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Plasticity responses in ultra-small confined cubes and films
Acta Materialia 54 (2006) 4515–4523 www.actamat-journals.com
Plasticity responses in ultra-small confined cubes and films M.J. Cordill *, M.D. Chambers, M.S. Lund, D.M. Hallman, C.R. Perrey, C.B. Carter, A. Bapat, U. Kortshagen, W.W. Gerberich Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, MN 55455, USA Received 23 September 2005; received in revised form 13 May 2006; accepted 16 May 2006 Available online 8 August 2006
Abstract Nanoindentation-induced dislocation emission at 5–7 nm displacements in ultra-thin films (12–33 nm) and nanocubes (40–60 nm) is used to examine deformation and plasticity models. Using the Tabor estimate, this displacement corresponds to a plastic strain of 3–5%. Load–displacement curves produced using nanoindentation show evidence of discretized, Burgers vector–length displacement steps, or excursions, which can be associated with individual dislocation emission events. Using these displacement steps and the residual plasticity present on unloading, theoretical hardening models are developed. Linear and parabolic hardening approaches are compared for ultrathin films of nickel, cobalt, and Permalloy (Ni80Fe20), and also for silicon nanocubes. It is determined that the linear hardening model can predict the early trends of the experimental data while parabolic hardening may be more appropriate at later stages. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Thin films; Nanoindentation; Plasticity; Dislocation; Deformation
1. Introduction Recently a number of models for dislocation nucleation and hardening of thin films or small volumes under contact have been proposed [1–10]. These models include stress or strain gradient effects [5,7], linear hardening [1,6,10], parabolic hardening [6,10], and various surface energy approaches [8–10]. It has been suggested by several studies, both experimental and theoretical [8,11], that strain gradient plasticity models are not accurate at length scales below 100 nm. As films in the range of 10–100 nm were evaluated in this investigation, a strain gradient plasticity model was not considered. To ascertain if other dislocation approaches might be appropriate, four different thin film/substrate combinations were investigated with two different models. As surface roughness was also a concern, some of the films were grown on c-axis sapphire (with a roughness of 0.1– 0.2 nm as determined using atomic force microscopy) using molecular beam epitaxy (MBE). Initially, the similarities *
Corresponding author. Tel.: +1 612 624 0515. E-mail address: [email protected] (M.J. Cordill).
and differences between the earliest stages of deformation of a very thin film of nickel (grown using MBE) and a silicon nanocube were observed. For these 30–50 nm size structures, the first 5–7 nm of displacement was the focal point. This represented of the order of 3.0–5.0% plastic strain using Tabor’s estimate [12]. From these results, it appears that one or more hardening mechanisms might be induced during this initial yielding. Two hardening processes are reviewed since the unloading curves clearly demonstrated permanent plastic residual strain. Following the theoretical developments, the loads representing individual dislocations being emitted, hardening rates in terms of dP/dd during loading, and the hardness as a function of displacement are compared to the theory. The comparisons are made for four ultra-thin films of Co (30 nm), Ni (30 nm), and Ni80Fe20 (Permalloy) (33 and 12.5 nm). 2. Linear hardening in small structures First, nanocubes, the simplest small structures for evaluating plastic strain hardening, are considered. Previously,
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.05.037
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hardening in nanospheres [13] and nanoboxes [6] was examined, but in the former, contact radius and orientation are difficult to determine with certainty, and with the latter, the degree of wall thickening is unknown. Fortunately, isolated silicon nanocubes of the type shown in Fig. 1a have been grown [14] not having the former complications. They are grown with {1 0 0} faces [15] as shown by the electron diffraction pattern (Fig. 1(b)). Typically, they are coated with a 2 nm oxide film, which inhibits dislocation egress when plastically deformed. Initially, the possibility of phase transformations associated with plastic deformation was considered, but recent work on similar, relatively unconstrained films [16], nanodots [6], and nanospheres [17] of silicon have all exhibited dislocation plasticity. After locating and compressing a similar silicon cube of height h0 between a 1 lm radius diamond tip and a sapphire sub-
strate, the load–displacement data (Fig. 1c) were collected. Two immediate observations supporting the above claim were the discretized displacement steps upon loading and the residual plasticity upon unloading. The dashed unloading line parallel to the data subtracts the ‘‘creep’’ plasticity at maximum load that occurred during a hold of 3 s. Considering that such small loads have not produced creep at similar loads in either diamond-like carbon or sapphire, this was not considered to be an artifact of the testing environment. The ‘‘creep’’ in strain represented by a displacement of 0.5 nm is 0.01. This is compared to the normal qs bs representation of dislocation produced strain where qs and s are the stored dislocation density and the average distance moved. With s ¼ 0:5 nm, this would require too many dislocations considering these cubes are initially defect free. Alternatively, one could consider that fewer
Fig. 1. (a) Bright-field transmission electron microscopy image of a highly oriented silicon nanoparticle. (b) Using the {1 0 0} selected-area diffraction pattern (bottom), the faces of the particle were found to be {1 0 0}. (c) Load–displacement curve corresponding to the compression of a single nanocube using a 1 lm conical-shaped diamond tip. Discretized displacement steps during loading are emphasized along with a linear hardening model (Eq. (7)). (d) Differential curve of the nanocube load–displacement data showing six peaks and valleys of dP/dd consistent with the six displacement excursions noted in (c).
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dislocations moved larger distances during the hold. Given nine dislocations created during rising load due to geometrically necessary dislocations at each excursion, these represent a density at maximum load of 3.3 · 1015/m2 if they have a length equal to h0, the cube dimension. During the hold if these moved an average distance of 13 nm, this would satisfy a qs bs criterion. This does not seem likely at this short-time hold given the large Peierls stress. Also possible is instrument drift, since 0.5 nm in 3 s is not out of the question. Nevertheless, this did not seem likely either as all holds in these experiments represented forward displacement. Statistically, thermal drift would be forward or backward. A second possibility during the hold is that two more dislocations were nucleated and the observed discretized steps were associated with individual dislocation emission. During rising load this is reasonably consistent with the average step size of 0.22 nm, which is very close to the Burgers vector of silicon (0.236 nm). Furthermore, the total residual displacement after unloading, minus that at hold, was 2.1 nm, which would translate to N = 9 dislocations if each step represented a full Burgers vector. While this is not exactly correct, given a {1 0 0} orientation, it is a reasonable first-order expectation given the statistical averaging procedure used to obtain the data (Fig. 1c). See Section 4 for further discussion. Examining Fig. 1c, it is seen that there is no linear elastic portion since it should be greater than the initial hardening slope observed. The latter is much less than the modulus of elasticity. This is understood in terms of how a 1 lm radius diamond tip contacts a 52 nm silicon cube. To measure an elastic slope, the diamond would have to be perfectly smooth and the load train would have to be in perfect alignment with the flat of the cube. The cube is deposited on a sapphire wafer that is not perfectly smooth. Even though the root mean square roughness may be 0.1–0.2 nm there are spots with greater individual peaks. Also, Fig. 1a shows the silicon cube surface to be neither straight nor perfectly smooth. When the cube is deposited it can easily achieve a tilt to the load axis because it is deposited on roughness or debris or due to its own roughness. A combination of a 1 misalignment and a 1 nm high spot on either the cube or the substrate would require 1–2 nm of displacement before the cube surface is in full contact. This obviates any initial appearance of elasticity since by the time the entire surface area is load bearing, local stress concentrations at the oxide/silicon interface are sufficient to nucleate dislocations. The dotted theoretical line represents a simple theoretical model as discussed below. This behavior (Fig. 1c) is, by inspection, clearly plastic and not elastic. From the steeper initial unloading slope one can calculate the modulus 1 dP h0 dd to be between 120 and 150 GPa, reasonably close to the modulus of bulk silicon. The rapid departure from linear unloading below 2 lN suggests the cube may not be under
