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Nanoindentation Characterization of Silicon and Other MEMS Materials
Appendix 2
Nanoindentation Characterization of Silicon and Other MEMS Materials P. Zachariasz1,2, K. Brudzinski ´ 2, J. Gronicz2, S. Nagao2 and R. Nowak1,2 1
Department of Materials Science and Engineering, Aalto University, Espoo, Finland,
2
The Nordic Hysitron Laboratory, Helsinki University of Technology, Espoo, Finland
A2.1 INTRODUCTION The very rapid and progressive process of miniaturization of electromechanical devices, which is appreciable over the last few decades, simultaneously demands a high standard for both materials and base components used in prefabrication process. The potential of integrating a micro electro-mechanical system (MEMS) device together with a complete control of electric circuits implemented at individual chip, is nowadays one of the leading ideas in the field of micro- and nanoscale systems. MEMS technology, which is commonly based on silicon material, either as a fundamental constructive element or as an excellent substrate for other products of microfabrication technology, supplies an excellent capability for that purpose. However, further miniaturization of MEMS involves multiple problems with regard to the physical and mechanical properties of these devices. Making a restriction only to the strictly mechanical parameters such as elasticity, hardness, as well as friction and adhesion coefficients, those parameters are strongly size dependent and have a great influence on the quality and efficiency of the designed microstructures. In that way, further design engineering of MEMS devices has to be closely related to the modern science based techniques adequately suited for new challenges coming from the IC industry.
A2.2 NANOINDENTATION METHOD Measurements of mechanical parameters involving reduced Young’s modulus and hardness of many types of
materials can be carried out by conventional techniques such as bending and scratching tests of investigated specimens. However, large scale integration and miniaturization of MEMS devices, as well as a demand for the exploration of new aspects of nature at increasingly growing scale, enforce employment of much more sophisticated experimental methods. Therefore, one of the powerful scientific tools to study nanostructured materials involving both imaging details of an investigation region, and determination of mechanical properties of a specimen is atomic force microscopy (AFM). The essential component of that method is a silicon or silicon nitride tip fixed at the end of a cantilever, continuously kept in contact with sample surfaces during all imaging process. The deflection of the tip is recorded in a very precise manner by means of analysis of forcedistance curves [1]. Another widely accepted technique useful for practical application of the standard tests for determining plastic and elastic properties of materials at micro- and nanometer scale is an indentation method [2,3]. Many types of naniondentation device have recently been developed in response to an increasing number of requirements for mechanical testing of crystalline thin films and multilayer structures, which are utilized in optoelectronics and microfabrication of nanostructures. The nanoindentation technique, also named the depthsensing indentation, is a method, where an indenter with a diamond tip goes deeply into the specimen after applying external force. The plastic-elastic behavior of the test material is evaluated through the type and a shape of the Ph curve. 771
772
Appendix 2: Nanoindentation Characterization of Silicon and Other MEMS Materials
is adequate for cylindrical and spherical indenter tips, as well as for conical ones, for which it was originally derived. Thus, for the OliverPharr analysis [2], the effective Young’s modulus is estimated according to the relation: pffiffiffi π S pffiffiffiffiffi ; Eeff 5 (A2.3) 2β Ac
=
where the coefficient β is the geometry correction factor of an indenter (β 5 0.034 for the standard Berkovich tip), and Eeff is defined as:
ε
1 1 2 ν 21 1 2 ν 22 5 1 ; Eeff E1 E2
ε
FIGURE A2.1 Schematic diagram of the indentation method with a typical loaddisplacement curve and parameters used in the Oliver and Pharr analysis.
