$h$-AlN-Mg(OH)$_{2}$ vdW Bilayer Heterostructure: Tuning the excitonic characteristics

TL;DR

This study explores the structural, electronic, and optical properties of a $h$-AlN-Mg(OH)$_{2}$ van der Waals heterostructure, revealing tunable excitonic features and potential for light harvesting applications.

Contribution

It provides a comprehensive theoretical analysis of the heterostructure's properties, highlighting the direct band gap, Type-II alignment, and excitonic behavior not previously detailed.

Findings

Heterostructure prefers AB' stacking with direct band gap at Gamma.

Presence of excitonic peaks only in the hetero-bilayer.

First exciton peak is a spatially indirect exciton with electron and hole localized in different layers.

Abstract

Motivated by recent studies that reported the successful synthesis of monolayer Mg(OH)$_{2}$ [Suslu \textit{et al.}, Sci. Rep. \textbf{6}, 20525 (2016)] and hexagonal (\textit{h}-)AlN [Tsipas \textit{et al}., Appl. Phys. Lett. \textbf{103}, 251605 (2013)], we investigate structural, electronic, and optical properties of vertically stacked $h$-AlN and Mg(OH)$_{2}$, through \textit{ab initio} density-functional theory (DFT), many-body quasi-particle calculations within the GW approximation, and the Bethe-Salpeter equation (BSE). It is obtained that the bilayer heterostructure prefers the $AB^{\prime}$ stacking having direct band gap at the $\Gamma$ 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 hetero-bilayer are investigated. The hetero-bilayer possesses excitonic peaks which appear only after the construction of the hetero-bilayer. The lowest three exciton peaks are detailedly analyzed by means of band decomposed charge density and the oscillator strength. Furthermore, the wave function calculation shows that the first peak of the hetero-bilayer originates from spatially indirect exciton where the electron and hole localized at $h$-AlN and Mg(OH)$_{2}$, respectively, which is important for the light harvesting applications.

Tables

Table 1. Table 1: Calculated parameters for monolayer h ℎ h -AlN and Mg(OH) 2 are the lattice constant in the lateral direction, a 𝑎 a ; the final charges of Al, N, Mg, O, and H, ρ Al subscript 𝜌 Al \rho_{\text{Al}} , ρ N subscript 𝜌 N \rho_{\text{N}} , ρ Mg subscript 𝜌 Mg \rho_{\text{Mg}} , ρ O subscript 𝜌 O \rho_{\text{O}} , ρ H subscript 𝜌 H \rho_{\text{H}} , respectively; the work function Φ Φ \Phi ; the cohesive energy, E coh ; and the spin-orbit splitting, Δ S ​ O subscript Δ 𝑆 𝑂 \Delta_{SO} . E GGA g superscript subscript absent 𝑔 GGA {}_{g}^{\text{GGA}} and E GW 0 g superscript subscript absent 𝑔 GW 0 {}_{g}^{\text{GW${}_{0}$}} are the energy band gap values within GGA+SOC and GGA+GW 0 , respectively.

	a	${\rho }_{\text{Al}}$	${\rho }_{\text{N}}$	${\rho }_{\text{Mg}}$	${\rho }_{\text{O}}$	${\rho }_{\text{H}}$	$\Phi$	Ecoh	E${}_{}{}^{\text{GGA}}{}_g{}^{}$	${\Delta }_{SO}$	$\Gamma ⟶\Gamma$	$K⟶K$	E${}_{}{}^{\text{GW}\text{<sub><span>0</span></sub>}}{}_g{}^{}$
	(Å)	($e^{-}$)	($e^{-}$)	($e^{-}$)	($e^{-}$)	($e^{-}$)	(eV)	(eV)	(eV)	(meV)	(eV)	(eV)	(eV)
$h$-AlN	3.13	0.7	7.3	-	-	-	5.12	5.35	2.90	15	3.62	6.02	5.37 ($K-\Gamma$)
Mg(OH)2	3.13	-	-	0.3	7.4	0.4	4.20	4.38	3.23	25	3.23	8.92	6.51 ($\Gamma -\Gamma$)

Table 2. Table 2: The stacking labels are given in the first column. Calculated parameters for bilayer heterostructure of h ℎ h -AlN and Mg(OH) 2 are; the interlayer distance, d L − L subscript 𝑑 𝐿 𝐿 d_{L-L} ; the energy difference between the ground state, Δ E subscript Δ 𝐸 \Delta_{E} ; the interaction energy between the layers, E int ; and the work function Φ Φ \Phi . E GGA g superscript subscript absent 𝑔 GGA {}_{g}^{\text{GGA}} are the energy band gap values within GGA+SOC.

