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A convenient method to determine the bulk modulus of nanowires and its temperature dependence based on X-ray diffraction measurement
Solid State Communications 150 (2010) 1117–1119
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
A convenient method to determine the bulk modulus of nanowires and its temperature dependence based on X-ray diffraction measurement Xin Feng Li, Guang Tao Fei ∗ , Wen Fei Zhou, Li De Zhang Key Laboratory of Materials Physics, Institute of Solid State Physics, Hefei Institutes of Physical Science, Chinese Academy of Sciences, P.O. Box 1129, Hefei 230031, People’s Republic of China Anhui Key Laboratory of Nanomaterials and Nanostructures, Institute of Solid State Physics, Hefei Institutes of Physical Science, Chinese Academy of Sciences, P.O. Box 1129, Hefei 230031, People’s Republic of China
article
info
Article history: Received 23 September 2009 Received in revised form 4 March 2010 Accepted 24 March 2010 by E.G. Wang Available online 1 April 2010
abstract The bulk modulus of nanowires (NWs) and its temperature dependence were determined by a simple and convenient method based on temperature-dependent X-ray diffraction (XRD) measurement. It was found that the bulk moduli for Ni, Cu, and Ag NWs were much higher than that for their counterpart bulk materials in the temperature range from 25 °C to 800 °C and the influence of temperature on the bulk modulus for NWs was stronger than that for their counterpart bulk materials. A surface bond contraction model and the force–interatomic-distance curves were introduced to explain the experimental results. © 2010 Elsevier Ltd. All rights reserved.
Keywords: A. Nanowire D. Bulk modulus D. Coefficient of thermal expansion
1. Introduction
2. Methods
Nanowires (NWs) have attracted considerable interest due to their unique properties and potential applications [1–4]. As one of the fundamental properties, mechanical properties are of great importance for application of NWs as the basic building block in nano-electromechanical systems (NEMS) or other systems [5,6]. So the research on the mechanical properties of NWs is vital for their applications. In the past few years, atomic force microscopy (AFM) [7–9], electric-field-induced mechanical resonance in transmission electron microscopy (TEM) [10,11], and scanning electron microscopy (SEM) have been developed to characterize the mechanical properties of NWs. Although these methods can give the mechanical parameters of NWs, they suffer from the difficulties of sample manipulation and testing due to the small size of the objects. Therefore, a simple and easy method to obtain the mechanical properties of NWs is still a challenge. It is known that there exists an inherent relation between thermal expansion and the bulk modulus since both of them are related to interatomic interaction. Here in this paper, we determine the bulk modulus, one kind of mechanical property, and its variation of NWs as a function of temperature based on XRD heating measurement.
As is well known, the bulk modulus can be expressed as the second derivative of the total energy of the crystal versus volume [12]. Here, in order to calculate crystal energy, a Morse potential function [13] is adopted to simulate the interatomic interaction, which is given by
∗
Corresponding author. Tel.: +86 551 5591453; fax: +86 551 5591434. E-mail address: [email protected] (G.T. Fei).
0038-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2010.03.027
u(r ) = D[e−2α(r −r0 ) − 2e−α(r −r0 ) ],
(1)
where D is the dissociation energy, α is an inverse width of the potential, and r0 is the nearest equilibrium distance of atoms. The total energy of the crystal φ is expressed as the sum of Eq. (1) over the entire crystal
φ=
1X 2
u(rij ) =
i,j
1 2
ND
X
−α(ri −r0 )
[e−2α(ri −r0 ) − 2e
],
(2)
i
where rij is the distance from the ith atom to the jth atom and N is the total number of atoms in the crystal. The bulk modulus can be given by [12]
B = V0
d2 φ dV 2
.
(3)
r =r 0
Once the total energy φ is known, it is possible to calculate the bulk modulus from Eq. (3). It can be seen from Eq. (2) that the total energy φ can be easily obtained after the parameters D and α are known.
