A Tight-Binding Analysis of Models, Sheet Width, and Strain for 2D Monoatomic Germanium Sheets

Materials Express

A tight-binding analysis of models, sheet width, and strain for 2D monoatomic germanium sheets

Aashka S. Bhandari and Amit Verma∗ Electrical Engineering and Computer Science Department, Texas A&M University – Kingsville, Kingsville, Texas, 78363, United States Communication ABSTRACT 2D Germanium (Ge) has attracted significant research interest in the short time it was first synthesized and reported. There currently exist significant disagreements on the exact electronic structure of these novel 2D materials between theoretical calculations and experiment, particularly in terms of both the nature and value of the bandgap. In this work we report on a detailed tight-binding (TB) analysis of the electronic structure of Hydrogen (H) passivated 2D monoatomicIP: 192.168.39.151 Ge sheets On: Sat, (Germanene). 25 Sep 2021 We 07:16:34 also explore different sets of nearest tight-binding (TB) parameters—sp3Copyright:and sp3d American5s∗. Spin-orbit Scientific coupling Publishers is included in the calculations, and the effects of same-atom off-diagonal parametersDelivered are explored. by Ingenta Using the models, we also explore the effects of sheet-width (pseudo 1D structures or nanoribbons), axis orientation, and uniaxial strain. The physical structures for the unit-cells for the nanoribbons are determined after full-geometry optimization in the DFT package Gaussian. Therefore, our TB models account for the deviation from the idealized structures because of deformation perpendicular to the axis of these structures. All TB models are able to match the nature of the experimentally reported bandgap, while the more detailed sp3d5s∗ TB model provides a relatively close match with the experimentally reported bandgap value. We also propose a small modification of the existing Ge–H TB parameters to improve the accuracy of the bandgap value match. Keywords: Tight Binding Model, Nanosheets, Strain, Width, Band Structure, Bandgap.

1. INTRODUCTION semiconductor band structure with band gap of 0.99 eV Germanene has a hexagonal (honeycomb) structure and is (in the same work it was seen that fluorine (F) passi- composed of two vertically displaced lattices.1 In a ger- vated structure reported an indirect gap semiconductor manene sheet, a germanium atom is connected to 3 other with 0.19 eV band gap, and a direct gap semiconduc- germanium atoms to create a germanium nanosheet. The tor with 0.18 eV band gap for chlorine (Cl) passivated material has been of significant interest since it was first structure. 4 This work focuses on H-passivated sheets). reported.2 Density functional theory (DFT) calculations Another DFT study using the same GGA-PBE functional using Vienna ab initio package (VASP), show that ger- by Trivedi et al. using Atomistix Toolkit-Virtual Nanolab manene has a direct band gap of 1.53 eV3 and excep- (ATK-VNL) provided by Quantumwise, reported a direct tional electron mobility, while experimental result lead to band gap of 1.8 eV, 5 while a DFT study by Yoon et al. an indirect band gap of 1.59 eV.3 DFT computational using ABINIT code reported a direct band gap of 1.5 eV. 6 calculations by Liang et al. using VASP with GGA-PBE Wang et al. have reported a band gap of 0.32 eV for functional on H passivated sheet reported a direct gap bilayer germanene using GW approximation with VASP package.7 Table I summarizes the above results. ∗Author to whom correspondence should be addressed. Tight-binding (TB) method has been used extensively Email: [email protected] over the past several years to determine the electronic

Mater. Express, Vol. 9, No. 3, 2019 273 Materials Express A TB analysis of models, sheet width, and strain for 2D monoatomic germanium sheets Bhandari and Verma

Table I. Inﬁnite 2D germanene band gap results using different methods.