	$d_{L-L}$	${\Delta }_E$	Eint	$\Phi$	E${}_{}{}^{\text{GGA}}{}_g{}^{}$
	(Å)	(meV)	(meV)	(eV)	(eV)
$A{B}^{\prime }$	2.13	0	237	4.31	2.30
$AA$	2.12	6	231	4.32	2.31
$AC$	2.06	66	171	4.70	2.65
$A{C}^{\prime }$	2.10	76	161	4.69	2.61
$A{A}^{\prime }$	2.75	162	75	4.35	2.14
$AB$	2.75	164	73	4.29	2.14

Full text

I Introduction

After the discovery of grapheneNovoselov2 , interest in atomically thin materialsNovoselov1 has grown rapidly due to their extraordinary physical properties.Butler ; Chhowalla In the last decade, several 2D materials have been synthesized and theoretically predicted, such as silicene,Cahangirov ; Kara germanene,Cahangirov stanene,Guzman ; Bechstedt transition metal dichalcogenides (TMDs such as MoS2, WS2),Gordon ; Coleman ; Wang1 ; Ross ; Sahin2 ; Tongay ; Horzum ; Chen3 and III-V binary compounds (e.g. $h$-BN, $h$-AlN).Sahin3 ; Wang2 ; Kim ; Tsipas ; Bacaksiz Beside the single crystal of 2D materials, recently emerging field is their vertically stacked heterostructures.Geim Because of the van der Waals type weak interlayer interaction, the synthesis of heterostructures is not restricted with the lattice matching of the each layers. This provides a wide variety of combinations of layers which exhibit different electronic and optical properties.Britnell ; Fang ; Lee2 ; Hunt ; Hong ; Ferrari ; Tan

Stable hexagonal crystalline structure of AlN was first theoretically predicted by Sahin et al.Sahin3 and experimentally synthesized by Tsipas et al.Tsipas They found that, differing from its bulk structure which is an insulator, monolayer $h$-AlN is a semiconductor with the indirect band gap where the valance band maximum and conduction band minimum at the K and $\Gamma$ points, respectively. Almeida et al.Almeida investigated properties of defects, such as vacancies, antisites, and impurities, in $h$-AlN. It was reported that N vacancies and Si impurities lead to the breaking of the planar symmetry and cause significant changes in the electronic properties. Shi et al.Shi calculated the magnetic properties of bare and transition-metal (TM) doped AlN nanosheets by using first-principles calculations. They reported that nonmagnetic $h$-AlN can be magnetized upon a single TM atom. Moreover, the electronic structures of nanoribbon of AlN were investigated by Zheng et al.Zheng and it is predicted that zig-zag edge nanoribbons have an indirect band gap while armchair edge nanoribbons have a direct band gap, and band gaps of both zig-zag and armchair decrease monotonically with increasing ribbon width. Furthermore, Bacaksiz et al.Bacaksiz reported that the electronic band structure changes significantly when the number of layer increases and the structure having 10 or more layers exhibits direct bandgap as bulk form. More recently, Kecik et al.Kecik investigated the optical properties of mono- and few-layer $h$-AlN under strain. They reported that the absorption peaks stand outside the visible-light regime, on the other hand, the applied tensile strain gradually redshifts the optical spectra.

