The mechanical properties and thermal stability of ultrathin germanium nanowires

Abstract

The most stable structures of four ultrathin germanium nanowires (GeNWs) were predicted by a simulated annealing basin-hopping method (SABH) with a Stillinger–Weber (SW) potential, including helix, pentagon, hexagon and 7-1 nanowires. The size and temperature dependence of the tensile behavior and mechanical properties are investigated to approach a real environment. The ultimate tensile strength, strain at failure and Young's modulus are evaluated. All the mechanical properties of nanowires are severely reduced when temperature increases from 20 K to 180 K, but become less severe at high temperature. At room temperature (300 K), the yielding stress and Young's modulus of all nanowires are higher than bulk, and the pentagonal NW exhibits the best mechanical properties among these three GeNWs. This study also demonstrates that the mechanical properties are not proportional to the size or radius of ultrathin GeNWs, a phenomenon different from that in the bulk. In addition, the phonon density of states and thermal stability of GeNWs are also discussed in this study.

---

Introduction

In recent years, semiconductor device fabrication has been scaled down to the nanoscale regime, which has attracted much attention to nanomaterials. When one or more dimensions of a bulk material is reduced to only several nanometers, its material properties, including electrical conductivity,¹ mechanical properties,² and thermal conductivity,³ become significantly different from those of its bulk counterpart. This phenomenon creates some problems in semiconductor fabrication. For example, silicon in transistors exhibits severe leakage due to the quantum tunneling effect.⁴ Therefore, development of new materials or skeletons for nano-electronic devices is a crucial issue for the manufacturing industry.

For nano-electronic devices, most new nanomaterials can effectively reduce energy consumption for the electronic components and extend the available use time for portable devices as well as achieving a lower fabrication cost.⁵ In recent years, nanowires have been investigated widely because they possess broad applications in different areas, such as nano-mechanical^6–10 and nano-electronic devices.^11–16 Furthermore, semiconductor nanowires have been used for nanoscale field-effect transistors (FETs) with performance comparable to or exceeding that of single-crystal materials.^17,18 Consequently, germanium (Ge) has been considered as one possible element to replace silicon in the complementary metal oxide semiconductor (CMOS) for its high carrier mobility, higher electrical conductivity, and smaller band gap than that of Si.^19–23 According to a previous comparison of bulk properties,¹⁹ the electron and hole mobilities of germanium are 2.6 and 4.2 times faster than silicon, which compensates for the energy loss from the leakage phenomenon. Currently, there are widely proposed applications for the germanium nanowire (GeNW). For example, Chan et al. showed that GeNWs have a high electric capacitance and excellent cycling performance for Li-ion battery anodes.²⁴ Kennedy et al. fabricated the high density germanium nanowire arrays for lithium-ion battery anodes, which also show high-performance and high-capacity for energy storage.²⁵ In Mary's study, they conducted a performance analysis on the silicon and germanium nanowire transistors, and found that the transistors built using GeNWs provided a higher electric performance than the SiNWs.²⁶ Thus, future CMOS devices will be aiming at higher device drive current and faster operation speed.

Since the studies of GeNW mechanical properties are still lacking and the temperature effect of GeNW is the key limitation for which application, this study focuses on this cause and effect relationship. In addition, some previous studies have indicated that the electric conductivity of GeNW may be enhanced when under high tensile strain.¹² Four different sizes of GeNW were constructed and their thermal stabilities were analyzed. The first three type single-shell GeNWs are named helix, pentagon and hexagon GeNW, with radii of 1.8 Å, 2.0 Å and 2.4 Å, respectively. There is also a one core–shell GeNW named 7-1 GeNW which demonstrates 7 Ge atoms in its outer shell, 1 Ge atom in the core and a radius of 2.9 Å. The size and temperature dependence of the tensile behavior and mechanical properties are investigated to approach a real environment. The ultimate tensile strength, strain at failure and Young's moduli are evaluated. In addition, the phonon density of states and thermal stability of GeNWs are also discussed in this study.

