- Email: [email protected]
Size effects on strength and plasticity of vanadium nanopillars
Available online at www.sciencedirect.com
Scripta Materialia 63 (2010) 1153–1156 www.elsevier.com/locate/scriptamat
Size effects on strength and plasticity of vanadium nanopillars Seung Min Han,a,⇑ Tara Bozorg-Grayeli,b James R. Grovesb and William D. Nixb a
Graduate School of EEWS, Korea Advanced Institute of Science and Technology, 373-1 Guseong Dong, Yuseong Gu, Daejeon 305-701, Republic of Korea b Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, USA Received 23 June 2010; revised 5 August 2010; accepted 6 August 2010 Available online 11 August 2010
A size effect study was conducted on [0 0 1] oriented vanadium nanopillars that were synthesized from both a thin film and a bulk crystal. The results indicate that a size-dependent deformation behavior exists for vanadium; the smaller nanopillars displayed discrete strain bursts and higher stresses during deformation. The size effect exponent is found to be 0.79, and the results are compared with previous reports on other body-centered cubic (bcc) metals: Nb, Ta, Mo and W. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Bcc metal; Size-dependent plasticity; Microcompression; Nanopillar
Mechanical properties of small scale structures are of technological importance in various applications, including microelectronic devices, and MEMS/NEMS devices. Since the microcompression methodology was introduced by Uchic et al. [1], there has been an explosion of research on strength and plasticity of metallic nanopillars made out of face-centered cubic (fcc) single crystals [1–3]. Through extensive study of fcc nanopillar compression, there is wide agreement on the basic results: the pillars display discrete flow behavior with frequent strain bursts and an increase in the yield strength (rYS ) with decrease in the sample diameter (d) according to rYS / d n . The size effect exponent n ranges between 0.6 and 1.0 for fcc metals. However, interpretation of the observed behavior is still under debate. There are two dominant competing theories that aim to explain the observed size-dependent properties in fcc nanopillars. One is the single-armed source theory, as proposed by Parthasarathy et al. [4], and the other is the dislocation starvation/nucleation theory, as proposed by Greer et al. [2]. In both single-armed source and starvation theories, it is important to understand the dislocation multiplication process responsible for the observed plasticity. The multiplication process is achieved by continued operation of a truncated source or by surface nucleation in the single-armed source and starvation theories, respectively. In this regard, studying body-centered cubic (bcc) metals and comparing the results to those for fcc metals might provide insight into
⇑ Corresponding author. E-mail: [email protected]
the operative multiplication processes. BCC metals have higher Peierls stresses and exhibit easier cross-slip of the slower screw dislocations compared to fcc metals. These factors may result in more dislocation interactions that, in turn, may cause more dislocations to be retained in the nanopillar during deformation. In a recent study, Weinberger and Cai showed by dislocation dynamics (DD) simulation that the slower screw components of dislocations are retained during deformation and can cross-slip to form a loop that eventually divides into two separate dislocations [5]. Their work indicated that the multiplication of dislocations in bcc pillars may occur through this mechanism. Weinberger and Cai reported that this mechanism occurs above a critical applied compressive stress, and that this critical stress is pillar size dependent; the larger the pillar dimension, the smaller the critical stress for dislocation multiplication. More recently, a significant body of results for bcc nanopillars has been emerging. Bei et al. have shown that their alloyed Mo pillars, uniquely synthesized from directional solidification, deform at close to the theoretical strength and do not display any size-dependent strength [6]. On the other hand, Nb, Ta, Mo, and W nanopillars synthesized using focused ion beam milling show a size effect that differs for the different metals [7–13]. Molybdenum nanopillars showed size-dependent strength with n varying between 0.3 and 0.5 [7–9,12]. However, the size effect exponent for Nb was observed to be higher (n = 0.48– 1.06) than that for Mo nanopillars [10,11,13]. Kim et al. and Schneider et al. proposed that the difference is due to the different Peierls stresses for these metals at room
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.08.011
