Electronic and optical properties of pristine and oxidized borophene

The lattice parameters of the optimized rectangular unit cell of borophene are displayed in figures 1(a)–(c), i.e. (a = 1.6212; b = 2.8699) Å with a buckling height of 0.89 Å between bottom and top atoms (B1 and B2 respectively). The calculated cohesive energy (${E}_{{\rm{c}}}$) is −6.707 eV/atom (see equation (5)) while the cohesive energy of the bulk crystal of boron (α-rhombohedral with 12 B atoms per unit cell) is −7.262 eV/atom, i.e. borophene is metastable by 0.555 eV/atom with respect to bulk phase.

$$\begin{eqnarray}&&{E}_{{\rm{c}}}={E}_{\mathrm{borophene}}/{N}_{{\rm{B}}}\,-\,{E}_{{\rm{B}}},\end{eqnarray} \tag{ 5 }$$

where ${E}_{{\rm{B}}}$ and ${E}_{\mathrm{borophene}}$ are the total energies of an isolated B atom and of the borophene layer containing ${N}_{{\rm{B}}}$ boron atoms. Other boron monolayer structures have been reported with differences in total energies with respect to bulk phase being in the order 0.32–0.40 eV/atom [2, 3]. This means that the present borophene structure (buckled closed-packed triangular) is not the most stable 2D phase but may be favored by the underlying Ag surface on which it is grown.

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The electronic band structure (figure 1(d)) exhibits a priori a strongly anisotropic metallic character as the bands are found to be highly dispersive in the k_x direction (Γ–X and Y–S) with relatively high group velocity of 6.6 10⁵ m s⁻¹, while it appears that no bands cross the Fermi level in the k_y direction (Γ–Y and X–S) as reported previously [8, 10]. Notwithstanding, one also notes that dispersion occurs along the oblique Γ–S path. This reveals that bands can actually cross the Fermi level parallel to the k_y direction in some region of the 2D Brillouin zone. In order to have better insight on the exact anisotropy of the electronic structure it is thus important to plot the electronic band structure in the whole 2D Brillouin zone and identify the Fermi surface (which in 2D actually reduces into a Fermi line) (see figure 1(e)). While electronic bands always cross the Fermi level for any line parallel to k_x, this is not the case for lines parallel to k_y. Therefore, it is expected that transport will occur only for selective wave vectors k whose k_x component belongs to the allowed region (figure 1(e)). This sets a maximal angle (k${}_{\theta }^{\max }$) for the wave vectors k = (k_x, k_y) ≡ (k_r, k${}_{\theta }$). One also notes, that the two bands crossing the Fermi level become exactly degenerate at the edge of the Brillouin zone along the S–Y path.

Then, the DOS around the Fermi level (figure 1(d)) exhibits an almost planar shape, typical of 2D free electron gas, except when approaching an energy equal to E_F−1 eV, where a van Hove singularity is observed. This energy point corresponds to a saddle point of an electronic band at the Γ point (figures 1(d) and (e)). The presence of this peak generates a native asymmetry between electrons and holes which will add to the above mentioned asymmetry between transport directions. Finally, the band structure of borophene was recalculated within a LDA approach using ABINIT package and then corrected by applying a one shot G₀W₀ approximation to the quasi particle problem. No significant changes were observed between the Kohn–Sham structure and the G₀W₀ one. One can only notice an increase of the local band gap at the Γ point, and generally speaking an increase of the Fermi velocity.

Experimental measurements indicate that bare borophene on a Ag substrate is partially oxidized within several hours unless it is capped with amorphous silicon/silicon oxide [8]. Even after encapsulation and after several weeks, residual oxygen bound to borophene has been found [8]. Therefore, the effect of oxygen on the electronic and optical properties of borophene cannot be ignored and should be addressed. Due to computational limitations, however, one-side oxidation of the free standing borophene is considered rather than the surface oxidation of the full borophene-substrate system. It is worth noting that the detached free standing borophene can experience strong structural changes before the oxidation takes place, due to the aforementioned small phonon instabilities. Nevertheless, our simulations aim to describe the properties of oxidation of the supported borophene albeit with the assumption of a limited impact of the underneath substrate.

