Single-layer Black Phosphorus Research Articles

Thermally stimulated nonlinear vibration of rectangular single-layered black phosphorus

Black phosphorus (BP), a novel material with wide potential applications, has recently attracted considerable attention. In this study, the nonlinear thermal vibrational behavior of single-layered BP (SLBP) is investigated using a nonlinear orthotropic plate model (OPM) and molecular dynamics (MD) simulations. The dynamic equation for the nonlinear orthotropic plate considering large deflections and thermal stress is obtained. The stationary probability density of the nonlinear thermally stimulated vibrational behavior of SLBP is presented based on the nonlinear OPM for both the pre-buckling and post-buckling cases. For the post-buckling case, the time histories of SLBP with different damping ratios are obtained by nonlinear OPM using the fourth-order Runge–Kutta algorithm. As the damping ratio increases, one equilibrium position jumps to the other more frequently. The time histories of the SLBP at different temperatures are gotten by nonlinear OPM and MD simulations. The MD results indicate that the nonlinear OPM can predict the nonlinear thermally stimulated vibrational behavior of SLBP well.

Finite temperature stability of single-layer black and blue phosphorus adsorbed on Au(1 1 1): a first-principles study

The finite-temperature stability of single-layer black (BP) and blue (bP) phosphorus deposited on a gold substrate is investigated using first-principles calculations. In contrast to previous studies, a density functional theory (DFT) treatment including van der Waals (vdW) corrections predicts that the thermo-structural properties do not lead to a phase transition from BP to bP. To account for entropic (i.e. finite temperature) effects on the stability of materials deposited on a substrate within a first-principles framework, we develop an algorithm to investigate phonons properties at the interface between the phosphorus layers and a metallic substrate. This approach greatly reduces the computational cost and makes it possible to use DFT to model vibrational properties of systems with hundreds of atoms, and especially those that can be separated into weakly interacting sub-systems. It also allows for the description of interfacial shear and breathing-like modes, which enter in the evaluation of finite temperature effects via the Helmholtz free energy. In contrast to the freestanding case, we find that bP is energetically more stable than BP adsorbed on Au(1 1 1) with an energy difference between these two phases equal to 46 meV/atom at room temperature.

Exploring the Limits of Super-Planckian Far-Field Radiative Heat Transfer Using 2D Materials

Very recently it has been predicted that the far-field radiative heat transfer between two macroscopic systems can largely overcome the limit set by Planck’s law if one of their dimensions becomes much smaller than the thermal wavelength (λTh ≈ 10 μm at room temperature). To explore the ultimate limit of the far-field violation of Planck’s law, here we present a theoretical study of the radiative heat transfer between two-dimensional (2D) materials. We show that the far-field thermal radiation exchanged by two coplanar systems with a one-atom-thick geometrical cross section can be more than 7 orders of magnitude larger than the theoretical limit set by Planck’s law for blackbodies and can be comparable to the heat transfer of two parallel sheets at the same distance. In particular, we illustrate this phenomenon with different materials such as graphene, where the radiation can also be tuned by a external gate, and single-layer black phosphorus. In both cases the far-field radiative heat transfer is domina...

Anisotropic electronic heat capacity and electrical conductivity of monolayer biased impurity-infected black phosphorus

We theoretically investigate the electron-impurity interaction effects on the direction-dependent electronic heat capacity (EHC) and electrical conductivity (EC) of single-layer biased black phosphorus (BP) as a function of temperature. By means of the Born approximation besides the continuum approximation of the tight-binding Hamiltonian and the Green's function approach, we obtain a significant difference in the x− and y−direction EHC and EC. Inherent higher value of the y−direction density of states around the Fermi energy in the valence band exhibits interesting anisotropic EHC and EC results. The Schottky anomaly in EHC quantity decreases slightly with impurity in x−direction of unbiased BP, whereas there is no change in y−direction. On the other hand, the probability of transition between energy levels provided by the decreased critical temperature decreases with impurity in x−direction EC and similarly there is no alteration in y−direction EC. We show that the Drude weight in EC appears with impurity in both directions. Furthermore, we study the influence of bias voltage on the EHC and EC of impurity-infected BP. Our findings showcase the increased (decreased) EHC (EC) of disordered BP and disappeared the Drude weight of EC in the presence of bias voltage.

