  year = {2014},
  publisher = {American Physical Society},
  journal = {Phys. Rev. Lett.},
  volume = {113},
  pages = {097001},
  issue = {9},
  numpages = {5},
  doi = {10.1103/PhysRevLett.113.097001},
  url = {https://link.aps.org/doi/10.1103/PhysRevLett.113.097001},
  abstract = {Molybdenum disulphide (MoS2) has attracted much interest in recent years due to its potential applications in a new generation of electronic devices. Recently, it was shown that thin films of MoS2 can become superconducting with a highest Tc of 10 K when the material is heavily gated to the conducting regime. In this work, using the group theoretical approach, we determine the possible pairing symmetries of heavily gated MoS2. Depending on the electron-electron interactions and Rashba spin-orbit coupling, the material can support an exotic spin-singlet p+ip-wavelike, an exotic spin-triplet s-wavelike, and a conventional spin-triplet p-wave pairing phase. Importantly, the exotic spin-singlet p+ip-wave phase is a topological superconducting phase that breaks time-reversal symmetry spontaneously and possesses nonzero Chern numbers where the Chern number determines the number of branches of chiral Majorana edge states.}
}

@article{Mak27062014,
  author = {Mak, K. F. and McGill, K. L. and Park, J. and McEuen, P. L.},
  title = {{The valley Hall effect in ${\mathrm{MoS}}_{2}$ transistors}},
  year = {2014},
  month = jun,
  publisher = {American Association for the Advancement of Science},
  journal = {Science},
  volume = {344},
  number = {6191},
  pages = {1489-1492},
  doi = {10.1126/science.1250140},
  url = {https://www.sciencemag.org/content/344/6191/1489.abstract},
  abstract ={Electrons in two-dimensional crystals with a honeycomb lattice structure possess a valley degree of freedom (DOF) in addition to charge and spin. These systems are predicted to exhibit an anomalous Hall effect whose sign depends on the valley index. Here, we report the observation of this so-called valley Hall effect (VHE). Monolayer MoS2 transistors are illuminated with circularly polarized light, which preferentially excites electrons into a specific valley, causing a finite anomalous Hall voltage whose sign is controlled by the helicity of the light. No anomalous Hall effect is observed in bilayer devices, which have crystal inversion symmetry. Our observation of the VHE opens up new possibilities for using the valley DOF as an information carrier in next-generation electronics and optoelectronics.}
}

@article{Xu2014,
  author = {Xu, Xiaodong and Yao, Wang and Xiao, Di and Heinz, Tony F.},
  title = {{Spin and pseudospins in layered transition metal dichalcogenides}},
  year = {2014},
  month = may,
  publisher = {Nature Publishing Group},
  journal = {Nat Phys},
  volume = {10},
  number = {5},
  pages = {343-350},
  note = {Review},
  issn = {1745-2473},
  url = {https://dx.doi.org/10.1038/nphys2942},
  abstract = {The recent emergence of two-dimensional layered materials — in particular the transition metal dichalcogenides — provides a new laboratory for exploring the internal quantum degrees of freedom of electrons and their potential for new electronics. These degrees of freedom are the real electron spin, the layer pseudospin, and the valley pseudospin. New methods for the quantum control of the spin and these pseudospins arise from the existence of Berry phase-related physical properties and strong spin–orbit coupling. The former leads to the versatile control of the valley pseudospin, whereas the latter gives rise to an interplay between the spin and the pseudospins. Here, we provide a brief review of both theoretical and experimental advances in this field.}
}

@article{PhysRevB.88.054515,
  author = {Rold\'an, R. and Cappelluti, E. and Guinea, F.},
  title = {{Interactions and superconductivity in heavily doped ${\mathrm{MoS}}_{2}$}},
  year = {2013},
  month = aug,
  publisher = {American Physical Society},
  journal = {Phys. Rev. B},
  volume = {88},
  pages = {054515},
  issue = {5},
  numpages = {5},
  doi = {10.1103/PhysRevB.88.054515},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.88.054515},
  abstract = {We analyze the microscopic origin and the physical properties of the superconducting phase recently observed in MoS2. We show how the combination of the valley structure of the conduction band, the density dependence of the screening of the long-range Coulomb interactions, the short-range electronic repulsion, and the relative weakness of the electron-phonon interactions makes possible the existence of a phase where the superconducting order parameter has opposite signs in different valleys, resembling the superconductivity found in the pnictides and cuprates.}
}

