Indenter Size Effect on Stress Relaxation Behaviors of Surface-modified Silicon:A Molecular Dynamics Study

CHEN Juan ,FANG Liang ,CHEN Huiqin＊ ,SUN Kun ,HAN Jing

(1.School of Materials Science and Engineering,Taiyuan University of Science and Technology,Taiyuan 030024,China;2.School of Mechanical &Electrical Engineering &Xiamen University Tan Kah Kee College,Zhangzhou 363105,China;3.China State Key Laboratory for Mechanical Behavior of Materials,Xi’an Jiaotong University,Xi’an 710049,China;4.School of Mechanical and Electrical Engineering,China University of Mining and Technology,Xuzhou 221116,China)

Abstract: Long-lasting constant loading commonly exists in silicon-based microelectronic contact and can lead to the appearance of plastic deformation.Stress relaxation behaviors of monocrystalline silicon coated with amorphous SiO2 film during nanoindentation are probed using molecular dynamics simulation by varying the indenter’s size.The results show that the indentation force (stress) declines sharply at the initial and decreases almost linearly toward the end of holding for tested samples.The amount of stress relaxation of SiO2/Si samples indented with different indenters during holding increases with growing indenter size,and the corresponding plastic deformation characteristics are carefully analyzed.The deformation mechanism for confined amorphous SiO2 film is depicted based on the amorphous plasticity theories,revealing that the more activated shear transformation zones(STZs) and free volume within indented SiO2 film promote stress relaxation.The phase transformation takes place to monocrystalline silicon,the generated atoms of Si-II and bct-5 phases within monocrystalline silicon substrate during holding are much higher than those for smaller indenter.

Key words: SiO2/Si bilayer composite;stress relaxation behaviors;plastic deformation;molecular dynamics simulation;phase transformation

1 Introduction

Monocrystalline silicon,as one of the most popular semiconductor materials,plays an important role in the manufacture of micro-electro-mechanical systems (MEMS),precision optics elements and electronic products.An amorphous SiOfilm served as a dielectric layer and mechanical component locates on the top surface of monocrystalline silicon,affecting the time-dependent mechanical properties of Si-based devices.The effects of loading time and holding period at the maximum load on the hardness and modulus were systematically investigated by indentation creep analysis for different materals.It was reported that hardness,Young’s modulus and strain rate decrease with increasing holding time through various metallic glasses (MG) samples.The stress exponent (

n

) can provide useful information on the mechanism of the time dependence of plastic deformation,from which it was found that the mechanism for the indentation creep varies as the indent size increases,from linear diffusional flow to climb-controlled,even to eventually glide-controlled process for crystalline metals.Grain boundary diffusion along grain boundary sliding serves as creep mechanisms under tension creep for nanocrystalline fcc metal.For amorphous materials,an interface diffusion mechanism dominates the creep deformation at shallow indentation depth region,while the intrinsic creep behaviors for Zr-based MG at deep depth region were observed.This deformation mechanism was depicted by the“shear transformation zones (STZs)”and“free volume”models based on the amorphous plasticity theories.The STZs are small clusters of randomly close-packed atoms that spontaneously and cooperatively reorganize as applied shear stress exceeds yield stress.The operation of STZs creates localized distortion of the surrounding material and triggers the autocatalytic formation of shear bands,resulting in the accumulation of free volume during localized shearing process.Amorphous materials (

e g

metallic glasses) deform inhomogeneously and are confined to a small number of thin shear bands at low temperature and high stress,while the homogeneous deformation occurs through shear-mediated strain accommodation by operation of STZs at higher temperature and lower stress.Much more free volume and activated STZs are thought to be generated at higher load and/or faster loading,causing better atomic mobility in metallic glasses (MG).Thus more free volume leads to a larger number of STZs operation and homogeneous flow for the following holding,similar result was also observed in Fe-based MG indentation creep,while an inverse trend occurs to Ta amorphous film .For amorphous SiO,less attention has been paid.Caodepicted the deformation mechanism of nanoindentation creep for plasma-enhanced chemical vapor deposited silicon oxide thin film (SiO) and reported that the indentation size effect of

n

is much more pronounced in the asdeposited state because of its higher defect density and weaker shear resistance of STZs.For monocrystalline silicon,the primary creep mechanism is dislocation glide at elevated temperatures (from 800 to 1 300 ℃)and low stress (from 2 to 150 MPa) through uniaxial compression and bending tests.However,the phase transformation accompanied by the dislocation activity dominates the deformation mechanism under nanoindentation at room temperature because of the much higher stress and highly localized strains in small deformed volumes.The portion of high-pressure phases found in the residual indents after longer indentation holding periods arises,showing that the Si-III and Si-XII phases increase during holding time at the expense of Si-II phase.Despite much efforts have been paid to amorphous materials and crystalline silicon,the specific time-dependent characteristics of confined film and bilayer composites during nanoindentation process,especially at atomic scale,are still unclear because of limitation of experimental conditions.

