We report the quasiparticle band gap, excitons, and highly anisotropic optical responses of few-layer black phosphorous (phosphorene). It is shown that these new materials exhibit unique many-electron effects; the electronic structures are dispersive essentially along one dimension, leading to particularly enhanced self-energy corrections and excitonic effects. Additionally, within a wide energy range, including infrared light and part of visible light, few-layer black phosphorous absorbs light polarized along the structure's armchair direction and is transparent to light polarized along the zigzag direction, making them viable linear polarizers for applications. Finally, the number of phosphorene layers included in the stack controls the material's band gap, optical absorption spectrum, and anisotropic polarization energy-window across a wide range.
Anisotropic in-plane thermal conductivity observed in few-layer black phosphorus
黑磷作为一种新型的二维材料，在电子学和光电子学领域有着潜在的应用前景。在这里，我们报告的各向异性面内导热系数的悬浮少层黑磷测量的显微拉曼光谱。对于厚度大于15 nm的黑磷膜，扶手椅型和锯齿型热导率分别为~20和~40 W m $^{-1}$ K $^{-1}$，并且随着膜厚度减小，热导率降低至~10和~20 W m $^{-1}$ K $^{-1}$，表现出显著的各向异性。发现对于厚的黑磷膜，热导率各向异性比为~2，对于最薄的9.5 nm厚的膜，热导率各向异性比下降到~1.5。理论模型表明，所观察到的各向异性主要与各向异性声子色散有关，而本征声子散射率被发现是类似的扶手椅和锯齿形方向。黑磷薄膜中的表面散射强烈抑制长平均自由程声学声子的贡献。
Using the density functional theory of electronic structure, we compute the anisotropic dielectric response of bulk black phosphorus subject to strain. Employing the obtained permittivity tensor, we solve Maxwell's equations and study the electromagnetic response of a layered structure comprising a film of black phosphorus stacked on a metallic substrate. Our results reveal that a small compressive or tensile strain, $\sim 4\%$, exerted either perpendicular or in the plane to the black phosphorus growth direction, efficiently controls the epsilon-near-zero response, and allows a perfect absorp…
When a two-dimensional material, adhered onto a compliant substrate, is subjected to compression it can undertake a buckling instability yielding to a periodic rippling. Interestingly, when black phosphorus flakes are compressed along the zig-zag crystal direction the flake buckles forming ripples with a 40% longer period than that obtained when the compression is applied along the armchair direction. This anisotropic buckling stems from the puckered honeycomb crystal structure of black phosphorus and a quantitative analysis of the ripple period allows us to determine the Youngs's modulus of f…
Tuning band gaps in two-dimensional (2D) materials is of great interest in the fundamental and practical aspects of contemporary material sciences. Recently, black phosphorus (BP) consisting of stacked layers of phosphorene was experimentally observed to show a widely tunable band gap by means of the deposition of potassium (K) atoms on the surface, thereby allowing great flexibility in design and optimization of electronic and optoelectronic devices. Here, based on the density-functional theory calculations, we demonstrates that the donated electrons from K dopants are mostly localized at the…
Resistivity measurements of a few-layer black phosphorus (bP) crystal in parallel magnetic fields up to 45 T are reported as a function of the angle between the in-plane field and the source-drain (S-D) axis of the device. The crystallographic directions of the bP crystal were determined by Raman spectroscopy, with the zigzag axis found within 5° of the S-D axis, and the armchair axis in the orthogonal planar direction. A transverse magneto-resistance (TMR) as well as a classically-forbidden longitudinal magneto-resistance (LMR) are observed. Both are found to be strongly anisotropic and non-m…
Black phosphorus has attracted great research interest due to its numerous applications in electronic devices, optoelectronic devices, energy storages and so on. Compared with the majority of two-dimensional materials, black phosphorus possesses a unique property, i.e. the strong in-plane anisotropy. All the properties reported so far, including its effective mass, electron mobility, light absorption, thermal conductivity and so on, have shown great anisotropy in the basal plane. This property renders black phosphorus unique applications not achievable with other two-dimensional materials. In …
