Nanoindentation pop‐in in oxides at room temperature: Dislocation activation or crack formation?

The scale effect (or “size effect”) on the brittleness of solid materials (mainly for brittle solids such as ceramic, glasses, etc.) was first appreciated by Lawn et al.,14 who defined the brittleness incorporating the ratio of hardness and fracture toughness. Later, Gerberich et al.5,6 revealed and discussed the scaledependent brittleness transition in silicon. However, the exact role played by dislocations, the onedimensional defects which are one of the main carriers of plastic deformation in crystals, on the scaledependent “brittleductile transition” was insufficiently discussed for brittle oxides. Contrasting the conventional belief that the majority of oxide ceramics are brittle macroscopically at room temperature,7 there is abundant evidence for dislocation motion at room temperatures in various ceramics.812 Furthermore, the dislocationbased functionality in oxides is witnessing a research upsurge in recent years. Dislocations are introduced into various oxides to harvest the functional properties. For instance, dislocations have been engineered Received: 31 January 2021 | Revised: 18 March 2021 | Accepted: 18 March 2021 DOI: 10.1111/jace.17806

There are mainly three approaches to introduce dislocations into oxide ceramics. First, using sintering techniques such as flash sintering on TiO 2 21 and yttria-stabilized ZrO 2 , 22 or field-assisted hot pressing on SrTiO 3 . 23 The shortcoming in these sintering processes is that the dislocation structures are poorly controlled. Second, well-aligned dislocations can be produced by bonding crystals such as bi-crystal fabrication, [24][25][26][27] which requires strict control of fabrication parameters (temperature, pressure, impurity of the crystals, etc.) for bi-crystals of high quality. Third, dislocations can be introduced into samples by mechanical deformation, for example, via bulk compression at room temperature [28][29][30][31][32] or high temperature, 13,15,33 by near surface mechanical treatment using nanoindentation at a small scale, [34][35][36][37][38][39][40] micro-scratching/machining, 41 and so on. While the first two approaches mainly involve processing and fabrication, the third approach relies on the dislocation mechanics of the target ceramic materials under mechanical loading. Mechanical deformation on bulk samples offers the opportunity to well align the dislocations in the sample on their slip planes. 15,28,33 Nevertheless, one of the most critical issues in bulk deformation of oxide ceramics at room temperature is the crack formation due to the much higher population of pre-existing flaws in the bulk volume 32 in comparison to the small volumes probed at a small scale. Studying the dislocation and crack behavior in these oxides is pertinent as it is desirable to introduce dislocations without cracks in order to harvest the dislocation-based functionalities. In addition, understanding the dislocation-governed mechanics will complement the assessment of the mechanical reliability of such functional materials and devices that are designed to incorporate dislocations.
Many oxides are capable of dislocation-mediated plastic deformation at room temperature at small scales. Page et al. 42 were among the first to point out that deformation in ceramics at very low load (nano)indentation could be mainly dominated by dislocation-mediated plastic deformation, as in the case of sapphire. Further examples are available from nanoindentation tests on MgO, 37,43,44 ZrO 2 , 39 ZnO, 45 and SrTiO 3 . 32,34,35,46 In addition to the examples in conventional instrumented nanoindentation studies with post-mortem surface characterization, the advanced in situ TEM indentation study has been employed to directly reveal the dislocation activation in oxides without inducing cracks. For instance, Kondo et al. 27,47 have demonstrated that the dislocations can be induced in SrTiO 3 prior to crack formation. Hockey et al. [48][49][50] investigated the plastic deformation of brittle ceramics such as Al 2 O 3 and SiC using indentation and TEM characterization, although the deformation was very often accompanied by crack formation.
