Exploring the origins of the indentation size effect at submicron scales

Significance

Downscaling material structure into ever-decreasing levels for modern technological advancements demands small-scale mechanical characterizations of materials, such as nanoindentation. Thus, a comprehensive understanding of indentation size effect (ISE) has a far-reaching impact, but such understanding remains incomplete at the submicron scale despite a decades-long quest since the 1980s. Here, we explored the origins of this effect experimentally by linking dislocation behavior evolution with nanoindentation mechanical data at various depths. Importantly, the dislocation behavior is found to be strongly depth dependent and gives rise to subgrain formation progressively. Aided by subgrain boundaries, the ISE mechanism transitions from dislocation source starvation to dislocation interaction as indentation depth increases. The critical mechanism transition elucidates the distinctive ISE behavior at the submicron scale.

Abstract

The origin of the indentation size effect has been extensively researched over the last three decades, following the establishment of nanoindentation as a broadly used small-scale mechanical testing technique that enables hardness measurements at submicrometer scales. However, a mechanistic understanding of the indentation size effect based on direct experimental observations at the dislocation level remains limited due to difficulties in observing and quantifying the dislocation structures that form underneath indents using conventional microscopy techniques. Here, we employ precession electron beam diffraction microscopy to “look beneath the surface,” revealing the dislocation characteristics (e.g., distribution and total length) as a function of indentation depth for a single crystal of nickel. At smaller depths, individual dislocation lines can be resolved, and the dislocation distribution is quite diffuse. The indentation size effect deviates from the Nix–Gao model and is controlled by dislocation source starvation, as the dislocations are very mobile and glide away from the indented zone, leaving behind a relatively low dislocation density in the plastically deformed volume. At larger depths, dislocations become highly entangled and self-arrange to form subgrain boundaries. In this depth range, the Nix–Gao model provides a rational description because the entanglements and subgrain boundaries effectively confine dislocation movement to a small hemispherical volume beneath the contact impression, leading to dislocation interaction hardening. The work highlights the critical role of dislocation structural development in the small-scale mechanistic transition in indentation size effect and its importance in understanding the plastic deformation of materials at the submicron scale.

Driven by an ever-increasing interest in how materials behave in the world of the small, research on physical phenomena at the micrometer and submicrometer scales has proliferated over the past few decades (1–5). Size and scale effects have been observed and documented in numerous studies, often demonstrating that material properties (mechanical, electrical, magnetic, etc.) can vary significantly as the sample dimension or volume tested is reduced (6–8). For indentation testing, it is often observed that hardness increases as the impression size or indentation depth reduces, a phenomenon referred to as the indentation size effect (ISE) (9–13).

At the micrometer scale, the ISE has been well reasoned by a widely used mechanistic model developed by Nix and Gao, where the theory of strain gradient plasticity was employed to account for the increasing hardness (H) with decreasing depth (h) for geometrically self-similar indenters (14). The basis of the model is that, as the indent becomes smaller, the geometrically necessary dislocation (GND) density rises, leading to a higher hardness based on the dislocation interaction through the Taylor hardening mechanism. As a result, a linear relationship can be predicted between the square of hardness (H²) and the inverse of depth (1/h). An alternative rationale was also developed to account for the linearity based on dislocation pile-up, another type of dislocation interaction, in the vicinity of microscale indents (15). However, nanoindentation studies with very small depths (submicron and nanometer scale) often reveal a breakdown of such linearity between the hardness–depth relationship (H² versus 1/h) (16–18). A fundamental understanding of the mechanisms responsible for this breakdown has not been fully established.