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temperature [10,13]. It is well known that low temperature deformation in bcc metals is strongly dependent on temperature, and there is a critical temperature above which the screw dislocations can glide easily due to thermal activation over the Peierls barriers [14,15]. This critical temperature for Nb is 350 K, whereas for Mo the critical temperature is much higher, 480 K [15,16]. Schneider et al. and Kim et al. explored two additional bcc metals (Ta, W in addition to Nb, Mo), and they both reported that the size-dependent strength depends on the critical temperatures for each of these materials [11,13]. Schneider et al. reported that the exponent scales well with Tc, while Kim et al. reported that the size exponents are approaching those of the fcc metals when tested at or above Tc but do not scale with Tc. In the present study, we report on the size-dependent strength of vanadium single crystal nanopillars. This study of vanadium provides useful insights into the deformation mechanisms of bcc pillars in two respects. First, vanadium offers a unique opportunity to explore in situ TEM compression testing to reveal the dislocation mechanisms that are responsible for the observed size effects in bcc crystals. Vanadium is a good choice for this purpose because it is relatively electron transparent due to its low atomic number, compared to other bcc metals such as Mo, Nb, and Ta. Although in situ TEM testing of nanopillars is a powerful technique, a major disadvantage of the technique is that a wide range of pillar sizes cannot be studied because the sample dimensions must be limited to achieve electron transparency. Therefore, we have conducted ex situ microcompression tests on V nanopillars to obtain a more complete picture of size effects for V. Another objective of this work is to compare the size effect study of the V, which has Tc = 380 K [17], to those for other bcc metals with different Tc to potentially shed light on to the Tc dependent size-dependent plasticity. Two vanadium samples were used for the microcompression tests. The first sample was a bulk single crystal of vanadium with a (1 0 0) surface that had been electro-polished and well-annealed to remove the surface defects arising from the polishing process. The second sample was a vanadium thin film that had been grown epitaxially onto a (1 0 0) MgO substrate using e-beam evaporation. The (1 0 0) MgO substrate was initially subjected to a light ion etch (50 V for 1 min with Ar ions) to remove any surface contaminants. Vanadium was then evaporated onto the substrate at 650 °C at a deposition ˚ s1. The growth process was monitored using rate of 1 A reflection high energy electron diffraction (RHEED) to ensure epitaxy, and X-ray diffraction was used to confirm the epitaxy after the film had been deposited. The total thickness of the vanadium film was 1.2 lm. Nanopillars were then machined using the focused ion beam (FIB) from both of these samples to produce a wide range of sample diameters, all of which were kept close to a length to diameter ratio of 3:1. Thin film pillars were limited in length by the film thickness, but they have the advantage that the length of the pillar can be determined precisely. The bulk sample, on the other hand, was useful for producing a wide range of sample sizes. The FIB synthesized pillars were then deformed using two different nanoindentation systems, the Agilent NanoXP and the Hysitron Ti-750, both fitted with a diamond flat punch tip.
Compression tests were performed at a constant nominal displacement rate of 2 nm s1. Since bcc materials are known to be strain-rate-dependent, a few tests at nominally constant strain rates were also performed. A series of scanning electron microscopy (SEM) images of the V pillars made from both the bulk single crystal and the thin film, before and after deformation, are shown in Figure 1. These images clearly indicate that there are multiple slip events that lead to inhomogeneous deformation behavior. This behavior is more apparent for the sub-micrometer scale pillars compared to the larger pillars. The SEM image in Figure 1b shows a thin film nanopillar after compression. From this image, the angle between the top surface of the pillar and the slip plane was calculated to be 40.4°. This is in approximate agreement with {1 1 0} slip of a (0 0 1) orientated bcc pillar. The conventional constant volume, homogeneous deformation model was used to calculate the true stress–true strain curves. It should be noted that the computed true stresses and true strains are only approximate since the deformation geometry is not uniform. However, we minimize the error by working with stress vs. strain up to 5% strain where inhomogeneous deformation is minimized. First, assuming that the pillar deforms homogeneously during compression such that the pillar becomes shorter and wider uniformly, the instantaneous cross-sectional area at the top during compression was estimated using ð1Þ A0 L0 ¼ Ap Lp ! Ap ¼ A0 L0 =Lp where A0, L0 are initial top cross-sectional area and length of the sample, and Ap, Lp are the top area and length of the sample during deformation. To determine Lp, the plastic displacement was first defined by subtracting the compliances associated with Sneddon sinkin and compression of the substrate pedestal: ð2Þ up ¼ utot uSneddon uSubs where uSneddon ¼ uSubs ¼
P k Smed
P k Subs
¼
P v2Subs
2ESubs = 1 P ¼ ESubs Abottom =LSubs
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Abottom =p ð3Þ
The Sneddon compliance arises due to the elastic punching effect of the pillar (pedestal), and the substrate compliance refers to the MgO pedestal’s finite compliance for the pillars made from the thin film. For the case of the
Figure 1. Before and after deformation SEM images of V pillars. (a)– (c) are from V thin film and (d)–(e) are from bulk V.
