Refined DFT recipe and renormalisation of band-edge parameters for electrons in monolayer MoS$_2$ informed by the measured spin-orbit splitting

TL;DR

Addressing the mismatch between experimental and DFT-predicted conduction-band spin-orbit splitting in MoS2, the study uses magnetotransport to measure the threshold for filling the upper SOS band and applies a screened-exchange analysis to compute the SOS renormalization. By combining a screened-exchange framework with a DFT+U+V scheme that tunes Mo–S orbital hybridization, the authors link the measured threshold density to a bare SOS and determine a consistent set of band-edge parameters. The key results yield a bare SOS $\Delta_0$ around 8 meV, a valence-band SOS $\Delta_v$ around 155 meV, and a quasiparticle gap $E_g$ around 1.90 eV, with effective masses $m_{u0}$ and $m_{l0}$ close to experimental values. This approach demonstrates that a carefully parameterized DFT+U+V method can reproduce both conduction- and valence-band spin-orbit splittings and gaps, offering a computationally efficient route to accurate band-edge parameters for MoS2 and related TMDs without requiring GW calculations.

Abstract

Conduction band-edge spin-orbit splitting (SOS) in monolayer transition metal dichalcogenides determines a competition between bright and dark excitons and sets conditions for spintronics applications of these semiconductors. Here, we report the SOS measurement for electrons in monolayer MoS$_2$, found from the threshold density, $n_*$, for the upper spin-orbit-split band population, which exceeds by an order of magnitude the values expected from the conventional density functional theory (DFT). Theoretically, half of the observed value can be attributed to the exchange enhancement of SOS in a finite-density electron gas, but explaining the rest requires refining the DFT approach. As the conduction band SOS in MoS$_2$ is set by a delicate balance between the contribution of sulphur $p_x$ and $p_y$ orbitals and $d_{z^2}-d_{xz}$ and $d_{z^2}-d_{yz}$ mixing in molybdenum, we use a DFT+U+V framework for fine-tuning the orbital composition of the relevant band-edge states. An optimised choice of Hubbard U/V parameters produces close agreement with the experimentally observed conduction band SOS in MoS$_2$, simultaneously resulting in the valence-band SOS and the quasi-particle band gap which are closer to their values established in the earlier-published experiments.

Figures (3)

• Figure 1: Comparison of experimental data and theoretical predictions for spin-split band fillings. (a) Four-terminal resistance $R_{xx}$ as a function of the bottom gate voltage $V_{BG}$ and magnetic field $B$. (b) Band diagram indicating SOS conduction band of MoS$_2$ monolayer, effective masses $m_u$ and $m_l$ (red: spin down, blue: spin up) at the $\pm K$ valleys. The bare SOS $\Delta_0$ (dotted) increases to $\Delta$ (solid) by exchange interaction. The dashed horizontal line marks the Fermi level $E_F$ at the upper band filling threshold density, $n_*$. (c) Density, $n$, dependence of the Fourier spectrum of Shubnikov-de Haas oscillations with the frequency $f*2e/h$ rescaled to represent electron sheet densities in the lower ($n_l$) and upper ($n_u$) spin-split bands, along with their combinations. The threshold density $n_*=4.2\times10^{12}$ cm$^{-2}$ marks onset of populating the upper spin-split band. At high-density end, $n_l$ and $n_u$ change almost linearly with $n$; from their slopes we estimate $m_u/m_l\approx1.3$ as the ratio between the upper and lower spin-split band masses. One may notice deviations from linear behaviour of $n_u$ near $n_*$, which is attributed to the electron-electron interaction as discussed in the Supplemental Information supp and with the calculated results shown by dashed lines. (d) The threshold density $n_*$ (blue surface) calculated from Eq. (\ref{['eqnstar']}) as a function of the lower band effective mass $m_l$ (normalised by the DFT-computed value $m_{l0} \approx 0.43 m_e$) and bare SOS, $\Delta_0$, in comparison to threshold density $n_0$ evaluated without exchange renormalisation of SOS (green surface).
• Figure 2: (a) Set of diagrams contributing to the polarisability. (b) Calculated mass renormalisation. The dashed blue and red lines show the mass renormalisation using a static first- and second-order correction to the polarisability, respectively. Red solid line is calculated with the median $\Pi(q,\omega)=\Pi_0(q,\omega)+(0.29r_s-0.04r_s^2)\frac{\pi\hbar^2}{m_{l0}}$; the top and bottom axis show the electron density and corresponding Wigner-Seitz radius $r_s$, vertical dashed line corresponds to $n_*=4.2\times 10^{12} \text{ cm}^{-2}, \,r_s(n_*)=4.9$.
• Figure 3: (a) Sketch illustrating how SOS is generated for electrons at the K-valley conduction band edge of MoS$_2$ monolayer. The contributing molybdenum $d$ and sulphur $p$ orbitals are indicated together with their out-of-plane angular momentum projections. Green arrows represent the kinematic angular momentum $\kappa_z$ arising from the Bloch phase factor. Intralayer coupling between conduction and upper conduction band (denoted as c and c+1, respectively) arises from the part of the atomic spin–orbit operator, $J_{\mathrm{Mo}} L_- S_+$, where $J_{\mathrm{Mo}}$ is the atomic spin–orbit constant of the molybdenum atom. (b) Hubbard $U$ and $V$ parameters yielding the range of the conduction-band SOS, $\Delta_0$, (orange area) and valence band SOS, $\Delta_{v}$, (green area). Red star marks the optimal parameters $U_*=0.59$ eV and $V_*=0.5$ eV, corresponding to SOS $\Delta_{0} = 8.1$ meV, $\Delta_{v} = 155$ meV.