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Anisotropy effects on the reliability of single-crystal silicon
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Scripta Materialia 63 (2010) 997–1000 www.elsevier.com/locate/scriptamat
Anisotropy effects on the reliability of single-crystal silicon Oscar Borrero-Lo´pez,a,b,* Tania Vodenitcharovab and Mark Hoffmanb a
Departamento de Ingenierı´a Meca´nica, Energe´tica y de los Materiales, Universidad de Extremadura, 06071 Badajoz, Spain b School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia Received 31 May 2010; revised 17 July 2010; accepted 21 July 2010 Available online 24 July 2010
Anisotropy effects on the reliability of single-crystal silicon were investigated by means of scratch tests along [1 1 0] and [1 0 0] crystallographic directions. It was found that fracture (partial cone cracks) starts along favoured {1 1 0} and {1 1 1} cleavage planes, with crack orientation varying upon the scratching direction. Moreover, the [1 0 0] direction was found to be twice as reliable as the [1 1 0] direction. Stress and phase analyses were carried out to explain this effect, which has implications for the design of siliconbased devices. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Scratch test; Fracture; Elemental semiconductors; Finite element analysis
Owing to its unique electronic properties, silicon is the most important semiconductor material and the base material in photovoltaic devices and microelectromechanical systems. However, its inherent brittleness impairs the mechanical reliability of silicon-based devices. Thus, studies investigating the fracture strength of silicon have recently attracted a great deal of attention [1–5]. From those works, it is found that the fracture strength of pristine silicon can be quite high; however, even subthreshold defects (i.e., without well-developed cracks) can lead to its degradation. More recently, it has been suggested that bulk silicon is also susceptible to fracture-controlled fatigue [6]. While fracture arising from normal contacts has been extensively investigated [1,7], sliding contacts (scratch) have attracted less attention, despite being a configuration of great practical importance during manufacturing and service. Moreover, scratch tests offer a unique opportunity to investigate anisotropy effects, as the sliding direction can be easily modified. Driven by an interest in ductile machining, most of the previous studies on scratch of silicon have focused on deformation and metallization processes induced by both sharp [8,9] and spherical tools [8]. The scratch fracture of silicon has been studied by Leu and Scattergood [10], who investigated applied load, sliding speed and coefficient of friction effects on the crack geometry and scratch width. * Corresponding author at: Departamento de Ingenierı´a Meca´nica, Energe´tica y de los Materiales, Universidad de Extremadura, 06071 Badajoz, Spain; e-mail: [email protected]
However, statistical analysis of the reliability, a crucial parameter for safe design, is still unavailable. This study aims to reveal the effect of anisotropy on the mechanical reliability of single-crystal silicon, through scratch tests. Figure 1 shows representative optical micrographs of the surface fracture patterns that arise upon scratching pristine (1 0 0) silicon wafers with a 200 lm spherical tip, along the [1 1 0] (Fig. 1a) and [1 0 0] (Fig. 1b) directions. For comparative purposes, the fracture pattern in an isotropic material (soda-lime glass) is also shown (Fig. 1c). It can be observed that, in all cases, fracture takes place in the form of partial cone cracks, characteristic of brittle materials under sliding contacts. The high tensile stress at the trailing edge of the contact is responsible for crack initiation [11]. While the surface cracks in soda-lime glass are circular, typical for a Hertzian stress field [12], the {1 1 1} preferred cleavage planes in a diamond structure distort this geometry, resulting in somewhat square-shaped cracks in single-crystal silicon [7]. Moreover, there is a strong effect of the crystal orientation. Indeed, when sliding is performed along the [1 0 0] direction, the surface cracks appear to be rotated 45° compared to the cracks which appear upon sliding along the [1 1 0] direction. Underlying this effect is the different intersection of the easy cleavage planes with the surface, depending on the sliding direction [10]. In order to investigate the stochastics of the scratch fracture in single-crystal silicon along the two crystallographic directions considered, a series of scratch tests at progressively increasing normal loads (initial load 1 N;
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.07.024
998
O. Borrero-Lo´pez et al. / Scripta Materialia 63 (2010) 997–1000
Figure 1. Characteristic surface fracture patterns upon scratching of: (a) single-crystal silicon along a [1 1 0] direction; (b) single-crystal silicon along a [1 0 0] direction; (c) soda-lime glass.
