Direct measurements of irradiation-induced creep in micropillars of amorphous Cu₅₆Ti₃₈Ag₆, Zr₅₂Ni₄₈, Si, and SiO₂

Amorphous materials are known to experience Newtonian flow under combined stress and ion irradiation,^1,2 with irradiation-induced creep (IIC) rates being orders of magnitude higher than those in polycrystalline materials.³ An understanding of this phenomenon is important for applications in ion implantation^4,5 and materials used in nuclear power generation.⁶ Relatively few IIC measurements have been performed on amorphous materials, and of these, most have relied on stress relaxation measurements through wafer curvature or beam bending methods.^1,2,7–10 These measurements, moreover, have concentrated on amorphous (a-) SiO₂,^7–9 although some additional work has been performed on metallic glasses¹⁰ and a-Si.² While stress relaxation measurements are very convenient and accurate, they suffer from a number of problems. Most significantly, the measurements integrate the stress relaxation over the entire range of the projectile thereby complicating the analysis, particularly when more than one mechanism is taking place such as in a-SiO₂ which also undergoes phase changes (densification) and anisotropic deformation.^8,11 A second difficulty concerns the effect of the end of range damage. Here, the implantation ion is added, the damage is very high, and an interaction with the undamaged substrate can become important.¹⁰ It can be challenging to interpret the results of such experiments.¹¹ 

Direct tensile creep measurements have the benefit of easier data interpretation; however, these measurements can be difficult to perform. Direct tensile creep measurements generally require relatively thick specimens, up to tens of microns, and as a consequence, only ions with large penetration depths can be used, such as light ions^12–17 and protons^18–24 in the MeV energy range or swift heavy ions^25,26 at much higher energies. The former, however, are limited to low damage levels, owing to their small cross sections for defect production, while the latter probes physics not associated with ion implantation or defect production in reactor materials. Bulge tests on thin films have recently been shown useful for this purpose,^27,28 and can use MeV heavy ions, although presently they are limited to thin film specimens. Recently, we have developed an in situ compression creep apparatus for use with submicron-sized specimens.²⁹ This method is also compatible with use of MeV heavy ions and high dpa levels, and it is not restricted to thin films. In the present work, we demonstrate the versatility of this apparatus by measuring the creep properties of three different classes of amorphous materials: metallic glasses, a-Si, and a-SiO₂. We show that the creep compliances of these three glasses, arising from nuclear stopping, are virtually the same when normalized by damage energy, or dpa (displacements per atom), but for a-SiO₂ a second contribution to the creep compliance arising from electronic stopping must also be included in the total creep response.

Figure 1 describes the measurement apparatus, which consists of a nanopositioner, an interferometric laser displacement sensor, and a doubly clamped silicon beam (the transducer).²⁹ The micropillar sample is mounted on the nanopositioner, which pushes the micropillar against the compliant transducer. A laser displacement sensor measures the deflection at the center of the transducer. Figures 1(a)–1(c) show a schematic of the testing procedure. Initially, the micropillar is disengaged from the transducer (Fig. 1(a)). Then, the micropillar is moved by the nanopositioner, and it deflects the transducer, resulting in compressive stress in the micropillar (Fig. 1(b)). The micropillar creeps under combined stress and ion bombardment, and its deformation corresponds to a change in the deflection of the transducer, which is measured by the displacement sensor, relative to the nanopositioner displacement (Fig. 1(c)). The micropillar stress was calculated in these experiments by using the average deflection of the transducer over the compression period, and measuring the spring constant of the transducer, and diameter of the micropillar. Strain rate was calculated from the change in the displacement (deflection) reading of the laser sensor during compression and height of the micropillar. Thermal drift from beam heating of the apparatus was limited by placing a 1 mm aperture in front of the sample. The compression experiment, in any case, was initiated only after this drift became negligible. The experiments were performed in an irradiation chamber with a vacuum level <1 × 10⁻⁷Torr. An HVE Van de Graaff accelerator provided 1.8–2.1 MeV Ne⁺, Ar⁺, and Kr⁺ ion beams at ≈7 × 10¹⁵ ions/(m²·s). More complete details of the apparatus and the measurement principles, including effects of damage inhomogeneity and beam heating, are provided elsewhere.²⁹ 

