Strain-dependent electronic and optical properties of boron-phosphide and germanium-carbide hetero-bilayer: A first-principles study Open Access

A new era in the field of material science engineering has been opened up owing to the increasing number of explorations of novel two-dimensional (2D) materials having remarkable optoelectronic properties. Developments in the field of graphene have attracted the study of further graphene-like 2D materials and their hetero-bilayers (HBLs) due to having incomparable physical properties.¹ During the last few years, the interest on hetero-bilayers, especially van der Waals hetero-bilayers (vdW HBLs), has been remarkable as a consequence of their modulating nature in electronic properties, efficient carrier separation, and type-II band alignment.^2–6 In these hetero-bilayers, the bonding between monolayers (MLs) exists due to the vdW interaction. Besides, the structures show electronic and optical properties which are different from their bulk materials. It also indicates that the tunability in the bandgap due to stacking patterns, interlayer distances, cross-plane electric field, and biaxial self-induced strain had arisen from the lattice mismatch.^7–14 These unique features provoke theoretical and experimental realization of novel 2D layered materials for the ensuing nano-scaled optoelectronic devices. Recently, graphene-like boron phosphide (BP) having promising physical properties shows a lot of attention.^15–19 It has direct bandgap semiconducting properties with modulating features with the variation of stacking, interlayer distances, and external electric fields.^20,21 There are many experimental reports on the growth of BP.^22–24 Woo et al. reported that the optical bandgap of undoped and Sn-doped BP is ∼2.2 eV and 1.74 eV, respectively.²² Padavala et al. suggested that for growing high-quality BP epitaxial films, AlN can be a perfect substrate.²⁴ The n-type BP films show a high-electron mobility of 37.8 cm²/V s and the low-carrier concentration of 3.15 × 10¹⁸ cm⁻³. Mogulkoc et al. theoretically reported the stacking-dependent modulation in the bandgap for BP/blue phosphorene.¹⁰ Additionally, the modulation of optical and electronic properties was observed owing to the stacking patterns and cross-plane electric field in the hetero-bilayer material. Mohanta and De Sarkar suggested effective carrier separation in BP stacked BP/MoS2 hetero-bilayers for potential applications in piezo-electronic applications.¹¹ Another promising material is germanium carbide (GeC), which has a wide electronic bandgap of 2.09 eV (3.45 eV) calculated by density functional theory (DFT) (G₀W₀).^20,25 This group IV based 2D material has profound physical properties making it a suitable substrate material in various hetero-bilayer structures. This monolayer has tunable electronic properties due to stacking patterns as well. Some recent studies are available regarding the photocatalytic and photoelectronic applications of this material. A recent study showed that the GeC/GaS hetero-bilayer has type-II band alignment with enhanced carrier mobility.²⁶ 

Now-a-days, GeC considered on other 2D layered materials, such as GeC/MoS₂,²⁷ GeC/MSSe (M = Mo, W), GeC/ZnO,^28,29 blue phosphorus/GeC,³⁰ and GeC/WS₂,³¹ has also been studied due to their amazing opto-electronic properties. Specifically, promising type-II band edge positions and high-spin smearing in the GeC/MSSe (M = W, Mo) hetero-bilayers are shown by Din et al.,²⁸ which turn out to be potential materials for spintronics and future photovoltaic devices. Furthermore, the GeC/ZnO vdW HBL depicts an outstanding absorption coefficient in the visible spectrum, predicting potential candidates for photocatalytic energy conversion, which was investigated by Gao et al.²⁹ However, still, there are many unsolved issues in these areas starting from new material exploration to experimental implementation. As far as we know, there is still no work that is instigated on vertically stacked BP and GeC monolayers and their intrinsic electronic and optical properties, as well as the modulation of the properties due to stacking and biaxial compressive and tensile strains. Therefore, detailed understanding and proper inclusion are immensely important to unleash the potential photocatalytic and photovoltaic application of the layered materials.

In this study, we have unveiled the underlying structural, electronic, and optical properties of the BP/GeC hetero-bilayer for various stacking arrangements utilizing the density functional theory (DFT) framework. The effects of homogeneous biaxial strains, both tensile and compressive, on the basic properties of materials such as electronic properties, electron effective mass, and optical absorption are explored. Besides, the electron density difference, electrostatic potential difference, and work functions along with intrinsic electronic properties have been calculated for gaining more insight into carrier dynamics. The optical properties of the proposed hetero-bilayer systems are also studied for the fabrication of the next-generation optoelectronic devices.