@article{PhysRevB.88.085433,
  author = {Gui-Bin Liu and Wen-Yu Shan and Yugui Yao and Wang Yao and Di Xiao},
  title = {{Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides}},
  year = {2013},
  month = aug,
  publisher = {American Physical Society},
  journal = {Phys. Rev. B},
  volume = {88},
  pages = {085433},
  issue = {8},
  numpages = {10},
  doi = {10.1103/PhysRevB.88.085433},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.88.085433},
  abstract = {We present a three-band tight-binding (TB) model for describing the low-energy physics in monolayers of group-VIB transition metal dichalcogenides MX2 (M=Mo, W; X=S, Se, Te). As the conduction- and valence-band edges are predominantly contributed by the dz2, dxy, and dx2−y2 orbitals of M atoms, the TB model is constructed using these three orbitals based on the symmetries of the monolayers. Parameters of the TB model are fitted from the first-principles energy bands for all MX2 monolayers. The TB model involving only the nearest-neighbor M-M hoppings is sufficient to capture the band-edge properties in the ±K valleys, including the energy dispersions as well as the Berry curvatures. The TB model involving up to the third-nearest-neighbor M-M hoppings can well reproduce the energy bands in the entire Brillouin zone. Spin-orbit coupling in valence bands is well accounted for by including the on-site spin-orbit interactions of M atoms. The conduction band also exhibits a small valley-dependent spin splitting which has an overall sign difference between MoX2 and WX2. We discuss the origins of these corrections to the three-band model. The three-band TB model developed here is efficient to account for low-energy physics in MX2 monolayers, and its simplicity can be particularly useful in the study of many-body physics and physics of edge states.}
}

@article{PhysRevB.88.075409,
  author = {Cappelluti, E. and Rold\'an, R. and Silva-Guill\'en, J. A. and Ordej\'on, P. and Guinea, F.},
  title = {{Tight-binding model and direct-gap/indirect-gap transition in single-layer and multilayer ${\mathrm{MoS}}_{2}$}},
  year = {2013},
  month = aug,
  journal = {Phys. Rev. B},
  volume = {88},
  pages = {075409},
  issue = {7},
  numpages = {18},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevB.88.075409},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.88.075409},
  abstract = {In this paper we present a paradigmatic tight-binding model for single-layer as well as multilayered semiconducting MoS2 and similar transition metal dichalcogenides. We show that the electronic properties of multilayer systems can be reproduced in terms of a tight-binding modeling of the single-layer hopping terms by simply adding the proper interlayer hoppings ruled by the chalcogenide atoms. We show that such a tight-binding model makes it possible to understand and control in a natural way the transition between a direct-gap band structure, in single-layer systems, and an indirect gap in multilayer compounds in terms of a momentum/orbital selective interlayer splitting of the relevant valence and conduction bands. The model represents also a suitable playground to investigate in an analytical way strain and finite-size effects.}
}

@article{PhysRevB.88.045416,
  author = {Korm\'anyos, Andor and Z\'olyomi, Viktor and Drummond, Neil D. and Rakyta, P\'eter and Burkard, Guido and Fal'ko, Vladimir I.},
  title = {{Monolayer ${\mathrm{MoS}}_{2}$: Trigonal warping, the $\ensuremath{\Gamma}$ valley, and spin-orbit coupling effects}},
  year = {2013},
  month = jul,
  publisher = {American Physical Society},
  journal = {Phys. Rev. B},
  volume = {88},
  pages = {045416},
  issue = {4},
  numpages = {8},
  doi = {10.1103/PhysRevB.88.045416},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.88.045416},
  abstract = {We use a combined ab initio calculations and k⋅p theory based approach to derive a low-energy effective Hamiltonian for monolayer MoS2 at the K point of the Brillouin zone. It captures the features which are present in first-principles calculations but not explained by the theory of Xiao et al. [Phys Rev Lett 108, 196802 (2012)], namely the trigonal warping of the valence and conduction bands, the electron-hole symmetry breaking, and the spin splitting of the conduction band. We also consider other points in the Brillouin zone which might be important for transport properties. Our findings lead to a more quantitative understanding of the properties of this material in the ballistic limit.}
}

