Effects of Structure, Temperature, and Strain Rate on Mechanical

Feb 24, 2010 - The effects of structure, temperature, and strain rate on mechanical properties of all the SiGe nanotubes in armchair and zigzag struct...
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J. Phys. Chem. C 2010, 114, 4309–4316

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Effects of Structure, Temperature, and Strain Rate on Mechanical Properties of SiGe Nanotubes Xin Liu,† Dapeng Cao,*,† and Aiping Yu‡ DiVision of Molecular and Materials Simulation, Key Lab for Nanomaterials, Ministry of Education, Beijing UniVersity of Chemical Technology, Beijing 100029, People’s Republic of China, and Department of Chemical Engineering, UniVersity of Waterloo, Ontario, Canada N2L3G1 ReceiVed: NoVember 01, 2009; ReVised Manuscript ReceiVed: February 02, 2010

The effects of structure, temperature, and strain rate on mechanical properties of all the SiGe nanotubes in armchair and zigzag structures (n ) 4-13) in two atomic arrangement types are investigated by classical molecular dynamics simulation. During the extending tests, we observe three structural transformations from initial structure, tensile structure, to critical structure deformation. The simulation results indicate that the Young’s modulus of nanotubes is closely dependent on their diameter, chirality, and arrangement structure. The type 1 (alternating atom arrangement type) armchair SiGe nanotube exhibits the largest Young’s modulus, compared with other nanotubes with the same index n. By exploring the effects of temperature and strain rate on mechanical properties of SiGe nanotubes, it is found that the higher temperature and lower strain rate lead to the lower critical strain and tensile strength. Furthermore, it is also found that the critical strains for both armchair and zigzag nanotubes in two arrangement types are significantly dependent on the tube diameter and chirality. The armchair type 1 nanotube exhibits the highest mechanical critical strain and tensile strength among all these nanotubes with the same index n. On the basis of the transition-state theory model, we predict that the critical strain of the SiGe (6,6) type 1 nanotube at 300 K, stretched with a strain rate of 5%/h, is about 3.38%, which is in good agreement with the recent experimental results. Our results might provide potential applications in manipulating mechanical and electromechanical properties of the nanostructures suitable for electronic devices. 1. Introduction The discovery of one-dimensional nanotubes (NTs) and nanowires (NWs) has opened a challenging new field in nanoscale materials, attracting widespread interest in the research community.1-3 Intensive experimental and theoretical studies have clearly shown that the nanotube and nanowire conformations can also be formed by many other elements and compounds, such as BN,4 GeC,5 GaN,6 GaSe,7 NiCl,8 NiCl2,9 MoTe2,10 and SiC.11,12 Besides, carbon, silicon, and germanium also reside in the column IV of the periodic table and are isoelectronic to carbon, which indicates that these nanomaterials might also be promising candidates for future nanoelectronics-based technologies.13,14 Thus, progressively increasing investigations have been reported on one-dimensional Si, Ge, and SiGe nanomaterials. The prototype silicon nanotubes have been demonstrated to hold unique structural and electronic properties,15 such as high mechanical strength, great thermal and chemical stabilities, and excellent heat conduction.16,17 Therefore, silicon nanotubes can be of technological importance in many fields, such as disease detectors, biochemical sensors, and nanoelectronics.18,19 Compared to previously mentioned SiNTs, only several investigations have been reported on one-dimensional Ge nanotubes (GeNT). The computational results indicated that germanium nanotubes possess unusual photoluminescence behavior and that GeH20 and GeC5,21 nanotubes are semiconducting. Very recently, one* To whom correspondence should be addressed. E-mail: [email protected], [email protected]. Fax: +8610-64427616. † Beijing University of Chemical Technology. ‡ University of Waterloo.

dimensional structures of SiGe nanotubes have also been the focus of extensive research. Experimental methods have been developed to fabricate SiGe compounds.22,23 For example, Schmidt et al.14 fabricated SiGe nanotubes experimentally by using the method of the thin-film bending mechanism. Besides the experimental research, theoretical and computational predictions are also useful to study the SiGe nanomaterials. By theoretical analysis and molecular dynamics (MD) simulation, Zang et al.24 found that a bilayer SiGe nanofilm could bend into a SiGe nanotube with Ge as the inner layer. Using hybrid density functional theory (DFT), Rathi et al.25 found that the SiGe nanotubes are semiconducting in nature, with a wide spectrum of band gaps. In addition, our previous work26 indicated that alternating atom arrangement type SiGe nanotubes have a higher thermal durability than other prototype SiGe nanotubes, which provides a great potential in the hightemperature applications. Generally speaking, SiGe nanomaterials have been regarded as one of the most promising building blocks for future development of functional nanostructures.26-28 Accurate characterization of nanomechanics in elastic and plastic regimes, therefore, is highly desirable for any applications in nanocomposites and devices.29 Preliminary evidence of excellent mechanical properties for C,30-33 Si,16,34 and SiC35,36 nanotubes suggests that they can be used as nanodrillers, nanotweezers, and microscopy tips.17 Thus, the mechanical properties of SiGe nanotubes are very important because potential applications depend on the stability and stiffness of the nanotubes. However, few studies contributed to the detailed study on mechanical properties of the full range of SiGeNTs.