The DFT based plane-wave pseudopotential methods are carried out for the simulations of the BP/GeC hetero-bilayer employing the Cambridge Serial Total Energy Package (CASTEP).³² The generalized gradient approximation (GGA), which has been introduced by Perdew, Burke, and Ernzerhof (PBE), is applied for the exchange-correlation interactions of electrons.³³ Usually, the traditional PBE functional always undervalues the bandgap for semiconductors. To the obtain appropriate bandgap, the Heyd–Scusena–Ernzerhof (HSE 06) hybrid functional is also applied in the calculation of electronic structures.³⁴ The ultra-soft pseudopotentials (USPs) are incorporated for the detailed understanding of electron–ion interactions.³⁵ The semi-empirical Grimme dispersion corrected DFT (DFT-D) is employed to consider for the van der Waals (vdW) interactions between vertically stacked layers.³⁶ 

The vdW hetero-bilayer models are built to simulate the electronic and optical properties from 2 × 2 supercells of BP and GeC monolayers. A vacuum slab model having 20 Å spacing along the z-direction is considered to avoid the interactions between proximal structural entities. All geometry relaxations are done using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm until the values of energy convergence, maximum force, maximum stress, and maximum displacement reach 5 × 10⁻⁶ eV/atom, 0.01 eV/Å, 0.02 GPa, and 5 × 10⁻⁶ Å, respectively. The kinetic energy cutoff is 310 eV. The Monkhorst–Pack k-point mesh of 6 × 6 × 1, 9 × 9 × 1, and 12 × 12 × 1 is considered for the calculation of electronic dispersion, projected density of states (PDOS), and optical properties, respectively. The binding energies, depicting the stability of structures, are calculated by the following equation:⁵ 

where E_{heterobilayer} depicts the total energy of the hetero-bilayer, while E_i(layer_i) refers to the energy of i-th freestanding monolayers comprising the hetero-bilayer. Meanwhile, the difference of hetero-interface charge densities is calculated by the following equation [Eq. (2)]:.

where ρ_{heterobilayer} is the charge density of the respective hetero-bilayer systems and ρ_i(layer) is the charge density of constituent free-standing monolayers. To demonstrate the dynamical stability of the proposed hetero-bilayer models, phonon spectra are computed applying density functional perturbation theory (DFPT) with 2 × 2 supercells.

The biaxial homogeneous strains we have adopted for altering the electronic properties are given by the following equation [Eq. (3)]:

where a_strained and a_unstrained are the corresponding lattice constants for biaxial strained structural models and unstrained preliminary structural models in our calculation. Relevant compressive and tensile strains are denoted by −e and +e, respectively.

The response due to electromagnetic illumination in a semiconductor is referred to by the complex dielectric function as

In Eq. (4), the ε₂(ω) portion of the dielectric function, directly correlated with electronic inter-band transitions energies of electronic dispersion, is derived from the momentum of filled and unfilled states by the following equation [Eq. (5)]:

where e is defined as the electronic charge and $ψkc$ and $ψkv$ are referred to as the conduction band (CB) and valence band (VB) wave-function at wavevector, k, respectively.

For seeking the dynamically stable hetero-bilayer derived from BP and GeC monolayers, the structural parameters, along with phonon dispersions, are studied here. The crystal structures of BP and GeC monolayers are hexagonal (the P3M1 space group for both). The geometry relaxed free-standing monolayers are shown in Figs. 1(a) and 1(b). It is found that both monolayers have planner non-buckling honeycomb structures. The structural parameters after geometry relaxation are lattice constant a = 3.212 Å with a bond length of 1.855 Å for the BP monolayer and lattice constant a = 3.268 Å with a bond length of 1.887 Å for the GeC monolayer. These structural parameters are in good harmony with the reported values.^37–39 The 2 × 2 supercells of the GeC (a = b = 6.536 Å) monolayer are taken as the substrate layer in our hetero-bilayer (HBL) systems. The 2 × 2 BP supercells (a = b = 6.424 Å) are vertically stacked on the substrate layer in six different ways. Owing to similar lattice constants, the lattice mismatch found between monolayers is 1.71%, which is quite smaller compared with other vdW hetero-bilayer systems.^40,41 For the vertically stacked hetero-bilayer system, six models are proposed: HBL 1, boron atoms (B) are right on top of germanium atoms (Ge) and phosphorous atoms (P) are in hollow; HBL 2, P atoms are exactly on top of Ge atoms and B atoms are in hollow; HBL 3, boron atoms (B) are directly on top of carbon atoms (C) and phosphorous atoms (P) are in hollow; HBL 4, B and P atoms are directly on top of Ge and C atoms, respectively; HBL 5, P atoms are directly on top of C atoms and B atoms are in hollow; and finally, in HBL 6, P and B atoms are directly on top of Ge and C atoms. For finding the dynamical stability of the hetero-bilayer models as described, DFPT based phonon spectra are considered. The HBLs 4, 5, and 6 are found to be dynamically unstable due to the negative frequencies in phonon dispersions [shown in Figs. 1(a)–1(h) of the supplementary material], which results in the exclusion of the structural models from further studies.

