Crucial role of interfacial interaction in 2D polar SiGe/GeC heterostructures

The planar charge transfer is a distinctive characteristic of the two-dimensional (2D) polar materials. When such 2D polar materials are involved in vertical heterostructures (VHs), in addition to the van der Waals (vdW) interlayer interaction, the interfacial interaction triggered by the in-plane charge transfer will play a crucial role. To deeply understand such mechanism, we conducted a comprehensive theoretical study focusing on the structural stability and electronic properties of 2D polar VHs built by commensurate SiGe/GeC bilayers with four species ordering patterns (classified as a C-group with patterns I and II and a Ge-group with patterns III and IV, respectively). It was found that the commensurate SiGe/GeC VHs are mainly stabilized by interfacial interactions (including the electrostatic interlayer bonding, the vdW force, as well as the sp 2/sp 3 orbital hybridization), with the Ge-group being the most energetically favorable than the C-group. A net charge redistribution occurs between adjacent layers, which is significant (∼0.23–0.25 e cell−1) in patterns II and IV, but slightly small (∼0.05–0.09 e cell−1) in patterns I and III, respectively, forming spontaneous p–n heterojunctions. Such interlayer charge transfer could also lead to a polarization in the interfacial region, with the electron depletion (accumulation) close to the GeC layer and the electron accumulation (depletion) close to the SiGe layer in the C-group (the Ge-group). This type of interface dipoles could induce a built-in electric field and help to promote photogenerated electrons (holes) migration. Furthermore, a semi-metal nature with a tiny direct band gap at the SiGe layer and a semiconducting nature at the GeC layer indicate that the commensurate SiG/GeC VHs possess a type-I band alignment of heterojunction and have a wide spectrum of light absorption capabilities, indicating its promising applications for enhancing light-matter interaction and interfacial engineering.

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Introduction
Recently, layer-by-layer stacking of two-dimensional (2D) materials to form vertical heterostructures (VHs) governed by weak van der Waals (vdW) force between adjacent layers has attracted tremendous attention due to their various inherent advantages [1][2][3][4][5][6][7].Stacking monolayers in the form of such vdW-VHs is one of the most efficient methods to tune or exploit the combined properties of their parent monolayers.On the other hand, the lattice mismatch, which is the main source of the strain induced effect, can also be tunable by changing the way of assembly, for it can be maximized and spread on the surface to introduce a strain all over the surface and can enhance the electrons' confinement into different layers of 2D vdW-VHs.Such 2D vdW-VHs appear to have appealing performance in electronic and optoelectronic applications besides their novel phenomena, such as ultrafast charge transfer in MoS 2 /WS 2 heterostructures [8] and self-similar Hofstadter butterfly states in graphene/h-BN heterostructures, etc [9].Another interesting finding is that the physical and electronic properties of vdW-VHs are sensitive and differ radically with any change in the number of stacking layers, stacking pattern, strain level, applied electric field, etc [10][11][12][13].They have been extensively applied to vital applications such as nanoelectronics, excitonic solar cells, digital data storage, optoelectronics, spintronics, optospintronics, ferro-electronics, energy storage, sensors, photocatalysis, electrocatalysis, heat transfer, brain-inspired computing, etc [14][15][16][17][18][19][20][21][22].
Different from 2D vdW materials, there are other types of 2D materials that have no layered vdW bulk counterparts and possess a chemical bonding nature like their bulk counterparts, such as the binary SiC, GeC, and SiGe monolayers.Due to a charge transfer between different species, they are classified as 2D polar materials (e.g. the SiC monolayer is considered to have more ionic-like covalent bonding, while the GeC monolayer is considered to have less ioniclike covalent bonding [23][24][25]).Apparently, when such 2D polar materials are stacked to build 2D polar-VHs, the electrostatic force triggered by the in-plane charge transfer will has a significant impact on the structural and physical properties through interfacial hybridization.Thus, for 2D polar-VHs, the stacking arrangement between layers, the artificial strain induced by the lattice mismatch, and the electrostatic interlayer bonding induced by the out-of-plane species ordering must be considered simultaneously to fully understand the physical aspect of 2D polar-VHs.Inspired by this interesting premise, we conducted a comprehensive first-principles study focusing on the structural stability and electronic properties of 2D polar-VHs built by SiGe and GeC monolayers (i.e.SiGe/GeC-VHs) and aimed to deeply understand the role played by the interfacial hybridization (induced by vdW and electrostatic interlayer bonding) and its influence in stabilizing such hybrid heterostructures and in modulating their electronic properties.