The above-reviewed literature of indentation studies on ceramics at a small scale has more or less provided proof that sharp tips such as Berkovich tip or spherical indenter with small tip radius promote dislocation generation while crack formation is suppressed to a certain extent due to locally high hydrostatic compressive stresses. However, there is a lack of systematic and quantitative analysis of the critical criterion to discern these two events. In instrumented nanoindentation, one of the specific focuses in recent years was placed on the nanoindentation pop-in behavior in the load-displacement curve. Pop-in in load-controlled nanoindentation refers to the sudden jump of the displacement during a specific threshold load, signifying the elasticplastic transition. The nanoindentation pop-in behavior has been extensively studied in recent years in metallic materials [51][52][53][54][55][56][57][58] but has been barely investigated in ceramics materials until very recently. 12,34,59,60 In metallic materials (without thin oxide film formed on the surface 61 ) the pop-in indicates dislocation activation. Dislocation activation is believed to be caused either by homogeneous dislocation nucleation 62 under the sharp tip from a volume where no pre-existing defects or dislocations are present or by heterogeneous dislocation nucleation/dislocation multiplication if pre-existing defects or dislocations are available. 51 On the contrary, in ceramics, the corresponding dislocationgoverned plasticity during indentation pop-in seems elusive in the shadow of crack formation. Both dislocations and cracks are observed after indentation, making it is difficult to state if crack formation or dislocation-mediated plasticity is dominating. To carefully separate them during dynamic deformation is a challenge where it is particularly difficult to induce dislocations without crack formation. A critical tip radius, R c , below which the pop-in event correlates only to dislocations without crack formation, 63 is of great interest but yet is ambiguously defined.
In this work, we start by asking the question: for a given oxide ceramic which is ideally free of pre-existing dislocations or surface cracks, which one of the following events occurs first when the near-surface volume is stressed in instrumented indentation: the incipient dislocation nucleation or the incipient crack initiation? The follow-up question is: What is the critical condition/tip radius for the transition to happen between dislocation activation (e.g., dislocation nucleation, multiplication, and motion) and crack formation during indentation pop-in?
With these two questions, we aim to elucidate the competition between the incipient dislocation-based plasticity and incipient crack formation during indentation tests. Many advanced ceramic oxides are being used for MEMS and NEMS devices, meanwhile dislocations are being engineered into oxides to tune the functionalities as briefly introduced before.
It is thus of paramount importance to identify the deformation sequence between dislocation-governed plasticity and crack initiation in ceramic oxides. Such studies would unambiguously contribute to the understanding of dislocation-based mechanics and material's failure in a small-scale deformation of brittle ceramics oxides.
Here, we use the nanoindentation method to investigate the incipient deformation behavior in SrTiO 3 , which is a prototype perovskite and a candidate for harvesting the functionality 23 and is also a well-known "ductile" oxide which can be plastically deformed even up to about 20% plastic strain at room temperature during bulk compression tests. 64 Three other representative and advanced technology-relevant oxides, that is, Al 2 O 3 , BaTiO 3 , and TiO 2 are further examined to validate the generality derived from SrTiO 3 . α-Al 2 O 3 (sapphire) is the hardest oxide used for various semiconductor applications. 65 BaTiO 3 is one of the most widely studied ferroelectric oxides, and TiO 2 is widely used as functional oxide for gas sensors, photocatalysis, and photoelectrical devices. 16,66 The following analyses and results are presented on SrTiO 3 as a model system, while the accompanying measurements of the other three oxides (see Supplementary Materials) provide the generality of the derived concept.
To be specific, the first pop-in event in the nanoindentation load-displacement curve is adopted as the critical point for interpreting the competing deformation process, dislocation-governed plasticity and crack formation (initiation and/or propagation) in ceramic oxides. In Section 2, we first elucidate the theoretical considerations on nanoindentation pop-in, followed by experimental validation on various advanced oxide ceramics in Sections 3 and 4. Phase transformation will not be the focus in this work, but a short discussion on this issue regarding the incipient plasticity and pop-in will be presented in Section 5 by briefly combing through the literature.

