A Stillinger–Weber potential for single-layered black phosphorus, and the importance of cross-pucker interactions for a negative Poisson's ratio and edge stress-induced bending

Jiang, Jin-Wu; Rabczuk, Timon; Park, Harold S.

doi:10.1039/c4nr07341j

Cited by 89 publications

(88 citation statements)

References 26 publications

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“…For phosphorene, we show (see Supplementary Material) that its anisotropic mechanical properties [3,36] can only be obtained by considering proper strain-displacement relations. Furthermore, phosphorene is considered to be very interesting from a strain-engineering perspective as it displays a strain dependent band-gap [37][38][39].…”

Section: Discussionmentioning

confidence: 99%

Strain–displacement relations for strain engineering in single-layer 2d materials

Midtvedt

Lewenkopf

2016

2D Mater.

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We investigate the electromechanical coupling in 2d materials. For non-Bravais lattices, we find important corrections to the standard macroscopic strain -microscopic atomic-displacement theory. We put forward a general and systematic approach to calculate strain-displacement relations for several classes of 2d materials. We apply our findings to graphene as a study case, by combining a tight binding and a valence force-field model to calculate electronic and mechanical properties of graphene nanoribbons under strain. The results show good agreement with the predictions of the Dirac equation coupled to continuum mechanics. For this long wave-limit effective theory, we find that the strain-displacement relations lead to a renormalization correction to the strain-induced pseudo-magnetic fields. Implications for nanomechanical properties and electromechanical coupling in 2d materials are discussed.

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Section: Discussionmentioning

confidence: 99%

Strain–displacement relations for strain engineering in single-layer 2d materials

Midtvedt

Lewenkopf

2016

2D Mater.

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“…[122] Consequently,these effects render the magnetic properties highly anisotropic,a sb oth diamagnetic and paramagnetic behavior can be observed depending on the orientation in the magnetic field. [138,139] It was recently reported that the Youngsm odulus of few-layer BP was about 58 and 27 GPa in the zigzag and armchair directions,r espectively,w ith highly anisotropic transport behavior in BP and ad erived conductance anisotropy of 63.7 %. 1TPa for graphene).…”

Section: Physical Propertiesmentioning

confidence: 99%

Black Phosphorus Rediscovered: From Bulk Material to Monolayers

2017

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Phosphorus is a nonmetal with several allotropes, from the highly reactive white phosphorus to the thermodynamically stable black phosphorus (BP) with a puckered orthorhombic layered structure. The bulk form of BP was first synthesized in 1914, but received little attention until it was rediscovered in 2014 as a member of the new wave of 2D layered nanomaterials. BP can be exfoliated to a single sheet that acts as a semiconductor with a tunable direct band gap, a high carrier mobility at room temperature, and an in-plane anisotropy. The development of BP applications is hampered by surface degradation, thus efforts to achieve effective BP passivation are ongoing, such as its integration in van der Waals heterostructures. BP has been tested as a novel nanomaterial in batteries, transistors, sensors, and photonics. This Review begins with the origin of the BP story, following the path from a bulk material to modern few/single layers. The physical and chemical properties are summarized, and the state-of-the-art of BP applications highlighted.

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“…Additionally, the intragroup and intergroup angles are, respectively, given by θ 213 = 96.359 ∘ and θ 214 = 102.09 ∘ . In the linear regime, the total strain energy of atom 1 can be expressed as

U = \frac{1}{2} K_{r} {(r_{12} - r_{120})}^{2} + \frac{1}{2} K_{r} {(r_{13} - r_{130})}^{2} + \frac{1}{2} K_{r} {(r_{14} - r_{140})}^{2} + \frac{1}{2} K_{θ} {(θ_{213} - θ_{2130})}^{2} + \frac{1}{2} K_{θ} {(θ_{214} - θ_{2140})}^{2} + \frac{1}{2} K_{θ} {(θ_{314} - θ_{3140})}^{2}

where the former three terms describe the VFF potentials steming from the deformations of related bond lengths, and the later three terms parameterize the VFF potentils resulting from the variations of associated bond angles. The parameters K r and K θ are two VFF force constants, and the symbols containing subscript “0” denote the corresponding quantities of a undeformed monolayer BP.…”

Section: Principlementioning

confidence: 99%

“…Additionally, the intragroup and intergroup angles are, respectively, given by θ 213 = 96.359 ∘ and θ 214 = 102.09 ∘ . [21,22] In the linear regime, the total strain energy of atom 1 can be expressed as [23]…”

Section: Principlementioning

confidence: 99%

Strain engineering of the lattice vibration modes in monolayer black phosphorus

Zhang

Sun

et al. 2019

J Raman Spectroscopy

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Strain engineering can be served as an efficient tool for tuning physical properties of two‐dimentional (2D) materials. However, the quantifying strain and characterizing its spatially inhomogeneous distribution are always challenging. Here, we report the strain‐induced lattice vibration mode variations in monolayer black phosphorus (BP) and clarify the underlying mechanism resorting to the bond relaxation theory based on the combination of mass‐spring model and Lagrangian dynamics. We reveal that the Ag2 and B2g modes exhibit blue (red) shifts while the Ag1 mode shows a red (blue) shift through applying an armchair (zigzag) strain, which can be ascribed to the orientation‐dependent lattice vibration mode variations resulting from the different responses of force constants. Moreover, it is found that a negative Poisson's ratio along the out‐of‐plane direction emerges when the strain is applied in the zigzag direction. Our results offer intriguing insights into the underlying physics of vibration mode variations in strained BP‐like 2D layered materials.

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A Stillinger–Weber potential for single-layered black phosphorus, and the importance of cross-pucker interactions for a negative Poisson's ratio and edge stress-induced bending

Cited by 89 publications

References 26 publications

Strain–displacement relations for strain engineering in single-layer 2d materials

Strain–displacement relations for strain engineering in single-layer 2d materials

Black Phosphorus Rediscovered: From Bulk Material to Monolayers

Strain engineering of the lattice vibration modes in monolayer black phosphorus

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