Structural and electronic properties of hydrogen-functionalized
armchair germanene nanoribbons: A first-principles study
Nguyen Thanh Phuong1 and Nguyen Duy Khanh1,*
1

Information Technology Center, Thu Dau Mot University, Binh Duong Province, Vietnam
*

Corresponding at

Abstract
Structural and electronic properties of hydrogen-functionalized armchair germanene nanoribbons
(AGeNR) are investigated using the first-principles calculations. The critical physical quantities
to analyze the structural and electronic properties are fully developed through the first-principles
calculations, including the functionalization energy, optimal structural parameters, orbital- and
atom-decomposed electronic band structures and density of states, charge density, and charge
density difference. Under hydrogen functionalization, the functionalization energy is achieved
around -2.59 eV, and the structural parameters are slightly distorted as compared to the pristine
system. This evidences for good structural stability of the functionalized system. Besides, the
very strong H-Ge bonds are created by the strong charge transfer of electrons from Ge atoms to
H atoms that generates free holes in the functionalized system, which can be considered as ptype doping. As a result, the π bonds of Ge-4pz orbitals at low-lying energy are fully terminated
by the strong H-Ge covalent bonds, in which the strong hybridizations of H-1s and Ge-(4s, 4px,
4py, and 4pz) orbitals are occurred at deep valence band. The termination of π bonds leads to the
opened bandgap of 2.01 eV in the hydrogen-functionalized AGeNR that belongs to the p-type
semiconductor. The feature-rich electronic properties of the hydrogen-functionalized AGeNR
identify that the hydrogen-functionalized AGeNR will be the very potential 1D semiconductor
for high-performance optoelectronic applications.
Keywords: graphene, germanene, armchair germanene nanoribbons, hydrogen functionalization,
DFT calculation, band structure, and charge transfer.

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TĨM TẮT
Các đặc tính cấu trúc và điện tử của các dải nano germanene cạnh ghế bành chức hóa hydro
(AGeNR) được nghiên cứu bằng cách sử dụng các tính tốn nguyên lý đầu tiên. Các đại lượng
vật lý quan trọng để phân tích các đặc tính cấu trúc và điện tử được phát triển đầy đủ thơng qua
các tính tốn nguyên lý đầu tiên, bao gồm năng lượng chức năng hóa, các thơng số cấu trúc tối
ưu, cấu trúc vùng điện tử phân tách theo quỹ đạo và nguyên tử và mật độ trạng thái, mật độ điện
tích và sai khác mật độ điện tích. Trong q trình chức hóa hydro, năng lượng chức năng hóa đạt
được khoảng -2,59 eV, và các thông số cấu trúc bị biến dạng rất ít so với hệ nguyên sơ. Đây là
bằng chứng cho sự ổn định cấu trúc tốt của hệ thống được chức năng hóa. Bên cạnh đó, các liên
kết H-Ge rất bền được tạo ra do sự chuyển điện tích mạnh của các electron từ nguyên tử Ge sang
nguyên tử H tạo ra các lỗ trống tự do trong hệ cơ năng, có thể được coi là sự pha tạp loại p. Kết
quả là, các liên kết π của các obitan Ge-4pz ở năng lượng thấp bị kết thúc hoàn toàn bằng các liên
kết cộng hóa trị mạnh H-Ge, trong đó các liên kết lai hóa mạnh của H-1s và Ge- (4s,4px, 4py và
4pz) các quỹ đạo xuất hiện ở vùng hóa trị sâu. Sự kết thúc của liên kết π dẫn đến độ rộng vùng
cấm mở ra là 2,01 eV trong AGeNR chức hydro thuộc bán dẫn loại p. Các đặc tính điện tử phong
phú của AGeNR chức năng hydro xác định rằng AGeNR chức năng hydro sẽ là chất bán dẫn 1D
rất tiềm năng cho các ứng dụng quang điện tử hiệu suất cao.
Từ khóa: graphene, germanene, dải nano germanene cạnh ghế bành, chức năng hóa hydro, tính
tốn DFT, cấu trúc dải vùng điện tử và chuyển điện tích.