| Indentation pop-in
The instrumented nanoindentation pop-in event signifies the elastic-plastic transition during the deformation process. In Figure 1 a representative pop-in event (indicated by the black arrow) is shown on single-crystal SrTiO 3 . In the following, we assume elastic, isotropic behavior for simplicity, and the elastic behavior prior to pop-in can be described by the Hertzian theory 67 : where P is the load, R is tip radius, E r is the reduced Young's modulus, and h is the indentation displacement. E r is calculated from the elastic constants of the indenter and the specimen by 54 : where E is Young's modulus, and v is Poisson's ratio. The subscripts i and s indicate indenter and sample, respectively.
On the one hand, the maximum shear stress, max , induced during the indentation process at the point of pop-in is located about 0.5a (a being the contact radius) beneath the surface along the indentation axis, and can be expressed by 67 : where the tip radius R, pop-in load P 0 , and reduced modulus E r are obtained in Equations (1-2).
On the other hand, as mode I fracture dominates in brittle solids, we attend to the maximum tensile stress that is available beneath the indenter at the point of pop-in, which is found to be at the contact ring and is usually considered to result in cone crack ("Hertzian crack") for a blunt spherical tip on brittle solids. 68 This maximum tensile stress is 67,68 : where the contact radius at the onset of pop-in is expressed as . 67 Hence we have: Combining Equations (3) and (5), it yields: where the Poisson's ratio v = 0.3 is used. It is interesting to note the tip radius does not appear in Equation (6), which holds true up to the elastic limit, i.e., at the onset of pop-in. Throughout this paper, we adopt the onset of the pop-in as the elastic limit, after which the material will go through incipient inelastic deformation caused by dislocations and/or cracks (depending on the tip radius as will be shown later). In order to clarify the competition between the incipient dislocation-mediated plasticity and the crack-dominated irreversible deformation, the critical stress beneath the indenter tip depending on the tip radius as well as the threshold for shear strength and the fracture strength is explained in the following section.

| Incipient plasticity and incipient cracking
By incipient plasticity or cracking, we refer to the inelastic deformation events occurring at the onset and during the elasto-plastic transition (e.g., pop-in). Consider a defect-free crystal, the incipient plasticity will be dominated by dislocation nucleation, with probable concurrent occurrence of dislocation multiplication and/or glide motion at room temperature; the incipient cracking will be induced by breaking the atomic bonds and creating new surfaces. Therefore, the competition between the incipient dislocation-mediated plasticity and the crack formation will be determined by the critical stress level reached beneath the indenter tip, with respect to the material intrinsic resistance to plastic deformation (theoretical shear strength being the upper bound) and cracking (theoretical fracture strength being the upper bound). In what follows, we first address these two theoretical upper bounds.

| Theoretical shear strength
Due to the conventional processing of ceramics at high temperatures, it yields very often samples with very low dislocation density. Therefore, in indentation tests with a small indenter tip radius (e.g. R = 90 nm 34 ), the maximums shear stress max in Equation (3)  In this case, if a crack occurs during indentation, it will be initiated from the pristine "perfect" surface or near-surface region, and the crack initiation will only be caused by surpassing the material's fracture strength, 70,71 in this case, the cohesive stress, c−th , that supplies the bonding of the atoms, which can be estimated by 72 : where E is Young's modulus, s the surface energy and x 0 the equilibrium spacing being approximately the distance from two adjacent atomic planes. In what follows we have adopted a much simpler and reasonable estimation of the theoretical fracture strength in the absence of plasticity,that is, c−th = E∕10. 69,73 Later in Section 4, it is demonstrated on SrTiO 3 as an example that the estimation of E/10 is even more conservative than Equation (8).
Combining Equations (7) and (8), the ratio between the two theoretical strength is obtained: By comparing Equations (6) and (9), it suggests that when the surface (in an ideal case without pre-existing flaws) is indented with a spherical indenter regardless of the tip radius, the maximum shear stress to reach the theoretical strength will always be first achieved prior to the maximum tensile stress to reach the fracture strength, as illustrated in Figure 2. For linear elastic consideration, the above theoretical prediction holds true for sharp indenter tip probing into sample surfaces with low near-surface imperfection density as in most of the state-of-the-art single-crystal oxides. In such cases, the ideal fracture strength easily exceeds the ideal yield strength, promoting dislocation plasticity prior to crack formation, as will be experimentally validated later in Section 3 using sharp indenter tips on various singlecrystal oxides.
Care must be taken, however, when indenters with large tip radii are used, for example, from tens of micrometers for large spherical indenters or Vickers indenters (at the apex), the deformation complexity increases for the stress analysis due to the higher probability of probing pre-existing flaws (e.g., pre-existing dislocations or cracks) in a much bigger volume that is stressed. The pre-existing dislocations lower the maximum shear stress for dislocation activation (e.g., heterogeneous dislocation nucleation or motion of the preexisting dislocations to assist the dislocation plasticity). 32 This may well occur in ceramics fabricated with novel sintering techniques such as flash sintering with a high density of dislocations. 21 On the other hand, the pre-existing cracks would propagate when the applied stress intensity factor reaches the critical value. The presence of pre-existing cracks also lowers the maximum tensile stress required to drive the crack propagation in mode I fracture. These changes involving the flaws will alter Equation (6) since the theoretical values are no longer valid for comparison. The effect of preexisting defects is discussed later in Section 5.