Computational details
The structures of SiGe/GeC-VHs were constructed from optimized pristine SiGe and GeC monolayers.These combined systems were then fully relaxed, and their dynamic stability, structural properties, and electronic properties were analyzed by employing the Vienna Ab-initio Simulation Package [26] within the first-principles framework [27,28].Specifically, the projector augmented plane-wave approach [29], the Perdew-Burke-Ernzerhof (PBE) [30] potential function through the generalized gradient approximation technique [31], and the zero damping DFT-D3 method proposed by Grimme et al [32][33][34] were adopted to characterize the corevalence electron interaction, the exchange-correlation functional, and the vdW interactions, respectively.A vacuum region of 25 Å is set to prevent the contact generated by the periodic boundary condition to retain the 2D frame during the simulations.Additionally, a gamma-centered 5 × 5 × 2 (9 × 9 × 2) k-point mesh based on the Monkhorst technique [35] was used to sample the reciprocal space for the structural relaxation (the electronic density of states calculations).A value of 500 eV was adopted as the cutoff energy in all computations.The conjugate gradient algorithm [36] was applied without limitations on the cell volume, the cell shape, and the atomic locations in each relaxation step to achieve a full structural relaxation.The convergence conditions for the total energy and the forces on atoms were established as 10 −5 eV and 10 −3 eV Å −1 , respectively.
By calculating the force constants or Hessian matrix, the dynamic stability is investigated from the vibration frequencies at the gamma point [37].The binding energy (E b ), which is defined as E b = E total -E SiGe -E GeC , was used to analyze the role of the electrostatic interlayer interaction versus the vdW interlayer interaction in stabilizing the commensurate SiGe/GeC-VHs.The energies of the combined SiGe/GeC bilayer, the pristine SiGe monolayer, and the pristine GeC monolayer are represented by the terms of E total , E SiGe , and E GeC , respectively.The energies with vdW correction (E vdW ) and without vdW correction (E no-vdW ) were used to define the energy difference (E diff = E vdW −E no-vdW ), and the interlayer distances with vdW correction (d vdW ) and without vdW correction (d no-vdW ) were used to define the difference of interlayer separation (d diff = d vdW −d no-vdW ), which was used to analyze the contributions from the vdW and electrostatic interlayer interactions.
The charge redistribution of the combined systems was further examined.Specifically, the Bader analysis scheme [38,39], an intuitive approach of distinguishing the charge linked to each atom, was applied to evaluate the charge transfer between atoms, and the differential electron charge density (DCD), defined as ∆ρ = ρ total −ρ SiGe −ρ GeC , was used to analyze the charge redistribution and electronic properties at the interface region, where ρ total is the total electron charge density of the combined system, and ρ GeC (ρ SiGe ) is the electron charge density associated with the GeC (SiGe) layer in the combined system which is carried out by extracting the GeC (SiGe) layer from the relaxed combined system and computing the density of states in the absence of additional relaxing.Therefore, the DCD can extract the information about which atoms are involved in the interaction and to what extent they interact.
In agreement with previous density functional theory (DFT) calculations [20,24,[40][41][42][43][44][45], the optimized lattice constants are 3.951 Å for a buckled SiGe monolayer (with a buckling of 0.59 Å) and 3.267 Å for a flat GeC monolayer, respectively.When the two parent monolayers are stacked vertically, the large lattice mismatch (∼17%) between their optimum lattice constants will result in a significant artificial strain or defects at the interface.To minimize such large lattice mismatch and reduce this type of artificial strain, a commensurate SiGe/GeC-VH with a small compression (∼0.8%) of a 6 × 6 supercell of the optimized GeC monolayer on the top of a 5 × 5 supercell of the optimized SiGe monolayer was built using CellMatch code [46].Corresponding commensurate joint supercell has total 122 atoms (including 36 C, 61 Ge, and 25 Si atoms) and two specific atomic pairs between the adjacent layers, forming by an atom on the top layer directly facing an atom on the bottom layer.As shown by the blackdashed circles in figure S1, the first specific pair is situated at the corner of the supercell with a long interlayer distance, while the second specific pair is situated at the center of a hexagon with a short interlayer distance (denoted by blacksolid circles in figure S1).Furthermore, due to the charge distribution induced by the in-plane charge transfer on each polar monolayer, there will several possible types of out-ofplane species arrangements in constructing such commensurate SiGe/GeC polar VHs (see figure S1).These arrangements will be crucial for the interfacial hybridization as they directly depend on the distribution of species forming the two specific pairs between layers.They are classified into two groups and four patterns.In pattern I, a C atom on the flat GeC sheet is stacked on the top of a Ge (Si) atom on the buckled SiGe sheet forming a short (long) interlayer distance with a C-Ge (C-Si) specific pair at the center of a hexagon (the corner of the supercell), while in pattern II, a C atom on the flat GeC sheet is stacked on the top of a Si (Ge) atom on the buckled SiGe sheet forming a short (long) interlayer distance with a C-Si (C-Ge) specific pair at the center of a hexagon (the corner of the supercell).So, patterns I and II are referred to C-group.In pattern III, on the other hand, a Ge atom on the flat GeC sheet is stacked on the top of a Ge (Si) atom on the buckled SiGe sheet forming a short (long) interlayer distance with a Ge-Ge (Ge-Si) specific pair at the center of a hexagon (the corner of the supercell), and in pattern IV, a Ge atom on the flat GeC sheet is stacked on the top of a Si (Ge) atom on the buckled SiGe sheet forming a short (long) interlayer distance with a Ge-Si (Ge-Ge) specific pair at the center of a hexagon (the corner of the supercell).So, patterns III and IV are referred to Ge-group.Thus, the interlayer bonding varies between different atoms in different patterns and may inspire the physical properties of the hybrid system, especially those with short interlayer distance (e.g. with a C-Ge specific pair in pattern I, a C-Si specific pair in pattern II, a Ge-Ge specific pair in pattern III, and a Ge-Si specific pair in pattern IV, respectively), which will be addressed in section 3.

