Chemical Functionalization of Pentagermanene Leads to Stabilization and Tunable Electronic Properties by External Tensile Strain.

Inspired by the unique geometry and novel properties of a newly proposed two-dimensional (2D) carbon allotrope called pentagraphene, we have performed first-principles calculations to study the structural stability and electronic properties of pentagermanene (pGe) modulated by chemical functionalization and biaxial tensile strain. It is observed that the 2D pGe is energetically unfavorable. However, the 2D pentagonal nanosheets can be stabilized by both hydrogenation and fluorination. Phonon dispersion spectrum and ab initio molecular dynamics simulations demonstrated that the dynamic and thermal stabilities of the two functionalized pGe nanostructures can be maintained even under a high temperature of 500 K. Our calculations revealed that both hydrogenated and fluorinated-pentagonal germanenes are semiconductors with indirect band gaps of 1.92 and 1.39 eV (2.60 and 2.09 eV by the hybrid functional), respectively. The electronic structures of the functionalized pGes can be effectively modulated by biaxial tensile strain, and an indirect to direct gap transition can be achieved for the hydrogenated pGe sheet by 6% biaxial strain. Moreover, the band gap of the hydrogenated pGe could be further tailored from 0.71 to 3.46 eV (1.16-4.35 eV by the hybrid functional) by heteroatom doping (C/Si/Sn/Pb), suggesting the semiconductor-insulator transition for differently doped nanostructures. As a result, the functionalized pGes are expected to have promising applications in nanoelectronics and nanomechanics.

Introduction

Graphene, presenting exceptional properties derived from the truly two-dimensional (2D) nanostructure, has rapidly and continually attracted enormous interests of researchers in various disciplines since its fabrication in 2004.[1] Inspired by the exotic properties originating from the 2D nanostructure, group-IV (silicon, germanium, and tin) analogues have been theoretically and experimentally investigated.[2,3] The 2D silicene, germanene, and stannum sheets with low-buckled structures are predicted to be stable at first through first-principles calculations,[4−6] and then they have been experimentally fabricated on a metal surface[7−12] Furthermore, the characterized electronic properties of the group-IV analogues in experiments are in good agreement with the computational modeling reported previously. It is noticed that most inorganic nanosheets possess hexagonal lattices similar to those of graphene, such as binary BN nanosheets with flat honeycomb[13] and transition-metal dichalcogenides (MoS2, WS2, and MoSe2) with buckled honeycomb.[14,15] In addition, the recently fabricated black-phosphorene nanosheets with puckered nanostructure can also be regarded as akin to hexagonal lattices.[16,17]

More recently, a new 2D carbon allotrope named pentagraphene, presenting a tetragonal lattice with threefold and fourfold coordinated hybrid carbon sheets, was proposed by computational modeling.[18] The pristine pentagraphene is found to have negative Poisson’s ratio, ultrahigh strength, and reduced thermal conductivity compared to those of graphene.[19] More importantly, first-principles calculations have revealed that pentagraphene is a semiconductor with a quasidirect band gap of 3.25 eV, suggesting that it could be a promising candidate for possible applications in the field of nanoelectronics and optoelectronics.[20,21] In addition, some pentagraphene-like analogues have also been proposed and predicted to be stable at room temperature.[22−24] Meanwhile, surface decoration can be utilized to improve the stability, electronic structure, and thermal conductivity of 2D nanomaterials.[25−28] Previous studies have shown that the energetic and mechanical stabilities of pentasilicene can be greatly improved via hydrogenation,[29] the electronic and mechanical properties of pentagraphene can also be effectively modulated by hydrogenation and fluorination,[30] and unexpected enhancement in the thermal conductivity can be obtained for the hydrogenated pentagraphene.[31]

Recently, Goldberger’s research group has fabricated free-standing monolayers of hydrogenated germanium (GeH) and methylated germanium (GeCH3) by a chemical method, and their characterizations on these monolayer germananes have shown that they have large direct band gaps and high electron mobilities.[32−34] More importantly, previous studies have shown that the structural stability, electronic structure, and transport properties of Ge-based 2D nanomaterials can be improved by covalent chemistry decoration of different surface ligands,[35−38] allowing us to tailor desirable properties to achieve practical applications in the fields of electronics and optoelectronics.[39−42]

