#### 4.1. Stress Comparison and Activation of the Slip Systems

In order to compare the similarities and differences in the dislocation-mediated plasticity, we first compare the slip systems that can be activated during deformation. For simplicity, we consider in both tests the loading direction along the [001] direction. In bulk deformation illustrated in

Figure 4a, only two representative slip planes are presented while there are four equivalent slip planes (all of which are inclined 45° to the [001] loading direction) that can be activated. Contrasting the bulk deformation, two additional slip planes that are 90° inclined to the surface can be activated (two red planes in

Figure 4b) during indentation due to the non-zero resolved shear stress on these two planes. A detailed analysis is presented later. These additionally activated slip planes comprise one of the major differences between the bulk uniaxial deformation and the indentation tests.

The onset of the plastic deformation can be correlated to dislocation nucleation, multiplication and glide motion at room temperature. Homogeneous dislocation nucleation in a perfect crystal requires that the maximum shear stress reaches the theoretical shear strength, which is about G/2π, with G being the shear modulus. Heterogeneous dislocation nucleation (i.e., dislocation nucleation from pre-existing defects), dislocation multiplication and dislocation glide motion can be activated at a much lower stress as in the case of bulk deformation.

The different stress distribution as well as the stress states in bulk and indentation deformation result in different mechanical responses with respect to the slip plane activation shown in

Figure 4. In uniaxial bulk compression, the maximum shear stress is expressed with respect to the normal stress by

${\tau}_{\mathrm{max}}^{bulk}=\frac{{\sigma}_{uniaxial}}{2}$, with

${\sigma}_{uniaxial}$ being the uniaxial stress along the compression direction. In the case of loading in [001] direction (with a Schmid factor of 0.5), the maximum shear stress in bulk compression, in an ideal case, lies along the 45° planes inclined to the [001] loading direction and is equal to the critically resolved shear stress

${\tau}_{CRSS}^{bulk}$, as has been validated by Patterson et al. [

29].

In indentation, the maximum shear stress is

${\tau}_{\mathrm{max}}^{indent}=0.31{(\frac{6{E}_{r}{}^{2}}{{\pi}^{3}{R}^{2}}{P}_{pop-in})}^{1/3}$ [

35]. Here,

R is the effective tip radius, and

P_{pop−in} corresponds to the load at the onset of the first pop-in (the elastic limit). The reduced modulus

${E}_{r}$ is calculated from the elastic constants of the indenter and the specimen by

$\frac{1}{{E}_{r}}=\frac{1-{v}_{i}{}^{2}}{{E}_{i}}+\frac{1-{v}_{s}{}^{2}}{{E}_{s}}$. With

${E}_{i}$ = 1140 GPa and

${v}_{i}$ = 0.07 for the diamond tip,

${E}_{s}$ = 264 GPa and

${v}_{s}$ = 0.237 for SrTiO

_{3} [

29], it gives

E_{r} = 224 GPa. Following Swain et al. [

38], the resolved maximum shear stress on the {110}-45° slip planes is of 0.46

p_{0} (with

p_{0} being the mean pressure) at the position of about 0.5

a (

a is the contact radius) beneath the indentation surface along the central axis, while the resolved maximum shear stress on the {110}-90° slip planes is 0.33

p_{0} at a position approximately 0.5

a directly below the circle of the contact. Hence, the activation of the {110}-45° slip planes occurs prior to the {110}-90° slip planes [

39], and the dislocations on the {110}-90° slip planes are all initiated from the edge of the contact circle and travel away from the indenter in <110> direction, as illustrated in

Figure 3b.

For comparison, the

${\tau}_{CRSS}^{bulk}$ is the stress for dislocation glide in bulk deformation and should be close to the lattice friction stress,

${\tau}_{f}$, at the yield of plastic deformation (incipient plasticity in bulk), where the effect of dislocation-dislocation hardening (in later stage with large strain) does not need to be regarded due to a very low dislocation density at this stage in single crystal [

29]. In indentation, however, the friction stress

${\tau}_{f}$ must not be correlated to the resolved maximum shear stress at pop-in (as it correlates mainly to dislocation nucleation, being either homogeneous or heterogeneous), but rather can be estimated from the dislocation pile-ups [

9,

40] as revealed by the etch pit patterns (

Figure 3b). A nice correlation between

${\tau}_{CRSS}^{bulk}$ and

${\tau}_{f}^{indent}$ has been found in single crystal SrTiO

_{3} [

29,

40], which gives a value of about 60–90 MPa at room temperature. Consider

${\tau}_{CRSS}^{bulk}={\tau}_{\mathrm{max}}^{bulk}=\frac{{\sigma}_{uniaxial}}{2}$ and the obtained yield stress from

Figure 2, a good agreement is confirmed by our experiment as well.