@article{1.4804936,
  author = {Zahid, Ferdows and Liu, Lei and Zhu, Yu and Wang, Jian and Guo, Hong},
  title = {{A generic tight-binding model for monolayer, bilayer and bulk ${\mathrm{MoS}}_{2}$}},
  year = {2013},
  month = may,
  journal = {AIP Advances},
  volume = {3},
  number = {5},
  eid = 052111,
  doi = {10.1063/1.4804936},
  url = {https://scitation.aip.org/content/aip/journal/adva/3/5/10.1063/1.4804936},
  abstract = {Molybdenum disulfide (MoS2) is a layered semiconductor which has become very important recently as an emerging electronic device material. Being an intrinsic semiconductor the two-dimensional MoS2 has major advantages as the channel material in field-effect transistors. In this work we determine the electronic structures of MoS2 with the highly accurate screened hybrid functional within the density functional theory (DFT) including the spin-orbit coupling. Using the DFT electronic structures as target, we have developed a single generic tight-binding (TB) model that accurately produces the electronic structures for three different forms of MoS2 - bulk, bilayer and monolayer. Our TB model is based on the Slater-Koster method with non-orthogonal sp3d5 orbitals, nearest-neighbor interactions and spin-orbit coupling. The TB model is useful for atomistic modeling of quantum transport in MoS2 based electronic devices.}
}

@article{Bao2013,
  author = {Bao, Wenzhong and Cai, Xinghan and Kim, Dohun and Sridhara, Karthik and Fuhrer, Michael S.},
  title = {{High mobility ambipolar ${\mathrm{MoS}}_{2}$ field-effect transistors: Substrate and dielectric effects}},
  year = {2013},
  month = jan,
  journal = {Applied Physics Letters},
  volume = {102},
  number = {4},
  eid = 042104,
  doi = {10.1063/1.4789365},
  url = {https://scitation.aip.org/content/aip/journal/apl/102/4/10.1063/1.4789365},
  abstract = {We fabricate MoS2 field effect transistors on both SiO2 and polymethyl methacrylate (PMMA) dielectrics and measure charge carrier mobility in a four-probe configuration. For multilayer MoS2 on SiO2, the mobility is 30–60 cm2/Vs, relatively independent of thickness (15–90 nm), and most devices exhibit unipolar n-type behavior. In contrast, multilayer MoS2 on PMMA shows mobility increasing with thickness, up to 470 cm2/Vs (electrons) and 480 cm2/Vs (holes) at thickness ∼50 nm. The dependence of the mobility on thickness points to a long-range dielectric effect of the bulk MoS2 in increasing mobility.}
}

@article{Ye30112012,
  author = {Ye, J. T. and Zhang, Y. J. and Akashi, R. and Bahramy, M. S. and Arita, R. and Iwasa, Y.},
  title = {{Superconducting Dome in a Gate-Tuned Band Insulator}},
  year = {2012},
  month = nov,
  journal = {Science},
  volume = {338},
  number = {6111},
  pages = {1193-1196},
  doi = {10.1126/science.1228006},
  url = {https://www.sciencemag.org/content/338/6111/1193.abstract},
  abstract ={A dome-shaped superconducting region appears in the phase diagrams of many unconventional superconductors. In doped band insulators, however, reaching optimal superconductivity by the fine-tuning of carriers has seldom been seen. We report the observation of a superconducting dome in the temperature–carrier density phase diagram of MoS2, an archetypal band insulator. By quasi-continuous electrostatic carrier doping achieved through a combination of liquid and solid gating, we revealed a large enhancement in the transition temperature Tc occurring at optimal doping in the chemically inaccessible low–carrier density regime. This observation indicates that the superconducting dome may arise even in doped band insulators.}
}