| Samples
All the oxides used in this study are single crystals. The un-doped SrTiO 3 crystals (Shinkosha Co., Ltd.), grown by the Verneuil method with high-purity SrTiO 3 powder (99.9 wt% and Sr/Ti = 1.00) and high-purity SrCO 3 powder (99.99 wt%), were used for indentation tests on the (001) surface. The sample surfaces were polished by diamond abrasives first and then finished by vibrational polishing with colloidal silica to remove the surface mechanical deformation layer, where an extremely low dislocation density of ~10 10 m −2 is shown by the etch pit study ( Figure S1). The surface roughness is checked using atomic force microscopy (AFM, Vecco, Plainview, NY, USA) to ensure a small average surface roughness (e.g., <1 nm for SrTiO 3 34 ) before the indentation tests. Details on sample preparation of the other three oxides (Al 2 O 3 , BaTiO 3 , and TiO 2 ) for the purpose of validation are provided in the Supplementary Materials.

| Indentation tests
All indentation tests were performed at room temperature (T = 21℃, with a relative humidity of 33%) on G200 nanoindenter (Keysight Technologies

| Surface structure characterization
The dislocations and cracks in the near-surface region after indentation tests are revealed using high-resolution electron channeling contrast imaging (ECCI), scanning electron microscopy (SEM), and the etch pit method. For SrTiO 3 , after the indentation tests, the (001)