S. M. Han et al. / Scripta Materialia 63 (2010) 1153–1156
True Stress (MPa)
1800 1600 1400
Bulk and Thin Film Vanadium Pillars Tested Using Hysitron Ti-750
b
Thin Film 183 nm Thin Film 374 nm Thin Film 509 nm Bulk 1103 nm Bulk 3420 nm
Bulk Vanadium Pillars using Agilent XP 1000
1200 1000 800 600
800
247 nm 387 nm 780 nm 1990 nm 2790 nm
600 400 200
400 200 0 0.00
0.01
0.02
0.03
True Strain
0.04
0.05
0 0.00
1000
100 Thin Film Bulk Bulk Const Strain Rate=0.002
10
100
1000
Pillar Diameter (nm)
Figure 3. Log–log plot of 2% flow stress plotted against pillar diameter. The slope of the plot or the size effect exponent was calculated to be 0.79. a Flow Stress (MPa)
2000
True Stress (MPa)
a
Vanadium Size Effect Plot
b
Size Effect Plot of BCC Pillars 10000
1000
100
V (this work) Nb (Kim) Ta (Kim) Mo (Kim) W (Kim)
10
0.01
0.02
0.03
0.04
0.05
True Strain
Figure 2. True stress vs. true strain of bulk and vanadium pillars tested with (a) Hysitron Ti-750 and (b) Agilent NanoXP.
Size Effect Plot of BCC Pillars
10000
Flow Stress (MPa)
Representative true stress vs. true strain plots for V pillars with different diameters are reported in Figure 2. Figure 2a shows results of pillars made from both thin film and bulk V tested using the Hysitron Ti-750 and Figure 2b shows results of V bulk pillars tested with the Agilent NanoXP. For smaller pillars with diameters <1 lm, discrete strain bursts, similar to those for fcc pillars, are observed. Pillars tested using the Hysitron Ti750 nanoindenter, Figure 2a, show unloads associated with discrete bursts indicating that the testing system is able to quickly remove the load to keep up with the specified constant displacement rate. Pillars tested using the Agilent NanoXP, Figure 2b, show displacement bursts at constant loads, indicating that this system is not able to remove the load quickly to maintain constant displacement rate. However, the overall deformation behavior, excluding the response during the burst events, is similar for pillars tested with the two systems. We note that the initial loading slopes of the stress vs. strain plot vary between different sized pillars, and the values also do not correspond to the V modulus, which is 128 GPa. There are two reasons why the loading slopes may differ from the modulus. First of all, any tapering that will cause an increase in the cross-sectional areas during compression results in an overestimation of the stresses since we are using the initial top diameters for the stress calculations. Smaller pillars tend to have more tapering of the side walls, and therefore the loading slopes appear bigger than those of larger pillars. The second reason why the loading slopes are not the same as the V modulus is that the punch is not perfectly aligned with the top surface of the pillar, and this results in gradual contact and reduced initial slopes. However, if we take the slope from the unloads that occur during the strain bursts, then the modulus is closer to the actual modulus of the sample. For example, the slope taken from the unload at a strain of 0.015 from the 1103 nm bulk V pillar from Figure 2a is 128 GPa, which is the expected modulus for V. As shown in Figure 2, there clearly exists a size dependence in the deformation behavior, with the smaller diameter pillars have the higher yield and flow stresses. We also note that pillars with diameter less than 1 lm experience obvious discrete strain bursts, whereas
