5
Investigation of plasticity in silicon nanowires by molecular dynamics simulations. J. Guénolé, J. Godet a , and S. Brochard Institut Pprime Département de Physique et Mécanique des matériaux CNRS - Université de Poitiers – ENSMA UPR 3346 SP2MI - BP 30179 86962 Futuroscope Chasseneuil Cedex, FRANCE a email: [email protected] Keywords: Semiconductor, silicon, plasticity, nanowire, dislocation, molecular dynamics Abstract. We have performed molecular dynamics simulations on silicon nanowires (Si-NW) with [001] axis and square section. The forces are modeled by well-tested semi-empirical potentials. First we investigated the edge reconstruction of Si nanowires. Then, we studied the behavior of the NW when submitted to compression stresses along its axis. At low temperature (300K), we observed the formation of dislocation loops with a Burgers vector 1/2 [10-1]. These dislocations slip in the unexpected {101} planes having the largest Schmid factor. Introduction Since it is possible to build Si-NWs and nanopillars, numerous experimental investigations have been realized to understand the plasticity of such objects. Surprisingly many results reveal an elastic limit close to the theoretical yield stress of bulk silicon [1]. A second aspect is the competition between fracture and plastic deformation at such small scale. Indeed, it is well known that bulk silicon is brittle at low temperature; however it can be plastically deformed when it is confined under very high confining pressure to avoid fracture [2]. The same phenomenon appears in nano object where plasticity could appear before fracture [3]. Here we aim at studying the plastic behavior of Si-NWs by molecular dynamics simulations, a first necessary step being the determination of the stable undeformed state of Si-NWs. I – Simulation cell and technique We used Si-NWs orientated along the [001] direction. The section of the nanowire is square and it is delimited by (100) and (010) surfaces. In this study the size of the silicon nanowire is 20 a 0 along the [001] direction and 5 to 10 a 0 along [100] and [010] directions, where a 0 is the cell parameter of silicon (a 0 =5.43 Å). An infinite nanowire is obtained by using periodic conditions along the nanowire axis [001]. To investigate the onset of plasticity we applied a uniaxial stress along the Si-NW axis. We strained the system in compression with a constant strain rate about 10 7 s -1 , the lateral surfaces being free. This strain rate is many order higher than the experimental ones, but it is a common value used in simulations to keep reasonable CPU time [4]. The atomic interactions are modelled by the semi-empirical potential of Stillinger-Weber [5] which is quite robust [6] and has already produced good results on the onset of plasticity in silicon from Key Engineering Materials Vol. 465 (2011) pp 89-92 Online available since 2011/Jan/20 at www.scientific.net © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/KEM.465.89 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 129.93.16.3, University of Nebraska-Lincoln, Lincoln, USA-06/11/14,01:45:01)

Investigation of Plasticity in Silicon Nanowires by Molecular Dynamics Simulations

Embed Size (px)

Citation preview

Page 1: Investigation of Plasticity in Silicon Nanowires by Molecular Dynamics Simulations

Investigation of plasticity in silicon nanowires by molecular dynamics

simulations.

J. Guénolé, J. Godeta, and S. Brochard

Institut Pprime Département de Physique et Mécanique des matériaux

CNRS - Université de Poitiers – ENSMA UPR 3346 SP2MI - BP 30179

86962 Futuroscope Chasseneuil Cedex, FRANCE

aemail: [email protected]

Keywords: Semiconductor, silicon, plasticity, nanowire, dislocation, molecular dynamics

Abstract.

We have performed molecular dynamics simulations on silicon nanowires (Si-NW) with [001] axis and square section. The forces are modeled by well-tested semi-empirical potentials. First we investigated the edge reconstruction of Si nanowires. Then, we studied the behavior of the NW when submitted to compression stresses along its axis. At low temperature (300K), we observed the formation of dislocation loops with a Burgers vector 1/2 [10-1]. These dislocations slip in the unexpected {101} planes having the largest Schmid factor.

Introduction

Since it is possible to build Si-NWs and nanopillars, numerous experimental investigations have been realized to understand the plasticity of such objects. Surprisingly many results reveal an elastic limit close to the theoretical yield stress of bulk silicon [1]. A second aspect is the competition between fracture and plastic deformation at such small scale. Indeed, it is well known that bulk silicon is brittle at low temperature; however it can be plastically deformed when it is confined under very high confining pressure to avoid fracture [2]. The same phenomenon appears in nano object where plasticity could appear before fracture [3]. Here we aim at studying the plastic behavior of Si-NWs by molecular dynamics simulations, a first necessary step being the determination of the stable undeformed state of Si-NWs.