Simulation model

In the present study, the ultrathin GeNWs were constructed by a simulated annealing basin-hopping (SABH) global optimization algorithm. For the initial geometries of our basin-hoping (BH) search process,²⁷ all the Ge atoms were randomly distributed in a very small space with very high repulsive interatomic energies, leading to the evolution of atoms by their repulsive forces between atoms. This stage is so called “atom explosion” in the “big bang” global minimum search method proposed by Jackson.²⁸ In the basin-hopping (BH) method a conjugate gradient method was used to search the local minimum geometry. The simulated annealing (SA) method²⁹ was also implemented with the BH method to be a SABH method, which includes a wider searching domain within the potential energy surface and obtain the local minimum.

For the Ge element, the Stillinger–Weber (SW) potential³⁰ was used to predict the structure of the GeNWs. The SW potential possesses a simple formula and saves computational time when compared to other many-body potentials. The SW style computes a three body potential for the energy E of a system of atoms as:

V₂ = εA(Bσ^pr_ij^−p − σ^qr_ij^−q)exp[σ(r_ij − aσ)⁻¹]

V₃ = εA exp[γσ(r_ij − aσ)⁻¹ + γσ(r_jk − aσ)⁻¹](cos θ_ijk − cos θ₀)²

where V₂ is a two-body term and V₃ is a three-body term. The summations in the formula are over all neighbors j and k of atom i within a cutoff distance aσ. The parameters of SW potential relating to Ge are listed in Table 1.³¹ Because the searched configurations are nearly spherical in the original SABH method, the SABH method combined with the penalty function was used to force a structural evolution along the axial direction of the nanowire. The penalty function has the Lennard-Jones form, which describes the interaction between the atom and the pseudo repulsive cylindrical wall for constraining all atoms within a one-dimensional cylinder space during the geometrical optimization process. The repulsive potential is expressed as the following form:
where r is the distance from the atom to the repulsive wall, and r_c is the cutoff distance at which the interaction between the atom and repulsive wall is not considered. The values of ε and σ are 1 eV and 1 Å. To produce a repulsive interaction between atoms and the virtual cylindrical surface, the r_c value is set to be smaller than 1 Å.

Table 1 The Stillinger–Weber potential parameters of germanium

Scaling		Two-body part				Three-body part		Cutoff
ε (eV)	σ (Å)	A	B	p	q	λ	γ	a
1.93	2.181	7.049556277	0.6022245584	4	0	19.5	1.19	1.8

---

The stability and reliability of all predicted nanowires were further checked by density functional theory (DFT) calculation. The DFT geometrical optimization was carried out by the DMol³ package,³² which employs the all electron calculation with double numerical plus polarization (DNP) basis sets. The generalized gradient approximation (GGA)³³ with Perdew–Burke–Ernzerhof (PBE) correction functional³⁴ was adopted. For Ge nanowires and bulk Ge unit cell, the (6 × 1 × 1) and (4 × 4 × 4) Monkhorst–Pack mesh k-points were used, respectively. Spin-unrestricted treatment was considered and the convergence condition of electronic self-consistent field (SCF) was set as 10⁻⁵ hartree. Table 2 lists the lattice constant and binding energy of bulk Ge obtained by the experimental approach³⁵ and the DFT prediction. The binding energy is defined as following equation:

where E_bulk is the total energy of lattice Ge, E_atom is the atomic energy of Ge atom and n is the number of Ge atom in lattice. These DFT calculation results are in good agreement with those by experimental measurement, indicating the current DFT setting can correctly describe the material behaviors of the Ge element.