1. Introduction
Since the first two-dimensional (2D) graphene monolayer made of carbon elements
arranging in a planar hexagonal lattice has been successfully synthesized by Novoselov et al
through the top-down approach in 2004 [1], it strongly motivates for many studies of graphenelike 2D materials owing to its novel and unique properties to significantly enhance performance
for applications as compaered with the traditional bulk materials [2-5]. As a close anolog of
graphene, germanene made of germanium (Ge) elements arranging in a low-buckled hexagonal
lattice has attracted plenty of efforts because the Ge constituents have a good compatibility with
the silicon elements in the current semiconducting industry and the low-buckled structure of

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germanene also possesses better stability than the planar graphene in devices [6-8]. However, the
zero-gap feature of 2D germanene is a critical drawback that prevents great potential of
germanene for electronic applications. Thus, opening bandgap for germanene is an essential
issue for practical applications that has recently drawn much attention in scientific community
[9]. Various approaches have been utilized to open bandgap for germanene, including
functionalization [10], adsorption [11], substitution [12], inducing defects [13], forming stacking
configurations [14], applying external electric or magnetic fields [15], and creating finte-size
confinements [16]. Among these methods, the finite-size confinements of the 2D germanene
resulting in the one-dimensional (1D) germanene that can enhance bandgap without any serious
deformations in geometries is the very effective way to open bandgap for germanene.
This 1D germanene is termed as germanene nanoribbons (GeNRs), in which the different
edge terminations can create two typical germanene nanoribbons are armchair (AGeNR) and
zigzag (ZGeNR) germanene nanoribbons [17, 18]. It should be noted that the bandgaps of
GeNRs strongly depend on its widths, and the different edge configurations exhibit different
electronic and magnetic behaviors. The AGeNR presents the direct bandgap at band-edge states
and exhibits non-magnetic states. Meanwhile, the ZGeNR displays the direct bandgap at far
band-edge states and presents the anti-ferromagnetic states across the edges, in which each
zigzag edge presents opposite ferromagnetic states. Nevertheless, the opened bandgaps of
GeNRs are too narrow to have a good compatibility with the optoelectronic applications that
needs bandgaps larger than 0.7 eV [19]. Thus, enhancing the bandgap of GeNRs is an essential
topic for electronic and optoelectronic applications that has interested in many recent studies.
Many methods to enhance bandgap for GeNRs has been studied, including the inducing defects
[20], external fields [21], stacking configurations [22], atom dopings [23], and edge or surface
functionalizations [24, 25]. Among these methods, the surface functionalizations by hydrogen
atoms can significantly enhance bandgap of GeNR with good structural stability owing to strong
H-Ge bonds that is worthy for a detailed investigation. In this work, the structural and electronic
properties of hydrogen-functionalized AGeNRs are thoroughly studied using the first-principles

calculations. Through the first-principles calculations, a generalized theoretical framework to
determine the studying properties are fully developed, including the functionalization energies,
optimal lattice parameters, atom- and orbital-projected electronic band structures and density of
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states, charge density, and charge denstiy difference. The developed first-principles theoretical
framework can be fully generalized to other functionalization systems.

2. Computational detials
Structural and electronic properties of hydrogen-functionalized armchair germanene
nanoribbons are investigated using the density functional theory (DFT) method, implemented in
Vienna Ab Initio Simulation Package (VASP) [26]. In VASP calculations, the electron-electron
Coulomb interactions coming from the many-body exchange and correlation energies are
calculated using the Perdew-Burke-Ernzerhof (PBE) functional under the generalized gradient
approximation [27]. The intrinsic electron-ion interactions are evaluated by the projectoraugmented wave (PAW) pseudopotentials. As to the complete set of plane waves, the kinetic
energy cutoff is set to be 500 eV, being suitable for evaluating Bloch wave functions and
electronic energy spectra. A vacuum space of 20 Å is used to create the free-standing monolayer.
The first Brillouin zone is sampled by 1×1×12 and 1×1×100 k-point meshes within the
Monkhorst-Pack scheme for geometric optimizations and electronic structure calculations,
respectively. The energy convergence is equal to 10-5 eV between two consecutive steps, and the
maximum Hellman-Feynman force acting on each atom is less than 0.01 eV/Å during the ionic
relaxations.