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X.F. Li et al. / Solid State Communications 150 (2010) 1117–1119
In general, the value D can be obtained from cohesive energy. Gülseren et al. [14] have studied the cohesive energy of Al and Pb NWs. Their results show that NWs’ cohesive energies are smaller than their corresponding bulk materials’ because of the appearance of surface energy. However, their results show that for NWs with diameter larger than 20 nm, the difference of cohesive energy between bulk and NWs can be ignored. Therefore, it is rational to use the cohesive energy of bulk material in calculating the value of D for NWs with diameter bigger than 20 nm. For simplicity, we only consider the nearest-neighbor interaction in the calculation of Eq. (2). The other parameter α can be obtained by the following process. Assuming that metallic atoms behave like some anharmonic Morse oscillators and do not couple with each other, and adopting Debye model and Boltzmann distribution law, Ruffa [15] obtained an analytic formula for coefficient of thermal expansion (CTE), βCTE , which can be given by
βCTE =
3k
2α Dr
T
3
θD
g (xD ).
(4)
a
b
c
Therefore, the parameter α can be expressed as
α=
3k 2βCTE Dr
T
3
θD
g (xD ),
(5)
dx.
(6)
where g ( xD ) =
xD
Z 0
x 4 ex
(ex − 1)2 θ
¯ω , x x = hkT = TD , h¯ , ω, k and θD are Planck constant, the D frequency of oscillator, Boltzmann constant and Debye temperature, respectively. Substituting experimental βCTE into Eq. (5), the value α can be calculated. However, due to the limit of the Morse potential, this method is valid only for simple cubic metals rather than semiconductors because of different bond types. This method also fails at low temperature. When the temperature is low enough, based on Grüneisen law, the effect of electron on specific heat and further on the thermal expansion can not be ignored compared with the effect of lattice vibration. Therefore, the method can not be used at low temperature because the Morse potential can not accurately depict the electronic effect.
3. Results and discussion Based on our previous experimental results [16–18], the βCTE curves of NWs and their corresponding bulk materials are plotted in Fig. 1. It can be seen clearly that the CTEs of NWs increase from 3.95 × 10−6 /K at 25 °C to 10.52 × 10−6 /K at 800 °C for Ni, from 1.86 × 10−6 /K to 8.11 × 10−6 /K for Cu, and from 1.33 × 10−6 /K to 10.47 × 10−6 /K for Ag, all of which are smaller than the values of their counterpart bulk materials. Fig. 2(a), (b) and (c) are the relations of the bulk modulus versus temperature for Ni, Cu and Ag NWs with diameter of 45 nm, 30 nm and 38 nm, respectively. It can be seen that the calculated bulk moduli decrease dramatically from 2510 GPa at 25 °C to 431 GPa at 800 °C for Ni NWs, from 14 667 GPa to 864 GPa for Cu NWs, and from 24 801 GPa to 425 GPa for Ag NWs. The insets of Fig. 2(a), (b) and (c) are relationships between the bulk modulus and temperature for bulk Ni, Cu and Ag, respectively. At 25 °C, the calculated bulk moduli are 209.9 GPa for bulk Ni, 177 GPa for bulk Cu and 125.08 GPa for bulk Ag, which are close to the previously reported values of 186 GPa for Ni, 137 GPa for Cu and 100.7 GPa for Ag [12], indicating availability of our method. Fig. 2 shows that the calculated bulk moduli of NWs are much higher than that of bulk materials over the range of entire experimental temperatures, and the influence of temperature is more significant on NWs than that on bulk materials. The similar result was also reported by Gu
Fig. 1. Temperature-dependent thermal expansion of (a) Ni NWs, (b) Cu NWs, (c) Ag and their corresponding bulk materials in the temperature range of 25–800 ° C.
a
b
c
Fig. 2. Variation of the bulk modulus B of (a) Ni, (b) Cu and (c) Ag NWs as a function temperature T . The upper insets are variance of the bulk modulus with temperature corresponding to their bulk materials.