Table 2 The binding energy and lattice constant of bulk germanium from experiments and DFT calculations

	Method	Binding energy (eV)	Lattice constant (Å)
Gebulk	GGA/PBE/all electron	3.829	5.734
	Exp.^48	3.850	5.658
	Error (%)	0.54	1.52

---

For determine the phonon density of states (PDOS) of GeNW, the relative phonon intensity, integrated across the Brillouin zone, versus the continuous frequency range are demonstrated. The phonon modes were determined by calculate the force constant matrix, given by the second derivative with respect to the atoms in space. All the PDOS profiles were determined by molecular statics calculation with SW potential and performed by the GULP package.^36,37

For the tensile test simulation, the loading state of the molecular dynamics simulation for the tensile test is presented as follows: the loading is applied along the axis of the nanowire, with top and bottom layers set as fixed layers and others as free layers. Before the tensile test, the pressure on the axial direction of the nanowire was also relaxed by the Nosé–Hoover barostat to remove the residual stress in the nanowire. To keep the system temperature constant during simulation, the Nosé–Hoover thermostat is adopted to ensure a constant system temperature during the simulation process. The tensor stress for atom i is given by the following formula:

The first term is a kinetic energy contribution for atom i. The second term is a pairwise energy contribution where n loops over the N_p neighbors of atom i, r₁ and r₂ are the positions of the two atoms in the pairwise interaction, and F₁ and F₂ are the forces on the two atoms resulting from the pairwise interaction. The N_S is the number of atoms. The V_i is the partial volume for atom i, which is derived by Srolovitz³⁸ and shown as the following equation.

The a_i is the atomic average radius and the r_ij is the average distance between atom i and neighbor atom j. The normal strain in the axial direction ε is calculated as:

where is the average length in the axial direction and l_z(o) is its average initial length. Thus, the stress-strain relationship of the ultrathin GeNW can be found by means of eqn (2) and (4). To precisely monitor the local structural deformation, the atomic local shear strain η^Mises_i calculated by OVITO package³⁹ is presented. The η^Mises_i of an individual atom, introduced by Shimizu et al.,⁴⁰ was originally used to monitor the development of shear transition zones (STZ) and the formation of the shear band within the bulk metallic glass. The detailed definition of η^Mises_i can be found in ref. 41 and is therefore not introduced here.

Results and discussion

After the global minimum search by the SABH method with the penalty function, three types of single shell and one type of multi-shell nanowire can be found. These GeNWs in Fig. 1(a)–(d) are labeled as helix, pentagon, hexagon and 7-1 nanowires, respectively. The tube-like structures are composed of penta-rings or hexa-rings in a parallel arrangement.

Fig. 1 The four types of germanium nanowires.

To check the stability of all GeNW structures predicted by the SABH method with the SW potential and penalty function, a geometric optimization by the DFT calculation was conducted. The bond and angle distributions between the DFT optimized GeNW structures and those from the BH process were used to evaluate the similarity of GeNWs after optimized by the DFT method. The results display both bond and angle distributions are very close, so the DFT optimized structures are very similar to the corresponding structures determined from the SABH method. Table 3 lists the binding energies of these four GeNWs, and one can see these energies are slightly lower than that of bulk Ge.

Table 3 The binding energy of bulk germanium and four germanium nanowires in this work

Type	Binding energy (eV)	Radius (Å)
Helix	3.342	1.8
Pentagon	3.438	2.0
Hexagon	3.388	2.4
7-1	3.571	2.9

---

For the thermal stability observation, a temperature elevation process from 1 to 800 K was carried out. The system was relaxed for 100 ps before applying the subsequent temperature increment of 1 K, and the Nosé–Hoover thermostat was adopted to ensure a constant system temperature during the simulation process. Similarly, stress on the GeNW axial direction was relaxed by the Nosé–Hoover barostat to remove the residual stress during the temperature elevation process. To indicate the temperature at which the nanowires undergoes a serious structural deformation, a parameter, delta R, was used, and is defined as:

where r_ij is the distance between atoms i and j, and n is the total number of atoms. The variation of this parameter is very sensitive to the structural change and any distinct increase or decrease in delta R indicates the structure is undergoing a considerable deformation. In Fig. 2, the difference of total energy during the temperature elevation process is applied to get the melting point of GeNWs. The melting of GeNW usually exhibits the local structural deformation, so analyzing the total energy variation is a sensitive way to obtain the melting point of GeNWs. The relative energy at temperature T is defined as