A schematic loading-unloading diagram describing the response h of a specimen recorded as a function of load P is presented in Figure A2.1, where hmax, Pmax indicate the maximum penetration depth at the maximum load, and the other parameters are described as follows: hr-residual indentation depth, hc at contact, S 5 dP/dH-stiffness, defined as the slope of the unloading curve at maximum load. Using the OliverPharr procedure [2], it is possible to determine from the Ph curve the effective Young’s modulus Eeff and a hardness H. The hardness H is a measure of the resistance of a material to plastic deformation, and is simply defined as an applied force (“mean pressure” P) acting on the area Ac: H5
Pmax ; Ac
(A2.1)
where Ac denotes a projected area of a contact between the probing tip and the specimen at the maximum load Pmax, and is estimated from the geometry of the tip during the procedure of the area function calibration (Ac 5 f(dc)). This is the basis of the OliverPharr method, where the momentary contact area between the tip and the specimen is simply related to the actual depth of the indenter. On the other hand, with respect to the range of completely elastic behavior, the effective Young’s modulus is described as a measure of uniaxial strain under a given uniaxial stress. It was proved by Bulychev et al. [4] that the relationship between the stiffness S, effective elastic modulus Eeff and the contact area Ac, defined as: pffiffiffiffiffi dP 2 5 pffiffiffi Eeff Ac (A2.2) S5 dH h5hmax π
(A2.4)
where Et and ν i are elastic moduli and the Poisson’s ratios of the sample and the indenter tip, respectively. A very important criterion determining the quality of indentation experiments is the type and shape of the probing tip (ε parameter on Figure A2.1). A Berkovich threesided pyramidal tip, as well as other conical indenters, are very useful for making deep indentations with comparatively small residual impressions, while the relatively large spherical tips are generally used for crystalline materials revealing a big elastic-plastic anisotropy and for brittle materials. There are many scientific dissertations concerning interactions between a sharp tip and a small part of a specimen being found underneath a tip-sample contact [5]. All of them consider the size of residual impression, rather than a direct measurement of the contact area, since the dimensions of nanoindentations are inadequately suited for classical and optical techniques.
A2.3 INDENTATION IN SILICON Silicon is a hard, brittle material containing many types of structural defects related to particular methods of crystal growing, and different foreign atoms being introduced into the silicon structure during the processes of silicon wafer fabrication. Therefore, there is a quite essential detail in any experimental route, especially with respect to thin films and extremely thin monolayers based on silicon and silica, since any defect on the surface may cause permanent cracking of the investigation materials. Additionaly, very tenuous MEMS devices, which nowadays are widely produced with the close cooperation of the IC industry, have challenged scientists to employ much more sophisticated methods and techniques to determine physical properties of these materials. In large scale production of MEMS devices, a typical 550 µm thick, p-type (100) oriented silicon wafer is commonly used as a prefabricated element for further steps
Appendix 2: Nanoindentation Characterization of Silicon and Other MEMS Materials
of nanodevice manufacturing. Hence, the indentation of silicon and related microstructural systems based on silicon have been extensively carried out during the last few decades. Many scientific papers have reported interesting information on the physical properties of that type of material, as well as reporting data involving phase transitions and creation of new phases during nanoindentation tests. Therefore, nanoindentation method is a very fast and useful technique for the investigation of prefabricated silicon devices, nevertheless in compliance with silicon structure modifications, one should be very careful, especially with the interpretation of the results arising from that method.
A2.3.1 Structure Transition In general, material deformation under an indentation process might involve such competing mechanisms as brittle macro- and microfracture, plasticity dislocation, defect formation and structural transitions. It was reported [6], that during nanoindentation, silicon shows small or almost no dislocation activity but highly disturbed structures. In an earlier work Pharr et al. [7] showed that during an indenter loading into a silicon specimen, permanent pressure treated as a locally acting hydrostatic stress just beneath the tip can involve a tetragonal phase formation. A very similar phenomenon was also registered upon removing the applied force, where a phase transition to bcc structure was delivered [8]. Those observations prompted further investigations, hence numerous indentation experiments have been done, showing a great variety of different phase transitions occuring in silicon structures with respect to various pressure conditions. In 1997 Gogotsi et al. [9] methodically considered deformations in silicon wafers induced by indentation measurements. They adapted a hardness indentation technique and micro-Raman spectroscopy to show many interesting phenomena involving phase transitions and other microcracking processes. The conclusion was that silicon undergoes a transition from the cubic-diamond structure (so called Si-I) into β-tin phase (Si-II) at the loading stage of indentation, between 9 and 16 GPa of hydrostatic pressure with a significant decrease of the unit cell volume by about 22%. Further increase of pressure can induce consecutive phase transitions into Si-V, Si-VI and Si-X structures, respectively. On the other hand, decreasing the hydrostatic pressure transforming the silicon material into the rhombohedral phase (Si-XII) starting at around 9 GPa of contact pressure and the Si-II to Si-XII transition results in about 9%
773
volume expansion. This phase was found to be unstable at ambient pressure, although its existence was proved from postindentation analysis. However, Si-XII can transform to the bcc phase (SiIII), upon further unloading. It was shown that transition between the Si-XII and Si-III phases is martensitic, as well as reversible and occurs at around 2 GPa of pressure. The final Si-III structure was found to be about 9% denser than the initial Si-I phase and 2% less dense than Si-XII. It is also possible to obtain an amorphous structure of silicon (a-Si), if the unloading rate is slow enough. The presented description of silicon deformations is only a simplification of the structural changes arising from that system. Up to now, at least 12 different crystallographic structures of silicon have been distinguished. All recognized and well established silicon structures are collected in Table A2.1 with respect to their occurrence and the pressure conditions. A schematic diagram of hydrostatic stress as a function of volumetric strain is presented in Figure A2.2. More detailed information including particular lattice parameters of development structures, and also a complete collection of publication references are included in Hull [10].
