Stark shift of excitons and trions in two-dimensional materials

The effect of an external in-plane electric field on neutral and charged exciton states in two-dimensional (2D) materials is theoretically investigated. These states are argued to be strongly bound, so that electron-hole dissociation is not observed up to high electric field intensities. Trions in the anisotropic case of monolayer phosphorene are demonstrated to be especially robust under electric fields, so that fields as high as 100 kV/cm yield no significant effect on the trion binding energy or probability density distribution. Polarizabilities of excitons are obtained from the parabolicity of numerically calculated Stark shifts. For trions, a fourth order Stark shift is observed, which enables the experimental verification of hyperpolarizability in 2D materials, as observed in the highly excited states of the Rydberg series of atoms and ions.

Figure 1

(a) Sketch of the top view of the TMDC and BP crystal lattices, along with the direction of $F_x$ and $F_y$ components of the applied field. Stark shift of exciton binding energies in (b) ${\mathrm{MoS}}_2$, in the suspended, on-substrate, and encapsulated cases, and in (c) and (d) suspended phosphorene with different numbers of layers. In the latter, results are shown for fields applied in the $y$ and $x$ direction, respectively (c) and (d). The values of polarizability (fitting parameters) are presented in Table 1.

Figure 2

(a) Numerically obtained (symbols) field dependence of the trion binding energy $E_b$ in monolayer ${\mathrm{MoS}}_2$ in the suspended case, on a BN substrate, and encapsulated by BN. The values of polarizability and hyperpolarizability (fitting parameters for the curves) are presented in Table 3. (b) Contour map of the square modulus of the trion wave function in suspended ${\mathrm{MoS}}_2$. The left (right) column represents the exciton's (trion's) center of mass wave function in the absence of electric field in the first row, and for a $F=60$ kV/cm field applied in the $x$ direction in the second row. The color scale goes from 0 (blue, MIN) to 0.012 (red, MAX).

Figure 3

Comparison between fitting curves of second and fourth order corrections (lines) to the numerically obtained Stark shift of trions (symbols) in suspended ${\mathrm{MoS}}_2$. For the former, two cases are considered, namely with and without a first order term in the fitting expression (see text).

Figure 4

(a) Numerically obtained (symbols) electric field dependence of the binding energy $E_b$ for trions in phosphorene with a different number of layers for a field applied in the $y$ (left) and $x$ (right) directions. The values of polarizability and hyperpolarizability (fitting parameters for the curves) are presented in Table 3. (b) Contour maps of the square modulus of the trion wave function in suspended monolayer BP. The left (right) column represents the exciton (trion) center-of-mass wave function in the absence of electric field in the first row, for a $F=60$ kV/cm field applied in the $x$ direction in the second row, and $F=100$ kV/cm applied in the $y$ direction in the third row. The color scale ranges from 0 (blue, MIN) to 0.018 (red, MAX).