final load 16 N) were conducted. The critical normal load at which the first crack appears on the surface, PC, was calculated post mortem from the optical micrographs, by employing the loading rate and sliding speed values (20 N min1 and 10 mm min1, respectively). PC was used as a measure of the fracture strength, since the coefficient of friction was found to be independent of the scratch direction – an average value of 0.029 ± 0.005 was obtained at crack initiation. Average PC values of 9 ± 4 N and 8 ± 2 N were obtained for scratches along the [1 1 0] direction and the [1 0 0] direction, respectively. The scatter in the fracture data is best represented in a Weibull plot (Fig. 2), which is constructed by ranking the critical load values in ascending order (scratches 1– 25), and subsequently assigning a probability of failure, Pf, to each of them.1 The Weibull modulus values, obtained by fitting a Weibull probability function [13], are 2.3 ± 0.1 for [1 1 0] scratches vs. 4.9 ± 0.3 for [1 0 0] scratches. Thus, there is a clear crystallographic effect on the reliability of single-crystal silicon, being the [1 0 0] direction significantly more reliable (by more than a factor 2) compared to the [1 1 0] direction. Fracture of a brittle material is caused by a combination of defects and a high tensile stress. Assuming criti1
Typically, for the ith value, Pf = (i 0.5)/N, where N is the total number of measurements [13]].
Figure 2. Weibull plot (probability of failure, Pf, vs. failure load, PC) for [1 1 0] and [1 0 0] scratches. The points correspond to experimental data obtained by scratch tests. The solid lines are the best linear fits to the experimental data. Regions 1 and 2 represent low- and highstrength scratches, respectively.
cal defects of the same size in all orientations, the observed differences in failure loads must be related to a difference in the stresses and surface energies on the cleavage planes (i.e., anisotropy effects). To ascertain this, finite element (FE) simulations were performed using the general-purpose software ANSYS for the two loading cases. Three-dimensional linear solid
O. Borrero-Lo´pez et al. / Scripta Materialia 63 (2010) 997–1000
elements were utilized to represent the silicon wafer (SOLID45, 15942 elements all together with the finest elements of 1.5 lm 1.5 lm 2.2 lm in the vicinity of the indenter tip). The indenter was considered as rigid and was meshed with 3D elements (TARGET170). The top surface of the wafer was overlaid by contact elements (CONTACT173). Linear static analysis was performed as the indenter was forced into the wafer along the sliding direction, and perpendicular to it, at a ratio of depth of penetration to sliding penetration similar to the experiment. Silicon was simulated as elastic and orthotropic. The non-zero elastic constants Cmn in the cubic crystal coordinate system are: C11 = 165.7 GPa, C12 = 63.9 GPa and C44 = 79.6 GPa2 [14]. The material properties (Eij, Gij, lij) for the [1 0 0] scratches were determined by applying the general Hooke’s law to Cmn: EXX = EYY = EZZ = 130.1 GPa, GXY = GXZ = GYZ = 79.6 GPa, and lXY = lXZ = lYZ = 0.278, where X is the [1 0 0] direction and Y and Z are perpendicular to it. Rotational transformation of Cmn and the Hooke’s law provided the material properties for the [1 1 0] scratches: EX0 X0 = EY0 Y0 = 169.1 GPa, EZZ = 130.1 GPa, lX0 Y0 = 0.062, lX0 Z = lY0 Z = 0.362, GX0 Y0 = 50.9 GPa and GX0 Z = GY0 Z = 79.6 GPa, where X0 is the [1 1 0] direction and Y0 and Z are perpendicular to it. In the same way, the Young’s modulus in the direction [1 1 1] was found to be E[111] = 187.9 GPa. The coefficient of friction was assumed to be 0.029, as measured from the scratch tests. For a simulated normal load of 3.64 N, all stress components were outputted, and the normal stress rn at each node and on each cleavage plane n passing through the node calculated by 3D stress transformation. A fracture criterion based on Griffith’s equation was applied [15], according to which mode I fracture will initiate at the node with the smallest critical defect size: ac ¼
2Enn cn 1:122 pr2n
ð1Þ
where cn is the surface energy on the cleavage plane [7]. The simulations for a [1 1 0] sliding direction found the smallest critical defect (i.e., the fracture origin) on the wafer surface, at the trailing edge along the sliding axis, located on a (1 1 0) plane (r(110) = 1356 MPa). For a [1 0 0] sliding direction, the smallest critical defect was found to be also on the wafer surface, at the trailing edge at approximately 45° from the sliding axis, on a (1 1 1) plane ðrð111Þ ¼ 1244 MPaÞ. Note that, in the [1 1 0] sliding case, fracture starts from favourably oriented {1 1 0} planes, despite a higher surface energy compared to {1 1 1} planes. The intersection of {1 1 0} planes with the wafer surface is the same as that of {1 1 1} planes, hence the fourfold symmetry of the observed surface cracks, typical of {1 1 1} cleavage [7,10]. Moreover, the stress normal to the cleavage planes is higher when sliding takes place along the [1 1 0] direction for the simulated normal load, and the relative differences are expected to increase for higher loads.