I – Simulation cell and technique

We used Si-NWs orientated along the [001] direction. The section of the nanowire is square and it is delimited by (100) and (010) surfaces. In this study the size of the silicon nanowire is 20 a0 along the [001] direction and 5 to 10 a0 along [100] and [010] directions, where a0 is the cell parameter of silicon (a0=5.43 Å). An infinite nanowire is obtained by using periodic conditions along the nanowire axis [001]. To investigate the onset of plasticity we applied a uniaxial stress along the Si-NW axis. We strained the system in compression with a constant strain rate about 107 s-1, the lateral surfaces being free. This strain rate is many order higher than the experimental ones, but it is a common value used in simulations to keep reasonable CPU time [4]. The atomic interactions are modelled by the semi-empirical potential of Stillinger-Weber [5] which is quite robust [6] and has already produced good results on the onset of plasticity in silicon from

Key Engineering Materials Vol. 465 (2011) pp 89-92Online available since 2011/Jan/20 at www.scientific.net© (2011) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/KEM.465.89

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 129.93.16.3, University of Nebraska-Lincoln, Lincoln, USA-06/11/14,01:45:01)

Page 2: Investigation of Plasticity in Silicon Nanowires by Molecular Dynamics Simulations

surface defect [7, 8]. We studied the system either at 0 K through an algorithm of conjugate gradients, or either at finite temperature by performing molecular dynamics simulations. We used the LAMMPS code [9] for our simulation and we controlled the temperature thanks to the Nose-Hoover thermostat. II – Edge geometry optimization

In this part we investigate the edge reconstruction of undeformed squared nanowires in order to determine the optimum geometry. There are several possibilities for etching a nanowire in silicon bulk. One of these gives one-fold coordinated silicon atoms on the four edges of the structure (Fig.1 -a, -b). This non reconstructed configuration is noted A1. To decrease the energy of the Si-NW, we looked for configurations where atoms were more coordinated. A simple way was to make two bonds more by approaching the edge atom to the surface leading to three atoms three-fold coordinated on Si-NW edges (Fig.1.-c). This configuration is noted A2. Another way was to remove all atoms with only one bond to make the Si-NW more stable (Fig.1 -d, -e). We note this configuration A3. A volume atom then becomes a surface atom with a dangling bond i.e. three-fold coordinated.

Figure 1. (a) and (b) atomic con-figuration A1 of non reconstructed Si-NW, with one-fold coordinated atoms on edges. (c) Same configuration with reconstructed edge noted A2. (d) and (e) atomic configuration of Si-NW noted A3, obtained by removing the one-fold coordinated edge atoms of configuration A1.

A second possibility to construct a square Si-NW is to shift the origin of the NW by a vector of ¼ [010] inside the silicon matrix (Fig. 2). We then obtain a Si-NW without one-fold coordinated atoms. In that case the edge atoms are two fold coordinated. This configuration is noted B1.

Figure 2. Second way for cutting a silicon NW in bulk matrix: configuration B1.

The comparison of the different Si-NWs is not trivial due to their different sizes and their number of atoms that varies between configurations. The Si-NW energy ENW can be decomposed in two parts

[100]

[110]

[001]

[010]

81810880018

(a) (b)

[100]

[110]

[001]

[010]

81810880018

(a) (b)(c)

(a) (b)

[100]

[110]

[001]

[010]

81810880018

(a) (b)(d) (e)

[100]

[110]

[001]

[010]

81810880018

(a) (b)

A3

B1

90 Materials Structure & Micromechanics of Fracture VI

Page 3: Investigation of Plasticity in Silicon Nanowires by Molecular Dynamics Simulations

[10]: one for the volume EV and one for the perimeter EP. EP encompasses the energy contribution from the surface ES and from the edges EE which are difficult to decorrelate (Ep=ES+EE). All of these quantities are in eV/atom. When the section of the nanowire goes to infinity, the perimeter energy EP becomes negligible with respect to the volume energy EV. We plot the perimeter energy of several Si-NWs as a function of the inverse of the perimeter for a better visualization (Fig. 3). We note that the real value of the Si-NW perimeter cannot be easily obtained. We then took the square root of the number of atoms contained in the section of the Si-NW, this value being proportional to the perimeter length.