Several hypothetical explanations have been provided to understand the small-depth ISE by modifying or removing some of the underlying assumptions in the Nix–Gao model (19–22). For example, Huang et al. suggested that below a critical depth, the GND density reaches an upper limit and the hardness gradually approaches an upper saturated value (19). Kreuzer and Pippan examined the discrete nature of dislocation-mediated plasticity when considering the ISE at very small depths (20). More recently, Liu et al. argued that the elastic contributions to deformation become more pronounced with decreasing depth to address the ISE departure from the classical Nix–Gao model (21). Despite numerous modeling efforts, experimental evidence for the mechanisms that control the ISE remains elusive because of difficulties in imaging and characterizing dislocation structures beneath indents and thereby establishing how the microstructure develops and evolves. Part of the experimental challenge lies in the limitations of the microscopy techniques that have been employed to date (13). Electron backscatter diffraction (EBSD) can readily delineate the overall lattice orientation change in the area of interest, but EBSD has difficulty revealing single dislocations in a discrete manner and suffers from relatively poor spatial resolution. Conventional transmission electron microscopy (TEM) enables direct visualization of dislocations, but dislocation imaging becomes challenging due to dynamical contrast when the dislocation density is relatively high, as is often the case for the deformed region beneath an indentation. Moreover, conventional TEM usually fails to deliver a statistical picture of lattice misorientations. Both EBSD and TEM have been separately employed to probe the deformation phenomena beneath indents but have rarely been used in a combined way (15, 23–27). Nevertheless, combining the overall lattice orientation and the discrete dislocation structure is expected to provide unique insights into the origins of ISE at the submicron scale by examining the underlying dislocation evolution in detail and testing some of the key assumptions in the prevailing mechanistic descriptions.

In this study, we take advantage of a powerful TEM technique that emerged in the last three decades—precession electron diffraction (PED) (28)—to explore the evolution of both lattice orientations and dislocation configurations beneath the submicron indents in single-crystal nickel. PED was introduced by Vincent and Midgley in 1994 as a way of recording diffraction intensity that better suits structure determination (29). It then began to gain momentum in the microscopic exploration of deformation behavior of materials due to its crystal orientation indexability and improved spatial resolution compared with EBSD (30–33). For example, Vetterick et al. directly observed grain boundary-mediated plasticity at a crack tip in nanocrystalline Fe using PED (30). Leff, Weinberger, and Taheri demonstrated the viability and utility of estimating dislocation densities using PED misorientation data and discussed the interpretation of dislocation contents under different acquisition parameters (31). When imaging dislocations, PED can mitigate dynamical effects (i.e., achieving quasi-kinematical diffraction conditions) during the electron–specimen interaction to enable a sharper and clearer dislocation contrast than conventional techniques (34). In the present work, by correlating the microscopic observations with the nanoindentation depth-hardness data, we detail the microstructural evolution underlying the transition of ISE at the submicron scale where the deviation from the Nix–Gao model commences. It is shown that dislocation structural development at small depths leads to entanglements and subgrains at a critical depth. The subgrain boundaries play a pivotal role in shifting the ISE mechanism from dislocation source starvation at the very small depths to dislocation interaction hardening beyond the critical depth. The transition in mechanism is further supported by indentation tests in single crystals with different initial dislocation densities.

Results and Discussion

An array of Berkovich indentations with different depths was produced using a commercial nanoindentation system on a fully annealed single crystal of nickel with the indentation direction perpendicular to the (001) planes of the crystal and one of the three indenter sides parallel to the <100> crystallographic direction (see Fig. 1A and details in Materials and Methods). A representative indentation load-depth curve is presented in Fig. 1B, where pop-in behavior indicating the initiation of dislocation activity was observed at a depth around 20 nm. The depth of interest in this study ranges from 100 nm to 3 µm but mainly focuses on the submicron range. Depths below 100 nm were excluded because information extracted from indentations in this depth range can be significantly influenced by tip blunting and surface quality (35–37). The depth dependence of the hardness (H-h) in the range of 100 nm to 3 µm is presented Fig. 1C, where a strong ISE is observed similar to that in many previous indentation studies on different materials (13, 38, 39). Converting the H versus h plot to the H² versus 1/h plot (Fig. 1D) does not produce a perfect linear relationship, suggesting that the Nix–Gao model does not accurately predict the whole depth range. The larger-depth data appear rather linear and can be fit by the Nix–Gao model (${\left(\frac{H}{H_0}\right)}^2=1+\frac{h^{\ast }}{h}$, where $H_0$ and $h^{\ast }$ are fitting constants). However, at smaller depths (larger 1/h), the experimental data exhibit a breakdown of perceived linearity at a depth in the range of 250 to 400 nm (the yellow shaded region in Fig. 1D). We also employed a recent model from Liu et al. (21) to assess the role of elastic deformation in affecting the H²-1/h relationship (SI Appendix, Fig. S1). It was found that elastic deformation does not account for the breakdown at this range of depth. Thus, the breakdown (i.e., ISE transition) at 250 to 400 nm is intrinsically associated with a different mechanism that we focus on here.