A 3 × 3 borophene supercell is first considered with four adsorption sites which correspond to a position on top of a boron atom (B1 or B2), a bridge position either in between first nearest-neighbor atoms (along the x-direction) or in between third nearest neighbor atoms (along the y-direction) of the top layer (B2–B2) (see supplementary figure 7). The last case is found to be the most stable adsorption site with a relatively high adsorption energy of −4.548 eV. The adsorption energy of the other configurations can be in found in supplementary table 1. The adsorption energy (${E}_{\mathrm{ad}}$) has been computed with respect to total energies of isolated borophene and molecular oxygen (O₂) as

$$\begin{eqnarray}&&{E}_{\mathrm{ad}}={E}_{\mathrm{borophene}+{\rm{O}}}\,-\left({E}_{\mathrm{borophene}}+\displaystyle \frac{{E}_{{{\rm{O}}}_{2}}}{2}\right).\end{eqnarray} \tag{ 6 }$$

A negative value of ${E}_{\mathrm{ad}}$ means that the adsorption is favorable.

During the adsorption, the two B2 atoms bound to the O atom are slightly displaced out of the top layer and get closer to each other as depicted in figures 2(a)–(c). The corresponding folded electronic band structure (compared to the band structure of pristine borophene), as well as the DOS and the average velocities (v_x and v_y), are plotted in figure 2(d). From the band structure, one notes that many band crossings and band degeneracies are avoided and lifted up, respectively, but it remains difficult to clearly assess the impact of oxygen adatom because of the folding. However, the impact on the DOS is more obvious. Several new peaks appear and are well distributed in energy which makes the DOS curve much more noisy than in pristine case but still non-zero at the Fermi level (no gap opening). The band structures and DOS for the other adsorption sites are displayed in supplementary figure 8. Even more obvious than the impact on the DOS is the impact on velocities. Indeed, as it can be seen in figure 2(d), right to the DOS, the computed average velocity along the x- and y-direction largely decreases when the O adatom is incorporated compared to the pristine case. One also notes that the anisotropy has almost disappeared. Finally, charge transfer between borophene and O adatom has been investigated using Voronoi and Hirshfeld methods. The O adatom is found to loose 0.15–0.17 $| e|$ in bridge positions while it looses 0.23–0.25 $| e|$ in top positions. The electronic charge gained by borophene is spread over the system.

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To provide a better understanding of the modifications induced by O adatoms, it is also essential to investigate its dependence as a function of the concentration. For this purpose, in addition to the previous 3 × 3 supercell, larger supercells have been considered i.e. a 4 × 4, a 5 × 5, and a 6 × 6 (supplementary figure 9). The corresponding oxygen surface density are n${}_{{\rm{O}}}$ = $2.34\,{10}^{14}$, $1.31\,{10}^{14}$, $8.41\,{10}^{13}$, and $5.84\,{10}^{13}$ cm⁻². The dependence on the DOS is illustrated in figure 3. As it can be seen, the DOS remains always non-zero and thus the oxidation preserves a priori the metallic character. However, due to the presence of numerous DOS peaks and the velocity decreasing one can anticipate the presence of localized states around the defects and thus strong scattering mechanisms which should largely degrade the electronic conductivity and mobility in oxidized borophene.

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Beside its potentially interesting anisotropic conduction properties, pristine borophene is expected to be highly transparent owing to its extremely small thickness. This could make the material interesting for photovoltaic applications and touch screens. A transparent conductor is a priori a counter-intuitive property since common bulk metals are usually not transparent but instead looks gray and shiny to human eyes. It is because a large part of light is reflected at the interface. It is however well known that beyond the plasmon frequency, i.e. for photon energy larger than ${E}_{{\rm{p}}}={\hslash }{\omega }_{{\rm{p}}}$, the reflectance can strongly decrease and possibly render the metal transparent. For most of metals the plasmon frequency is in the ultra-violet (UV) which means they indeed reflect light in the visible region. However, when the thickness is reduced to few (or even one) atomic layers (the so-called thin film limit; TFL), the reflectance drops to nearly zero and the transmission increases to almost 100% and their variations are actually mostly related to the optical conductivity. In the following one assesses carefully the optical properties of borophene accounting correctly for the 2D nature, in contrast to a recent work [10] where the 3D bulk formula was considered, showing relatively high reflectance. We first discuss the optical conductivity and second the transmittance (the reflectance plots can be found in supplementary material).