The amorphous (a-) Cu₅₆Ti₃₈Ag₆ cylinders were prepared by ball milling followed by hot compression. The cylinders were cut in the form of small pins and mechanically polished to obtain a sharp tip. The tips were then ion milled, using an FEI Helios Nanolab 600i focused ion beam, to obtain micropillars approximately 1 μm in diameter and 2 μm in height. a-Si samples were prepared by first microfabricating 50 μm diameter, 150 μm tall, single-crystal Si posts on 3 mm × 3 mm wafer pieces using deep reactive-ion etching (DRIE). The posts were then milled using the focused ion beam to obtain the micropillars. The Si micropillars were given a pre-irradiation, amorphization treatment, using ion doses greater than 0.4 displacements per atom (dpa),³⁰ prior to creep testing. The a-SiO₂ samples were prepared by plasma-enhanced chemical vapor deposition (PECVD) onto the microfabricated Si posts. The 3 μm thick a-SiO₂ layers on the posts were then milled using the focused ion beam to obtain the micropillars. a-Zr₅₂Ni₄₈ was prepared similarly, but it was deposited on the Si posts using magnetron sputtering. Amorphous structures of the samples were verified by X-ray diffraction (XRD) and selected area electron diffraction (SAED) analyses. Figure 2 shows the XRD data of a-Cu₅₆Ti₃₈Ag₆,²⁹ a-Zr₅₂Ni₄₈, and a-SiO₂ samples before irradiation and SAED pattern of an a-Si micropillar after irradiation. Inset of Fig. 2(d) shows literature data of a-SiO₂ after Ref. 31.

Figure 3 shows SEM images of a-Cu₅₆Ti₃₈Ag₆,²⁹ a-Zr₅₂Ni₄₈, a-Si, and a-SiO₂ micropillars before and after the creep test. Initial experiments with a-Si samples resulted in bending of the micropillar, which was attributed to the inhomogeneous amorphization at the base region of the micropillar due to the large nonuniformity in the displacement damage at that location. The inhomogeneous amorphization was eliminated by placing a ≈1 μm thick platinum protection layer at the base of the micropillar, using focused ion beam-assisted chemical vapor deposition. The penetration depth of 1.8 MeV Ar⁺ in platinum is ≈800 nm;³² therefore, the platinum layer completely shields the base of the micropillar from irradiation damage. No bending was observed in the a-Si micropillars that had this protection layer; a similar protection layer was utilized for the a-SiO₂ micropillars.

The a-SiO₂ micropillars were observed to become thinner along the ion beam direction following irradiation, presumably due to irradiation-induced anisotropic deformation.⁸ In order to eliminate the uncertainties that would result from the compression of elongated micropillars, the pre-irradiation of the SiO₂ micropillars was limited to ≈7 × 10¹⁴ ions/cm², minimizing the change in shape. For all the ions used, this damage level is beyond the saturation dose required for the irradiation-induced densification of SiO₂,⁸ ≈0.05 dpa. The other samples, a-Cu₅₆Ti₃₈Ag₆, a-Zr₅₂Ni₄₈, and a-Si micropillars did not show any observable anisotropic deformation.

Figure 4 shows the deformation and strain of a-Cu₅₆Ti₃₈Ag₆,²⁹ a-Zr₅₂Ni₄₈, a-Si, and a-SiO₂ micropillars as a function of time and displacement damage. Each micropillar was loaded three times with different initial transducer deflections, resulting in different stress levels. In the intervals where no deformation is taking place, the micropillar is disengaged from the transducer, i.e., no loading, but there is irradiation. Thus, the measured deformation is not due to beam heating or some other artifact associated with the irradiation, per se. For each loading, the deformation increases approximately linearly with time, and the deformation rate is proportional to the applied stress. These are the consequences of the Newtonian nature of the IIC in amorphous materials, in agreement with previous findings.^2,8,33 Deformation jumps are seen at the onset of each loading; we attribute these artifacts to the combined effects of repositioning of the transducer after each engagement and the finite compliance of the frame of the apparatus.