After the geometry optimization of dynamically stable HBL models, the optimized lattice constants are found to be 3.2321 Å, 3.2328 Å, and 3.2337 Å for HBL 1, HBL 2, and HBL 3, respectively. The values are in between the lattice parameters of the GeC and BP monolayer, which is as desired. The HBLs are planar-zero-buckling, where HBL 1 has the lowest interlayer distance of ∼3.442 Å, while the highest interlayer distance of ∼3.573 Å has been found for HBL 3.

To intuitively find out the energetically favorable and comparatively stable model, the binding energy of dynamically stable hetero-bilayer models is calculated with varying interlayer spacing using the following equation [Eq. (6)], a modified form of Eq. (1) mentioned in Sec. II:

where E_{BP/GeC heterobilayer} denotes the total energy of the hetero-bilayer structure, E_BP denotes the total energy of the BP monolayer, and E_GeC denotes the total energy of the GeC monolayer. The binding energy curves for HBL 1, HBL 2, and HBL 3 are demonstrated in Figs. 2(a)–2(c) of the supplementary material, respectively. The calculated minimum binding energy, respective interlayer distances, modified lattice constants, and bond lengths are summarized in Table I. The significance of binding energy is that the lower the energy, the more energetically favorable the hetero-bilayer model is. Compared with the binding energy of three proposed models, HBL 1 is found to be the most energetically favorable, indicating its experimental feasibility. Meanwhile, HBL 3 has the highest binding energy, which is consistent with the highest interlayer distance. These structural differences appear due to the stacking pattern resulting from a different apparent position of four constituting atoms of the HBL unit cells. In this case, the smallest interlayer distance is 3.442 Å in HBL 1, which is higher than the accumulated covalent radii of Ge and B atoms, i.e., 2.04 Å, suggesting the absence of covalent bonding between layers. While the interlayer distance value is within the sum of the vdW radii of Ge and B atoms, i.e., 4.03 Å, it means that the vdW interaction increases the binding energy.⁴² 

Initially, a similar band structure using PBE and HSE 06 non-local hybrid functionals of BP and GeC monolayers is calculated. The bandgap values calculated by the PBE (HSE 06) functional of GeC and BP monolayers are found to be 0.82 eV (1.4 eV) and 2.04 eV (2.45 eV), respectively [as shown in Figs. 3(a) and 3(b) of the supplementary material]. The bandgap values show a good consonance with other theoretical studies.^23,39 Then, the HSE 06 band structure values of HBL 1, HBL 2, and HBL 3 are depicted in Figs. 2(a)–2(c), respectively. It reveals that the bandgap value is favorably sensitive to the stacking patterns of the hetero-bilayer models. As demonstrated in Fig. 2(a), HBL 1 has an indirect bandgap electronic property. At the same time, the other two dynamically stable hetero-bilayer models are found to have direct bandgap semiconducting electronic properties, highly required for photovoltaic applications. The bandgap values of all the three hetero-bilayer models in this study are enlisted in Table I.