E_relative(T) = E_total(T) − E_total(1 K) (6)

where E_total(T) is the total energy of GeNW at temperature T, E_total(1 K) is the total energy of GeNW at temperature of 1 K. Fig. 2(a)–(d) shows the profiles of relative energy and delta R during the temperature elevation process for four GeNWs, with corresponding snapshots at 1 K, at the temperature just before breakage, and at the temperature just after breakage also shown. For these four GeNWs, it can be seen the relative energies generally display a linear increase with increasing temperature until the breakages occur at certain temperatures. Within these temperature ranges, the delta R values of all cases almost remains constant, indicating that the local structures are not deformed. After the temperatures exceed 435, 450, 385, and 230 K for helix, pentagon, hexagon and 7-1 nanowires, both relative energy and delta R profiles drop abruptly, indicating that the local structures undergo significant change. These temperatures are defined as the melting points of these four GeNWs, which are much lower than the melting temperature of bulk Ge, both the 1211 K as determined by experimental measurement⁴² as well as 1360 K, the value predicted by the current SW potential.³¹ From the snapshots of seriously deformed structures, one can see most local structures rearrange to sp³ structures as the local arrangement in bulk Ge. For the 7-1 GeNW, the Ge atoms of the one-atom chain surrounded by 7 neighbor Ge atoms more easily to form the sp³ local structures during the temperature elevation process, which leads to the significant deformation of the 7-1 GeNW at a relatively lower temperature compared to those of the other three GeNWs. Since the 7-1 GeNW structure has seriously deformed at room temperature, which could limit its possible applications within the broader temperature range, only the predicted material properties of the other three GeNWs are further considered in the current study.

Fig. 2 The relative energy and delta R difference of (a) helix, (b) pentagon, (c) hexagon and (d) 7-1 nanowires in a simulated heating process. The inset snapshot corresponds to the structure of the nanowire at 1 K as well as immediately before and after the melting point.

GeNW structures directly affect their phonon density of states (PDOS), the PDOS of bulk Ge and GeNWs are shown in Fig. 3. For bulk Ge, the four characteristic PDOS peaks which appear at 3.32, 6.26, 7.48, and 9.22 THz are in good agreement with the corresponding experimental data,⁴³ implying these PDOS calculations by the SW potential are reliable. For these three GeNWs, the main PDOS peaks are different from the bulk one due to their specific local structures. In bulk Ge, the main vibrational mode within the highest frequencies ranging from 8.27 to 10.94 THz is the bond stretching along two covalently bonded Ge atoms. The PDOS profiles of the three GeNWs are significantly different from that of bulk, because all characteristic peaks red shift to the lower frequency range. Due to the specific bonding modes in GeNW, the main peaks of PDOS profiles distribute among the wider frequency ranges. These phenomena indicate that the heat may be transferred by various phonon modes in GeNWs. The reduction of PDOS intensity implies that the thermal conductivities are smaller than bulk, which is consistent with previous experimental results.⁴⁴

Fig. 3 The phonon density of states (PDOS) of bulk Ge and the three GeNWs (helix, pentagon, hexagon).

The stress–strain profiles of helix, pentagon and hexagon GeNWs obtained by the uniaxial tensile test at different temperatures lower than their corresponding melting temperatures are shown in Fig. 4(a)–(c). For all GeNWs at different temperatures, it can be seen the stress profiles increase linearly with increasing tensile strain until exceeding certain values, at which the stress profiles abruptly drop from maximum stresses. Accordingly, the strain at maximum stress can be regarded as the yielding strain, and the corresponding stress is the yielding stress or strength of the GeNW.