| RESULTS AND ANALYSES
In what follows, the experimental results obtained on the (001) surface of single-crystal SrTiO 3 are presented and analyzed for representative purposes. The generality of the analysis derived from SrTiO 3 is further validated on the other three oxides (Al 2 O 3 , BaTiO 3 , and TiO 2 ), with the detailed results and analyses given in Supplementary Materials. The post-mortem surface features using etch pits method after indentation tests with different tip radii have been captured using SEM imaging in Figure 3. The fresh dislocations in Figure 3 are induced by deformation as the pre-existing dislocation density is very low ( Figure S1). The corresponding load-displacement curves for different tip radii are displayed in Figure 4. For indentation with Berkovich indenter, a higher load than the pop-in load (~0.1 mN at a displacement of ~10 nm) is used to increase the plastic zone size to assist the location of these indents later in SEM. Both Berkovich tip and spherical tip (R = 2 μm) reveal only dislocation activities without any cracks detected, as shown in Figure 3A-B.
Consider the case for the smallest tip radius, R = 100 nm, where no surface imperfections were present. Using Young´s modulus E = 264 GPa 28 and the simple estimation of c−th ≈ E∕10, we have the theoretical fracture strength c−th ≈ 26.4 GPa in an ideal case. Moreover, on the one hand, the load-displacement curves give an average pop-in load of p = ~0.1 mN at a displacement of ~10 nm, corresponding to a contact radius of a = ~40 nm. With v = 0.237 28 and Equation (4), it yields a maximum tensile stress max−tensile ≈ 5.2 GPa, which is far below the fracture strength to induce crack formation. On the other hand, for the same boundary condition at pop-in, the maximum shear stress is calculated by Equation (3) to be max ≈ 16.8 GPa, which is identical to the theoretical shear strength G/2π (~17 GPa for SrTiO 3 ) in Equation (7).
In this case, it is evident that for the tip with an effective tip radius of R = 100 nm, the critical condition of shear strength is reached first, therefore the dislocations will be nucleated prior to crack formation, which is shown in a more straightforward manner by Kondo et al. 27,47 using in situ TEM, where a wedge-shaped indenter (with a tip radius of about R = 100 nm) was used to induce a large number of dislocations near the surface in SrTiO 3 without forming cracks.
The above evaluation, however, does not hold true for a large tip radius, e.g., R ≥ 5 μm in the case of SrTiO 3 . For instance, Figure 3C,D shows that both cracks and dislocations are present right after the pop-in events for indenting with tip radii R = 5 μm and R = 25 μm. Similar results concerning the competition between purely dislocation-dominated process and concurrent dislocation-crack formation depending on the indenter tip radius have been observed in singlecrystals Al 2 O 3 , BaTiO 3 , and TiO 2 (see Supplementary Materials), demonstrating the generality of the analysis based on SrTiO 3 . This size effect on the competing effect of dislocation plasticity and crack formation is considered here as a new indentation size effect (ISE), different from the ISE on depth-dependent hardness, 75 or the size-dependent pop-in phenomena that involve only dislocation activities in metallic materials. 54,56 The ISE on pop-in presented here seems to be exclusively for brittle crystalline ceramics. A detailed discussion on this size effect is presented in Section 5.

| Size effect on the crack formation during indentation pop-in
The pop-in size effect in brittle oxides described in the preceding section can be attributed to several influencing factors. Of most interest is the crack formation during indentation pop-in under larger indenter tips, as evidenced directly after the pop-in stop tests. In brittle solids, there are several prevailing theories concerning the crack formation during indentation. Here we first briefly summarize these theories and then a correlation between these theories and our experimental results will be made.

| Crack propagation from pre-existing cracks
In brittle solids, the pre-existing cracks or flaws are detrimental as they can serve as stress concentrators. When the applied stress intensity factor reaches the critical value, the cracks propagate. Take mode I fracture in brittle solids as an example, the maximum tensile stress at the contact edge during indentation, when meeting locally pre-existing crack(s), would promote the crack propagation (forming circular or cone cracks) when the stress intensity factor from external loading (K a ) reaches the critical value (K Ic ), as illustrated by case 2 in Figure 5. The crack length, in an ideal case, when the crack orients itself perpendicular to the maximum tensile stress, can be estimated by the following equation: where K Ic is fracture toughness, c is the crack length, Y is a geometry factor and for simplicity is taken as Y = √ . Recalling the expression for the maximum tensile stress in terms of the pop-in load, reduced modulus, and tip radius in Equation (5), it yields: Therefore the critical crack length, c 0 , can be estimated by the following equation: is the contact radius. Equation (12) is equivalent to the Griffith energy approach 63 if K 2 Ic = 2E s is taken for brittle solids. Note this holds true for linear elastic fracture in the absence of crack tip blunting, as in the case of most ceramics where no crack tip dislocation emission is observed at room temperature.
Take single-crystal SrTiO 3 as an example, the sample is indented with tip radius R = 25 μm, P ≈ 300 mN at the pop-in event, v = 0.237, E r = 225 GPa, K Ic = 1 MPa·m 1/2 , 76 we can estimate the crack length c to be about 45 nm, meaning that a minimum crack length of 45 nm is required for a crack to propagate when the pop-in occurs. Such cracks seem not to exist for single-crystal SrTiO 3 of high-quality surface as no such features were detected in the AFM images. The argument based on Equations (11)(12) applies for other brittle oxides as long as the corresponding material parameters are provided. Note that if the pre-existing cracks are responsible for the crack formation, due to the maximum tensile stress distribution and a higher chance of probing pre-existing cracks, the cracks are most likely to be evidenced as circular cracks (or cone cracks/Hertzian cracks 68,73 ) under large indenter tips, although the formation of median and radial cracks could complicate the isolation of the circular cracks, as shown in Figure  3C,D. In this case, the density and distribution of the preexisting cracks on the surface will also be critical. 77 However, at this stage, it is remains uncertain whether the presence of pre-existing cracks and their influence for large spherical tip can be completely ruled out, especially for hard materials such as Al 2 O 3 . For a more comprehensive discussion, we also include the existence of pre-existing cracks in Figure 5.