Electro/mechanical mutable properties of black phosphorene by electric field and strain engineering

Phosphorene a two dimensional (2D) arrangement of black phosphorus atoms has greatly manifested extraordinary properties. In this work we investigated the electronic and mechanical properties of a deformed structural (i.e. modified two characteristic angles) single-layer black phosphorus using tight-binding (TB) Hamiltonian and Green’s function approach, where a planar and a most puckered structure have been created by modifying the angle between two nearest neighboring P–P bond from two sublayers. Moreover, by using density functional theory (DFT), we show that the applied strain in the x, y axes and electric field can be used to tune the electronic properties of black phosphorene. We also provide a comparison of phonon vibrational modes of planar and puckered phosphorene to compare their structure stability. It was found that by structural deformation such as modifying the angle between two nearest neighboring P–P bond from two sublayers, consequently having a planar and a most puckered structure, no change can be seen for the effective masses along the y (zigzag) direction, but totally affected along the x (armchair) direction leading to heavier electrons and holes. Furthermore, 2D phosphorene exhibits different stiffness for applied strain in the x, y direction. Such rich variety of electronic, mechanical and structural transformations provides 2D phosphorene as a unique material for fundamental physics studies and applicable in nanoelectronic and nanophotonics devices.

Direction-dependent electronic thermal conductivity and thermopower of single-layer black phosphorus in the presence of bias voltage and dilute charged impurity

By using the Green's function and the tight-binding model within the Born approximation, we obtain the electronic thermal conductivity (TC) and Seebeck coefficient [or thermopower (TP)] of monolayer black phosphorus (BP) under charged impurity doping. The results show that the TC and TP along the y−direction are larger than the x−direction in clean BP due to the inherent higher value of the electronic density of states in y−direction. Furthermore, the TC linearly decreases with low impurity concentrations in both directions, whereas it decreases and does not change with low impurity scattering potentials in x− and y−direction, respectively. On the other hand, TP increases (decreases) with very dilute impurity concentrations and scattering potentials in x−(y−)direction. To be a complete work, we also studied these properties in impurity-infected biased BP. We found that the TC of BP in the presence of impurity gradually increases with bias voltage in both directions and TP has a decreasing (increasing) treatment in x−(y−)direction. Our findings move the thermoelectric (TE) applications of BP into promising TE materials.

Thermal vibration of rectangular single-layered black phosphorus predicted by orthotropic plate model

Vibrational behavior is very important for nanostructure-based resonators. In this work, an orthotropic plate model together with a molecular dynamics (MD) simulation is used to investigate the thermal vibration of rectangular single-layered black phosphorus (SLBP). Two bending stiffness, two Poisson's ratios, and one shear modulus of SLBP are calculated using the MD simulation. The natural frequency of the SLBP predicted by the orthotropic plate model agrees with the one obtained from the MD simulation very well. The root of mean squared (RMS) amplitude of the SLBP is obtained by MD simulation and the orthotropic plate model considering the law of energy equipartition. The RMS amplitude of the thermal vibration of the SLBP is predicted well by the orthotropic plate model compared to the MD results. Furthermore, the thermal vibration of the SLBP with an initial stress is also well-described by the orthotropic plate model.

Tight binding parametrization of few-layer black phosphorus from first-principles calculations

We employ a tight-binding parametrization based on the Slater Koster model in order to fit the band structures of single-layer, bilayer and bulk black phosphorus obtained from first-principles calculations. We find that our model, which includes 9 or 17 parameters depending on whether overlap is included or not, reproduces quite well the ab initio band structures over a wide energy range, especially the occupied bands. We also find that the Inclusion of overlap parameters improves the quality of the fit for the conduction bands. On the other hand, hopping and on-site energies are consistent throughout the different systems, which is an indication that our model is suitable for calculations on multilayer black phosphorus and more complex situations in which first-principles calculations become prohibitive, such as disordered systems and heterostructures with a large lattice mismatch. We also discuss the limitations of the model and how the fit procedure can be improved for a more accurate description of bands in the vicinity of the Fermi energy.