Fig. 1.

Nanoindentation testing on single-crystal Ni specimen. (A) Schematics of the side and top views of Berkovich indentations in the single-crystal specimen. The indentation direction is parallel to one of the <001> axes of the crystal. TEM foils were lifted out from the center of the imprint along the yellow line from one edge to the opposite face of the indent, parallel to the <010> axis of the crystal. (B) A representative indentation load-depth profile. The inset magnifies the small-depth region to reveal the pop-in event. (C) Hardness–depth (H-h) relationship in the well-annealed single-crystal specimen. (D) H²-1/h plot of the data with the shaded yellow region highlighting the depth where the experimental data starts to deviate from the Nix–Gao model. The fitting results for the linearity at large depth are also shown.

To uncover the microstructure details, TEM-PED characterization was conducted on foil specimens lifted out from the center of five indentations (Fig. 1A) with depths of 100, 150, 200, 400, and 800 nm, respectively. The PED bright-field images constructed by integrating the diffraction intensity of the direct beam are provided in Fig. 2 A–E. The crystal is tilted to a <001> zone axis so that all full dislocations (Burgers vector of a/2[011]) become visible under this imaging condition (SI Appendix, Table S1). Strikingly, the dislocation configurations for different depths are not self-similar despite the geometric self-similarity of the indenter and the self-similar plastic strain it should produce regardless of depth if continuum principles apply. Indeed, the dislocation configuration at the submicron scale strongly depends on the indentation depth, breaking the self-similarity. Most notably, there is a significant microstructural transition from the indent at 200 nm to that at 400 nm. Specifically, dislocation entanglements and cell structures emerge, and subgrains with lattice orientation different from the parent crystal form as the indent proceeds to 400 nm and beyond, as reflected by the diffraction contrast (yellow arrow in Fig. 2 D and E). Subgrain boundaries appear below the face side of indents first (400 nm indent), then form throughout the deformed volume as indentation proceeds. Markedly, for the 800 nm indent, a number of subgrains develop in the plastic zone. This renders a stark contrast to the much less distorted but still dislocation-containing crystal indented to a depth of 200 nm. To better illustrate this critical change with respect to indentation depth, we mapped the lattice orientation of the same areas using a modified color key to highlight slight misorientation information (Fig. 2 F–J). The point-to-origin misorientation profiles along the blue lines in each map are provided in Fig. 2 K–O. As the indent depth increases from 200 nm to 400 nm and beyond, there is a dramatic change in dislocation arrangement with the emergence of subgrain boundaries across which the lattice misorientation is more than 3 but less than 15 degrees. Importantly, the depth range (200 to 400 nm) for this prominent transition coincides with where deviation from the Nix–Gao model commences in Fig. 1D. Hence, the formation of subgrain boundaries appears to be closely associated with the intrinsic mechanism transition of the ISE.

Fig. 2.

TEM-PED characterization of the microstructure evolution as a function of indentation depth. (A–E) PED bright-field micrographs of the microstructure underneath the indents of various depth (100 to 800 nm). At low depths, the nanoscale speckles are FIB-induced damage (vacancy clusters and small dislocation loops), which are not counted in the indentation-induced dislocation quantification. At large depth, arrows point to the emerging subgrains. (F–J) Corresponding lattice orientation mappings of the first row to further confirm the emergence of the subgrain boundaries by stark color contrast, viewed from the horizontal in-plane direction of the foils. To the right is the customized color legend. (K–O) Point-to-origin lattice misorientation of the corresponding blue line profiles marked in F through J, highlighting the dramatic changes in lattice misorientation at larger depths. Note E and J were created by montage (combining two adjacent scans due to the limited PED scan areas) to reflect the microstructure under the whole indent impression.