2 , epij , pt, pc, R) function [16,17] Figure 3. Difference between the f(rm, r along the [1 1 0] and the [1 0 0] directions (f = f[110] f[100]), for a normal load of 3.64 N, in the vicinity of the indenter tip in the plane of symmetry Y = 0. The X-direction represents the sliding direction. The indenter (not at scale) is at X = 0, Y = 0, Z = 0.2 mm. The points are the simulated values, and the black marks are their projections on the bottom plane.
This results in the lower PC values observed in [1 1 0] scratches compared to [1 0 0] ones (Region 1 in Fig. 2 – low strength scratches). The differences in the elastic modulii for the two sliding directions, reported above, are ultimately responsible for the obtained differences in stress. If a critical defect is not encountered at low loads, a pressure-induced phase transformation will take place prior to fracture [8], which dissipates the tensile stress at the contact, thus delaying the onset of cracking (high-strength scratches – region 2 in Fig. 2). To assess solid phase transformations in pressure-sensitive materials, such as single-crystal silicon, a phase transformation 2 , epij , pt, pc, R) = 03 has been proposed surface f(rm, r [16,17], with positive f values indicating the occurrence of the transformations. The function f takes into account both the hydrostatic pressure and the equivalent stress and hence is a measure of the level of stress at a given point – higher values of the function mean that the material point is stressed more and that the load is closer to the value for phase transformation. The values of f were calculated in the present case at each loading step and for each scratching direction from the FE simulations. Figure 3 shows the difference between the f function along [1 1 0] and [1 0 0] directions (f = f[110] f[100]) for a normal load of 3.64 N. It can 3
2
Thus, the Zener anisotropy ratio of silicon is A = 2C 44 / (C11 C12) = 1.56. Since A is not close to 1, silicon cannot be considered as isotropic.
999
is the equivalent stress, ei jp are the rm is the hydrostatic pressure; r plastic strain components, pt and pc are the tensile and compressive phase transformation thresholds (for diamond silicon ? b silicon (which on unloading transforms into amorphous silicon)), and R is the varying radius of the phase transformation surface.
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Figure 4. Raman spectra (Ar laser; 514.5 nm) at the first crack in selected scratches. Upper row, left to right: scratches number 5, 14 and 25, from the Weibull plot along [1 1 0]; lower row, left to right: scratches number 5, 14 and 25, from the Weibull plot along [1 0 0]. The horizontal axis scale is 100– 600 cm1. Critical load values, PC, are indicated in each case.
be observed that f > 0 for all the points in the neighbourhood of the indenter, suggesting that phase transformation is more likely to occur – i.e., start at lower loads – when sliding is performed along the [1 1 0] direction. As a result, the trend observed in Region 1 in Figure 2 reverses, and [1 1 0] scratches fail at higher PC values than [1 0 0] scratches in Region 2, thereby leading to a higher scatter and, in turn, to the observed lower Weibull modulus. Again, the differences in the elastic modulii for the two sliding directions are deemed responsible for the differences in stress, and in turn, for the obtained differences in the function f. Evidence of differences in phase transformations depending on the sliding direction is shown in Figure 4, which shows the Raman spectra (Ar laser; 514.5 nm) at the first crack in selected scratches. It can be observed that, along the [1 1 0] direction, traces of amorphous silicon (a-Si) can be detected at the surface from relatively low loads, whereas a-Si does not appear until high load values for [1 0 0] scratches, consistent with the FE simulations. In summary, the anisotropy in the structure of single-crystal silicon leads to differences in the stress state for a given loading condition. This results in variations in the extent of phase transformations that occur during scratching, depending on the sliding direction, and in turn in differences in the critical loads for the onset of fracture. The results are expected to provide useful insight for the design of reliable silicon-based devices.