3. Results and Discussions
3.1. Structural properties
The atomic structures of pristine and hydrogen-functionalized armchair germanene
nanoribbons (AGeNR) displayed in top-view and side-view are shown in Figs. 1(a) and 1(b),
respectively, in which the ribbon width of six dimer lines is used in the calculations. The
dangling bonds along armchair edges are eliminated by passivation of the hydrogen atoms at the

edges. Under the optimal calculations, the hydrogen atoms are flavorably functionalized at the
top sites of AGeNR among other sites, and the only double-side functionalization of AGeNR can
lead to the stable structure that is evaluated by the functionalized energy (E func), while the singleside functionalization generates the unstable structure. The Efunc is calculated as follows:
E func = ( Etot − E pris − nEH ) / n

(1)

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Whereas Etot, Epris, and EH are the ground-state energy of the hydrogen-functionalized AGeN,
pristine AGeNR, and isolated hydrogen atoms, respectively; and n is the total number of
functionalized hydrogen atoms. The calculated Efunc is valued at -2.59 eV as shown in Table 1.
This Efunc value is large enough to form the stable functionalized structure. Under the double-side
functionalization effect, it creates very short H-Ge bond lengths of 1.55 Å that is very shorter
than Ge-Ge bond lengths of 2.37 Å shown in Table 1. This means that the generated H-Ge
covalent bonds are very stronger than the Ge-Ge bonds. Due to effect of finite-size termination,
the Ge-Ge bond lengths near edges (1st Ge Ge) are larger than the Ge-Ge bond lengths far edges
(2nd Ge-Ge) as shown in Table 1, in which the 1st Ge-Ge of 2.39 Å and 2.37 Å and the 2nd GeGe of 2.36 Å and 2.35 Å correspond to the pristine and H-functionalized AGeNRs. Also, it can
be identified that the Ge-Ge bond lengths of the H-functionalized system are slightly shorter than
that of the pristine system. The buckling height of 0.91 Å in the H-functionalized system is
shorter than that of 1.08 Å in the pristine system, confirming that the buckling is reduced under
the H functionalization. The shorter bond lengths and buckling result in the larger Ge-Ge-Ge
bond angle of 114.25 9 (˚) in the H-functionalized system. This implies that the double-side H
functionalization can create better symmetric structure. It should be mentioned that the pristine
AGeNR exhibits a mix of sp2/sp3 hybridization in buckled Ge-Ge bonds, in which it exists the
strong bonded σ network of Ge-(4s, 4px, and 4py) orbitals and the weak π bonds of Ge-4pz
orbitals. Such weak π bonds are fully terminated under the double-side H functionalization that
creates the hybridization mechanism of 1s-sp3 in H-Ge bonds.
Table 1: Functionalization energy [Efunc(eV)], Ge-Ge bond lengths near edges [1st Ge-Ge] and far edges

[2nd Ge-Ge], H-Ge bond length (Å), buckling height (Å), Ge-Ge-Ge bonds (˚), and bandgap [Eg (eV)] of
the pristine and hydrogen-fucntionalized armchair germanene nanoribbons at the with of six dimer lines.

Configurations

Efunc (eV)

pristine 7AGeNR
H-6AGeNR

X
-2.59360

1st Ge-Ge
(Å)
2.396
2.378

2nd Ge-Ge
(Å)
2.362
2.355

79

H-Ge
(Å)
X
1.55


Ge-Ge-Ge
angle (˚)
107.74
114.25

Buckling
(Å)
1.087
0.919

Eg (eV)
0.23
2.01


Figure 1: Atomic models of the pristine AGeNR and hydrogen-functionalized AGeNR shown in top-view
and side-view.

3.2.

Electronic properties

The 1D electronic band structures of the pristine and H-functionalized AGeNR systems
are presented in Fig. 2, in which the Fermi level is set at the zero energy to determine the energy
gaps and electronic states illustrated by short dot black lines. As for the band structure of the
pristine AGeNR in Fig. 2(a), it presents a direct bandgap of 0.23 eV (shown in Table 1) that is
determined by the highest occupied vanlence band and lowest unoccupied conduction band at Г
point. All 1D subbands are asymmetric from the Fermi level, and they belong to the anti-crossing
weakly dispersed bands. Different dominations of Ge-orbitals in 1D subbands are identified by
the orbital-decomposed band structure. Specifically, Ge-4pz orbitals shown by red circles in Fig.