Structural properties and energetics
Figure 1 visualizes the optimized SiGe/GeC-VHs with four species ordering patterns obtained through a full relaxation process described in section 2. The artificial strain associated with the lattice mismatch in the optimized commensurate structures is indeed small, e.g. about 0.03% (∼0.8%) on the GeC (the SiGe) layer in the C-group and ∼0.1% (∼0.9%) on the GeC (the SiGe) layer in the Ge-group, respectively (the 4th column in table 1).The optimized lattice constants (a) of the commensurate joint supercells with or without vdW correction were found in the range from 19.577 Å to 19.598 Å (see the 3rd column in table 1).Consequently, the difference in a between various patterns is exceedingly small (within ∼ 0.02 Å), which indicates that the out-of-plane species ordering and the vdW interlayer interaction have very weak effects on the lattice constant in such a commensurate system.However, the optimized short interlayer distance (d) between Ge and Si (Ge) specific pairs in the Ge-group or between C and Ge (Si) specific pairs in the C-group changes from 2.88 to 3.535 Å with a difference of ∼0.655 Å (see the 5th column in table 1).The difference in d is due to the factor that Ge atoms at the GeC layer tend to form sp 3 hybridization with Ge and Si atom at the SiGe layer, whereas C atoms at the GeC layer tend to form sp 2 hybridization with Ge and Si at the SiGe layer.These results show a signature of the interlayer distance dependence on the species ordering and interlayer bonding.
The energetics of the optimized commensurate SiGe/GeC-VHs have been analyzed in terms of the binding energy (E b ) and are listed in the 6th column of table 1.It was found that values of E b in the optimized commensurate SiGe/GeC-VHs with four species ordering patterns are all negative, indicating that the attractive interaction between layers is strong enough to make the optimized commensurate SiGe/GeC-VHs structurally stable.It is also found that the combined system is energetically preferential to stay in the form of the Ge-group with pattern IV (with the lowest value in E b ) rather than in the form of the C-group, due to the strong sp 3 type of interfacial hybridization between Ge and Si atoms.Furthermore, all E b values are within the range of −43 and −51 meV atom −1 , which are almost as twice as those of the vdW-VHs (e.g.−27.08 meV atom −1 in bilayer graphene [47] and −20.75 meV atom −1 in graphene/h-BN-VH [28]) and comparable to other heterostructures stacked by 2D polar materials (e.g.∼−41.55 meV atom −1 in SiC/GeC-VHs [48], ∼−48 meV atom −1 in AlAs/germanene-VH [29], and ∼−48.9/54.5 meV atom −1 in SiC(GeC)/MoS 2 -VHs [49]).This interesting finding implies that the existence of the electrostatic interlayer interaction in hybrid 2D polar-VHs, together with the vdW interaction, plays a key role in stabilizing the combined 2D polar bilayers.
The dynamic stability of the optimized commensurate SiGe/GeC VHs was examined from the density of lattice vibration frequency (FDOS).Except small imaginary frequencies (|f | < 17 cm −1 ) for the long wavelength acoustic modes at Gamma point, all the frequencies of optical lattice  vibration modes are positive, indicating that the combined system with the four species ordering patterns is dynamically stable and can be realized experimentally.The small imaginary frequencies of the long wavelength acoustic modes at Gamma point, on the other hand, were further checked using the continuous symmetry measurement approach [50][51][52] and confirmed as numerical fault, which come from a common shortcoming of ab initio phonon calculations in most of the 2D materials [53] and have been removed.Comparing the total and partial FDOS (figure S2), it was found that the higher phonon frequencies (above 800 cm −1 ) in all patterns are contributed by the flat GeC layer, while the lower phonon frequencies (around 400 cm −1 ) in all patterns come from the buckled SiGe layer, respectively.The fluctuation of the peak positions of FDOS for all patterns of SiGe/GeC-VHs is quite small, indicating that phonon modes are not sensitive to the species ordering.These findings are a consequence of the small compressive strain on SiGe and GeC layers and the small symmetry breaking induced by interlayer coupling.