The real part of optical conductivity (${\sigma }_{1}(E)\ =E\,{\varepsilon }_{0}{\varepsilon }_{2}(E)/{\hslash }$, where ${\varepsilon }_{0}$ is the vacuum dielectric constant) of borophene is first investigated and is displayed in figures 4(a)–(d). Different polarizations of light are considered, i.e. an electric field along the x direction, along the y direction, and average of both directions (i.e. a light propagating along the z direction). In figure 4(a), the scattering time parameter originating from the Drude model is fixed to τ = 1.21 10⁻¹⁸ s (Γ = 20 Ha) which is a very small value describing thus a 'bad' metal (actually a mean free path in the order of a picometer). At the same time one can see, from equations (2) and (3), that for such a large value of Γ, the Drude contributions ($\propto 1/{{\rm{\Gamma }}}^{2}$) becomes negligible, just as if one was using only the usual interband transitions to describe the optical properties. In such a situation, the optical conductivity is found to be almost zero for low energy radiation up to 3 eV (i.e. violet, close to the end of visible light). Then in the UV part, the optical conductivity increases slowly up to 6.5 eV before exhibiting an enhanced signal composed of a series of four peaks at 6.5, 8.2, 8.5 and 9.6 eV. The low optical absorption of borophene is even better highlighted by comparing it with the one of monolayer graphene (gray dashed lines in figure 4) whose optical conductivity is between 1 and 2 ${\sigma }_{0}$ (with ${\sigma }_{0}={e}^{2}\pi /2h$) up to 3 eV. The fact that borophene appears to be transparent in the infrared-visible region is related here to the fact that it has been considered as a bad metal (and acts thus as an insulator) and that interband transitions occur only above 3 eV. One also notes the anisotropy of the optical absorption between x and y directions. In particular, the optical conductivity remains very small up to 6.5 eV for polarization along the x direction.

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When going now to the case of a 'good' metal, that is increasing τ up to 1.21 10⁻¹³ s (mean free path in the order of 10² nm) (figures 4(b)–(d)), one observes first an almost constant increase of the optical conductivity forming a plateau, and then a divergence at low energy going towards a delta-Dirac function at zero energy which finally restores a very low optical conductivity in the visible region, i.e. similar to the one observed in figure 4(a) except for the zero energy limit. This is the expected behavior for ${\sigma }_{1}$ of a perfect metal in the Drude model, i.e. a delta-Dirac function at zero energy. This reflects the fact that in the static limit, i.e. for a constant electric field, the dc conductivity tends to infinity (${\sigma }_{1}(0)\to \infty$) as the scattering vanishes and hence does not damp the acceleration of electrons caused by the electric field. For an alternating field the real part of the ac conductivity is zero (${\sigma }_{1}(\omega )=0$), and the ac conductivity becomes actually purely imaginary (${\sigma }_{2}(\omega )=\tfrac{n\,{e}^{2}}{m\,\omega }$). Hence, in both limits of a bad and good metal, borophene would exhibit nearly zero conductivity in the visible region. As mentioned previously, the fact that the optical conductivity is zero in the visible region implies, following the TFL, that the transmittance goes to 100% and that the material should be fully transparent.

The actual transmittance and reflectance have been computed following both rigorous Fresnel equations, and TFL equations (that are obtained from the former ones by taking the limit of small thickness). In contrast to the bulk case where reflectance accounts only for one interface (typically between vacuum and the material), the reflectance in thin films accounts for the presence of a back side interface which produces secondary reflections. As most ab initio codes do not have this particular treatment for 2D thin systems, it is important to carefully compute optical properties using general Fresnel equations or at least TFL equations. Although small differences are observed in reflectance curves, both approaches yield very similar results for transmittance validating therefore the TFL in case of graphene and borophene. The reflectance (R), the transmittance (T), and the absorbance (A), in TFL are given by [25, 26]