Motivated by the advance of chemically functionalized Ge-based 2D nanomaterials and the penta structure with tetragonal lattice, researchers would wonder the possibility of experimental realization of pentagermanene (pGe) and the influences of chemical functionalization on the stability and electronic structures of pGe. Thus, in this article, we have carried out first-principles calculations to study the structural stability of pristine pGes and chemically functionalized pGes by analyzing the phonon dispersion and adsorption energy. It is found that pristine pGe is dynamically unstable; however, the Ge-based pentagonal structure can be stabilized by hydrogenation and fluorination. Furthermore, the influences of biaxial strain on the electronic properties of hydrogenated and fluorinated pGes have been investigated in detail. The band gap of the hydrogenated pGe could be further tailored by heteroatom doping (C/Si/Sn/Pb). Our results could be helpful in understanding the electronic properties of Ge-based nanomaterials and may be useful for the development of Ge-based electronics.

Results and Discussion

Figure a shows the atomic structural geometry of pristine pGe, presenting a tetragonal lattice consisting of four pentagons in the primitive cell. The optimized lattice constant of the pristine pGe is a = 5.668 Å, and the structure possesses the P4̅21m symmetry (space group no. 113). The unit cell comprises two tetracoordinated Ge atoms and four tricoordinated Ge atoms, and for convenience, they are labeled as Ge1 and Ge2, respectively. Furthermore, we label the pristine pGe as pGe in the following discussions. For the pGe nanostructure, the bond length of Ge1–Ge2 atoms is R1 = 2.508 Å and that of Ge2–Ge2 atoms is R2 = 2.505 Å. The basal plane of pGe is puckered, and the buckling distance (determined by the vertical height difference between the Ge1 and Ge2 layers shown in Figure a) is h = 1.308 Å, which is much larger than the corresponding values of pentagraphene[18] and pentasilicene.[29] This is attributed to the largest atomic radii of germanium atoms. Moreover, the bond angle of Ge2–Ge1–Ge2 is θ1 = 105.78°, and the bond angle of Ge1–Ge2–Ge1 is θ2 = 106.07°, indicating distinct sp3 hybridized bonds. To assess the dynamic stability of the 2D pGe, we have measured the phonon spectrum of the pentagonal structure, as shown at the bottom of Figure a. The presence of imaginary vibration in the phonon dispersion shows that the pGe structure is dynamically unstable. The detailed analysis shows that the two soft modes around the Γ-point are associated with the tricoordinated Ge2 atoms because the Ge2 atoms are arranged in an unfavorable, highly buckled configuration and they prefer transforming into sp2 hybridization, causing soft modes to collapse the 2D sheet. Overall, the Ge2 atoms tend to distort the pentagonal structure, leading to the dynamical instability.

Figure 1

Top and side views of the optimized structures along with their phonon dispersions of the (a) pristine (pGe) and (b) hydrogenated (H–pGe–H) and (c) fluorinated (F–pGe–F) pGes. The squares marked by blue dashed lines denote the unit cell. The highlighted yellow spheres refer to the fourfold coordinated Ge1 atoms in each system.