[1] R.F. Cook, J. Mater. Sci. 41 (2006) 841. [2] A. Pajares, M. Chumakov, B.R. Lawn, J. Mater. Res. 19 (2004) 657. [3] E.E. Noonan, Y. Peles, C.S. Protz, S.M. Spearing, Scripta Mater. 59 (2008) 927. [4] O. Borrero-Lo´pez, T. Vodenitcharova, M. Hoffman, A.J. Leo, J. Am. Ceram. Soc. 92 (2009) 2713. [5] K.S. Chen, A. Ayon, S.M. Spearing, Scripta Mater. 83 (2000) 1476. [6] S. Bhowmick, J.J. Mele´ndez-Martı´nez, B.R. Lawn, Appl. Phys. Lett. 91 (2007) 201902. [7] B.R. Lawn, J. Appl. Phys. 39 (1968) 4828. [8] Y. Gogotsi, G. Zhou, S.-S. Ku, S. Cetinkunt, Semicond. Sci. Technol. 16 (2001) 345. [9] R. Gassiloud, C. Ballif, P. Gassez, G. Buerki, J. Michler, Phys. Status Solidi A 202 (2005) 2858. [10] H.J. Leu, R.O. Scattergood, J. Mater. Sci. 23 (1988) 3006. [11] B.R. Lawn, Proc. Roy. Soc. Lond. 299 (1967) 307. [12] B.R. Lawn, J. Am. Ceram. Soc. 81 (1998) 1977. [13] Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics. American Society for Testing and Materials, Philadelphia PA, USA, C1239-07. [14] D.J. Green, An Introduction to the Mechanical Properties of Ceramics, Cambridge University Press, Cambridge, UK, 1998. [15] T. Vodenitcharova, L.C. Zhang, I. Zarudi, Y. Yin, H. Domyo, T. Ho, M. Sato, J. Mater. Process. Technol. 194 (2007) 52. [16] T. Vodenitcharova, L.C. Zhang, Int. J. Solids Struct. 40 (2003) 2989. [17] T. Vodenitcharova, L.C. Zhang, Int. J. Solids Struct. 41 (2004) 5411.
Scripta Materialia 63 (2010) 997–1000 www.elsevier.com/locate/scriptamat
Anisotropy effects on the reliability of single-crystal silicon Oscar Borrero-Lo´pez,a,b,* Tania Vodenitcharovab and Mark Hoffmanb a
Departamento de Ingenierı´a Meca´nica, Energe´tica y de los Materiales, Universidad de Extremadura, 06071 Badajoz, Spain b School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia Received 31 May 2010; revised 17 July 2010; accepted 21 July 2010 Available online 24 July 2010
Anisotropy effects on the reliability of single-crystal silicon were investigated by means of scratch tests along [1 1 0] and [1 0 0] crystallographic directions. It was found that fracture (partial cone cracks) starts along favoured {1 1 0} and {1 1 1} cleavage planes, with crack orientation varying upon the scratching direction. Moreover, the [1 0 0] direction was found to be twice as reliable as the [1 1 0] direction. Stress and phase analyses were carried out to explain this effect, which has implications for the design of siliconbased devices. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Scratch test; Fracture; Elemental semiconductors; Finite element analysis
Owing to its unique electronic properties, silicon is the most important semiconductor material and the base material in photovoltaic devices and microelectromechanical systems. However, its inherent brittleness impairs the mechanical reliability of silicon-based devices. Thus, studies investigating the fracture strength of silicon have recently attracted a great deal of attention [1–5]. From those works, it is found that the fracture strength of pristine silicon can be quite high; however, even subthreshold defects (i.e., without well-developed cracks) can lead to its degradation. More recently, it has been suggested that bulk silicon is also susceptible to fracture-controlled fatigue [6]. While fracture arising from normal contacts has been extensively investigated [1,7], sliding contacts (scratch) have attracted less attention, despite being a configuration of great practical importance during manufacturing and service. Moreover, scratch tests offer a unique opportunity to investigate anisotropy effects, as the sliding direction can be easily modified. Driven by an interest in ductile machining, most of the previous studies on scratch of silicon have focused on deformation and metallization processes induced by both sharp [8,9] and spherical tools [8]. The scratch fracture of silicon has been studied by Leu and Scattergood [10], who investigated applied load, sliding speed and coefficient of friction effects on the crack geometry and scratch width. * Corresponding author at: Departamento de Ingenierı´a Meca´nica, Energe´tica y de los Materiales, Universidad de Extremadura, 06071 Badajoz, Spain; e-mail: [email protected]
