Abstract
Quantum Monte Carlo (QMC) is applied to obtain the fundamental (quasiparticle) electronic band gap, $\Delta_f$, of a semiconducting two-dimensional (2D) phosphorene whose optical and electronic properties fill the void between graphene and 2D transition metal dichalcogenides. Similarly to other 2D materials, the electronic structure of phosphorene is strongly influenced by reduced screening, making it challenging to obtain reliable predictions by single-particle density functional methods. Advanced GW techniques, which include many-body effects as perturbative corrections, are hardly consistent with each other, predicting the band gap of phosphorene with a spread of almost 1 eV, from 1.6 to 2.4 eV. Our QMC results, from infinite periodic superlattices as well as from finite clusters, predict $\Delta_f$ to be about 2.4 eV, indicating that available GW results are systematically underestimating the gap. Using the recently uncovered universal scaling between the exciton binding energy and $\Delta_f$, we predict the optical gap of 1.75 eV that can be directly related to measurements even on encapsulated samples due to its robustness against dielectric environment. The QMC gaps are indeed consistent with recent experiments based on optical absorption and photoluminescence excitation spectroscopy. We also predict the cohesion of phosphorene to be only slightly smaller than that of the bulk crystal. Our investigations not only benchmark GW methods and experiments, but also open the field of 2D electronic structure to computationally intensive but highly predictive QMC methods which include many-body effects such as electronic correlations and van der Waals interactions explicitly.
Highlights
Two-dimensional materials have already revolutionized science, and have the potential to revolutionize technology due to their unique electronic, optical, thermal, spin, and magnetic properties [1,2,3,4,5,6,7]
Our results are consistent with available optical absorption and photoluminescence emission spectroscopy experiments
We argue that previous calculations based on GW underestimate the quasiparticle gap and do not give consistent predictions for phosphorene
Summary
Two-dimensional materials have already revolutionized science, and have the potential to revolutionize technology due to their unique electronic, optical, thermal, spin, and magnetic properties [1,2,3,4,5,6,7]. (b) Recently, a universal linear scaling between the exciton binding energy and the fundamental gap, Δb ≈ 0.27Δf, was predicted based on many examples from the 2D realm [26] (see the predecessor [27]), including phosphorene Combining these two observations, (a) and (b), allows us to estimate Δo from a calculation of Δf on a pristine 2D material, namely Δo ≈ 0.73Δf, and compare with experimental Δo obtained from an encapsulated or capped sample. Given the large parameter space for tuning the electronic structure of 2D systems, such as layer and strain engineering or dielectric embedding and the large scatter of results produced by experimental methods and customary computational techniques, the robust QMC method lends itself naturally for benchmarking purposes and validation of the different methods in the realm of 2D materials. We believe that our adaptation of QMC methods will open the way for this powerful technique to investigations of electronic structures of 2D systems, which are inherently prone to strong interactions and require careful considerations
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