A2.3.2 Size, Substrate, and Other Effects in Silicon The miniaturization process of MEMS devices is bringing to pass multiple problems regarding their mechanical properties. As was introduced by Li et al. [11] the size effect cannot be neglected, especially when the technology goes into the scale of nanometers. The differences between the values of mechanical properties of a single crystal of silicon obtained in macroand nanoscale are definitely large. Many measurements carried out for bulk specimens could not be applied to microdevices due to the scale dependence of the investigated material. Also, many physical results coming from different experiments are not consistent with each other, which could be very problematic for designers of MEMS devices. Other independent parameters could also influence values of mechanical properties of silicon structures. An excellent example of that relation is the variation of hardness as a function of temperature. As one can see in Figure A2.3, the hardness H is not a constant value, but could vary with respect to temperature conditions. For extremely small prefabricated elements one has to consider an effect coming from the substrate, of which mechanical parameters and time-dependent behavior under specific conditions might play a dominant role. Nowadays, thin silicon microstructures are fabricated, for example, on glass substrates by hot-wire chemical vapor deposition, however the similar process of silicon microstructures
774
Appendix 2: Nanoindentation Characterization of Silicon and Other MEMS Materials
TABLE A2.1 Structural Data for Crystalline Phases of Silicon [10] Designation
Space Group
Crystallographic Structure
Pressure Range (GPa)
Si-I
Fd3m
Diamond cubic
012.5
Si-II
I41/ amd
Body centered tetragonal
8.816
Si-III
la3
Body centered cubic
2.10
Si-IV
P63/ mmc
Diamond hexagonal
Si-V
P6/ mmm
Primitive hexagonal
1435
Si-VI
Unidentified
3440
Si-VII
P63/ mmc
Hexagonal close packed
4078.3
Si-VIII
P41212
Tetragonal
14.80
Si-IX
P422
Tetragonal
120
Si-X
Fm3m
Face centered cubic
78.3230
Si-XI
Imma
Body centered orthorhombic
13/15
Si-XII
R3
Rhombohedral
122
FIGURE A2.2 Demonstrative diagram of hydrostatic stress P in function of volumetric strain εv for mono-crystalline silicon. Only phases appearing during loading stage are presented.
spanning could also be maintained on Si substrate. Nevertheless, even in the case of full compatibility of component materials, the above effects have to be taken into consideration during the MEMS project implementation.
FIGURE A2.3 Temperature dependence of hardness H for monocrystalline silicon measured by means of Vickers indentation method [12,13].
A2.3.3 Analytical and Mathematical Modeling Aspects Alongside experimental efforts, many theoretical approaches have been done in order to describe structural changes, as well as the dynamic processes occuring in the specimen during indentation tests. Nanoindentation produces a very high contact pressure between the diamond tip and the specimen, and brings about deformations of the material due to dislocation gliding, twinning or phase transitions. The still growing importance of computers in science gives great opportunities for employing mathematical formalism through a variety of computational techniques, which are very useful tools for modeling, making simulations and predictions of many physical properties and phenomena occuring in different materials. With molecular dynamic analysis, the phase transition from Si-I to Si-II structures was simulated by Cheong and Zhang [14]. The simulation results revealed that, at the maximum load some atoms just beneath the indenter tip appear to have a different coordination number than the initial structure. Investigations of the bond length between atoms showed that this coordination complied with β-tin structure, and after a very fast unloading, an amorphous structure was reproduced. Thus, the experimental results have been confirmed by theoretical predictions. Interesting work was done by Vodenitcharova et al. [15], who developed a constitutive model for multi-phase transitions in mono-crystalline silicon. The background of that model was based on the combination of physical mechanisms observed experimentally, the finite element method and the theory of plasticity. The developed model was successfully applied to predict correct results involving the elastic-plastic response of silicon subjected to hydrostatic pressure and nanoindentation tests, also including the phenomena of pop-in and pop-out events.