X.F. Li et al. / Solid State Communications 150 (2010) 1117–1119
1119
4. Conclusion In conclusion, the bulk moduli of NWs were determined by a simple and convenient method based on temperature-dependent XRD measurement. The results show that the NWs’ bulk moduli are much higher than their counterpart bulk materials’ in the temperature range of 25–800 °C and the influence of temperature on the bulk modulus for NWs is stronger than that for bulk materials. A surface bond contraction model and force–interatomic distance curves are introduced to explain the experimental results. Acknowledgements
Fig. 3. Plot of curves of force–interatomic distance for NWs and bulk materials.
et al. [19]. Their results show that the bulk moduli of Au and Ag nanoparticles with diameter of 30 nm and 10 nm are about 290 GPa and 139 GPa, respectively, which are higher than 172.5 GPa and 116 GPa for bulk Au and Ag. Chen et al. [11] revealed that the Young’s modulus of ZnO NWs with diameter of 17 nm was about 220 GPa, which was bigger than 140 GPa for bulk ZnO. It is expected that surface effect is the reason for the bulk modulus enhancement for NWs. As coordination numbers of surface atoms reduction, the surface atoms have a lower electron density than the bulk atoms. In order to increase surface local electron density, the surface bond length will contract, which will results in generation of surface stress f due to the different configuration of surface atoms compared with bulk atoms [20–23]. The existence of surface stress will lead to surface pressure P on interior atoms and additional elastic energy, according to Laplace–Young equation, P =
2f
(7)
r
where r is the radius of a spherical particle. For NWs, due to their large surface-to-volume ratio, the surface stress and surface pressure will play a dominant role in the bulk modulus enhancement, so the bulk modulus will enhance. The temperature effect on the bulk modulus can be explained by force–interatomic distance curves [24], as shown in Fig. 3. The bulk modulus is proportional to the slope of force curve, B∝−
dF dr
.
(8)
As the interatomic distance increases with the temperature rising, the absolute value of the slope dF around r0 decreases, thus the dr bulk modulus decreases according to Eq. (8). In Fig. 3, the force curve of NWs is steeper than that of bulk, which means that the slopes of curves at different point change faster for NWs than that for bulk. As a result, the bulk moduli of NWs decrease more dramatically than that of bulk materials with temperature rising.
We are grateful to Professor L.J. Zou and to Professor C.S. Liu for some interesting discussions. This work has been supported by the National Natural Science Foundation of China (Nos. 50671099, 50172048, 10374090 and 10274085), Ministry of Science and Technology of China (No. 2005CB623603), and Hundred Talent Program of Chinese Academy of Sciences. References [1] B. Tian, X. Zheng, T.J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, C.M. Lieber, Nature 449 (2007) 885–889. [2] A. Javey, S. Nam, R.S. Friedman, H. Yan, C.M. Lieber, Nano Lett. 7 (2007) 773–777. [3] A. Kolmakov, Y. Zhang, G. Cheng, M. Moskovits, Adv. Mater. 15 (2003) 997–1000. [4] Q. Wan, Q.H. Li, Y.J. Chen, T.H. Wang, X.L. He, J.P. Li, C.L. Lin, Appl. Phys. Lett. 84 (2004) 3654–3656. [5] K.L. Ekinci, M.L. Roukes, Rev. Sci. Instrum. 