2D Monoelements. Группа авторов
to two types of bonding, namely, the in-plane bond length d1 is about 2.224 Å , and the out-of-plane bond length d2 that is 2.244 Å, as illustrated in Figure 1.1b. The binding angles y and x are 96.3° and 102.095°, respectively. The height difference between the two half-layers is dz = 2.10 Å .
1.2.2 Physical Properties
1.2.2.1 Anisotropic Eectronic Behavior
Pristine phosphorene is a p-type semiconductor with a direct band gap [13, 46–48]. By using polarization-resolved photoluminescence excitation spectroscopy at room temperature, the quasi-particle band gap of phosphorene is measured to be 2.2 eV [49]. The same value is observed with the typical tunneling spectra of U-shaped electronic spectra [48].
Pure phosphorene has no spin polarization, which is confirmed by symmetrical density of spin-up and -down states displayed in Figure 1.2b. Meanwhile, Figure 1.2a shows that the band dispersion is highly anisotropic around the electronic gap. Indeed, one can observe a much greater dispersion along the Γ-X direction for CBM and VBM with respect to the vertical bands in Γ-Y region. The partial density of states (p-DOS) plots clearly show that px orbital contribute mainly in the states of the unfilled C-band, while the pz orbital of phosphorus dominates the valence band states [50]. The number of layers mainly affects the gap energy [24]. For instance, it takes the values of 1.51, 0.59, and 0.3 eV for the monolayer, the five layers, and the bulk black phosphorus [51]. Furthermore, the gap energy decreases with increasing the magnitude of external electrical field, which breaks the out-of-plane symmetry. Under biaxial strain and the two possible uniaxial strains, the deformed phosphorene shows a transition from the semiconducting to metallic phase [23].
Figure 1.2 Graph of electronic features corresponding to 2D BP. (a) the band structure, (b) represents the total and partial density of states.
1.2.2.2 Optical Properties
The puckered structure of phosphorene attributes it interesting optical properties. Phosphorene absorbs transverse radiation along its AC-axis, while it highly transmitted light along ZZ-axis [19, 52]. The photoluminescence excitation spectroscopy (PLE) measures an optical band gap of 1.31 eV owing to the exciton binding energy as discussed in [19, 49] and measured in [53]. Notice that the theoretical values are larger than the measured amounts because of the increased screening from the dielectric substrate that reduces the quasi-particle band gap and consequently the exciton binding energy [54]. Furthermore, phosphorene can absorbs the visible light since its optical absorption peak is located at 1.6 eV. All these features suggest phosphorene as a promising optoelectronic device for future applications.
The absorption peak can be tunable via strain as displayed in Figure 1.3. For deformed phosphorene, the absorption peak ranges from 0.38 to 2.07 eV under compressive and tensile strain revealing that the material absorbs both infrared and visible light. For the electric field vector E⊥, the graph displaying the imaginary part of the dielectric function E2(ω) shows a considerable shift of the first peak towards high energies when including quasi-particle corrections. However, the shape of E2(ω) spectrum changes considerably when taking into account the electron-hole correlations (BSE) [19]. The exciton binding energy is 0.818 eV for the first active one and 0.66 eV for the first dark one. As displayed in Figures 1.4a and b, the dielectric screening enlarges both the gap and binding energy at ω = 0. Furthermore, a large excitonic wave function distribution is observed for the first bright exciton in Figure 1.4c whose peak emerges in the IR part as shown in Figure 1.4d. The maximum reflectivity Rmax(ω) of 38% occurs in the IR range while it did not exceed 22% for the visible light (see Figure 1.4e). The electron energy loss spectra in Figure 1.4f reveal that the first plasmon peaks in phosphorene sheet has a height of 11.003 dispersed in the IR range of the spectrum.
Figure 1.3 Absorption spectra for undeformed monolayer phosphorene with 0% strain and deformed mono-layer under compressive –8%, –6%, and –4% and tensile strain of 4% and 5.5%. The curves are obtained through the GW+BSE method.
Figure 1.4 (a) and (b) dielectric function, (c) absorption coefficient, (d) reflectivity function, and (e) EELs function, obtained by using three approximations GW-BSE, GW-RPA, and GGA-RPA. (f) Represents the wave function of electron-hole.
In phosphorene multi-layers, the photoluminescence depends significantly on N. The first absorption peak is shifting to lower energies with increasing N. Consequently, the optical absorption coefficient ranges in the interval [0.3–1.2] eV [19], which means that the absorption radiation spectrum include IR and part of visible [52].
1.2.2.3 Elastic Parameters
The elastic properties of phosphorene are also very anisotropic since the Young values in AC-axis (x-direction) is four times lower than the one along the zigzag axis (y-axis), as indicated by the polar diagrams of Υ(θ) illustrated in Figure 1.5a [23, 55]. Notice that the weakest P-P bond strength is the main cause of these small values of the Young parameter [22], compared to 1 TPa and 270 GPa reported for graphene and MoS2, respectively. Furthermore, the Poisson’s ratio, namely, 0.73 and 0.165 in the AC and ZZ directions, respectively, confirms also the high anisotropy in phosphorene [56]. However, the negative value observed in the small interval [5π, 10/3π] as displayed in Figure 1.5b reveals that the material is auxetic. More precisely, for some particular stretch, phosphorene shows a lateral extension instead of longitudinal elongation as it is the case for conventional materials [57, 58]. Besides this, both the shear and compressional acoustic waves propagate more rapidly in the ZZ-axis as it is clearly deduced from the polar plot of the speed of sound in Figure 1.5c. Same anisotropic behavior is found for Debey temperature that is half times lower (see Figure 1.5d).
Besides the strong anisotropy of the elastic parameters, phosphorene exhibits a high elasticity compared with other monolayer materials such