Tunable bright interlayer excitons in few-layer black phosphorus based van der Waals heterostructures

Few-layer black phosphorus (BP) is a direct band gap material with large exciton binding energies, and shows great promise in optoelectronic applications. Here, we study the excitons in BP-based heterostructures with encapsulation and spacer 2D layers, using first principles GW and Bethe–Salpeter equation (BSE) methods. The 2D layers chosen are germanium sulfide (GeS) and hexagonal boron nitride (hBN), representing respectively strong and weak hybridization with BP. Except for hBN-encapsulated BP, all systems host bright interlayer (or indirect) excitons. In contrast to 2D indirect gap heterostructures, the interlayer excitons here are much brighter. Strong hybridization between GeS and BP increases the effective mass and room temperature exciton lifetimes. In contrast, the hBN spacer layer decouples the BP monolayers in BP/hBN/BP, resulting in the lowest energy exciton being dark. Surprisingly, however, BP/hBN/BP hosts interlayer BP excitons that are even brighter than those in bilayer BP. This lowest energy bright exciton lies very close in energy to the dark state, resulting in an increased effective lifetime. Our work uncovers the interplay between interlayer interactions and the physics of interlayer excitons, and paves the way for the use of bottom-up materials design to optimize the dipole oscillator strengths and lifetimes of interlayer excitons for excitonic device applications.


Introduction
Layered two-dimensional (2D) materials possess novel excitonic properties due to reduced spatial dimensions and enhanced Coulomb interactions [1][2][3]. Depending on the application, the essential parameters important in the physics of excitons are the exciton binding energy, the spatial extent, the optical dipole oscillator strength and the radiative lifetime. Compared with semiconducting quantum wells, reduced screening in 2D leads to much larger exciton binding energies (BE) in 2D materials [4], which opens up opportunities for their observation and manipulation at room temperature. Stacking different 2D layered materials together forms socalled van der Waals heterostructures (vdWHs) [5,6], enabling the bottom-up design of material properties without the need for explicit lattice matching. Recent experimental advancements have even enabled the control of stacking twist angle to within a few degrees [7,8]. Just as in bulk quantum well structures [4], 2D vdWHs can also host interlayer (or indirect) excitons, where the electron and hole reside on different layers [9][10][11][12]. These excitons are of particular interest because they can be controlled by electric fields, and also because of their prolonged lifetime, paving the way for interesting applications such as excitonic devices, high temperature superfluidity, etc [13][14][15][16][17]. However, the larger spatial spread of the interlayer exciton also results in reduced oscillator strengths [18,19], which are undesirable for light emitting diode and excitonic device applications. This is particularly notable in indirect band gap vdWHs (e.g. multilayer transition metal dichalcogenide (TMD) systems). The bottom-up process of forming vdWHs opens up opportunities to design 2D heterostructures that host strongly bound excitons with the optimal balance between lifetime and oscillator strength.
Few layer black phosphorus (BP) is a recent member of the family of 2D materials from group V [28][29][30][31]. Its tunable band gap size of 0.3-2.0 eV falls in the range between the band gaps of graphene and TMDs [22,32,33]. Importantly, few-layer BP is a direct band gap semiconductor for both mono layers and multilayers [34], and therefore should exhibit stronger optical absorption and emission spectra compared to the indirect band gap TMD multilayers. Furthermore, the intrinsic anisotropic properties and relatively high carrier mobilities of BP are also promising for novel applications [21,[35][36][37][38]. In contrast to TMDs, graphene and many other 2D materials, the interaction between BP layers is not completely van der Waals (vdW) in nature, due to the electron lone pairs on phosphorus atoms [39,40]. Yet, the interlayer interactions are also significantly weaker than the covalent interactions in bulk quantum well structures. The interlayer interactions in BP-based heterostructures can be tuned by choice of stacking material, and present additional opportunities for the tuning of exciton properties by materials design.
Here, we use DFT+GW/BSE to predict the excitonic properties of BP-based 2D heterostructures, specifically, encapsulated BP monolayers and BP bilayers with spacer layers in between. GeS and hBN are chosen as the encapsulation/spacer layers (figure 1), representing respectively the limits of strong and weak hybridization with BP. These BP-based heterostructures all have direct band gaps. Except for hBN-encapsulated structures, all systems host bright interlayer excitons, which can be tuned by the spacer material and thickness, and by the interlayer interactions. The strong hybridization between GeS and BP contributes to changes in the exciton effective mass, which increases the room temperature lifetime of excitons in GeS/BP-based heterostructures significantly. In BP/hBN/BP, low energy dark excitons also exist, because the hBN decouples the BP layers and reorders the BP states. The lowest energy dark exciton lies close in energy to the lowest energy bright exciton (an interlayer BP exciton), resulting in an increased effective lifetime. Our results reveal the nature of excitons in BP-based heterostructures in the limit of strong and weak hybridization with the encapsulation/spacer layers, and show how exciton energies, lifetimes and oscillator strengths can be tuned by materials design in direct band gap BP-based heterostructures.