@article{PhysRevLett.108.196802,
  author = {Di Xiao and Gui-Bin Liu and Wanxiang Feng and Xiaodong Xu and Wang Yao},
  title = {{Coupled Spin and Valley Physics in Monolayers of ${\mathrm{MoS}}_{2}$ and Other Group-VI Dichalcogenides}},
  year = {2012},
  month = may,
  publisher = {American Physical Society},
  journal = {Phys. Rev. Lett.},
  volume = {108},
  pages = {196802},
  issue = {19},
  numpages = {5},
  doi = {10.1103/PhysRevLett.108.196802},
  url = {https://link.aps.org/doi/10.1103/PhysRevLett.108.196802},
  abstract = {We show that inversion symmetry breaking together with spin-orbit coupling leads to coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides, making possible controls of spin and valley in these 2D materials. The spin-valley coupling at the valence-band edges suppresses spin and valley relaxation, as flip of each index alone is forbidden by the valley-contrasting spin splitting. Valley Hall and spin Hall effects coexist in both electron-doped and hole-doped systems. Optical interband transitions have frequency-dependent polarization selection rules which allow selective photoexcitation of carriers with various combination of valley and spin indices. Photoinduced spin Hall and valley Hall effects can generate long lived spin and valley accumulations on sample boundaries. The physics discussed here provides a route towards the integration of valleytronics and spintronics in multivalley materials with strong spin-orbit coupling and inversion symmetry breaking.}
}

@article{Wang2012,
  author = {Wang, Qing Hua and Kalantar-Zadeh, Kourosh and Kis, Andras and Coleman, Jonathan N. and Strano, Michael S.},
  title = {{Electronics and optoelectronics of two-dimensional transition metal dichalcogenides}},
  year = {2012},
  month = nov,
  publisher = {Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.},
  journal = {Nat Nano},
  volume = {7},
  number = {11},
  pages = {699-712},
  doi = {10.1038/nnano.2012.193},
  issn = {1748-3387},
  url = {https://dx.doi.org/10.1038/nnano.2012.193},
  abstract = {The remarkable properties of graphene have renewed interest in inorganic, two-dimensional materials with unique electronic and optical attributes. Transition metal dichalcogenides (TMDCs) are layered materials with strong in-plane bonding and weak out-of-plane interactions enabling exfoliation into two-dimensional layers of single unit cell thickness. Although TMDCs have been studied for decades, recent advances in nanoscale materials characterization and device fabrication have opened up new opportunities for two-dimensional layers of thin TMDCs in nanoelectronics and optoelectronics. TMDCs such as MoS2, MoSe2, WS2 and WSe2 have sizable bandgaps that change from indirect to direct in single layers, allowing applications such as transistors, photodetectors and electroluminescent devices. We review the historical development of TMDCs, methods for preparing atomically thin layers, their electronic and optical properties, and prospects for future advances in electronics and optoelectronics.}
}

@article{doi:10.1021/nl2021575,
  author = {Yijin Zhang and Jianting Ye and Yusuke Matsuhashi and Yoshihiro Iwasa},
  title = {{Ambipolar ${\mathrm{MoS}}_{2}$ Thin Flake Transistors}},
  year = {2012},
  month = jan,
  journal = {Nano Letters},
  volume = {12},
  number = {3},
  pages = {1136-1140},
  doi = {10.1021/nl2021575},
  note = {PMID: 22276648},
  url = {https://dx.doi.org/10.1021/nl2021575},
  abstract = {Field effect transistors (FETs) made of thin flake single crystals isolated from layered materials have attracted growing interest since the success of graphene. Here, we report the fabrication of an electric double layer transistor (EDLT, a FET gated by ionic liquids) using a thin flake of MoS2, a member of the transition metal dichalcogenides, an archetypal layered material. The EDLT of the thin flake MoS2 unambiguously displayed ambipolar operation, in contrast to its commonly known bulk property as an n-type semiconductor. High-performance transistor operation characterized by a large “ON” state conductivity in the order of ∼mS and a high on/off ratio >102 was realized for both hole and electron transport. Hall effect measurements revealed mobility of 44 and 86 cm2 V–1 s–1 for electron and hole, respectively. The hole mobility is twice the value of the electron mobility, and the density of accumulated carrier reached 1 × 1014 cm–2, which is 1 order of magnitude larger than conventional FETs with solid dielectrics. The high-density carriers of both holes and electrons can create metallic transport in the MoS2 channel. The present result is not only important for device applications with new functionalities, but the method itself would also act as a protocol to study this class of material for a broader scope of possibilities in accessing their unexplored properties.}
}