Appendix 2: Nanoindentation Characterization of Silicon and Other MEMS Materials
REFERENCES [1] B. Cappella, G. Dieter, Surf. Sci. Rep. 34 (1999) 1. [2] W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 (1992) 1564. [3] M.F. Doerner, D.S. Gardner, W.D. Nix, J. Mater. Res. 1 (1986) 601. [4] S.I. Bulychev, V.P. Alekhin, M.K. Shorshorov, A.P. Ternovskii, G.D. Shnyrev, Industrial Lab. 41 (1975) 1409. [5] A.C. Fischer-Cripps, Nanoindentation, Springer, 2004. [6] T.F. Page, W.C. Oliver, C.J. McHargue, J. Mater. Res. 7 (1992) 450. [7] G.M. Pharr, W.C. Oliver, D.S. Harding, J. Mater. Res. 6 (1991) 1129.
775
[8] J.Z. Hu, L.D. Merkle, C.S. Menoni, I.L. Spain, Phys. Rev. B 34 (1986) 4679. [9] A. Kailer, Y.G. Gogotsi, K.G. Nickel, J. Appl. Phys. 81 (1997) 3057. [10] A. George, High pressure phases of c-Si, in: R. Hull (Ed.), Properties of Crystalline Silicon, twentieth ed., mspec, The Institution of Electrical Engineers, London, 1999, pp. 104107. [11] X. Li, B. Bhushan, K. Takashima, C.W. Baek, Y.K. Kim, Ultramicroscopy 97 (2003) 481. [12] V.I. Trefilov, Y.V. Milman, Sov. Phys.-Dokl. 8 (1964) 1240. [13] I. Yonenaga, T. Hoshi, A. Usui, J. Phys.: Condens. Matter 12 (2000) 10319. [14] W.C.D. Cheong, L.C. Zhang, Nanotechnology 11 (2000) 173. [15] T. Vodenitcharova, L.C. Zhang, Int. J. Solids Struct. 41 (2004) 5411.
Nanoindentation Characterization of Silicon and Other MEMS Materials P. Zachariasz1,2, K. Brudzinski ´ 2, J. Gronicz2, S. Nagao2 and R. Nowak1,2 1
Department of Materials Science and Engineering, Aalto University, Espoo, Finland,
2
The Nordic Hysitron Laboratory, Helsinki University of Technology, Espoo, Finland
A2.1 INTRODUCTION The very rapid and progressive process of miniaturization of electromechanical devices, which is appreciable over the last few decades, simultaneously demands a high standard for both materials and base components used in prefabrication process. The potential of integrating a micro electro-mechanical system (MEMS) device together with a complete control of electric circuits implemented at individual chip, is nowadays one of the leading ideas in the field of micro- and nanoscale systems. MEMS technology, which is commonly based on silicon material, either as a fundamental constructive element or as an excellent substrate for other products of microfabrication technology, supplies an excellent capability for that purpose. However, further miniaturization of MEMS involves multiple problems with regard to the physical and mechanical properties of these devices. Making a restriction only to the strictly mechanical parameters such as elasticity, hardness, as well as friction and adhesion coefficients, those parameters are strongly size dependent and have a great influence on the quality and efficiency of the designed microstructures. In that way, further design engineering of MEMS devices has to be closely related to the modern science based techniques adequately suited for new challenges coming from the IC industry.