To further understand the dislocation evolution and how it gives rise to the emergence of subgrain boundaries, we quantified the total length and the density distributions of dislocations by tracing them in views like the one shown in Fig. 3 A and B (Materials and Methods and SI Appendix, Fig. S2 for the detailed approach). The semicircle in light yellow corresponds to the perceived minimum hemispherical plastic zone (radius of R_min = 4.35 h, Fig. 3B) beneath the impression whose size is determined by the contact radius (see geometry of Berkovich indenter in Fig. 3C). SI Appendix, Fig. S3 provides similar image processing for the 100 and 150 nm indents. Fig. 3D shows the depth dependence of the total dislocation length in the whole view of the indented volume and compares it to that predicted by the Nix–Gao model (see Materials and Methods). Clearly, the total dislocation length monotonically increases as indents proceed. Both the measured and the predicted values are the same order of magnitude, and their trends as a function of depth are essentially the same. Fitting the data to a power-law relation yields the power-law exponents close to 2 for both cases. Hence, the increase of total dislocation length can be rationalized by the strain gradient theory and the Nix–Gao model. As an aside, we note the dislocation lengths measured from experiments are about one-third of the Nix–Gao predictions. This discrepancy may be caused by dislocation escape at the free surface driven by surface proximity and image forces (40). However, provided that the missing dislocations are proportional to the total quantity, the agreement between the measured and the predicted trends is reasonable, at least to first-order.

Fig. 3.

Dislocation length and density analysis for depths before the formation of subgrains. (A) A raw PED bright-field micrograph of a 200 nm indent. (B) The sketched dislocation structure underneath the indent. The transparent yellow semicircle depicts the assumed plastic zone for dislocation density quantification in E. The transparent red region is an example of a shell used in the calculation of the shell dislocation density in F. (C) Schematics of a Berkovich indent to determine the geometrical parameters for dislocation density quantification. (D) Measured and Nix–Gao predicted total dislocation lengths as a function of the indentation depth. (E) Predicted and measured total dislocation density within the hemispherical plastic zone for indents with 100- to 200-nm depths. (F) Shell dislocation density as a function of the shell locations (measured by R/h) for indents with 100- to 200-nm depths.

With respect to the dislocation density in the hemispherical zone for small-depth indents, the experimental observations are not in agreement with the Nix–Gao model in two important ways. Firstly, as shown in the bar graphs in Fig. 3E, the dislocation density resulting from the Nix–Gao assumption that all dislocations reside within the hemispherical zone is two orders of magnitude higher than the measured values. This disparity suggests that most of the injected dislocations have moved beyond the boundary of the hemispherical zone. Such an extended dislocation distribution is consistent with the observations of Durst et al. that the plastic zone radius is greater than the contact radius (41, 42). Secondly, the measured dislocation density within the hemispherical plastic volume appears to increase slightly as the indentation depth increases, in direct opposition to the decreasing trend predicted by the Nix–Gao model (Fig. 3E). Overall, it thus appears dislocations spread out over a much larger distance at very small depths. The shell dislocation density distribution given in Fig. 3F further demonstrates how far dislocations travel away from the indents. When plotting the dislocation density in each incremental shell as a function of the distance to the indent center (an example of such shell is highlighted by light red in Fig. 3B), we observed distinct distribution profiles at different indentation depths. The profile is rather flat for the 100 nm indent but exhibits an apparent decreasing trend for the 150 and 200 nm indents. In other words, dislocations are relatively more far-reaching in shallower indents and become gradually confined as indentation depth increases.

The observed depth-dependent dislocation behavior is responsible for the eventual development of dislocation entanglements that subsequently self-organize into subgrain walls. When the crystal is indented at very small depths (<100 nm), the injected dislocations readily move either to the surface or far into the bulk. Two factors make them highly mobile. First, the starting crystal microstructure is rather pristine, leaving few or no obstacles to impede dislocation motion. Second, the strong repulsive forces between GNDs with the same Burgers vector could enhance their spreading. As a result, the dislocations easily spread out to produce a relatively uniform density profile near the indented volume. As indentation proceeds, the dislocations gradually become entangled and relatively less mobile. Eventually, the entanglement becomes so severe that dislocation activity is limited to the small volume around the hardness impression, rendering a confined dislocation configuration in the vicinity of the indented volume. This progressive confinement and accumulation of dislocations lead to an increasing density of remnant dislocations beneath the indent (Fig. 3E). Once a threshold density is achieved, the remnant dislocations rearrange locally, enabling the formation of dislocation cell boundaries and subgrains. The subgrain boundaries are likely more effective than discrete dislocations in constraining newly injected dislocations.