Fig. 4 The stress–strain curve of (a) helix, (b) pentagon and (c) hexagon nanowires at various temperatures.

The distribution of local shear strain and corresponding snapshots of three GeNWs at different strains during the tensile process can be seen in Fig. 5(a)–(c). The GeNW deformation behaviours at different temperatures are very similar, so only the tensile snapshots at 300 K are shown. The color bar from blue to red is corresponding to η^Mises_i value from 0 to 1, a large η^Mises_i value indicates atom i is under local plastic and shear deformation, whereas a small η^Mises_i value implies atom i undergoes a small amount of movement relative to all its first neighbor atoms or atom i is under local elastic deformation.

Fig. 5 The distribution of atomic local shear strain in structure of (a) helix, (b) pentagon and (c) hexagon nanowires at strain of 0, at yielding strain, and at strain of 0.1. The color bar from blue to red corresponds to η^Mises_i value from 0 to 1.

In Fig. 5(a) for the helix GeNW at the yielding strain 0.049, the local structure with higher η^Mises_i value displays significant deformation. Then the deformed region at the yielding strain evolves into the one-atom chain structure when the strain becomes larger. It is obvious that the damaged part of the helix GeNW at strains over yielding is very localized while the other local structures still remain the helix arrangement. For pentagon and hexagon GeNWs shown in Fig. 5(b) and (c), after the strains reach the corresponding yielding strains of 0.054 and 0.050, necking of local structures appear. As the strain becomes larger, the local necking parts in both GeNWs induce subsequent deformation of non-deformed parts.

For a detailed investigation of the local structural change during the tensile process, the variations of bond length and bending angle at different strains are presented. Fig. 6 shows two different types of bonds and three types of angles in each GeNW at different strains during the tensile process. The bonds which align more along the axial direction are designated as bond type 1 and the bonds vertical to bond type 1 are designated as bond type 2. The angles formed by two connected type 1 bonds, one type 1 bond and one type 2 bond, two connected type 2 bonds are designated as angle type 1, angle type 2 and angle type 3, respectively.

Fig. 6 Definition of bond types in (a) helix, (b) pentagon and (c) hexagon nanowires.

Fig. 7(a)–(c) displays the averaged bond lengths and angles of different types for all GeNWs at 300 K during the tensile test. The averaged bond lengths and angles of the helix GeNW are shown in Fig. 7(a). In the elastic region, the average length of bond type 1 slightly increases with increasing strain, whereas the average length of bond type 2 decreases with increasing strain. This is because the necking of the nanowire gradually appears. For the bending angles, the variation of angle type 1 displays a relatively larger fluctuation than other two angle types. When the strain reaches the yielding strain (0.049), the average lengths of both bond type 1 and type 2 increase dramatically with a further increase in the strain. For the bending angle variations, one can see the angles of all types of helix GeNW only display a slight change at the yielding strain of 0.049, but change more dramatically at a strain of about 0.065. For pentagon and hexagon GeNWs, shown in Fig. 7(b) and (c), the bond length variations with the strain are very similar to those of helix GeNW. For bending angles, the changes at the yielding strains are very significant for these two GeNWs. At strain of 0.1, it should be noted that the type 1 bending angles of pentagon and hexagon GeNWs change more considerably as compared with their angels at strain of 0, indicating the deformation of helix GeNW is more localized, as shown in Fig. 5(a).

Fig. 7 The average bond length and bond angle of (a) helix, (b) pentagon and (c) hexagon nanowires at different tensile strain.