A2.2 NANOINDENTATION METHOD Measurements of mechanical parameters involving reduced Young’s modulus and hardness of many types of
materials can be carried out by conventional techniques such as bending and scratching tests of investigated specimens. However, large scale integration and miniaturization of MEMS devices, as well as a demand for the exploration of new aspects of nature at increasingly growing scale, enforce employment of much more sophisticated experimental methods. Therefore, one of the powerful scientific tools to study nanostructured materials involving both imaging details of an investigation region, and determination of mechanical properties of a specimen is atomic force microscopy (AFM). The essential component of that method is a silicon or silicon nitride tip fixed at the end of a cantilever, continuously kept in contact with sample surfaces during all imaging process. The deflection of the tip is recorded in a very precise manner by means of analysis of forcedistance curves [1]. Another widely accepted technique useful for practical application of the standard tests for determining plastic and elastic properties of materials at micro- and nanometer scale is an indentation method [2,3]. Many types of naniondentation device have recently been developed in response to an increasing number of requirements for mechanical testing of crystalline thin films and multilayer structures, which are utilized in optoelectronics and microfabrication of nanostructures. The nanoindentation technique, also named the depthsensing indentation, is a method, where an indenter with a diamond tip goes deeply into the specimen after applying external force. The plastic-elastic behavior of the test material is evaluated through the type and a shape of the Ph curve. 771
772
Appendix 2: Nanoindentation Characterization of Silicon and Other MEMS Materials
is adequate for cylindrical and spherical indenter tips, as well as for conical ones, for which it was originally derived. Thus, for the OliverPharr analysis [2], the effective Young’s modulus is estimated according to the relation: pffiffiffi π S pffiffiffiffiffi ; Eeff 5 (A2.3) 2β Ac
=
where the coefficient β is the geometry correction factor of an indenter (β 5 0.034 for the standard Berkovich tip), and Eeff is defined as:
ε
1 1 2 ν 21 1 2 ν 22 5 1 ; Eeff E1 E2
ε
FIGURE A2.1 Schematic diagram of the indentation method with a typical loaddisplacement curve and parameters used in the Oliver and Pharr analysis.
A schematic loading-unloading diagram describing the response h of a specimen recorded as a function of load P is presented in Figure A2.1, where hmax, Pmax indicate the maximum penetration depth at the maximum load, and the other parameters are described as follows: hr-residual indentation depth, hc at contact, S 5 dP/dH-stiffness, defined as the slope of the unloading curve at maximum load. Using the OliverPharr procedure [2], it is possible to determine from the Ph curve the effective Young’s modulus Eeff and a hardness H. The hardness H is a measure of the resistance of a material to plastic deformation, and is simply defined as an applied force (“mean pressure” P) acting on the area Ac: H5
Pmax ; Ac
(A2.1)
where Ac denotes a projected area of a contact between the probing tip and the specimen at the maximum load Pmax, and is estimated from the geometry of the tip during the procedure of the area function calibration (Ac 5 f(dc)). This is the basis of the OliverPharr method, where the momentary contact area between the tip and the specimen is simply related to the actual depth of the indenter. On the other hand, with respect to the range of completely elastic behavior, the effective Young’s modulus is described as a measure of uniaxial strain under a given uniaxial stress. It was proved by Bulychev et al. [4] that the relationship between the stiffness S, effective elastic modulus Eeff and the contact area Ac, defined as: pffiffiffiffiffi dP 2 5 pffiffiffi Eeff Ac (A2.2) S5 dH h5hmax π
(A2.4)
where Et and ν i are elastic moduli and the Poisson’s ratios of the sample and the indenter tip, respectively. A very important criterion determining the quality of indentation experiments is the type and shape of the probing tip (ε parameter on Figure A2.1). A Berkovich threesided pyramidal tip, as well as other conical indenters, are very useful for making deep indentations with comparatively small residual impressions, while the relatively large spherical tips are generally used for crystalline materials revealing a big elastic-plastic anisotropy and for brittle materials. There are many scientific dissertations concerning interactions between a sharp tip and a small part of a specimen being found underneath a tip-sample contact [5]. All of them consider the size of residual impression, rather than a direct measurement of the contact area, since the dimensions of nanoindentations are inadequately suited for classical and optical techniques.
A2.3 INDENTATION IN SILICON Silicon is a hard, brittle material containing many types of structural defects related to particular methods of crystal growing, and different foreign atoms being introduced into the silicon structure during the processes of silicon wafer fabrication. Therefore, there is a quite essential detail in any experimental route, especially with respect to thin films and extremely thin monolayers based on silicon and silica, since any defect on the surface may cause permanent cracking of the investigation materials. Additionaly, very tenuous MEMS devices, which nowadays are widely produced with the close cooperation of the IC industry, have challenged scientists to employ much more sophisticated methods and techniques to determine physical properties of these materials. In large scale production of MEMS devices, a typical 550 µm thick, p-type (100) oriented silicon wafer is commonly used as a prefabricated element for further steps
Appendix 2: Nanoindentation Characterization of Silicon and Other MEMS Materials
of nanodevice manufacturing. Hence, the indentation of silicon and related microstructural systems based on silicon have been extensively carried out during the last few decades. Many scientific papers have reported interesting information on the physical properties of that type of material, as well as reporting data involving phase transitions and creation of new phases during nanoindentation tests. Therefore, nanoindentation method is a very fast and useful technique for the investigation of prefabricated silicon devices, nevertheless in compliance with silicon structure modifications, one should be very careful, especially with the interpretation of the results arising from that method.