Our experimental observations of the subgrain emergence and the depth-dependent dislocation behavior at the submicron scale offer insights into the origins of the ISE at different stages. In Stage I from pop-in to <∼100 nm, the hardness ISE is dominated by the “dislocation source starvation” mechanism. The pop-in stress is known to be governed by dislocation source density (12, 43). Shortly after pop-in, initial dislocations are highly mobile, given no or few obstacles in the undeformed crystal. When they have reached far beyond the indented volume or slipped onto the material surface, subsequent plastic strain must be accommodated by the nucleation and motion of new dislocations, where the stress required for the nucleation or source activations dictates the hardness. Increasing the depth/contact area adds the likelihood of activating a dislocation source, thus lowering the hardness. For Stage II, from ∼100 nm to <400 nm, the hardness ISE undergoes a mixed-mode of “dislocation source starvation” and “dislocation interaction hardening” mechanisms, but the former still prevails. The dislocation nucleation or activations stress continues to govern the hardness due to the prevalent need for generating new dislocations. Meanwhile, dislocation interaction hardening is in its infancy, with dislocations being immobilized near the indent but not entirely confined (Fig. 3). At this stage, dislocation interaction hardening plays a minor role. For Stage III, from 400 nm and beyond, the hardness ISE is dominated by the dislocation interaction hardening and in good agreement with the linearity in the Nix–Gao model. The formation of subgrain boundaries plays a key role in the transition from Stage II to III, since they serve as major obstacles to dislocation motions. Dislocations then become largely constrained in a well-defined plastic volume that likely grows in proportion to the indentation depth, giving rise to an ISE dominated by dislocation interaction hardening.

To further test our hypothesis of multistage ISE mechanisms at the submicron scale, we conducted an additional set of indentation tests on another nickel specimen with everything the same but with a higher initial dislocation density under the unannealed condition (Fig. 4B). We then compared the results to the well-annealed specimen with the lower initial dislocation density (Fig. 4A). The ISE for both specimens is shown in Fig. 4C (linear fitting results of H²-1/h in both datasets are provided in SI Appendix, Fig. S4). One should take note of the fact that the scatter bars of the measurements overlap due to the intrinsically increasing stochasticity for small-depth indentation (44). However, having >25 indents for each measurement should lend some credibility to the average values. When overlaying the ISE results from two specimens, one could infer the higher initial dislocation density lowers the hardness for Stage I and II, as it would provide more dislocation sources. Meanwhile, the higher initial dislocation density would increase the hardness for Stage III due to the additional dislocation interaction hardening it provides. This is indeed what is observed in Fig. 4C, with the crossover possibly representing the transition from Stage II to III. Notably, the crossover (450 nm) is close to 400 nm where subgrains emerge in the well-annealed crystal. Comparing these two sets of nanoindentation data thus lends further support to our proposed ISE mechanism transition that is aided by the emergence of subgrains.

Fig. 4.

Comparison of ISE in specimens with different initial dislocation densities. (A and B) Conventional bright-field TEM micrographs of the well-annealed (lower initial dislocation density) and the unannealed specimens (higher initial dislocation density), respectively. (C) Overlaying the ISE data from two specimens that reveals a crossover in their H-h profiles.

The experimental observation of the ISE mechanism transition has significant implications. Firstly, we reinterrogate the deviation from the Nix–Gao model at very small indentation depths. While the model captured the trend of total dislocation content, it overlooks the possibility of pronounced dislocation motion out of the predicted plastic zone. This results in a lower remnant dislocation density right below smaller indents, invalidating the linear relationship between H² and 1/h by altering the hardening mechanism. Secondly, it manifests the “smaller is stronger” phenomena—small-size crystals exhibit higher strength than their large-size counterparts—in many other small-scale mechanical tests. It becomes clear that, instead of solely attributed to the strain gradient, the ISE also resembles other size-dependent mechanical behaviors as size decreases to a critical level, transitioning from dislocation–dislocation interaction and rearrangement to dislocation source starvation. Thirdly, with the dislocation source starvation mechanism being operative, ISE appears stronger for metals with lower dislocation contents, also shown in Fig. 4C. This could be important when designing device components with potential submicron-scale contact in service.