Relative band edge position calculation is a significant tool to predict photocatalytic activities. To reveal the band alignment, the HSE 06 bandgap values are used involving the equation $ECB=X−Ee−(EgHSE062)$ for the conduction band edge and $EVB=X−Ee+(EgHSE062)$ for the valance band edge. Here, X represents the geometric mean of Mulliken electronegativities of the comprising atoms of the corresponding monolayers and hetero-bilayers and E_e represents the energy (hydrogen scale) of the free electrons (4.5 eV). In our calculation, the Mulliken electronegativities for BP and GeC monolayers are 4.89 eV and 5.45 eV, respectively. Figure 1(d) demonstrates the relative band edge positions of GeC, BP, and our proposed hetero-bilayers. It shows that the valence band maxima are contributed by the BP monolayer, while conduction band minima (CBM) are dominated by the GeC monolayer, stemming type-II band orientation. We have also shown the (2H⁺/H₂) reduction potential at the zero energy level and (H₂O/O₂) oxidation potential at the 1.23 eV energy level to explore the probable photo-catalytic over-potential of the hetero-bilayers. It turns out that the HBL 1 and HBL 2 do not have sufficient kinetic over-potential for photocatalysis. Alternatively, HBL 3 has sufficient dynamic over-potential to initiate photocatalysis as the conduction band edge (valance band edge) has lower (higher) energy than the reduction (oxidation) potential. The valance band offset (VBO) and conduction band offset (CBO) are 0.1 eV and 1.1 eV, respectively, suggesting effective carrier separation, which is highly necessitated for photocatalysis and photovoltaic applications.

To further investigate the atomic contribution in the band edges, total and atom-projected densities of states are shown in Figs. 3(a)–3(c). In HBL 1, HBL 2, and HBL 3, the conduction band minima (CBM) are mostly contributed by the B atom of the BP monolayer, while valance band maxima (VBM) are dominantly contributed by the Ge atom of the GeC monolayer, which results in type-II (staggered) band alignment in both models, confirming what we have found in band alignment calculations.

The work function calculation is added to get the tendency of electron transfer in BP/GeC HBLs. It is equated as W = E_vaccum − E_Fermi, i.e., the energy required by the electron to reach vacuum minus the energy of the Fermi level. Usually, the work function of the GeC monolayer is 4.75 eV, while that of the BP monolayer is 4.98 eV.

Having the low work function value of GeC relative to BP, it can also be concluded that the electrons direct from the GeC to BP monolayer when hetero-bilayers are to be formed until equilibrium is reached between the Fermi levels at the interface. It also supports the type-II band alignment calculation. The work functions of HBL 1, HBL 2, and HBL 3 are 4.905 eV, 4.66 eV, and 4.907 eV, respectively. The values infer that the work function is highly responsive to the stacking patterns of the hetero-bilayer models.

Apart from this, the charge (electron) density difference of different HBLs is demonstrated in Figs. 4(a)–4(c) considering the following equation: Δρ = ρ_HBLs − ρ_{ML GeC} − P_{ML BP}, where ρ_HBLs is the charge density of the hetero-bilayer system, ρ_{ML GeC} refers to the charge density of the GeC monolayer, and P_{ML BP} depicts the charge density of the BP monolayer. The blue and yellow zones signify the mushrooming and diminishing of electrons, respectively. Here, the top BP monolayer dominantly accumulates the charges depleted from the bottom GeC substrate layer. In HBL 1, the B atoms accumulate the charge in the top layer, while Ge atoms mainly deplete the charges. In the HBL 2 system, on the other hand, charges accumulate in the interfacial region depleted by P atoms mainly. In the HBL 3 system, the P atoms accumulate charges, while the C atoms deplete the charges. These natures are also confirmed by the charge density difference at the interface of HBLs, as shown in Figs. 4(d)–4(f). In addition, a stronger interfacial linkage of hetero-bilayer systems can be suggested by the order of interfacial charge distribution. The result indicates that the highest charge redistribution occurs in HBL 1, confirming its low-interlayer distance and strong binding energy. Following the charge density difference, the electrostatic potentials of the HBLs using the Poisson equation are shown in Figs. 5(a)–5(c). ML BP has a lower potential than ML GeC, which is consistent with the charge transfer as well. A significant potential drop of ∼11.5 eV is achieved in our calculation, which is higher compared with the other hetero-bilayer structures.^2,3,43 This high-potential difference refers to a strong electrostatic electric field, which can efficiently separate the charge carriers, suggesting the low-probability of recombination, highly effective for photovoltaic applications.

