Introduction

Monocrystalline silicon, as one of the most popular semiconductor materials, is widely used in the manufacture of micro-electro-mechanical systems (MEMS), precision optics elements, and electronic products [1]. An amorphous SiO2 film served as a dielectric layer and mechanical component locates on the top surface of monocrystalline silicon [2], affecting the time-dependent mechanical properties (creep or stress relaxation) of Si-based devices [3]. The stress exponent n at steady-state creep can reveal the mechanisms of the time-dependent plastic deformation. An interface diffusion mechanism dominates the creep deformation at shallow indentation depth region for amorphous materials, while the intrinsic creep behaviors for Zr-based metallic glasses (MGs) at deep depth region were observed [4]. This deformation mechanism was depicted by the “STZs” and “free volume” models based on the amorphous plasticity theories [5,6,7]. The STZs are small clusters of randomly close-packed atoms that spontaneously and cooperatively reorganize as applied shear stress exceeds its yield stress [8]. The operation of STZs creates a localized distortion of the surrounding materials and triggers the autocatalytic formation of shear bands, resulting in the accumulation of free volume during the localized shearing process. This deformation mechanism is different from what takes place within crystalline materials, i.e., grain boundary diffusion [9], grain sliding [10], and the dislocation governed plasticity [11].

Instantaneous plastic deformation at the onset of holding affects subsequent creep flow [12,13,14]. At higher load and/or faster loading, much more activated STZs and accumulated free volume are thought to be generated, leading to better atomic mobility, homogeneous deformation in MG [15], and lower stress exponent n [4]. The mechanism of nanoindentation creep for plasma-enhanced chemical vapor deposited silicon oxide thin film (SiOx) was investigated and reported that as-deposited state exhibited a strong indentation size effect of n [16], which was much reduced after rapid thermal annealing [17]. For monocrystalline silicon, the dislocation glide is the primary creep mechanism at elevated temperatures (from 800 to 1300 °C) and low stress (from 2 to 150 MPa) through uniaxial compression and bending tests [18]. Creep testing of fully lithiated silicon nanowires was performed at fixed force levels above and below the yield stress, reporting that the creep mechanism transformed from diffusional to dislocation-mediated in nature [19]. Electron-beam irradiation greatly enhanced creep deformation of amorphous silicon nano-cantilever and resulted in the reduction of n [20]. The size effects of silicon micro-samples were studied; the creep results of silicon diaphragms annealed at 900 °C under atmospheric pressure showed that large diaphragms have a higher risk of creep. No creep effects were visible when the diaphragm radius is below a certain value [21]. The creep rate of silicon cantilever microbeams increases at the temperature of 600 ~ 700 °C and under the stress of 235 ~ 501 MPa, and the duration of the steady-state creep decreases with the increasing temperature or stress [22]. The phase transformation coupled with the dislocation activities however dominates the deformation mechanism under nanoindentation at room temperature [23,24,25]. Though much efforts have been paid to amorphous materials and crystalline silicon, the specific time-dependent features of confined film and bilayer composites during nanoindentation process, especially at atomic scale, are still unclear because of limitation of experimental conditions.

In this study, the stress relaxation behaviors of monocrystalline silicon covered with an amorphous SiO2 film are studied through nanoindentation using molecular dynamics (MD) simulation by varying the loading speed. Focuses are put on the time-dependent behaviors of SiO2/Si bilayer composite and plastic deformation characteristics of both amorphous SiO2 film and monocrystalline silicon. This study may contribute to a better understanding of stress relaxation of confined amorphous SiO2 film and guide MEMS design.