10.1021/jp910423k  2010 American Chemical Society Published on Web 02/24/2010

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In this work, we employ the classical MD simulation based on the Tersoff potential function to comprehensively investigate the mechanical properties of various SiGe nanotubes. The structural characteristics and mechanical behavior during the tensing process of the SiGe nanotubes at two atomic arrangement types (alternating and layered configurations) are addressed. The effects of structure, temperature, strain rate, and compound concentration on the tensile behavior of SiGe nanotubes are also discussed. 2. Methodology For the Si-Si, Si-Ge, and Ge-Ge interactions, the present study adopts the Tersoff potential,37,38 which was fitted to the lattice constant and binding energy of a number of silicon lattices as well as the elastic constants and vacancy formation energies of the diamond structure. The Tersoff potential has been proven to be fruitful in these studies of Si, C, and Si-C39-41nanotubes. This empirical potential was used successfully by Jeng et al.16 to investigate the mechanical properties of the silicon-based nanotubes. Moreover, the Tersoff potential gives satisfactory results in these studies of the axial compression behavior of Si nanotubes34,42 and mechanical properties of SiC,43 SiGe44 nanowires, and SiC36 nanotubes. More importantly, the Tersoff potential has also been successfully applied in our previous work to study the thermal behavior of the Si-Ge system.26 The parameters and formulas of the Tersoff potential used in this work can be found in the literature.45 The optimal atomic configurations of the SiGe nanotubes used in the MD simulations were obtained from our previous work,26 where a zero-K MD simulation based on the Tersoff potential was performed to optimize these structures based on the DFT calculation. In our previous work, we compared the results obtained from the Tersoff potential with those obtained from the DFT and found that the structural parameters of SiGe nanotubes from the Tersoff potential are in good agreement with those from the DFT. According to the DFT calculation, the tube diameter, unit cell length, Si-Ge bond distance, and the cohesive energy of the armchair (6,6) type 1 SiGe nanotube are 13.00, 3.96, and 2.33-2.36 Å and 2.73 eV/atom, respectively. After the Tersoff-potential-based zero-K MD simulation, the tube diameter, unit cell length, and Si-Ge bond distance are 13.215, 3.60, and 2.366-2.369 Å and 2.78 eV/atom, respectively. Obviously, the satisfactory agreement gives us justification that the Tersoff potential can be efficiently applied to the investigation of SiGe nanotubes. The DFT calculation, which is known as time-consuming and considers more details about the electrons and the ions, can give more electronic interactions, corrections, and structural details, which are disregarded in classical MD. Considering these corrections, the simulated structures of nanotubes would be more credible, but it can only simulate the static state.46,47 In this case, using MD simulation to investigate the mechanical property of large-scale nanotubes can get more benefits, such as keeping a balance between efficiency and precision and giving a thoroughly dynamic display during the tensile process. In this work,

Figure 1. Structural transformation of SiGe nanotubes for a tensile test: (a) indicates the initial configuration before the tensile test, (b) is the elongated configuration under the tensile state, and (c) shows the critical configuration and SW defects under the yielding state in detail.

we investigate the mechanical properties of both armchair and zigzag SiGe nanotubes with the index of n ) 4-13 by adopting the MD method. All MD runs were carried out by using the large-scale atomic/molecular massively parallel simulator (LAMMPS) developed by Sandia National Laboratory.48 Our MD simulations were performed in NPT ensemble, and the temperature of the systems was kept constant via the Langevin thermostat with a time step of 1 fs and 200 000 time steps, leading to the total time of 2000 ps. Periodic boundary conditions are imposed in the x and y directions, and the tensile process involves continuous stretching of nanotubes by extending the two end atoms axially at the appropriate temperature and strain rate; the atoms in the x and y directions are free to relax during the extending process. A 0.1% tensile strain is applied in a single MD step, and the strain rate is maintained by the time interval for the structural relaxation before the next 0.1% strain is applied. To investigate the relative influences of the structure, temperature, and strain rates on the mechanical properties of the SiGe nanotubes in tension conditions, we simulate these cases of four distinctive prototypes of SiGe nanotubes, various temperature ranges of 300-1300 K, and with strain rates varying from 2.5 × 10-4/ps to 2.5 × 10-6/ps. As seen from Figure 1, a tensile test of SiGe nanotubes can be described by three tensile processes, that is, the structural transformation from the initial pre-yielding state (as shown in Figure 1a), the elongated structural state (Figure 1b), to the final yielding state (Figure 1c). To validate the applicability of the Tersoff potential and our method, we studied the mechanical properties of the armchair C (6,6), Si (6,6) and SiC (6,6) nanotubes and further compared our results with the ones reported in the literature. Different structures show significant effects on mechanical properties, especially for the Young’s modulus. As shown in Table I, our