As a constituent of hetero-bilayer, physical properties of Mg(OH)2 were investigated previously in several studies.Utamapanya ; Ding ; Sideris It is a layered metal hydroxide with a wide band gapMurakami . As a member of the alkaline-earth-metal hydroxides (AEMHs) family, Ca(OH)2 was predicted to be stable in the bilayer and monoayer forms by Aierken et al.Aierken In addition, Torun et al. investigated the optical properties of GaS-Ca(OH)2 bilayer heterostructure by using GW+BSE and reported that in spite of the similarities of electronic structures, the different stacking types have different optical spectra.Torun2 Recently, Tsukanov et al. investigated the interaction of organic anions with layered double hydroxide nanosheets consist of Mg and Al by using molecular dynamics simulations.Tsukanov In addition, very recently, the monolayer Mg(OH)2 was synthesized and reported that in spite of the optically inactive nature of Mg(OH)2, the photoluminescence intensity of monolayer MoS2 was assisted by Mg(OH)2 and enhancedSuslu . Most recently, Yagmurcukardes et al. investigated the hetero-bilayer of the Mg(OH)2 and WS2.Yagmurcukardes They reported that the lower energy optical spectrum of the Mg(OH)2-WS2 hetero-bilayer is dominated by the excitons originates WS2 layers. These two studies indicate that the Mg(OH)2 is a candidate for tuning the optical property of other monolayer materials.

In the present work, the heterostructure of two perfectly-matching monolayers of $h$-AlN, a member of III-V binary compounds, and Mg(OH)2, a member of alkaline-earth-metal hydroxide (AEMH), are considered. The similar lattice constants of the layers provide us to use smaller supercell which is important especially for the calculating the optical properties. We found that the vertically stacked heterostructure possesses excitonic peaks which appear only after the construction of the hetero-bilayer. More significantly, the wave function calculation shows that the first peak of the imaginary part of the dielectric function for the hetero-bilayer originates from spatially indirect exciton where the electron and hole localized at $h$-AlN and Mg(OH)2, respectively.

The paper is organized as follows: In Sec. II we give details of our computational methodology. In Sec. III we present a brief overview of the structural and electronic properties of monolayer $h$-AlN and Mg(OH)2. In Sec. IV different stacking orders of bilayer heterostructure of $h$-AlN and Mg(OH)2 and also optical property of the energetically favorable stacking order are investigated in detail. Finally, we present our conclusion in Sec. V.

II Computational Methodology

Employing (DFT)-based methods, we investigated the the structural, electronic and optical properties of monolayer $h$-AlN and Mg(OH)2 and their hetero-bilayers. We used the Vienna ab-initio simulation package VASPvasp1 ; vasp2 ; vasp3 which solves the Kohn-Sham equations iteratively using a plane-wave basis set. To describe electron exchange and correlation, the Perdew-Burke-Ernzerhof (PBE) form of the generalized gradient approximation (GGA)GGA-PBE was adopted. The van der Waals (vdW) forces which is important for layered materials was taken into account by using the DFT-D2 method of Grimme.vdW1 ; vdW2 To obtain partial charge on the atoms, a Bader charge analysis was used.Bader1

Structural optimizations were performed with the following parameters. The kinetic energy cut-off of the plane-wave basis set was 500 eV in all calculations. The total energy difference between the sequential steps in the iterations was taken 10*-5* units as convergence criterion. The convergence for the Hellmann-Feynman forces per unit cell atom was taken to be 10*-4* eV/Å. Gaussian smearing of 0.05 eV was used and the pressures on the unit cell were decreased to a value less then 1.0 kB in all three directions. For the determination of accurate charge densities, Brillouin zone integration was performed using a $35\times 35\times 1$ $\Gamma$-centered mesh for the primitive unit cell. To avoid interactions between adjacent monolayer and hetero-bilayers, our calculations were performed with a large unit cell including $\sim$18 Å vacuum space. We also calculated the cohesive energy ($E_{coh}$), which was formulated as $E_{coh}=[(\sum\limits_{i}n_{i}E_{i})-E_{T}]/\sum\limits_{i}n_{i}$ where $i$ stands for the atoms which contract the structure; $n_{i}$, $E_{i}$, and $E_{T}$ are the number of $i$ atom in the unit cell, the energy of free $i$ atom, and the total energy per unit cell.

