By performing ab initio calculations for one-to four-layer black phosphorus within the GW approximation, we obtain a significant difference in the band gap (∼1.5 eV), which is in line with recent experimental data. The results are analyzed in terms of the constructed four-band tightbinding model, which gives accurate descriptions of the mono-and bilayer band structure near the band gap, and reveal an important role of the interlayer hoppings, which are largely responsible for the obtained gap difference.

Recently, several research groups have reported the growth of germanene, a new member of the graphene family. Germanene is in many aspects very similar to graphene, but in contrast to the planar graphene lattice, the germanene honeycomb lattice is buckled and composed of two vertically displaced sub-lattices. Density functional theory calculations have revealed that free-standing germanene is a 2D Dirac fermion system, i.e. the electrons behave as massless relativistic particles that are described by the Dirac equation, which is the relativistic variant of the Schrödinger equation. Germanene is a very appealing 2D material. The spin-orbit gap in germanene (~24 meV) is much larger than in graphene (<0.05 meV), which makes germanene the ideal candidate to exhibit the quantum spin Hall effect at experimentally accessible temperatures. Additionally, the germanene lattice offers the possibility to open a band gap via for instance an externally applied electrical field, adsorption of foreign atoms or coupling with a substrate. This opening of the band gap paves the way to the realization of germanene based field-effect devices. In this topical review we will (1) address the various methods to synthesize germanene (2) provide a brief overview of the key results that have been obtained by density functional theory calculations and (3) discuss the potential of germanene for future applications as well for fundamentally oriented studies.

Utilizing a combination of low-temperature scanning tunneling microscopy/spectroscopy (STM/STS) and electronic structure calculations, we characterize the structural and electronic properties of single atomic vacancies within several monolayers of the surface of black phosphorus. We illustrate, with experimental analysis and tight-binding calculations, that we can depth profile these vacancies and assign them to specific sublattices within the unit cell. Measurements reveal that the single vacancies exhibit strongly anisotropic and highly delocalized charge density, laterally extended up to 20 atomic unit cells. The vacancies are then studied with STS, which reveals in-gap resonance states near the valence band edge and a strong p-doping of the bulk black phosphorus crystal. Finally, quasiparticle interference generated near these vacancies enables the direct visualization of the anisotropic band structure of black phosphorus.

We study the electronic and optical properties of single-and bilayer black phosphorus with short-and long-range defects by using the tight-binding propagation method. Both types of defect states are localized and induce a strong scattering of conduction states, reducing significantly the charge carrier mobility. In contrast to properties of pristine samples, the anisotropy of defect-induced optical excitations is suppressed due to the isotropic nature of the defects. We also investigate the Landau level spectrum and magneto-optical conductivity and find that the discrete Landau levels are sublinearly dependent on the magnetic field and energy level index, even at low defect concentrations.

We provide a tight-binding model parametrization for black phosphorus (BP) with an arbitrary number of layers. The model is derived from partially self-consistent GW0 approach, where the screened Coulomb interaction W0 is calculated within the random phase approximation on the basis of density functional theory. We thoroughly validate the model by performing a series of benchmark calculations, and determine the limits of its applicability. The application of the model to the calculations of electronic and optical properties of multilayer BP demonstrates good quantitative agreement with ab initio results in a wide energy range. We also show that the proposed model can be easily extended for the case of external fields, yielding the results consistent with those obtained from first principles. The model is expected to be suitable for a variety of realistic problems related to the electronic properties of multilayer BP including different kinds of disorder, external fields, and many-body effects.

