Electronic structure and optic absorption of phosphorene under strain

We studied the electronic structure and optic absorption of phosphorene (monolayer of black phosphorus) under strain. Strain was found to be a powerful tool for the band structure engineering. The in-plane strain in armchair or zigzag direction changes the effective mass components along both directions, while the vertical strain only has significant effect on the effective mass in the armchair direction. The band gap is narrowed by compressive in-plane strain and tensile vertical strain. Under certain strain configurations, the gap is closed and the energy band evolves to the semi-Dirac type: the dispersion is linear in the armchair direction and is gapless quadratic in the zigzag direction. The band-edge optic absorption is completely polarized along the armchair direction, and the polarization rate is reduced when the photon energy increases. Strain not only changes the absorption edge (the smallest photon energy for electron transition), but also the absorption polarization.     

Double band-inversions of bilayer phosphorene under strain and their effects on optical absorption

Strain is a powerful tool to engineer the band structure of bilayer phosphorene.The band gap can be decreased by vertical tensile strain or in-plane compressive strain.At a critical strain,the gap is closed and the bilayer phosphorene is turn to be a semi-Dirac semimetal material.If the strain is stronger than the criterion,a band-inversion occurs and it re-happens when the strain is larger than another certain value.For the zigzag bilayer phosphorene ribbon,there are two edge band dispersions and each dispersion curve represents two degenerate edge bands.When the first band-inversion happens,one of the edge band dispersion disappears between the band-cross points while the other survives,and the latter will be eliminated between another pair of band-cross points of the second band-inversion.The optical absorption of bilayer phosphorene is highly polarized along armchair direction.When the strain is turn on,the optical absorption edge changes.The absorption rate for armchair polarized light is decreased by gap shrinking,while that for zigzag polarized light increases.The bandtouch and band-inversion respectively result in the sublinear and linear of absorption curve versus light frequency in low frequency limit.     

Mechanical properties of kirigami phosphorene via molecular dynamics simulation

The newly discovered two-dimensional phosphorene suffers low stretchability which limits its application in flexible devices. Herein we employ kirigami technique to overcome this limitation. Molecular dynamics simulation is employed to investigate the mechanical properties of kirigami-phosphorene under shear and tensile loadings. Our simulation results show that loading type, intrinsic structural anisotropy, and the height of middle cuts are three key factors that govern the mechanical response of kirigami-phosphorene. Under the tensile loading along the armchair direction, phosphorene exhibits a considerable increase in its tensile strain. By contrast, phosphorene is too weak to stand any structural modification induced by kirigami in the zigzag direction. Under shear loading, there is merely no improvement in the shear properties of kirigami-phosphorene. Our results demonstrate the prospective applications of kirigami-phosphorene along the armchair direction in modern wearable, and stretchable electronics and optoelectronics devices.     

Band gap of carbon nanotubes under combined uniaxial–torsional strain

Within tight-binding model, the band gaps of armchair and zigzag carbon nanotubes (CNTs) under both uniaxial tensile and torsional strains have been studied. It is found that the changes in band gaps of CNTs depend strongly on the strain type. The torsional strain can induce a band gap for armchair CNTs, but it has little effect on band gap of the zigzag CNTs. While the tensile strain has great effect on band gap of zigzag CNTs, but it has no effect on that of the armchair CNTs. More importantly, when both the tensile and torsional strains are simultaneously applied to the CNTs, the band gap changes of armchair CNTs are not equal to a simple sum over those induced separately by uniaxial tensile and torsional strains. There exists a cooperative effect between two kinds of strains on band gap changes of armchair CNTs. But for zigzag CNTs, the cooperative effect was not found. Analytical expressions for the band gaps of armchair and zigzag CNTs under combined uniaxial–torsional strains have been derived, which agree well with the numerical results.     

Anomalous strain effect on the thermal conductivity of borophene: a reactive molecular dynamics study