Band Band Methods Functional Package gap (eV) Ge termination gap type

Experimental – 159 H Indirect3 Theoretical PBE VASP 153 H Direct3 GGA-PBE VASP 099 H Direct4 GGA-PBE VASP 019 F Indirect4 GGA-PBE VASP 018 Cl Direct4 GGA-PBE ATK-VNL 18 H Direct5 LDA ABINIT code 15 H Direct6 GW VASP 032 – Direct7 Tight-binding sp3 (Hattori) This work. Codes 327 H Indirect model (this developed in work) MATLAB, and available upon request. sp3d5s∗ (Niquet) without d and s∗ H passivation & 1305 H Indirect without off-diagonal same-atom matrix elements sp3d5s∗ (Niquet) with d and s∗ H passivation & 148 H Indirect without off-diagonal same-atom matrix elements sp3d5s∗ (Niquet) without d and s∗ H passivation & 144 H Indirect with off-diagonal same-atom matrix elements sp3d5s∗ (Niquet) with d and s∗ H passivation & 1496 H Indirect with off-diagonal same-atom matrix elements sp3d5s∗ (Tan) without d and s∗ H passivation & 088 H Indirect without off-diagonal same-atom matrix elements sp3d5s∗ (Tan) with d and s∗ H passivation & 114 H Indirect without off-diagonal same-atom matrix elements sp3d5s∗ (Tan) without d and s∗ H passivation & 082 H Indirect with off-diagonal same-atom matrix elements sp3d5s∗ (Tan) withIP: d and 192.168.39.151 s∗ H passivation & On: with Sat, 25 Sep 2021 07:16:34 1065 H Indirect off-diagonal same-atomCopyright: matrix elements American Scientific Publishers Delivered by Ingenta

properties of different 2D materials. This has included has an error, and we selected the later in this work. The 8–10 work on graphene, silicene, and MoS2 sheets. In this effect of inclusion of off-diagonal same-atom matrix ele- work, we investigate the electronic structures of hydro- ments between two orbitals of the same Ge atom due to genated 2D monoatomic germanium sheets in different nearest neighbor Ge atoms is also investigated.12 15 configurations—infinite 2D sheets, infinite pseudo 1D The importance of treating surface passivation (Ge–H sheets (or nanoribbons)—TB method. This also includes

Communication in this case) accurately is well known and has been fur- 3 3 5 ∗ comparing the existing sp and sp d s TB models. The ther highlighted in recent works16 17 for different systems. effects of sheet widths, axis orientation, as well as uniaxial In this work, we suggest a modification of the Ge–H TB strain on the band structure are also investigated. parameters for the sp3d5s∗ models. These modified parameters were obtained through empirically matching the TB 2. CALCULATIONS band gap with the reported experimental band gap. Multiple parameters and aspects are investigated in this work. This arises from different TB parameters that have 3. RESULTS AND DISCUSSION been utilized over the years for Ge–Ge and Ge–H bonds. Figures 1(a), (b) shows the calculated band structure for One set of TB parameters for the Ge–Ge sp3 model was 11 a free-standing, ideal 2D infinite monoatomic Ge sheet given by Hattori et al. The other, for the more involved 3 5 ∗ 12 ∗ sp d s sp3d5s model, was given by Niquet et al. 12 Very recent using the TB model by Niquet et al., includ- sp3d5s∗ TB parameters were given by Tan et al.13 There ing spin-orbit coupling. Figure 1(c) shows the infinite Ge appears to be an agreement on the p-orbital spin-orbit cou- sheet where each Ge atom makes a bond with two other pling parameter,12 which is what we considered in the Ge atoms, forming a repetitive hexagonal structure. All sp3d5s∗ model. We also investigated two sets of parameters dangling bonds are H-terminated. Therefore, there are as for the Ge–H bond. One set of parameters14 gave disper- many Ge atoms as H atoms in the structure. The Ge atoms sion less bands very close to E = 0 eV for some of the are not all on the same plane, with the Ge–Ge bond angle structures, corresponding to the H atoms. The other set11 on the plane perpendicular to the sheet being 103 , while did not show such a behavior. Unless future experiments the bond-length taken was 2.44 Å.18 Most authors refer demonstrate otherwise, it is possible that the former set to this as buckling of the sheet.