@article{RevModPhys.83.1057,
  author = {Qi, Xiao-Liang and Zhang, Shou-Cheng},
  title = {{Topological insulators and superconductors}},
  year = {2011},
  month = oct,
  publisher = {American Physical Society},
  journal = {Rev. Mod. Phys.},
  volume = {83},
  pages = {1057--1110},
  issue = {4},
  numpages = {0},
  doi = {10.1103/RevModPhys.83.1057},
  url = {https://link.aps.org/doi/10.1103/RevModPhys.83.1057},
  abstract = {Topological insulators are new states of quantum matter which cannot be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time-reversal symmetry. These topological materials have been theoretically predicted and experimentally observed in a variety of systems, including HgTe quantum wells, BiSb alloys, and Bi2Te3 and Bi2Se3 crystals. Theoretical models, materials properties, and experimental results on two-dimensional and three-dimensional topological insulators are reviewed, and both the topological band theory and the topological field theory are discussed. Topological superconductors have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions. The theory of topological superconductors is reviewed, in close analogy to the theory of topological insulators.}
}

@article{PhysRevB.84.153402,
  author = {Zhu, Z. Y. and Cheng, Y. C. and Schwingenschl\"ogl, U.},
  title = {{Giant spin-orbit-induced spin splitting in two-dimensional transition-metal dichalcogenide semiconductors}},
  year = {2011},
  month = oct,
  publisher = {American Physical Society},
  journal = {Phys. Rev. B},
  volume = {84},
  pages = {153402},
  issue = {15},
  numpages = {5},
  doi = {10.1103/PhysRevB.84.153402},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.84.153402},
  abstract = {Fully relativistic first-principles calculations based on density functional theory are performed to study the spin-orbit-induced spin splitting in monolayer systems of the transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2. All these systems are identified as direct-band-gap semiconductors. Giant spin splittings of 148–456 meV result from missing inversion symmetry. Full out-of-plane spin polarization is due to the two-dimensional nature of the electron motion and the potential gradient asymmetry. By suppression of the Dyakonov-Perel spin relaxation, spin lifetimes are expected to be very long. Because of the giant spin splittings, the studied materials have great potential in spintronics applications.}
}

@article{RadisavljevicB.2011,
  author = {B. Radisavljevic and A. Radenovic and J. Brivio and V. Giacometti and A. Kis},
  title = {{Single-layer ${\mathrm{MoS}}_{2}$ transistors}},
  year = {2011},
  month = mar,
  publisher = {Nature Publishing Group},
  journal = {Nat Nano},
  volume = {6},
  number = {3},
  pages = {147-150},
  doi = {10.1038/nnano.2010.279},
  issn = {1748-3387},
  url = {https://dx.doi.org/10.1038/nnano.2010.279},
  abstract = {Two-dimensional materials are attractive for use in next-generation nanoelectronic devices because, compared to one-dimensional materials, it is relatively easy to fabricate complex structures from them. The most widely studied two-dimensional material is graphene, both because of its rich physics and its high mobility. However, pristine graphene does not have a bandgap, a property that is essential for many applications, including transistors. Engineering a graphene bandgap increases fabrication complexity and either reduces mobilities to the level of strained silicon films or requires high voltages. Although single layers of MoS2 have a large intrinsic bandgap of 1.8 eV (ref. 16), previously reported mobilities in the 0.5–3 cm2 V−1 s−1 range are too low for practical devices. Here, we use a halfnium oxide gate dielectric to demonstrate a room-temperature single-layer MoS2 mobility of at least 200 cm2 V−1 s−1, similar to that of graphene nanoribbons, and demonstrate transistors with room-temperature current on/off ratios of 1 × 108 and ultralow standby power dissipation. Because monolayer MoS2 has a direct bandgap, it can be used to construct interband tunnel FETs, which offer lower power consumption than classical transistors. Monolayer MoS2 could also complement graphene in applications that require thin transparent semiconductors, such as optoelectronics and energy harvesting.}
}