In addition, the quasi-particle (QP) energies were calculated within the GW0 approximation where the single-particle Green function part (G) was iterated and the screened Coulomb interaction part (W) was fixed. On top of GW0 approximation, the energies of two particle system of quasi-electron and quasi-hole (exciton) were calculated by solving BSEbse ; Hanke . To obtain accurate QP states, 320 bands were considered in the GW0 calculations. Our convergence tests showed that the $12\times 12\times 1$ k-point sampling and the vacuum spacing of $\sim$28 Å well approximates the excitonic properties of vdw hetero-bilayer structure (See Appendix). The 8 highest valence bands and lowest conduction bands were considered as a basis for the excitonic states.

III Single Layer $h$-AlN and Mg(OH)2

Before analysis of bilayer heterostructure of $h$-AlN and Mg(OH)2, the monolayer constituents are discussed. Firstly, the monolayer structure of $h$-AlN (see Fig. 1(a)) belongs to space group $P6_{3}/mmc$ with the lattice constant 3.13 Å which is consistent with that of few-layer hexagonal AlN that reported by Tsipas and the co-workers.Tsipas In Table 1, the number of valance electrons of the isolated single atom and of the one in crystal are given. It is seen that Al donates 2.3 of 3.0 $e^{-}$ to N which indicates that the bond between Al and N have strong ionic character. The work function is found to be 5.12 eV. As shown in Fig. 1(c), $h$-AlN has an indirect band gap of 2.9 eV where the valance band maximum (VBM) and the conduction band minimum (CBM) are at the $\Gamma$ and $K$ points, respectively. GW0 band gap is calculated to be 5.30 eV. In addition, differing from similar TMDs, the spin-orbit (SO) splitting at the $\Gamma$ point is quite small (15 meV).Mak

Structural and electronic properties of monolayer Mg(OH)2 which is another building-block of hetero-bilayer are also presented in Figs. 1(b) and (d). The lattice parameter of Mg(OH) which belongs to $P\bar{3}m1$ space group is found to be 3.13 Å. Therefore, it perfectly matches to $h$-AlN. Upon the formation of this crystalline structure In this structure: (i) Mg donates almost all valance electrons, 1.8 of 2.0 $e^{-}$, and (ii) H donates 0.6 of 1.0 $e^{-}$ to O atom. The work function is calculated to be 4.20 eV. Mg(OH)2 has direct band gap of 3.23 eV, 6.51 within GGA and GGA+GW0, respectively. The wide GW0 band gap is close to the value which is previously obtained.Suslu The SO splitting at the $\Gamma$ is 25 meV which is slightly larger than that of $h$-AlN (15 meV). This small difference can be understood from the difference between the atomic radius of N and O atoms.

In Fig. 2, the energy dependency of the imaginary part of the dielectric function of pristine $h$-AlN and Mg(OH)2 are shown. We name the peaks as I, II, and III (first, second and third, respectively) with the subscript that specify the corresponding individual layer. The IAlN appears at 4.12 eV with the exciton binding energy of 1.88 eV. The IIAlN and IIIAlN appear at 4.97 and 5.4 eV, respectively. The IIIAlN consists of 4 exciton levels. The I${}_{Mg(OH)_{2}}$ and II${}_{Mg(OH)_{2}}$ appear at 4.18 and 5.20 eV, respectively. The excitonic binding energy of the I${}_{Mg(OH)_{2}}$ is found to be 2.33 eV. The peaks of the $h$-AlN are larger than those of Mg(OH)2. The reason is that all the peaks shown in Fig. 2 originate from the excitation at the $\Gamma$, hence the overlapping of electron and hole states are larger for the $h$-AlN which is clearly seen at the charge density of band edges in Fig. 1. In addition, the excitonic binding energies are larger compared to the bulk materials. Such binding energies were previously reported and discussed.Wirtz ; Pishtshev ; Cudazzo

IV Bilayer Heterostructure

In this section, we give an analysis of structural, electronic and optical properties of bilayer heterostructure of vertically stacked monolayer $h$-AlN and Mg(OH)2.