2(a) significantly dominate at the long-range energies near the Fermi level from -2.6 eV to 2.6
eV that indicates for long-range π bands, in which the co-domination of Ge-4pz and Ge-(4px+4py)
orbitals shown by the blue circles exists at the valence band from -2.6 eV to the Fermi level, and
the strongest domination of Ge-4pz is from the lowest unoccupied conduction band to 2.6 eV.
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From the middle valence band of -2.6 eV to deeper range energies, the Ge-(4px+4py) orbitals are
strongly dominated and co-dominated with the Ge-4s orbital shown by the green circles at the
deepest range energies, whereas the domination of Ge-4pz orbitals is disappeared. It should be
noted that the co-domination of Ge-4s and Ge-(4px+4py) orbitals implies for sp2 hybridization
and the co-domination of the Ge-4s, Ge-(4px+4py), and Ge-4pz orbitals illustrates for the sp3
hybridizaton. This can clarify that the hybridization mechanism in AGeNR is a mix of sp2/sp3,
which is responsile for the weak separation of σ and π bands. Under the hydrogen
functionalization, the 1D band structure of the pristine AGeNR is dramatically changed, as
shown in Fig. 2(a), in which the bandgap is much enlarged at 2.01 eV. The Ge atoms fully
dominate at long-range energies from -3.2 eV to 4 eV illustrated by pink circles in Fig. 2(a),
which determine the opened bandgap, while the hydrogen atoms strongly dominate at deep
valence energies from -3.2 eV to -5 eV shown by the cyan circles. This is due to that the H-Ge
bond lengths are very shorter than the Ge-Ge bond lengths.

Figure 2: Orbital- and atom-projected band structures of (a) pristine AGeNR and (b) hydrogenfunctionalized AGeNR.

81


The orbital-projected density of states (DOSs) is utilized to verify all main features of the
electronic band structures. The DOSs of the pristine and H-functionalized AGeNRs are presented
in Figs. 3(a) and 3(b), respectively, in which the Fermi energy is illustrated by the black short dot
lines. As for the DOSs of the pristine AGeNR shown in Fig. 3(a), the vacant region between the

highest occupied valence and lowest unoccupied conduction peaks is responsible for the energy
gap. The dominant peaks made of Ge-4pz orbitals (red curves)) at long-range energies near the
Fermi level are resulted from the long-range π bands, in which the 4pz peaks are stronger than
the peaks made of Ge-4s (green curves) and Ge-(4px+4py) orbitals (blue curves) above the Fermi
level, while the Ge-(4px+4py) peaks become more dominant than the 3pz peaks below the Fermi
level; however, the peaks of Ge-4s, Ge-4pz, Ge-(4px+4py) orbitals are simutaneously appeared in
the long-range π regions. This confirms for the sp3 hybridization. Below the middle valence
energy of -2.6 eV, the 3pz peaks are almost disappeared, and there only exist the merged peaks of
Ge-4s and Ge-(4px+4py) orbitals, whereas the second one is more dominant than the first one,
indicating for the sp2 hybridization. In the whole energy range, it can show that there is a mix of
sp2/sp3 hybridizations. Under the H functionalization, the DOSs is fully reshaped as shown in
Fig. 3(b). The appearance of the strong H-Ge bonds creates the very strong peaks at the deep
valence band merged by the H-1s (cyan shot dot curves) and Ge-4s (green curves), Ge- 4px+4py
(blue curves), and 4pz (red curves), in which the H-1s peaks are stronger than the other peaks.
This illustrates for the hybridization of 1s-sp3 in H-Ge bonds. The π peaks at long-range energies
in Fig. 3(a) are fully destroyed due full termination of π bonds in Fig. 3(b), whereas the Ge-4s
and Ge-(4px+4py) peaks fully dominate in the long-range energies.
The bonding magnitute and hybridization mechanism are verified through the charge
density distribution. The charge density distribution of the pristine and H-functionalized
AGeNRs is presented in Figs. 4(a) and 4(b), respectively. The highest and lowest charge
densitties are used to display for strongest and weakest bonds, as illustrated by the red and blue
colors, respectively. As for the charge density of the pristine AGeNR in Fig. 4(a), the charge
density is strongly distributed between two Ge atoms in (x,y) plane illustrated by the red region
that is due to the strong σ Ge-Ge bonds. Meanwhile, the charge density is much lower along z
direction displayed by the green region between two Ge atoms that is owing to the weak π Ge-Ge
82


bonds. From the clear information in the charge density, it can be mentioned that the σ bonds are
very stronger than the π bonds, and the strong σ bonds form the stable monolayer structure. The

H functionalization causes much change in the charge density as shown in Fig. 4(b), in which the
π bonds of Ge-Ge are fully terminated by forming in the very strong H-Ge covalent bonds, and
the H-Ge bonds (dark red region) are stronger than the Ge-Ge bonds since the H-Ge bond lengths
are very shorter than the Ge-Ge bonds as indentified in Table 1, while the σ bonds of Ge-Ge in
the H-functionalized system are enhanced as compared with the pristine σ bonds owing to the
Ge-Ge bond lengths shortened under the H-functionalization as identified in Table 1. To observe
the charge transfer mechanism, the charge density difference of the H-functionalized AGeNR is
presented in Fig. 4(c), whereas the red and blue regions are responsible for vanished and gained
electrons, respectively. This indicates that the electrons are transferred from Ge atoms to H
atoms to generate the very strong H-Ge bonds. This charge transfer process creates the free holes
in the functionalized system that can be regarded as the p-type doping.