@article{RevModPhys.82.3045,
  author = {Hasan, M. Z. and Kane, C. L.},
  title = {{\textit{Colloquium}: Topological insulators}},
  year = {2010},
  month = nov,
  publisher = {American Physical Society},
  journal = {Rev. Mod. Phys.},
  volume = {82},
  pages = {3045--3067},
  issue = {4},
  numpages = {0},
  doi = {10.1103/RevModPhys.82.3045},
  url = {https://link.aps.org/doi/10.1103/RevModPhys.82.3045},
  abstract = {Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducting states on their edge or surface. These states are possible due to the combination of spin-orbit interactions and time-reversal symmetry. The two-dimensional (2D) topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional (3D) topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. In this Colloquium the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topological insulators have been observed. Transport experiments on HgTe∕CdTe quantum wells are described that demonstrate the existence of the edge states predicted for the quantum spin Hall insulator. Experiments on Bi1−xSbx, Bi2Se3, Bi2Te3, and Sb2Te3 are then discussed that establish these materials as 3D topological insulators and directly probe the topology of their surface states. Exotic states are described that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation. Prospects for observing these exotic states are also discussed, as well as other potential device applications of topological insulators.}
}

@article{PhysRevLett.105.136805,
  author = {Mak, Kin Fai and Lee, Changgu and Hone, James and Shan, Jie and Heinz, Tony F.},
  title = {{Atomically Thin ${\mathrm{MoS}}_{2}$: A New Direct-Gap Semiconductor}},
  year = {2010},
  month = sep,
  publisher = {American Physical Society},
  journal = {Phys. Rev. Lett.},
  volume = {105},
  pages = {136805},
  issue = {13},
  numpages = {4},
  doi = {10.1103/PhysRevLett.105.136805},
  url = {https://link.aps.org/doi/10.1103/PhysRevLett.105.136805},
  abstract = {The electronic properties of ultrathin crystals of molybdenum disulfide consisting of N=1,2,…,6 S-Mo-S monolayers have been investigated by optical spectroscopy. Through characterization by absorption, photoluminescence, and photoconductivity spectroscopy, we trace the effect of quantum confinement on the material’s electronic structure. With decreasing thickness, the indirect band gap, which lies below the direct gap in the bulk material, shifts upwards in energy by more than 0.6 eV. This leads to a crossover to a direct-gap material in the limit of the single monolayer. Unlike the bulk material, the MoS2 monolayer emits light strongly. The freestanding monolayer exhibits an increase in luminescence quantum efficiency by more than a factor of 104 compared with the bulk material.}
}

@article{RevModPhys.82.1959,
  author = {Xiao, Di and Chang, Ming-Che and Niu, Qian},
  title = {{Berry phase effects on electronic properties}},
  year = {2010},
  month = jul,
  publisher = {American Physical Society},
  journal = {Rev. Mod. Phys.},
  volume = {82},
  pages = {1959--2007},
  issue = {3},
  numpages = {0},
  doi = {10.1103/RevModPhys.82.1959},
  url = {https://link.aps.org/doi/10.1103/RevModPhys.82.1959},
  abstract = {Ever since its discovery the notion of Berry phase has permeated through all branches of physics. Over the past three decades it was gradually realized that the Berry phase of the electronic wave function can have a profound effect on material properties and is responsible for a spectrum of phenomena, such as polarization, orbital magnetism, various (quantum, anomalous, or spin) Hall effects, and quantum charge pumping. This progress is summarized in a pedagogical manner in this review. A brief summary of necessary background is given and a detailed discussion of the Berry phase effect in a variety of solid-state applications. A common thread of the review is the semiclassical formulation of electron dynamics, which is a versatile tool in the study of electron dynamics in the presence of electromagnetic fields and more general perturbations. Finally, a requantization method is demonstrated that converts a semiclassical theory to an effective quantum theory. It is clear that the Berry phase should be added as an essential ingredient to our understanding of basic material properties.}
}