Figure 3. Perimeter energy (eV/atom) of several Si-NWs as a function of the inverse of the perimeter. Lines are guides for the eyes. As expected the perimeter energy reaches zero when the perimeter goes to infinity (i.e. when P-1 goes to 0). However for Si-NWs with small section, the perimeter energy becomes significant with values varying from 1.0 to 1.5 eV/atom. The smallest perimeters correspond to section sides about 1 or 2 cell parameters. The non reconstructed edge (configuration A1) is effectively the most energetic due to the

edge atoms under coordinated with only one bond. Obviously the Si-NW with two-fold coordinated edge atoms (configuration B1) is the second most energetic structure, the most stable being configurations A2 and A3 with three-fold coordinated edge atoms. Despite the slight distortion of the atomic bond, the greater stability of A2 could be explained by a largest number of three-fold coordinated atoms on the edge than in the configuration A3.

We then took into account the effect of surface reconstruction on the Si-NW stability. We considered the simple 2x1 reconstruction of the {100} surfaces [11]. As expected the perimeter energy is decreased for all configurations without changing the respective stability. In consequence we used the most stable configuration A2 with 2x1 surface reconstruction for the study of the onset of plasticity in Si-NWs.

III – Onset of plasticity in Si-NW in compression

Figure 4. Nucleation of dislocation loop from the Si-NW surfaces, when submitted to large compression stress at 300K. (a) to (d) evolution of the dislocation loop as a function of time. The upper panel is the top view of the Si-NW projected along the [001] direction, the bottom is a side view projected along the [100] direction. We used the Von Mises criterion implemented in the atomeye viewer [12] for defect visualization.

Key Engineering Materials Vol. 465 91

Page 4: Investigation of Plasticity in Silicon Nanowires by Molecular Dynamics Simulations

We then investigated the onset of plastic behavior of the optimized Si-NW in compression at 300K. During the first part of the compression only surfaces show various bond formations and bond ruptures due to thermal agitation. When sufficiently high strain level is reached (about 13.2%), we can observe the formation of dislocation embryos from Si-NW surfaces (Fig.4 -a). While this strain level is quite large, it remains well bellow the elasticity limit of the system calculated at 0K (around 24.5 %) . When embryos are large enough, a dislocation loop is nucleated and shears the Si-NW, relaxing the applied stress. The analysis of the plastic event reveals a Burgers vector equal to ½ a0 [10-1], but amazingly the slip plane is (101) whereas the usual one is (111) in silicon. Nevertheless because of the geometry of the Si-NW, the resolved shear stress is maximum in the (101) slip plane with a Schmid factor of 0.5. Note that the activation of this slip plane has already been observed in a similar study [13]. We performed the same calculation with a more robust MEAM (modified embedded atomic method) potential [14], and we obtained similar plastic events with glide in the (101) plane. Finally we think that the stress orientation applied on our Si-NWs does not favor any usual slip system, which could lead to the activation of this unexpected slip plane.

Conclusion

As a first step for the study of Si-NWs plasticity, we determined the optimum edge reconstruction for a Si-NW with square basis. It appears that the most stable configuration correspond to this one with the higher number of three-fold coordinated atoms in corners, despite the distorted bond arrangement introduced. The optimized Si-NW is then submitted to large uniaxial stress, and dislocations formation from surfaces is observed. Unexpectedly, the dislocations are nucleated in a (101) plane, whatever the potential used. This slip system, though not being usual in the zinc-blende structure, is the one with the maximum Schmid factor. More investigations are currently in progress in our group to better understand this astonishing plasticity in Si-NWs.