To improve the structural stability of pGe, the key is to stabilize the buckled structure of Ge2 atoms. Thus, a powerful route to enhance the stability is chemical functionalization, which has been widely used to tune the properties of low-dimensional nanostructures,[39,44] including the Ge-based nanomaterials.[37,45] Indeed, previous experimental studies have shown that both hydrogenated- and methylated-germanene sheets with graphene-like lattices are stable at room temperature.[33,36,42] The fluorinated-germanene sheet and various chemically functionalized group-IV 2D nanosheets hereafter are suggested to be topological insulators.[3,46−48] The methylated germanene can also be driven into a topologically nontrivial state by biaxial strain.[49,50] Therefore, we explore the structural stability of the hydrogenated and fluorinated pGes. Thus, for pGe sheets, considering that there are four unsaturated Ge2 atoms per unit cell, we decorate the top of Ge2 atoms by hydrogenation and fluorination, where all Ge atoms become tetracoordinated, as shown in Figure b,c. They are labeled as H–pGe–H and F–pGe–H, respectively. Correspondingly, the hydrogenated and fluorinated hexagonal germanenes are labeled as H–Ge–H and F–Ge–F for comparison, respectively. For the nanostructures of the hydrogenated and the fluorinated pGe, it is noted that the ratio of Ge/H or Ge/F is 3:2. The structure parameters of the hydrogenated and fluorinated germanenes and the pGe are listed in Table . As can be seen clearly, hydrogenation in H–pGe–H results in a decrease in the bond length, which is due to a significant reduction in the lattice constant. In contrast, the structural changes in fluorinated F–pGe–F are subtle. It is noticed that the buckling distances of the hydrogenated and fluorinated pGe’s have experienced a little change when compared with those in the pristine pGe sheet. This is in sharp contrast to the cases of pentagraphene and the corresponding results of hydrogenated and fluorinated pentagraphenes.[30] To examine the dynamical stability of the hydrogenated and fluorinated nanostructures, we have performed phonon calculations of H–pGe–H and F–pGe–F sheets, as shown at the bottom of Figure b,c. Obviously, there are no soft modes throughout the whole Brillouin zone (BZ), confirming the dynamical stabilities of the two functionalized pGe sheets.

Table 1

Structural Parameters and Energy Gaps of the Optimized Germanene-Based 2D Nanomaterials Chemically Functionalized by Hydrogenation and Fluorinationa

conformation	R1 (Å)	R2 (Å)	R3 (Å)	θ1/θ2 (deg)	h (Å)	a (Å)	Eb	Eg-PBE	Eg-HSE06
H–pGe–H	2.48	2.47	1.56	106.3/105.2	1.32	5.57	0.52	1.92	2.60
F–pGe–F	2.51	2.53	1.78	105.4/106.9	1.29	5.69	3.47	1.39	2.09
H–Ge–H[14]		2.47	1.56		0.73	4.09	0.25	0.95	1.62
F–Ge–F[43]		2.54	1.79		0.58	4.29	3.21	0.00	0.17

R1 (Ge1–Ge2 bond length in Å), R2 (Ge2–Ge2 bond length in Å), R3 (Ge–H/Ge–F bond length in Å), θ1/θ2 (Ge2–Ge1–Ge2/Ge1–Ge2–Ge1 angle in deg), h (buckling distance in Å), a (lattice constant in Å), Eb (binding energy in eV/per atom), Eg-PBE (in eV), and Eg-HSE06 (in eV)

To examine the energetic stability of the hydrogen and fluorine atoms with respect to pGe, we have calculated the binding energy, Eb, defined as Eb = −(Etotal – EpGe – 4Eadatom), where Etotal, EpGe, and Eadatom denote the total energies of the H–pGe–H and F–pGe–F sheets, pGe sheet, and isolated hydrogen and fluorine atoms. Considering that there are four hydrogen and fluorine adatoms per unit cell, we average the binding energy on each decorated atom. The binding energy denotes the interaction strength between the adsorption atom and the undecorated penetagermanene and germanene sheets. The binding energy of fluorination is found to be larger than that of hydrogenation for both pentagonal and hexagonal structures because fluorine has superior negativity compared to that of hydrogen. It is worth mentioning that the binding energies of hydrogen and fluorine with hexagonal germanene sheets are 0.25 eV/H and 3.21 eV/F,[35,38,43] which are smaller than the corresponding values for pGe sheets. The larger binding energies of hydrogen and fluorine with the pGe sheet demonstrate that these chemical functionalizations are energetically more favorable than those of the 2D hexagonal sheet. These results can be understood from the differences in the tricoordinated Ge2 atoms with nearly sp3 hybridization in the pGe sheet and the partially delocalized π orbital arising from sp3-like hybridization. Therefore, the Ge2 atoms in the 2D pGe sheet are chemically more active than the Ge atoms in the hexagonal germanene.