Electrostatic vs vdW interlayer interactions
There is an in-plane charge transfer between Si and Ge atoms in buckled SiGe monolayer and between Ge and C atoms in flat GeC monolayer, respectively, due to their different electronegativities, which causes an accumulation of a net positive charge around Si (Ge) atoms and a net negative charge around Ge (C) atoms in the pristine SiGe (GeC) monolayer.When such 2D polar materials are stacked to form VHs, a charge redistribution will not only modulate electrostatic intralayer-bonding but also electrostatic interlayer-bonding between adjacent layers.Thus, both the electrostatic interlayer bonding and the vdW interaction will play the key role in stabilizing the interlayer distance of the commensurate SiGe/GeC-VHs.
To clearly understand how these two types of interlayer forces interact with atoms on the adjacent layers, we calculated the relative energies as a function of d (i.e.E(d)-E(d → ∞)) with and without vdW correction, which can be traced back to analyze the contributions of electrostatic and vdW interlayer interactions in stabilizing the combined systems.The results are shown in figure 2, with black (red) curves representing the relative energy per atom without (with) the vdW correction.It was found that for any species ordering, the relative energy always exhibits a minimum even without the vdW correction, revealing that such bilayer system can be stabilized within a certain interlayer space under the attractive electrostatic interlayer bonding.In this circumstance, the minimum relative energy in the Ge-group is slightly deeper (∼−9.0 -−17.5 meV atom −1 ), and d is shorter (∼3.0 Å), as compared to slightly shallow relative energy and longer d in the C-group (∼−6.2 to −7.3 meV atom −1 ; ∼3.7 Å), which clearly indicates that the strength of the electrostatic interlayer force strongly depends on the arrangement of net positive and negative charges between adjacent layers.Thus, the contribution of the electrostatic interlayer forces in stabilizing the stacked two layers is crucial.The relative energy minimum of the combined system is further deepened by ∼37.6-40.4meV atom −1 after considering the vdW interlayer corrections.The equilibrium interlayer distance, on the other hand, is almost unchanged in the Ge-group and is further shortened by ∼0.17-0.23 Å in the C-group (see the positions of red/black-dashed lines in figure 2).Comparing the relative energy with/without vdW interactions (i.e.E diff in the 7th column of table 1) and the difference in the equilibrium interlayer distance (i.e.d diff in the 8th column of table 1 and guided by the dashed lines in figure 2), it is found that the electrostatic interlayer force, triggered by the in-plane charge transfer, plays a significant role in forming the interlayer bonding and acts as a driving force to stabilize the system, while the vdW interaction plays an important role in stabilizing the system by making the system attains a lower binding energy.