Auxetics and Other Systems of Anomalous Characteristics

Auxetics and Other Systems of Anomalous Characteristics

Directional mechanical and thermal properties of single-layer black phosphorus by classical molecular dynamics

Black phosphorus (BP) has received attention due to its own higher carrier mobility and layer dependent electronic properties, such as direct band gap. Interestingly, the single layer black phosphorus (SLBP) has had large popularity in applications related to thermoelectric, optoelectronic, and electronic devices. Here, we investigate the phonon spectrum, thermal conductivities, and stress strain effects. Robust anisotropy was mainly observed in the thermal conductivities together with the alongside zigzag (ZZ) direction value, compared to the armchair (AC) directions. We also investigated the attitude of stress that was anisotropic in both directions, and the stress effects were two times greater across the ZZ path than those in the AC direction at a low temperature. We obtained a Young’s modulus of 63.77 and 20.74 GPa in the AC and ZZ directions, respectively, for a strain range of 0.01. These results had good agreement with first principle calculations. Our study here is useful and significant for the thermal tuning of phosphorus-based nanoelectronics and thermalelectric applications of phosphorus.

One-dimensional phosphorus chain and two-dimensional blue phosphorene grown on Au(111) by molecular-beam epitaxy

Single layer (SL) phosphorus (phosphorene) has drawn considerable research attention recently as a two-dimensional (2D) material for application promises. It is a semiconductor showing superior transport and optical properties. Few-layer or SL black phosphorus has been successfully isolated by exfoliation from bulk crystals and extensively studied thereof for its electronic and optical properties. Blue phosphorus (blueP), an allotrope of black phosphorus where atoms are arranged in a more flat atomic configuration, has been recently suggested by theory to exist in the SL form on some substrates. In this work, we report the formation of a blueP-like epilayer on Au(111) by molecular-beam epitaxy. In particular, we uncover by scanning tunneling microscopy (STM) one-dimensional (1D) atomic chains at low coverage, which develop into more compact islands or patches of $(\sqrt{3}\times\sqrt{3})R30^\circ$ structure with increasing coverage before blueP-like islands nucleate and grow. We also note an interesting growth characteristic where the $(\sqrt{3}\times\sqrt{3})R30^\circ$ surface at intermediate coverage tends to phase-separate into locally low-coverage 1D chain and high-coverage blueP-like structures, respectively. This experiment thus not only lends a support of the recently proposed half-layer by half-layer (HLBHL) growth mechanism but also reveals the kinetic details of blueP growth processes.

Negative In‐Plane Poisson's Ratio for Single Layer Black Phosphorus: An Atomistic Simulation Study

We utilized molecular statics (MS) simulations to investigate the auxeticity of single layer black phosphorus (SLBP). Our simulation results show that the SLBP has a negative in‐plane Poisson's ratio in the zigzag direction when the applied strain along the armchair direction exceeds 0.018. We show that the interplay between bond stretching and bond rotating modes determines the in‐plane Poisson's ratio behavior. While the bond stretching mode always tends to increase the in‐plane auxeticity, the bond rotating mode might increase or decrease the in‐plane auxeticity. Furthermore, we show that graphite also exhibits an in‐plane negative Poisson's ratio at finite strains due to a similar mechanism.

Coherent nonlinear optical response of single-layer black phosphorus: third-harmonic generation

We theoretically calculate the nonlinear optical (NLO) response of phosphorene (a black phosphorus monolayer) to a normally incident and linearly polarized coherent laser radiation of frequency ω, resulting in the generation of radiation at frequency 3ω. We derive explicit analytic expressions for four independent nonvanishing elements of the third-order NLO susceptibility tensor, describing the third-harmonic generation (THG) from phosphorene. The final formulas are numerically evaluated for typical values of the system’s parameters to explore how the efficiency of the THG varies with both the frequency and the polarization direction of the incident radiation. The results obtained show a resonant enhancement of the THG efficiency when the pump photon energy ℏω approaches a value of one third of the bandgap energy E g (≈1.5 eV) of phosphorene. It is also shown that the THG efficiency exhibits a specific polarization dependence, allowing the THG to be used for determining the orientation of phosphorene’s crystallographic axes. Our findings highlight the material’s potential for practical application in nanoscale photonic devices such as frequency convertors operating in the near-infrared spectral range.