#### 4.2. Strain Comparison

The strain in bulk deformation can be directly read from the stress-strain curves, for instance, in

Figure 1 and

Figure 2. The strain analysis in indentation, however, is less straightforward. The elastic strain under spherical indentation can be estimated according to Field et al. [

41] via

$\epsilon ={0.2}^{}a/R$. At the critical condition of pop-in occurrence, there is:

where

R is the effective tip radius,

a_{c} is the contact radius at pop-in and is estimated by [

41]:

With

h_{c} = 60 nm being the indentation depth at pop-in and the effective tip radius

R = 1.4 μm obtained in

Figure 3a, this gives the estimation of the strain at the pop-in:

This strain corresponds to the elastic limit in indentation test and is much larger than the elastic limit in bulk deformation, which is smaller than 0.5% in

Figure 1 and

Figure 2.

The estimation of plastic strain is more complicated under spherical indentation depending on the deformation stage [

41] as well as the tip size with respect to the defect density being probed (

Section 4.3). For simplicity, however, we still adopt

$\epsilon ={0.2}^{}a/R$ as an upper bound for the estimation of the plastic strain beyond the pop-in. In this case, we take the post-mortem SEM image in

Figure 3b and determine the ultimate contact radius

a = 500 nm, with

R = 1.4 μm it gives

${\epsilon}_{p}\approx 7\%$. It is noteworthy that both {110}-45° and {110}-90° slip planes (

Figure 4b) have been activated at this plastic strain during indentation, while only the {110}-45° slip planes were activated during bulk compression tests although with a much higher plastic strain (13.6% and 13.1% in

Figure 2).

#### 4.3. Indentation Pop-in Related to Defect Population

In comparison to bulk deformation, the indentation pop-in has been frequently used as a powerful tool for understanding the incipient plasticity at micro-/nanoscale, with a focus on the dislocation nucleation as well as multiplication and motion of pre-existing dislocations, as has been extensively studied in metallic materials [

42]. In contrast, the pop-in mechanisms in ceramics have been less addressed. Therefore, a detailed discussion on the indentation pop-in is made here.

Considering single crystal ceramic or semiconductor materials, the most relevant defects for the crystal plasticity are the pre-existing dislocations and point defects prior to the mechanical loading. It remains yet a challenging topic to quantify the impact of the defects individually from the pop-in statistics. Nevertheless, these defects present in ceramics, independent of whether they are pre-existing dislocations or point defects, are very often rather far away from the indenter tip and therefore only need rather low stresses in comparison to the homogeneous dislocation nucleation (G/2π), which as discussed above, occurs only at nanoscale testing such as using a sharp indenter [

27,

43]. As a result, these defects can most likely still be activated before the stress for homogeneous dislocation nucleation is reached underneath the indenter. However, regularly the

x-axis of a pop-in statistical distribution is specified by the maximum shear stress available beneath the indenter tip even though this is not actually the critical stress level for the relevant defects to be activated [

27,

44]. Recent models in metallic materials have been suggested, which effectively and accurately convert pop-in statistics into a defect strength and density [

42,

44,

45]. While these approaches are accurate, their up-front time investment makes them less convenient to accompany the development of understanding.

Instead, we suggest to start with a simple consideration with the basic question: How many defects will be in the volume underneath the indenter? Therefore, both volumetric defect density as well as the relevant volume need to be known. Defect density is either directly specified in volumetric units or can be directly converted to it when approximating line defects, such as dislocations, as point defects with Equation (4).

It is tempting to use the areal dislocation density and contrast it to the contact area, which we avoid for two reasons. One, using a volumetric density allows using the approach for all types of defects, which can be helpful later. Two, it is more difficult to reasonably approximate a representative area underneath the indent. In particular, a representative area is not the contact area of the indenter. Instead, it is a much larger area where the stress is sufficient to activate defects. The radius of a reasonable area can be approximated from the half sphere discussed below where we believe it is more intuitive to adopt the volumetric perspective.

Calculating the volume, which is stressed above a certain value, is a bit more cumbersome. It will be approximated here as a half sphere with a radius

r, which is defined as the distance from the tip where the stress is lower than the stress required to activate a defect. Here,

${\tau}_{critical}$ is the stress required to activate a defect,

${\tau}_{local}$ is the stress in the distance from the tip

r, and

a is the contact radius. The maximum stress underneath the indenter is labeled as

${\tau}_{max}$, where either the experimentally observed maximum stress can be inserted or the theoretical shear stress used, depending on the individual needs.