76 (2005) 061101. [6] A. Husain, J. Hone, H.W.C. Postma, X.M.H. Huang, T. Drake, M. Barbic, A. Scherer, M.L. Roukes, Appl. Phys. Lett. 83 (2003) 1240–1242. [7] B. Wu, A. Heidelberg, J.J. Boland, Nature Mater. 4 (2005) 525–529. [8] G. Stan, C.V. Ciobanu, P.M. Parthangal, R.F. Cook, Nano Lett. 7 (2007) 3691–3697. [9] M.F. Yu, O. Lourie, M.J. Dyer, K. Moloni, T.F. Kelly, R.S. Ruoff, Science 287 (2000) 637–640. [10] R.P. Gao, Z.L. Wang, Z.G. Bai, W.A. de Heer, L.M. Dai, M. Gao, Phys. Rev. Lett. 85 (2000) 622. [11] C.Q. Chen, Y. Shi, Y.S. Zhang, J. Zhu, Y.J. Yan, Phys. Rev. Lett. 96 (2006) 075505. [12] C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, New York, 1976, pp. 84, 85, 74. [13] P.M. Morse, Phys. Rev. 34 (1929) 57–64. [14] O. Gülseren, F. Ercolessi, E. Tosatti, Phys. Rev. Lett. 80 (1998) 3775. [15] A.R. Ruffa, Phys. Rev. B 16 (1977) 2504–2514. [16] X.W. Wang, G.T. Fei, B. Wang, M. Wang, L.D. Zhang, Solid State Sci. 10 (2008) 1185–1188. [17] W.F. Zhou, G.T. Fei, X.F. Li, S.H. Xu, L. Chen, B. Wu, L.D. Zhang, J. Phys. Chem. C 113 (2009) 9568–9572. [18] X.J. Xu, G.T. Fei, W.H. Yu, L.D. Zhang, X. Ju, X.P. Hao, D.N. Wang, B.Y. Wang, Appl. Phys. Lett. 89 (2006) 181914. [19] Q.F. Gu, G. Krauss, W. Steurer, F. Gramm, A. Cervellino, Phys. Rev. Lett. 100 (2008) 045502. [20] C.Q. Sun, B.K. Tay, X.T. Zeng, S. Li, T.P. Chen, J. Zhou, H.L. Bai, E.Y. Jiang, J. Phys.: Condens. Matter 14 (2002) 7781–7795. [21] X.J. Liu, J.W. Li, Z.F. Zhou, L.W. Yang, Z.S. Ma, G.F. Xie, Y. Pan, C.Q. Sun, Appl. Phys. Lett. 94 (2009) 131902. [22] G. Wang, X. Li, J. Appl. Phys. 104 (2008) 113517. [23] R.C. Cammarata, K. Sieradzki, Annu. Rev. Mater. Sci. 24 (1994) 215–234. [24] A.J. Walton, Three Phases of Matter, Oxford Press, Oxford, 1983, p. 282.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
A convenient method to determine the bulk modulus of nanowires and its temperature dependence based on X-ray diffraction measurement Xin Feng Li, Guang Tao Fei ∗ , Wen Fei Zhou, Li De Zhang Key Laboratory of Materials Physics, Institute of Solid State Physics, Hefei Institutes of Physical Science, Chinese Academy of Sciences, P.O. Box 1129, Hefei 230031, People’s Republic of China Anhui Key Laboratory of Nanomaterials and Nanostructures, Institute of Solid State Physics, Hefei Institutes of Physical Science, Chinese Academy of Sciences, P.O. Box 1129, Hefei 230031, People’s Republic of China
article
info
Article history: Received 23 September 2009 Received in revised form 4 March 2010 Accepted 24 March 2010 by E.G. Wang Available online 1 April 2010
abstract The bulk modulus of nanowires (NWs) and its temperature dependence were determined by a simple and convenient method based on temperature-dependent X-ray diffraction (XRD) measurement. It was found that the bulk moduli for Ni, Cu, and Ag NWs were much higher than that for their counterpart bulk materials in the temperature range from 25 °C to 800 °C and the influence of temperature on the bulk modulus for NWs was stronger than that for their counterpart bulk materials. A surface bond contraction model and the force–interatomic-distance curves were introduced to explain the experimental results. © 2010 Elsevier Ltd. All rights reserved.