Electronic properties
To reveal the electronic properties of the hybrid heterostructure, we calculated the band structure and analyzed the nature of the band gap. Figure 3 presents the weighted band structures contributed by different elements on the layers of the commensurate SiGe/GeC-VHs for the four patterns.The larger the size of the balls, the bigger the contribution to the band structures.It was found that the commensurate SiGe/GeC-VHs show semi-metal behavior with a tiny direct band gap (∼12-67 meV) located at K point (also listed in table 2), where the C-group has a slightly larger band gap than that of the Ge-group.The main contributions to the band structure near the fermi level come from the buckled SiGe layer (pink and blue balls in figure 3), showing a semi-metal nature.While the contributions from the flat GeC layer (yellow and red balls in figure 3) appear in the deeper valence bands and upper conduction bands, respectively, showing a semiconducting nature.In patterns I and IV, the valence band maximum (VBM) is mainly dominated by the Si atoms on the SiGe layer, and the conduction band (CBM) is mainly dominated by the Ge atoms on the SiGe layer.In pattern II, it is just the opposite, and in pattern III, all Si atoms contribute to both.The partial density of states (figure S3) show that the main contribution near the Fermi level region comes from p z orbitals, indicating the strong out of plane hybridization.These results demonstrate that the band structures of the commensurate SiGe/GeC-VHs on each layer maintain the basic electronic natures of corresponding pristine SiGe and GeC monolayers (see figure S4) besides the existence of the strong correlation of the out-ofplane hybridization.Such a feature implies that commensurate SiGe/GeC-VHs form semi-metal/semiconductor heterojunctions and possess type-I band alignment [54][55][56][57][58].Note that DFT calculations usually underestimate bandgaps, and such underestimate values can be corrected by HSE06 or GW method.Such calculations, however, are especially expensive for the commensurate SiGe/GeC-VHs as the size of the combined systems is too large (122 atoms per cell) to perform the HSE06 or GW calculation.Alternatively, considering the bandgap of commensurate SiGe/GeC-VHs is dominated by the semi-metal SiGe layer, and the bandgap of the pristine SiGe monolayer (∼0.07 eV at DFT-PBE level) is opened up to 0.32 eV at the HSE06 level (∼4.5 times of DFT-PBE value, see figure S5), it is expected that the bandgap of commensurate SiGe/GeC-VHs should be opened up by about the same order at HSE06 level.

Charge redistribution and interlayer hybridization
It is found that the large difference (∼0.65) in the electronegativities of Ge and C elements leads to a large charge transfer (∼1.38 e atom −1 ) between Ge and C atoms in the pristine GeC monolayer, and the small difference (∼0.11) in the electronegativities of Si and Ge elements causes a tiny chare transfer (∼0.06 e atom −1 ) between Ge and Si in the pristine SiGe monolayer.When such 2D polar SiGe and GeC monolayers are stacked to form commensurate SiGe/GeC-VHs, it is expected that the effects from the lattice mismatch induced strain, the electrostatic interlayer bonding, and the sp 3 hybridization will have a significant impact on the charge transfer between atoms.It was found that after a full relaxation, there is indeed a net charge redistribution on all atoms of the optimized commensurate SiGe/GeC-VHs.For instance, Ge (Si) atoms on the SiGe layer gained (lost) more charges compared to Ge (Si) atoms on the pristine SiGe monolayer, while C (Ge) atoms on the Ge-C layer gained (lost) less charges compared to C (Ge) atoms on the pristine GeC layer (see the 3-6th columns Table 2. Band gap at DFT-PBE level (2nd column with the VBM/CBM at the high symmetric K point in the parentheses) and the charge transfer (positive (negative) values mean gain (loose) electrons) projected on Si, Ge, and C atoms in the pristine SiGe and GeC monolayers and the commensurate SiGe/GeC-VHs with four species ordering patterns (3-6th columns), respectively.There are two types of Ge atoms: one is at the SiGe layer (denoted by Ge (SiGe)), and the other is at the GeC layer (denoted by Ge (GeC)).The net charges distributed on SiGe and GeC layers per supercell are listed in the 7th and 8th columns, respectively.