Furthermore, the observed ISE mechanism transition also raises several fundamental questions of how materials’ intrinsic parameters could influence it, which should be addressed in future studies. First, crystal structure effects—how does the transition occur in materials with other crystal structures, such as hexagonal close-packed or body-center cubic materials, since they have distinctive dislocation types and mobilities? Second, microstructural feature effects—how would existing dislocations, grain boundaries, or precipitates affect the ISE mechanism transition by hindering the motion of the injected dislocations at very small depths? Third, chemistry effects—alloying elements are known to modify the energy landscapes for dislocation motion; how would alloying (or even multiprincipal elements composition) affect the ISE mechanism transition? These issues are worthy of further investigation, both experimentally and computationally.

Conclusions

In summary, a combination of nanoindentation testing and TEM-PED characterization of the microstructure beneath a series of indents in single-crystalline nickel has provided insights into the mechanistic origins of the ISE at submicron scales. Specifically, it has been shown that at very small indentation depths, ∼100 nm, the dislocations produced during indentation spread in an uninhibited way rather than staying localized in a hemispherical region underneath the contact impression. This gives rise to a size effect at small depths that is distinctly different from that modeled by the Nix–Gao mechanism. However, subsequent dislocation evolution leads to dislocation entanglement and eventually the emergence of subgrain boundaries as indentation proceeds to the large depth. This serves to constrain the volume of dislocation activity to a hemispherical region beneath the indentation that could be geometrically self-similar for large depths. The subgrain boundaries are thought to play a pivotal role in the ISE mechanism transition from the dislocation source starvation to dislocation interaction hardening, as they represent strong obstacles to dislocation mediated plasticity. The subgrain-aided mechanism transition is further corroborated by comparing nanoindentation data from specimens with different initial dislocation densities. The experimental observations provide a more robust understanding of the mechanisms controlling the ISE at the submicron scales.

Materials and Methods

Materials.

An (001)-oriented single-crystal Ni cylinder (99.999%) with a diameter of 12 mm and a length of 10 mm was purchased from Goodfellow Cambridge, Ltd. The samples for nanoindentation and electron microscopy were sliced from the cylinder using a diamond wire saw, heat-treated at 1,473 K for 24 h to anneal out dislocations using a tube furnace with Ar flow protection, mechanically ground using silicon carbide papers, and electrochemically polished using a DC power supply with an electrolyte consisting of 40% sulfuric acid and 60% deionized water at room temperature. A specimen that was not annealed was used to explore the behavior of a relatively higher dislocation density. Surface quality of the electrochemically polished specimens was carefully examined to ensure no significant remnant of artifact dislocations on the topmost layer to be indented (SI Appendix, Fig. S5).

Nanoindentation.

All nanoindentation experiments were performed using a KLA iMicro nanoindentation testing system (KLA Instruments). Indentations were made using a diamond indenter with triangular pyramid Berkovich geometry, whose area function was determined by calibrating indents on a fused silica standard specimen. All indents were made using the constant strain rate test method in which the load increases as an exponential function of time to keep the ratio of loading rate to load (Ṗ/P) constant at 0.2/s. During testing, continuous stiffness measurement (CSM) was used to determine hardness and modulus at the peak load. More information regarding CSM testing can be found elsewhere (45, 46). Indentation experimental parameters used were as follows: Ṗ/P value of 0.2/s, CSM harmonic frequency of 110 Hz, and a root mean squared displacement amplitude of 2 nm was used to load to a prescribed target depth ranging from 100 nm to 3 µm where the sample was immediately unloaded. More than 25 indents were made at each depth in arrays with an appropriate spacing of over 20 times indentation depth in order to avoid indent-to-indent interaction.

Electron Microscopy.

The TEM samples were prepared using a focus ion beam system (FIB) to lift out a foil from the strict centers of the indents (Fig. 1A) using an FEI Helios NanoLab DualBeam microscope. All indents were contained in a single lift-out sample and thinned concurrently to ensure a similar thickness throughout the TEM foil for subsequent quantitative dislocation analysis. The TEM characterizations were performed in an FEI Tecnai TEM operated at 200 keV, and an FEI Titan Themis S/TEM operated at 300 keV and equipped with a NanoMEGAS ASTAR system for PED experiments (28). TEM-PED scans were performed on indents with depths of 100, 150, 200, 400, and 800 nm, respectively. The aberration-corrected microscope provided the high spatial resolution needed to image dislocations for small-depth indents, and the ASTAR system (PED) enabled the quantitative analysis of crystal orientation and misorientation. Advanced characterization techniques like these allow an experimental understanding of the microstructure evolution during the indentation process at ever-decreasing length scales and ever-increasing precision.