Keywords: A. Nanowire D. Bulk modulus D. Coefficient of thermal expansion
1. Introduction
2. Methods
Nanowires (NWs) have attracted considerable interest due to their unique properties and potential applications [1–4]. As one of the fundamental properties, mechanical properties are of great importance for application of NWs as the basic building block in nano-electromechanical systems (NEMS) or other systems [5,6]. So the research on the mechanical properties of NWs is vital for their applications. In the past few years, atomic force microscopy (AFM) [7–9], electric-field-induced mechanical resonance in transmission electron microscopy (TEM) [10,11], and scanning electron microscopy (SEM) have been developed to characterize the mechanical properties of NWs. Although these methods can give the mechanical parameters of NWs, they suffer from the difficulties of sample manipulation and testing due to the small size of the objects. Therefore, a simple and easy method to obtain the mechanical properties of NWs is still a challenge. It is known that there exists an inherent relation between thermal expansion and the bulk modulus since both of them are related to interatomic interaction. Here in this paper, we determine the bulk modulus, one kind of mechanical property, and its variation of NWs as a function of temperature based on XRD heating measurement.
As is well known, the bulk modulus can be expressed as the second derivative of the total energy of the crystal versus volume [12]. Here, in order to calculate crystal energy, a Morse potential function [13] is adopted to simulate the interatomic interaction, which is given by
∗
Corresponding author. Tel.: +86 551 5591453; fax: +86 551 5591434. E-mail address: [email protected] (G.T. Fei).
0038-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2010.03.027
u(r ) = D[e−2α(r −r0 ) − 2e−α(r −r0 ) ],
(1)
where D is the dissociation energy, α is an inverse width of the potential, and r0 is the nearest equilibrium distance of atoms. The total energy of the crystal φ is expressed as the sum of Eq. (1) over the entire crystal
φ=
1X 2
u(rij ) =
i,j
1 2
ND
X
−α(ri −r0 )
[e−2α(ri −r0 ) − 2e
],
(2)
i
where rij is the distance from the ith atom to the jth atom and N is the total number of atoms in the crystal. The bulk modulus can be given by [12]
B = V0
d2 φ dV 2
.
(3)
r =r 0
Once the total energy φ is known, it is possible to calculate the bulk modulus from Eq. (3). It can be seen from Eq. (2) that the total energy φ can be easily obtained after the parameters D and α are known.
1118
X.F. Li et al. / Solid State Communications 150 (2010) 1117–1119
In general, the value D can be obtained from cohesive energy. Gülseren et al. [14] have studied the cohesive energy of Al and Pb NWs. Their results show that NWs’ cohesive energies are smaller than their corresponding bulk materials’ because of the appearance of surface energy. However, their results show that for NWs with diameter larger than 20 nm, the difference of cohesive energy between bulk and NWs can be ignored. Therefore, it is rational to use the cohesive energy of bulk material in calculating the value of D for NWs with diameter bigger than 20 nm. For simplicity, we only consider the nearest-neighbor interaction in the calculation of Eq. (2). The other parameter α can be obtained by the following process. Assuming that metallic atoms behave like some anharmonic Morse oscillators and do not couple with each other, and adopting Debye model and Boltzmann distribution law, Ruffa [15] obtained an analytic formula for coefficient of thermal expansion (CTE), βCTE , which can be given by
βCTE =
3k
2α Dr
T
3
θD
g (xD ).
(4)
a
b
c
Therefore, the parameter α can be expressed as
α=
3k 2βCTE Dr
T
3
θD
g (xD ),
(5)
dx.
(6)
where g ( xD ) =
xD
Z 0
x 4 ex
(ex − 1)2 θ
¯ω , x x = hkT = TD , h¯ , ω, k and θD are Planck constant, the D frequency of oscillator, Boltzmann constant and Debye temperature, respectively. Substituting experimental βCTE into Eq. (5), the value α can be calculated. However, due to the limit of the Morse potential, this method is valid only for simple cubic metals rather than semiconductors because of different bond types. This method also fails at low temperature. When the temperature is low enough, based on Grüneisen law, the effect of electron on specific heat and further on the thermal expansion can not be ignored compared with the effect of lattice vibration. Therefore, the method can not be used at low temperature because the Morse potential can not accurately depict the electronic effect.