274 Mater. Express, Vol. 9, 2019 A TB analysis of models, sheet width, and strain for 2D monoatomic germanium sheets Materials Express Bhandari and Verma

(a) (b)

Fig. 1. Band structure of free-standing,IP: ideal 192.168.39.151 2D inﬁnite monoatomic On:Ge Sat, sheet 25 using Sepsp 32021d5s∗ TB 07:16:34 model (a) without and (b) with off-diagonal same atom matrix elements and passivated d and s∗ Copyright:orbitals of Ge American atoms in the Ge–HScientific bonds, Publishers and (c) structure of ideal 2D inﬁnite monoatomic Ge sheet, where the grey colored atom is Ge and white colored atom isDelivered H; the rectangle by Ingenta shows the unit cell used.

The germanene sheet without the off-diagonal same Table II. Ge–H TB parameters. atom matrix elements and unpassivated d and s∗ orbitals 2009 Niquet TB 2016 Tan TB of Ge atoms in the Ge–H bonds shows an indirect band Parameters Parameter values (eV) Parameter values (eV) gap of 1.30557 eV, with a partial band structure depicted E − − ss (from Ref. [11]) 3 29 3 29 in Figure 1(a). The indirect nature of the band gap was E sp (from Ref. [11]) 2 66 2 66 also observed in the reported experimental work, with a E − − sd (this work) 8 01 6 1 3 E − − band gap of 1.59 eV. The TB bandgap value increased sS (this work) 3 11 3 51 to 1.44 eV when off-diagonal same-atom matrix elements were included in the calculations. The difference between the two results, with and without same-atom off-diagonal matrix elements, is consistent between, for the largest reported so far, but within margins of estimated example, what has been computed by Niquet et al.,12 and expected band gap in some reported DFT works dis- and other theoretical and experimental works in bulk cussed above based on an underestimation of DFT com- Ge system.19–21 Using the same approach and utilizing puted band gaps. 5 6 Given the wide difference between the most recently reported Ge sp3d5s∗ TB parameters,13 the computed results with this model, and the reported which were obtained through fitting with reported ab ini- experimental result, the reported Ge sp3 TB parameters tio results, we obtained an indirect band gap of 0.88 eV, appear to be unsuitable for the 2D sheets. without the inclusion of off-diagonal same-atom matrix DFT results, on the other hand, as depicted in Table I elements. This difference may be attributed to the later TB have consistently shown a direct band gap. This under- parameters parametrized with hybrid functional calcula- scores the need for further experimental work to verify tions and fitted to reproduce results for superlattices under the theoretical models. Once ab initio models have been various strained conditions, while the Niquet parameters verified, it is relatively simple to improve the accuracy of have found broader usage. the existing TB models since a significant number of TB The sp3 TB model in our work for the same sheet gave parameters are obtained after fitting the TB band structures an indirect band gap of 3.27 eV. This is clearly larger than with DFT band structures.13 16 17

Mater. Express, Vol. 9, 2019 275 Materials Express A TB analysis of models, sheet width, and strain for 2D monoatomic germanium sheets Bhandari and Verma

In order to empirically match the band gap we obtained gap. However, using the 2016 Tan TB parameters, and for the sp3d5s∗ cases with the experimentally determined again focusing on the d and s∗ orbitals for Ge in the Ge–H band gap, we focused on the d and s∗ orbitals for the bonds, the largest band gap we could obtain was 1.065 eV Ge–H bonds, while keeping the other reported TB param- when same-atom off-diagonal parameters were included. eters as the same. This is because the existing parameters This is significantly smaller than the experimentally deter- have been extensively utilized, and we try to avoid mak- mined band gap. Table II lists the Ge–H TB parameters ing significant changes in the scenario that currently exists. for the two schemes. To the best of our knowledge, there have not been any The TB calculations were extended to germanene sheets reported TB parameters for the d and s∗ Ge–H bonds till confined to one direction—i.e., nanoribbons. Three spe- date. Doing so resulted in an indirect band gap of 1.496 eV cific cases were considered—a nanoribbon approximately as shown in Figure 1(b) for the 2009 reported Niquet 0.77 nm wide, and a nanoribbon 3.85 nm wide, [1120]¯ axi- TB parameters. In this case, the off-diagonal same-atom ally aligned. In the third case, the nanoribbon was [1010]¯ matrix elements were included and d and s∗ orbitals of Ge axially aligned, with a width of 1.08 nm. sp3d5s∗ (Niquet) atoms in the Ge–H bonds were passivated. This band gap TB models were considered, with spin-orbit coupling is approximately 6% smaller than the experimental band utilized and without same atom off-diagonal elements.

