k · p theory for two-dimensional transition metal dichalcogenide semiconductors

Abstract

We present k· p Hamiltonians parametrised by ab initio density functional theory calculations to describe the dispersion of the valence and conduction bands at their extrema (the K, Q, Γ, and M points of the hexagonal Brillouin zone) in atomic crystals of semiconducting monolayer transition metal dichalcogenides. We discuss the parametrisation of the essential parts of the k· p Hamiltonians for MoS 2 , MoSe 2 , MoTe 2 , WS 2 , WSe 2 , and WTe 2 , including the spin-splitting and spin-polarisation of the bands, and we briefly review the vibrational properties of these materials. We then use k· p theory to analyse optical transitions in two-dimensional transition metal dichalcogenides over a broad spectral range that covers the Van Hove singularities in the band structure (the M points). We also discuss the visualisation of scanning tunnelling microscopy maps. PACS numbers:Contents 1 Introduction 2 2 Lattice parameters, band-structure calculations and vibrational properties 43 Band-edge energy differences and spin-splittings 7 4 Valence band width D vb 9 arXiv:1410.6666v3 [cond-mat.mes-hall] 6 Apr 2015 k · p theory for 2D TMDCs to be studied without constructing slabs in three-dimensionally periodic cells and the resulting electronic spectra are free of plane-wave continua. All our fleur calculations were carried out with a cut-off k max of 10.6 eV −1 for the plane-wave basis set and 144 k points corresponding to a 12 × 12 × 1 Monkhorst-Pack grid in the irreducible wedge of the BZ. Muffin-tin radii of 1.0, 1.21, 1.27, 1.27, and 1.27Å were used for S, Se, Te, Mo, and W, respectively. We note that considering local orbitals for Mo (s, p), Se (s, p, d), and W (s, p, f ) to improve the linearised augmented plane-wave basis proved to be crucial for a correct description of the excited states. We used the Perdew-Burke-Ernzerhof (PBE) generalised gradient approximation [83] to the exchange-correlation potential. The structures were relaxed (with the effects of SOC included) until the forces were less than 0.0005 eV/Å. The calculated values of a 0 and d S−S for monolayer TMDCs are shown in Table 1 and compared to measured values for the corresponding bulk materials. The lattice parameters obtained from the first of the DFT approaches described above are shown in the rows labelled by "(HSE)", the ones from the second approach are in the rows labelled by "(PBE)". "(Exp)" indicates experimental results found in the literature. Although there is some scatter in the experimental data, Table 1 suggests that using the HSE06 functional to relax the monolayer crystal structure leads to a good agreement with the room-temperature empirical bulk a 0 values. On the other hand, the PBE functional seems to slightly overestimates a 0 . However, the situation is less clear in the case of d X−X . We note that both the HSE06 and the PBE results are in good agreement with Reference [84].Recent experiments show that the energy of the photoluminescence peak is quite sensitive to the temperature [5,85,86], which can be understood in terms of th...

High-temperature superfluidity of the two-component Bose gas in a transition metal dichalcogenide bilayer

“…Let us mention that the order of types of different TMDC materials with respect to the increase of T c is different from the order of these materials with respect to the exciton effective masses, presented in Table I, and the parameters a, t, and the separation between X planes d X−X taken from Ref. 29.…”

“…According to Figure 4 in Ref. 29, the spin-orbit splitting in the valence band is much larger than in the conduction band. In the valence bands of both MoX 2 and WX 2 , the energy for spin-down electrons is larger than for spin-up electrons.…”

“…Monolayers of TMDC such as MoS 2 , MoSe 2 , MoTe 2 , WS 2 , WSe 2 , and WTe 2 are 2D semiconductors, (below for TMDC monolayer we use the chemical formula MX 2 , where M denotes a transition metal M = Mo or W, and X denotes a chalcogenide, X = S, Se or Te) which have a variety of applications in electronics and opto-electronics [29]. The strong interest in TMDC monolayers is motivated by the following properties: a semiconductor band structure characterized by a direct gap in the single particle spectrum [30][31][32][33], the existence of excitonic valley physics [34,35], and the demonstration of electrically tunable, strong light-matter interactions [36,37].…”

“…In the valence bands of both MoX 2 and WX 2 , the energy for spin-down electrons is larger than for spin-up electrons. This spin-orbit spitting causes the experimentally observed energy difference between the A and B excitons [29]. Two-photon spectroscopy of excitons in monolayer TMDCs was studied using the BSE [52].…”

Direct Observation of Ultrafast Exciton Formation in a Monolayer of WSe2

“…The corresponding single-particle transitions occur at the K and K′ points of the Brillouin zone 42 (Figure 1c). Figure 2a (upper panel) shows the waveform of the transmitted probe pulse E ref (t EOS ) in absence of excitation (black curve) together with the pump-induced change ΔE(t EOS , t PP = 75 fs) (red curve).…”