3. Results and discussion Based on our previous experimental results [16–18], the βCTE curves of NWs and their corresponding bulk materials are plotted in Fig. 1. It can be seen clearly that the CTEs of NWs increase from 3.95 × 10−6 /K at 25 °C to 10.52 × 10−6 /K at 800 °C for Ni, from 1.86 × 10−6 /K to 8.11 × 10−6 /K for Cu, and from 1.33 × 10−6 /K to 10.47 × 10−6 /K for Ag, all of which are smaller than the values of their counterpart bulk materials. Fig. 2(a), (b) and (c) are the relations of the bulk modulus versus temperature for Ni, Cu and Ag NWs with diameter of 45 nm, 30 nm and 38 nm, respectively. It can be seen that the calculated bulk moduli decrease dramatically from 2510 GPa at 25 °C to 431 GPa at 800 °C for Ni NWs, from 14 667 GPa to 864 GPa for Cu NWs, and from 24 801 GPa to 425 GPa for Ag NWs. The insets of Fig. 2(a), (b) and (c) are relationships between the bulk modulus and temperature for bulk Ni, Cu and Ag, respectively. At 25 °C, the calculated bulk moduli are 209.9 GPa for bulk Ni, 177 GPa for bulk Cu and 125.08 GPa for bulk Ag, which are close to the previously reported values of 186 GPa for Ni, 137 GPa for Cu and 100.7 GPa for Ag [12], indicating availability of our method. Fig. 2 shows that the calculated bulk moduli of NWs are much higher than that of bulk materials over the range of entire experimental temperatures, and the influence of temperature is more significant on NWs than that on bulk materials. The similar result was also reported by Gu
Fig. 1. Temperature-dependent thermal expansion of (a) Ni NWs, (b) Cu NWs, (c) Ag and their corresponding bulk materials in the temperature range of 25–800 ° C.
a
b
c
Fig. 2. Variation of the bulk modulus B of (a) Ni, (b) Cu and (c) Ag NWs as a function temperature T . The upper insets are variance of the bulk modulus with temperature corresponding to their bulk materials.
X.F. Li et al. / Solid State Communications 150 (2010) 1117–1119
1119
4. Conclusion In conclusion, the bulk moduli of NWs were determined by a simple and convenient method based on temperature-dependent XRD measurement. The results show that the NWs’ bulk moduli are much higher than their counterpart bulk materials’ in the temperature range of 25–800 °C and the influence of temperature on the bulk modulus for NWs is stronger than that for bulk materials. A surface bond contraction model and force–interatomic distance curves are introduced to explain the experimental results. Acknowledgements
Fig. 3. Plot of curves of force–interatomic distance for NWs and bulk materials.
et al. [19]. Their results show that the bulk moduli of Au and Ag nanoparticles with diameter of 30 nm and 10 nm are about 290 GPa and 139 GPa, respectively, which are higher than 172.5 GPa and 116 GPa for bulk Au and Ag. Chen et al. [11] revealed that the Young’s modulus of ZnO NWs with diameter of 17 nm was about 220 GPa, which was bigger than 140 GPa for bulk ZnO. It is expected that surface effect is the reason for the bulk modulus enhancement for NWs. As coordination numbers of surface atoms reduction, the surface atoms have a lower electron density than the bulk atoms. In order to increase surface local electron density, the surface bond length will contract, which will results in generation of surface stress f due to the different configuration of surface atoms compared with bulk atoms [20–23]. The existence of surface stress will lead to surface pressure P on interior atoms and additional elastic energy, according to Laplace–Young equation, P =
2f
(7)
r
where r is the radius of a spherical particle. For NWs, due to their large surface-to-volume ratio, the surface stress and surface pressure will play a dominant role in the bulk modulus enhancement, so the bulk modulus will enhance. The temperature effect on the bulk modulus can be explained by force–interatomic distance curves [24], as shown in Fig. 3. The bulk modulus is proportional to the slope of force curve, B∝−
dF dr
.
(8)
As the interatomic distance increases with the temperature rising, the absolute value of the slope dF around r0 decreases, thus the dr bulk modulus decreases according to Eq. (8). In Fig. 3, the force curve of NWs is steeper than that of bulk, which means that the slopes of curves at different point change faster for NWs than that for bulk. As a result, the bulk moduli of NWs decrease more dramatically than that of bulk materials with temperature rising.