To address the enhancement in the thermal stability induced by hydrogenated and fluorinated functionalization, we have further performed ab initio molecular dynamics (AIMD) simulation at 300 and 500 K, respectively. The 2D chemically functionalized sheet is expanded to a 3 × 3 supercell consisting of 90 atoms. Figure presents the total energy fluctuations and the final structures of the functionalized H–pGe–H and F–pGe–F sheets at the end of AIMD simulations, and the corresponding results for the pristine pGe sheet are also shown for comparison. The time step is set to be 2 fs, and 2.5 and 5 ps AIMD simulations are carried out for the pristine pGe and functionalized pGe nanostructures, respectively. As clearly shown in Figure a, the average values of the total energies of the pGe sheet at 300 and 500 K are decreased. As a consequence, its final nanostructure has been broken almost completely after a period of 2.5 ps. In contrast, the functionalized pGe sheets are found to sustain their integrated nanostructures during the AIMD simulations at 300 and 500 K. The observed total energy fluctuations of both hydrogenated and fluorinated pGes are almost zero, as evidenced by the zoomed insets in Figure b,c. Furthermore, it is noted that the total energy fluctuations of the hydrogenated and fluorinated pentagraphenes at room temperature are much larger than those of the corresponding functionalized pGe cases,[30] indicating that the functionalized pGe could be energetically more favorable. Taking the calculated results shown above into account, hydrogenation and fluorination of 2D pGe sheets lead to not only dynamical stability but also thermal stability at room temperature and even at a temperature of 500 K. Next, we will focus on the electronic structures of the hydrogenated and fluorinated pGes, and the electronic properties of the pristine pGe will not be considered because of its dynamically unstable nanostructure.

Figure 2

Snapshots from the AIMD simulations of the structures of (a) pGe, (b) H–pGe–H, and (c) F–pGe–F sheets at 300 and 500 K at the end of simulation. The zoomed figures of the total energy fluctuations at different temperatures are shown in each inset.

The calculated electronic structures and the corresponding partial density of states (PDOSs) of the hydrogenated and fluorinated pGes are shown in Figure . Clearly, H–pGe–H is a semiconductor with an indirect band gap of 1.92 eV at Perdew–Burke–Ernzerhof (PBE) level. Its valence bond maximum (VBM) is located at the M-point, and its conducting bond minimum (CBM) is located at about 1/2 along the Γ – M path in the momentum space. This feature is different from that of the hydrogenated pentasilicene[29] and hydrogenated pentagraphene,[30] whose CBMs are located at the Γ-point. The underestimation of electronic band gap in the framework of PBE functional is well-known; thus, we have carried out Heyd–Scuseria–Ernzerhof (HSE)06 functional calculations to obtain more accurate results. The electronic structures obtained by PBE and HSE06 are analogous, whereas HSE06 gives significant modification on band gap to 2.60 eV, showing that hydrogenation can tune the electronic structure of pGe from metal to semiconductor. Furthermore, we found that both PBE and HSE06 band gaps are approximately 1 eV larger than the corresponding value for germanane.[32,51] On the basis of the analysis of charge density distribution, as shown in Figure a, the VBM of H–pGe–H is mainly contributed by the Ge2–Ge1 bonding state and partially by the Ge2–Ge2 bonding state, indicating the σpp characters. In contrast, its CBM is composed of the Ge2–H antibonding state, indicating the σsp* characters.

Figure 3

Electronic band structures obtained on the basis of the PBE and HSE06 functionals and the PDOSs of (a) H–pGe–H and (b) F–pGe–F. Isosurface plots of the band-decomposed charge density distributions corresponding to VBM and CBM are displayed on the right side of the PDOS result. The isovalue is 0.0085e/b3. The Fermi level is set to zero and indicated by the gray dashed line.

For the fluorinated pGe, it is also an indirect band gap semiconductor. The band structure calculation at PBE level reports a band gap of 1.39 eV, and the HSE06-calculated result is essentially identical to the PBE one, except for the increased band gap of 2.09 eV. The electronic structures of H–pGe–H and F–pGe–F shown in Figure are very similar. However, hydrogenated and fluorinated decorations in pGe give rise to distinct influences on the characters of VBM and CBM states. For instance, the charge density of F–pGe–F reveals that both Ge and F atoms have significant contributions to the VBM state, which is verified by the PDOS result, and its CBM state is originated from strong covalent bonds formed between Ge2 and F atoms. The differences between the PDOSs of hydrogenated and fluorinated decorations are attributed to the considerably different electronegativities of the adatoms. The electronegativity is increased from H to Ge and F atoms. The Ge2 atoms are regarded as acceptors in the hydrogenated case of H–pGe–H nanostructure, whereas these become donors in the fluorinated case of F–pGe–F nanostructure. More importantly, the characters of VBM and CBM states, arising from different chemical functionalizations, can be modified by external tensile strains.