Methods
Our calculations were performed using Quantum-ESPRESSO [41] and BerkeleyGW [42] software packages. Norm conserving pseudopotentials with a kinetic energy cutoff of 55 Ry were used, together with a vacuum height of at least 12 Å between periodic 2D slabs. A dielectric matrix cutoff of 15 Ry was used in the one-shot G 0 W 0 calculations. Monkhorst-Pack k-mesh densities of at least 14 × 10 × 1 and 56 × 40 × 1 for a 1 × 1 BP primitive cell were used in the GW and BSE calculations, respectively. Further calculation details are available in the supplementary information (SI) (stacks.iop.org/TDM/5/045031/mmedia).
Following [43], we define the dimensionless optical oscillator strength per phosphorus atom as: where v is the velocity operator, Ω S is the exciton frequency. BP has anisotropic optical properties, with the lowest lying bright excitons arising from light polarized in the armchair direction [22]. We therefore focus on the excitonic properties arising from light polarized in the armchair direction. Using Fermi's Golden Rule [44,45], we derive expressions for the radiative lifetimes of such excitons to be (2) Here, ∆ S = E 2 S (0)/2M S c 2 is the maximum kinetic energy, where M S = m * e + m * h is the exciton mass (the electron and hole effective masses were fitted from DFT-PBE bands near Γ along the armchair direction). Note that the second term in (2) is much smaller than the first term, so that the lifetime has a ∼ √ T dependence on temperature, similar to that in one-dimensional materials [44]. The lifetime at 0 K, τ S (0) , is defined as where A uc is the area of the unit cell, and is the square of the dipole matrix element. The expression for τ S (0) is similar to that in a 2D isotropic material, but larger by a factor of 2 [45]. Details of the derivation are presented in the SI.
We define an effective radiative lifetime by averaging the decay rates over the lowest energy bright and dark excitons:

Monolayer BP and GeS
Monolayer (ML) BP has a puckered orthorhombic geometry with armchair and zigzag directions as shown in figure 1(a). Our GW band gap and exciton binding energy (table 1) are in good agreement with previous theoretical and experimental results [22,46,47]. We also compute a large oscillator strength of 0.068 and an intrinsic radiative lifetime of 2.0 ps at room temperature (RT; 300 K).
Few-layer GeS is a readily available material that does not degrade in ambient conditions [48]. GeS is a group IV monochalcogenide layer with a similar atomic structure to BP (figure 1) and a relatively small lattice mismatch with BP. Like BP, it has anisotropic optical properties, with low energy optical excitations along the armchair direction (figure S2). Compared to ML BP, the exciton oscillator strength is weaker (0.039) and the RT lifetime is longer (4.5 ps). This longer RT lifetime is partly due to a larger effective mass (table 1).

Bilayer BP/GeS and BP/hBN
For the relaxed AB stacked BP/GeS bilayer structure, the strain on BP (GeS) is −2.4% (3.8%) along armchair and 3.9% (−7.2%) along zigzag directions. The puckered structure of GeS implies that the interaction between GeS and BP will resemble that between BP layers themselves. Indeed, the DFT projected density of states (PDOS) in figure 2(a) shows significant hybridization between BP and GeS in the BP/GeS bilayer (BL). We note that while it is not clear whether the −7.2% strain for GeS in the zigzag direction can be achieved in experiment, this does not change the conclusions in the manuscript significantly, because we are focusing on the low energy excitations along the armchair direction in BP.
In contrast to GeS, ML hBN has an optical band gap larger than 5.5 eV and does not hybridize significantly with BP ( figure 2(b)). We note that for the bilayer BP/hBN supercell, we have interfaced a 1 × 3 supercell of BP with a 1 × 4 supercell of an orthorhombic hBN primitive cell, similar to [49,50]. We note that BP can sustain a relatively large strain along the armchair In addition to these structures, we also consider BP/hBN/BP systems where BP is AB stacked, as well as BP/3L-GeS/BP and GeS/BP/GeS systems with AB-stacking between layers. Here orange, purple, yellow, grey, light blue balls denote P, Ge, S, B, N atoms, respectively. direction at low energy cost [51], and the relaxed structure has −6.2% (+0.3%) strain on BP along the armchair (zigzag) direction. The optical absorption spectra for BL BP/GeS and BL BP/hBN are anisotropic (figure S4), and the spectra for light polarized in the armchair direction of BP are shown in figures 2(c) and (d). The lowest energy bright excitons are both 1s excitons, but the exciton in BP/GeS is a mixed interlayer exciton [27], whereas that in BP/ hBN is an intralayer one localized on BP (figures 2(e) and (f)). The contrasting nature of the exciton wavefunctions is consistent with the significant hybridization between BP and GeS, and the clear Type I alignment in BP/hBN.