TABLE I: Cohesive Energy, Tube Diameter, Bond Length, Critical Strain, Modified Young’s Modulus, Young’s Modulus, and Literature Data for Si (6,6), C (6,6), and SiC (6,6) Nanotubes

nanotubes

cohesive energy (eV/atom)

tube diameter (Å)

bond length (Å)

critical strain (%)

modified Young’s modulus (GPa)

Young’s modulus (GPa)

theoretical data (GPa)

experimental data (GPa)

SiNT CNT SiCNT

-3.212 -6.203 -4.64

13.465 8.190 10.330

2.30-2.33 1.31-1.32 1.760-1.766

15.25 18.75 17.25

24.39 331.67 163.32

79.8 975.5 623.42

97316 10249 61535

970 ( 16050 95-12542 534-75051

Effects on Mechanical Properties of SiGe Nanotubes

J. Phys. Chem. C, Vol. 114, No. 10, 2010 4311 theoretical yield strain of the carbon nanotube at T ) 800 K with the strain rate of 2.5 × 10-6/ps was 16.25%, and at much higher temperature and lower strain rate, a larger kinetic energy increases the process rate to overcome the barrier and the simulated yield strain at T ) 2400 K approximates the experimental value of about 5%.53,54 Obviously, the satisfactory agreement above gives us justification that our methodology is efficient for the calculation of the mechanical properties of nanotubes. Also, as seen from Table I, the critical strain of the Si (6,6) nanotube at T ) 800 K is about 15.25%, which is in good agreement with the theoretical values of 14.87-21.33%16 and with experimental data of 10% for Si nanotubes among various other products (mostly silicon oxide nanoparticles).55 This illustrates that the Si (6,6) nanotube has a lower tensile strength than SiC and C nanotubes.

Figure 2. Strain energy per atom as a function of axial strain for C (6,6), SiC (6,6), and Si (6,6) nanotubes at T ) 800 K and T ) 2400 K with a strain rate of 2.5 × 10-6/ps.

calculated Young’s modulus of C, Si, and SiC nanotubes with highly symmetrical structures are 975.5, 79.8, and 624.42 GPa respectively, which are in good agreement with recent theoretical data of 973,16 102,49 and 61535 GPa, and are also in agreement with the experimental data of 970 ( 160 GPa for C nanotubes,50 95-125 GPa for Si nanotubes,42 and 534-750 GPa for SiC nanowires and nanotubes.51 Figure 2 presents the strain energy per atom of C (6,6), SiC (6,6), and Si (6,6) nanotubes as a function of the tensile strain with a strain rate of 2.5 × 10-6/ps at T ) 800 K and T ) 2400 K. The strain energy per atom is represented by the difference between the total energies per atom in the strained and unstrained conditions. The abrupt deviation of the strain energy from the elastic behavior at high tensile strain is the symbolization of structural collapse, and we define such strain as the critical strain or failure strain. It is noted that the critical strains of C (6,6) and SiC (6,6) nanotubes at T ) 800 K are 18.75, 17.25, and 15.25%, and the critical strain of C (6,6) at T ) 2400 K is 10.75%. Wei et al.52 reported that the

3. Results and Discussion 3.1. Mechanical Properties of SiGe Nanotubes. The structures of SiGe nanotubes with the index of n ) 4-10 have two unique optimized geometries and two distinct atomic arrangement types, as seen in Figure 3. The ratio of Si and Ge in the two structures is 1:1. Figure 3a shows the top views of four fascinating prototype SiGe nanotubes. It is found in Figure 3a that both the (6,0) type 1 (alternating atom arrangement type) and the type 2 (layered atom arrangement type) zigzag nanotubes exhibit the gearlike configuration, whereas both the (6,6) type 1 and the type 2 armchair ones possess the puckering configuration. Moreover, Figure 3b shows the side view of type 1, which consists of alternating Si and Ge atoms with each Si atom connecting to three neighboring Ge atoms and vice versa, whereas Figure 3c gives the side view of another arrangement named type 2, in which the nearest neighbors of each Si atom consist of two Ge atoms and one Si atom. The detailed structural information of these nanotubes is summarized in Table II. Some experiment findings14,56 indicate that the C- and Ge-substituted Si nanotube has a 3C structure (i.e., the type 1 arrangement shown in Figure 3b) and bending the Si-Ge bilayer graphene can yield the layered atom arrangement type (the type 2 arrangement shown in Figure 3c). In addition, our DFT-