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full contact at the lower loads as discussed above. For the purposes of calculation, however, the cube is assumed to be under full contact to allow for linear hardening analysis. A single-ended pile-up of edge dislocations is assumed with the length of the pile-up equal to the height of the cube, h0. This gives a shear stress of [18] s¼
lbN pð1 mÞh0
ð1Þ
where lb is the shear modulus times the Burgers vector, N is the number of dislocations, and m is Poisson’s ratio. The only adjustable assumption is that N = d/2b where d is the total displacement. This implies that half the displacement is elastic and half plastic, which is not inconsistent with Fig. 1c. With a uniform compressive stress, r, approximately twice the shear stress (one could use 2.45 if {1 1 0} h1 1 0i slip were identified) combined with Eq. (1), these give r¼
ld Ed ¼ pð1 mÞh0 2pð1 m2 Þh0
ð2Þ
Regarding a detailed crystallographic assessment of dislocations involved, this has not been accomplished since it is not clear what slip system is involved. For example, Rabier et al. [19] have found that silicon undergoes dislocation emission under high pressure at low temperatures with full Burgers vectors of the step-shuffle variety rather than dissociated dislocations. This equation assumes an isotropic relationship between l and E and can be used to evaluate the data by finding the load and loading slope from Eq. (2) to be P ¼ rh20 ¼
Edh0 2pð1 m2 Þ
dP Eh0 ¼ dd 2pð1 m2 Þ
ð3aÞ ð3bÞ
For the isotropic values of E = 160 GPa and m = 0.218 and the cube height of 52 nm, dP/dd is calculated to be 1390 N/m. This calculation can be compared to the experimental values found from the linear segments between steps indicated in Fig. 1(c). These averaged 1310 N/ m ± 15% and are seen to be parallel to the calculated dashed line from Eq. (3b). This is a strong indicator that linear hardening can play a major role in the plasticity of small, confined structures. As applied to thin films, starting with a model recently determined for linear hardening [20], the hardness is H¼
ld pð1 vÞh0
ð4Þ
which appears inconsistent with Eq. (1) since hardness is expected to represent a greater mean pressure than a uniform compressive stress. However, a nanocube is small in three dimensions confining the slip band length to h0 or slightly larger while a thin film which is only small in one dimension can spread the plastic zone to be much greater
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than h0 as discussed elsewhere [4]. Using the same estimate as above, that half the total displacement is plastic giving d = 2Nb, combined with Eq. (4) gives H¼
2Nbl pð1 mÞh0
ð5Þ
With hardness or a mean contact pressure defined by P/pa2, the load can be found from P¼
2Nblpa2 4dRNbl ¼ pð1 mÞh0 ð1 mÞh0
ð6Þ
the latter using a2 = 2dR for a geometric contact. Using N = d/2b as above, we also find P¼
2lRd2 ð1 mÞh0
ð7Þ
which gives the loading slope for linear hardening to be dP 4lRd ¼ dd ð1 mÞh0
ð8Þ
Compared to Eq. (3b) for the loading slope, which is independent of displacement, Eq. (8) indicates an increasing slope with d. This is seen experimentally in Fig. 2, for a 303 nm radius diamond tip indenting a 30 nm thick MBE-grown nickel film on c-axis sapphire. Here the loading curve based on Eq. (7) using l = 951 GPa and m = 0.31 for nickel gives a reasonable but low accounting of the linear hardening (Fig. 2) at small displacements and an over-prediction at larger displacements. Due to the overprediction (Fig. 1) for cubes and the mixed prediction (Fig. 2) for films, a more thorough examination of the hardening mechanisms for very thin films is necessary. 3. Theoretical approaches As initial contact in the thin film may be elastic, a Hertzian contact approximation can be used for the 303 nm spherical tip of the present study. This is followed by linear and parabolic hardening which sequentially follow the initial Hertzian approximation. Fig. 2. (a) Load–displacement data for 30 nm Ni film. The linear hardening curve is based on Eq. (7). (b) Differential curve of the 30 nm Ni film data showing six peaks and valleys of dP/dd reasonably consistent with the six displacement steps noted in (a).
3.1. Hertzian contact The earliest stages of contact may be elastic but, even if not, this is a good comparison point. Load–displacement is given by 4 P ¼ E R1=2 d3=2 3
ð9Þ
where the reduced modulus, E*, is normally taken as 1 1 m2i 1 m2f þ ¼ E Ei Ef
ð10Þ
1 This seemingly high shear modulus for nickel is a composite modulus as discussed in Section 3.
with mi, Ei representing Poisson’s ratio and modulus for the diamond and mf, Ef being those of the film. This, however, ignores any effect from the substrate. Since these 12.5– 33 nm thick films are so thin, even 5–10 nm displacements can result in a substantial composite film/substrate effect. Here, the model of Gao et al. [21] for indentation into multilayers was initially used giving 1 ms ðmf ms ÞI 1 ha 1m ¼ ð11Þ l ls þ ðlf ls ÞI 0 ha eff where ms, mf are Poisson’s ratios and ls, lf are shear moduli of the film and substrate, respectively. This requires a
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numerical analysis but Gao et al. [21] have given graphical solutions for the weight functions I1 and I0 which vary as a function of the film thickness to contact radius ratio. Taking a typical contact displacement of 7.5 nm for these thinfilm measurements gives a contact radius of 57.7 nm for the 303 nm radius diamond tip. For the 30 nm nickel film deposited, this gives h/a = 0.52. From Fig. 3 of Gao et al. [21] for I0 and I1 and the following film/substrate properties, one can determine both l and E. As an upper bound for these very thin films, one might use a flat punch approximation with film modulus properties for EÆ111æ. This, however, gave an even larger value than expected which is probably not representative of the nanocrystalline film/substrate combination. As discussed in Section 5, there is evidence that such films have a considerably lower modulus than might be expected if these were single-crystal nickel. Bearing this in mind, the following properties were used to determine NiÆ111æAl2O3Æ0001æ composite modulus: h ENi ¼ 200 GPa; mNi ¼ 0:30; lNi ¼ 76:9 GPa; I 0 ¼ 0:25 a EAl2 O3 ¼ 432 GPa; mAl2 O3 ¼ 0:2; h ¼ 0:45 lAl2 O3 ¼ 180 GPa; I 1 a Insertion of these into Eq. (11) gives Eeff = 340 GPa which is considerably greater than the average reduced modulus of 248þ25 14 GPa as determined in the standard way by nanoindentation from four tests. This degree of fit is not too unexpected given the numerical solution of Chen and Vlassak [22] which shows that the weight function at Eq. (11) is larger than Gao’s result. As the term in the denominator is dominant, this would reduce the calculated modulus. They obtained a discrepancy for I0(0.52) of about 33% based on a modulus ratio of 1.8. Here, the modulus ratio of sapphire to nanocrystalline nickel film is about 2.16 which could result in further deviation. The end result is that Gao’s analysis over-predicted the composite modulus effect. Qualitatively, the predicted moduli of 340 and 346 GPa for the nominal 30 and 12.5 nm Permalloy films are in the right direction but over-predicting. Still, this gives confidence in using the measured moduli values for both the nickel and Permalloy films in subsequent calculations. The values of E* for the Permalloy gave 246 ± 18 GPa and 252 ± 10 GPa for the 33 nm (7 tests) and 12.5 nm (4 tests) thick films, respectively.
Differentiating Eq. (9) to represent the elastic load– displacement slope gives dP ¼ 2E R1=2 d1=2 dd
ð12Þ
noting this has a displacement dependence between Eqs. (3b) and (8) and is indirectly dependent on thickness through the modulus dependence of Eq. (11). 3.2. Linear hardening For single crystals or nanocubes and nanospheres with very few or no grown-in dislocations, linear hardening can apply. It also applies to thin films grown by molecular beam epitaxy or perhaps even to sputtered films where the film thickness is no larger than the grain size. Therefore Eqs. (4)–(8) are used to evaluate hardness and load– displacement curves. It is also important to notice that the hardness and loading slope here (Eqs. (4) and (8)) are proportional to d/h0. This represents a larger thickness dependence on the loading slope than would be anticipated from elasticity (Eqs. (11) and (12)). 3.3. Parabolic hardening Forest dislocation interactions or Taylor hardening may control the deformation in highly dislocated structures. Additionally the latter stages of thin-film deformation, initially controlled by linear hardening aspects, might undergo a transition to parabolic hardening. Here an initial hardness, H0, is assumed, followed by Taylor hardening and a compressive flow stress equal to twice the shear giving pffiffiffi H ¼ H 0 þ 6aT lb q ð13Þ where aT is the Taylor coefficient and q is the dislocation density. This is partially based on the hardness dependence on flow stress where H 3rflow [23]. As schematically shown in Fig. 3, and observed experimentally, the assumption is that the plastic zone can extend in size well beyond the film thickness. Generously, c a is used to estimate the largest possible q for Taylor hardening based on line length per unit volume. Line length is estimated as a number of loops, on average, being 2pa in length with the volume being pc2h (Fig. 3). With N = d/2b as used above and a contact geometry of a2 = 2dR, we find sequentially 1=2 line length N 2pa 2N 1 d ¼ 2 ’ q¼ ’ ð14Þ volume pc h0 4ah0 bh0 2R Combined with Eq. (13) this gives 1=2 1=4 b d H ¼ H 0 þ a0T l h0 R
Fig. 3. Schematic of relatively unconstrained (in two dimensions) plastic zone, 2c, much larger than the contact diameter, 2a.