@article{doi:10.1021/nl903868w,
  title = {{Emerging Photoluminescence in Monolayer ${\mathrm{MoS}}_{2}$}},
  author = {Andrea Splendiani and Liang Sun and Yuanbo Zhang and Tianshu Li and Jonghwan Kim and Chi-Yung Chim and Giulia Galli and Feng Wang},
  year = {2010},
  month = mar,
  journal = {Nano Letters},
  volume = {10},
  number = {4},
  pages = {1271-1275},
  note ={PMID: 20229981},
  doi = {10.1021/nl903868w},
  url = {https://dx.doi.org/10.1021/nl903868w},
  abstract = {Novel physical phenomena can emerge in low-dimensional nanomaterials. Bulk MoS2, a prototypical metal dichalcogenide, is an indirect bandgap semiconductor with negligible photoluminescence. When the MoS2 crystal is thinned to monolayer, however, a strong photoluminescence emerges, indicating an indirect to direct bandgap transition in this d-electron system. This observation shows that quantum confinement in layered d-electron materials like MoS2 provides new opportunities for engineering the electronic structure of matter at the nanoscale.}
}

@article{Konig02112007,
  author = {K\"{o}nig, Markus and Wiedmann, Steffen and Br\"{u}ne, Christoph and Roth, Andreas and Buhmann, Hartmut and Molenkamp, Laurens W. and Qi, Xiao-Liang and Zhang, Shou-Cheng},
  title = {{Quantum Spin Hall Insulator State in HgTe Quantum Wells}},
  journal = {Science},
  volume = {318},
  number = {5851},
  pages = {766-770},
  year = {2007},
  doi = {10.1126/science.1148047},
  url = {http://www.sciencemag.org/content/318/5851/766.abstract},
  abstract = {Recent theory predicted that the quantum spin Hall effect, a fundamentally new quantum state of matter that exists at zero external magnetic field, may be realized in HgTe/(Hg,Cd)Te quantum wells. We fabricated such sample structures with low density and high mobility in which we could tune, through an external gate voltage, the carrier conduction from n-type to p-type, passing through an insulating regime. For thin quantum wells with well width d < 6.3 nanometers, the insulating regime showed the conventional behavior of vanishingly small conductance at low temperature. However, for thicker quantum wells (d > 6.3 nanometers), the nominally insulating regime showed a plateau of residual conductance close to 2e2/h, where e is the electron charge and h is Planck's constant. The residual conductance was independent of the sample width, indicating that it is caused by edge states. Furthermore, the residual conductance was destroyed by a small external magnetic field. The quantum phase transition at the critical thickness, d = 6.3 nanometers, was also independently determined from the magnetic field–induced insulator-to-metal transition. These observations provide experimental evidence of the quantum spin Hall effect.}
}

@article{PhysRevLett.96.106802,
  author = {Bernevig, B. Andrei and Zhang, Shou-Cheng},
  title = {{Quantum Spin Hall Effect}},
  year = {2006},
  month = mar,
  publisher = {American Physical Society},
  journal = {Phys. Rev. Lett.},
  volume = {96},
  pages = {106802},
  issue = {10},
  numpages = {4},
  doi = {10.1103/PhysRevLett.96.106802},
  url = {https://link.aps.org/doi/10.1103/PhysRevLett.96.106802},
  abstract = {The quantum Hall liquid is a novel state of matter with profound emergent properties such as fractional charge and statistics. The existence of the quantum Hall effect requires breaking of the time reversal symmetry caused by an external magnetic field. In this work, we predict a quantized spin Hall effect in the absence of any magnetic field, where the intrinsic spin Hall conductance is quantized in units of 2e4π. The degenerate quantum Landau levels are created by the spin-orbit coupling in conventional semiconductors in the presence of a strain gradient. This new state of matter has many profound correlated properties described by a topological field theory.}
}

@article{PhysRevLett.95.226801,
  author = {Kane, C. L. and Mele, E. J.},
  title = {{Quantum Spin Hall Effect in Graphene}},
  year = {2005},
  month = nov,
  publisher = {American Physical Society},
  journal = {Phys. Rev. Lett.},
  volume = {95},
  pages = {226801},
  issue = {22},
  numpages = {4},
  doi = {10.1103/PhysRevLett.95.226801},
  url = {https://link.aps.org/doi/10.1103/PhysRevLett.95.226801},
  abstract = {We study the effects of spin orbit interactions on the low energy electronic structure of a single plane of graphene. We find that in an experimentally accessible low temperature regime the symmetry allowed spin orbit potential converts graphene from an ideal two-dimensional semimetallic state to a quantum spin Hall insulator. This novel electronic state of matter is gapped in the bulk and supports the transport of spin and charge in gapless edge states that propagate at the sample boundaries. The edge states are nonchiral, but they are insensitive to disorder because their directionality is correlated with spin. The spin and charge conductances in these edge states are calculated and the effects of temperature, chemical potential, Rashba coupling, disorder, and symmetry breaking fields are discussed.}
}