To date germanene has only been synthesized on metallic substrates. A metallic substrate is usually detrimental for the two-dimensional Dirac nature of germanene because the important electronic states near the Fermi level of germanene can hybridize with the electronic states of the metallic substrate. Here we report the successful synthesis of germanene on molybdenum disulfide (MoS2), a band gap material. Pre-existing defects in the MoS2 surface act as preferential nucleation sites for the germanene islands. The lattice constant of the germanene layer (3.8 ± 0.2Å) is about 20% larger than the lattice constant of the MoS2 substrate (3.16Å). Scanning tunneling spectroscopy measurements and density functional theory calculations reveal that there are, besides the linearly dispersing bands at the K points, two parabolic bands that cross the Fermi level at the Γ point. The discovery that graphene, a single layer of sp 2 hybridized carbon atoms arranged in a honeycomb registry, is stable has resulted in numerous intriguing and exciting scientific breakthroughs [1,2]. The electrons in graphene behave as relativistic massless fermions that are described by the Dirac equation, i.e. the relativistic variant of the Schrödinger equation. One might anticipate that elements with a similar electronic configuration, such as silicon (Si), germanium (Ge) and tin (Sn), also have a "graphene-like" allotrope. Unfortunately, silicene (the silicon analogue of graphene), germanene (the germanium analogue of graphene) and stanene (the tin analogue of graphene) have not been found in nature and therefore these two-dimensional (2D) materials have to be synthesized. Theoretical calculations have revealed that the honeycomb lattices of the "graphene-like" allotropes of silicon and germanium are not fully planar, but slightly buckled [3,4]. The honeycomb lattices of these 2D materials consist of two triangular sub-lattices that are slightly displaced with respect to each other in a direction normal to the honeycomb lattice. Despite this buckling the 2D Dirac nature of the electrons is predicted to be preserved [3,4]. Another salient difference with graphene is that silicene and germanene have a substantially larger spin-orbit gap than graphene (<0.05 meV). Silicene's spin-orbit gap is predicted to be 1.55 meV, whereas the predicted spin-orbit gap of germanene is even 23.9 meV. This is very interesting because graphene and also silicene and germanene are in principle 2D topological insulators and thus ideal candidates to exhibit the quantum spin Hall effect. The interior of a 2D topological insulator exhibits a spin-orbit gap, whereas topologically protected helical edge modes exist at the edges of the material [5,6]. The two topologically protected spin-polarized edge modes have opposite propagation directions and therefore the charge conductance vanishes, whereas the spin conductance has a non-zero value.In the past few years various groups have successfully synthesized silicene [7][8][9] and germanene [10-13] on a variety of substrates. T...

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We present a theory for single-and two-phonon charge carrier scattering in anisotropic twodimensional semiconductors applied to single-layer black phosphorus (BP). We show that in contrast to graphene, where two-phonon processes due to the scattering by flexural phonons dominate at any practically relevant temperatures and are independent of the carrier concentration n, two-phonon scattering in BP is less important and can be considered negligible at n 10 13 cm −2 . At smaller n, however, phonons enter in the essentially anharmonic regime. Compared to the hole mobility, which does not exhibit strong anisotropy between the principal directions of BP (µxx/µyy ∼ 1.4 at n = 10 13 cm −2 and T = 300 K), the electron mobility is found to be significantly more anisotropic (µxx/µyy ∼ 6.2). Absolute values of µxx do not exceed 250 (700) cm 2 V −1 s −1 for holes (electrons), which can be considered as an upper limit for the mobility in BP at room temperature.Electron-phonon scattering is considered to be the main factor limiting intrinsic charge-carrier mobility in graphene [1][2][3][4][5][6][7]. Flexural phonons (out-of-plane vibrations) are especially important in this respect because they provide the dominant contribution to the resistivity at room temperature [6,7]. Recently, many new two-dimensional (2D) materials have attracted attention [8], such as hexagonal boron nitride [9], stoichiometric graphene derivatives [10,11], transition-metal dichalcogenides [9,12], and black phosphorus (BP) [13]. All these materials are typically more defective than graphene and are characterized by significantly smaller electron mobility; therefore, much less is known experimentally on their intrinsic transport properties [14][15][16][17][18][19][20][21].Comprehensive theories have been developed to describe the mechanism of phonon scattering in graphene [7]. The application of those is, however, not straightforward to systems with reduced symmetries that give rise to anisotropy of electronic and vibrational properties. At the same time, anisotropy of 2D materials in not uncommon. It can naturally arise in finite-size samples and be governed by the shape (e.g., nanoribbons) [22] or can be determined by external conditions such as defects [23], mechanical strain [24], or contact potentials [25]. Few-layer black phosphorus is the most prominent example among 2D materials with inherent anisotropy [13]. Early attempts to describe intrinsic mobility in ultrathin BP were based on isotropic transport theory and were focused on single-phonon processes only [26,27].In this Letter, we develop a theory for phonon-limited transport in anisotropic 2D semiconductors. We obtain general expressions for the scattering matrix of singleand two-phonon processes, where both in-plane and flexural acoustic phonons are included. The theory is applied to monolayer black phosphorus, for which the relevant parameters are estimated from first principles.For isotropic materials, phonon limited dc conductivity is usually calculated using the standard semiclassical...