Modeling and methods

In this work, the simulations are conducted using LAMMPS open-source software [26]. The nanoindentation system of stress relaxation for SiO2/Si bilayer composite is composed of a spherical diamond indenter and a monocrystalline silicon covered with an amorphous SiO2 film (SiO2/Si bilayer composite). The radius of spherical indenter is 6 nm. The silicon substrate has a size of 30 × 30 × 25 nm3; it contains 1,130,685 atoms. Its crystallographic orientation is [100], [010], and [001] along X, Y, and Z axes, respectively. The amorphous SiO2 film, with a size of 30 × 30 × 1 nm3, is set on the top surface of silicon. The film is prepared by quenching melted beta-cristobalite similar to the work of Chowdhury [27]. In this work, the atoms within the bilayer composite are divided into three areas, the frozen area, the thermostatic area, and the newton area. The frozen area locates at the bottom to stabilize the sample. The thermostatic area adjacent to the frozen area is set to maintain a constant temperature during the whole nanoindentation. The thickness of the frozen and thermostatic areas is 1.0 and 1.5 nm, respectively. The remaining are newton atoms, which can integrate according to Newton’s second law; the illustration refers to ref. [34]. The periodic boundary conditions are imposed in X and Y axes, while non-periodic boundary conditions are applied in Z axis.

To depict the interatomic relations between Si–Si, O–O, and Si–O atoms within the bilayer substrate, the extended Tersoff potential based on ab initio calculations by Munetoh [28] was employed. This potential has been successfully adopted to describe interactions of amorphous SiO2 [29, 30]. The interactions between Si atoms and C atoms of diamond indenter are described by the widely used Morse potential [31], the atomic potential energy expressed as:

$$E=\sum_{ij}{D}_{0}\left[{e}^{-2\alpha \left(r-{r}_{0}\right)}-{2e}^{-\alpha \left(r-{r}_{0}\right)}\right]$$
(1)

where \(D_{0}\) = 0.435 eV, α = 4.6487 Å−1, r0 = 1.9475 Å, and r represent cohesive energy, elastic modulus, interatomic equilibrium distance, and instantaneous distance, respectively. Lennard–Jones potential with the parameters ε = 0.1 eV and σ = 3.275 Å is utilized to describe the interactions of C–O atoms.

The interactions of C–C atoms are neglected due to its rigid body characteristic.

The process of stress relaxation simulation through nanoindentation involves three periods: relaxation, loading, and holding. To stabilize the simulation system, relaxation is conducted using NVT ensemble at 300 K for 90 ps. The nanoindentation loading is performed along the –Z direction by controlling the displacement of the indenter at a constant rate of 25 ms−1 until the indentation depth reaches 5.6 nm. Then the indenter remains unchanged for 200 ps during holding, as shown in Fig. 1. During the whole indentation process, the 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.

Fig. 1
figure 1

Schematic of MD simulation for nanoindentation stress relaxation

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Results and discussions

Stress relaxation features of SiO 2 /Si bilayer composite

The time-dependent stress relaxation behaviors of SiO2/Si bilayer composite are performed through nanoindentation with a spherical diamond indenter. The indenter penetrates the bilayer sample at various speeds (10, 25, 100, and 200 ms−1) separately until the indentation depth grows to 5.6 nm, then the indenter remains unchanged for 200 ps. The indentation force during holding is monitored, as shown in Fig. 2. The indentation force declines dramatically at the beginning and almost linearly towards the end of holding. It is also found that the reduced rate of indentation force at the initial of holding increases with increasing loading speed; it reverses however as the holding time is larger than 50 ps. The decreased indentation force during holding is calculated in Fig. 2b, showing the higher loading speed enhances the stress relaxation of tested samples, which is consistent with Wang and Huang’s results [4, 32].

Fig. 2
figure 2

Changes of indentation force for SiO2/Si samples indented at different speeds. a Variations of indentation force versus time. b Changings of indentation force versus speed

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It is worth noting from Fig. 2a that the indentation force at the initial of holding differs at various loading speeds, which results from the response of SiO2/Si bilayer composite to different loading speeds. The indentation force variations as a function of indentation depth at different speeds are illustrated in Fig. 3; the curves of indentation force and depth almost overlap with each other and a slight increase appears in indentation force at higher speed from 0.0 to 5.0 nm, then the indentation force drops at different degrees depending upon various loading speeds (100 ms−1 and 200 ms−1). The higher the loading speed, the more indentation force declines. This is because the higher speed leads to the appearance of higher strain energy and activated STZs within amorphous SiO2 films. The operation of activated STZs generates the free volume, and the accumulated free volume in turn favors the operation of STZs, resulting in an increasing number of defects and a decrease in mechanical stiffness of bilayer composite at higher loading speed [33].