The IIC of an amorphous material under uniaxial stress can be described by

where $ε̇$ is strain rate, $ϕ̇$ is ion flux, σ is the applied stress, η is viscosity, and $H=1/ηϕ̇$ is the irradiation-induced fluidity. For point-defect mediated creep, it is more convenient to define $Hdpa=1/ηϕ̇dpa$⁠, where H_dpa (Pa⁻¹ dpa⁻¹) is the fluidity normalized by the displacement damage rate, $ϕ̇dpa$ (dpa/s), which is calculated using TRIM;³² dpa refers to displacements per atom and is a normalized unit of dose. Figure 5 shows the measured strain rates normalized by $ϕ̇dpa$ as a function of absolute value of the micropillar stress. The slopes of the data in these plots yield H_dpa. Figure 5(a) shows our data for a-Cu₅₆Ti₃₈Ag₆,²⁹ a-Zr₅₂Ni₄₈, and a-Si, as well as literature values for: 6.3 MeV proton irradiated a-Ni₇₈B₁₄Si₈,⁶ 700 keV Kr⁺ irradiated a-Zr₆₅Cu_27.5Al_7.5,¹⁰ and 800 keV Kr⁺ irradiated a-Zr₆₅Cu₃₅.¹⁰ The solid line is the prediction of a point defect model³⁴ which is described in Sec. IV. The dashed line is a fit to the a-Cu₅₆Ti₃₈Ag₆ and a-Zr₅₂Ni₄₈ data, and the dotted line is a fit to the a-Si data. For the stress relaxation measurements,¹⁰ an average stress is used for locating the data point (assuming Newtonian flow) and biaxial strain rate is multiplied by two in order to obtain the corresponding uniaxial strain rate for comparison.

Figure 5(b) shows our data for a-SiO₂ bombarded by 1.8 MeV Ne⁺, Ar⁺, and Kr⁺. Dashed lines are fits to the Ne⁺ and Ar⁺ data, and dotted line indicates the average of stress relaxation measurements for 0.25–1.8 MeV Ne⁺ and 0.25–4 MeV Xe⁺ irradiation on a-SiO₂.⁸ Several differences are observed in comparison to the other amorphous materials. First, the fluidities for the different ions no longer scale with dpa; the fluidities, moreover, tend to be greater than those for a-Cu₅₆Ti₃₈Ag₆, a-Zr₅₂Ni₄₈ and a-Si, except for the Kr⁺ irradiation, which is similar to these in magnitude. Several measurements using stress relaxation have previously been performed on a-SiO₂.^7–9,11 These data representing various ions and energies (dotted line in Fig. 5(b)) all fall close to the line passing through our Kr⁺ irradiation, and consequently they are also similar to the a-metals and a-Si in Fig. 5(a).

The irradiation-induced fluidities (IIF) in three very different types of amorphous materials: metallic glasses, a-Si, and a-SiO₂, arising from nuclear stopping, are very nearly the same when normalized by dpa. While we have used dpa for normalization, we have chosen the same displacement energy for all materials, 10 eV, and so scaling with damage energy works equally well. By using dpa, however, comparison can be made directly with molecular dynamics (MD) simulation and good agreement is obtained as discussed below. The damage energy is the total projectile energy less that going into electronic excitation. Previous work using stress relaxation measurements showed that in a-SiO₂, IIF scaled with the maximum nuclear stopping power of the projectile.⁸ Indeed, several later studies using stress relaxation measurements also indicated this same behavior.^9,11 While those results are in good agreement with our results for 1.8 MeV Kr⁺ irradiation of a-SiO₂, they are not, at first glance, in agreement with our results using Ne⁺ and Ar⁺ irradiations. This is surprising, since the previous work included Ne⁺ irradiation in the same energy range.⁸ The present work shows, however, that the scaling of IIF in a-SiO₂ with damage energy is an approximation that only holds when the electronic stopping power can be neglected. We discuss these findings in terms of current models of irradiation-induced fluidity and offer an explanation for the different results obtained using stress relaxation or micropillar creep experiments.

We first discuss the thermal spike model of IIF by Trinkaus.³⁵ The Trinkaus model assumes that energy deposition during the slowing of an ion leads to local heating, and if the energy density is sufficiently high in these regions, the local temperature can rise above the glass temperature. Elastic relaxations around these thermal spikes then lead to permanent plastic deformation when the melt region refreezes.^35–39 The Trinkaus model distinguishes between spherical spikes and cylindrical spikes, although more important for the present work are the causes of the heat spikes. In metals and Si, electronic excitation plays no significant role since the electronic system carries away the heat before it can effectively couple locally to the phonon system. It is only the energy transferred directly to the phonon systems that can cause local heating, i.e., nuclear stopping. Thermal spikes arising from nuclear stopping tend to be spherical. In insulators, electron excitations remain localized, eventually coupling to the phonon system.⁴⁰ Since the electron stopping is more or less continuous along the track of the ion, the thermal spike is cylindrical.