Borophene, an atomically thin, corrugated, crystalline two-dimensional boron sheet, has been recently synthesized. Here we investigate mechanical properties and lattice thermal conductivity of borophene using reactive molecular dynamics simulations. We performed uniaxial tensile strain simulations at room temperature along in-plane directions, and found 2D elastic moduli of 188 N m⁻¹ and 403 N m⁻¹ along zigzag and armchair directions, respectively. This anisotropy is attributed to the buckling of the borophene structure along the zigzag direction. We also performed non-equilibrium molecular dynamics to calculate the lattice thermal conductivity. Considering its size-dependence, we predict room-temperature lattice thermal conductivities of 75.9 ± 5.0 W m⁻¹ K⁻¹ and 147 ± 7.3 W m⁻¹ K⁻¹, respectively, and estimate effective phonon mean free paths of 16.7 ± 1.7 nm and 21.4 ± 1.0 nm for the zigzag and armchair directions. In this case, the anisotropy is attributed to differences in the density of states of low-frequency phonons, with lower group velocities and possibly shorten phonon lifetimes along the zigzag direction. We also observe that when borophene is strained along the armchair direction there is a significant increase in thermal conductivity along that direction. Meanwhile, when the sample is strained along the zigzag direction there is a much smaller increase in thermal conductivity along that direction. For a strain of 8% along the armchair direction the thermal conductivity increases by a factor of 3.5 (250%), whereas for the same amount of strain along the zigzag direction the increase is only by a factor of 1.2 (20%). Our predictions are in agreement with recent first principles results, at a fraction of the computational cost. The simulations shall serve as a guide for experiments concerning mechanical and thermal properties of borophene and related 2D materials.     

Structure,Electronic, and Mechanical Properties of Three Fully Hydrogenation h-BN:Theoretical Investigations

The structural, electronic, elastic, mechanical properties and stress-strain relationship of chair, boat, and stirrup conformers of fully hydrogenated h-BN(fh-BN) are investigated in this work using the Perdew-Burke-Ernzerhof(PBE) function in the frame of density functional theory. The achieved results for the lattice parameters and band gaps of three conformers in this research are in good accordance with other theoretical results. The band structures of three conformers show that the chair, boat, and stirrup are direct band gap with a band gaps of(3.12, 4.95, and4.95 e V), respectively. To regulate the band structures of fh-BN, we employ a hybrid functional of Heyd-ScuseriaErnzerhof(HSE06) calculations and the band gaps are 3.84(chair), 6.12(boat), and 6.18 e V(stirrup), respectively.The boat and stirrup fh-BN exhibits varying degrees of mechanical anisotropic properties with respect to the Young's modulus and Poisson's ratio, while the chair fh-BN exhibits the mechanical isotropic properties. Furthermore, tensile strains are applied in the armchair and zigzag directions related to tensile deformation of zigzag and armchair nanotubes,respectively. We find that the ultimate strains in zigzag and armchair deformations in stirrup conformer are 0.34 and0.25, respectively, larger than the strains of zigzag(0.29) and armchair(0.18) deformations in h-BN although h-BN can surstain a surface tension up to the maximum stresses higher than those of three conformers of fh-BN. Furthermore, the band gap energies in three conformers can be modulated effectively with the biaxial tensile strain.     

Molecular dynamics simulation of compression of single-layer graphene

The compression of a single-layer graphene sheet in the “zigzag” and “armchair” directions has been investigated using the molecular dynamics method. The distributions of the xy and yx stress components are calculated for atomic chains forming the graphene sheet. A graphene sheet stands significant compressive stresses in the “zigzag” direction and retains its integrity even at a strain of ～0.35. At the same time, the stresses which accompany the compressive deformation of single-layer graphene in the “armchair” direction are more than an order in magnitude lower than corresponding characteristics for the “zigzag” direction. A compressive strain of ～0.35 in the “armchair” direction fractures the graphene sheet into two parts.     

Velocities of sound and the densities of phonon states in a uniformly strained flat graphene sheet

The effect of elastic strain on the mechanical and physical properties of graphene has been intensively studied in recent years. Using the molecular dynamics method, a surface has been built in the three-dimensional space of components of the plane strain tensor bounds the region of the structural stability of a flat graphene sheet without considering thermal vibrations and the influence of boundary conditions. The velocities of sound and the densities of phonon states in graphene subjected to an elastic strain within the region of the structural stability have been calculated. It has been shown that one of the velocities of sound becomes zero near the stability boundary of a flat graphene sheet. During biaxial tension of graphene, there is no gap in its phonon spectrum; however, it forms under uniaxial tension along the zigzag or armchair directions and also under combined tensile and compressive strains.     

Decay and the double-decay properties of edge bands of phosphorene ribbons

Phosphorene (a monolayer of black phosphorus) recently spurred much attention due to its potential for application. We notice there are two types of zigzag edge and two types of armchair edge for phosphorene lattice. We study the winding number of various types of edge of phosphorene ribbons and conclude that, besides on the typical zigzag edge, the flat zero-energy edge band can be found in the ribbon of another nontypical armchair edge. The localization of these edge bands is investigated analytically. We find every single edge state of the atypical armchair edge decays to the bulk at two different decay rates.     