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ð15Þ
where a0T ¼ 2:5 if the Taylor coefficient is taken as 0.50. It is seen that hardness now has an even weaker dependence on indentation depth than the linear hardening mechanism given by Eq. (4) to be d/h0.
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With these three representations of the loading slope and hardening mechanisms, four thin-film structures in the 12.5–33 nm thickness regimes are compared. 4. Experimental procedures Single-crystal silicon nanocubes were deposited onto caxis sapphire substrates using the constrictive mode capacitive plasma technique. The deposition took 90 s when 2.5 sccm of silane (SiH4) and 3 sccm of argon (Ar) were used in the apparatus with a chamber pressure of 1.5 Torr. Particles were extracted through a 1 mm orifice forming a broad beam of particles. The beam then impacted a substrate and the particles adhered. Constriction of the flow from a 50 mm diameter tube to a 1 mm diameter hole accelerated the flow and the particles formed a particle beam. Estimated particle velocities are of the order of 200 m/s. The nanocubes formed a 1–2 nm thick native oxide layer on the exposure to air. The Ni80Fe20 and Ni samples were deposited in an ultrahigh-vacuum chamber by electron beam evaporation with a base pressure of 1011 Torr. The Co sample was deposited by DC magnetron sputtering with a base pressure in the low 109 Torr range and a sputtering pressure of 3 mTorr of ultra-pure Ar gas. For the Ni80Fe20 and Ni films, deposition at a rate of 0.01 nm/s at room temperature onto chemically cleaned c-axis sapphire substrates was utilized. The Co film was deposited at a rate of 0.09 nm/s onto a Si(1 0 0)/SiO2 (native oxide) substrate, chemically cleaned and degassed at 100 C. To determine layer thicknesses, grazing-incidence X-ray reflectivity (GIXR) was used for all of the films. Using GIXR and atomic force microscopy (AFM), a surface roughness range of 0.5–0.8 nm is insufficient to invalidate Hertzian contact theory. Nanoindentation was utilized to compress the silicon nanocubes and to determine the properties of the thin films. As-deposited films and nanocubes were evaluated in air over a time period where the room temperature was controlled to 22 ± 1 C and the relative humidity was controlled to 30 ± 5%. Individual nanocubes were located using scanning probe microscopy (SPM) instrument-based nanoindenter (Hysitron TriboScope, Minneapolis, MN), centered in a 2 · 2 lm scan area and repeatedly scanned to ensure that the scanning itself did not initiate permanent deformation. A 1 lm conical tip was used to compress and image the silicon nanocubes. Regarding statistical averaging, for the loading curves averages of every seven points were collected and reported in the figures. This was also accomplished for the unloading curve of Fig. 1(c) but not the unloading curves (there are two) of Fig. 2. The particle height was recorded, followed by the compression at a constant loading and unloading rate of 2 lN/s. The hold time at maximum load was 3 s for all loads. After each compression, the structure was again imaged and its height was measured. The thin films of Permalloy, cobalt, and nickel were indented using a Hysitron Triboindenter (Edina, MN) and
a Berkovich tip. The tip had a radius of about 303 nm. The films also had an oxide film of about 2–3 nm, which acted as a constraint, keeping dislocation egress to the surface at a minimum. To validate the steps indicated in Figs. 1, 2 and 4, the first two curves were numerically differentiated in the following manner. A least squares slope was determined for each series of three data points in Figs. 1c and 2a. Each one of these points is an average of seven raw data points. In effect this gives a first-order least squares slope of 21 measurements for each point in Figs. 1d and 2b removing much of the statistical scatter due to vibrational or electrical transients. As seen in Figs. 1d and 2b the number of displacement discontinuities do produce an equal number of slope changes. While inherent material conditions and/or machine- and environment-produced noise are still present,
Fig. 4. (a) Load–displacement curve for a 33 nm Ni80Fe20 (Permalloy) film. Discretized Burgers vector displacement steps and the hardening slopes are shown. (b) Load–displacement curve for 12.5 nm Ni80Fe20 (Permalloy) film. Greater hardening slopes as well as an indication of Hertzian behavior (lighter curve) (Eq. (9)) are shown.
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the differential fluctuations are reasonably coincident with the placement of the steps. The actual load response, even if it is a single dislocation event in a two-dimensional depiction, is complicated by the relative dislocation velocity to spring response of the loading device even in a load-controlled setting. Also, in a three-dimensional sense dislocation emissions at one site along the periphery will, with some small delay, induce an emission at another site. Further complications arise in the averaging procedure of each point during loading in Figs. 1c, 2a and 4a and b representing seven raw data points to reduce noise. While on a perfect surface, with dislocation emission around the periphery occurring simultaneously, one might expect a negative slope as is occasionally observed in Figs. 1b and 2b, other factors have normally interceded. 5. Experimental comparison First, consider the Hertzian baseline predictor of the load and the load–displacement slope, dP/dd, for two film thicknesses of Permalloy. Just the slopes are shown in Fig. 4a for the 33 nm film, but both the hardening slopes and the Hertzian curve from Eq. (9) are indicated in Fig. 4b for the 12.5 nm film. Calculations were based on a measured reduced modulus of 252 GPa. The Hertzian curve for both (Fig. 4a and b) lies on top of the experimental data for both the 33 and 12.5 nm thick film. A similar fit results for the MBE Ni of Fig. 2. Similar to what was shown for the MBE Ni, the discretized steps for both the 33 and 12.5 nm Permalloy films suggest dislocation hardening. The 9 steps observed in the thinner film multiplied by bNi equal to 0.25 nm exactly matches the 2.25 nm residual displacement observed. This was not the case, however, for the thicker 33 nm film, as the approximate 12 steps would translate to 3.0 nm whereas a residual displacement of 5.5 nm was observed. Considering this difference, other sources of plasticity rather than just the discretized steps are indicated for the thicker film. Using the experimental hardening slopes found in between discretized steps, the result (Fig. 5) is calculated. The theoretical slopes from Eq. (8), using a shear modulus of 95 GPa determined from measured indentation modulus, bracket the observed values and give an appropriate trend. More importantly, the hardening slopes do increase linearly with displacement as predicted, and the separation between the two is consistent with the linear hardening model. The fact that hardening slopes for both theory and experiment are greater than the load–displacement slope for Hertzian elastic behavior was initially both counterintuitive and troublesome. There are two factors at work here. First, consider the composite modulus of the thinnest film, which increased from 246 to 252 GPa. As only one curve based on 250 GPa is given in Fig. 5 there could be a slightly closer match between Hertzian and the thinner film data shown in Fig. 5. Second, considering the contact area measurements, the hardening slopes are measured based on a shifting abscissa (e.g. Fig. 4a and b) due to
Fig. 5. Theoretical hardening slopes (Eq. (8)) compared to the data in Fig. 4 and the Hertzian prediction of Eq. (12).
the plastic displacement excursions. This results in two effects. First, thepcontact for these geometffiffiffiffiffiffiffiffi radiuspisffiffiffiffiffigreater ffi ric contacts: 2dR versus dR for the Hertz contact radius. This means the hardening slope could be about a factor of two greater due to the increase in contact area proportional to a2. In addition, pile-up which easily occurs in the indentation of very thin films could increase the contact area even more as indicated in Fig. 3. As such, these factors easily account for the fact that the elastic slopes from Eq. (12) are less than what might be expected compared to such measurements that are clearly inelastic. Having said that it is still astonishing that the load– displacement is ‘‘elastic-like’’ even after the dislocations, albeit few of them, have been nucleated in these very thin films. Second, consider the supporting load for all materials (Fig. 6) compared to Hertzian elastic and linear hardening behavior. For the nominal thickness of 30 nm two curves are shown using measured values of l = 95 GPa for the Ni and Permalloy and l = 66.4 GPa for the Co/Si films. The data generally lie on the elastic curves in Fig. 6a. The linear hardening curve from Eq. (7) follows the Hertzian curve of Eq. (9) closely for the first 6 nm of penetration in Fig. 6a. For the thinner film (Fig. 6b) it is seen that the load-bearing capacity is over-predicted, similar to what might be anticipated from Fig. 5. Still there is an increase in the hardness from the 33 to the 12.5 nm thick films consistent with linear hardening. Finally, while Taylor hardening has not been discussed and is not considered to be a viable candidate by itself, it is useful to illustrate the Permalloy data based upon Eq. (15). Assuming a0T ¼ 2:5 and the initial film hardness to be 1 GPa, a hardness comparison (Fig. 7) does demonstrate a good correspondence to the 33 nm thick film data, but the data points for the thinner film are too scattered to allow for any definitive conclusions about the slope. The
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Fig. 7. Ni80Fe20 (Permalloy) hardness data modeled with a Taylor hardening approach (Eq. (15)). For baseline hardness, H0 is arbitrarily set at 1 GPa.