Encapsulated monolayer BP
The optical absorption spectra for ML BP encapsulated with GeS and hBN MLs are shown in figures 3(a) and (b), respectively, for light polarized in the armchair direction (zigzag direction: figure S5). These systems all have direct band gaps (figure S6), with GW quasiparticle (QP) gaps that are smaller than those in ML BP, because of increased screening from the encapsulation layers, as well as wavefunction hybridization in the case of GeS/BP/GeS. (We note that the GW gap in hBN/BP/hBN is larger than that in BL BP/hBN (table 1), because of the interaction between p z orbitals in BP and hBN [50]). The lowest energy absorption peaks both correspond to 1s excitons, but that in GeS/BP/GeS is a mixed interlayer exciton (figure 3(a) inset) while that in hBN/BP/hBN is an intralayer exciton on BP. Figure 3(c) compares the exciton BE in different systems (showing excitons with oscillator strengths that are at least 5% of the maximum oscillator strength in the respective systems).
We first focus on comparing the BE for excitons in trilayer (TL) hBN/BP/hBN, BL BP/hBN and in ML BP. Although no interlayer excitons are present here, these systems are relevant because of the use of hBN for encapsulation of BP [52][53][54][55]. hBN does not change the exciton spectrum significantly. However, similar to the QP gap, the BE are shifted to smaller values due to elec- tronic screening from the hBN layers (the exciton BE is a measure of the electron-hole interaction strength, and it depends crucially on quantum confinement and environmental screening effects [25]). As the exciton energy is given by the difference between the QP gap and the exciton BE, there is negligible change in the exciton energy (table 1). Interestingly, we see that just one layer of hBN is sufficient to reduce the BE of the 1s exciton by 19%, whereas adding another hBN ML on the other side of BP has relatively little additional effect on the exciton BE. These results point toward non-trivial screening from the atomically thin 2D hBN monolayer, despite its large QP gap. The RT intrinsic radiative lifetime of the 1s exciton increases slightly from ~2 ps in ML BP to ~3 ps in BL BP/hBN and TL hBN/BP/hBN (table 1).
In contrast to hBN-encapsulated BP, there are significant changes to the exciton levels in GeS/BP systems in figure 3(c), compared to ML BP. There are a larger number of bound excitons with oscillator strengths within the range we have considered here. The exciton BE of the lowest lying excitons are sig- nificantly smaller than those in BP. Besides increased screening from GeS, the smaller BE arise from hybridization between GeS and BP, giving a larger spatial spread of the interlayer excitons, which reduces the effective electron-hole interaction. For GeS/BP/GeS, the levels LX 1 and LX 2 in figure 3(c) have weaker oscillator strengths and result in a low energy shoulder in the optical absorption spectra ( figure 3(a)). The binding energy of the brightest exciton (figure 3(a) inset) is 0.20 eV, ~4 times smaller than that in ML BP, but still an order of magnitude larger than those in bulk semiconductor quant um wells [4]. Interestingly, compared to the 1s exciton in ML BP, the RT lifetime of this bright interlayer 1s exciton is 28.92 ps, almost 15 times longer (table 2), while its oscillator strength is only 5 times smaller. The enhanced lifetime at RT arises from the larger effective mass of the exciton (table 2; see equation (2)), which in turn results from a combination of lattice strain and hybridization with GeS, the latter having larger effective masses. Our results suggest that GeS can function as an electronically active encapsulation layer and also point to the possibility of using hybridization and lattice strain to increase the exciton effective mass, and thereby increase the RT exciton lifetime without compromising the oscillator strength.