Strain Effects on Properties of Phosphorene and Phosphorene Nanoribbons: a DFT and Tight Binding Study

We perform comprehensive density functional theory calculations of strain effect on electronic structure of black phosphorus(BP) and on BP nanoribbons. Both uniaxial and biaxial strain are applied, and the dramatic change of BP's band structure is observed. Under 0-8% uniaxial strain, the band gap can be modulated in the range of 0.55-1.06 eV, and a direct-indirect band gap transition causes strain over 4% in the y direction. Under 0-8% biaxial strain, the band gap can be modulated in the range of 0.35-1.09 eV, and the band gap maintains directly.Applying strain to BP nanoribbon, the band gap value reduces or enlarges markedly either zigzag nanoribbon or armchair nanoribbon. Analyzing the orbital composition and using a tight-binding model we ascribe this band gap behavior to the competition between effects of different bond lengths on band gap. These results would enhance our understanding on strain effects on properties of BP and phosphorene nanoribbon.     

应变对二维蜂巢晶格光子晶体能带结构的影响

基于平面波法,本论文对应变引起的二维蜂巢晶格光子晶体的能带结构进行了数值计算。选取的两个方向分别是锯齿型边界(zigzag)方向和扶手椅型边界(armchair)方向,在这两个典型方向上对二维蜂巢晶格进行了正负各20%的单轴应变。由于应变导致的对称性破缺,能带结构会有显著的变化。在沿锯齿型边界方向上,TE模带隙随着晶格被拉伸逐渐减小,TM模带隙在应变量大于16%时消失。对于沿扶手椅型边界方向,TE模带隙在压缩15%以上时逐渐减小,在其他应变量的情况下几乎保持不变;TM模带隙在应变量大于18%时消失。这些结果对于完善应力工程和设计二维光子晶体器件有重要的指导意义。     

Thermal transport in phosphorene and phosphorene-based materials: A review on numerical studies

The recently discovered two-dimensional(2D) layered material phosphorene has attracted considerable interest as a promising p-type semiconducting material. In this article, we review the recent advances in numerical studies of the thermal properties of monolayer phosphorene and phosphorene-based heterostructures. We first briefly review the commonly used first-principles and molecular dynamics(MD) approaches to evaluate the thermal conductivity and interfacial thermal resistance of 2D phosphorene. Principles of different steady-state and transient MD techniques have been elaborated on in detail. Next, we discuss the anisotropic thermal transport of phosphorene in zigzag and armchair chiral directions. Subsequently, the in-plane and cross-plane thermal transport in phosphorene-based heterostructures such as phosphorene/silicon and phosphorene/graphene is summarized. Finally, the numerical research in the field of thermal transport in 2D phosphorene is highlighted along with our perspective of potentials and opportunities of 2D phosphorenes in electronic applications such as photodetectors, field-effect transistors, lithium ion batteries, sodium ion batteries, and thermoelectric devices.     

Discrete breathers in deformed graphene

The linear and nonlinear dynamics of elastically deformed graphene have been studied. The region of the stability of a planar graphene sheet has been represented in the space of the two-dimensional strain (? _xx , ? _yy ) with the x and y axes oriented in the zigzag and armchair directions, respectively. It has been shown that the gap in the phonon spectrum appears in graphene under uniaxial deformation in the zigzag or armchair direction, while the gap is not formed under a hydrostatic load. It has been found that graphene deformed uniaxially in the zigzag direction supports the existence of spatially localized nonlinear modes in the form of discrete breathers, the frequency of which decreases with an increase in the amplitude. This indicates soft nonlinearity in the system. It is unusual that discrete breather has frequency within the phonon spectrum of graphene. This is explained by the fact that the oscillation of the discrete breather is polarized in the plane of the graphene sheet, while the phonon spectral band where the discrete breather frequency is located contains phonons oscillating out of plane. The stability of the discrete breather with respect to the small out-of-plane perturbation of the graphene sheet has been demonstrated.     

Electronic structures for armchair-edge graphene nanoribbons under a small uniaxial strain

We theoretically investigate the electronic structures for armchair-edge graphene nanoribbons (AGNRs) under a small in-plane uniaxial strain along armchair (longitudinal) and zigzag (transversal) direction, respectively. We demonstrate that, by both the tight-binding calculation and first-principles study, the applying of a small asymmetrical strain results in variation of energy subband spacing, which opens a band gap for metallic AGNRs and modifies the band gaps for semiconducting AGNRs near the Fermi level. It is believed that these results are of importance in the band gap engineering and electromechanical applications of graphene-nanoribbon-based devices.     