IV.1 Determination of Stacking Type

First of all, to determine energetically favorable structure, six different stacking orders are considered. As shown in Fig. 3(a), possible stacking orders are: $AA$ (Al on Mg, N on H), $AA^{\prime}$ (N on Mg, Al on H), $AB$ (Al on H, N at the midpoint of three H), $AB^{\prime}$ (N on H, Al at the midpoint of three H), $AC$ (Al and N on top of midpoint of H triangles and N on Mg), and $AC^{\prime}$ (switched of Al and N form of $AC$). The energy difference between different stackings are given in Table 2. In general, the hetero-bilayer has three type of properties with respect to localization of Al or N on trigonal H surface of Mg(OH)2; when N atom is on top of H atom, which means the $AB^{\prime}$ and $AA$ stacking types, the hetero-bilayers posses the lowest energies with highest binding energies. They have also similar band gap of 2.30 and 2.31 eV, respectively. The work functions are found to be 4.31 and 4.32 eV which are very close to each other. Instead of these energetically favorable forms, the minimum interlayer distances are obtained when the Al and N atoms coincide to the midpoint H triangles ($AC^{\prime}$ and $AC^{\prime}$). The band gap and work function values of the $AC^{\prime}$ and $AC^{\prime}$ stacking are found to be similar as well. When Al is on top of H , the structures have the highest interlayer distances highest energy and lowest binding energy. The $AA$ with $AB^{\prime}$, $AA^{\prime}$ with $AB$, and $AC$ with $AC^{\prime}$ have similar parameters.

The minimum energy of the hetero-bilayer is obtained when the layer are stacked in the form of $AB^{\prime}$ stacking which exhibits the maximum interlayer interaction of 237 meV. Interlayer distance, which is defined as the perpendicular distance from surface H atom to the $h$-AlN plane, is 2.13 Å. The work function of Mg(OH)2 side is calculated to be 4.31 eV. The VBM and CBM are at $\Gamma$ as in monolayer Mg(OH)2, however the band gap values of 2.30 eV within GGA which are significantly low as compared to that of monolayer of $h$-AlN and Mg(OH)2. In addition, the $AA$ stacking order is energetically very close to $AB^{\prime}$. For both $AB^{\prime}$ and $AA$, the layer-layer interaction, interlayer distances, and also band gap values are almost same.

In addition, considering vacuum level positions, formation of a staggered gap, with a mismatch of 1.02 eV at the intimate contact point of monolayers is predicted. Moreover, Since the minimum energy difference between the VB and CB is at $\Gamma$ point (see Fig. 3 (b)), the lower energy optical activity takes place at this high symmetry point of Brillouin Zone. Such a contact between two atomically-flat surfaces forms an ultra-clean Type-II heterojunction.

IV.2 Origin of Excitonic States of Bilayer Heterostructure

In this part, the origin of the prominent excitonic peaks in the AlN/Mg(OH)2 hetero-bilayer structure, which corresponds to the AB*′* stacking, are investigated in details. There are two effects that determine characteristic properties of exciton in two-dimensional materials: the dielectric screening and the structural confinement. In general, when the crystal structure of a usual semiconductor is reduced from 3D to 2D, the dielectric screening between electron and hole takes place only inside layer. Therefore, as a consequence of the dimensional reduction, the screening effect on electron-hole pair is lower for 2D crystals as compared to the bulk.

In Fig. 4, we show interlayer-spacing-dependent excitonic properties to investigate the origin and evolution of the excitons by tracking their peak positions and corresponding oscillator strengths in the energy spectrum.

As shown in Fig. 4 (a), starting from the interlayer spacing of 6.0 Å (equilibrium distance is set to 0 Å), the first peak of the isolated Mg(OH)2 splits into two different peaks due to the broken symmetry of the top and bottom surface charge densities. It is seen that the first peak of the bilayer structure, Ibil, appears at the lower energy level than the first peak of both individual $h$-AlN and Mg(OH)2. As calculated in the previous section, the VBM and CBM electronic states are mainly formed by the O and N, respectively. Therefore, the first peak of the bilayer heterostructure, Ibil, corresponds to a spatially indirect exciton in which the electron localizes at the N of $h$-AlN and the hole is at O of Mg(OH)2. As shown in Fig. 4 (b), the reduction in the oscillator strength of Ibil (minimum of all) at around 3.0 Å, reveals that electronic state of Mg(OH)2 at the intimate contact surface overlaps with $h$-AlN layer as the distance decreases. However, at the optimum interlayer distance, other Mg(OH)2-originated exciton state evolves into the second peak of the heterostructure labeled as IIbil. Apparently, the electronic state of IIbil exciton peak originates from the outer surface state of Mg(OH)2.