However, statistical analysis of the reliability, a crucial parameter for safe design, is still unavailable. This study aims to reveal the effect of anisotropy on the mechanical reliability of single-crystal silicon, through scratch tests. Figure 1 shows representative optical micrographs of the surface fracture patterns that arise upon scratching pristine (1 0 0) silicon wafers with a 200 lm spherical tip, along the [1 1 0] (Fig. 1a) and [1 0 0] (Fig. 1b) directions. For comparative purposes, the fracture pattern in an isotropic material (soda-lime glass) is also shown (Fig. 1c). It can be observed that, in all cases, fracture takes place in the form of partial cone cracks, characteristic of brittle materials under sliding contacts. The high tensile stress at the trailing edge of the contact is responsible for crack initiation [11]. While the surface cracks in soda-lime glass are circular, typical for a Hertzian stress field [12], the {1 1 1} preferred cleavage planes in a diamond structure distort this geometry, resulting in somewhat square-shaped cracks in single-crystal silicon [7]. Moreover, there is a strong effect of the crystal orientation. Indeed, when sliding is performed along the [1 0 0] direction, the surface cracks appear to be rotated 45° compared to the cracks which appear upon sliding along the [1 1 0] direction. Underlying this effect is the different intersection of the easy cleavage planes with the surface, depending on the sliding direction [10]. In order to investigate the stochastics of the scratch fracture in single-crystal silicon along the two crystallographic directions considered, a series of scratch tests at progressively increasing normal loads (initial load 1 N;
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.07.024
998
O. Borrero-Lo´pez et al. / Scripta Materialia 63 (2010) 997–1000
Figure 1. Characteristic surface fracture patterns upon scratching of: (a) single-crystal silicon along a [1 1 0] direction; (b) single-crystal silicon along a [1 0 0] direction; (c) soda-lime glass.
final load 16 N) were conducted. The critical normal load at which the first crack appears on the surface, PC, was calculated post mortem from the optical micrographs, by employing the loading rate and sliding speed values (20 N min1 and 10 mm min1, respectively). PC was used as a measure of the fracture strength, since the coefficient of friction was found to be independent of the scratch direction – an average value of 0.029 ± 0.005 was obtained at crack initiation. Average PC values of 9 ± 4 N and 8 ± 2 N were obtained for scratches along the [1 1 0] direction and the [1 0 0] direction, respectively. The scatter in the fracture data is best represented in a Weibull plot (Fig. 2), which is constructed by ranking the critical load values in ascending order (scratches 1– 25), and subsequently assigning a probability of failure, Pf, to each of them.1 The Weibull modulus values, obtained by fitting a Weibull probability function [13], are 2.3 ± 0.1 for [1 1 0] scratches vs. 4.9 ± 0.3 for [1 0 0] scratches. Thus, there is a clear crystallographic effect on the reliability of single-crystal silicon, being the [1 0 0] direction significantly more reliable (by more than a factor 2) compared to the [1 1 0] direction. Fracture of a brittle material is caused by a combination of defects and a high tensile stress. Assuming criti1
Typically, for the ith value, Pf = (i 0.5)/N, where N is the total number of measurements [13]].
Figure 2. Weibull plot (probability of failure, Pf, vs. failure load, PC) for [1 1 0] and [1 0 0] scratches. The points correspond to experimental data obtained by scratch tests. The solid lines are the best linear fits to the experimental data. Regions 1 and 2 represent low- and highstrength scratches, respectively.