A second model of IIF due to nuclear stopping derives from MD simulations by Mayr et al.,³⁴ i.e., the point defect model. The point defect in this context refers to a Frenkel pair-like defect generated by the displacement of an atom from its local site to a new high energy “interstitial” site, leaving an excess volume, or “vacancy” behind.³⁴ Local relaxations around these defect sites are biased by an applied stress, and this leads to creep deformation. Such local relaxations are akin to shear transformation zones (STZ) in supercooled liquids, although under irradiation their concentration is determined by defect production rather than thermal excitations. The model differs from the thermal spike model in that spatial location of the defects, and therefore energy distribution in the solid, has no direct effect on IIF. Consequently, the near universality of IIF is naturally explained within this model, since the number of defects created scales with damage energy, i.e., the displacement energy for defect production does not greatly vary from one material to the next.⁴¹ As shown in Fig. 5(a), this model agrees very well quantitatively with experiments, with H_dpa ≈ 3 GPa⁻¹ dpa⁻¹. The defect model does not account for electronic excitation. Additional details of these models are not important for the current discussion and will not be provided here. The interested reader, however, is referred to Refs. 34 and 37.

The total irradiation-induced fluidity can be estimated by H_total = H_e + H_n, where H_e and H_n are the fluidities due to electronic and nuclear stopping, respectively. We separate the contributions of nuclear and electronic excitation since the nuclear contribution is caused almost entirely by recoils while the electronic component results from the thermal spike along the track of the irradiation particle. Fluidity due to nuclear stopping will be taken as H_dpa,n = 3 GPa⁻¹ dpa⁻¹ as determined by experiments (or the defect model). Since the spherical thermal spike regions and the displacement cascades leading to point defects mostly overlap, we ignore the contribution from spherical thermal spikes to avoid relaxing the same volume twice. Fluidity due to electronic stopping can be estimated using the expression³⁶ 

where K is the efficiency of the cylindrical spikes, μ is the shear modulus, B is a geometric factor, S_e is the electronic stopping power excluding recoils, V is the volume that goes through stress relaxation per spike, and Q is the energy deposited per spike. For uniaxial loading perpendicular to the ion beam axis,^36,$Be=4(1−ν)$⁠, where ν = 0.17 for a-SiO₂.^9,42 The value of $V/Q$ for the cylindrical spikes is given by the expression³⁶ 

where ρ is the density (2.23 g/cm³ for a-SiO₂), C is the specific heat capacity (1250 J/(kg·K)^9,41 for a-SiO₂), T_f is the flow temperature (we use T_f = 2000 K, in accordance with Refs. 9 and 38), and T₀ is the temperature of the irradiated specimen (273 K).

Below an electronic stopping power threshold, the energy deposited along the ion track does not induce effective thermal spikes that contribute to IIF in a-SiO₂. Effective thermal spikes also result in anisotropic deformation,^7,43 which in fact were used to determine the threshold value. Van Dillen et al.,⁴⁴ for example, have measured the dimensional changes of spherical silica colloids due to irradiation as a function of S_e in the range 0.5–4 keV/nm. Their data suggest a threshold value in the range 0.5–1.5 keV/nm. Benyagoub et al.,⁴⁵ on the other hand, have suggested a value of 2 keV/nm. We use a threshold of 1 keV/nm for calculations below; this value is in the middle of the range of possible values. Therefore, the efficiency factor, K = 0 for S_e < 1 keV/nm. Due to the presence of this threshold value, the electronic stopping along the paths of the relatively low-energy recoils does not contribute to the fluidity; therefore, S_e does not include the stopping power from recoils in our calculations. For S_e > 1 keV/nm, we take K = 0.21 following the experimental findings of Brongersma et al.⁹ K being equal to unity corresponds to the limiting case where all of the electronic stopping power is spent on perfectly efficient cylindrical spikes.