The temperature effects on the Young's modulus and the yielding stress are shown in Fig. 8 for helix, pentagon, and hexagon GeNWs. The Young's modulus of each GeNW can be determined from the slope of stress–strain curve at strains under 0.02 and the yielding stress is the maximal stress of the stress–strain curve. For each GeNW, the lower system temperature is 20 K and an increment of 20 K was applied. Both yielding stress and Young's modulus of these three GeNWs at different temperatures can be described by the logarithmic relationship in the following formula:

E = E₀ − E_T ln T (7)

Y = Y₀ − Y_T ln T (8)

where E and Y are the Young's modulus and yielding stress at temperature T K. E₀ and Y₀ are the Young's modulus and yielding stress at 0 K, and E_T and Y_T are the temperature factors in eqn (7) and (8). The fitted parameters are shown in Table 5 and a fitted curve can also be seen in Fig. 8. One can see that both yielding stress and Young's modulus decrease severely when the temperature increases from 20 K to 180 K, but the extent of yielding stress and Young's modulus reductions with temperature become less significant at higher temperatures.

Fig. 8 The yielding stress and Young's modulus of (a) helix, (b) pentagon and (c) hexagon nanowires at different temperatures.

The mechanical properties of bulk Ge, helix, pentagon, and hexagon GeNWs at room temperature as well as those of GeNWs in previous empirical and theoretical studies are displayed in Table 4. For bulk Ge, the Young's modulus predicted by the SW potential is in good agreement with the experimental value. For the GeNWs, the Young's moduli reported in the previous experimental measurements and the theoretical results range from 30 to 260 GPa.^6,10,45,46 One can see the predicted Young's moduli of three ultrathin GeNWs are much higher than those in previous works, which was also reported in previous experimental studies.¹⁰ Because of the small size involved, GeNWs with such small radii have not been successfully synthesized in the experiments. Using a reliably SW potential, which can accurately describe the hybridization of Ge bonding, this work is a numerical study to predict the most stable GeNWs configurations as well as their thermal and mechanical properties prior to the related experiments.

Table 4 The mechanical properties of bulk Ge, helix, pentagon, and hexagon GeNWs as well as those of GeNWs in previous empirical and theoretical studies

	Young's modulus (GPa)	Yielding stress (GPa)	Yielding strain
Bulk (Exp.^49)	103	—	—
Bulk (MD simulation by SW potential)	115	—	—
Ge nanowire (Exp.^50)	112	—	—
Ge nanowire (Exp.^51)	30–180
Ge nanowire (Exp.^52)	40–260
Ge nanowire (Cal.^50)	160
Ge nanowire (Cal.^50)	125
Helix	565	26.507	0.048
Pentagon	613	34.097	0.053
Hexagon	504	26.638	0.049

---

Table 5 The fitted function for Young's modulus (E) and yielding stress (Y) of Ge nanowires versus temperature (T) in Fig. 5

Ge nanowire	Fitted parameter
	E0	ET	Y0	YT
Helix	847.9	54.6	86.2	10.0
Pentagon	881.1	46.5	92.5	10.3
Hexagon	846.7	63.3	65.5	6.2

---

Conclusions

In this work, the simulated annealing basin-hopping method with Stillinger–Weber potential was used to construct four types of GeNW: helix, pentagon, hexagon and a 7-1 nanowire. The results indicate that all the nanowires are stable in their DFT calculation, but the 7-1 nanowire was determined by an MD simulation thermal stability test to be unstable at room temperature. Further simulated tensile tests are performed on the three other GeNWs using MD simulation. All the mechanical properties of nanowires are severely reduced when temperature increases from 20 K to 180 K, but becoming less severe at high temperature. At room temperature (300 K), the yielding stress and Young's modulus of all nanowires are higher than bulk, and the pentagonal NW exhibits the best mechanical properties among these three GeNWs. This study also determines that the mechanical properties are not proportional to the size or radius of ultrathin GeNWs, a phenomenon different from that in bulk. It has been reported that the GeNW could be a promising nanomaterial, which increase the Li ion diffusion rate for Li-ion battery.^24,47 Since the most stable structures of three ultrathin GeNWs have been obtained in the current study, using DFT calculation for exploring the Li diffusion mechanism on GeNWs is our future study.