cal defects of the same size in all orientations, the observed differences in failure loads must be related to a difference in the stresses and surface energies on the cleavage planes (i.e., anisotropy effects). To ascertain this, finite element (FE) simulations were performed using the general-purpose software ANSYS for the two loading cases. Three-dimensional linear solid
O. Borrero-Lo´pez et al. / Scripta Materialia 63 (2010) 997–1000
elements were utilized to represent the silicon wafer (SOLID45, 15942 elements all together with the finest elements of 1.5 lm 1.5 lm 2.2 lm in the vicinity of the indenter tip). The indenter was considered as rigid and was meshed with 3D elements (TARGET170). The top surface of the wafer was overlaid by contact elements (CONTACT173). Linear static analysis was performed as the indenter was forced into the wafer along the sliding direction, and perpendicular to it, at a ratio of depth of penetration to sliding penetration similar to the experiment. Silicon was simulated as elastic and orthotropic. The non-zero elastic constants Cmn in the cubic crystal coordinate system are: C11 = 165.7 GPa, C12 = 63.9 GPa and C44 = 79.6 GPa2 [14]. The material properties (Eij, Gij, lij) for the [1 0 0] scratches were determined by applying the general Hooke’s law to Cmn: EXX = EYY = EZZ = 130.1 GPa, GXY = GXZ = GYZ = 79.6 GPa, and lXY = lXZ = lYZ = 0.278, where X is the [1 0 0] direction and Y and Z are perpendicular to it. Rotational transformation of Cmn and the Hooke’s law provided the material properties for the [1 1 0] scratches: EX0 X0 = EY0 Y0 = 169.1 GPa, EZZ = 130.1 GPa, lX0 Y0 = 0.062, lX0 Z = lY0 Z = 0.362, GX0 Y0 = 50.9 GPa and GX0 Z = GY0 Z = 79.6 GPa, where X0 is the [1 1 0] direction and Y0 and Z are perpendicular to it. In the same way, the Young’s modulus in the direction [1 1 1] was found to be E[111] = 187.9 GPa. The coefficient of friction was assumed to be 0.029, as measured from the scratch tests. For a simulated normal load of 3.64 N, all stress components were outputted, and the normal stress rn at each node and on each cleavage plane n passing through the node calculated by 3D stress transformation. A fracture criterion based on Griffith’s equation was applied [15], according to which mode I fracture will initiate at the node with the smallest critical defect size: ac ¼
2Enn cn 1:122 pr2n
ð1Þ
where cn is the surface energy on the cleavage plane [7]. The simulations for a [1 1 0] sliding direction found the smallest critical defect (i.e., the fracture origin) on the wafer surface, at the trailing edge along the sliding axis, located on a (1 1 0) plane (r(110) = 1356 MPa). For a [1 0 0] sliding direction, the smallest critical defect was found to be also on the wafer surface, at the trailing edge at approximately 45° from the sliding axis, on a (1 1 1) plane ðrð111Þ ¼ 1244 MPaÞ. Note that, in the [1 1 0] sliding case, fracture starts from favourably oriented {1 1 0} planes, despite a higher surface energy compared to {1 1 1} planes. The intersection of {1 1 0} planes with the wafer surface is the same as that of {1 1 1} planes, hence the fourfold symmetry of the observed surface cracks, typical of {1 1 1} cleavage [7,10]. Moreover, the stress normal to the cleavage planes is higher when sliding takes place along the [1 1 0] direction for the simulated normal load, and the relative differences are expected to increase for higher loads.
2 , epij , pt, pc, R) function [16,17] Figure 3. Difference between the f(rm, r along the [1 1 0] and the [1 0 0] directions (f = f[110] f[100]), for a normal load of 3.64 N, in the vicinity of the indenter tip in the plane of symmetry Y = 0. The X-direction represents the sliding direction. The indenter (not at scale) is at X = 0, Y = 0, Z = 0.2 mm. The points are the simulated values, and the black marks are their projections on the bottom plane.
This results in the lower PC values observed in [1 1 0] scratches compared to [1 0 0] ones (Region 1 in Fig. 2 – low strength scratches). The differences in the elastic modulii for the two sliding directions, reported above, are ultimately responsible for the obtained differences in stress. If a critical defect is not encountered at low loads, a pressure-induced phase transformation will take place prior to fracture [8], which dissipates the tensile stress at the contact, thus delaying the onset of cracking (high-strength scratches – region 2 in Fig. 2). To assess solid phase transformations in pressure-sensitive materials, such as single-crystal silicon, a phase transformation 2 , epij , pt, pc, R) = 03 has been proposed surface f(rm, r [16,17], with positive f values indicating the occurrence of the transformations. The function f takes into account both the hydrostatic pressure and the equivalent stress and hence is a measure of the level of stress at a given point – higher values of the function mean that the material point is stressed more and that the load is closer to the value for phase transformation. The values of f were calculated in the present case at each loading step and for each scratching direction from the FE simulations. Figure 3 shows the difference between the f function along [1 1 0] and [1 0 0] directions (f = f[110] f[100]) for a normal load of 3.64 N. It can 3
2
Thus, the Zener anisotropy ratio of silicon is A = 2C 44 / (C11 C12) = 1.56. Since A is not close to 1, silicon cannot be considered as isotropic.