pillars with diameters of 2–3 lm show more traditional, continuous plastic flow with strain hardening. For all pillar compression results, flow stresses taken at 2% plastic strain were determined and plotted against their diameters as shown in Figure 3. This log–log plot quantifies the size effect; the slope of the line is 0.79. Our results for V can be compared with microcompression studies for other bcc metals. The 2% flow stresses from our V study are first compared with the 8% flow stresses of Nb, Ta, Mo, W from Kim et al. [11] in Figure 4a and with the 5% flow stresses for these same bcc metals reported by Schneider et al. [13] in Figure 4b. The resulting size exponents from these studies are summarized in Table 1 together with the respective critical temperatures (Tc). Collected results for different bcc metal nanopillars are analyzed with respect to their Tc to determine whether the Tc dependent size effect theory proposed by Schneider et al. and Kim et al. [10,13] is consistent with these results. The Tc dependent size effect theory argues that a bcc metal would have a higher Peierls stress below Tc, but that it would behave more like an fcc metal above Tc, at which point the dislocations would glide more easily, resulting in fewer interactions and less multiplications. According to this picture, above the Tc, the mechanism for dislocation multiplication in nanopillars, as described by Weinberger and Cai [5], may not occur, and the nanopillars may approach a starvation state instead. Based on this Tc theory, we would then expect the V to behave more like the fcc metals since the Tc of 380 K is closer to the testing temperature of 298 K (room temperature). Thus, the size effect exponent might be expected to be close to the values for fcc metals, but may still be slightly smaller than that for bcc Nb, which has a smaller Tc than the V. As can be seen in Figure 4a, the present results are in good agreement with the data reported by Kim et al.
2% Flow Stress (MPa)
bulk sample, uSubs is zero and only the Sneddon compliance displacement was subtracted from total displacement. The plastic length, Lp, was then determined by Lp ¼ L0 up . Finally, Lp was inserted to Eq. (1) to calculate Ap, which was used to calculate the true stress and true strain as stated below: r ¼ P =Ap ; e ¼ eel þ ep ¼ P =ðEAp Þ þ lnðL0 =Lp Þ ð4Þ
1155
1000
V (this work) Nb (Schneider) Ta (Schneider) Mo (Schneider) W (Schneider)
100
10 100
1000
Size (nm)
10000
100
1000
10000
Size (nm)
Figure 4. V size effect plot compared with Nb, Mo, Ta, W by (a) Kim et al. [11] and (b) Schneider et al. [13].
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S. M. Han et al. / Scripta Materialia 63 (2010) 1153–1156
Table 1. Critical temperature and size effect exponent reported for different bcc nanopillars [11,13,17]. Material
Nb V (this work) Ta
Critical temperature (K) 350 380 n (with Kim et al. [11]) 0.93 0.79 n (with Schneider et al. [13]) 0.48 0.79
Mo W
450 480 800 0.43 0.44 0.44 0.41 0.38 0.21
[11], where our size effect exponent (0.79) is comparable to but slightly smaller than that of Nb (0.93), as expected. As also expected, the V exponent is larger than that for Ta (0.43), Mo (0.44), and W (0.44) reported by Kim et al., and therefore is consistent with the Tc theory. Our results are also compared with Schneider et al.’s Nb, Ta, Mo, and W results in Figure 4b [13]. The V size exponent is larger than those from Schneider et al.’s Ta (0.41), Mo (0.38), and W (0.21), and this trend is in accordance with the Tc theory. However, there is a discrepancy between our results and Schneider et al.’s Nb results; our V exponent is larger than their exponent for Nb (0.48), even though V has a higher Tc. The reported flow stresses for the Nb nanopillars of Schneider et al. are generally higher for larger diameters and this resulted in a smaller size effect when compared to the present V results. A possible explanation for higher flow stresses at larger diameters might be the presence of impurities in the bcc bulk crystal, which can be sensitive to impurity concentrations. Another possible explanation might involve different surface defect concentrations introduced during polishing steps. Alternatively, the present experiments might be underestimating the flow stresses at large diameters because testing at a constant nominal displacement rate gives lower strain rates for larger pillars and possibly a lower flow stress due to strain rate effects. We discuss this possibility below. The effect of strain rate on the flow stress of V nanopillars was also explored. Bcc metals