Acknowledgements

The authors would like to thank the Ministry of Science and Technology, R.O.C for funding under contract number MOST 104-2221-E-110-008, and is also grateful for the computational time, resources, and facilities from the National Center for High-Performance Computing, Taiwan, and support from the National Center for Theoretical Sciences, Taiwan.

References

1. A. I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.-K. Yu, W. A. Goddard Iii and J. R. Heath, Nature, 2008, 451, 168–171.
2. B. Wu, A. Heidelberg and J. J. Boland, Nat. Mater., 2005, 4, 525–529.
3. D. Li, Y. Wu, P. Kim, L. Shi, P. Yang and A. Majumdar, Appl. Phys. Lett., 2003, 83, 2934–2936.
4. R. Chau, B. Boyanov, B. Doyle, M. Doczy, S. Datta, S. Hareland, B. Jin, J. Kavalieros and M. Metz, Phys. E, 2003, 19, 1–5.
5. D. Goldhaber-Gordon, M. S. Montemerlo, J. C. Love, G. J. Opiteck and J. C. Ellenbogen, Proc. IEEE, 1997, 85, 521–540.
6. D. A. Smith, V. C. Holmberg and B. A. Korgel, ACS Nano, 2010, 4, 2356–2362.
7. J. Chen, G. Zhang and B. Li, Nano Lett., 2012, 12, 2826–2832.
8. L. Vincent, G. Patriarche, G. Hallais, C. Renard, C. Gardès, D. Troadec and D. Bouchier, Nano Lett., 2014, 14, 4828–4836.
9. J. Greil, A. Lugstein, C. Zeiner, G. Strasser and E. Bertagnolli, Nano Lett., 2012, 12, 6230–6234.
10. L. T. Ngo, D. Almécija, J. E. Sader, B. Daly, N. Petkov, J. D. Holmes, D. Erts and J. J. Boland, Nano Lett., 2006, 6, 2964–2968.
11. D. Wang, Y.-L. Chang, Q. Wang, J. Cao, D. B. Farmer, R. G. Gordon and H. Dai, J. Am. Chem. Soc., 2004, 126, 11602–11611.
12. Y. Li, R. Clady, J. Park, S. V. Thombare, T. W. Schmidt, M. L. Brongersma and P. C. McIntyre, Nano Lett., 2014, 14, 3427–3431.
13. Y.-B. Zhang, L. Sun, H. Xu, Y.-Q. Xia, Y. Wang and S.-D. Zhang, Jpn. J. Appl. Phys., 2014, 53, 04EN03.
14. J. Lee and M. Shin, arXiv preprint arXiv:1304.5125, 2013.
15. H. Jung, P. K. Allan, Y.-Y. Hu, O. J. Borkiewicz, X.-L. Wang, W.-Q. Han, L.-S. Du, C. J. Pickard, P. J. Chupas and K. W. Chapman, Chem. Mater., 2015, 27, 1031–1041.
16. C. Wang, J. Ju, Y. Yang, Y. Tang, J. Lin, Z. Shi, R. P. Han and F. Huang, J. Mater. Chem. A, 2013, 1, 8897–8902.
17. B. Yu and M. Meyyappan, Solid-State Electron., 2006, 50, 536–544.
18. Y. Cui, Z. Zhong, D. Wang, W. U. Wang and C. M. Lieber, Nano Lett., 2003, 3, 149–152.
19. B. Yu, X. Sun, G. Calebotta, G. Dholakia and M. Meyyappan, J. Cluster Sci., 2006, 17, 579–597.
20. T. Krishnamohan, D. Kim, C. D. Nguyen, C. Jungemann, Y. Nishi and K. C. Saraswat, IEEE Trans. Electron Devices, 2006, 53, 1000–1009.
21. T. Maeda, K. Ikeda, S. Nakaharai, T. Tezuka, N. Sugiyama, Y. Moriyama and S. Takagi, IEEE Electron Device Lett., 2005, 26, 102–104.
