Band structure and spin–orbit coupling engineering in transition‐metal dichalcogenides

Abstract

Graphene is a fascinating material, but it cannot compete with silicon for its niche in digital information technology due to one important reason: unlike silicon, graphene has no band gap. Among others, this fact is behind the push to find and investigate novel two-dimensional materials that would have a direct semiconducting gap, allowing for field-effect transistor action with large ON/OFF current ratio as well as for optoelectronic applications. Perhaps we could even have two-dimensional materials with a tunable band gap! That could lead to multifunctional and reconfigurable logical devices, only a few atomic layers thick. Further prospects for two-dimensional materials are in the field of spintronics 1. Here it is the electron spin, not only the electron charge, that determines the outcome of the device. Graphene itself appears ideal for spintronics, since its spin–orbit coupling is weak 2, and the main isotope of carbon has no nuclear spin which normally causes spin decoherence in confined structures. However, for some spintronic effects, such as spin Hall 3, spin–orbit coupling should be relatively large. For graphene this can be achieved by adding adatoms 4. Two-dimensional materials with heavier atoms than carbon have naturally larger spin–orbit coupling. While we have a reasonable understanding of the essential electronic effects with spin–orbit coupling in these novel materials, the systematic study of spin transport and spin–orbit effects in transport and optics is only starting. The two theory papers 5 and 6 in this special issue deal with electronic and spin aspects of two-dimensional transition-metal (di)chalcogenides (TMC), materials perhaps most studied after graphene. Monolayer TMC show a direct band gap in the 1–2 eV window, a very comfortable semiconducting range for optoelectronic and solar cell applications. The gap forms at the K point, also familiar from graphene physics. The optics of TMC is strongly influenced by many-body effects 7. Going to TMC bilayers, the band gap becomes indirect due to the shift of the valence band maximum to the Γ point. A direct band gap at K appears in heterobilayers combining two different TMC. Several studies have indicated that the electronic structure of monolayer TMC makes a transition from the direct to indirect gap, and ultimately to band closing (metallic regime) upon strain. Such strain engineering of the electronic band structure could open new possibilities for controlling electronic and optical properties, further boosting the prospects of TMC for applications. Indeed, the carrier density, effective masses, recombination and optical absorption rates, are all different for direct and indirect gap regimes. In the indirect gap regime, for example, the optical transitions are accompanied by phonons to preserve momentum. The transitions are thus much weaker than for the direct gap, limiting optical applications. Wang et al. 5 look at the electronic structure of MoS2 and WS2. They start with density functional calculations and include various many-body effects (GW correction, excitonic and trionic effects on a phenomenological footing) and spin–orbit coupling, to make rather fine predictions for the strain effects on the electronic structure of the two materials. They confirm that the two-dimensional semiconductors undergo a transition from the direct gap, located at K, to an indirect gap, with the conduction band maximum shifting from K to the Γ point. The authors argue that the critical strain to see the transition is higher than previously predicted values of 1%, precisely due to the inclusion of many-body corrections and spin–orbit coupling. These corrections, as a whole, have not been included in previous calculations. Wang et al. instead predict critical strain values of 2.7% for MoS2 and 3.9% for WS2. Although experiments appear to observe the direct-to-indirect gap transition at smaller values of strain, about 1%, the authors give arguments that these experiments are yet inconclusive. However, phonon renormalization of the electronic gaps has not yet been taken into consideration in the calculations. This may be the reason why the calculated gaps reported in 5 are about 10% above the experimental optical gaps. What is important, the authors argue, based on their study, is that the net strain effects are accurately included already at the DFT level without additional corrections. The correct band gap for the strain-free material is needed; the rate of change of the gap is accurately described by DFT only, without the need for many-body effects. This fact significantly lowers the computational efforts to obtain strain effects. In addition to affecting the band gap character, strain is also found to affect the electron and hole effective masses: with increasing strain the masses at K and Γ points decrease. This would further affect the mobilities of the charge carriers. TMC monolayers have no space inversion symmetry. This means that in the presence of spin–orbit coupling their electronic states are in general spin split 8: spin up and spin down electrons of the same momentum have different energies. The spin is quantized along a fixed direction, determined by the momentum. This is the Dresselhaus-Rashba effect 9, 10, exhaustively studied in conventional zinc-blende semiconductors. The spin splitting is effectively described by spin–orbit fields (effective momentum-dependent magnetic fields), which can lead to many interesting effects. In particular, they can cause a rapid spin relaxation even in rather clean systems 1. Combined with large spin–orbit coupling, we also expect interesting spin optoelectronic properties 1, such as optical spin orientation and spin pumping as well as spin-valley locking. Zibouche et al. 6 make a systematic investigation of the spin–orbit coupling in the whole set of transition-metal dichalogenides, MoS2, MoSe2, MoTe2, and WS2, WSe2, WTe2. They perform density-functional calculations on the optimized lattice structures. The spin–orbit splitting increases with the atomic number, as expected. It is smallest for MoS2, for which the valence and conduction band splittings (for the PBE functional) are about 150 and 10 meV, respectively; the splitting is largest for WTe2, reaching about 480 meV for the valence and 30 meV for the conduction band. (The authors find that the splittings depend somewhat on the functional used, making it difficult to make accurate quantitative predictions.) In TMC bilayers, which are centrosymmetric, the spin–orbit splitting is absent; there are no spin–orbit fields. However, this absence of the Dresselhaus-Rashba effect does not mean that TMC bilayers would not be useful for spintronics. On the contrary: The intrinsic spin–orbit coupling, which is always present, would yield longer spin relaxation times in clean systems of bilayer TMC, so spin transport could be of a longer range, and spin phenomena such as spin Hall effects can still be observed. However, optical applications of bilayers may be limited due to their indirect band gap. When bilayers are composed of different monolayers, as in MoS2–MoSe2, the inversion symmetry is again absent and the spin–orbit fields reappear. The spin–orbit splitting of the bands is found in 6 to be similar to the splitting in the heavier monolayer. For example, in the MS2–WSe2 bilayer the spin–orbit splitting of the valence band is calculated to be about 420 meV, close to the value for monolayer WSe2 which is 450 meV. In addition to the richer spin physics, heterobilayers have a direct band gap which makes them more suitable for applications. Another important finding is the strain control of the spin–orbit splitting. Zibouche et al. 6 calculated TMC monolayers under tensile strain. They give an example of a WS2 monolayer whose spin–orbit splitting of the valence band increases by 90 meV, and the conduction band by 110 meV for the 5% strain. At 10% strain WS2 becomes a metal with emerging Dirac cones below the Fermi level. Similarly to what is found in Wang et al. 5, the strain induces a direct-to-indirect band gap transition due to the relative increase of the valence band maximum at Γ versus at K. The further prospects of the research covered in 5 and 6 will certainly depend on the experimental progress. Both strain and spin physics should be investigated in transport and optical experiments. Spin injection, spin relaxation 11, optical orientation, spin–valley coupling, strain engineering, electric field effects on the band structure and spin–orbit splittings in TMC mono- and bilayers—these are just a few interesting topics to be covered by future research on these novel materials.