Anisotropic thermo-mechanical response of layered hexagonal boron nitride and black phosphorus: application as a simultaneous pressure and temperature sensor
In this work we briefly review the studies of the optical constants of monolayer transition metal dichalcogenides and few layer black phosphorus, with particular emphasis to the complex dielectric function and refractive index. Specifically, we give an estimate of the complex index of refraction of phosphorene and few-layer black phosphorus. We extracted the complex index of refraction of this material from differential reflectance data reported in literature by employing a constrained Kramers-Kronig analysis. Finally, we studied the linear optical response of multilayer systems embedding phos…
Black phosphorus (BP) is a two-dimensional layered material composed of phosphorus atoms. Recently, it was demonstrated that external perturbations such as an electric field close the band gap in few-layer BP, and can even induce a band inversion, resulting in an insulator phase with a finite energy gap or a Dirac semimetal phase characterized by two separate Dirac nodes. At the transition between the two phases, a semi-Dirac state appears in which energy disperses linearly along one direction and quadratically along the other. In this work, we study the optical conductivity of few-layer BP us…
In this study, we develop a systematic weak localization/antilocalization theory fully considering the anisotropy and Berry phase of the system, and apply it to various phases of few-layer black phosphorus (BP), which has a highly anisotropic electronic structure with an electronic gap size tunable even to a negative value. The derivation of a Cooperon ansatz for the Bethe-Salpeter equation in a general anisotropic system is presented, revealing the existence of various quantum interference effects in different phases of few-layer BP, including a crossover from weak localization to antilocaliz…
The effect of anisotropy in the energy spectrum on the binding energy and structural properties of excitons, trions, and biexcitons is investigated. To this end we employ the stochastic variational method with a correlated Gaussian basis. We present results for the binding energy of different excitonic complexes in black phosphorus (bP) and TiS$_3$ and compare them with recent results in the literature when available, for which we find good agreement. The binding energies of excitonic complexes in bP are larger than those in TiS$_3$. We calculate the different average interparticle distances i…
We theoretically study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in two-dimensional black phosphorus, phosphorene. The RKKY interaction enhances significantly for the low levels of hole doping owing to the nearly valence flat band. Remarkably, for the hole-doped phosphorene, the highest RKKY interaction occurs when two impurities are located along the zigzag direction and it tends to a minimum value with changing the direction from the zigzag to the armchair direction. We show that the interaction is highly anisotropic and the magnetic ground-state of two magnetic adatoms can be tun…
We combine femtosecond electron diffuse scattering experiments and first-principles calculations of the coupled electron-phonon dynamics to provide a detailed momentum-resolved picture of the ultrafast lattice thermalization in a thin film of black phosphorus. The measurements reveal the emergence of highly anisotropic non-thermal phonon populations which persist for several picoseconds following excitation of the electrons with a light pulse. Combining ultrafast dynamics simulations based on the time-dependent Boltzmann formalism and calculations of the structure factor, we reproduce the expe…
We review the basic optical, electronic, optoelectronic, thermoelectric and mechanical properties of few-layer black phosphorus (BP), a layered semiconductor that can be exfoliated from bulk BP, the most stable allotrope of phosphorus. The distinguishing trait of BP is its highly anisotropic crystal structure, which leads to strong optical linear dichroism, elliptical plasmonics energy surfaces, anisotropic electronic and thermal conductivities and elasticity. We provide tutorial-like discussion of these phenomena and their theoretical models.