On the other hand, the origin and interlayer-spacing-dependence of IIIbil, differs from those of Ibil and IIbil. Firstly, in Fig. 4, it is evident that by decreasing interlayer spacing the first excitonic peak of $h$-AlN crystal gradually evolves into third state of hetero-bilayer, labeled as IIIbil. It is seen from Figs. 4 (a) and (b) that the shape and the oscillator strength of the IIIbil do not change significantly. Therefore, one can conclude that the characteristic properties of the IIIbil are almost same with the first excitonic state of $h$-AlN monolayer crystal. When the distance decrease under 4.0 Å, a significant charge distribution appears on the O atom. It appears that the screening on the $h$-AlN increases when the Mg(OH)2 layers become closer. Therefore as a consequence of decrease in exciton binding energy corresponding peak shows a significant blue shift.

For a deeper understanding of nature of excitonic states of the vdW Mg(OH)2/$h$-AlN heterostructure we also investigate the band decomposed charge densities and exctionic wave functions. In Fig. 5 (a), the interlayer-spacing-dependent band decomposed charge densities of the first and second lowest levels of the CB at $\Gamma$ point are shown. The lowest and second lowest levels are labeled as CB-$\Gamma_{1}$ and CB-$\Gamma_{2}$, respectively. The changes in the charge densities of these two levels contain remarkable informations about the evolution of the electrons involved in the states Ibil, IIbil, and IIIbil.

Firstly, when the distance is 6.0 Å, the charge density shown in CB-$\Gamma_{2}$ part of the Fig. 5 (a) corresponds to the charge density at the CBM of individual Mg(OH)2. It is seen that this density appears mostly on O atoms and also at the surfaces of the Mg(OH)2. As the interlayer distance decreases, the charge density vanishes at the inner surface and increases at the outer surface. On the other hand, the density shown in the CB-$\Gamma_{1}$ for the interlayer distance of 6.0 Å corresponds to the charge density at the CBM of individual $h$-AlN. As the distance decreases, charge changes negligibly at the vicinity of $h$-AlN but an additional charge density appears on inner O atom when the distance is less than 4.0 Å. It appears that when the layers approach each other, the inner surface electron of the Mg(OH)2 is transfered to the $h$-AlN and the outer surface electron states of the Mg(OH)2 remains at its original position.

Furthermore, we also calculate the real space exciton localizations by using the methodology given in Appendix B. The wave functions of the Ibil, IIbil, and IIIbil are shown in Fig. 5 (b) where the yellow regions correspond to electrons and the black points stand for holes. As shown in the lower panel, it is evident that the Ibil is the spatially indirect exciton in which the electron and hole localize on the $h$-AlN and Mg(OH)2, respectively. The IIbil localizes at the outer side of the Mg(OH)2 as shown in middle panel of the Fig. 5 (b). It seems that the IIbil is not affected by the $h$-AlN. The IIIbil shown in upper panel of the Fig. 5 (b) stays mostly on the $h$-AlN. A small portion of the IIIbil appears on the hydroxide regions of the Mg(OH)2.

V Conclusion

In this study, structural, electronic, and optical properties of the recently synthesized novel 2D materials of $h$-AlN and Mg(OH)2 and their bilayer heterostructure are investigated by performing DFT calculation. The excitonic states are also calculated by solving BSE over the $GW_{0}$ approximation on top of DFT. Our investigation revealed that: (i) the individual monolayer of the $h$-AlN and Mg(OH)2 can form a vertically stacked hetero-bilayer, (ii) when the hetero-bilayer formed, although the interlayer interaction is weak novel exciton states appear, (iii) while $h$-AlN-states have weak spacing dependence, Mg(OH)2 states are strongly affected by the presence of a neighboring layer, (iv) exponential increase in screening of Mg(OH)2 on $h$-AlN-states was also predicted, (v) while the first exciton peak is a spatially indirect one, the second and third exciton states are spatially direct states.