We have also considered the electron effective mass of our proposed monolayer and hetero-bilayer systems to have further insight into carrier dynamics using parabolic band approximation, which is tabulated in Table II. The results suggest that the electron effective masses are also dependent on stacking patterns. In the case of HBL 1, the effective electron mass is lower than that of the derived monolayers, i.e., BP and GeC monolayers. On the contrary, for HBL 2 and HBL 3, the values are low (high) compared to the GeC (BP) monolayer. Thus, it indicates the high-carrier mobility in the HBLs compared with the monolayers.

Applying strain is one of the profound strategies to modulate the electronic properties of hetero-bilayers as it is self-originated due to the lattice mismatch, defect, and impurities during the growth. Hence, peripheral biaxial strain is pertained to uncover in what way it affects the intrinsic electronic properties of BP/GeC HBL systems. The modulated bandgap values with compressive (negative) and tensile (positive) strains are shown in Figs. 6(a)–6(f). The electronic dispersions are highly sensitive to the applied biaxial strain in both their bandgap values and direct to indirect transitions. For HBL 1, the unstrained band structure is an indirect gap at the K^* to K point. When tensile strain is implied, it remains an indirect bandgap at the K^* to K point specifically for 2%, 4%, and 6% of tensile strain, as depicted in Fig. 6(a). When compressive strain is applied, it still remains indirect for 2%, while it transits from indirect to direct at the K^* to K^* point for 4% and 6% of compressive strains, as demonstrated in Fig. 6(b). However, for 2% compressive strain, it becomes an indirect bandgap, while for 4% and 6% of compressive strain, the indirect to direct transition of the bandgap occurs, as shown in Fig. 6(d). In case of HBL 3, the unstrained direct bandgap hetero-bilayer structure ceases the feature and turns out to be an indirect bandgap due to the application of tensile strain, as demonstrated in Fig. 6(e). However, the application of compressive strain again makes it a direct bandgap at K to K, as shown in Fig. 6(f). These direct to indirect transition natures of the bandgap are also compiled in Fig. 7(a). We have also calculated the electron effective mass with varying biaxial compressive and tensile strain, as summarized in Fig. 7(b). Due to the pertaining tensile strain in the HBLs, the electron effective mass increases due to high-bandgaps and flattening of the band edges. However, the highest electron effective mass is attained in HBL 3 at 4% tensile strain, which is related to the highest bandgap value of HBL 3 and causes the destruction of the Dirac point. The lowest electron effective mass is acquired in HBL 1 under unstrained conditions, consistent with the bandgap value.

Following the electronic properties and their modulations due to biaxial strain, the optical properties, precisely the dielectric functions and absorption co-efficient with different photon energies, are calculated using the GGA-PBE exchange-correlation function. The complex dielectric function [ε(ω) = ε₁(ω) + iε₂(ω)] demonstrates the retort of the semiconducting medium due to the incident light depending on frequency (ω). Dielectric physics depicts that the real part ε₁(ω) [imaginary part, ε₂(ω)] is intimately relayed on the absorption loss. The conforming absorption coefficient with varying photon energy can also be computed via the real and imaginary parts of the dielectric function as follows:

where α_a represents the absorption co-efficient and ω represents the photon frequency of incident light. The real and imaginary portion of the complex dielectric function and absorption coefficient with photon energy ranging from 0 eV to 5 eV is demonstrated in Figs. 8(a)–8(c). Surprisingly, the real portion of the dielectric function for HBL 1 is negative within the range of 0.4 eV–0.8 eV, suggesting the semi-metallic nature of the hetero-bilayer. This feature can reveal the potential application of a nano-coating material in the infrared region of HBL 1. Other structures in our study show non-negative values in the real part of the dielectric function, referring to their high-refractive and semiconducting nature. However, in the imaginary part of the dielectric function, the origins and peaks are also calculated, congruent with other theoretical works.⁴⁴ The origins of the absorption peaks in the imaginary part of the dielectric function, as depicted in Fig. 8(b), are around 0.43 eV, 0.7 eV, and 1.1 eV for HBL 1, HBL 2, and HBL 3, respectively, agreeable with the bandgap values calculated within the PBE-GGA DFT framework. Moreover, after the beginning of the peaks, HBL 1 and HBL 2 have three consecutive peaks, while HBL 3 has two successive peaks in the imaginary dielectric function. It is worth mentioning that, in the visible region, all the HBLs follow the peaks of the BP monolayer. Among them, interestingly, the HBL 1 and HBL 2 are blue-shifted, while HBL 3 is redshifted. We further studied the optical absorption, as shown in Fig. 8(c), to predict the efficient photovoltaic material in the solar spectra among the HBLs. The results suggest that all the HBLs have higher absorption in the solar spectra than its constituting BP and GeC monolayers. The highest absorption we have achieved is on the order of 5 × 10⁵ cm⁻¹ for HBL 2, comparable with the perovskite material, which is considered as efficient for optoelectronic devices.⁴⁵ As the increasing amount of electron–hole pair generations is related to the increasing of absorption peaks, it suggests that these HBLs are highly potential for photovoltaic applications. Moreover, the larger number of peaks suggest that more carrier transitions occur in HBL 1 and HBL 2 compared to HBL 3, which is also consistent with the charge transfer calculation.