We report the realization of novel symmetry-protected Dirac fermions in a surface-doped two-dimensional (2D) semiconductor, black phosphorus. The widely tunable band gap of black phosphorus by the surface Stark effect is employed to achieve a surprisingly large band inversion up to ~0.6 eV. High-resolution angle-resolved photoemission spectra directly reveal the pair creation of Dirac points and their moving along the axis of the glide-mirror symmetry. Unlike graphene, the Dirac point of black phosphorus is stable, as protected by spacetime inversion symmetry, even in the presence of spin-orbi…
The incorporation of electrically tunable materials into photonic structures such as waveguides and metasurfaces enables dynamic control of light propagation by an applied potential. While many materials have been shown to exhibit electrically tunable permittivity and dispersion, including transparent conducting oxides (TCOs) and III-V semiconductors and quantum wells, these materials are all optically isotropic in the propagation plane. In this work, we report the first known example of electrically tunable linear dichroism, observed here in few-layer black phosphorus (BP), which is a promisi…
Black phosphorus (BP), a two-dimensional (2D) van der Waals layered material composed of phosphorus atoms, has been one of the most actively studied 2D materials in recent years due to its tunable energy band gap (tunable even to a negative value) and its highly anisotropic electronic structure. Depending on the sign of the band gap tuning parameter, few-layer BP can be in a gapped insulator phase, gapless Dirac semimetal phase, or gapless semi-Dirac transition point between the two phases. Using the fully anisotropic multiband Boltzmann transport theory, we systematically study the dc conduct…
We investigate the role of pseudospin structure of few-layer black phosphorus (BP) in interband tunneling properties in lateral BP junctions. We find that interband tunneling is critically dependent on junction directions because of the anisotropic pseudospin structure of BP. When the armchair direction of BP is normal to the interface, pseudospins of incident and transmitted carriers are nearly aligned so that interband tunneling is highly effective, analogous to the Klein tunneling in graphene. However, when the zigzag direction is normal to the interface, interband tunneling is suppressed b…
Screening in reduced dimensions has strong consequences on the electronic properties in van der Waals semiconductors, impacting the quasiparticle band gap and exciton binding energy. Screening in these materials is typically treated isotropically, yet black phosphorus exhibits in-plane electronic anisotropy seen in its effective mass, carrier mobility, excitonic wavefunctions, and plasmonic dispersion. Here, we use the adsorption of individual potassium atoms on the surface of black phosphorus to vary the near-surface doping over a wide range, while simultaneously probing the dielectric screen…
The electronic transport properties of Janus monolayer black arsenic phosphorus (b-AsP) nanoribbons have been investigated utilizing the tight-binding approach. The dependence of electronic structure on edge structures is systematically investigated. (1,3)nb and (3,1)nb b-AsP nanoribbons exhibit flatter edge bands than zigzag and armchair counterparts. The edge band of the armchair ones show double degeneracy. Further, the calculated results show the band gap of the zigzag ribbon with different boundary morphology is widely tunable by transverse electric field. A critical electric field can fu…
We use an atomistic approach to study the electronic properties of monolayer and bilayer black phosphorus in the vicinity of a charged defect. In particular, we combine screened defect potentials obtained from first-principles linear response theory with large-scale tight-binding simulations to calculate the wavefunctions and energies of bound acceptor and donor states. As a consequence of the anisotropic band structure, the defect states in these systems form distorted hydrogenic orbitals with a different ordering than in isotropic materials. For the monolayer, we study the dependence of the …