| Shear band/fault formation as crack nuclei
Besides the crack propagation from the pre-existing flaws, it was proposed that the formation of shear fault (especially in glasses), which is akin to a shear crack, could serve as crack nuclei and propagate into the subsurface under critical conditions, so as to relax the contact pressure underneath indentation. 3,4,[78][79][80][81] It was discussed by Lawn et al. 4 that the shear fault surfaces would follow the curved shear stress trajectories in isotropic materials such as glasses or fine-grain polycrystals, while the faults would follow more along the weak crystallographic planes in single crystals. Due to the limited slip systems and a rather high lattice friction stress in most ceramic oxides, the fault patterns in such materials tend to be strongly localized around the indentation imprints. The interactions between different shear fault surfaces result in local high stress concentration sites which are in favor of the initiation of ensuing radial cracks, as experimentally shown in Vickers indentation induced cracks in glasses. 4,80 Similar operating mechanisms have been reported in nanoindentation tests in semiconductor materials such as Si and GaAs 82,83 combined with post-mortem TEM examination, but median cracks were more often induced.
It is worthy of note that in crystalline materials such as oxides and semiconductors, the shear fault surfaces correspond to the slip planes. Therefore, a more detailed examination of the dislocation interactions within these slip planes is of great interest to understand the dislocation-based crack formation and propagation. 63

| Crack initiation from dislocation pileup
In addition to the above two mechanisms, compelling experimental observation has suggested crack initiation from dislocation pileup in ceramic materials. In comparison to metals, most ceramic materials have only limited independent slip systems (very often only two 84 ) at room temperature. A direct consequence of limited-slip systems in oxides is that the allowed arbitrary plastic strain without change in volume is greatly suppressed. 85 This means that in the highly confined local deformed region in nanoindentation test, cracks may be formed due to dislocation pileups. For instance, based on the cross-sectional observation of the crack formation and dislocation slip patterns, Hagan 79 proposed the Vickers indentation induced median crack formation in LiF was due to the intersecting of the dislocations on the {110} 45° slip planes, where the angle 45° indicates the inclination of the slip planes with respect to the plane being indented. Farber et al. 86 performed elevated-temperature indentation on {111} planes in 21 mol% Y 2 O 3 fully stabilized ZrO 2 and investigated the Lomer-type dislocation pileups using TEM. They observed crack formation from the interaction between the slip bands. Crack arising from slip bands was also observed in MgO single crystals in bulk compression tests by Stokes et al. 87 and Argon et al. 88 Very recently, Fang et al. 63 studied the dislocation pileups induced by spherical indenter tip on (001) plane of single-crystal SrTiO 3 . By choosing a proper spherical indenter tip radius (R = 2 μm), they captured the crack initiation due to dislocation pileups using the etch pit method. The number of dislocations was quantitatively evaluated for crack initiation based on the theory of Zener-Stroh crack formation. 89 The correlation between the theory and experimental observation suggested that a pileup of about 20 edge dislocations is sufficient to initiate a crack along the {110} plane.
In short, the discussion in this section suggests that in oxides that exhibit dislocation plasticity, the radial cracks are most likely formed due to dislocation pileup, whereas the circular cracks have a more complicated source, which could be either the pre-existing cracks or the dislocation pileups beneath the indenter. Moreover, median cracks have not been observed in "ductile" ceramics such as SrTiO 3 by successive polishing and etching 35 after nanoindentation tests.