Figure 3. Top view of optimized structures of (a) SiGe (6,0) type 1, SiGe (6,0) type 2, SiGe (6,6) type 1, and SiGe (6,6) type 2 nanotubes. Side view of the atomic arrangement of (b) type 1 and (c) type 2 SiGe nanotubes (green atoms, Ge; yellow atoms, Si).

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TABLE II: Tube Diameter (R) and the Unit Cell Length of the Supercell in the Axial Direction (C) for Type 1 and Type 2 SiGe Nanotubes nanotubes

(4,4)

(5,5)

(6,6)

(7,7)

(8,8)

(9,9)

(10,10) (13,13) (4,0) (5,0) (6,0) (7,0)

(8,0)

(9,0)

(10,0) (13,0)

type 1 8.74 10.85 13.00 15.16 17.10 19.20

21.38

23.65

5.07

6.00

7.63

8.46 10.08 11.00 12.60

14.78

type 2 8.46 10.85 12.90 14.98 17.02 19.08 unit cell length type 1 3.93 3.95 3.96 3.96 3.98 4.12 C (Å) type 2 3.94 3.95 3.95 3.96 3.98 4.12

21.27 4.12

23.96 4.13

5.14 6.74

6.04 6.81

7.65 6.81

8.50 10.18 11.12 12.70 6.82 6.82 6.84 6.84

15.02 6.84

4.12

4.12

6.74

6.80

6.80

6.81

tube diameter R (Å)

optimized structures are also in good agreement with the experimental results and other theoretical simulations.25 Figure 4 presents the snapshots of structural transformation during the tensile process and the cohesive energy per atom as a function of tensile strain of the SiGe (6,6) type 1 nanotube. The cohesive energy is represented by the difference between the total energy of the SiGe system and the sum of the separate atomic cohesive energies. It is found in Figure 4 that three structural transformations during the extending process, corresponding to the initial, extending, and critical structural processes, are marked as stage (1), stage (2), and stage (3), respectively. Numerical simulation and theoretical studies have shown that Stone-Wales (SW) bond rotations lead to the formation of defects on the nanotubes.57-59 The configuration in Figure 1c shows a group of pentagon and heptagon resulting from several SW rotations in the connected region, which indicates that the start of breaking of the SiGe (6,6) type 1 nanotube. For mechanical properties of SiGe nanotubes, we investigate the Young’s modulus, which is described as the second derivative of the strain energy with respect to the strain at the equilibrium configuration10

Ys )

( )

1 ∂2E V0 ∂ε2

(1)

where V0 is the relaxed equilibrium volume, ε is the axial strain, and E is the strain energy per unit cell. For nanotubes, V0 can be defined as the equilibrium volume

V0 ) 2πc0Rl

(2)

where R is the radius and l is the tube thickness. In our calculations, we adopted the same definition as that in the literature;60 that is, the interlayer distance of graphite is used as the tube thickness of C, Si, SiC, and SiGe single-walled

Figure 4. Snapshots and cohesive energy per atom of SiGe (6,6) type 1 nanotubes as a function of axial strain at T ) 800 K at a strain rate of 2.5 × 10-5/ps.

6.82

6.84

6.85

6.85

nanotubes, and they are 3.4, 4.1, 3.8, and 4.4 Å, respectively. The calculated Young’s modulus was plotted as a function of the tube diameter for all four sets of SiGe nanotubes, as shown in Figure 5. The respective values are summarized in Table III. As expected, the Young’s modulus is closely dependent on its diameter and chirality and increases monotonically with the increase of the diameter. Interestingly, the armchair type 1 nanotubes have the largest Young’s modulus among these tubular systems with the same index n, which indicates that the geometric structure alters not only the electronic and thermal properties but also the mechanical properties of the nanotubes. As seen from Figure 5, the differences of Young’s moduli between armchair and zigzag configurations are obvious. Impressively, for all these nanotubes, the Young’s modulus reaches saturation at the tube diameters of D > 15 nm for zigzag nanotubes and D > 20 nm for armchair nanotubes. The saturation values of the Young’s modulus are 66.14 GPa for armchair nanotubes and 66.03 GPa for zigzag nanotubes. This observation is in good agreement with other investigations of C, SiC, BN, and BeO nanotubes.35,61 In addition, compared with the Young’s modulus of C, SiC, and Si nanotubes mentioned above, SiGe nanotubes are much softer because the strain energy per atom of the SiGe nanotube is smaller than these of the C, SiC, and Si nanotubes with the same chirality. From the elastic potential energy E(∆x) ) k∆x2/2,34 where k is the spring constant and ∆x is the displacement, E(ε) is calculated by