The external strain has been widely used to tune the electronic structure and thermal properties of materials,[52] and previous studies have shown that a topologically nontrivial phase can be achieved in the Ge-based 2D nanomaterials by tensile strain.[49] Thus, we investigate the effects of biaxial strain on the electronic structures of hydrogenated and fluorinated pGes and hope that the indirect–direct band gap transition can be obtained for these functionalized pGes by tensile strain. In this study, the biaxial strain is stimulated by varying the in-plane lattice to a series of values, which are larger/smaller than those of the equilibrium structure. The strain imposed on the structure is defined as ε = (a – a0)/a0, where a0 and a denote the lattice constants of the unstained and strained systems, respectively. Considering that the differences in the electronic structures obtained by PBE and HSE06 are concentrated on band gap values, we calculate the electronic structures at PBE level, as shown in Figure . Correspondingly, the HSE06 band gap values can be roughly estimated by scissor operation.

Figure 4

Effects of tensile strain on the total energies and electronic structures of the hydrogenated and fluorinated pGes. The electronic band structures of (a) the hydrogenated pGe and (d) the fluorinated pGe under different tensile strains. The band structures shown here are calculated by the PBE functional. Biaxial strain dependence of the band gap (Eg) of (b) the hydrogenated pGe and (e) the fluorinated pGe. The total energy and buckling distance (h) variations as a function of biaxial strain for (c) the hydrogenated pGe and (f) the fluorinated pGe, confirming that the unstrained nanostructure corresponds to the lowest total energy. Band structures based on the PBE functional under certain external tensile strains are listed in the figures.

As explicitly shown in Figure a, a direct band gap at the M-point can be obtained for the H–pGe–H 2D sheet when ε = 6% biaxial strain is applied. The presence of the indirect–direct semiconductor transition is favorable for its potential applications, and this phenomenon can be understood from the charge transfer between Ge2 and Ge1 atoms. In detail, once a tensile stretch is applied on the nanostructure, the Ge1–Ge2 bond length is increased from 2.479 to 2.553 Å when ε = 6% and the buckling distance (h) is also monotonously decreased. As a result, Ge1 has superior electronegativity compared to that of the Ge2 atoms and charge transfers from Ge2 to Ge1 atoms, which is confirmed by its CBM charge density and PDOS results shown in Figure S1. As the tensile stretch increased monotonously, the Ge1 atom’s induced energy level with the σpp* character is shifted downward at the M-point and the Ge2 atom’s induced energy level (at 1/2 of M−Γ path) with the σsp* character is shifted upward. Once the critical strain of ε = 6% is achieved, the σpp*-characterized state eventually becomes CBM to replace the original σsp* state, leading to the indirect–direct transition for the hydrogenated pGe. Generally, the band gap of H–pGe–H is decreased in a linear way with a continuous increase in stretch strain. When a compressing tensile is applied, conversely, the σpp*-characterized level is shifted upward and the σpp*-characterized one is shifted downward. As a consequence, the indirect band gap is reduced as a compressing strain is exerted on the H–pGe–H nanosheet, as shown in Figure b. For the 2D H–pGe–H nanosheet, it is interesting to find that not only its band gap can be tuned in a wide range of 1.25–1.92 eV (1.95–2.62 eV at HSE06 level according to scissor operation), but also the indirect to direct band gap transition can be obtained for this semiconductor. Most importantly, the strain range calculated in our study is generally achievable for experimental realization.[53] For instance, the 2D MoS2 nanomaterial can be subjected to an external strain of 11%.[54] Specifically, we noticed that the lattice constant of the zinc blende MgSe film is 5.89 Å, which can be served as a substrate for applying the epitaxial strain by molecular beam epitaxy.[55] Owing to the interaction between the film and the substrate via the weak van der Waals interactions, the biaxially strained H–pGe–H with ε = 6% can be realized similar to the 2D sheet grown on the BN substrate.[50,56] Therefore, the hydrogenated pGe is expected to have practical applications in the field of nanoelectronics.