Homogeneous biaxial external strain can also modify the absorption spectrum as the electronic dispersion changes resulting in modulating inter-band transitions. With this sight, we have also performed the calculation of the biaxial strain-dependent absorption coefficient with varying wavelengths so that we can unleash the absorption increment–decrement effect due to external strain. The photon wavelength-dependent absorption coefficients are illustrated in Figs. 8(d)–8(f) for HBL 1, HBL 2, and HBL 3, respectively. These results indicate that with increasing biaxial-tensile strains, the absorption coefficients are decreasing for all the HBLs, which is also inconsistent with the strain-dependent electronic properties. The reasons behind it are as follows: (1) tensile strains increase the inter-band forbidden gap energies and (2) the HBLs, specifically HBL 1 and HBL 2, have become indirect bandgaps. Interestingly, compressive biaxial stresses improve the absorption coefficients, and the highest value is attained around 6.5 × 10⁵ cm⁻¹ in HBL 2 at 4% compressive strain among all the proposed HBLs in the visible spectrum. Moreover, at 4% and 6% compressive strain, the absorption is the highest for all the HBLs. These results also indicate that the GeC/BP hetero-bilayer can be an effective material for photovoltaic applications.

In summary, a first-principles calculation based on DFT with the vdW corrected Grimme scheme are involved in this study of strain-mediated electronic and optical properties of novel BP/GeC hetero-bilayers. The proposed HBL models have a low-lattice mismatch along with the chemical and dynamical stability, which is confirmed by binding energy calculation and phonon spectra, respectively. The electronic properties of the HBLs are highly responsive to stacking patterns. Among them, only HBL 1 is an indirect bandgap, while the other two, i.e., HBL 2 and HBL 3, have the direct bandgap electronic property. The charge transfer is highly responsive to the stacking pattern and the highest in HBL 1 while the lowest in HBL 3. Besides, a large electrostatic potential difference of about 11.5 eV is attained in the HBL models, which may considerably affect the charge transfer and carrier separation dynamics. Upon pertaining homogeneous biaxial compressive and tensile strain, the modulation of the electronic band structure and effective electron mass are extracted. In addition, all the hetero-bilayers become a direct bandgap at 4% and 6% of compressive biaxial strain. The effective electron mass is significantly small in the HBLs when compared with their comprising BP and GeC monolayers, which is the indicator of the improving carrier mobility. Effective mass is also sensitive to stacking patterns and biaxial strain. Finally, the optical properties, precisely the complex dielectric function and absorption coefficient, are inspected. It is found that HBLs are highly receptive to stacking patterns and show distinctive peaks in an imaginary dielectric function with the peaks of the constituent BP monolayers being tracked. Significant optical absorption on the order of 5 × 10⁵ cm⁻¹ is attained, which is similar to perovskite materials, unleashing the prospective of photovoltaics. Moreover, strain-dependent absorption coefficients are calculated for the proposed HBLs and they showed improvement due to compressive biaxial strain. These results are highly suggestive for revealing the intrinsic and tuning electronic and optical properties of the BP/GeC hetero-bilayer, which is substantially required for the imminent use of our proposed HBLs in 2D electronics and photovoltaics.

See the supplementary material for phonon dispersion and the interlayer dependent binding energy of different HBLs. Additionally, electronic band structures are shown for both GeC and BP monolayers.

The data that support the findings of this study are available within the article and its supplementary material.

There are no conflicts to declare.