Fig. 3
figure 3

The relationships between force and indentation depth during loading

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Plastic deformation of SiO 2 film

To deeply understand the plastic deformation of amorphous SiO2 film, its deformation features during the loading period are first probed and the number of coordinated-x silicon atoms (NCNx) of SiO2 film is therefore employed. The variations of NCNx and Si–O bond as a function of indentation depth are illustrated in Fig. 4; it is observed that the NCN2 and NCN3 silicon atoms decrease as indentation depth varies from 0.0 to 4.2 nm, then increase sharply. The corresponding values of NCN2 and NCN3 are higher at a higher speed than those at a lower speed at the same indentation depth. The NCN4 decreases monotonously during the whole loading period with a lower decrease rate at shallow indentation and a higher decrease rate as the indentation depth exceeds 4.2 nm. Furthermore, the decrease rate increases with increasing loading speed. The NCN5 grows steadily and its values are lower at 200 ms−1 at the same indentation depth. 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. 4d. It shows that the number of Si–O bonds increases with growing indentation depth until the depth is more than 4.2 nm, then it declines dramatically. The corresponding values are much higher at a lower speed than those at higher speed at the same indentation depth, implying the amorphous SiO2 film experiences a larger degree of densification at lower loading speed, while the film at higher speed exhibits less densification and tends to fracture preferentially. The reason for this is that much more STZs are activated and the resultant accumulated free volume within indented SiO2 film favors the further deformation. On the other hand, there is not enough time for generated free volume to annihilate at a higher speed, increasing in accumulated free volume.

Fig. 4
figure 4

Number variations of coordinated silicon atom and Si–O bond within SiO2 film during loading. a NCN2. b NCN3. c NCN4 and NCN5. d number of Si–O bonds

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The partial atomic configuration of the film after loading is shown in Fig. 5; it is observed that the films are thinned to almost a single atomic layer and some small cavities exist within the films. The cavities within the film at 200 ms−1 are significantly larger, compared with those at 10 ms−1, demonstrating the film at higher speed experiences larger plastic deformation and tends to fracture preferentially.

Fig. 5
figure 5

The atomic configuration of indented SiO2 after loading at different speeds. a SiO2 film at 10 ms−1. b SiO2 film at 200 ms−1

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The plastic deformation of the indented films during holding are also studied in Fig. 6; the NCN2 atoms oscillate around zero at lower speeds (10 ms−1 and 25 ms−1), while the number at higher speed (100 ms−1 and 200 ms−1) declines sharply and then slowly as the relaxation time exceeds 50 ps. The increased NCN3 atoms decreases at lower speed and its values are slightly less at higher speed at the same time, except that at 200 ms−1, where the NCN3 atoms increase and saturate around 40. The NCN4 decreases while NCN5 increases at a high rate as the time varies from 0 to 50 ps, then the NCN4 and NCN5 saturate at certain values. The absolute values of decreased NCN4 and increased NCN5 are significantly enhanced at a higher speed than those at a lower speed. The number variations of Si–O bonds during holding are also analyzed in Fig. 6d, showing that the values of Si–O bonds increase quickly at the initial and then steadily towards the end of holding. Its values grow with the increasing load speed, except those at 200 ms−1, where the values are slightly less than those at 100 ms−1. The decrease in NCN2, NCN3, and NCN4 atoms and increase in NCN5 atoms and Si–O bond number imply that the indented films are further densification during holding according to our previous work [34,35,36]. The degree of densification is strengthened with the increasing loading speed, except that at 200 ms−1. This is contributed from the plastic deformation behaviors after loading; the higher loading speed promotes much more activated STZs and accumulated free volume within films, which favors the further deformation during the subsequent stress relaxation period. For the film at 200 ms−1, further densification coupled with some elastic recovery occurs during holding.