Fig. 6. (a) Supporting loads and displacements for the descretized displacement steps are shown along with Hertzian (Eq. (9), dashed curve) and linear hardening (Eq. (7), solid curve) models for the 30 nm films (Ni, Co, Ni80Fe20). (b) Loads and displacements of the descretized displacement steps for Ni80Fe20 (Permalloy) films (12.5 and 33 nm).
algorithm does, however, tend to bracket the data for the two film thicknesses. With the Hertzian behavior and the two hardening models represented by Eqs. (3)–(15) compared to the four material conditions of interest, additional discussion follows. Here, the possible existence of a dominant hardening mechanism is investigated. 6. Dominant deformation mechanisms First, elastic behavior during loading beyond a few nanometers is not found in the Si cubes or in any of the
ultra-thin films. For example (Figs. 1, 2 and 4), where total indentation depth is generally less than 10 nm, there was always a residual displacement on unloading indicative of permanent plastic deformation. It is clear that one might think these load–displacement curves are elastic since the loading profile fits all of the P d results for the three nominally 30 nm thick films (Fig. 6a). However, both the residual displacements and the discretized nature of displacement jumps (Figs. 1–4) speak of plasticity, not to elastic behavior. The question then remains as to why Hertzian agreement (Fig. 6a) appears possible. As suggested before [24,25], and now shown with more precise information, a material can load elastically, undergo a displacement jump, and then continue to load nearly elastically. Previously this was on a more macroscopic scale associated with staircase yielding. With the present observations at the nanometer scale the following is proposed in the earliest stages of deformation. Dislocations are nucleated by some heterogeneous mechanism involving ledges or grain boundaries near the very high contact stresses at the edge of the indenter. A dislocation displacement jump then occurs and achieves equilibrium at some distance below the free surface [17]. The increased contact area and the internal stress created by the dislocation then require a higher load to allow emission of the next dislocation. One way of looking at this is that the dislocation motion is so limited after equilibrium that the material appears to be behaving elastically between jumps. For the small displacements of this investigation, this gives the appearance of elastic behavior shown (Fig. 6a). One can rightfully argue that the data are too scattered in Fig. 2, one of our best results, to determine the functionality of P = f(d) in between excursions, as is it could be either a d3/2 or d dependence.
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This leads to a second way of looking at the results. In Fig. 1b for the cube, and in Figs. 2, 4a and b for the films, the loading slopes in between jumps are consistent with a dislocation-based, linear hardening model. Further confirmation comes from the slopes being independent of displacement for the cube, but dependent on displacement for the spherical contacts into the thin films. Additionally, linear hardening is reinforced by the separation of the data for the two film thicknesses of Permalloy (Fig. 5) and the relative similarities predicted by Eq. (8). Even though linear hardening predicts the loads reasonably well (Fig. 6a and b), it over-predicts the film thickness effect. The last way of looking at these results applies a forest dislocation approach and parabolic hardening. The agreement in Fig. 7 between the data and Eq. (15) is acceptable but the choice of the plastic zone being equal to the indenter contact radius is not. It is known from previous AFM experiments of contacts into thin films that pile-up and plastic deformation outside the contact area exist making c/a unity unrealistic for the present study. Also, one can show that the hardening slope would only increase with a d1/4 functionality, too small to predict the dependence shown (Fig. 5) for the thinner film. On the other hand, the fit to the 33 nm film would be very good. 7. Summary To summarize, it is clear that these small, constrained volumes are not behaving elastically. It is equally clear that a few nucleated dislocations trapped between an indenter tip and relatively rigid substrates engender a very large hardening response. Neither linear hardening nor parabolic hardening alone appears to capture the plasticity observations. At this nascent development stage, this study proposes that the hardening observed could represent a transition between linear and parabolic hardening or some other mechanism not discussed here. Whatever the mechanism, it appears that these thin films approach elastic-like behavior during the very early stages of deformation by indentation due to the constraining influences of tip and substrate on emitted dislocations.
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Acknowledgements This work was supported by the National Science Foundation under grants DMI 0103169, CMS-0322436, an NSF-IGERT program through grant DGE-0114372 and the United States Department of Energy Office of Science, DE-AC04-94AL85000. References [1] Gerberich WW, Venkataraman SK, Huang H, Harvey SE, Kohlstedt DL. Acta Metall Mater 1995;43:1569. [2] Kiely JD, Houston JE. Phys Rev B 1998;57:12588. [3] Gouldstone A, Koh HJ, Zeng KY, Giannakoupoulos AE, Suresh S. Acta Mater 2000;48:2277. [4] Kramer DE, Yoder KB, Gerberich WW. Philos Mag A 2001;81:2033. [5] Lilleodden ET, Zimmerman JA, Foiles SM, Nix WD. J Mech Phys Solids 2003;51:901. [6] Mook WM, Jungk JM, Cordill MJ, Moody NR, Sun Y, Xia Y, et al. Z Metallkd 2004;95:416. [7] Miller RE, Acharya AA. J Mech Phys Solids 2004;52:1507. [8] Gerberich WW, Tymiak NI, Grunlan JC, Horstemeyer MF, Baskes MI. J Appl Mech 2002;69:433. [9] Horstemeyer MF, Baskes MI, Plimpton SJ. Acta Mater 2001;49:4363. [10] Gerberich WW, Cordill MJ, Mook WM, Moody NR, Perrey CR, Carter CB, et al. Acta Mater 2005;53:2215. [11] Huang Y, Ou S, Hwang KC, Li M, Gao HA. Int J Plast 2004;20:753. [12] Tabor D. The hardness of metals. Oxford Press: Oxford, UK; 1951. [13] Gerberich WW, Mook WM, Perrey CR, Carter CB, Baskes MI, Mukherjee R, et al. J Mech Phys Solids 2003;51:979. [14] Dong Y, Bapat A, Hilchie S, Kortshagen U, Campbell SA. J Vac Sci Technol B 2004;22:1923. [15] Perrey CR, Deneen JM, Carter CB. Mater Res Soc Symp Proc 2004;818:259. [16] Minor AM, Lilleodden ET, Jin M, Stach EA, Chrzan DC, Morris JW. Philos Mag 2005;85:323. [17] Gerberich WW, Mook WM, Cordill MJ, Carter CB, Perrey CR, Heberlein JV, et al. Int J Plast 2005;21:2391. [18] Eshelby JD, Frank FC, Nabarro FRN. Philos Mag 1959;42:351. [19] Rabier J, Cordier P, Tondellier T, Demret JL, Garem H. J Phys Condens Matter 2000;12:10059. [20] Jungk JM, Mook WM, Cordill MJ, Bahr DF, Moody NR, Hoehn J, et al. J Mater Res 2004;19:2812. [21] Gao H, Chiu CH, Lee J. Int J Solids Struct 1992;29:2471. [22] Chen X, Vlassak JJ. J Mater Res 2001;16:2974. [23] Johnson KL. Contact mechanics. Cambridge, UK: Cambridge University Press; 1985. [24] Bahr DF, Kramer DE, Gerberich WW. Acta Mater 1998;46:3605. [25] Suresh S, Nieh TG, Choi BW. Scripta Mater 1999;41:951.
