Electrons and holes in phosphorene

We present a symmetry analysis of electronic band structure including spin-orbit interaction close to the insulating gap edge in monolayer black phosphorus (“phosphorene”). Expressions for energy dispersion relation and spin-dependent eigenstates for electrons and holes are found via simplification of a perturbative expansion in wave vector $k$ away from the zone center using elementary group theory. Importantly, we expose the underlying symmetries giving rise to substantial anisotropy in optical absorption, charge, and spin transport properties, and reveal the mechanism responsible for valence band distortion and possible lack of a true direct gap.

Figure 1

Orthorhombic lattice of phosphorene. (a) In real space, two types of bonds between neighboring atoms are indicated by $d_1$ and $d_2$. The origin of the Cartesian coordinates (red arrows) is chosen to be at the center of a $d_2$ bond. (b) Top view, showing the two lattice vectors (blue arrows). Inset: 2D projection of the reciprocal lattice Brillouin zone, with high-symmetry points indicated.

Figure 2

Four configurations of the $p_y$ orbitals within a unit cell, together with their associated IRs, all of which are odd under the reflection operator $R_y$. They represent the four zone-center eigenstates composed solely from the pure $p_y$ atomic orbitals. From left to right, the eigenenergies of the four orbital states decrease $\left[E\left({\Gamma }_3^{+}\right)>E\left({\Gamma }_2^{-}\right)>E\left({\Gamma }_4^{+}\right)>E\left({\Gamma }_1^{-}\right)\right]$, according to the bonding or antibonding nature of the covalent bonds between neighboring atoms. Note that the ${\Gamma }_2^{-}$ configuration also represents the long-wavelength acoustic phonon mode where all atoms move in phase along the zigzag direction.

Figure 3

Same as Fig. 2, here for (a) $p_x$, (b) $p_z$, and (c) $s$ orbitals that are even under the reflection operator $R_y$. IRs in each row are listed from left to right in descending order of total covalent bond energy. Similar to ${\Gamma }_2^{-}$ in Fig. 2, here we see configurations corresponding to the remaining two long-wavelength acoustic phonon modes: ${\Gamma }_3^{-}$ in (a) corresponds to in-plane motion along the armchair direction while ${\Gamma }_4^{-}$ in (b) corresponds to out-of-plane (flexural) phonons.

Figure 4

Free electron band structure, including symmetry labels (yellow) of zone-center eigenstates formed from superpositions of plane waves centered at the reciprocal lattice points indicated (purple).

Figure 5

Schematic band structure of phosphorene near the zone center, illustrating the relevant perturbative interactions allowed by symmetry. Red double arrows indicate dominant coupling terms, and pink arrows highlight important non-negligible additional interactions. The gray region represents the forbidden gap. See Eq. (19) and related text for conditions resulting in indirect gap ($E_0,k_0\ne 0$). Not to scale.