Figure 3: Orbital-projected density of states of (a) pristine AGeNR and (b) hydrogen-functionalized
AGeNR.

83


Figure 4: Charge density distributions of (a) pristine AGeNR and (b) hydrogen-functionalized AGeNR;
and charge density difference distribution of (c) hydrogen-functionalized AGeNR.

4. Conclusion
The structural and electronic properties of the pristine and H-functionalized AGeNRs are
fully revealed in the physical quantities developed under the DFT calculations, including the
functionalization energies, optimal structural parameters, atom- and orbital-decomposed
electronic band structures, density of states (DOSs), charge density, and charge density
difference. The calculated functionalization energies demonstrate that the H functionalization
can create the stable 1D structure. The created H-Ge bonds are very stronger than the Ge-Ge
bonds that fully terminate the π bonds and slightly distort the σ bonds. The formation of strong
H-Ge bonds is due to the transfer of electrons from Ge atoms to H adatoms that create the 1s-sp3

hybridization mechanism. A a close relationship, the significantly change in geometric structure
in the H-functionalized system results in their enriched electronic properties. The bandgap of
2.01 eV is opended in the H-functionalized AGeNR, and this bandgap is determined the σ
orbitals of Ge atoms. The enriched properties under the H functionalization effect are very
potential for optoelectronic applications that requires bandgap larger than 0.7 eV.

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Acknowledgments
This research is funded by Thu Dau Mot University, Binh Duong Province, Vietnam under grant
number DA.21.1-003, and this research used resources of the high-performance computer cluster
(HPCC) at Thu Dau Mot University, Binh Duong Province, Vietnam.
Reference
[1] Novoselov, K. S., Geim, A. K., Morozov, S. V., Jiang, D. E., Zhang, Y., Dubonos, S. V &
Firsov, A. A. (2004). Electric field effect in atomically thin carbon films. Science, 306(5696),
666-669.
[2] Nguyen, D. K., Tran, N. T. T., Chiu, Y. H., Gumbs, G., & Lin, M. F. (2020). Rich essential
properties of Si-doped graphene. Scientific Reports, 10(1), 1-16.
[3] Nguyen, D. K., Tran, N. T. T., Chiu, Y. H., & Lin, M. F. (2019). Concentration-diversified
magnetic and electronic properties of halogen-adsorbed silicene. Scientific Reports, 9(1), 1-15.
[4] Acun, A., Zhang, L., Bampoulis, P., Farmanbar, M. V., van Houselt, A., Rudenko, A. N &
Zandvliet, H. J. (2015). Germanene: the germanium analogue of graphene. Journal of Physics:
Condensed Matter, 27(44), 443002.
[5] Balendhran, S., Walia, S., Nili, H., Sriram, S., & Bhaskaran, M. (2015). Elemental analogues
of graphene: silicene, germanene, stanene, and phosphorene. Small, 11(6), 640-652.
[6] Nijamudheen, A., Bhattacharjee, R., Choudhury, S., & Datta, A. (2015). Electronic and
chemical properties of germanene: the crucial role of buckling. The Journal of Physical
Chemistry C, 119(7), 3802-3809.
[7] Zhang, L., Bampoulis, P., Rudenko, A. N., Yao, Q. V., Van Houselt, A., Poelsema, B., &