In this work,the nanoindentation stress relaxation behaviors of monocrystalline silicon coated with an amorphous SiOfilm are conducted using molecular dynamics simulation by varying the size of the indenter.Emphasis is put on the time-dependent behaviors of SiO/Si composite and deformation mechanisms of amorphous SiOfilm and silicon substrate during stress relaxation.This work may contribute to a better understanding of stress relaxation of confined amorphous SiOfilm and provide new insights into deformation of amorphous film/ crystalline substrate composite.

2 Modeling and simulation methods

The system of nanoindentation stress relaxation consists of a spherical diamond indenter and a monocrystalline silicon substrate coated with an amorphous SiOfilm (SiO/Si bilayer composite).The silicon substrate,with a size of 30 nm×30 nm×25 nm,contains 1 130 685 atoms.Its crystallographic orientation is [100],[010]and [001]along

X

,

Y

and

Z

axis,respectively.The amorphous SiOfilm is prepared by quenching melted beta-cristobalite,similar to the methods reported by Chowdhury,and its size is 30 nm×30nm×1 nm,containing 58 996 atoms.In the simulation,the bilayer composite is divided into three areas,the fixed area,the thermostatic area and newton area.The fixed area,with a thickness of 1.0 nm,is located at the bottom to stabilize the sample.The thermostatic area next to the fixed area remains a constant temperature during the whole nanoindentation.The remaining is the newton area,where the motion of atoms obeys Newton’s second law.The periodic boundary conditions are imposed in

X

and

Y

axis.The Tersoff potential,which was proposed to study different phases and deformation behaviors of siliconand extended for Si-O system by Munetohbased on

ab initio

calculations,is employed to depict the interatomic relations between Si-Si,O-O and Si-O atoms within the bilayer substrate.This extended potential has been successfully used to describe interatomic interactions of amorphous SiO.The interactions between Si atoms and C atoms of diamond indenter are described by the widely used Morse potential,the atomic potential energy is expressed as:

where

D

(=0.435 eV),α (=4.648 7 Å),

r

(=1.947 5 Å),and

r

represent the cohesive energy,elastic modulus,interatomic equilibrium distance,and instantaneous distance,respectively.The interactions between C atoms of the indenter and O atoms within amorphous SiOfilm are depicted by Lennard-Jones potential with the parameters

ε

=0.1 eV and

σ

=3.275 Å.The interactions between C atoms within diamond indenter are neglected due to its rigid body characteristic.The stress relaxation simulation through nanoindentation consists of four periods:relaxation,loading,holding and unloading.The simulation system first experiences local potential energy minimization at 300 K,and then the system is relaxed to reach global stability configuration using NVT ensemble for 90 ps.After equilibrium,the nanoindentation is performed through the indenter pressing into bilayer samples along -

Z

direction at a constant rate of 25 msduring loading,and then the indenter remains unchanged at the indentation depth of 5.6 nm for a constant period of 200 ps during holding.Finally the indenter releases in +

Z

direction.During the whole indentation process,NVE ensemble with Langevin thermostat is adopted to control the simulation of 300 K,and the motion of Newtonian atoms is integrated with a velocity-Verlet algorithm with a time step of 0.5 fs.The simulations are conducted using the large-scale atomic/molecular massively parallel simulator(LAMMPS).

3 Results and discussion

The time-dependent stress relaxation behaviors of SiO/Si bilayer composite are conducted through nanoindentation with spherical diamond indenter by varying indenter’s radius (6.0,7.4 and 8.8 nm).The indenter penetrates the composite at the speed of 25 msuntil the indentation depth increases to 5.6 nm,as shown in Fig.1.Then the indenter remains unchanged for some time (200 ps),and the corresponding indention force is monitored and shown in Fig.2(a).It is found that the stress relaxation of the SiO/Si samples takes place,the indentation force decreases sharply at the initial and decreases steadily towards the end of holding for the tested samples.The decreased amount of indentation force during holding is plotted in Fig.2(b).It is observed that the amount of stress relaxation increases with the increasing indenter size,which is induced by the plastic deformation of samples during holding.