| Critical tip radius R c
It is of great interest for us to theoretically determine or estimate the critical tip radius R c at which the concurrent crack formation and dislocation-based plasticity occur during the pop-in event. As below this critical tip radius, it is expected that purely dislocation-governed plastic deformation prevails, avoiding the complications of crack formation so that efforts of functionality measurement (e.g., electrical conductivity) can be carried out purely on dislocations. Yet it is no easy task to determine the R c as the boundary conditions would have to involve and must be dependent on the target material (different deformation mechanisms, slip systems, different lattice friction stress, Burgers vector), sample surface condition (distribution and density of pre-existing defects due to sample preparation 90 ), chemical environment (humidity, surface chemistry, reducing atmosphere, etc.), loading rate, tip material (harder diamond tip or softer sapphire tips), and machine dynamics 63,91 in load-controlled machines. In addition, the complexity lies in the concurrent plastic deformation and crack formation during the pop-in event for large indenter tip, making it challenging to discern the sequence of these two deformation processes at the on-set of pop-in. Out of the four oxides we were only able to discern the deformation sequence in SrTiO 3 . 63 We intend to address these complexities on the critical tip radius in future works with other well-defined materials, controllable surface quality (e.g., various previous dislocation density and crack density), and test conditions.
The discussions above are summarized in Figure 6. It has now become clear that for tips smaller than R c , the nanoindentation pop-in corresponds to purely dislocationgoverned plasticity, as schematically shown in Figure 6A,B. Noticeably, for extremely small tips (R << R c ), a critical load P c (larger than the pop-in load P 0 ) exists at which cracks start to initiate from the dislocation pileup. For instance, for a sharp Berkovich indenter (with an effective tip radius of R ≈ 100 nm), the pop-in load is about P 0 ≈ 0.1 mN, while the load for inducing cracks can be as high as P c ≈ 3 mN. 35 For spherical tip (with an effective tip radius of R = 2 μm), the pop-in load is about P 0 ≈ 6~7 mN, while the load for inducing cracks is P c ≈ 11 mN. 63 On the other hand, when the tip radius is larger than R c , concurrent crack formation and dislocation activities are expected during the pop-in event. For larger tips, it is worth noticing that the pop-in distance is also much larger (e.g., Figure 4 and Figure S4, Figure 6). Such a large jump distance, as clearly shown by the post-mortem surface characterization, cannot purely be accommodated by dislocation activities alone due to the limited plasticity. Concurrent crack formation is triggered to accommodate the large pop-in jump. It is also worth pointing out that, no pop-in will be observed if the pre-existing dislocation/defect density in the near-surface region is too high, which is beyond the scope of this study.

| Generality of the theory: sizedependent brittle-to-ductile transition at room temperature
The generality of the analysis based on SrTiO 3 has been validated on Al 2 O 3 , BaTiO 3 , and TiO 2 . Now we turn to a much broader view of ceramic materials. Previous studies using instrumented nanoindentation on various ceramic oxides such as MgO, 37,43,44 ZrO 2 , 39 and ZnO 45 have provided experimental evidence that even brittle oxides deform purely plastically with dislocations under sharp indenter tips (Berkovich tips or small spherical tips). Many other types of ceramics, such as alkali halides, for example, LiF 92 and CaF 2 , 40 carbides such as SiC, 42,93 borides such as ZrB 2 , 94 sulfides such as ZnS, 12 exhibit also dislocation-governed plasticity during tests with sharp indenters, although a densification process accompanying dislocation activation was sometimes observed. Yet, this dislocation/crack competition-related indentation size effect in pop-in was not explicitly discussed in detail in literature. In the majority of these mentioned studies, cracks were very often found to accompany the deformation process due to the fact that the indentation load was set to a very large value, or indenters with a large radius (>R c ) were used, for example, in LiNbO 3 and LiTaO 3 . 95 Our experimental and theoretical approach based on ceramic oxides is expected to apply for other types of ceramic materials, as have been mentioned previously on, e.g., carbides 93 and borides 94 at room temperature using nanoindentation tests. The essence of the crack-related indentation size effect or the indentation size-dependent brittle to ductile transition is that the shear strength is reached prior to the fracture strength at ever-decreasing small scale with lower flaw density. It is worth mentioning that the size-dependent ductile to brittle transition has been previously reported in Si using nano-/micro pillar compression at room temperature by Östlund et al., 96 who showed that as the diameter of the pillar decreases below a critical size (310 to 400 nm), the pillar deforms fully ductile without cracking, which is comparable to that of metals. In contrast, cracks were always present when the pillar diameter is larger than 400 nm. A model based on dislocation shielding was proposed to explain this transition. 96 Based on the above discussion, it is suggested that with a small load and a small indenter tip radius, there is a window to introduce only dislocations with forming cracks in the ceramic oxides. Within this window, it calls for further investigation in both dislocation-based mechanics and functionalities in oxides.