Mechanical properties of prestrained single-layer black phosphorus: effect of thermal environment

Molecular dynamics simulation with the Stillinger–Weber potential is employed to study the mechanical behaviors of single-layer black phosphorus (SLBP) at different temperatures. The SLBP is initially loaded by a uniaxial tensile/compressive prestrain in either armchair or zigzag direction then stretched or compressed in the direction that is orthogonal to the direction of the prestrain. It is found that mechanical properties such as tensile/compressive Young’s modulus, tensile/compressive strength and yield strength are enhanced by a compressive prestrain but are reduced by a tensile prestrain, and that an armchair-oriented prestrain can improve mechanical properties more significantly than a zigzag-oriented prestrain. The enhancement in mechanical properties by a compressive prestrain becomes less significant as the temperature increases. In addition, both yield and compressive strengths of SLBP are found to be highly sensitive to its size while this effect on the modulus and tensile strengths is quite limited. The mechanisms of prestrain underlying the mechanical behaviors of SLBP are also clarified in detail. Numerical results indicate that the application of a compressive prestrain is an effective way to achieve improved modulus and strength for two-dimensional materials.

Self-assembly of a parallelogram black phosphorus ribbon into a nanotube

A nanotube from single-layer black phosphorus (BP) has never been discovered in experiments. The present study proposed a method for the fabrication of a BP nanotube (BPNT) from a parallelogram nanoribbon self-assembled on a carbon nanotube (CNT). The nanoribbon has a pair of opposite sides along the third principal direction. According to the numerical simulation via molecular dynamics approach, we discover that a wider BP nanoribbon can form into a series of chiral nanotube by self-assembly upon CNTs with different radii. The radius of a BPNT from the same ribbon has a wide range, and depends on both geometry of the ribbon and the CNT. One can obtain a BPNT with the specified radius by placing the ribbon nearby a given CNT. The method provides a clue for potential fabrication of BPNTs.

Normal and skewed phosphorene nanoribbons in combined magnetic and electric fields

The energy spectrum and eigenstates of single-layer black phosphorus nanoribbons in the presence of a perpendicular magnetic field and an in-plane transverse electric field are investigated by means of a tight-binding method, and the effect of different types of edges is examined analytically. A description based on a continuum model is proposed using an expansion of the tight-binding model in the long-wavelength limit. The wave functions corresponding to the flatband part of the spectrum are obtained analytically and are shown to agree well with the numerical results from the tight-binding method for both narrow (10 nm) and wide (100 nm) nanoribbons. Analytical expressions for the critical magnetic field at which Landau levels are formed and the ranges of wave numbers in the dispersionless flatband segments in the energy spectra are derived. We examine the evolution of the Landau levels when an in-plane lateral electric field is applied, and we determine analytically how the edge states shift with magnetic field. For wider nanoribbons, the conductance is shown to have a characteristic staircase shape in combined magnetic and electric fields. Some of the stairs in zigzag and skewed armchair nanoribbons originate from edge states that are found in the band gap.

Ab-initio quantum transport simulation of self-heating in single-layer 2-D materials

Through advanced quantum mechanical simulations combining electron transport and phonon transport from first-principles, self-heating effects are investigated in n-type transistors with single-layer MoS2, WS2, and black phosphorus as channel materials. The selected 2-D crystals all exhibit different phonon-limited mobility values, as well as electron and phonon properties, which have a direct influence on the increase in their lattice temperature and on the power dissipated inside their channel as a function of the applied gate voltage and electrical current magnitude. This computational study reveals (i) that self-heating plays a much more important role in 2-D materials than in Si nanowires, (ii) that it could severely limit the performance of 2-D devices at high current densities, and (iii) that black phosphorus appears less sensitive to this phenomenon than transition metal dichalcogenides.