Bilayer BP with GeS and hBN spacers
Experimental advancements in the synthesis of 2D layered heterostructures, enabling control over both the stacking sequence and stacking angle [7,8], imply that bottom-up design of 2D layered heterostructures can be used to tailor the properties of interlayer excitons. In particular, it is interesting to ask how a spacer of 2D layers between two BP MLs would change the excitonic properties. Would these systems host interlayer excitons involving the BP layers? How would the oscillator strength, lifetime and BE of these excitons depend on the thickness and nature of the interlayers? We present here the optical properties of ABA-stacked TL BP/GeS/BP ( figure 1(d)), ABABA-stacked BP/3L-GeS/BP (figure 4(e)), and TL BP/hBN/BP (figure 1(e)) with AA stacking between the BP layers. TL BP/hBN/ BP with AB stacking between the BP layers have similar results (see SI). All these systems have direct band gaps (figure S8).
Compared to BP MLs, the low energy excitons in BL BPs all have strong interlayer character and the exciton BE are reduced to ~0.5 eV (tables 2 and S1, also see figure S7). AA-and AB-stacked BP bilayers have similar optical properties (tables 2 and S1; figure S7). First, we consider the effect of GeS spacers. There is significant hybridization between GeS and BP wavefunctions, but the VBM and CBM wavefunctions of TL BP/GeS/BP are predominantly of BP character ( figure 4(a)). Compared to AA-BL BP, the direct band gap in BP/GeS/BP is larger (1.54 eV versus 1.27 eV), and the exciton BE is slightly reduced from 0.54 eV in AA-BL BP to 0.48 eV in BP/GeS/BP (figure 4(g), table 2). More significant changes are observed in the oscillator strengths and lifetimes. The oscillator strength of the lowest lying 1s exciton is reduced by ~50%, while the zero temperature lifetime is doubled. Insertion of GeS increases the exciton effective mass, which leads to a larger increase in the RT lifetime of the exciton, from 3.1 ps in AA-BL BP to 8.4 ps in BP/ GeS/BP. The interlayer nature of this 1s exciton is clearly seen in figure 4(d).
Even when the GeS thickness is increased to three layers, the BP/3L-GeS/BP system still hosts bound interlayer excitons (figure 4(e)) involving the two BP layers, with a large exciton BE of 0.37 eV and a reasonably large oscillator strength (only three times smaller than the 1s exciton in AB-BL BP, and four times smaller than that in AA-BL BP). The RT lifetime of this 1s exciton is 15.93 ps, five times that in AA-BL BP. We note that taking into account the darker exciton close in energy to this 1s exciton (LX; figure 4(b)), the effective exciton lifetime is significantly longer-65.6 ps. These results indicate that GeS spacer layers allow one to tune the properties of interlayer BP excitons, enhancing their lifetimes while still maintaining significant oscillator strength and exciton BEs.
BP/hBN/BP systems also host bright interlayer excitons (figure 4(f)). Similar interlayer excitons were also observed in TMD heterostructions with hBN interlayers [13]. The hBN layer decouples the two BP layers. As a result, the GW band gap is larger than that of bilayer BP, and the VBM and VBM-1 bands are reordered, changing the low energy exciton spectrum considerably (figures 4(c) and (g)). The reordering of bands makes the optical transition between VBM and CBM optically forbidden (DX), while the trans itions from VBM-1 to CBM (IX 1 ) and VBM to CBM+1 (IX 2 ) become bright (figures 4(c) and (g)), in contrast to bilayer BP (see figure S9 for the Γ-point wavefunctions) [25]. DX, IX 1 and IX 2 all have 1s character. From the exciton wavefunction distribution (figure 4(f)), we see that IX 1 has a larger intralayer character, with 65% of the electron wavefunction residing in the same layer as the hole, while IX 2 has a larger interlayer character.
Because of this distinction, we expect that the relative energies of IX 1 and IX 2 can be tuned by an external vertical electric field [27]. Surprisingly, the oscillator strength of IX 1 is in fact slightly larger than that of the 1s exciton in AA-BL BP (table 2). With the larger oscillator strength and similar exciton effective mass, the RT lifetime of the IX 1 exciton is slightly shorter than those of the 1s excitons in BL BP. However, the low energy dark exciton DX serves to increase the effective RT lifetime to 17.3 ps.

Conclusion
There is an obvious trade-off between strong optical oscillator strength of interlayer excitons and their prolonged lifetime. While in type-II TMD heterostructures, interlayer exciton lifetime up to nanoseconds at low temperature have been observed [9], their oscillator strength is rather low [18,19].
Here, we show that bright interlayer excitons exist in BP-based heterostructures with a reasonable balance between the oscillator strength and lifetimes. Compared to bulk quantum well systems, the BE of these excitons are much larger due to reduced screening and quantum confinement. We further show that non-trivial interlayer interactions between BP and other 2D materials can be used to tune excitonic properties from the bottom up-such tuning is not possible in traditional TMD materials with only vdW interactions. Specifically, for BP interfaced with monochalcogenides, hybridization and strain increase the RT lifetime significantly (BP/GeS systems; tables 1 and 2). On the other hand, the weak coupling between BP and hBN result in additional low energy excitons with very small oscillator strengths in BP/hBN/BP, enhancing the effective lifetimes (table 2). These tunable interlayer interactions, as well as layer material and thickness, can be used to optimize the exciton properties in BP-based heterostructures for various optoelectronic and excitonic device applications, motivating bottom-up experimental efforts toward this direction.