Figure 6(a) shows the electronic stopping and nuclear stopping power of ions as a function of depth into the sample for all three experimental conditions of a-SiO₂ micropillars. The inset shows the same information for a-Cu₅₆Ti₃₈Ag₆, a-Zr₅₂Ni₄₈, and a-Si micropillars. Electronic stopping due to recoils is not included in the plot for the reasons noted above. As mentioned above, the IIF associated with nuclear stopping is known, and since it scales with dpa, we use such scaling for calculations discussed below. The vertical line indicates the average “thickness” of the 1 μm diameter micropillars as viewed by the ion beam. The electronic stopping decreases and the nuclear stopping increases as the ions penetrate through the micropillar specimen. For the ion energies and targets selected, nuclear stopping is higher and electronic stopping is lower for heavier ions. The horizontal line of 1 keV/nm is the estimated threshold electronic stopping power for efficient thermal spikes noted above.⁴⁴ For 1.8 MeV Kr⁺ on a-SiO₂, the electronic stopping is below the threshold value throughout the micropillar. We thus see why IIF for 1.8 MeV Kr⁺ irradiated a-SiO₂ is similar to a-metals and a-Si.

By using the information in Fig. 6(a), the irradiation-induced fluidity, H, due to each mechanism, and the total fluidity, H_total = H_e + H_n can now be determined. As noted above for a-Cu₅₆Ti₃₈Ag₆, a-Zr₅₂Ni₄₈, and a-Si, high electron mobility results in rapid dissipation of electronic excitations, therefore cylindrical spikes are not effective, H_e ≈ 0. The lack of visible anisotropic deformation in these samples supports this assumption since anisotropic deformation takes place only due to cylindrical spikes and not due to spherical spikes.³⁹ 

Before discussing the predicted irradiation-induced fluidity values in a-SiO₂, we briefly comment on the effect of anisotropic deformation on the compression creep measurements. Anisotropic deformation refers to the change in shape of an amorphous material due to ion irradiation, and it arises from the cylindrical thermal spikes. It results in shrinkage in the direction parallel to the path of the projectile and expansion normal to it. The governing equation that relates the irradiation-induced creep and the anisotropic deformation is given by⁴⁶ 

where ε_ij is the strain tensor, σ_ij is the stress tensor, G is the shear modulus, ν is Poisson's ratio, δ_ij is Kronecker delta, A_ij is the anisotropic deformation tensor, and H_ij is the fluidity tensor. For the constant applied stress in these experiments, the first term can be neglected. The anisotropic deformation is linearly related to the electronic stopping power¹¹ and one can write

where c = 3.38 × 10⁻²⁴ m²nm/eV¹¹ and we take S_threshold as 1 keV/nm. Considering the case of the most pronounced anisotropic deformation, 1.8 MeV Ne⁺ on a-SiO₂, S_e = 1.2 keV/nm, averaged over the thickness of the micropillar, and A = 6.8 × 10⁻²² m²/ion. The lowest stress applied in our measurements is σ ≈ 50 MPa, and corresponding normalized strain rate, $ε̇/ϕ̇$ = 1.3 × 10⁻²⁰ m²/ion. Therefore, for the parameters of the current experiment, the irradiation-induced creep dominates over anisotropic deformation during loading, and therefore we neglect the effect of anisotropic deformation in these creep measurements.

Figure 6(b) shows the irradiation-induced fluidity due to cylindrical spikes (H_e) and from point defect creation (H_n), and the total value (H_total) as a function of sample depth for the a-SiO₂ micropillars. The same information is shown in the inset for a-Cu₅₆Ti₃₈Ag₆, a-Zr₅₂Ni₄₈, and a-Si. For a-Cu₅₆Ti₃₈Ag₆, a-Zr₅₂Ni₄₈, and a-Si, H_total increases towards the backside of the micropillar, being proportional to the nuclear stopping (Fig. 6(a)). a-Si has higher H in comparison to a-Cu₅₆Ti₃₈Ag₆ and a-Zr₅₂Ni₄₈ due to the higher displacement damage per ion fluence.

The trends for a-SiO₂ are more complex due to opposing variations of nuclear and electronic stopping with the atomic weight of the bombarding ion. In the range 0–780 nm (the thickness of the micropillar), electronic stopping of 1.8 MeV Ar⁺ is ≈10% smaller than that for 1.8 MeV Ne⁺, whereas Ar⁺ has a larger H_n. Therefore, H_total is nearly the same for Ne⁺ and Ar⁺. In either case, however, the variation of H_total is dominated by electronic stopping, which decreases with depth and falls below the threshold toward the backside of the micropillar. For 1.8 MeV Kr⁺, the electronic stopping power never rises above the threshold (Fig. 6(a)), so for it, H_e ≈ 0. On the other hand, H_n is very high due to the large nuclear stopping power, and it acts toward balancing the loss of H_e contribution. Since H_total = H_n for Kr⁺ bombardment, it increases toward the backside of the micropillar.