Plasticity responses in ultra-small confined cubes and films M.J. Cordill *, M.D. Chambers, M.S. Lund, D.M. Hallman, C.R. Perrey, C.B. Carter, A. Bapat, U. Kortshagen, W.W. Gerberich Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, MN 55455, USA Received 23 September 2005; received in revised form 13 May 2006; accepted 16 May 2006 Available online 8 August 2006
Abstract Nanoindentation-induced dislocation emission at 5–7 nm displacements in ultra-thin films (12–33 nm) and nanocubes (40–60 nm) is used to examine deformation and plasticity models. Using the Tabor estimate, this displacement corresponds to a plastic strain of 3–5%. Load–displacement curves produced using nanoindentation show evidence of discretized, Burgers vector–length displacement steps, or excursions, which can be associated with individual dislocation emission events. Using these displacement steps and the residual plasticity present on unloading, theoretical hardening models are developed. Linear and parabolic hardening approaches are compared for ultrathin films of nickel, cobalt, and Permalloy (Ni80Fe20), and also for silicon nanocubes. It is determined that the linear hardening model can predict the early trends of the experimental data while parabolic hardening may be more appropriate at later stages. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Thin films; Nanoindentation; Plasticity; Dislocation; Deformation
1. Introduction Recently a number of models for dislocation nucleation and hardening of thin films or small volumes under contact have been proposed [1–10]. These models include stress or strain gradient effects [5,7], linear hardening [1,6,10], parabolic hardening [6,10], and various surface energy approaches [8–10]. It has been suggested by several studies, both experimental and theoretical [8,11], that strain gradient plasticity models are not accurate at length scales below 100 nm. As films in the range of 10–100 nm were evaluated in this investigation, a strain gradient plasticity model was not considered. To ascertain if other dislocation approaches might be appropriate, four different thin film/substrate combinations were investigated with two different models. As surface roughness was also a concern, some of the films were grown on c-axis sapphire (with a roughness of 0.1– 0.2 nm as determined using atomic force microscopy) using molecular beam epitaxy (MBE). Initially, the similarities *
Corresponding author. Tel.: +1 612 624 0515. E-mail address: [email protected] (M.J. Cordill).
and differences between the earliest stages of deformation of a very thin film of nickel (grown using MBE) and a silicon nanocube were observed. For these 30–50 nm size structures, the first 5–7 nm of displacement was the focal point. This represented of the order of 3.0–5.0% plastic strain using Tabor’s estimate [12]. From these results, it appears that one or more hardening mechanisms might be induced during this initial yielding. Two hardening processes are reviewed since the unloading curves clearly demonstrated permanent plastic residual strain. Following the theoretical developments, the loads representing individual dislocations being emitted, hardening rates in terms of dP/dd during loading, and the hardness as a function of displacement are compared to the theory. The comparisons are made for four ultra-thin films of Co (30 nm), Ni (30 nm), and Ni80Fe20 (Permalloy) (33 and 12.5 nm). 2. Linear hardening in small structures First, nanocubes, the simplest small structures for evaluating plastic strain hardening, are considered. Previously,
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.05.037
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hardening in nanospheres [13] and nanoboxes [6] was examined, but in the former, contact radius and orientation are difficult to determine with certainty, and with the latter, the degree of wall thickening is unknown. Fortunately, isolated silicon nanocubes of the type shown in Fig. 1a have been grown [14] not having the former complications. They are grown with {1 0 0} faces [15] as shown by the electron diffraction pattern (Fig. 1(b)). Typically, they are coated with a 2 nm oxide film, which inhibits dislocation egress when plastically deformed. Initially, the possibility of phase transformations associated with plastic deformation was considered, but recent work on similar, relatively unconstrained films [16], nanodots [6], and nanospheres [17] of silicon have all exhibited dislocation plasticity. After locating and compressing a similar silicon cube of height h0 between a 1 lm radius diamond tip and a sapphire sub-
strate, the load–displacement data (Fig. 1c) were collected. Two immediate observations supporting the above claim were the discretized displacement steps upon loading and the residual plasticity upon unloading. The dashed unloading line parallel to the data subtracts the ‘‘creep’’ plasticity at maximum load that occurred during a hold of 3 s. Considering that such small loads have not produced creep at similar loads in either diamond-like carbon or sapphire, this was not considered to be an artifact of the testing environment. The ‘‘creep’’ in strain represented by a displacement of 0.5 nm is 0.01. This is compared to the normal qs bs representation of dislocation produced strain where qs and s are the stored dislocation density and the average distance moved. With s ¼ 0:5 nm, this would require too many dislocations considering these cubes are initially defect free. Alternatively, one could consider that fewer
Fig. 1. (a) Bright-field transmission electron microscopy image of a highly oriented silicon nanoparticle. (b) Using the {1 0 0} selected-area diffraction pattern (bottom), the faces of the particle were found to be {1 0 0}. (c) Load–displacement curve corresponding to the compression of a single nanocube using a 1 lm conical-shaped diamond tip. Discretized displacement steps during loading are emphasized along with a linear hardening model (Eq. (7)). (d) Differential curve of the nanocube load–displacement data showing six peaks and valleys of dP/dd consistent with the six displacement excursions noted in (c).
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dislocations moved larger distances during the hold. Given nine dislocations created during rising load due to geometrically necessary dislocations at each excursion, these represent a density at maximum load of 3.3 · 1015/m2 if they have a length equal to h0, the cube dimension. During the hold if these moved an average distance of 13 nm, this would satisfy a qs bs criterion. This does not seem likely at this short-time hold given the large Peierls stress. Also possible is instrument drift, since 0.5 nm in 3 s is not out of the question. Nevertheless, this did not seem likely either as all holds in these experiments represented forward displacement. Statistically, thermal drift would be forward or backward. A second possibility during the hold is that two more dislocations were nucleated and the observed discretized steps were associated with individual dislocation emission. During rising load this is reasonably consistent with the average step size of 0.22 nm, which is very close to the Burgers vector of silicon (0.236 nm). Furthermore, the total residual displacement after unloading, minus that at hold, was 2.1 nm, which would translate to N = 9 dislocations if each step represented a full Burgers vector. While this is not exactly correct, given a {1 0 0} orientation, it is a reasonable first-order expectation given the statistical averaging procedure used to obtain the data (Fig. 1c). See Section 4 for further discussion. Examining Fig. 1c, it is seen that there is no linear elastic portion since it should be greater than the initial hardening slope observed. The latter is much less than the modulus of elasticity. This is understood in terms of how a 1 lm radius diamond tip contacts a 52 nm silicon cube. To measure an elastic slope, the diamond would have to be perfectly smooth and the load train would have to be in perfect alignment with the flat of the cube. The cube is deposited on a sapphire wafer that is not perfectly smooth. Even though the root mean square roughness may be 0.1–0.2 nm there are spots with greater individual peaks. Also, Fig. 1a shows the silicon cube surface to be neither straight nor perfectly smooth. When the cube is deposited it can easily achieve a tilt to the load axis because it is deposited on roughness or debris or due to its own roughness. A combination of a 1 misalignment and a 1 nm high spot on either the cube or the substrate would require 1–2 nm of displacement before the cube surface is in full contact. This obviates any initial appearance of elasticity since by the time the entire surface area is load bearing, local stress concentrations at the oxide/silicon interface are sufficient to nucleate dislocations. The dotted theoretical line represents a simple theoretical model as discussed below. This behavior (Fig. 1c) is, by inspection, clearly plastic and not elastic. From the steeper initial unloading slope one can calculate the modulus 1 dP h0 dd to be between 120 and 150 GPa, reasonably close to the modulus of bulk silicon. The rapid departure from linear unloading below 2 lN suggests the cube may not be under
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full contact at the lower loads as discussed above. For the purposes of calculation, however, the cube is assumed to be under full contact to allow for linear hardening analysis. A single-ended pile-up of edge dislocations is assumed with the length of the pile-up equal to the height of the cube, h0. This gives a shear stress of [18] s¼