1 1 ∆x2 E(∆x) ) k∆x2 ) k 2 x20 2 2 x 0 2 1 ∆x 1 2 ) K ) Kε ) E(ε) 2 x0 2

( )

(3)

where K ) kx20 is defined as the effective spring constant and x0 is the initial lattice constant in this work. The curve fitted by

Figure 5. Young’s modulus as a function of tube diameter for all SiGe nanotubes considered (n ) 4-10, 13).

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TABLE III: Calculated Young’s Modulus of the SiGe Nanotubes nanotubes

(4,4)

(5,5)

(6,6)

(7,7)

(8,8)

(9,9) (10,10) (13,13) (4,0)

Young’s modulus type 1 61.89 63.57 65.19 65.88 66.02 66.11 (Gpa) type 2 61.26 63.15 64.55 65.66 65.92 66.06

eq 3 shown in Figure 6 is E(ε) ) 21.3ε2/2. Therefore, K is 21.3 eV/atom for the SiGe (6,6) type 1 nanotube. The K values corresponding to C and Si nanotubes are around 5929,31,62 and 28.5 eV/atom,34 respectively, which are much higher than that of the SiGe nanotube obtained in this work. 3.2. Effects of Temperature, Strain Rate, Structure, and Size on Mechanical Properties of SiGe Nanotubes. To explore mechanisms of material deformation and important factors affecting the tube failure during the tensile process, we first investigate the effect of temperature. Figure 7 presents the strain energies of the SiGe (6,6) type 1 nanotube as a function of the tensile strain at different temperatures of T ) 300, 600, 900, and 1300 K and at the strain rate of 2.5 × 10-6/ps. The results clearly illustrate that, in this case, the critical strain and corresponding tensile strength decrease significantly at the higher temperatures. For instance, the critical strain of the SiGe (6,6) type 1 nanotube decreases from 13% at 900 K to 8.75% at 1300 K, which means that a higher temperature may result in a lower tensile strength. The reason for this phenomenon above is that a greater number of molecules have gained sufficient energy to escape and rearrange the initial configuration and these molecules could overcome the activation energy barrier when the temperature was increased, and consequently, the remarkable material deformation occurred (the breaking of two neighboring bonds and formation of pentagon-heptagon pair defects on nanotubes, shown in Figure 1c). It can be concluded that the mechanical properties of the nanotubes are strongly influenced by temperature. Previous theoretical investigations also reported some similar mechanical behavior about silicon-based16 and carbon nanotubes;52,63 that is, a thermally activated process plays an activating role in the complete elongation and compression of nanotubes. To get a comprehensive understanding of material deformation during the tensile process, we also investigate the effect of strain rate on mechanical properties of SiGe nanotubes. The calculated strain energy per atom of the SiGe (6,6) type 1 nanotube at T ) 800 K was plotted as a function of axial strain at different strain rates of 5 × 10-5/ps, 2.5 × 10-5/ps, 5 ×

Figure 6. Strain energy per atom dependent on tensile strain for the SiGe (6,6) type 1 nanotube.

(5,0)

(6,0)

(7,0)

(8,0)

(9,0) (10,0) (13,0)

66.12

66.16

60.26 62.78 64.03 64.85 65.48 65.63 65.86 66.04

66.12

66.13

66.02 62.54 63.98 64.77 65.36 65.47 65.80 66.03

10-6/ps, and 2.5 × 10-6/ps, as shown in Figure 8. It is noteworthy that the critical strain (corresponding to the tensile strength) increases at a quicker strain rate, which indicates that the mechanical properties are also sensitive to the strain rate. The SiGe (6,6) type 1 nanotube at 2.5 × 10-5/ps strain rate exhibits a 24% higher critical strain than at 2.5 × 10-6/ps. The reason for this observation above may be that during the tensile test, the strain or deformation does not tend to uniformly distribute within the whole SiGe nanotube, especially for a large strain. Consequently, some regions of the nanotube are subjected to larger stresses or strains than others, and it is within these regions that the defects will first become evident. In the present study, when a slower strain rate is applied, the SiGe nanotubes have more time to induce excessive local deformations, and hence, the onset of plastic deformation is accelerated. Therefore, in the SiGe nanotubes, a quicker strain rate results in a larger

Figure 7. Strain energy of the SiGe (6,6) type 1 nanotube as a function of axial strain for temperatures of T ) 300, 600, 900, and 1300 K at the strain rate of 2.5 × 10-6/ps.