@article{PhysRevB.64.235305,
  author = {B\"oker, Th. and Severin, R. and M\"uller, A. and Janowitz, C. and Manzke, R. and Vo\ss{}, D. and Kr\"uger, P. and Mazur, A. and Pollmann, J.},
  title = {{Band structure of ${\mathrm{MoS}}_{2}$, ${\mathrm{MoSe}}_{2}$, and $\ensuremath{\alpha}-{\mathrm{MoTe}}_{2}$ Angle-resolved photoelectron spectroscopy and \textit{ab initio} calculations}},
  year = {2001},
  month = nov,
  publisher = {American Physical Society},
  journal = {Phys. Rev. B},
  volume = {64},
  pages = {235305},
  issue = {23},
  numpages = {11},
  doi = {10.1103/PhysRevB.64.235305},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.64.235305},
  abstract = {In this work the complete valence-band structure of the molybdenum dichalcogenides MoS2, MoSe2, and α−MoTe2 is presented and discussed in comparison. The valence bands have been studied using both angle-resolved photoelectron spectroscopy (ARPES) with synchrotron radiation, as well as ab initio band-structure calculations. The ARPES measurements have been carried out in the constant-final-state (CFS) mode. The results of the calculations show in general very good agreement with the experimentally determined valence-band structures allowing for a clear identification of the observed features. The dispersion of the valence bands as a function of the perpendicular component k⊥ of the wave vector reveals a decreasing three-dimensional character from MoS2 to α−MoTe2 which is attributed to an increasing interlayer distance in the three compounds. The effect of this k⊥ dispersion on the determination of the exact dispersion of the individual states as a function of k‖ is discussed. By performing ARPES in the CFS mode the k‖ component for off-normal emission spectra can be determined. The corresponding k⊥ value is obtained from the symmetry of the spectra along the ΓA, KH, and ML lines, respectively.}
}

@article{PhysRevLett.61.2015,
  author = {Haldane, F. D. M.},
  title = {{Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly"}},
  year = {1988},
  month = oct,
  publisher = {American Physical Society},
  journal = {Phys. Rev. Lett.},
  volume = {61},
  pages = {2015--2018},
  issue = {18},
  numpages = {0},
  doi = {10.1103/PhysRevLett.61.2015},
  url = {https://link.aps.org/doi/10.1103/PhysRevLett.61.2015},
  abstract = {A two-dimensional condensed-matter lattice model is presented which exhibits a nonzero quantization of the Hall conductance σxy in the absence of an external magnetic field. Massless fermions without spectral doubling occur at critical values of the model parameters, and exhibit the so-called "parity anomaly" of (2+1)-dimensional field theories.}
}

@article{0022-3719-5-7-007,
  author = {R A Bromley and R B Murray and A D Yoffe},
  title = {{The band structures of some transition metal dichalcogenides. III. Group VIA: trigonal prism materials}},
  year = {1972},
  month = apr,
  journal = {Journal of Physics C: Solid State Physics},
  volume = {5},
  number = {7},
  pages = {759},
  url = {http://stacks.iop.org/0022-3719/5/i=7/a=007},
  abstract = {For pt. II see ibid., vol. 5, 746 (1972). The semiempirical tight binding method is applied to the calculation of the electronic band structures of MoS2, MoSe2, alpha -MoTe2, WS2 and WSe2 in a two dimensional approximation. After a review of the structure and the physical properties of these compounds, the symmetry of the relevant Brillouin zones, including the effects of spin, is discussed. With this material one can fit the calculated band structures of all five compounds to the experimental data. Some of the fine structure of the optical spectra can be interpreted by extending the work into three dimensions. It is also possible to explain some of the properties to which the calculations were not fitted explicitly.}
}