22. C. O. Chui, IEEE Electron Device Lett., 2002, 23, 476–478.
23. Y. Maeda, N. Tsukamoto, Y. Yazawa, Y. Kanemitsu and Y. Masumoto, Appl. Phys. Lett., 1991, 59, 3168–3170.
24. C. K. Chan, X. F. Zhang and Y. Cui, Nano Lett., 2008, 8, 307–309.
25. T. Kennedy, E. Mullane, H. Geaney, M. Osiak, C. O'Dwyer and K. M. Ryan, Nano Lett., 2014, 14, 716–723.
26. P. T. Mary and N. Balamurugan, International Journal of Advanced Computer Research, 2014, 4, 898–903.
27. D. J. Wales and J. P. Doye, J. Phys. Chem. A, 1997, 101, 5111–5116.
28. K. A. Jackson, M. Horoi, I. Chaudhuri, T. Frauenheim and A. A. Shvartsburg, Phys. Rev. Lett., 2004, 93, 013401.
29. S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, Science, 1983, 220, 671–680.
30. F. H. Stillinger and T. A. Weber, Phys. Rev. B: Condens. Matter Mater. Phys., 1985, 31, 5262.
31. M. Posselt and A. Gabriel, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 045202.
32. B. Delley, J. Chem. Phys., 2000, 113, 7756–7764.
33. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
34. M. Ernzerhof and G. E. Scuseria, J. Chem. Phys., 1999, 110, 5029–5036.
35. C. Kittel, University of Pennsylvania Law Review, 2005, vol. 154, p. 477.
36. J. D. Gale and A. L. Rohl, Mol. Simul., 2003, 29, 291–341.
37. J. D. Gale, J. Chem. Soc., Faraday Trans., 1997, 93, 629–637.
38. D. Srolovitz, K. Maeda, V. Vitek and T. Egami, Philos. Mag. A, 1981, 44, 847–866.
39. A. Stukowski, Modell. Simul. Mater. Sci. Eng., 2009, 18, 015012.
40. F. Shimizu, S. Ogata and J. Li, Mater. Trans., 2007, 48, 2923–2927.
41. J. Wang, P. D. Hodgson, J. Zhang, W. Yan and C. Yang, Comput. Mater. Sci., 2010, 50, 211–217.
42. D. R. Lide, CRC handbook of chemistry and physics, CRC press, 2004.
43. G. Nelin and G. Nilsson, Phys. Rev. B: Solid State, 1972, 5, 3151.
44. M. C. Wingert, Z. C. Chen, E. Dechaumphai, J. Moon, J.-H. Kim, J. Xiang and R. Chen, Nano Lett., 2011, 11, 5507–5513.
45. X. Wu, J. S. Kulkarni, G. Collins, N. Petkov, D. Almécija, J. J. Boland, D. Erts and J. D. Holmes, Chem. Mater., 2008, 20, 5954–5967.
46. D. A. Smith, V. C. Holmberg, D. C. Lee and B. A. Korgel, J. Phys. Chem. C, 2008, 112, 10725–10729.
47. E. Mullane, T. Kennedy, H. Geaney and K. M. Ryan, ACS Appl. Mater. Interfaces, 2014, 6, 18800–18807.
48. C. Kittel, Introduction to Solid State Physics, 8th edn, 2005.
49. J. J. Wortman and R. A. Evans, J. Appl. Phys., 1965, 36, 153.
50. X. Wu, J. Kulkarni and G. Collins, et al., Chem. Mater., 2008, 20, 5954–5967.
51. D. A. Smith, V. C. Holmberg and D. C. Lee, et al., J. Phys. Chem. C, 2008, 112, 10725–10729.
52. L. T. Ngo, D. Almecija and J. E. Sader, et al., Nano Lett., 2006, 6, 2964–2968.

---