An important aspect of the present irradiation-induced creep measurements is the effect of the inhomogeneity of the fluidity across the “thickness” of the specimen. In our previous work,²⁹ we have shown through finite element analysis that even for a damage variation on the order of 50%, the measured creep rate is within 30% of the hypothetical case of uniform damage for total strains up to 10%. This is due to the increased stress in the low fluidity regions of the micropillar, which acts towards balancing the damage inhomogeneity through Newtonian flow.

Table I shows a summary of H values averaged over the thickness of the specimen using the information in Fig. 6(b). The comparison between the predicted total fluidity, H_total, and measured fluidity H_exp shows very good agreement between model predictions and experimental data. The agreement for a-Cu₅₆Ti₃₈Ag₆, a-Zr₅₂Ni₄₈, a-Si, and 1.8 MeV Kr⁺ bombarded a-SiO₂ shows the accuracy of the point defect model by Mayr et al.,³⁴ whereas the agreement for 1.8 MeV Ne⁺ provides evidence about the validity of the Trinkaus model for electronic stopping.³⁶ It is interesting to note that the predicted fluidity, H_total (in units of m²/(Pa·ion)), remains almost the same for a-SiO₂ from Ne⁺ through Kr⁺ bombardment due to the two different mechanisms balancing each other. We note that the irradiation-induced fluidity for 1.8 MeV Ar⁺ on a-SiO₂ is somewhat larger than the theoretical estimate, for which we have no firm explanation. The deviation appears somewhat too large to be experimental error, or an effect on inhomogeneous damage. Possibly the difference derives from synergistic effects between the electronic and nuclear stopping. For Ne⁺, nuclear stopping is very small, whereas for Kr⁺, the electronic stopping is below the threshold. Therefore, Ar⁺ is the only ion that has significant contribution from both effects, and in the regions where the cylindrical and spherical spikes overlap, the stress relaxation might be more efficient. For example, MD simulations indicate that IIF in the point defect model increases by about a factor of three when the irradiation temperature is raised from 100 K to 400 K,³⁴ and so the thermal spike from electronic stopping may enhance the effect of point defects. Additional work will be necessary to clarify this point.

Finally, we comment why previous work on IIF of a-SiO₂ found that their experimental measurements could be fit very well by scaling IIF to nuclear stopping, without a contribution of electronic stopping. We believe the answer derives from using stress relaxation measurements where only the average stress in the sample is measured over the entire penetration depth of the ion. Since the ions in these experiments are stopped within the sample, it tends to diminish the role of the electronic excitation and increase that of the nuclear stopping. In the micropillar experiments, the projectile passes through the sample, diminishing the effect of the end of range damage, thus enhancing the effect of electronic excitation. This is seen in Fig. 6 which shows that the average nuclear stopping (and H_n) for given ion and energy is much smaller for the 1 μm-diameter micropillar compression case than the stress relaxation case. On the other hand, for Ne⁺ and Ar⁺ irradiation, the electronic stopping is less effective for stress relaxation case since the stopping power is below the threshold for most of the thickness of the irradiated region.

We have performed a comparative experimental analysis of irradiation-induced creep in a variety of amorphous materials with different types of bonding, using the same specimen geometry and testing conditions. The results show strong support for the point defect model of plastic flow in a-Cu₅₆Ti₃₈Ag₆, a-Zr₅₂Ni₄₈, and a-Si, in agreement with the predictions.³⁴ The model also works well for IIF in a-SiO₂, when electronic excitation is not too large. In general, however, IIF is larger in a-SiO₂ than in the other amorphous materials studied here, owing to stress relaxation in cylindrical thermal spikes along the track of the irradiation particle.³⁶ This is important, however, only when electronic stopping power exceeds ≈1 keV/nm.

This research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DEFG02-05ER46217. The work was carried out, in part, in the Frederick Seitz Materials Research Laboratory Central Facilities, University of Illinois. We thank Dr. M. T. Saif for useful discussions and S. Mao for TEM analysis of the Si micropillars.