We are grateful to Professor L.J. Zou and to Professor C.S. Liu for some interesting discussions. This work has been supported by the National Natural Science Foundation of China (Nos. 50671099, 50172048, 10374090 and 10274085), Ministry of Science and Technology of China (No. 2005CB623603), and Hundred Talent Program of Chinese Academy of Sciences. References [1] B. Tian, X. Zheng, T.J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, C.M. Lieber, Nature 449 (2007) 885–889. [2] A. Javey, S. Nam, R.S. Friedman, H. Yan, C.M. Lieber, Nano Lett. 7 (2007) 773–777. [3] A. Kolmakov, Y. Zhang, G. Cheng, M. Moskovits, Adv. Mater. 15 (2003) 997–1000. [4] Q. Wan, Q.H. Li, Y.J. Chen, T.H. Wang, X.L. He, J.P. Li, C.L. Lin, Appl. Phys. Lett. 84 (2004) 3654–3656. [5] K.L. Ekinci, M.L. Roukes, Rev. Sci. Instrum. 76 (2005) 061101. [6] A. Husain, J. Hone, H.W.C. Postma, X.M.H. Huang, T. Drake, M. Barbic, A. Scherer, M.L. Roukes, Appl. Phys. Lett. 83 (2003) 1240–1242. [7] B. Wu, A. Heidelberg, J.J. Boland, Nature Mater. 4 (2005) 525–529. [8] G. Stan, C.V. Ciobanu, P.M. Parthangal, R.F. Cook, Nano Lett. 7 (2007) 3691–3697. [9] M.F. Yu, O. Lourie, M.J. Dyer, K. Moloni, T.F. Kelly, R.S. Ruoff, Science 287 (2000) 637–640. [10] R.P. Gao, Z.L. Wang, Z.G. Bai, W.A. de Heer, L.M. Dai, M. Gao, Phys. Rev. Lett. 85 (2000) 622. [11] C.Q. Chen, Y. Shi, Y.S. Zhang, J. Zhu, Y.J. Yan, Phys. Rev. Lett. 96 (2006) 075505. [12] C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, New York, 1976, pp. 84, 85, 74. [13] P.M. Morse, Phys. Rev. 34 (1929) 57–64. [14] O. Gülseren, F. Ercolessi, E. Tosatti, Phys. Rev. Lett. 80 (1998) 3775. [15] A.R. Ruffa, Phys. Rev. B 16 (1977) 2504–2514. [16] X.W. Wang, G.T. Fei, B. Wang, M. Wang, L.D. Zhang, Solid State Sci. 10 (2008) 1185–1188. [17] W.F. Zhou, G.T. Fei, X.F. Li, S.H. Xu, L. Chen, B. Wu, L.D. Zhang, J. Phys. Chem. C 113 (2009) 9568–9572. [18] X.J. Xu, G.T. Fei, W.H. Yu, L.D. Zhang, X. Ju, X.P. Hao, D.N. Wang, B.Y. Wang, Appl. Phys. Lett. 89 (2006) 181914. [19] Q.F. Gu, G. Krauss, W. Steurer, F. Gramm, A. Cervellino, Phys. Rev. Lett. 100 (2008) 045502. [20] C.Q. Sun, B.K. Tay, X.T. Zeng, S. Li, T.P. Chen, J. Zhou, H.L. Bai, E.Y. Jiang, J. Phys.: Condens. Matter 14 (2002) 7781–7795. [21] X.J. Liu, J.W. Li, Z.F. Zhou, L.W. Yang, Z.S. Ma, G.F. Xie, Y. Pan, C.Q. Sun, Appl. Phys. Lett. 94 (2009) 131902. [22] G. Wang, X. Li, J. Appl. Phys. 104 (2008) 113517. [23] R.C. Cammarata, K. Sieradzki, Annu. Rev. Mater. Sci. 24 (1994) 215–234. [24] A.J. Walton, Three Phases of Matter, Oxford Press, Oxford, 1983, p. 282.