Hybrid $\mathbf{k}·\mathbf{p}$ tight-binding model for intersubband optics in atomically thin InSe films

We propose atomic films of $n$-doped $\gamma$-InSe as a platform for intersubband optics in the infrared and far-infrared range, coupled to out-of-plane polarized light. Depending on the film thickness (number of layers) and the amount of $n$-doping of the InSe film, these transitions span from $\sim 0.7$ eV for bilayer to $\sim 0.05$ eV for 15-layer InSe. We use a hybrid $\mathbf{k}·\mathbf{p}$ theory and tight-binding model, fully parametrized using density-functional theory, to predict their oscillator strengths and thermal linewidths at room temperature.

Figure 1

Intersubband energies for an allowed electric dipole transition for excitation from the lowest subband in weakly $n$-doped ($N=2$–15)-layer InSe. Transitions to the second lowest subband (marked in blue) are expected to be significantly stronger than transitions to higher subbands. The red line shows the $1|N\to 2|N$ intersubband transition energies in lightly $n$-doped films approximated by an asymptotic ($N\gg 1$) formula, $\hslash \omega \approx \frac{{\hslash }^2{\pi }^2}{2m_{Az}{a}_z^2}\frac{3}{{(N+2\nu )}^2}$, derived from Eq. (8). The lowest intersubband transition energy increases for heavily doped films (see Fig. 5 in Sec. 5).

Figure 2

$\mathrm{\Gamma }\text{−}A$ dispersion in bulk $\gamma$-InSe ($k_x=k_y=0$), from the two-band HkpTB model, Eq. (1). Zero of energy set to the conduction-band edge in a monolayer. Inset center: crystal structure of $\gamma$-InSe. The monolayer has a hexagonal structure, with point-group symmetry $D_{3h}$. The point group of the bulk crystal is $C_{3v}$, with each layer shifted with respect to the layer below such that selenium atoms in the upper layer lie above the indium atoms in the lower layer. $a_z=8.32$ Å is the experimentally known interlayer distance [19]. Inset bottom right: Brillouin zone of a conventional unit cell of InSe (three layers); bands plotted here have been unfolded.

Figure 3

Subbands of the conduction band in the $N=1-4$-layer InSe near the $\mathrm{\Gamma }$ point, from Eq. (1). 0 of energy set to the conduction-band minimum in the monolayer. Red arrows denote the strongest intersubband optical-absorption transitions, coupled to the out-of-plane electric dipole, while the dashed gray arrow for four-layer InSe indicates a much weaker transition. Arrows are labeled with the intersubband out-of-plane electric dipole moment of the transition, $d_z$ [Eq. (10)].

Figure 4

Intersubband line shapes (normalized to the $\mathrm{\Gamma }$-point transition in the bilayer) for $N$-layer InSe for excitation from lowest subband ($1|N$) to next-lowest subband ($2|N$), for a very light doping at $T=300$ K. Inset: thermal broadening ($T=300\mathrm{K}$, left-hand axis) of absorption lines at light doping due to the difference between subband effective masses (right-hand axis), Eq. (14). The red line was calculated using effective masses approximated by Eq. (9) for $N>4$.

Figure 5

Intersubband transition energies as a function of total gate-induced carrier density ($n_e$) for two- to six-layer InSe.