$h\text{-AlN-}\text{Mg}{\left(\text{OH}\right)}_2$ van der Waals bilayer heterostructure: Tuning the excitonic characteristics

Motivated by recent studies that reported the successful synthesis of monolayer $\mathrm{Mg}{\left(\mathrm{OH}\right)}_2$ [Suslu et al., Sci. Rep. 6, 20525 (2016)] and hexagonal $\left(h\text{−}\right)\text{AlN}$ [Tsipas et al., Appl. Phys. Lett. 103, 251605 (2013)], we investigate structural, electronic, and optical properties of vertically stacked $h$-AlN and $\mathrm{Mg}{\left(\mathrm{OH}\right)}_2$, through ab initio density-functional theory (DFT), many-body quasiparticle calculations within the GW approximation and the Bethe-Salpeter equation (BSE). It is obtained that the bilayer heterostructure prefers the $A{B}^{\prime }$ stacking having direct band gap at the $\mathrm{\Gamma }$ with Type-II band alignment in which the valance band maximum and conduction band minimum originate from different layer. Regarding the optical properties, the imaginary part of the dielectric function of the individual layers and heterobilayer are investigated. The heterobilayer possesses excitonic peaks, which appear only after the construction of the heterobilayer. The lowest three exciton peaks are analyzed in detail by means of band decomposed charge density and the oscillator strength. Furthermore, the wave function calculation shows that the first peak of the heterobilayer originates from spatially indirect exciton where the electron and hole localized at $h$-AlN and $\mathrm{Mg}{\left(\mathrm{OH}\right)}_2$, respectively, which is important for the light harvesting applications.

Figure 1

Upper panel illustrates the structure of monolayer $h$-AlN (left) and $\mathrm{Mg}{\left(\mathrm{OH}\right)}_2$ (right). (a) and (b) are the structures (top and side view), and the solid black parallelograms show the unit cell of the structure; (c) and (d) are the band structures of $h$-AlN and $\mathrm{Mg}{\left(\mathrm{OH}\right)}_2$, respectively. The blue and light-red curves with square are for GGA+SOC and GGA+${\mathrm{GW}}_0$, respectively. The dashed vertical line is shows the Fermi energy. In the band diagram, the charge densities of the VBM and CBM and the spin-orbit splitting at the $\mathrm{\Gamma }$ point.

Figure 2

The imaginary part of the dielectric function of $h$-AlN (blue curve) and $\mathrm{Mg}{\left(\mathrm{OH}\right)}_2$ (red curve) with oscillator strength of excitonic states (vertical green lines). The first, second, and third excitons are labeled as I, II, III with the subscript of the corresponding individual layer, respectively. The quasiparticle band gap of $h$-AlN (blue dashed) and $\mathrm{Mg}{\left(\mathrm{OH}\right)}_2$ (red dashed) at $\mathrm{\Gamma }$ and exciton binding energies $\left({\mathrm{E}}_{\text{bind}}^{\text{exc}.}\right)$ are given.

Figure 3

(a) Top view of six different stacking orders, (b) Electronic band dispersion (left panel) and the partial density of state (right panel) of $A{B}^{\prime }$ stacking order which is the most favorable stacking order. The $\text{CB-}{\mathrm{\Gamma }}_1$ and $\text{CB-}{\mathrm{\Gamma }}_2$ label the first and second lowest levels of the CB at $\mathrm{\Gamma }$ point, respectively.

Figure 4

(a) The imaginary part of the dielectric functions of the $A{B}^{\prime }$ stacked heterobilayer are shown for different interlayer distances starting from ground state distance of 2.13 Å. $\mathrm{\Delta }d$ refers to interlayer distance while the ground state distance is set to 0 and labeled as 0.0 Å. The lowest three exciton peaks are labeled as ${\mathrm{I}}_{\text{bil}}$, ${\mathrm{II}}_{\text{bil}}$, and ${\mathrm{III}}_{\text{bil}}$. (b) The distance dependent oscillator strengths of lowest three excitons are shown. Color code is given. The position of the prominent peaks are illustrated by shaded curves. The length of the out-of-plane lattice vector is fixed for different interlayer distance.

Figure 5

(a) Layer-layer distance dependent band decomposed charge densities at the $\mathrm{\Gamma }$ point for the lowest two bands $(\text{CB-}{\mathrm{\Gamma }}_1$ and $\text{CB-}{\mathrm{\Gamma }}_2)$ are presented. The numbers on the upper part of the figure are distances between the layers in the unit of Å. The optimized layer-layer distance is set to 0.0 Å. (b) The wave function of the lowest three excitons, which are labeled as ${\mathrm{I}}_{\text{bil}}$, ${\mathrm{II}}_{\text{bil}}$, and ${\mathrm{III}}_{\text{bil}}$. The yellow regions are the probability of the electron localization when the hole is at a specific point. The position of holes are shown as a black point on the structures for all wave functions. The methodology for the determination of excitonic wave functions is given in Appendix pp2.

Figure 6

The results of the $k$-point sampling tests: (a, b) the imaginary part of the dielectric functions of the monolayer $h$-AlN and $\mathrm{Mg}{\left(\mathrm{OH}\right)}_2$ for different k-point samplings, respectively; (c) the first exciton peak positions are shown as the blue and red squares for the monolayer $h$-AlN and $\mathrm{Mg}{\left(\mathrm{OH}\right)}_2$, respectively.

Figure 7

The results of vacuum spacing tests: (a, b) the imaginary part of the dielectric functions of the monolayer $h$-AlN and $\mathrm{Mg}{\left(\mathrm{OH}\right)}_2$ for different vacuum spacing values, respectively; (c) the first exciton peak positions are shown. The blue and red curves correspond to the monolayer $h$-AlN and $\mathrm{Mg}{\left(\mathrm{OH}\right)}_2$, respectively.