lbN pð1 mÞh0
ð1Þ
where lb is the shear modulus times the Burgers vector, N is the number of dislocations, and m is Poisson’s ratio. The only adjustable assumption is that N = d/2b where d is the total displacement. This implies that half the displacement is elastic and half plastic, which is not inconsistent with Fig. 1c. With a uniform compressive stress, r, approximately twice the shear stress (one could use 2.45 if {1 1 0} h1 1 0i slip were identified) combined with Eq. (1), these give r¼
ld Ed ¼ pð1 mÞh0 2pð1 m2 Þh0
ð2Þ
Regarding a detailed crystallographic assessment of dislocations involved, this has not been accomplished since it is not clear what slip system is involved. For example, Rabier et al. [19] have found that silicon undergoes dislocation emission under high pressure at low temperatures with full Burgers vectors of the step-shuffle variety rather than dissociated dislocations. This equation assumes an isotropic relationship between l and E and can be used to evaluate the data by finding the load and loading slope from Eq. (2) to be P ¼ rh20 ¼
Edh0 2pð1 m2 Þ
dP Eh0 ¼ dd 2pð1 m2 Þ
ð3aÞ ð3bÞ
For the isotropic values of E = 160 GPa and m = 0.218 and the cube height of 52 nm, dP/dd is calculated to be 1390 N/m. This calculation can be compared to the experimental values found from the linear segments between steps indicated in Fig. 1(c). These averaged 1310 N/ m ± 15% and are seen to be parallel to the calculated dashed line from Eq. (3b). This is a strong indicator that linear hardening can play a major role in the plasticity of small, confined structures. As applied to thin films, starting with a model recently determined for linear hardening [20], the hardness is H¼
ld pð1 vÞh0
ð4Þ
which appears inconsistent with Eq. (1) since hardness is expected to represent a greater mean pressure than a uniform compressive stress. However, a nanocube is small in three dimensions confining the slip band length to h0 or slightly larger while a thin film which is only small in one dimension can spread the plastic zone to be much greater
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than h0 as discussed elsewhere [4]. Using the same estimate as above, that half the total displacement is plastic giving d = 2Nb, combined with Eq. (4) gives H¼
2Nbl pð1 mÞh0
ð5Þ
With hardness or a mean contact pressure defined by P/pa2, the load can be found from P¼
2Nblpa2 4dRNbl ¼ pð1 mÞh0 ð1 mÞh0
ð6Þ
the latter using a2 = 2dR for a geometric contact. Using N = d/2b as above, we also find P¼
2lRd2 ð1 mÞh0
ð7Þ
which gives the loading slope for linear hardening to be dP 4lRd ¼ dd ð1 mÞh0
ð8Þ
Compared to Eq. (3b) for the loading slope, which is independent of displacement, Eq. (8) indicates an increasing slope with d. This is seen experimentally in Fig. 2, for a 303 nm radius diamond tip indenting a 30 nm thick MBE-grown nickel film on c-axis sapphire. Here the loading curve based on Eq. (7) using l = 951 GPa and m = 0.31 for nickel gives a reasonable but low accounting of the linear hardening (Fig. 2) at small displacements and an over-prediction at larger displacements. Due to the overprediction (Fig. 1) for cubes and the mixed prediction (Fig. 2) for films, a more thorough examination of the hardening mechanisms for very thin films is necessary. 3. Theoretical approaches As initial contact in the thin film may be elastic, a Hertzian contact approximation can be used for the 303 nm spherical tip of the present study. This is followed by linear and parabolic hardening which sequentially follow the initial Hertzian approximation. Fig. 2. (a) Load–displacement data for 30 nm Ni film. The linear hardening curve is based on Eq. (7). (b) Differential curve of the 30 nm Ni film data showing six peaks and valleys of dP/dd reasonably consistent with the six displacement steps noted in (a).
3.1. Hertzian contact The earliest stages of contact may be elastic but, even if not, this is a good comparison point. Load–displacement is given by 4 P ¼ E R1=2 d3=2 3
ð9Þ
where the reduced modulus, E*, is normally taken as 1 1 m2i 1 m2f þ ¼ E Ei Ef
ð10Þ
1 This seemingly high shear modulus for nickel is a composite modulus as discussed in Section 3.
with mi, Ei representing Poisson’s ratio and modulus for the diamond and mf, Ef being those of the film. This, however, ignores any effect from the substrate. Since these 12.5– 33 nm thick films are so thin, even 5–10 nm displacements can result in a substantial composite film/substrate effect. Here, the model of Gao et al. [21] for indentation into multilayers was initially used giving 1 ms ðmf ms ÞI 1 ha 1m ¼ ð11Þ l ls þ ðlf ls ÞI 0 ha eff where ms, mf are Poisson’s ratios and ls, lf are shear moduli of the film and substrate, respectively. This requires a
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numerical analysis but Gao et al. [21] have given graphical solutions for the weight functions I1 and I0 which vary as a function of the film thickness to contact radius ratio. Taking a typical contact displacement of 7.5 nm for these thinfilm measurements gives a contact radius of 57.7 nm for the 303 nm radius diamond tip. For the 30 nm nickel film deposited, this gives h/a = 0.52. From Fig. 3 of Gao et al. [21] for I0 and I1 and the following film/substrate properties, one can determine both l and E. As an upper bound for these very thin films, one might use a flat punch approximation with film modulus properties for EÆ111æ. This, however, gave an even larger value than expected which is probably not representative of the nanocrystalline film/substrate combination. As discussed in Section 5, there is evidence that such films have a considerably lower modulus than might be expected if these were single-crystal nickel. Bearing this in mind, the following properties were used to determine NiÆ111æAl2O3Æ0001æ composite modulus: h ENi ¼ 200 GPa; mNi ¼ 0:30; lNi ¼ 76:9 GPa; I 0 ¼ 0:25 a EAl2 O3 ¼ 432 GPa; mAl2 O3 ¼ 0:2; h ¼ 0:45 lAl2 O3 ¼ 180 GPa; I 1 a Insertion of these into Eq. (11) gives Eeff = 340 GPa which is considerably greater than the average reduced modulus of 248þ25 14 GPa as determined in the standard way by nanoindentation from four tests. This degree of fit is not too unexpected given the numerical solution of Chen and Vlassak [22] which shows that the weight function at Eq. (11) is larger than Gao’s result. As the term in the denominator is dominant, this would reduce the calculated modulus. They obtained a discrepancy for I0(0.52) of about 33% based on a modulus ratio of 1.8. Here, the modulus ratio of sapphire to nanocrystalline nickel film is about 2.16 which could result in further deviation. The end result is that Gao’s analysis over-predicted the composite modulus effect. Qualitatively, the predicted moduli of 340 and 346 GPa for the nominal 30 and 12.5 nm Permalloy films are in the right direction but over-predicting. Still, this gives confidence in using the measured moduli values for both the nickel and Permalloy films in subsequent calculations. The values of E* for the Permalloy gave 246 ± 18 GPa and 252 ± 10 GPa for the 33 nm (7 tests) and 12.5 nm (4 tests) thick films, respectively.
Differentiating Eq. (9) to represent the elastic load– displacement slope gives dP ¼ 2E R1=2 d1=2 dd
ð12Þ
noting this has a displacement dependence between Eqs. (3b) and (8) and is indirectly dependent on thickness through the modulus dependence of Eq. (11). 3.2. Linear hardening For single crystals or nanocubes and nanospheres with very few or no grown-in dislocations, linear hardening can apply. It also applies to thin films grown by molecular beam epitaxy or perhaps even to sputtered films where the film thickness is no larger than the grain size. Therefore Eqs. (4)–(8) are used to evaluate hardness and load– displacement curves. It is also important to notice that the hardness and loading slope here (Eqs. (4) and (8)) are proportional to d/h0. This represents a larger thickness dependence on the loading slope than would be anticipated from elasticity (Eqs. (11) and (12)). 3.3. Parabolic hardening Forest dislocation interactions or Taylor hardening may control the deformation in highly dislocated structures. Additionally the latter stages of thin-film deformation, initially controlled by linear hardening aspects, might undergo a transition to parabolic hardening. Here an initial hardness, H0, is assumed, followed by Taylor hardening and a compressive flow stress equal to twice the shear giving pffiffiffi H ¼ H 0 þ 6aT lb q ð13Þ where aT is the Taylor coefficient and q is the dislocation density. This is partially based on the hardness dependence on flow stress where H 3rflow [23]. As schematically shown in Fig. 3, and observed experimentally, the assumption is that the plastic zone can extend in size well beyond the film thickness. Generously, c a is used to estimate the largest possible q for Taylor hardening based on line length per unit volume. Line length is estimated as a number of loops, on average, being 2pa in length with the volume being pc2h (Fig. 3). With N = d/2b as used above and a contact geometry of a2 = 2dR, we find sequentially 1=2 line length N 2pa 2N 1 d ¼ 2 ’ q¼ ’ ð14Þ volume pc h0 4ah0 bh0 2R Combined with Eq. (13) this gives 1=2 1=4 b d H ¼ H 0 þ a0T l h0 R
Fig. 3. Schematic of relatively unconstrained (in two dimensions) plastic zone, 2c, much larger than the contact diameter, 2a.