A2.3.1 Structure Transition In general, material deformation under an indentation process might involve such competing mechanisms as brittle macro- and microfracture, plasticity dislocation, defect formation and structural transitions. It was reported [6], that during nanoindentation, silicon shows small or almost no dislocation activity but highly disturbed structures. In an earlier work Pharr et al. [7] showed that during an indenter loading into a silicon specimen, permanent pressure treated as a locally acting hydrostatic stress just beneath the tip can involve a tetragonal phase formation. A very similar phenomenon was also registered upon removing the applied force, where a phase transition to bcc structure was delivered [8]. Those observations prompted further investigations, hence numerous indentation experiments have been done, showing a great variety of different phase transitions occuring in silicon structures with respect to various pressure conditions. In 1997 Gogotsi et al. [9] methodically considered deformations in silicon wafers induced by indentation measurements. They adapted a hardness indentation technique and micro-Raman spectroscopy to show many interesting phenomena involving phase transitions and other microcracking processes. The conclusion was that silicon undergoes a transition from the cubic-diamond structure (so called Si-I) into β-tin phase (Si-II) at the loading stage of indentation, between 9 and 16 GPa of hydrostatic pressure with a significant decrease of the unit cell volume by about 22%. Further increase of pressure can induce consecutive phase transitions into Si-V, Si-VI and Si-X structures, respectively. On the other hand, decreasing the hydrostatic pressure transforming the silicon material into the rhombohedral phase (Si-XII) starting at around 9 GPa of contact pressure and the Si-II to Si-XII transition results in about 9%
773
volume expansion. This phase was found to be unstable at ambient pressure, although its existence was proved from postindentation analysis. However, Si-XII can transform to the bcc phase (SiIII), upon further unloading. It was shown that transition between the Si-XII and Si-III phases is martensitic, as well as reversible and occurs at around 2 GPa of pressure. The final Si-III structure was found to be about 9% denser than the initial Si-I phase and 2% less dense than Si-XII. It is also possible to obtain an amorphous structure of silicon (a-Si), if the unloading rate is slow enough. The presented description of silicon deformations is only a simplification of the structural changes arising from that system. Up to now, at least 12 different crystallographic structures of silicon have been distinguished. All recognized and well established silicon structures are collected in Table A2.1 with respect to their occurrence and the pressure conditions. A schematic diagram of hydrostatic stress as a function of volumetric strain is presented in Figure A2.2. More detailed information including particular lattice parameters of development structures, and also a complete collection of publication references are included in Hull [10].
A2.3.2 Size, Substrate, and Other Effects in Silicon The miniaturization process of MEMS devices is bringing to pass multiple problems regarding their mechanical properties. As was introduced by Li et al. [11] the size effect cannot be neglected, especially when the technology goes into the scale of nanometers. The differences between the values of mechanical properties of a single crystal of silicon obtained in macroand nanoscale are definitely large. Many measurements carried out for bulk specimens could not be applied to microdevices due to the scale dependence of the investigated material. Also, many physical results coming from different experiments are not consistent with each other, which could be very problematic for designers of MEMS devices. Other independent parameters could also influence values of mechanical properties of silicon structures. An excellent example of that relation is the variation of hardness as a function of temperature. As one can see in Figure A2.3, the hardness H is not a constant value, but could vary with respect to temperature conditions. For extremely small prefabricated elements one has to consider an effect coming from the substrate, of which mechanical parameters and time-dependent behavior under specific conditions might play a dominant role. Nowadays, thin silicon microstructures are fabricated, for example, on glass substrates by hot-wire chemical vapor deposition, however the similar process of silicon microstructures
774
Appendix 2: Nanoindentation Characterization of Silicon and Other MEMS Materials
TABLE A2.1 Structural Data for Crystalline Phases of Silicon [10] Designation
Space Group
Crystallographic Structure
Pressure Range (GPa)
Si-I
Fd3m
Diamond cubic
012.5
Si-II
I41/ amd
Body centered tetragonal
8.816
Si-III
la3
Body centered cubic
2.10
Si-IV
P63/ mmc
Diamond hexagonal
Si-V
P6/ mmm
Primitive hexagonal
1435
Si-VI
Unidentified
3440
Si-VII
P63/ mmc
Hexagonal close packed
4078.3
Si-VIII
P41212
Tetragonal
14.80
Si-IX
P422
Tetragonal
120
Si-X
Fm3m
Face centered cubic
78.3230
Si-XI
Imma
Body centered orthorhombic
13/15
Si-XII
R3
Rhombohedral
122
FIGURE A2.2 Demonstrative diagram of hydrostatic stress P in function of volumetric strain εv for mono-crystalline silicon. Only phases appearing during loading stage are presented.