Strain effect on transport properties of hexagonal boronben nitride nanoribbons

The transport properties of hexagonal boron--nitride nanoribbons under the uniaxial strain are investigated by the Green's function method. We find that the transport properties of armchair boron--nitride nanoribbon strongly depend on the strain. In particular, the features of the conductance steps such as position and width are significantly changed by strain. As a strong tensile strain is exerted on the nanoribbon, the highest conductance step disappears and subsequently a dip emerges instead. The energy band structure and the local current density of armchair boron--nitride nanoribbon under strain are calculated and analysed in detail to explain these characteristics. In addition, the effect of strain on the conductance of zigzag boron--nitride nanoribbon is weaker than that of armchair boron nitride nanoribbon.     

MD investigation of the collective carbon atom behavior of a (17, 0) zigzag single wall carbon nanotube under axial tensile strain

The collective dynamic behavior of carbon atoms of a (17, 0) zigzag single wall carbon nanotube is investigated under tensile strains by molecular dynamics (MD) simulations. The “slip vector” parameter is used to study the collective motion of a group of atoms and the deformation behavior in three different directions (axial, radial, and tangential) of a (17, 0) carbon nanotube. The variations of radial slip vectors indicate almost all carbon atoms of the (17, 0) carbon nanotube will stay on the cylindrical surface before the yielding of the single wall carbon nanotube (SWNT). Furthermore, the tangential vectors show kinking deformation for the (17, 0) zigzag tube only rarely appears when the crack occurs. Non-symmetrical deformation around a carbon atom along the axial direction also can be found. The variations in the slip vector values of each atom display a symmetrical crack along the horizontal direction and normal to the tube axis. Chain-like structures with 3–4 atoms can be observed, with the number of chain-like structures decreasing before the breakage of the SWNT. The mechanical properties and dynamic behavior of a (17, 0) zigzag SWNT under tensile strain are also compared with that of a (10, 10) armchair tube in our previous study (Weng et al. 2009).     

Effects of strain on shot noise properties in graphene superlattices

In this paper the transmission and the shot noise properties through the strain-inducedgraphene superlattices are studied. It is found that for the zigzag direction strain theFano factor shows a peak at new Dirac-like point and the position of the new Dirac pointvaries against the strain. Also, Fano factor has an oscillatory behavior with respect tostrain strength and the oscillation period decreases by increasing the number of barriers.However, for the armchair direction strain the transmission can be blocked by the electricbarrier and the Fano factor approaches 1, this is different from the zigzag directionstrain.     

Strain-induced structural and direct-to-indirect band gap transition in ZnO nanotubes

First-principles calculations have been employed to investigate the structural transformation and direct to indirect band gap transition of ZnO nanotubes under uniaxial strain. The results show that armchair and zigzag nanotubes can be transformed to each other via unusual fourfold-coordinated structures under the applied strain. Both the armchair and zigzag nanotubes exhibit direct band gap while the unusual fourfold-coordinated ones display indirect band gap. The origin of such a direct-to-indirect band gap transition is explained based on the analyses of atomic orbital contributions.     

Gallium doped in armchair and zigzag models of boron phosphide nanotubes (BPNTs): A NMR study

The electrical properties and NMR parameters of the pristine and Ga-doped structures of two representative (8, 0) zigzag and (4, 4) armchair of boron phosphide nanotubes (BPNTs) have been investigated. The structural geometries of above nanotubes have been allowed to relax by optimization and then the isotropic and anisotropic chemical shielding parameters (CSI and CSA) of ¹¹B and ³¹P have been calculated based on DFT theory. The results reveal that the influence of Ga-doping was more significant on the geometries of the zigzag model than the armchair one. The difference of band gap energies between the pristine and Ga-doped armchair BPNTs was larger than the zigzag model. Significant differences of NMR parameters of those nuclei directly contributed to the Ga-doping atoms have been observed.     

Band gap engineering in silicene: A theoretical study of density functional tight-binding theory

In this work, we performed first principles calculations based on self-consistent charge density functional tight-binding to investigate different mechanisms of band gap tuning of silicene. We optimized structures of silicene sheet, functionalized silicene with H, CH₃ and F groups and nanoribbons with the edge of zigzag and armchair. Then we calculated electronic properties of silicene, functionalized silicene under uniaxial elastic strain, silicene nanoribbons and silicene under external electrical fields. It is found that the bond length and buckling value for relaxed silicene is agreeable with experimental and other theoretical values. Our results show that the band gap opens by functionalization of silicene. Also, we found that the direct band gap at K point for silicene changed to the direct band gap at the gamma point. Also, the functionalized silicene band gap decrease with increasing of the strain. For all sizes of the zigzag silicene nanoribbons, the band gap is near zero, while an oscillating decay occurs for the band gap of the armchair nanoribbons with increasing the nanoribbons width. At finally, it can be seen that the external electric field can open the band gap of silicene. We found that by increasing the electric field magnitude the band gap increases.