Pattern
Band  2).Clearly, the charge redistribution on the SiGe layer is ∼2-4 times more than that in the pristine SiGe layer which is mainly caused by the compression (∼0.8%-0.9%)induced a shortening in bond length, while it is about 15% less in GeC layer as compared to pristine GeC layer which is basically associated with the small compression (∼0.02%-0.1%).Furthermore, it was found that in patterns II and III, the charge transfers between Ge and Si on the SiGe layer are ∼two times as large as those in patterns I and IV.This phenomenon can be further interpreted in terms of the electrostatic interlayer bonding associated with the out-of-plane species ordering (e.g.characterized by the specific pairs with the short interlayer distance).As shown in figure 1, the out-of-plane species ordering in pattern II is characterized by the specific C-Si pair, while it is characterized by the specific Ge-Ge pair in pattern III.Such pairs are formed by the opposite sign of charges (see the 3rd-6th columns in table 2) and display an 'attractive' type interaction between layers, which could promote more charge transfer.While, in patterns I and IV, the out-of-plane species ordering are characterized by the specific C-Ge and Ge-Si pairs, respectively.In contrast to the cases of patterns II and III, these pairs are formed by the same sign of charges (see the 3rd-6th columns in table 2) and display a 'repulsive' type interaction between layers, which could weaken the charge transfer.The redistribution of the net charge transfer in the optimized commensurate SiGe/GeC-VHs was deeply examined by analyzing the net charge difference (Charge_ diff ), which is defined as Charge_ diff = Charge_ VH-Charge_ pristine , where Charge_ VH is the charge transfer of an atom in the commensurate SiGe/GeC-VHs, and Charge_ pristine is the charge transfer in the corresponding pristine monolayer, respectively.The results are depicted in figure S6 for all four patterns, where green and red dots represent gaining and losing charges on atoms, respectively.It is found that the redistribution of the net charge difference for each species ordering pattern is not uniform, which is mainly due to the distributions of different types of stackings (e.g.AA, AB, etc.) domains and species ordering throughout the commensurate joint supercell of SiGe/GeC-VHs.Overall, a total net charge transfer between adjacent layers was found from Bader analysis, where the charge flow is directed from the GeC layer to the SiGe layer in the C group, and from the SiGe layer to the GeC layer in the Gegroup, respectively (see the 7th and 8th columns in table 2 and the blue arrows in figure 4).This interlayer net charge transfer is slightly small in patterns I (∼0.09 e cell −1 ) and III (∼0.05 e cell −1 ), while it is significant in patterns II (∼0.25 e cell −1 ) and IV (∼0.23 e cell −1 ).Such a difference in the net charge transfer is quantitatively due to the difference in electronegativity and the strong interfacial hybridization triggered by electrostatic interaction between charged atoms in adjacent layers.Namely, the large electronegativity differences between Si at the buckled SiGe layer and C or Ge at the flat GeC layer count for the significant interlayer net charge transfer in patterns II and IV.However, the small electronegativity differences between Ge at the buckled SiGe layer and C or Ge at the flat GeC layer are the main reason that pattern I or III has small interlayer net charge transfer.The total net charge transfer between adjacent layers reveals that the GeC (SiGe) layer in the C-group behaves like p (n)-doping in the interfacial region.Oppositely, GeC (SiGe) layer in the Ge-group behaves like n (p)-doping in the interfacial region.Thus, the commensurate 2D polar SiGe/GeC-VHs can form spontaneous p-n heterojunctions.
To deeply understand the charge redistribution and interfacial hybridization, the DCD has been calculated and illustrated in figure 4 with an isosurface of 2.4 × 10 −4 e Å −3 .The yellow and blue contours represent the net electron accumulation and depletion, respectively.Clearly, there is in-plane orbital hybridization on each layer coming mainly from the strain induced effect (see the project views in figure 4), and it counts for the in-plane net charge redistribution.Most importantly, there is a significant interfacial orbital hybridization between layers (see the side views of figure 4), indicating strong outof-plane net charge redistribution.Inside the interfacial region, electron depletion mostly spreads at the bottom surface of the flat GeC layer and scatters near the top surface of the buckled SiGe layer, but the electrons mostly accumulate in the middle area between two layers.Furthermore, with the strong orbital hybridization at the Ge-Ge specific pair in pattern III and the Ge-Si specific pair in pattern IV, more electron depletion also occurs inside the middle region (see the side views in the bottom panels of figure 4).
For further understanding of the interfacial hybridization and net charge redistribution, the plane-averaged electron density difference (∆ρ) was examined along the vertical direction perpendicular to the surface of the commensurate SiGe/GeC-VHs (see figure 5), where the positive (negative)  value means electron accumulation (depletion), and the interfacial region in each pattern is indicated between the two bluedashed lines.It was found that in the interfacial region, more electron accumulation (depletion) was located near the SiGe (GeC) layer in the C-group, and more electron accumulation (depletion) was located near the GeC (SiGe) layer in the Gegroup, showing polarization between layers.Thus, a polarized dipole-like strip is produced, forming a p-type in GeC (SiGe) and an n-type in the SiGe (GeC) layer in the C-group (Gegroup).This pronounced polarization could lead to a built-in electric field (E in ) (indicated by the blue arrow) in the interfacial region (between the blue dashed lines).Such a built-in electric field could promote photogenerated holes (electrons) on the GeC (SiGe) layer migrating to the SiGe (GeC) layer in the C-group, and oppositely, in the Ge-group.
Moreover, the electrostatic potential across the interface of the heterojunctions is presented in figure S7, which was obtained by solving the Poisson equation [59].The potential around the bottom SiGe layer is slightly deeper than that around the top GeC layer due mainly to the in-plane charge redistribution and partially to the interfacial charge redistribution.The work functions and band edges of the commensurate SiGe/GeC-VHs and corresponding pristine GeC and SiGe monolayers were further evaluated from the electrostatic vacuum potentials (table S1).As discussed in the section 3.3, the commensurate SiGe/GeC-VHs possess the straddling gap showing the type-I band alignment (i.e. the feature of semimetal/semiconductor heterojunction), like the Graphene/h-BN heterostructure [60].Considering the calculated bandgaps of the pristine SiGe (∼0.32 eV at the HSE06 level) and GeC (∼2.84 eV at the HSE06 level [48]) monolayers, it is expected that commensurate SiGe/GeC-VHs, different from type-II heterojunctions, possess a wide spectrum of light absorption capabilities, spanning from ultraviolet to near infrared.The built-in electric field (E in ) in the interfacial region, on the other hand, plays a key role in carrier migration and enhance the efficiency of light-matter interactions.As illustrated in figure S8(a), the built-in electric field (E in ) in the C-group promotes (prevents) the migration of photogenerated holes (electrons) on the GeC layer to the SiGe layer, leading to an electron (hole)-rich environment on the GeC (SiGe) layer.Similarly, as shown in figure S8(b), the built-in electric field (E in ) in the Ge-group promotes (prevents) the migration of photogenerated electrons (holes) on the GeC layer to the SiGe layer, leading to a hole (electron)-rich environment on the GeC (SiGe) layer.Thus, the spatial distribution of the photogenerated electrons and holes can be selectively tuned by modulating the out-of-plane species ordering of the individual building blocks and reach the demand for designing solar cells.These novel properties open a new pathway for light-matter interaction by designing such type of heterostructures with tailored optoelectronic features [61].