Fig.1 The illustration of stress relaxation method

Fig.2 Changes of indentation force for SiO2/Si samples indented with different indenter during holding

Fig.3 The relationships between force and indentation depth during loading

It is worth noting in Fig.2(a) that the indentation force at the beginning of holding is different for different indenters,

e g

,3 000 and 4 580 nN for samples indented with 6.0 and 8.8 nm indenters,respectively.This is because the response of SiO/Si bilayer composite to the applied stress differs during loading periods of nanoindentation,as demonstrated in Fig.3.The indentation force increases with indentation depth significantly.Furthermore,there is a distinct increase in indentation force for samples indented with a larger indenter at the same indentation depth.The deformation features of composites during loading can influence the subsequent deformation behaviors during stress relaxation,which will be discussed later.

4 Plastic deformation behaviors during loading and holding

4.1 SiO2 film

The amorphous SiOfilm with a continuous random network structure consists of numerous SiOtetrahedra.In this section,to investigate the plastic deformation behaviors of SiOfilm during loading and holding,the nearest-neighbor radial distribution function (RDF) and the number of coordinated-

x

silicon atoms (CN

x

) of SiOfilm are used.The

x

means

x

oxygen atoms are bonding to one silicon atom.The RDF of Si-O and O-O atom pairs for SiOfilms penetrated with different indenters at the indentation depth of 5.6 nm during loading are obtained in Fig.4.RDF of as-prepared film is also contained for comparison.It shows that the height and width of Si-O and O-O main peaks grow with the increasing indenter size.This implies the SiOfilm penetrated with larger indenter experiences less plastic deformation based on our previous workthat the height of Si-O and O-O peaks decrease with the growing indentation depth(plastic deformation).The significant enlargement of O-O extra small peak in Fig.4(b) illustrates that the distance of more O-O atoms within the film indented with 8.8 nm indenter decreases due to the deformation and rotation of much more involved SiOtetrahedra,compared with the films indented with smaller indenters.These results imply more atoms participate in indentation for film deformed with larger indenter but its degree of plastic deformation is lower.

Fig.4 The RDF of atom pairs for original and indented SiO2 film

Fig.5 Number variations of coordinated silicon atom and Si-O bond during loading

The number evolution of coordinated silicon atoms within SiOfilm during loading is carefully analyzed,as shown in Fig.5.It is found that both the CN2 and CN3 silicon atoms first decrease and then raise sharply as the indentation depth exceeds 4.0 nm,and the corresponding values for SiOfilm indented with smaller indenter are higher than those with larger indenter at the same indentation depth.The CN4 decreases slowly at shallow depth and rapidly as the indentation depth is larger than 4.0 nm.The CN5 grows almost linearly during the whole loading period.The number variation of Si-O bond,which is a net result of the formation of Si-O bond due to densification and breakage owing to thinning,is analyzed within a cutoff of 0.2 nm during indentation in Fig.5 (d).The number of Si-O bonds increases up to its maximum value at the indentation depth of 4.0 nm,then decreases sharply except for SiOfilm deformed with a larger indenter(8.8 nm).These results imply that as the indentation proceeds,the SiOfilms indented with 6.0 nm and 7.4 nm indenters undergo densification at the shallow indentation depth (indentation depth <4.0 nm) and transition periods at large depth,and the degree of plastic deformation is more severe for films indented with smaller indenter.The SiOfilm beneath 8.8 nm indenter just experiences densification according to our previous workthat the plastic deformation of amorphous SiOfilm during nanoindentation exhibits three periods:densification,the transition from densification to rupture,and rupture periods.There is a decrease in CN2,CN3,CN4 and an increase in CN5,Si-O bond number during the densification period at the shallow indentation depth.As the depth increases and the film tends to rupture,the CN4 decreases while the CN2,CN3,CN5 and Si-O bond number increase during transition period.