Figure 8. Variation in strain energy with axial tensile strain for the SiGe (6,6) type 1 nanotube at different strain rates of 5 × 10-5/ps, 2.5 × 10-5/ps, 5 × 10-6/ps, and 2.5 × 10-6/ps.

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Figure 9. Critical strain changing with tube diameter for all the SiGe nanotubes considered (n ) 4-10).

Figure 10. (a) Critical strain of the SiGe (6,6) type 1 nanotube as a function of strain rate at different temperatures, which increases with the decrease of temperature and the increase of strain rate. (b) A(T) and B(T) are defined in eq 4 as functions of temperature.

critical strain (corresponding to tensile strength) during the tensing process. Our results are in good agreement with the previous calculations52,63 on the mechanical properties of Si and SiH nanotubes. To explore the structural and size effect on critical strain, we present in Figure 9 the dependence of critical strain on tube diameter for all the SiGe nanotubes with armchair and zigzag structures (n ) 5-10) in two arrangement types. It is clear that the critical strains for both armchair and zigzag nanotubes in two arrangement types are significantly dependent on the tube diameter and chirality. When the index n increases from 4 to 10, the critical strains for both armchair and zigzag nanotubes decrease with the increase of tube diameter. The similar behavior, that is, larger diameter tubes have lower critical strain and tensile strength, is also found in other investigations on elastic properties of single-walled carbon nanotubes. Moreover, for armchair and zigzag nanotubes, the critical strains of the type 1 structures are always much higher than those of the type 2 structures for all indices n. Obviously, the armchair type 1

nanotubes exhibit the highest mechanical critical strain and tensile strength among all these nanotubes with the same index n. The excellent mechanical property of armchair tubes is consistent with the Young’s modulus analysis studied above, suggesting the close relationship of the perfect structural character, high energetic stability, and prominent mechanical behavior. In our previous study, it was found that, for largediameter tubes, especially for the type 1 armchair nanotubes, the more planar tube surfaces and the overall symmetry configuration may be the key reasons for their extraordinary thermal durability. 3.3. Predictions of Mechanical Properties in Experimental Conditions. Some relatively straightforward theoretical (for example, transition-state theory (TST)) calculations are developed to enable the parametrization of the model at all experimentally feasible temperatures and strain rates. The TST model is valid for the yield strain prediction of nanotubes. According to the TST-based model,52 the yield strain for nanotubes can be described as

Effects on Mechanical Properties of SiGe Nanotubes

[

(

EV ε0nsite kBT ln VK VK N∆εstep ε )A(T) + B(T)ln ∆εstep

εY )

)]

+

k BT ε ln VK ∆εstep

J. Phys. Chem. C, Vol. 114, No. 10, 2010 4315

(4)

where V is the activation volume, K is the force constant, nsite is the number of sites available for state transition, ε is the strain rate, ∆εstep is the change of strain at each step used in the simulations, and ε0 is the intrinsic strain rate. In formula 4, A(T) j V/VK) - (kBT/VK)ln(nsiteε0/N∆εstep) and B(T) ) kBT/VK are ) (E identified as the intercept and slope, respectively, of the linear behavior shown in Figure 10b. The critical strain as a function of different strain rates and temperatures, is shown in Figure 10a. The critical strain of the SiGe nanotubes, under tensile stress with exprimentally feasible strain rates, can be predicted from this TST model. In our simulations, for the SiGe (6,6) type 1 nanotube, at T ) 300 K and at a strain rate of about 5%/h, the critical strain for a 10 nm long SiGe nanotube is about 3.38%, according to the equation of εY(%) ) 15.86 + 0.55 ln ε(ps-1), which is obtained from the data shown in Figure 10. Previous exprimental findings indicated that the yield strain and Young’s modulus of bulk Si are about 7% and 115 GPa, and these of bulk Ge are about 5% and 102 GPa,64 respectively. Because Si-Si bonds are stiffer than Si-Ge bonds and the bulk material exhibits a highly symmetrical structure and higher stability as well as Young’s modulus than nanotubes, it is rational that the yield strain of SiGe nanotubes under the experimental temperature and tensile strain rate is around 3.38%, according to our calculation. 4. Conclusions In summary, the effects of structure, temperature, and strain rate on mechanical properties of all the SiGe nanotubes in armchair and zigzag structures (n ) 4-13) in two atomic arrangement types were investigated by a classical MD simulation. During the extending test, we observed three structural transformations from initial structure, tensile structure, to critical structure deformation. It was found that Young’s modulus is significantly dependent on the diameter, chirality, and structure of the tube. In particular, the type 1 armchair SiGe nanotubes show the largest Young’s modulus, compared with other nanotubes with the same index n. It is noteworthy that the Young’s modulus reaches saturation at the tube diameters of D > 15 nm for zigzag nanotubes and D > 20 nm for armchair nanotubes. By exploring the effects of temperature and strain rate on mechanical properties of SiGe nanotubes, it is found that the higher temperature and lower strain rate lead to lower critical strain and tensile strength. The critical strains for both armchair and zigzag nanotubes in two arrangement types are significantly dependent on the tube diameter and chirality. The armchair type 1 nanotubes exhibit the highest mechanical critical strain and tensile strength among all the nanotubes with the same index n. Furthermore, the critical strain calculated from the TST (transition-state theory) model of the SiGe (6,6) type 1 nanotube at 300 K, stretched with a strain rate of 5%/h, is about 3.38%, which is in good agreement with the recent experimental data. In conclusion, the simulation results in this work illustrate that structural character, temperature, and strain rate are three primary governing factors in the mechanical behavior of SiGe nanotubes. By tuning and controlling these crucial variables, the SiGe nanotubes can open up many unique avenues in the fields of nanoelectronics devices, novel nanosolar cells, and photovoltaics, nanojunctions, and nanodevice components.