spanning could also be maintained on Si substrate. Nevertheless, even in the case of full compatibility of component materials, the above effects have to be taken into consideration during the MEMS project implementation.
FIGURE A2.3 Temperature dependence of hardness H for monocrystalline silicon measured by means of Vickers indentation method [12,13].
A2.3.3 Analytical and Mathematical Modeling Aspects Alongside experimental efforts, many theoretical approaches have been done in order to describe structural changes, as well as the dynamic processes occuring in the specimen during indentation tests. Nanoindentation produces a very high contact pressure between the diamond tip and the specimen, and brings about deformations of the material due to dislocation gliding, twinning or phase transitions. The still growing importance of computers in science gives great opportunities for employing mathematical formalism through a variety of computational techniques, which are very useful tools for modeling, making simulations and predictions of many physical properties and phenomena occuring in different materials. With molecular dynamic analysis, the phase transition from Si-I to Si-II structures was simulated by Cheong and Zhang [14]. The simulation results revealed that, at the maximum load some atoms just beneath the indenter tip appear to have a different coordination number than the initial structure. Investigations of the bond length between atoms showed that this coordination complied with β-tin structure, and after a very fast unloading, an amorphous structure was reproduced. Thus, the experimental results have been confirmed by theoretical predictions. Interesting work was done by Vodenitcharova et al. [15], who developed a constitutive model for multi-phase transitions in mono-crystalline silicon. The background of that model was based on the combination of physical mechanisms observed experimentally, the finite element method and the theory of plasticity. The developed model was successfully applied to predict correct results involving the elastic-plastic response of silicon subjected to hydrostatic pressure and nanoindentation tests, also including the phenomena of pop-in and pop-out events.
Appendix 2: Nanoindentation Characterization of Silicon and Other MEMS Materials
REFERENCES [1] B. Cappella, G. Dieter, Surf. Sci. Rep. 34 (1999) 1. [2] W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 (1992) 1564. [3] M.F. Doerner, D.S. Gardner, W.D. Nix, J. Mater. Res. 1 (1986) 601. [4] S.I. Bulychev, V.P. Alekhin, M.K. Shorshorov, A.P. Ternovskii, G.D. Shnyrev, Industrial Lab. 41 (1975) 1409. [5] A.C. Fischer-Cripps, Nanoindentation, Springer, 2004. [6] T.F. Page, W.C. Oliver, C.J. McHargue, J. Mater. Res. 7 (1992) 450. [7] G.M. Pharr, W.C. Oliver, D.S. Harding, J. Mater. Res. 6 (1991) 1129.
775
[8] J.Z. Hu, L.D. Merkle, C.S. Menoni, I.L. Spain, Phys. Rev. B 34 (1986) 4679. [9] A. Kailer, Y.G. Gogotsi, K.G. Nickel, J. Appl. Phys. 81 (1997) 3057. [10] A. George, High pressure phases of c-Si, in: R. Hull (Ed.), Properties of Crystalline Silicon, twentieth ed., mspec, The Institution of Electrical Engineers, London, 1999, pp. 104107. [11] X. Li, B. Bhushan, K. Takashima, C.W. Baek, Y.K. Kim, Ultramicroscopy 97 (2003) 481. [12] V.I. Trefilov, Y.V. Milman, Sov. Phys.-Dokl. 8 (1964) 1240. [13] I. Yonenaga, T. Hoshi, A. Usui, J. Phys.: Condens. Matter 12 (2000) 10319. [14] W.C.D. Cheong, L.C. Zhang, Nanotechnology 11 (2000) 173. [15] T. Vodenitcharova, L.C. Zhang, Int. J. Solids Struct. 41 (2004) 5411.