Conclusion
The roles of the interlayer bonding and the stacking species arrangement played on the structural stability and electronic properties of 2D polar VHs built by commensurate SiGe/GeC bilayers with four species stacking patterns (classified as the C-group and the Ge-group) have been addressed based on a comprehensive theoretical study within the DFT frame.It has been found that the commensurate SiGe/GeC-VHs with negligible strain are energetically and dynamically stable, with the Ge-group being the most energetically favorable than the C-group.The structural stability is mainly controlled by the interfacial interaction (including the electrostatic interlayer bonding triggered by the in-plane charge transfer, the vdW interaction, as well as the sp 2 /sp 3 hybridization).The larger binding energy (comparable with other vdW VHs) and its dependence on the species stacking pattern and interlayer interaction are proof of the effective influence of interfacial interaction in stabilizing the SiGe/GeC-VHs system.
Both in-plane and out of plane net charge redistributions were found in the commensurate SiGe/GeC-VHs, which are mainly due to the strong hybridization vertically and horizontally.Such phenomena are triggered by the electrostatic interaction between charged atoms.The charge transfers induced by such net charge redistribution between layers result in spontaneous p-n junctions, and the net charge redistribution at the interfacial region could lead to a polarized built-in electric field (E in ), which may affect charge carriers' dynamics and will certainly enhance photogenerated electrons (holes) flow.
In terms of their electronic properties, the commensurate SiGe/GeC-VHs possess a semi-metal nature with a tiny direct band gap at K point.Moreover, the natures of the VBM and the CBM are mainly dominated by the SiGe layer, while the contributions from the flat GeC layer appear in the deeper valence bands and upper conduction bands, which reflect a strong influence of the semi-metal nature of the SiGe monolayer over the semiconductor nature of the GeC monolayer.Thus, commensurate SiGe/GeC-VHs could form a type-I band alignment which is promising in applications for enhancing light-matter interaction (e.g. in solar cells, light emitting diodes, and laser applications).