Acknowledgment. This work is supported by the National Natural Science Foundation of China (No. 20736002), the ROCS Foundation (LX2007-02) and Novel Team (IRT0807) from MOE of China, and the “Chemical Grid Program” and Excellent Talent Funding of BUCT. References and Notes (1) Dalton, A.; Collins, S.; Razal, J. J. Mater. Chem. 2004, 14, 1. (2) Lau, K.; Hui, D. Composites, Part B 2002, 33, 263. (3) Qian, D.; Dickey, E.; Andrews, R.; Rantell, T. Appl. Phys. Lett. 2000, 76, 2868. (4) Cumings, J.; Aettl, A. Chem. Phys. Lett. 2000, 316, 211. (5) Rathi, S.; Ray, A. K. Nanotechnology 2008, 19, 335706. (6) Goldberger, J.; He, R.; Zhang, Y.; Lee, S.; Yan, H.; Chol, H.; Yang, P. Nature 2003, 422, 599. (7) Cote, M.; Cohen, M.; Chadi, D. J. Phys. ReV. B 1998, 58, 4277. (8) Hacohen, Y.; Grunbaum, E.; Tenne, R.; Sloan, J.; Hutchison, J. Nature 1998, 395, 336. (9) Hacohen, Y.; Popovia-Biro, R.; Grunbaum, E.; Prior, Y.; Tenne, R. AdV. Mater. 2002, 14, 1075. (10) Wu, X.; Xu, Z.; Zeng, X. Nano Lett. 2007, 7, 2987. (11) Alam, K. M.; Ray, A. K. Phys. ReV. B 2008, 77, 035436. (12) Alam, K. M.; Ray, A. K. Nanotechnology 2007, 18, 495706. (13) Barnard, A. S.; Russo, S. J. Phys. Chem. B 2003, 107, 7577. (14) Schmidt, O. G.; Eberl, K. Nature 2001, 410, 168. (15) Wang, N.; Tang, Y.; Zhang, Y.; Lee, S. C.; Lee, S. T. Phys. ReV. B 1998, 58, R16024. (16) Jeng, Y.; Tsai, P.; Fang, T. Phys. ReV. B 2005, 71, 085411. (17) Durgun, E.; Tongay, S.; Ciraci, S. Phys. ReV. B 2005, 72, 075420. (18) Hu, J.; Yang, M.; Yang, P.; Lieber, C. Nature 1999, 399, 48. (19) Margulis, L.; Salitra, G.; Tenne, R.; Tallanker, M. Nature 1993, 365, 113. (20) Rathi, S.; Ray, A. K. J. Comput. Theor. Nanosci. 2008, 5, 464. (21) Seifert, G.; Kohler, T.; Hajnal, Z.; Frauenheim, T. Solid State Commun. 2001, 119, 653. (22) Hartmann, J. M.; Burdin, M.; Rolland, G.; Billon, T. J. Cryst. Growth 2006, 294, 288. (23) Rastelli, A.; Kanel, H. V. Surf. Sci. 2003, 532, 769. (24) Zang, J.; Huang, M.; Liu, F. Phys. ReV. B 2007, 98, 146102. (25) Rathi, S.; Ray, A. K. Chem. Phys. Lett. 2008, 19, 79. (26) Liu, X.; Cheng, D. J.; Cao, D. P. Nanotechnology 2009, 20, 315705. (27) Musin, R. N.; Wang, X. O. Phys. ReV. B 2006, 74, 165308. (28) Musin, R. N.; Wang, X. O. Phys. ReV. B 2005, 71, 155318. (29) Srivastava, D.; Menon, M.; Cho, K. Phys. ReV. Lett. 1999, 83, 2973. (30) Wong, E. W.; Sheehan, P. E.; Lieber, C. M. Science 1997, 277, 1971. (31) Yakobson, B. I.; Barbec, C.; Bernholc, J. Phys. ReV. Lett. 1996, 76, 2411. (32) Treacy, M. M. J.; Ebbsen, T. W.; Gibson, J. M. Nature 1996, 381, 678. (33) Ruoff, R. S.; Lorents, D. C. 1995, 33, 925. (34) Kang, J. W.; Hwang, H. J. Nanotechnology 2003, 14, 402. (35) Baumeier, B.; Kruger, P.; Pollmann, J. Phys. ReV. B 2007, 76, 085407. (36) Moon, W. H.; Ham, J. K.; Hwang, H. J. Nonatech 2003, 3, 158. (37) Tersoff, J. Phys. ReV. B 1988, 38, 9902. (38) Tersoff, J. Phys. ReV. B 1988, 37, 6991. (39) Zhang, Y.; Huang, H. Comput. Mater. Sci. 2008, 10, 1016. (40) Tang, M. J.; Yip, S. Phys. ReV. B 1995, 52, 15150. (41) Robertson, D. H.; Brenner, D. W.; Mintmire, J. W. Phys. ReV. B 1992, 45, 12592. (42) Quitoriano, N. J.; Belov, M.; Evoy, S.; Kamins, T. I. Nano Lett. 2009, 9, 1511. (43) Zhang, K.; Stock, G. M.; Zhong, J. Nanotechnology 2007, 18, 285703. (44) Liu, X. W.; Hu, J.; Pan, B. C. Physica E 2008, 40, 3042. (45) Tersoff, J. Phys. ReV. B 1989, 39, 5566. (46) Zhang, J.; Yuan, Q.; Wang, F.; Zhao, Y. Comput. Mater. Sci. 2009, 46, 621. (47) Yao, Z.; Zhu, C.; Cheng, M.; Liu, J. Comput. Mater. Sci. 2001, 180. (48) http://lammps.sandia.gov/doc/Section_intro.html#1_5; Plimpton, S. J. J. Comput. Phys. 1995, 117, 1. (49) Palaria, A.; Klimeck, G.; Strachan, A. Phys. ReV. B 2008, 78, 205315. (50) Wu, Y.; Huang, M.; Wang, F.; Huang, X.; Rosenblatt, S.; Huang, L.; Yan, H.; Brien, S.; Hone, J.; Heinz, T. Nano Lett. 2008, 8, 4158. (51) Perisanu, S.; Gouttenoire, V.; Vincent, P.; Ayari, A.; Choueib, M.; Bechelany, M.; Cornu, D.; Purcell, S. T. Phys. ReV. B 2008, 77, 165434. (52) Wei, C.; Cho, K.; Srivastava, D. Phys. ReV. B 2003, 67, 115407.