Dislocation Length/Density Quantification from Electron Micrographs.

Dislocation lines were traced, and their lengths were extracted using first a Fast-Fourier Transform bandpass filter, then a local contrast and dimension threshold filter in MATLAB, and last linewidth normalization. SI Appendix, Fig. S2 shows such the processing procedure for the micrograph of 150 nm indent. The processed micrographs of all 100, 150, and 200 nm indents are presented in SI Appendix, Fig. S3. The dislocation density was determined using (47)

$$\rho =\left(\frac{4}{\pi }\right)\frac{L}{At},$$ [1]

where $L$ is the measured dislocation length; $A$ is the area of interest, either the semicircle (${\rho }_{\text{circle}}$) with a radius of R_min = 4.35 h (light yellow in Fig. 3B) or the incremental semishell (${\rho }_{\text{shell}}$) with a thickness of 0.25 h (an example in light red in Fig. 3B). $t$ is the TEM foil thickness determined by measuring the fringe spacings from converging backscatter electron diffraction (SI Appendix, Fig. S6A) and fitting them into the equation $\frac{s_i^2}{n_k^2}+\frac{1}{{\xi }_{\mathbf{g}}^2{n}_k^2}=\frac{1}{t^2}$, where $s_i$ is the deviation for the ith fringe measured by $s_i={\lambda }_e\frac{{\mathrm{\Delta }\mathrm{\theta }}_i}{2{\text{θ}}_B{d}^2}$, with ${\lambda }_e$ is the electron beam wavelength (2.51 pm), ${\mathrm{\Delta }\mathrm{\theta }}_i$ is the fringe spacing, ${\text{θ}}_B$ is the spacing between the direct beam disk and the diffracted beam disk (002), and$d$ is the interplanar spacing [0.175 nm for (002) crystallographic plane]. ${\xi }_{\mathbf{g}}$ is the extinction distance. $t$ is the specimen thickness which can be obtained from the fitted intercept. The fitting result is shown in SI Appendix, Fig. S6B, and the foil thickness was determined to be 157 nm.

The total dislocation length was further reckoned by integrating the density over the volume along the radius ($a$) using MATLAB by assuming the axial symmetry of dislocation density distribution:

$$\lambda =\underset{0}{\overset{a}{{\displaystyle \int }}}2\pi r^2{\rho }_{\text{shell}}\left(r\right)dr.$$ [2]

Dislocation Length/Density Quantification from the Nix–Gao Model.

The total dislocation length (${\lambda }_{\text{total}}$) and density (${\rho }_{\mathrm{total}}$) from the Nix–Gao mechanism was calculated by

$${\rho }_{\text{total}}={\rho }_{\text{GND}}+{\rho }_{\text{SSD}}=\frac{3{\alpha }^2}{2bh}+{\rho }_{\text{SSD}},$$ [3]

$${\lambda }_{\text{total}}={\lambda }_{\text{GND}}+{\lambda }_{\text{SSD}}=\frac{\pi }{b\alpha }{h}^2+\frac{2\pi {\rho }_{\text{SSD}}}{3{\alpha }^3}{h}^3,$$ [4]

where ${\rho }_{\text{GND}}$, ${\rho }_{\text{SSD}}$, ${\lambda }_{\text{GND}}$, and ${\lambda }_{\text{SSD}}$ are the density and length of GND and statistically stored dislocations (SSD), respectively; $b$ is Burgers vector; $\alpha$ is the geometrical factor for Berkovich indenter (0.358); ${\rho }_{\text{SSD}}$ is a result from the Nix–Gao model fitting in Fig. 1D and can be obtained by calculating

$${\rho }_{\text{SSD}}={\left(\frac{H_0}{3\sqrt{3}{\alpha }_0\mu b}\right)}^2,$$ [5]

where $H_0$ is the fitted result, 1.17 GPa; ${\alpha }_0$ is a constant taken as 0.5; $\mu$ is the shear modulus, 72 GPa; and $b$ is the Burgers vector, 0.249 nm. ${\rho }_{SSD}$ is calculated to be 3.94 × 10¹³ m⁻², which is much lower than ${\rho }_{GND}$ at smaller depths.

Data Availability

All study data are included in the article and/or SI Appendix.