$$\begin{eqnarray}\begin{array}{rcl}{R}_{\mathrm{TFL}}(E) & = & \displaystyle \frac{{({n}^{\mathrm{above}}-{n}^{\mathrm{below}}-\beta {n}_{2}^{\mathrm{film}})}^{2}}{{({n}^{\mathrm{above}}+{n}^{\mathrm{below}}+\beta {n}_{2}^{\mathrm{film}})}^{2}}\end{array}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}\,= & \displaystyle \frac{{\left(\tfrac{\beta {n}_{2}^{\mathrm{film}}}{2}\right)}^{2}}{{\left(1+\tfrac{\beta {n}_{2}^{\mathrm{film}}}{2}\right)}^{2}},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}\begin{array}{rcl}{T}_{\mathrm{TFL}}(E) & = & \displaystyle \frac{4{n}^{\mathrm{above}}{n}^{\mathrm{below}}}{{({n}^{\mathrm{above}}+{n}^{\mathrm{below}}+\beta {n}_{2}^{\mathrm{film}})}^{2}}\end{array}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}\,= & \displaystyle \frac{1}{{\left(1+\tfrac{\beta {n}_{2}^{\mathrm{film}}}{2}\right)}^{2}},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{TFL}}(E) & = & \displaystyle \frac{4{n}^{\mathrm{above}}\beta {n}_{2}^{\mathrm{film}}}{{({n}^{\mathrm{above}}+{n}^{\mathrm{below}}+\beta {n}_{2}^{\mathrm{film}})}^{2}}\end{array}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}\,= & \displaystyle \frac{\beta {n}_{2}^{\mathrm{film}}}{{\left(1+\tfrac{\beta {n}_{2}^{\mathrm{film}}}{2}\right)}^{2}},\end{eqnarray} \tag{ 12 }$$

where $n={n}_{1}+{{\rm{i}}{n}}_{2}=\sqrt{\varepsilon }$ is the complex refractive index for a non-magnetic material. The superindex indicates which medium it refers to, i.e. the medium above the film from where light comes, the medium below the film into which light is transmitted, and the film itself. Equations (8), (10) and (12) give the reduced expression of equations (7), (9) and (11) respectively, when above and below media are vacuum. $\beta =\tfrac{4\pi }{\lambda }\,{n}^{\mathrm{film}}\,{d}^{\mathrm{film}}=\tfrac{2E}{c{\hslash }}\,{n}^{\mathrm{film}}\,{d}^{\mathrm{film}}$, where ${d}^{\mathrm{film}}$ is the film thickness (or equivalently the polarizable electronic thickness defined as h in method section). (The fully rigorous Fresnel equations are also given in supplementary material, but as mentioned both sets of equations gives similar results.) The transmittance of borophene is displayed in figure 5. The scattering time parameter is first fixed to τ = 1.21 10⁻¹⁸ s (bad metal case) in figure 5(a) for which one observes high transparency up to 6 eV, and a complete transparency (100% transmission) up to 3 eV. High anisotropy is also found between light polarization with respect to crystal orientation. Then in figures 5(b)–(d) corresponding to these different polarization of the electric field, one varies τ up to 1.21 × 10⁻¹³ s (good metal case). In the visible range, T starts to decrease forming a plateau at 96%–98%, and then increase again to 100%. In the infrared range, T is first strongly reduced especially close to zero energy where T eventually drops to zero in a very good metal case. The reflectance curves can be found in supplementary figure 10 and corroborate results found for optical conductivities and transmittance. The reflectance is nearly zero in the infrared-visible range except when considering Drude model with long scattering time (τ = 1.21 × 10⁻¹³ s) especially in the infrared region.

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The optical conductivities of oxidized borophene systems are presented in figure 6 and correspond to the 3 × 3, 4 × 4, 5 × 5, and 6 × 6 supercells (depicted in supplementary figure 9). At low energy, prominent peaks are clearly observed in the infrared (IR) spectrum part. Optical conductivity is also significantly increased in the visible light region. Except for the 3 × 3 supercell, the optical conductivity curves of the larger oxidized borophene supercells are nicely reproduced using pristine borophene and a Drude term with τ = 1.21 × 10⁻¹⁵ s. Such a large increase in the infrared suggests that optical conductivity is highly sensitive to probably any surface modification, which makes optical conductivity/absorption measurement an interesting tool to investigate the cleanness of borophene surface. This result also implies that transparency of borophene will be slightly limited by surface chemical functionalization, although the absolute values in figure 6 indicate that even for reasonably high degree of oxygenation (10¹³ cm${}^{-2}\,\lt$ n${}_{{\rm{O}}}\lt {10}^{14}$ cm⁻²), the optical conductivity in the visible range is lower than in pristine graphene.

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