| Phase transformation and shear amorphization
So far, we have excluded phase transformation as it is irrelevant for the selected four oxides in this work. However, for the sake of completion, we note that phase transformation in brittle ceramic materials could also result in nanoindentation pop-in due to the sufficiently high levels of hydrostatic compression beneath the indenter. Examples of such have been reported in GaAs. 97,98 Another well-known example is Si, in which the incipient plasticity indicated by indentation pop-in has long been attributed to the phase transformation. 99,100 However, Minor et al. used in situ nanoindentation in TEM and confirmed that the incipient plasticity is dominated purely by dislocations in <100> n-type single-crystal Si samples that were fabricated lithographically. 101,102 It is also worth noting that phase transformation could also result in the so-called "pop-out" during unloading, as briefly reviewed on Si by Lawn et al. 4 In addition, a most recent work by Reddy et al. 59 verified that the nanoindentation popin for B 4 C with strong covalent bonds is resulted from the dislocation-mediated shear amorphization.

| CONCLUSION
Through the nanoindentation pop-in study with different tip radii on model material single-crystal SrTiO 3 , we demonstrate that the room-temperature incipient plastic deformation indicated by the indentation pop-in can be caused by purely dislocation-mediated plastic flow when the indenter tip radius is smaller than a critical value R c . When R c is exceeded, concurrent dislocation plasticity and crack formation will occur at such pop-in events. The theoretical justification and experimental validation suggest a new indentation size effect related to the size-dependent competition of dislocation/crack activities during pop-in events for brittle ceramic materials: 1. When the tip radius is sufficiently small (R < R c ), the maximum shear stress reaches the shear strength prior to the maximum tensile stress or mode I fracture, so that the dislocation activation (nucleation, multiplication, and motion) is favored. Therefore, purely dislocationmediated plasticity dominates during the indentation pop-in event, while crack formation is suppressed. By further increasing the load, cracks can be initiated by dislocation pileup beneath the indenter. 2. When the tip radius is sufficiently large (R > R c ), concurrent dislocation activity and crack formation would occur during the pop-in events due to the dynamic deformation process as well as increased probability of probing preexisting flaws in the near surface, which may result in a more favorable situation for crack formation.
With further preliminary validation on three other advanced ceramic oxides (α-Al 2 O 3 , BaTiO 3 , and TiO 2 ), the current study aims to serve as a guide for using nanoindentation as a feasible technique to study the dislocation-mediated incipient plasticity for brittle ceramic materials (beyond oxides) at micro-/nanoscale, including but not limited to quantification of pop-in statistics in ceramics 12,34 related to dislocation activation energy and activation volume, [103][104][105] and nanoindentation creep tests. 34 Such studies have not been comprehensively evaluated at room temperature or elevated temperatures due to the conventional belief that ceramics are brittle and susceptible to crack formation. Furthermore, our study also paves a path to introduce dislocations without forming cracks into oxides to facilitate the studies of dislocation-based functionalities.