Fig. 6
figure 6

Number variations of coordinated silicon atom and Si–O bond within SiO2 film during holding. a NCN2. b NCN3. c NCN4 and NCN5. d number of Si–O bonds

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Plastic deformation of monocrystalline silicon

The phase deformation occurs to monocrystalline silicon within the high-stressed regions as the film is indented. This follows the experimental researches and theory analysis that phase deformation dominates plasticity under point-contact conditions at room temperature. That is to say 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 [37,38,39,40]. Its deformation characteristics at different loading speeds are studied according to the coordinate number, as shown in Fig. 7. It is observed that the phase transformation atoms increase significantly with the growing indentation depth, and the curves of bct-5 phase atoms at different speeds almost overlap with each other, as well as the curves of Si-II phase atoms. In addition, the number of bct-5 silicon atoms is much higher than that of Si-II silicon at the same indentation depth; the phase distribution of Si(110) is shown in Fig. 8. It is found that the bct-5 and Si-II phase atoms at 10 ms−1 are much more than those at 200 ms−1, where more disordered silicon atoms appear under deformed surface of silicon substrate. This implies that the much higher loading speed promote amorphization.

Fig. 7
figure 7

Number variations of bct-5 and Si-II phase silicon atoms during loading

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Fig. 8
figure 8

The atomic configuration monocrystalline silicon after loading at different speeds. Silicon atoms are colored according the coordinate number. The dark green, light green, yellow, blue, and red atoms are Si-I, bct-5, Si-II, surface atoms, and other phase atoms, respectively. a Cross-section of Si (110) at 10 ms−1. b Cross-section of Si(110) at 200 ms−1

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Subsequently, the phase deformation of indented monocrystalline silicon during holding is analyzed in Fig. 9. It shows that the number of bct-5 silicon atoms raises sharply at the initial of holding, then it saturates as the time is larger than 50 ps. The values of generated bct-5 silicon atoms are much higher at higher speeds; e.g., they are 6000 and 1400 at 200 ms−1 and 10 ms−1 at the end of holding, respectively. The variations of Si-II atoms are somewhat similar to those of bct-5 atoms except those at 200 ms−1. The corresponding values increase and then decline with the increasing loading speed during the same time, as shown in Fig. 9b. The values of Si-II atoms at 200 ms−1 are almost unchanged during holding. The differences in phase deformation atoms originate from the nature of monocrystalline silicon and the effects of loading speed.

Fig. 9
figure 9

Phase transformation of monocrystalline silicon during holding. a Number of bct-5 atoms. b Number of Si-II atoms

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The atomic configuration of phase deformation distribution for monocrystalline silicon substrate after holding is shown in Fig. 10, finding that the Si-II phase atoms, which are surrounded by numerous bct-5 phase atoms, locate at the middle of the indent. The amorphous silicon atoms are beneath the indented top surface. It is also observed that the Si-II silicon atoms decrease while the bct-5 silicon atoms increase with the loading speed after holding. These results imply that the Si-II and bct-5 silicon trend to amorphization during time-dependent stress relaxation after loaded at higher speed (200 ms−1), where the applied shear stress is much higher compared with that at lower speed. Its values (30 GPa) exceed the transform stress from crystalline silicon to amorphous silicon (24 GPa) according to previous research [36, 41, 42]. This explains the number variations of bct-5 and Si-II phase atoms during holing.

Fig. 10
figure 10

The slice-section of the atomic configuration of phase deformation distribution of Si (110) after holding. The dark green, light green, yellow, blue, and red atoms are Si-I, bct-5, Si-II, surface atoms, and other phase atoms, respectively. a 10 ms−1. b 100 ms−1. c 200 ms−1

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Conclusions

The effects of loading speed on the stress relaxation behaviors of SiO2/Si bilayer composite are conducted using molecular dynamics simulation; the results are concluded as follows:

  1. (1)

    The indentation force decreases sharply at the initial and steadily towards the end of holding and the decreased indentation force increases with the increasing loading speed.

  2. (2)

    The amorphous SiO2 films are further densified during holding after loading at different speeds, and its degree of densification is enhanced at higher speed because much more activated STZs and accumulated free volume appear within films indented with higher speed.

  3. (3)

    The phase transformation of monocrystalline silicon occurs during the stress relaxation period, i.e., the Si-II and bct-5 silicon atoms increase with relaxation time and loading speed.