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(53) Yu, M. F.; Files, B. S.; Arepalli, S.; Ruoff, R. S. Phys. ReV. Lett. 2000, 84, 5552. (54) Walters, D. A. Appl. Phys. Lett. 1999, 74, 3803. (55) De Crescenzi, M.; Castrucci, P.; Scarcelli, M.; Diociauti, M.; Chaudhari, P. S.; Balasubramanian, C.; Bhave, T. M.; Bhoraskar, S. V. Appl. Phys. Lett. 2005, 86, 231901. (56) Harris, G. L. Properties of Silicon Carbide; Institution of Electrical Engineers: London, 1995. (57) Zhao, Q.; Nardelli, M. B.; Bernholc, J. Phys. ReV. B 2002, 65, 144105.

Liu et al. (58) Zhang, P.; Lammert, P. E.; Crespi, V. Phys. ReV. Lett. 1998, 81, 5346. (59) Nardelli, M. B.; Yakobson, B. I.; Bernholc, J. Phys. ReV. Lett. 1998, 81, 4656. (60) Lu, J. P. Phys. ReV. Lett. 1999, 79, 1297. (61) Xu, Z.; Wang, F.; Zheng, Q. Small 2008, 4, 733. (62) Ozaki, T.; Iwasa, Y.; Mitani, T. Phys. ReV. Lett. 2000, 83, 1712. (63) Zhang, C. L.; Shen, H. S. Appl. Phys. Lett. 2006, 89, 081904. (64) Wortman, J. J. J. Appl. Phys. 1965, 36, 153.

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