However, in the case of fluorinated pGe, we found that the indirect band gap character is maintained at the tensile range of ε = −8 to 10%. The absence of the indirect–direct band gap transition, in sharp contrast to that in H–pGe–H, is attributed to the high electronegativity of the fluorine atoms. The electronic structure evolutions of F–pGe–F under stretch strain can be described in the following steps: (1) At equilibrium position, its indirect band gap is determined by the σpp-characterized VBM at the M-point and the σpp*-characterized CBM at about 2/5 of the M−Γ path. (2) As a small stretch strain (ε = 2%) is applied, the σpp*-characterized CBM is shifted upward. We stress that the σpp*-characterized CBM is robust to the biaxial strain because the F atoms always serve as acceptors and the Ge2 atoms here are donors. The external tensile strain, including both stretch and compressing, can effectively tune the Ge1–Ge2 coupling, whereas the σpp*-featured CBM is less influenced and only relative shift is observed, as shown in Figure S2b. Meanwhile, the Ge1 atom’s induced energy level at the X-point is lifted toward the Fermi level, which is due to the increasing charge density around the Ge1 atoms. (3) When the stretch strain is further increased, the σpp* state governed by the Ge1 atoms becomes the VBM as ε = 4% is satisfied, leading to the change in VBM location. This phenomenon can also be understood from the significant increase in the calculated PDOSs of the Ge1 atoms, as demonstrated in Figure S2. Thus, the indirect band gap is determined by the VBM at the X-point and the CBM located at the M−Γ path. (4) As the stretch strain is further increased (ε > 4%), the VBM at the X-point is found to be more pronouncedly influenced, in contrast to the robust CBM. Therefore, the band gap is reduced with increasing biaxial strain. Additionally, the zinc blende CdSe with lattice constant a = 6.05 Å can be a potential substrate for the experimental realization of ε = 6% strain.[57] On the other hand, when the nanosheet is subjected to a compressing strain, its CBM is influenced and moves toward the Fermi level. The band gap of F–pGe–F is found to monotonously decreased, as shown in Figure e. Overall, the electronic structure of F–pGe–F is shown to be effectively modulated by the external strain.

Furthermore, we have considered that the tetracoordinated Ge atoms are likely to be substituted by other group-IV elemental atoms. The aim is to understand the influences of incorporation of C/Si/Sn/Pb substitutional atoms on the pGe structure and its electronic structures. For simplicity, we focus only on the hydrogenated pGe with substitutional doping, and the two Ge1 atoms are replaced by the C, Si, Sn, and Pb atoms, respectively. Hence, the four doped configurations, as schematically shown in Figure , are represented as H–pGe–C–H, H–pGe–Si–H, H–pGe–Sn–H, and H–pGe–Pb–H. These four doped H–pGe–H nanostructures are optimized, at first, with their structural parameters summarized in the Table , and the corresponding electronic structures are then calculated to assess the effects of various dopings. In addition, it is found that the symmetry of the doped nanostructures is still preserved (space group no. 113). From the viewpoint of energy in terms of Eb, the four substitutional dopings are all energetically favorable. Moreover, the lattice constant (a), the Ge2–X bond length (R1), and the buckling distance (h) increased as the atomic number increased from C to Pb. Correspondingly, the σpp*-featured CBM substantially reduced as R1 increased from C to Pb, and eventually, it almost vanished for Pb doping. The band gaps of the doped system can be varied in a wide range (1.16–4.35 eV by hybrid functional calculation), as shown in Figure , and Si, Sn, and Pb dopings give rise to high electron mobility at the M-point. In addition, it is worth mentioning that the electronic bands along the M−Γ path are doubly degenerate as a result of symmetry protection by the space group. This double degeneracy can be broken by semidoping, that is, only one Ge1 atom is replaced by the C atom, and the expected results have been confirmed by our calculation shown in Figure S3b. Thus, the substitutional dopings can be utilized to modulate the electronic structures of the H–pGe–H 2D sheet.