Group V materials: From bulk to monolayer

Since graphene has been fabricated successfully in 2004, two dimensional (2D) materials have received a great attention because of their exceptional properties and novel applications. However, there are problems exist in traditional 2D materials. Graphene, as the very first 2D material, is a zero band-gap semiconductor which results in low on-off ratio. Hexagonal boron nitride (hBN) has a large band-gap around 6 eV that can be considered as insulator. Monolayer MoS2 also is an important 2D material, but the relative low mobility limits its applications. In 2014, few and single layer black phosphorus (BP) was achieved by mechanical exfoliation. And p-type field effect transistors (FETs) based on few-layer phosphorene also have been fabricated successfully. The BP FETs exhibit high mobility value up to ~1000 cm2/(V s) at room temperature. Besides, a direct band-gap of ~2 eV indicates that monolayer BP possess a high on/off ratio. These two advantages make phosphorene a promising FET material which is the most desirable application of 2D materials. Despite the superior properties, phosphorene still has its problems. When exposure to air, phosphorene tends to degradation and breakdown which restrict their applications in electronic and optoelectronic devices. The production of monolayer BP provokes wondering if other elements from group V can be the potential elemental 2D crystals. Elements from group V beyond phosphorus should be more chemical steady than phosphorus. And like black phosphorus, grey arsenic, grey antimony and rhombohedral bismuth are the most stable phase of these three elements with layer structures. For these two reasons, there are many reports about arsenic, antimony and bismuth in 2D structures. In this paper, we review the development of group V elemental materials from three-dimensional (3D) to 2D. Bulk arsenic and antimony are narrow semiconductor and semimetal, respectively. Differ from black phosphorus with orthorhombic structure, grey arsenic and grey antimony possess rhombohedral structures. Through density functional theory (DFT) calculation, the phonon spectra proved the thermodynamics stabilities of monolayer As and Sb. And an intriguing indirect-to-direct band-gap transition at a relatively small critical strain has been reported. After that, a lot of properties of group V elemental 2D materials are reported. Theoretically, their properties can be manipulated by functionalization, strain and electric field to applying magnetic, electronic, topological and thermoelectrical materials. Experimentally, arsenene and antimonene can be produced via liquid exfoliation, van der Waals epitaxy, mechanical exfoliation, plasma-assisted process and molecular beam epitaxy. Based on the antimonene fabricated by liquid exfoliation, an unexpected nonlinear optical limiting property which is even better than graphene in the visible and near infrared region (532–2000 nm) and high transmission (more than 80%) when dispersed in solutions or high concentration doped in Ormosil gel glasses. This might lead to many promising applications in nonlinear optical fields such as laser protection.

An empirical description for the hinge-like mechanism in single-layer black phosphorus: The angle–angle cross interaction

The single-layer black phosphorus is characterized by its puckered configuration that possesses the hinge-like behavior, which leads to the highly anisotropic in-plane Poisson's ratios and the negative out-of-plane Poisson's ratio. We demonstrate that the hinge-like mechanism can be described by the angle–angle cross interaction, which, combined with the bond stretching and angle bending interactions, is able to provide a good description for the mechanical properties of single-layer black phosphorus. We also propose a nonlinear angle–angle cross interaction, which follows the form of Stillinger–Weber potential and can be advantageous for molecular dynamics simulations of single-layer black phosphorus under large deformation.

Continuum thin-shell model of the anisotropic two-dimensional materials: Single-layer black phosphorus

Using the geometric parameters and the elastic moduli determined by the density functional theory (DFT) calculations as the benchmarks, the original Stillinger–Weber (SW) potential of the single layer black phosphorus (SLBP) is revised. The revised SW potential can effectively describe the elastic behavior of SLBP, including both the elastic moduli and the stress–strain relationship under large deformation. Using the molecular dynamics (MD) simulations (based on the revised SW potential), it is found that the continuum thin-shell theory can effectively describe the bending and buckling behavior of SLBP: although the elastic behavior of SLBP is highly anisotropic, the thin-shell thicknesses along different directions determined from the continuum theory are very close to each other. In addition, the thin-shell thickness of SLBP is very close to its physical thickness, and the thin-shell thickness determined from the bending theory is also very close to that determined from its buckling behavior, which are highly different from the corresponding behaviors of graphene. Thus, the lower limit for effectively applying the continuum thin-shell theory to investigating the mechanical behavior of two-dimensional (2D) materials is that 2D-materials include at least two atomic layers.