Although the constituents have limited optical activity, the heterostructure shows unexpected optical properties. In particular, $h$-AlN-Mg(OH)2 hetero-bilayer exhibits spatially indirect excitons which is important for the optoelectronic application, especially based on photoexcited electron collecting.

VI Acknowledgments

The calculations were performed at TUBITAK ULAKBIM, High Performance and Grid Computing Center (TR-Grid e-Infrastructure). CB and RTS acknowledge the support from TUBITAK Project No 114F397. HS acknowledges support from Bilim Akademisi-The Science Academy, Turkey under the BAGEP program. HS acknowledges financial support from the Scientific and Technological Research Council of Turkey (TUBITAK) under the project number 116C073. AR and AD acknowledge financial support from the European Research Council(ERC-2015-AdG-694097), Spanish grant (FIS2013-46159-C3-1-P), Grupos Consolidados (IT578-13), and AFOSR Grant No. FA2386-15-1-0006 AOARD 144088, H2020-NMP-2014 project MOSTOPHOS (GA no. 646259) and COST Action MP1306 (EUSpec)

Appendix A Convergence Tests

Theoretical results on optical properties, especially many-body and excitonic effects, are strongly depend on computational parameters such as Brillouin Zone sampling and the vacuum spacing between the adjacent layers. The calculation with fine parameters, on the other hand, requires large computational resource and time. Therefore, we performed calculations to examine the convergence of the frequency dependent imaginary dielectric function with respect to k-point sampling and the vacuum spacing.

The imaginary part of the dielectric function for the monolayer h-AlN and Mg(OH)2 with respect to k-point sampling are shown in Fig. 6 (a) and (b), respectively. When the number of k-points increases, the spectrum of both monolayers are blue-shifted and rapidly converges by the 12$\times$12$\times$1 sampling (see Fig. 6 (c)). Therefore, using 12$\times$12$\times$1 for optical properties of monolayer h-AlN and Mg(OH)2, provides reliable results.

As shown in Fig. 7 (a) and (b), the effect of the vacuum spacing is also evident that the peak positions of both monolayers are red-shifted. For the h-AlN, the curves of 30 and 35 Å vacuum spacing values are almost the same. For Mg(OH)2, the curve does not change significantly when the vacuum spacing increases from 25 to 30 Å. Moreover, as shown in Fig. 7 (c), the difference between the first exciton energies of 30 and 35 Å vacuum spacing values is 11 meV for the monolayer h-AlN. The smaller energy difference of 5 meV is obtained between 25 and 30 Å vacuum spacing values for the monolayer Mg(OH)2. It is clearly seen that, the vacuum spacing of 30 and 25 Å are sufficient for the monolayers h-AlN and Mg(OH)2, respectively. Therefore, when we consider the hetero-bilayer, using the vacuum spacing of $\sim$28 Å for optical properties provides reliable results. After having convergence tests, the best parameter set of 12x12x1 kpoint and 28 Å vacuum spacing are used in our calculations.

Appendix B Methodology for Determination of Excitonic Wave Functions

For a deeper understanding of the origin of excitonic states and determination of their wave functions of the vdW hetero-bilayer, three lowest exciton wave functions were calculated by using BerkeleyGW packageDeslippe on top of QuantumEspresso code.Giannozzi The eigenvalues and the eigenfunctions in the independent particle picture were obtained by using PBE form of GGA for exchange and correlation. The quasi-particle energies were calculated within the GW approximation. We employed an energy cutoff of 160 Ry for the PBE calculations. For the computation of the dielectric matrix, we used 1987 conduction bands and G-vectors with energy up to 17 Ry, whereas the self-energy operator was computed using 1987 conduction bands and a G-vector cutoff of 17 and 160 Ry for the screened and bare Coulomb matrices, respectively. To sample the BZ we employed a $6\times 6\times 1$ k-point grid for the PBE and GW calculations, whereas for the solution of the BSE, we used a $18\times 18\times 3$ k-point grid with linearly interpolated GW quasi-particles energies. The 8 lowest conduction bands and 8 topmost valence bands were included to solve the BSE. After obtaining the excitonic states, the probability of localization of the electron is calculated for each excitonic level when the hole is fixed to a specific point. The result are given in Fig. 5 (b).