This equation can be re-arranged as:

The contact radius can be expressed in two ways. Firstly, it can be calculated according to

$a={\left(\frac{3PR}{4{E}_{r}}\right)}^{1/3}$ [

35], with the tip radius

R, reduced modulus

E_{r} and indentation load

P obtained from the experimental data. This expression has the clear advantage that it can be used for direct comparison with experimental data. Its disadvantage is, however, that it makes comparison between different tip radii difficult because the value of the load

P varies with tip radius. Regarding the load

P that is needed to reach a particular stress

$\tau $, which depends on the tip radius, it will be replaced with an expression that purely relies on tip radius. By combining

$a={\left(\frac{3PR}{4{E}_{r}}\right)}^{1/3}$ and

${\tau}_{\mathrm{max}}=0.31{(\frac{6{E}_{r}{}^{2}}{{\pi}^{3}{R}^{2}}P)}^{1/3}$, we have:

The contact radius a in dependence on the tip radius is retrieved by relating it to the tip radius R, and the reduced elastic modulus E_{r} = 224 GPa and the maximum shear stress ${\tau}_{\mathrm{max}}$ (which is the theoretical shear stress in the case of homogeneous dislocation nucleation) can be inserted.

Combining these equations, the volume can be calculated with an experimental load measured by Equation (8) and for purely theoretical analysis with Equation (9):

When the volume is known, the number n of expected defects can be estimated by multiplying the volume with the defect density, e.g., by $n=\rho V$.

As the probed volume is dependent on the tip radius by the third power, the range of the volume can be nicely tuned between e.g., in the range of 10

^{−3} µm

^{3} to 10

^{4} µm

^{3} when selecting readily available tip radii over a wide range [

27]. For 25 µm tip radius, the volume equals to 10

^{−14}–10

^{−13} m

^{3} while the volume for a 1.4 µm tip radius is only 10

^{−17}–10

^{−16} m

^{3}. Consider the case of pre-existing dislocation analysis, when contrasting the approximated volumetric density of ~10

^{15} m

^{3} for a dislocation density of ~10

^{10} m

^{2} by multiplying volume and density, it becomes clear that in one case dislocations should be readily detectable while in the other case next to no pre-existing dislocations are found. For a 25 µm tip radius, this approximation suggests to find

n = 10–100 dislocations, while for a 1.4 µm tip radius, only

n = 0.01–0.1 should be found. Hence, the pop-in behavior at 25 µm tip radius should show severe impact by pre-existing dislocations while the pop-in behavior at 1.4 µm tip radius should show next to no impact by pre-existing dislocations. However, it is worthy to note that in

Figure 5, the maximum shear stress for 1.4 µm tip radius is yet much lower than the theoretical value (blue line). This is most probably attributed to the free surface being probed by the indenter, and the surface imperfections can serve as heterogeneous dislocation nucleation sites (e.g., surface steps [

46]) to dramatically reduce the stress level. In the extreme case of a small volume (e.g., an effective tip radius

R = 90 nm [

27]), the chance to find a defect in the probed volume will be nearly zero and hence statistically irrelevant, meaning that predominantly homogeneous dislocation nucleation will be observed.

The overall distribution and transition of the maximum shear stress for different tip radii was experimentally demonstrated in

Figure 5 by our indentation experiments in single crystal SrTiO

_{3} on the (001) surface. Similar results were reported on metallic material such as Mo single crystal by Bei et al. [

45], where a series of different indenter sizes were tested. With the simple calculation suggested above, it is easy for the experimentalists to get a rough idea of which defect density can be best tested with what tip radius.

On the other hand, it should be noted that the maximum shear stress occurs at about 0.5

a below the indenter (

Figure 6a). When the tip is sharp and no defect is detected in the probed volume, then the maximum shear stress (

${\tau}_{\mathrm{max}}$) is responsible for nucleating dislocations homogeneously. In this case, the maximum shear stress is the critical shear stress (

${\tau}_{critical}$) for dislocation nucleation. However, when the number of defects increases as the probed volume becomes large underneath a larger indenter, it also becomes likely to find defects very close to the tip. Such defects would then already respond to smaller stresses (such as lattice friction stress

${\tau}_{friction}$) in order to become mobile (

Figure 6b). Thus, the observed force for the “pop-in” becomes low, as shown in

Figure 5 for the 25 µm tip radius. In consequence, the response for indentation test with a large tip radius and high density of defect would become very similar to a bulk compression test.