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ð15Þ
where a0T ¼ 2:5 if the Taylor coefficient is taken as 0.50. It is seen that hardness now has an even weaker dependence on indentation depth than the linear hardening mechanism given by Eq. (4) to be d/h0.
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With these three representations of the loading slope and hardening mechanisms, four thin-film structures in the 12.5–33 nm thickness regimes are compared. 4. Experimental procedures Single-crystal silicon nanocubes were deposited onto caxis sapphire substrates using the constrictive mode capacitive plasma technique. The deposition took 90 s when 2.5 sccm of silane (SiH4) and 3 sccm of argon (Ar) were used in the apparatus with a chamber pressure of 1.5 Torr. Particles were extracted through a 1 mm orifice forming a broad beam of particles. The beam then impacted a substrate and the particles adhered. Constriction of the flow from a 50 mm diameter tube to a 1 mm diameter hole accelerated the flow and the particles formed a particle beam. Estimated particle velocities are of the order of 200 m/s. The nanocubes formed a 1–2 nm thick native oxide layer on the exposure to air. The Ni80Fe20 and Ni samples were deposited in an ultrahigh-vacuum chamber by electron beam evaporation with a base pressure of 1011 Torr. The Co sample was deposited by DC magnetron sputtering with a base pressure in the low 109 Torr range and a sputtering pressure of 3 mTorr of ultra-pure Ar gas. For the Ni80Fe20 and Ni films, deposition at a rate of 0.01 nm/s at room temperature onto chemically cleaned c-axis sapphire substrates was utilized. The Co film was deposited at a rate of 0.09 nm/s onto a Si(1 0 0)/SiO2 (native oxide) substrate, chemically cleaned and degassed at 100 C. To determine layer thicknesses, grazing-incidence X-ray reflectivity (GIXR) was used for all of the films. Using GIXR and atomic force microscopy (AFM), a surface roughness range of 0.5–0.8 nm is insufficient to invalidate Hertzian contact theory. Nanoindentation was utilized to compress the silicon nanocubes and to determine the properties of the thin films. As-deposited films and nanocubes were evaluated in air over a time period where the room temperature was controlled to 22 ± 1 C and the relative humidity was controlled to 30 ± 5%. Individual nanocubes were located using scanning probe microscopy (SPM) instrument-based nanoindenter (Hysitron TriboScope, Minneapolis, MN), centered in a 2 · 2 lm scan area and repeatedly scanned to ensure that the scanning itself did not initiate permanent deformation. A 1 lm conical tip was used to compress and image the silicon nanocubes. Regarding statistical averaging, for the loading curves averages of every seven points were collected and reported in the figures. This was also accomplished for the unloading curve of Fig. 1(c) but not the unloading curves (there are two) of Fig. 2. The particle height was recorded, followed by the compression at a constant loading and unloading rate of 2 lN/s. The hold time at maximum load was 3 s for all loads. After each compression, the structure was again imaged and its height was measured. The thin films of Permalloy, cobalt, and nickel were indented using a Hysitron Triboindenter (Edina, MN) and
a Berkovich tip. The tip had a radius of about 303 nm. The films also had an oxide film of about 2–3 nm, which acted as a constraint, keeping dislocation egress to the surface at a minimum. To validate the steps indicated in Figs. 1, 2 and 4, the first two curves were numerically differentiated in the following manner. A least squares slope was determined for each series of three data points in Figs. 1c and 2a. Each one of these points is an average of seven raw data points. In effect this gives a first-order least squares slope of 21 measurements for each point in Figs. 1d and 2b removing much of the statistical scatter due to vibrational or electrical transients. As seen in Figs. 1d and 2b the number of displacement discontinuities do produce an equal number of slope changes. While inherent material conditions and/or machine- and environment-produced noise are still present,
Fig. 4. (a) Load–displacement curve for a 33 nm Ni80Fe20 (Permalloy) film. Discretized Burgers vector displacement steps and the hardening slopes are shown. (b) Load–displacement curve for 12.5 nm Ni80Fe20 (Permalloy) film. Greater hardening slopes as well as an indication of Hertzian behavior (lighter curve) (Eq. (9)) are shown.
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the differential fluctuations are reasonably coincident with the placement of the steps. The actual load response, even if it is a single dislocation event in a two-dimensional depiction, is complicated by the relative dislocation velocity to spring response of the loading device even in a load-controlled setting. Also, in a three-dimensional sense dislocation emissions at one site along the periphery will, with some small delay, induce an emission at another site. Further complications arise in the averaging procedure of each point during loading in Figs. 1c, 2a and 4a and b representing seven raw data points to reduce noise. While on a perfect surface, with dislocation emission around the periphery occurring simultaneously, one might expect a negative slope as is occasionally observed in Figs. 1b and 2b, other factors have normally interceded. 5. Experimental comparison First, consider the Hertzian baseline predictor of the load and the load–displacement slope, dP/dd, for two film thicknesses of Permalloy. Just the slopes are shown in Fig. 4a for the 33 nm film, but both the hardening slopes and the Hertzian curve from Eq. (9) are indicated in Fig. 4b for the 12.5 nm film. Calculations were based on a measured reduced modulus of 252 GPa. The Hertzian curve for both (Fig. 4a and b) lies on top of the experimental data for both the 33 and 12.5 nm thick film. A similar fit results for the MBE Ni of Fig. 2. Similar to what was shown for the MBE Ni, the discretized steps for both the 33 and 12.5 nm Permalloy films suggest dislocation hardening. The 9 steps observed in the thinner film multiplied by bNi equal to 0.25 nm exactly matches the 2.25 nm residual displacement observed. This was not the case, however, for the thicker 33 nm film, as the approximate 12 steps would translate to 3.0 nm whereas a residual displacement of 5.5 nm was observed. Considering this difference, other sources of plasticity rather than just the discretized steps are indicated for the thicker film. Using the experimental hardening slopes found in between discretized steps, the result (Fig. 5) is calculated. The theoretical slopes from Eq. (8), using a shear modulus of 95 GPa determined from measured indentation modulus, bracket the observed values and give an appropriate trend. More importantly, the hardening slopes do increase linearly with displacement as predicted, and the separation between the two is consistent with the linear hardening model. The fact that hardening slopes for both theory and experiment are greater than the load–displacement slope for Hertzian elastic behavior was initially both counterintuitive and troublesome. There are two factors at work here. First, consider the composite modulus of the thinnest film, which increased from 246 to 252 GPa. As only one curve based on 250 GPa is given in Fig. 5 there could be a slightly closer match between Hertzian and the thinner film data shown in Fig. 5. Second, considering the contact area measurements, the hardening slopes are measured based on a shifting abscissa (e.g. Fig. 4a and b) due to
Fig. 5. Theoretical hardening slopes (Eq. (8)) compared to the data in Fig. 4 and the Hertzian prediction of Eq. (12).
the plastic displacement excursions. This results in two effects. First, thepcontact for these geometffiffiffiffiffiffiffiffi radiuspisffiffiffiffiffigreater ffi ric contacts: 2dR versus dR for the Hertz contact radius. This means the hardening slope could be about a factor of two greater due to the increase in contact area proportional to a2. In addition, pile-up which easily occurs in the indentation of very thin films could increase the contact area even more as indicated in Fig. 3. As such, these factors easily account for the fact that the elastic slopes from Eq. (12) are less than what might be expected compared to such measurements that are clearly inelastic. Having said that it is still astonishing that the load– displacement is ‘‘elastic-like’’ even after the dislocations, albeit few of them, have been nucleated in these very thin films. Second, consider the supporting load for all materials (Fig. 6) compared to Hertzian elastic and linear hardening behavior. For the nominal thickness of 30 nm two curves are shown using measured values of l = 95 GPa for the Ni and Permalloy and l = 66.4 GPa for the Co/Si films. The data generally lie on the elastic curves in Fig. 6a. The linear hardening curve from Eq. (7) follows the Hertzian curve of Eq. (9) closely for the first 6 nm of penetration in Fig. 6a. For the thinner film (Fig. 6b) it is seen that the load-bearing capacity is over-predicted, similar to what might be anticipated from Fig. 5. Still there is an increase in the hardness from the 33 to the 12.5 nm thick films consistent with linear hardening. Finally, while Taylor hardening has not been discussed and is not considered to be a viable candidate by itself, it is useful to illustrate the Permalloy data based upon Eq. (15). Assuming a0T ¼ 2:5 and the initial film hardness to be 1 GPa, a hardness comparison (Fig. 7) does demonstrate a good correspondence to the 33 nm thick film data, but the data points for the thinner film are too scattered to allow for any definitive conclusions about the slope. The
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Fig. 7. Ni80Fe20 (Permalloy) hardness data modeled with a Taylor hardening approach (Eq. (15)). For baseline hardness, H0 is arbitrarily set at 1 GPa.
