Effects of static charging and exfoliation of layered crystals

Using a first-principle plane-wave method we investigate the effects of static charging on the structural, electronic, and magnetic properties of suspended, single-layer graphene, graphane, fluorographene, BN, and MoS${}_2$ in a honeycomb structure. The limitations of periodic boundary conditions in the treatment of negatively charged layers are clarified. Upon positive charging, the band gaps between the conduction and valence bands increase, but the single-layer nanostructures become metallic owing to the Fermi level dipping below the maximum of valence band. Moreover, their bond lengths increase, leading to phonon softening. As a result, the frequencies of Raman active modes are lowered. A high level of positive charging leads to structural instabilities in single-layer nanostructures, since their specific phonon modes attain imaginary frequencies. Similarly, excess positive charge is accumulated at the outermost layers of metallized BN and MoS${}_2$ sheets comprising a few layers. Once the charging exceeds a threshold value, the outermost layers are exfoliated. Charge relocation and repulsive force generation are in compliance with classical theories.

Figure 1

(a) Description of supercell geometry used to treat 2D single layer MoS${}_2$. $c$ and $s$ are the supercell lattice constant and vacuum spacing between adjacent layers, respectively. The $z$ axis is perpendicular to the layers. (b) Self-consistent potential energy of positively charged ($Q>0$ per cell), periodically repeating MoS${}_2$ single layers, which is planarly averaged along the $z$ direction. ${\overline{V}}_{el}\left(z\right)$ is calculated using different vacuum spacings $\mathit{\text{s}}$, as specified in the inset. The planarly averaged potential energy of a single and infinite MoS${}_2$ layer is schematically shown by linear dashed lines in the vacuum region. The zero of energy is set at the Fermi level, which is indicated by dash-dotted lines. (c) ${\overline{V}}_{el}\left(z\right)$ of negatively charged ($Q<0$ per cell) and periodically repeating MoS${}_2$ single layers. The averaged potential energy of an infinite MoS${}_2$ single layer is shown by the linear and dashed lines in the vacuum region. (d) Variation of ${\overline{V}}_{el}\left(z\right)$ and total potential energy including electronic and exchange-correlation potential, ${\overline{V}}_{el}\left(z\right)+{\overline{V}}_{xc}\left(z\right)$, between two negatively charged MoS${}_2$ layers corresponding to $Q=-0.110$ e/unit cell before the spilling of electrons into vacuum. The spacing between MoS${}_2$ layers is $s=20$ Å. (e) Same as (d) but $Q=-0.114$ e/unit cell, where the total potential energy dips below $E_F$ and hence excess electrons start to fill the states localized in the potential well between two MoS${}_2$ layers. (f) Corresponding planarly averaged charge density $\overline{\lambda }$. Accumulation of the charge at the center of $s$ is resolved in a fine scale. Arrows indicate the extremum points of ${\overline{V}}_{el}\left(z\right)$ in the vacuum region for $Q>0$ and $Q<0$ cases.

Figure 2

(a) Energy eigenvalues of the occupied electronic states, $E_i$ and corresponding $|{\Psi }_i{\left(z\right)|}^2$ are obtained by the numerical solution of the Schrödinger equation of a planarly averaged, 1D electronic potential energy of single-layer graphene for $s=12.5$ Å, 25 Å, and 50 Å shown by dashed lines. (b) Same as (a) for three-layer graphene. Zeros of $|{\Psi }_i{\left(z\right)|}^2$ at large $z$ are aligned with the corresponding energy eigenvalues.

Figure 3

Energy-band structures of a 2D single layer of graphene C, fluorographene CF, graphane CH, BN, and MoS${}_2$ calculated for $Q=+0.2$ e/cell, $Q=0$ (neutral), and $Q=-0.10$ e/cell. The zero of energy is set at the Fermi level, which is indicated by dash-dotted lines. The band gap is shaded. Note that band gap increases under positive charging. Parabolic bands descending and touching the Fermi level for $Q<0$ are free-electron-like bands. Band calculations are carried out for $s=20$ Å.

Figure 4

(a) Variation of the ratio of lattice constants $a$ of positively charged single-layer graphene, BN, CH, CF, and MoS${}_2$ to their neutral values $a_o$ with the average surface charge density, $\overline{\sigma }$. The unit cell and the lattice vectors are described by inset. (b) The charge density contour plots in a plane passing through a C–C bond. (c) Same as B–N bond.

Figure 5

Variation of planarly averaged positive excess charge $\overline{\lambda }\left(z\right)$ along $z$ axis perpendicular to the BN layers calculated for different $s$. As $s$ increases more excess charge is transferred from the center region to the surface planes.

Figure 6

Variation of cohesive energy and perpendicular force $F_{\perp }$ in three-layer BN slab as a function of the distance $L$ between the surfaces. Energies and forces are calculated for different levels of excess positive charge $Q$ (e/unit cell). Zero of energy corresponds to the energy as $L\to \infty$.

Figure 7

Exfoliation of outermost layers from layered BN and MoS${}_2$ slabs by positive charging of slabs. (a) Turquoise isosurfaces of excess positive charge density. (b) Change in total energy with excess surface charge density. (c) Variation of $L$ of slabs with charging.