Unified understanding to the rich electronic-structure evolutions of two-dimensional black phosphorus under pressure

Open Access

Unified understanding to the rich electronic-structure evolutions of two-dimensional black phosphorus under pressure

Yu-Meng Gao, Yue-Jiao Zhang, Xiao-Lin Zhao, Xin-Yu Li, Shu-Hui Wang, Chen-Dong Jin, Hu Zhang, Ru-Qian Lian, Rui-Ning Wang, Peng-Lai Gong, Jiang-Long Wang, and Xing-Qiang Shi

Phys. Rev. Research 6, 013267 – Published 11 March 2024

Abstract

The electronic-structure evolutions of few-layer black phosphorus (BP) under pressure shows a wealth of phenomena, such as the nonmonotonic change of direct gap at the $Γ$ point, the layer-number dependence, and the distinct responses to normal and hydrostatic pressures. A full and unified understanding to these rich phenomena remains lacking. Here, we provide a unified understanding from the competition between interlayer quasibonding (QB) interactions and intralayer chemical bonding interactions. The former decreases while the latter increases the band gap under pressure and the origin can be correlated to different combinations of inter- and intralayer antibonding or bonding interactions at the band edges. More interestingly, the interlayer QB interactions are a coexistence of two categories of interactions, namely, the coexistence of interactions between bands of the same occupancy (occupied-occupied and empty-empty interactions) and of different occupancies (occupied-empty interaction); and, the overall effect is a four-level interaction, which explains the anomalous interlayer-antibonding feature of the conduction band edge of bilayer BP. Our current study lays the foundation for the electronic-structure tuning of two-dimensional (2D) BP, and, our analysis method for multi-energy-level interactions can be applied to other 2D semiconductor homo- and heterostructures that have occupied-empty interlayer interactions.

Received 22 October 2023
Revised 15 January 2024
Accepted 14 February 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.013267

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Chemical bonding Electronic structure Electronic structure of atoms & molecules First-principles calculations Pressure effects Structural properties Surface & interfacial phenomena

2-dimensional systems Layered semiconductors Phosphorene Surfaces

Band structure methods Density functional theory Materials modeling

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yu-Meng Gao, Yue-Jiao Zhang, Xiao-Lin Zhao, Xin-Yu Li, Shu-Hui Wang, Chen-Dong Jin, Hu Zhang, Ru-Qian Lian, Rui-Ning Wang, Peng-Lai Gong^*, Jiang-Long Wang^†, and Xing-Qiang Shi ^‡

Key Laboratory of Optic-Electronic Information and Materials of Hebei Province, Hebei Research Center of the Basic Discipline for Computational Physics, College of Physics Science and Technology, Hebei University, Baoding 071002, People's Republic of China

^*gongpl@hbu.edu.cn
^†jlwang@hbu.edu.cn
^‡shixq20hbu@hbu.edu.cn

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Issue

Vol. 6, Iss. 1 — March - May 2024

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Images

Figure 1
Structure and direct-gap evolution under pressure. (a) Side and (b) top views of bulk BP, which also serves as the structural parents for few layers. In (a), $d$ and $L$ are the interlayer spacing and the cell length (or thickness of the bulk BP in one repeat unit) in the $z$ direction, respectively. In (a) and (b), the different P-P atom distances are labeled, including the interlayer P2-P3 across the vdW gap ( $R_{0}$ ), the intralayer P2-P1 along mainly the vertical direction ( $R_{1}$ ), and the intralayer P1-P4 along the horizontal direction ( $R_{2}$ ). $θ_{1}$ and $θ_{2}$ are the intralayer P-P-P bond angles. (c) Evolution of the direct gap under normal strain from monolayer to bulk. The line “bulk-h” means bulk BP under hydrostatic pressure. (d) The 2D Brillouin zone corresponding to the gray box in (b).
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Figure 2
Inter- and intralayer antibonding interactions of VB and CB edges for bilayer BP. (a) Band structure projected to atomic orbitals. The circle sizes are proportional to the weight of the projected orbitals. There is a color mixing for the $s + p_{y}$ orbirals as labeled at the upper left of (a). (b) The CB and VB wave functions at $Γ$ point. The two colors of wave functions denote the opposite signs; red (black) dashed lines indicate the interlayer (intralayer) interactions between the P2-P3 (P2-P1) atoms; for the P2-P3 atoms across the vdW gap in CB@ $Γ$ , the labeled numbers (1, 2, and $2^{'}$ ) denote the coexistence of bonding (1) and antibonding (2 and $2^{'}$ ) interlayer interactions.
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Figure 3
Schematic diagram for the four-level interaction from monolayer (1L) to bilayer (2L) BP. In addition to the two-level interactions of occupied-occupied and empty-empty levels in (a), there is occupied-empty interaction [62] in (b) since they have similar orbital characters ( $p_{z}$ , $p_{y}$ , and $s$ orbitals); and the overall effect is the four-level interaction in (c). For the bilayer energy levels, ${VB}_{2 L}^{(2)}$ and ${CB}_{2 L}^{(2)}$ denote the band edges from the (imagined) two-level interactions, and ${VB}_{2 L}^{(4)}$ and ${CB}_{2 L}^{(4)}$ indicate the final band edges from the four-level interaction. Panel (c) is a simplified description of the four-level interaction, namely, at least the indicated interactions are needed for understanding the antibonding feature of ${CB}_{2 L}^{(4)}$ . (d) The 2L CB wave function in real space with major contribution from 1L CB wave function (for the outer part and the interlayer region of 2L) and minor contribution from 1L VB (mainly for the interlayer region of 2L).
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Figure 4
Variation of the direct gap for BP under normal strain. For (a) bilayer, (c) trilayer, and (d) bulk BP, the overall gap change (the blue line) is decomposed into the effects of interlayer P2-P3 QB interactions (the red line) and intralayer P2-P1 chemical bonding interactions (the black line); both of them are from auxiliary calculations (see text for details). (b) The effects of inter- and intralayer interactions under strain on the evolution of band edges, which determines the variation of the direct gap in (a), and also (c) and (d).
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Figure 5
Direct-indirect band gap transition of CBM from the $Γ$ point to the $K_{1}$ point under normal strain. (a) and (b) show the band-structure change of bilayer BP from 0 to 1.03 GPa. (c),(d) Decompose the band-structure change to interlayer QB interactions (c) and intralayer chemical bonding interactions (d) at 1.03 GPa.
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Figure 6
Band gap change under hydrostatic pressure. (a) Evolution of the band gap under hydrostatic pressure from few layer to bulk. (b) The effects of interlayer interactions under pressure on the evolution of band edges, which finally determines the evolution of band gap in (a).
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