999
is the equivalent stress, ei jp are the rm is the hydrostatic pressure; r plastic strain components, pt and pc are the tensile and compressive phase transformation thresholds (for diamond silicon ? b silicon (which on unloading transforms into amorphous silicon)), and R is the varying radius of the phase transformation surface.
1000
O. Borrero-Lo´pez et al. / Scripta Materialia 63 (2010) 997–1000
Figure 4. Raman spectra (Ar laser; 514.5 nm) at the first crack in selected scratches. Upper row, left to right: scratches number 5, 14 and 25, from the Weibull plot along [1 1 0]; lower row, left to right: scratches number 5, 14 and 25, from the Weibull plot along [1 0 0]. The horizontal axis scale is 100– 600 cm1. Critical load values, PC, are indicated in each case.
be observed that f > 0 for all the points in the neighbourhood of the indenter, suggesting that phase transformation is more likely to occur – i.e., start at lower loads – when sliding is performed along the [1 1 0] direction. As a result, the trend observed in Region 1 in Figure 2 reverses, and [1 1 0] scratches fail at higher PC values than [1 0 0] scratches in Region 2, thereby leading to a higher scatter and, in turn, to the observed lower Weibull modulus. Again, the differences in the elastic modulii for the two sliding directions are deemed responsible for the differences in stress, and in turn, for the obtained differences in the function f. Evidence of differences in phase transformations depending on the sliding direction is shown in Figure 4, which shows the Raman spectra (Ar laser; 514.5 nm) at the first crack in selected scratches. It can be observed that, along the [1 1 0] direction, traces of amorphous silicon (a-Si) can be detected at the surface from relatively low loads, whereas a-Si does not appear until high load values for [1 0 0] scratches, consistent with the FE simulations. In summary, the anisotropy in the structure of single-crystal silicon leads to differences in the stress state for a given loading condition. This results in variations in the extent of phase transformations that occur during scratching, depending on the sliding direction, and in turn in differences in the critical loads for the onset of fracture. The results are expected to provide useful insight for the design of reliable silicon-based devices.
[1] R.F. Cook, J. Mater. Sci. 41 (2006) 841. [2] A. Pajares, M. Chumakov, B.R. Lawn, J. Mater. Res. 19 (2004) 657. [3] E.E. Noonan, Y. Peles, C.S. Protz, S.M. Spearing, Scripta Mater. 59 (2008) 927. [4] O. Borrero-Lo´pez, T. Vodenitcharova, M. Hoffman, A.J. Leo, J. Am. Ceram. Soc. 92 (2009) 2713. [5] K.S. Chen, A. Ayon, S.M. Spearing, Scripta Mater. 83 (2000) 1476. [6] S. Bhowmick, J.J. Mele´ndez-Martı´nez, B.R. Lawn, Appl. Phys. Lett. 91 (2007) 201902. [7] B.R. Lawn, J. Appl. Phys. 39 (1968) 4828. [8] Y. Gogotsi, G. Zhou, S.-S. Ku, S. Cetinkunt, Semicond. Sci. Technol. 16 (2001) 345. [9] R. Gassiloud, C. Ballif, P. Gassez, G. Buerki, J. Michler, Phys. Status Solidi A 202 (2005) 2858. [10] H.J. Leu, R.O. Scattergood, J. Mater. Sci. 23 (1988) 3006. [11] B.R. Lawn, Proc. Roy. Soc. Lond. 299 (1967) 307. [12] B.R. Lawn, J. Am. Ceram. Soc. 81 (1998) 1977. [13] Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics. American Society for Testing and Materials, Philadelphia PA, USA, C1239-07. [14] D.J. Green, An Introduction to the Mechanical Properties of Ceramics, Cambridge University Press, Cambridge, UK, 1998. [15] T. Vodenitcharova, L.C. Zhang, I. Zarudi, Y. Yin, H. Domyo, T. Ho, M. Sato, J. Mater. Process. Technol. 194 (2007) 52. [16] T. Vodenitcharova, L.C. Zhang, Int. J. Solids Struct. 40 (2003) 2989. [17] T. Vodenitcharova, L.C. Zhang, Int. J. Solids Struct. 41 (2004) 5411.