Zandvliet, H. J. W. (2016). Structural and electronic properties of germanene on MoS2. Physical
Review Letters, 116(25), 256804.
[8] Kaloni, T. P., & Schwingenschlögl, U. (2013). Stability of germanene under tensile
strain. Chemical Physics Letters, 583, 137-140.
[9] Yao, Q., Zhang, L., Kabanov, N. S., Rudenko, A. N., Arjmand, T., Rahimpour Soleimani, H.,
& Zandvliet, H. J. W. (2018). Bandgap opening in hydrogenated germanene. Applied Physics
Letters, 112(17), 171607.
[10] Pang, Q., Li, L., Zhang, L. L., Zhang, C. L., & Song, Y. L. (2015). Functionalization of
germanene by metal atoms adsorption: a first-principles study. Canadian Journal of
Physics, 93(11), 1310-1318.
[11] Hoat, D. M., Nguyen, D. K., Ponce-Pérez, R., Guerrero-Sanchez, J., Van On, V., RivasSilva, J. F., & Cocoletzi, G. H. (2021). Opening the germanene monolayer band gap using
85


halogen atoms: An efficient approach studied by first-principles calculations. Applied Surface
Science, 551, 149318.
[12] He, J., Liu, G., Wei, L., & Li, X. (2021). Effect of Al doping on the electronic structure and
optical properties of germanene. Molecular Physics, e2008540.
[13] Monshi, M. M., Aghaei, S. M., & Calizo, I. (2017). Doping and defect-induced germanene:
A superior media for sensing H2S, SO2, and CO2 gas molecules. Surface Science, 665, 96-102.
[14] Qin, Z., Pan, J., Lu, S., Shao, Y., Wang, Y., Du, S., & Cao, G. (2017). Direct evidence of
Dirac signature in bilayer germanene islands on Cu (111). Advanced Materials, 29(13), 1606046.
[15] Hattori, A., Yada, K., Araidai, M., Sato, M., Shiraishi, K., & Tanaka, Y. (2019). Influence
of edge magnetization and electric fields on zigzag silicene, germanene and stanene
nanoribbons. Journal of Physics: Condensed Matter, 31(10), 105302.
[16] Sharma, V., & Srivastava, P. (2021). Silicene and Germanene Nanoribbons for Interconnect
Applications. In Nanoelectronic Devices for Hardware and Software Security (pp. 85-100). CRC
Press.
[17] Shiraz, A. K., Goharrizi, A. Y., & Hamidi, S. M. (2019). The electronic and optical
properties of armchair germanene nanoribbons. Physica E: Low-dimensional Systems and

Nanostructures, 107, 150-153.
[18] Sharma, V., Srivastava, P., & Jaiswal, N. K. (2017). Prospects of asymmetrically Hterminated zigzag germanene nanoribbons for spintronic application. Applied Surface
Science, 396, 1352-1359.
[19] Pang, Q., Zhang, Y., Zhang, J. M., Ji, V., & Xu, K. W. (2011). Electronic and magnetic
properties of pristine and chemically functionalized germanene nanoribbons. Nanoscale, 3(10),
4330-4338.
[20] Samipour, A., Dideban, D., & Heidari, H. (2020). Impact of an antidote vacancy on the
electronic and transport properties of germanene nanoribbons: A first principles study. Journal of
Physics and Chemistry of Solids, 138, 109289.
[21] Matthes, L., & Bechstedt, F. (2014). Influence of edge and field effects on topological states
of germanene nanoribbons from self-consistent calculations. Physical Review B, 90(16), 165431.
[22] Arjmand, T., Tagani, M. B., & Soleimani, H. R. (2018). Buckling-dependent switching
behaviours in shifted bilayer germanene nanoribbons: A computational study. Superlattices and
Microstructures, 113, 657-666.
[23] Samipour, A., Dideban, D., & Heidari, H. (2020). Impact of substitutional metallic dopants
on the physical and electronic properties of germanene nanoribbons: A first principles
study. Results in Physics, 18, 103333.
86


[24] Sharma, V., Srivastava, P., & Jaiswal, N. K. (2018). Edge-oxidized germanene nanoribbons
for nanoscale metal interconnect applications. IEEE Transactions on Electron Devices, 65(9),
3893-3900.
[25] Liu, J., Yu, G., Shen, X., Zhang, H., Li, H., Huang, X., & Chen, W. (2017). The structures,
stabilities, electronic and magnetic properties of fully and partially hydrogenated germanene
nanoribbons: A first-principles investigation. Physica E: Low-dimensional Systems and
Nanostructures, 87, 27-36.
[26] Kresse, G., & Furthmüller, J. (1996). Efficient iterative schemes for ab initio total-energy
calculations using a plane-wave basis set. Physical Review B, 54(16), 11169.
[27] Perdew, J. P., Burke, K., & Ernzerhof, M. (1996). Generalized gradient approximation made

simple. Physical Review Letters, 77(18), 3865.

87