Figure 1 .
Figure 1.Schematic illustration of the top and side views of optimized commensurate SiGe/GeC-VHs with four species ordering patterns: the C-group with paterns I and II (top panels) and the Ge-group with patterns III and IV (bottom panels).The black circles (arrows) on the top (side) views denote the positions of species on the top/bottom layers with short interlayer distances indicated by C-Ge pair (pattern I), C-Si pair (pattern II), Ge-Ge pair (pattern III), and Ge-Si pair (pattern IV), respectively.Those short distance pairs are also indicated by the red color in the notations of each pattern.The black-dashed circles (arrows) in the top (side) views denote the positions of species on the top/bottom layers with long interlayer distances indicated by C-Si pair (pattern I), C-Ge pair (pattern II), Ge-Si pair (pattern III), and Ge-Ge pair (pattern IV), respectively.The optimized lattice constants of the commensurate joint supercell and short/long interlayer distances are denoted by the numbers, and the red, green, and brown balls represent Ge, Si, and C atom, respectively.

Figure 2 .
Figure 2. The relative energy per atom as a function of the interlayer distance (d) for the commensurate SiGe/GeC-VHs with (the red squares)/without vdW (the black circles) correction in patterns I (top left), II (top right), III (bottom left), and IV (bottom right), respectively.The spaces between dashed lines denote the difference (d diff ) of interlayer distance with (red dashed line)/without vdW (black dashed line) correction.The short distance pairs are indicated by the red color in the notations of each pattern.

Figure 3 .
Figure 3. Band structures of commensurate SiGe/GeC-VH with species ordering patterns I-IV.The numbers represent the band gaps, and the red circles point to the position of the VBM and the CBM.The yellow, red, purple, and blue balls represent the projected bands contributed by C, Ge (on the GeC layer), Ge (on the SiGe layer), and Si atoms, respectively.The Fermi level is set to be zero.The short distance pairs are indicated by the red color in the notations of each pattern.

Figure 4 .
Figure 4.The projected (upper) and side (bottom) views of DCD for commensurate SiGe/GeC-VHs with patterns I-IV, respectively.The electron accumulation and depletion are represented by the yellow and blue contours (with an isosurface of 2.4 × 10 −4 e Å −3 ).The directions of the net charge transferer from GeC to SiGe layers (C-group with patterns I and II) or from SiGe to GeC layers (Ge-group with patterns III and IV) are denoted by the blue arrows, and the corresponding values are indicated by the numbers.The C, Si, and Ge atoms are noted by brown, blue, and grey balls, respectively.The short distance pairs are indicated by the red color in the notations of each pattern.

Figure 5 .
Figure 5.The plane-averaged DCD (∆ρ) of commensurate SiGe/GeC-VHs for patterns I-IV, with the positive (negative) values indicating electron accumulation (depletion).The inserts show the side views of commensurate SiGe/GeC-VHs with blue-dashed lines representing the interfacial region, and the built-in electric field is indicated by the blue arrows.The short distance pairs are indicated by the red color in the notations of each pattern.

Table 1 .
The optimized lattice constant (a), strain on each layer (negative values indicating compression), optimized short interlayer distance (d), binding energy (E b ), the energy difference (E diff ), and the interlayer distance difference (d diff ) of the optimized commensurate SiGe/GeC bilayer with different species ordering patterns (see the 1st and 2nd columns where the stacking ordering of C-Ge (Ge-Ge) and C-Si (Ge-Si) belong to C-group (Ge-group), respectively).