Figure 5

Calculated electronic band structures based on the PBE/HSE06 functional of various doped H–pGe–H systems. (a) H–pGe–C–H, (b) H–pGe–Si–H, (c) H–pGe–Sn–H, and (d) H–pGe–Pb–H. The inset in each figure shows the optimized structure of each doped sheet. The Fermi level is set to zero and indicated by the gray dashed line. The inserted band gap values are calculated by the HSE06 functional.

Table 2

Structural Parameters and Energy Gaps of Various Doped H–pGe–H Nanostructures with Substitution X Consisting of C, Si, Sn, and Pba

conformation	R1 (Å)	R2 (Å)	R3 (Å)	θ1/θ2 (deg)	h (Å)	a (Å)	Eb	Eg-PBE	Eg-HSE06
H–pGe–C–H	2.01	2.45	1.56	107.5/108.3	1.11	4.62	0.52	3.46	4.35
H–pGe–Si–H	2.42	2.48	1.56	106.5/105.6	1.29	5.45	0.56	2.00	2.64
H–pGe–Sn–H	2.67	2.48	1.57	105.6/104.4	1.38	5.96	0.46	1.52	2.13
H–pGe–Pb–H	2.76	2.47	1.57	105.7/103.3	1.44	6.12	0.36	0.71	1.16

R1 (Ge2–X bond length in Å), R2 (Ge2–Ge2 bond length in Å), R3 (X–H bond length in Å), θ1/θ2 (Ge2–X–Ge2/X–Ge2–X angle in deg), h (buckling distance in Å), a (lattice constant in Å), Eb (binding energy in eV/per atom), Eg-PBE (in eV), and Eg-HSE06 (in eV).

Conclusions

In summary, we use first-principles calculations to study the structural stability and electronic properties of the pGe modulated by chemical functionalization. Although the pristine 2D pGe is energetically unfavorable, the functionalized pGes decorated by hydrogenation and fluorination are found to be dynamically and thermally stable. We have shown that the hydrogenated and fluorinated pGes are stable even at a high temperature of 500 K. Both hydrogenated and fluorinated-pentagonal germanenes are semiconductors with indirect band gaps of 1.92 and 1.39 eV (2.60 and 2.09 eV by the HSE06 functional), respectively. The electronic structures of the functionalized pGes can be effectively modulated by biaxial tensile strain. More importantly, our calculations reveal that the indirect to direct band gap transition can be achieved for the hydrogenated pGe sheet by 6% biaxial strain. Furthermore, we show that the band gap of H–pGe–H can be further tailored from 1.16 to 4.35 eV by heteroatom doping (C/Si/Sn/Pb). Therefore, the chemically functionalized pGes are expected to possess robust structural stability and excellent electronic properties, allowing Ge-based 2D nanosheets to have many potential applications in the future.

Computational Methods and Models

The first-principles calculations within density functional theory are utilized in the Vienna ab initio simulation package (VASP).[58,59] Projector-augmented wave pseudopotentials in the form of the PBE functional are used for electronic exchange-correlation potential.[60] The electronic wave functions of valence electrons are expanded using plane-wave basis sets with a cutoff energy of 500 eV. For structural optimizations, the exchange-correlation potential is described by the generalized gradient approximation. The PBE functional is used in most of our calculations,[60] and the hybrid HSE06 functional is also employed for more accurate electronic structure calculations.[61,62] The convergence criteria for total energy and force component are set to be 10–6 eV and 0.001 eV/Å. A vacuum layer of 30 Å is set in the plane normal direction to minimize image interactions. The Monkhorst–Pack set of 25 × 25 × 1 k-points for the 2D structures is used to sample the BZ for the geometry optimization, and a denser grid of 31 × 31 × 1 is used for electronic structure computations.[63] The phonon dispersions are calculated by density functional perturbation theory (DFPT) as implemented in the CASTEP code.[64,65] All AIMD simulations are performed in the canonical (NVT) ensemble via the VASP code. The temperature is controlled by the Nosé–Hoover algorithm at 300 and 500 K.[66] The equations of motion are integrated using the velocity Verlet algorithm.