Fig. 6. (a) Supporting loads and displacements for the descretized displacement steps are shown along with Hertzian (Eq. (9), dashed curve) and linear hardening (Eq. (7), solid curve) models for the 30 nm films (Ni, Co, Ni80Fe20). (b) Loads and displacements of the descretized displacement steps for Ni80Fe20 (Permalloy) films (12.5 and 33 nm).
algorithm does, however, tend to bracket the data for the two film thicknesses. With the Hertzian behavior and the two hardening models represented by Eqs. (3)–(15) compared to the four material conditions of interest, additional discussion follows. Here, the possible existence of a dominant hardening mechanism is investigated. 6. Dominant deformation mechanisms First, elastic behavior during loading beyond a few nanometers is not found in the Si cubes or in any of the
ultra-thin films. For example (Figs. 1, 2 and 4), where total indentation depth is generally less than 10 nm, there was always a residual displacement on unloading indicative of permanent plastic deformation. It is clear that one might think these load–displacement curves are elastic since the loading profile fits all of the P d results for the three nominally 30 nm thick films (Fig. 6a). However, both the residual displacements and the discretized nature of displacement jumps (Figs. 1–4) speak of plasticity, not to elastic behavior. The question then remains as to why Hertzian agreement (Fig. 6a) appears possible. As suggested before [24,25], and now shown with more precise information, a material can load elastically, undergo a displacement jump, and then continue to load nearly elastically. Previously this was on a more macroscopic scale associated with staircase yielding. With the present observations at the nanometer scale the following is proposed in the earliest stages of deformation. Dislocations are nucleated by some heterogeneous mechanism involving ledges or grain boundaries near the very high contact stresses at the edge of the indenter. A dislocation displacement jump then occurs and achieves equilibrium at some distance below the free surface [17]. The increased contact area and the internal stress created by the dislocation then require a higher load to allow emission of the next dislocation. One way of looking at this is that the dislocation motion is so limited after equilibrium that the material appears to be behaving elastically between jumps. For the small displacements of this investigation, this gives the appearance of elastic behavior shown (Fig. 6a). One can rightfully argue that the data are too scattered in Fig. 2, one of our best results, to determine the functionality of P = f(d) in between excursions, as is it could be either a d3/2 or d dependence.
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This leads to a second way of looking at the results. In Fig. 1b for the cube, and in Figs. 2, 4a and b for the films, the loading slopes in between jumps are consistent with a dislocation-based, linear hardening model. Further confirmation comes from the slopes being independent of displacement for the cube, but dependent on displacement for the spherical contacts into the thin films. Additionally, linear hardening is reinforced by the separation of the data for the two film thicknesses of Permalloy (Fig. 5) and the relative similarities predicted by Eq. (8). Even though linear hardening predicts the loads reasonably well (Fig. 6a and b), it over-predicts the film thickness effect. The last way of looking at these results applies a forest dislocation approach and parabolic hardening. The agreement in Fig. 7 between the data and Eq. (15) is acceptable but the choice of the plastic zone being equal to the indenter contact radius is not. It is known from previous AFM experiments of contacts into thin films that pile-up and plastic deformation outside the contact area exist making c/a unity unrealistic for the present study. Also, one can show that the hardening slope would only increase with a d1/4 functionality, too small to predict the dependence shown (Fig. 5) for the thinner film. On the other hand, the fit to the 33 nm film would be very good. 7. Summary To summarize, it is clear that these small, constrained volumes are not behaving elastically. It is equally clear that a few nucleated dislocations trapped between an indenter tip and relatively rigid substrates engender a very large hardening response. Neither linear hardening nor parabolic hardening alone appears to capture the plasticity observations. At this nascent development stage, this study proposes that the hardening observed could represent a transition between linear and parabolic hardening or some other mechanism not discussed here. Whatever the mechanism, it appears that these thin films approach elastic-like behavior during the very early stages of deformation by indentation due to the constraining influences of tip and substrate on emitted dislocations.
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Acknowledgements This work was supported by the National Science Foundation under grants DMI 0103169, CMS-0322436, an NSF-IGERT program through grant DGE-0114372 and the United States Department of Energy Office of Science, DE-AC04-94AL85000. References [1] Gerberich WW, Venkataraman SK, Huang H, Harvey SE, Kohlstedt DL. Acta Metall Mater 1995;43:1569. [2] Kiely JD, Houston JE. Phys Rev B 1998;57:12588. [3] Gouldstone A, Koh HJ, Zeng KY, Giannakoupoulos AE, Suresh S. Acta Mater 2000;48:2277. [4] Kramer DE, Yoder KB, Gerberich WW. Philos Mag A 2001;81:2033. [5] Lilleodden ET, Zimmerman JA, Foiles SM, Nix WD. J Mech Phys Solids 2003;51:901. [6] Mook WM, Jungk JM, Cordill MJ, Moody NR, Sun Y, Xia Y, et al. Z Metallkd 2004;95:416. [7] Miller RE, Acharya AA. J Mech Phys Solids 2004;52:1507. [8] Gerberich WW, Tymiak NI, Grunlan JC, Horstemeyer MF, Baskes MI. J Appl Mech 2002;69:433. [9] Horstemeyer MF, Baskes MI, Plimpton SJ. Acta Mater 2001;49:4363. [10] Gerberich WW, Cordill MJ, Mook WM, Moody NR, Perrey CR, Carter CB, et al. Acta Mater 2005;53:2215. [11] Huang Y, Ou S, Hwang KC, Li M, Gao HA. Int J Plast 2004;20:753. [12] Tabor D. The hardness of metals. Oxford Press: Oxford, UK; 1951. [13] Gerberich WW, Mook WM, Perrey CR, Carter CB, Baskes MI, Mukherjee R, et al. J Mech Phys Solids 2003;51:979. [14] Dong Y, Bapat A, Hilchie S, Kortshagen U, Campbell SA. J Vac Sci Technol B 2004;22:1923. [15] Perrey CR, Deneen JM, Carter CB. Mater Res Soc Symp Proc 2004;818:259. [16] Minor AM, Lilleodden ET, Jin M, Stach EA, Chrzan DC, Morris JW. Philos Mag 2005;85:323. [17] Gerberich WW, Mook WM, Cordill MJ, Carter CB, Perrey CR, Heberlein JV, et al. Int J Plast 2005;21:2391. [18] Eshelby JD, Frank FC, Nabarro FRN. Philos Mag 1959;42:351. [19] Rabier J, Cordier P, Tondellier T, Demret JL, Garem H. J Phys Condens Matter 2000;12:10059. [20] Jungk JM, Mook WM, Cordill MJ, Bahr DF, Moody NR, Hoehn J, et al. J Mater Res 2004;19:2812. [21] Gao H, Chiu CH, Lee J. Int J Solids Struct 1992;29:2471. [22] Chen X, Vlassak JJ. J Mater Res 2001;16:2974. [23] Johnson KL. Contact mechanics. Cambridge, UK: Cambridge University Press; 1985. [24] Bahr DF, Kramer DE, Gerberich WW. Acta Mater 1998;46:3605. [25] Suresh S, Nieh TG, Choi BW. Scripta Mater 1999;41:951.