Black phosphorus presents a very anisotropic crystal structure, making it a potential candidate for hyperbolic plasmonics, characterized by a permittivity tensor where one of the principal components is metallic and the other dielectric. Here we demonstrate that atomically thin black phosphorus can be engineered to be a hyperbolic material operating in a broad range of the electromagnetic spectrum from the entire visible spectrum to ultraviolet. With the introduction of an optical gain, a new hyperbolic region emerges in the infrared. The character of this hyperbolic plasmon depends on the int…
High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus
二维晶体是纳米电子学的新兴材料。该领域的发展需要候选系统具有高载流子迁移率和与石墨烯相反的足够大的电子带隙。在这里，我们提出了一个详细的理论研究的原子和电子结构的少层黑磷（BP），以预测其电学和光学性质。该系统具有直接带隙，可调从1.51 eV的单层到0.59 eV的5层样品。我们预测的迁移率是空穴占主导地位，相当高，高度各向异性。该单层具有极高的空穴迁移率（约10000 cm $^{2}$ V $^{-1}$ s $^{-1}$）和反转各向异性的异常弹性性质。光吸收光谱表明垂直面内方向之间的线性二向色性，这允许光学测定的晶体取向和光学激活的各向异性传输特性。这些结果使少层BP成为未来电子学的一个有前途的候选者。
Two-dimensional materials supporting deep-subwavelength plasmonic modes can also exhibit strong magneto-optical responses. Here, we theoretically investigate magnetoplasmons (MPs) in monolayer black phosphorus (BP) structures under moderate static magnetic fields. We consider three different structures, namely, a continuous BP monolayer, an edge formed by a semi-infinite sheet, and finally, a triangular wedge configuration. Each of these structures shows strongly anisotropic magneto-optical responses induced both by the external magnetic field and by the intrinsic anisotropy of the BP lattice.…
Single-Layered Hittorf's Phosphorus: A Wide-Bandgap High Mobility 2D Material
我们在这里提出了一个二维材料的基础上的一个单一的紫色或希托夫的磷层。使用第一性原理密度泛函理论，我们发现它是非常稳定的，与其他以前提出的单层磷结构。它只需要一个小的能量成本约$0.04~\text{eV/atom}$从它的本体结构，希托夫的磷，或结合能$0.3-0.4~\text{J/m}^2$每层创建，这表明在实验中剥离的可能性。我们发现单层Hittorf磷是一种宽带隙半导体，其直接带隙约为$2.5$ ~eV，并且我们的计算表明，它预计具有高且高度各向异性的空穴迁移率，其上限介于$3000-7000$ ~cm $^2$ V $^{-1}$ s $^{-1}$之间。这些综合性能使得单层Hittorf磷成为未来应用于各种技术的非常好的候选者，特别是用于高频电子学和在低波长蓝色范围内操作的光电子器件。
As a high mobility two-dimensional semiconductor with strong structural and electronic anisotropy, atomically thin black phosphorus (BP) provides a new playground for investigating the quantum Hall (QH) effect, including outstanding questions such as the functional dependence of Landau level (LL) gaps on magnetic field B, and possible anisotropic fractional QH states. Using encapsulating few-layer BP transistors with mobility up to 55,000 cm2/Vs, we extract LL gaps over an exceptionally wide range of B for QH states at filling factors ν=-1 to -4, which are determined to be linear in B, thus re…
The advent of black phosphorus field-effect transistors (FETs) has brought new possibilities in the study of two-dimensional (2D) electron systems. In a black phosphorus FET, the gate induces highly anisotropic 2D electron and hole gases. Although the 2D hole gas in black phosphorus has reached high carrier mobilities that led to the observation of the integer quantum Hall effect, the improvement in the sample quality of the 2D electron gas (2DEG) has however been only moderate; quantum Hall effect remained elusive. Here, we obtain high quality black phosphorus 2DEG by defining the 2DEG region…
We realize p-p'-p junctions in few-layer black phosphorus (BP) devices, and use magneto-transport measurements to study the equilibration and transmission of edge states at the interfaces of regions with different charge densities. We observe both full equilibration, where all edge channels equilibrate and are equally partitioned at the interfaces, and partial equilibration, where only equilibration only takes place among modes of the same spin polarization. Furthermore, the inner p'-region with low-doping level in the junction can function as a filter for highly doped p-regions which demonstr…
Effect of flexural phonons on the hole states in single-layer black phosphorus
晶体膜中的弯曲热涨落影响载流子的能带结构，这导致在未扰动的能带边缘之外的指数态密度（DOS）尾部。我们提出了一个理论描述这种尾巴的一个特定的情况下，在单层黑磷，材料表现出极其各向异性准一维色散（$m_y/m_x\gg1$），因此，增强范霍夫奇异性的价带顶部的空穴。通过{ \it ab initio}计算确定材料参数，然后用于定量估计双声子（弯曲）过程对载流子DOS的影响。结果表明，不同于各向同性的情况下，物理是由声子与波矢量的顺序为$q^*$，其中$q^*$决定的弯曲声子的谐波和非谐波行为之间的交叉。计算了有限温度下单层黑磷中空穴的谱密度。