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Fig. 2. Unit cells for (a) 0.77 nm [1120]¯ axially aligned, (b) 3.85 nm [1120]¯ axially aligned, and (c) 1.08 nm [1010]¯ axially aligned sheets, respectively.

276 Mater. Express, Vol. 9, 2019 A TB analysis of models, sheet width, and strain for 2D monoatomic germanium sheets Materials Express Bhandari and Verma

(a) (b)

Fig. 3. Band structure of optimized (a) 0.77 nm [1120] axially aligned and (b) 3.85 nm [1120]¯ axially aligned sheets respectively for the sp3d5s∗ model using spin-orbit coupling.

Geometry optimization was performed on structures using Density Functional Theory (DFT) method and 6-31G as a gradient minimization technique to ﬁnd the stable atomic basis set, and spin multiplicity considered as singlet. arrangement; the minima being characterized by having Figures 2(a)–(c) shows the unit cells for 3 cases after Communication zero gradient norms, and by diagonalizing the matrix of optimization; (a) 0.77 nm wide [1120]¯ axially aligned the second derivatives that give positive harmonic vibra- nanoribbon, with 13 Ge atoms and 22 H atoms, and a tional frequencies.22 The optimization was performed unit cell length along the axis of approximately 0.83 nm; in Gaussian 0923 using B3LYP24 25 functional as the (b) 3.85 nm wide [1120]¯ axially aligned nanoribbon, with

IP: 192.168.39.151 On: Sat, 25 Sep 2021 07:16:34 (a)Copyright: American (b)Scientific Publishers Delivered by Ingenta

Fig. 4. Band structure of optimized 3.85 nm [1120]¯ axially aligned sheets for the sp3d5s∗ model using spin-orbit coupling with (a) 5%, (b) 10%, (c) −5%, and (d) −10% strain, respectively.

Mater. Express, Vol. 9, 2019 277 Materials Express A TB analysis of models, sheet width, and strain for 2D monoatomic germanium sheets Bhandari and Verma

120

100

40 unstrained sheet 20

% of states strained sheet compared to –10–8–6–4–20246810 Strain applied (%)

Fig. 5. % of total states of strained sheet compared to unstrained optimized 3.85 nm [1120]¯ axially aligned sheet for the sp3d5s∗ model using spin-orbit coupling.