Fig.6 Number variations of coordinated silicon atom and Si-O bond during holding

The number variations of CN

x

and Si-O bond of loaded SiOfilms as a function of holding time are obtained in Fig.6.It is found that the CN2,CN3 and CN4 decrease while the CN5 increases for SiOfilms indented with indenters 6.0 and 8.8 nm.The distinct decrease in CN2,CN4 and increase in CN3,CN5 for film deformed with indenter with 7.4 nm indenter,along with those for 6.0 and 8.8 nm,indicating that the amorphous SiOfilms are further densified during the stress relaxation period.The absolute values of CN

x

(

x

=2 to 5) for the film indented with 7.4 nm indenter are much higher than those for the other films during the same holding time,implying the plastic deformation is more fierce for the film beneath 7.4 nm indenter.The increased number in Si-O bond as a function of holding time in Fig.6 (e) confirms the plastic deformation behaviors as well.

4.2 Monocrystalline silicon

The dominant plastic deformation mechanism for monocrystalline silicon under point contact conditions is phase transformation at room temperature.The original diamond cubic silicon (with 4-coordinated number) in silicon substrate transforms to metallic bct-5 (body-centered-tetragon,with 5-coordinated number) and Si-II (body-centered-tetragonal

β

-tin,with 6-coordinated number) phases in the nanoindentation by experimental researches and theory analysis.So the plastic deformation features of the monocrystalline silicon substrate in this work are analyzed by coordinate number.The phase deformation occurs as the indenter penetrates SiO/Si bilayer composite,as shown in Fig.7.The atom number of CN5 and CN6 silicon increase with the growing indentation depth during loading period when the indentation depth exceeds 2.0 nm.This is because the amorphous SiOfilm is first stressed and delays the deformation of the silicon substrate.The phase deformation atoms increase dramatically with the increasing indenter size at the same indentation depth.

Fig.7 Number of phase transformation atoms during loading

Fig.8 Phase transformation of monocrystalline silicon during holding

The deformation characteristics of monocrystalline silicon during loading can affect subsequent deformation behaviors during stress relaxation.Fig.8 shows the variations of phase transformation atoms as functions of holding time and spherical indenter size.It is observed that the CN5 and CN6 atoms raise quickly at the initial and linearly toward the end of holding in Fig.8 (a).The corresponding increased number of phase deformation atoms is calculated in Fig.8 (b),showing the increased atom numbers of CN5 and CN6 during holding are higher when the bilayer composites are deformed with larger indenter.Take the CN5 for example,its value is 1 800 for 6.0 nm indenter and 3 200 for 8.8 nm indenter.

Fig.9 Cross-section snapshots of silicon (1 0 0) Von Mises shear stress distribution for various indenters

The differences of plastic deformation for SiOfilms and silicon substrates indented with various indenters originate from the stress states of SiO/Si bilayer composites.The Von Mises stress distribution of samples after loading is shown in Fig.9.With the increasing indenter size,more atoms within SiOfilm involve in the nanoindentation,while the corresponding values of shear stress and areas of stress concentration decline.Therefore,the activation of STZs within the film is triggered and much more free volume generates within a certain area of the film deformed by 7.4 nm indenter as the applied stress exceeds yield stress.The accumulated free volume in turn favors the formation of STZs.This explains why the plastic deformation degree of the film deformed with 7.4 nm indenter is the highest and that for 8.8 nm is the lowest after loading.So as for stress relaxation behaviors of indented SiOfilm,more STZs and free volume homogeneously existed within the film enhance the plastic deformation during holding based on reported references.Thus,the film beneath the 7.4 nm indenter is the most severely densified for its higher Von Mises stress and more atoms involved in deformation.It is seen in Fig.9 that the shear stress of monocrystalline silicon substrate is almost the same for different indenters,but the areas of the stressed region increase significantly with the growing indenter size.This results in the higher phase deformation atoms (CN5 and CN6) transforming from original Si-I silicon after loading and subsequent stress relaxation periods for samples indented with 8.8 nm indenter.

5 Conclusions

The size effect of nanoindentation stress relaxation behaviors of SiO/Si bilayer composite is performed using molecular dynamics simulation by varying spherical indenter size.The conclusions are summarized as follows:

a) The amount of stress relaxation of SiO/Si sample increases with the increasing indenter size.The indentation force declines quickly at the initial and slowly toward the end of holding.

b) As the nanoindentation proceeds,the degree of plastic deformation for SiOfilms indented with different indenters varies.The film indented with a smaller indenter undergoes densification and transition periods,the film underneath larger indenter is densified during the whole loading period.

c) The deformed SiOfilms indented with different indenters are further densified during stress relaxation.The plastic deformation is the fiercest for the film (beneath 7.4 nm indenter) because of much more STZs and accumulated free volume.

d) The larger indenter favors the phase transformation of the monocrystalline silicon substrate during loading and subsequent holding.