References

[1] T. Kizuka, Y. Takatani, K. Asaka, and R. Yoshizaki: Phys. Rev. B Vol. 72 (2005), p. 035333 [2] J. Rabier, P. Cordier, T. Tondellier, J. L. Demenet, and H. Garem: J. Phys.: Condens. Matter

Vol. 12 (2000), p. 10059

[3] F. Ӧstlund et al.: Adv. Funct. Mater. Vol. 19 (2009), p. 1

[4] T. Zhu, J. Li, A. Samanta, A. Leach, and K. Gall:Phys. Rev. Lett. Vol. 100 (2008), p. 025502

[5] F. H. Stillinger and T. A. Weber: Phys. Rev. B Vol. 31 (1985), p. 5262 [6] J. Godet, L. Pizzagalli, S. Brochard and P. Beauchamp: J. Phys.: Condens. Matter Vol. 15

(2003), p.6943 [7] J. Godet, L. Pizzagalli, S. Brochard and P. Beauchamp: Phys. Rev. B Vol. 70 (2004),

p.054109 [8] J.Godet, P. Hirel, S. Brochard, and L. Pizzagalli: J. Appl. Phys. Vol. 105 (2009), p. 026104 [9] S. J. Plimpton: J. Comp. Phys. Vol. 117 (1995), p. 1 [10] J. F. Justo, R. D. Menezes, and L. V. C. Assali: Phys. Rev. B Vol. 75 (2007), p. 045303 [11] D. J. Chadi: Phys. Rev. Lett. Vol. 43 ( 1979), p. 43 [12] J. Li: Modelling Simul. Mater. Sci. Eng. Vol. 11 (2003), p. 173 [13] Z. Yang, Z. Lu, and Y.-P. Zhao: J. Appl. Phys. Vol. 106 (2009), p. 023537 [14] M.I. Baskes: Phys. Rev. B Vol 46 (1992), p.2727

92 Materials Structure & Micromechanics of Fracture VI

Page 5: Investigation of Plasticity in Silicon Nanowires by Molecular Dynamics Simulations

Materials Structure & Micromechanics of Fracture VI 10.4028/www.scientific.net/KEM.465 Investigation of Plasticity in Silicon Nanowires by Molecular Dynamics Simulations 10.4028/www.scientific.net/KEM.465.89

DOI References

[1] T. Kizuka, Y. Takatani, K. Asaka, and R. Yoshizaki: Phys. Rev. B Vol. 72 (2005), p. 035333

doi:10.1103/PhysRevB.72.035333 [2] J. Rabier, P. Cordier, T. Tondellier, J. L. Demenet, and H. Garem: J. Phys.: Condens. Matter ol. 12 (2000),

p. 10059

doi:10.1088/0953-8984/12/49/305 [4] T. Zhu, J. Li, A. Samanta, A. Leach, and K. Gall:Phys. Rev. Lett. Vol. 100 (2008), p. 25502

doi:10.1103/PhysRevLett.100.025502 [6] J. Godet, L. Pizzagalli, S. Brochard and P. Beauchamp: J. Phys.: Condens. Matter Vol. 15 2003), p.6943

doi:10.1088/0953-8984/15/41/004 [7] J. Godet, L. Pizzagalli, S. Brochard and P. Beauchamp: Phys. Rev. B Vol. 70 (2004), .054109

doi:10.1103/PhysRevB.70.054109 [9] S. J. Plimpton: J. Comp. Phys. Vol. 117 (1995), p. 1

doi:10.1006/jcph.1995.1039 [10] J. F. Justo, R. D. Menezes, and L. V. C. Assali: Phys. Rev. B Vol. 75 (2007), p. 045303

doi:10.1103/PhysRevB.75.045303 [13] Z. Yang, Z. Lu, and Y.-P. Zhao: J. Appl. Phys. Vol. 106 (2009), p. 023537

doi:10.1063/1.3186619 [14] M.I. Baskes: Phys. Rev. B Vol 46 (1992), p.2727

doi:10.1103/PhysRevB.46.2727 [2] J. Rabier, P. Cordier, T. Tondellier, J. L. Demenet, and H. Garem: J. Phys.: Condens. Matter Vol. 12

(2000), p. 10059

doi:10.1088/0953-8984/12/49/305 [4] T. Zhu, J. Li, A. Samanta, A. Leach, and K. Gall:Phys. Rev. Lett. Vol. 100 (2008), p. 025502

doi:10.1103/PhysRevLett.100.025502 [6] J. Godet, L. Pizzagalli, S. Brochard and P. Beauchamp: J. Phys.: Condens. Matter Vol. 15 (2003), p.6943

doi:10.1088/0953-8984/15/41/004 [7] J. Godet, L. Pizzagalli, S. Brochard and P. Beauchamp: Phys. Rev. B Vol. 70 (2004), p.054109

doi:10.1103/PhysRevB.70.054109