61 Ge atoms and 86 H atoms, and a unit cell length along depends on bond-lengths of the nearest neighbor parame- V d V d v the axis of approximately 0.83 nm; and (c) 1.08 nm wide ters, v and v 0 where and are two atomic [1010]¯ axially aligned nanoribbon, with 14 Ge atoms and orbitals on different atoms, which is given by the equation 24 H atoms, and a unit cell length along the axis of approx- below:12 d n imately 0.32 nm. In performing geometry optimization, in V d = V d 0 v v 0 each case, 5 unit cells were included, with the middle unit d cell incorporated in the TB calculations. As has been pre- d IP: 192.168.39.151 On: Sat,where 25 Sep 2021is the 07:16:34 distance between the two atoms in viously observed, finite germanene sheets show deforma- d Copyright: American Scientificthe strained Publishers crystal and 0 is the equilibrium distance. tion, resulting in deviations from the ideal 2D germaneneDelivered byThe Ingenta general trend in nanomaterials, particularly nanowires 26 structures. The deviations of the atomic positions from and nanotubes, has been one of decreasing band gap with the ideal structure, perpendicular to the sheet axis, were an increase in size. The results here, for two different included in our calculations (the deformation along the orientations, show no visible band gap, while the infinite infinitely long axial direction will not be present because germanene sheet does. This may likely be attributable to of the absence of a center point along an infinite axis). The the dispersion relation being strongly influenced by edge- effects of uniaxial strain, both tensile and compressive, up states and edge-reconstructions in such narrow structures, to 10% were also calculated. 27 Communication a phenomenon known in graphene nanoribbons. The ∗ In the sp3d5s TB model with spin-orbit coupling, the results may potentially also suggest the existence of a ¯ 0.77 nm wide [1120] axially aligned nanoribbon had a TB threshold sheet width above which germanene behaves as ¯ matrix size of 186 × 186, while the 3.85 nm wide [1120] a semiconductor, and below which it displays a metal- axially aligned nanoribbon had a matrix size of 858 × lic behavior. These possibilities will need determination ¯ 858. The 1.08 nm wide [1010] axially aligned nanoribbon through accurate DFT calculations and further experi- had a matrix size of 198 × 198. (Correspondingly, in the ments. The one consistent result for the size-limited sheets 3 ¯ sp TB model, the 0.77 nm wide [1120] axially aligned was the change in the total number of available states nanoribbon will have a TB matrix size of 69 × 69, while with the strain percent for energy values close to, and ¯ the 3.85 nm wide [1120] axially aligned nanoribbon will around 0 eV—Figure 5. The energy range considered was have a matrix size of 309 × 309). from −0.8 eV to 0.8 eV, with an energy grid spacing of The case for the more involved sp3d5s∗ Niquet TB 0.032 eV. Figure 5 points to a potential reduction in con- model with spin-orbit coupling actually shows no observ- ductance with strain. able band gap for both the structures as can be seen Many TB based studies over the years have incorporated in Figures 3(a), (b), as well as for the [1010]¯ axially the effects of strain not only through the power-law above, aligned 1.08 nm wide structure. In fact, under both positive but additionally also through the incorporation of on-site and negative uniaxial strains (dilational and compression, strain parameters (see for example Ref. [12]). The use of respectively), up to 10%, no observed band gap appeared, on-site parameters has shown to provide corrections to the as can be seen in Figures 4(a)–(d). The strain was applied results, where the corrections have been relatively small along the Y axis by scaling the two-center integrals using but nevertheless important. These on-site strain TB param- Harrison’s d−2 law. The effects of strain in this model eters, such as in Ref. [12] have been determined through

278 Mater. Express, Vol. 9, 2019 A TB analysis of models, sheet width, and strain for 2D monoatomic germanium sheets Materials Express Bhandari and Verma an empirical fit with DFT results. However, given the dis- 7. X. Wang and Z. Wu; Intrinsic magnetism and spontaneous band agreement among the reported studies in the case of ger- gap opening in bilayer silecene and germanene; Phys. Chem. Chem. manene, determination and utilization of existing on-site Phys., 19, 2148 (2017). 8. A. R. Botello-Mendez, J. C. Obeso-Jureidini, and G. G. Naumis; parameters should be undertaken with caution. We believe Toward an accurate tight-binding model of graphene’s electronic that the overall trend of decreasing total states with strain properties under strain; J. Phys. Chem. C 122, 15753 (2018). in Figure 5 may qualitatively be predictive of what may 9. C.-H. Wu; Tight-binding model and ab initio calculation of silicene be observed. This qualitative aspect may not change with with strong spin-orbit coupling in low-energy limit (2018). the inclusion of on-site strain effects. 10. F. Zahid, L. Liu, Y. Zhu, J. Wang, and H. Guo; A generic tight- binding model for monolayer, bilayer and bulk MoS2; AIP Advances 3, 052111 (2013). 4. CONCLUSION 11. A. Hattori, S. Tanaya, K. Yada, M. Araidai, M. Sato, Y. Hatsugai, K. Shiraishi, and Y. Tanaka; Edge states of hydrogen terminated A thorough tight-binding (TB) computation effort was monolayer materials: Silicene, germanene and stanene ribbons; undertaken to investigate the electronic structure of hydro- J. Phys.: Cond. Matter. 29, 115302 (2017). gen (H) passivated monoatomic germanium (Ge) sheets 12. Y. Niquet, D. Rideau, C. Tavernier, H. Jaouen, and X. Blasé; Onsite (germanene)—both infinite 2D, as well as pseudo-1D, with matrix elements of the tight-binding Hamiltonian of a strained crystal: Application to silicon, germanium, and their alloys; Phys. Rev. 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Received: 30 January 2019. Revised/Accepted: 30 March 2019.

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