are known to display strong strain rate effects, with higher flow stresses at higher strain rates. The strain rate effect on Mo bcc nanopillars was investigated by Schneider et al. and Kim et al.; Schneider et al. reported a 50% increase in flow stresses at 2.5% strain when the strain rate was increased by a factor of 100 [12], and Kim et al. reported a factor of 3 increase in yield strengths when the strain rate was increased by the same amount for 400 nm Mo pillars [12]. As noted above, most of our tests were performed at a fixed displacement rate, which resulted in different strain rates for differently sized pillars. The nominal engineering strain rate can be calculated as _ 0 , where u_ is the nominal displacement rate and e_ ¼ u=L L0 is the initial length of the pillar. Since we keep the aspect ratio, L0/D0, close to 3, then the strain rate is given _ by e_ ¼ u=3D 0 , where D0 is the initial diameter. Our smallest pillar has a diameter of 200 nm and that results in a strain rate of 3.3 103; one of our largest pillars has a diameter of 2000 nm, which results in a strain rate of 3.3 104 when tested at the same constant displacement rate of 2 nm s1. This strain rate ratio is smaller than that used by Schneider et al. in their strain rate sensitivity test, and therefore the difference in resulting flow stresses is expected to be smaller. Nevertheless, we tested three pillars with large diameters at a faster constant strain rate of 2 103, which is comparable to the rate
used for the smaller pillars. The resulting flow stresses are plotted in Figure 3, together with other data collected at a constant displacement rate. As can be seen in Figure 3, the flow stresses for the pillars tested at the constant strain rate (higher displacement rate for larger pillars) are close to the overall trend of the rest of data, and the calculated size effect exponent remains at 0.79. In summary, a study of size-dependent plasticity was conducted for bcc vanadium using both a thin film and a bulk crystal with the (1 0 0) orientation. Microcompression tests revealed a clear size-dependent deformation behavior with a size effect exponent of 0.79. Our results are in general agreement with other size effect studies of different bcc metals such as Nb, Ta, W and Mo, and the expected trend based on the critical temperature dependent plasticity theory was observed. Vanadium has a Tc of 380 K which is higher than that for Ta, Mo, and W, and the size effect exponent for V was correspondingly highest. The observed size effect exponent is close to those found for fcc metals, but the mechanism for deformation is still expected to differ from that for fcc metals because of the increased Peierls stress and corresponding easiness for cross-slip for bcc V. In situ TEM deformation of V nanopillars will be critical in developing a full understanding of the deformation mechanism for bcc metals. The authors thank A.S. Schneider and P.A. Gruber for providing the V bulk single crystal and for insightful discussions. This research was supported by the Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under Contract No. DE-FG02-04ER46163. [1] M.D. Uchic, D.M. Dimiduk, J. Florando, W.D. Nix, Science 305 (2004) 986. [2] J.R. Greer, W.C. Oliver, W.D. Nix, Acta Mater. 53 (2005) 1821. [3] C.A. Volkert, E.T. Lilleodeen, Philos. Mag. 86 (2006) 5567. [4] T.A. Parthasarathy, S.I. Rao, D.M. Dimiduk, M.D. Uchic, D.R. Trinkle, Scripta Mater. 56 (2007) 313. [5] C.R. Weinberger, W. Cai, Proc. Natl. Acad. Sci. USA 105 (38) (2008) 14304. [6] H. Bei, S. Shim, E.P. George, M.K. Miller, E.G. Herbert, G.M. Pharr, Scripta Mater. 57 (2007) 397. [7] J.R. Greer, C.R. Weinberger, W. Cai, Proc. Natl. Acad. Sci. USA 493 (2008) 21. [8] J.Y. Kim, J.R. Greer, Appl. Phys. Lett. 93 (10) (2008) 101916. [9] J.Y. Kim, J.R. Greer, Acta Mater. 57 (17) (2009) 5245. [10] J.Y. Kim, D.C. Jang, J.R. Greer, Scripta Mater. 61 (3) (2009) 300. [11] J.Y. Kim, D.C. Jang, J.R. Greer, Acta Mater. 58 (2010) 2355. [12] A.S. Schneider, B.G. Clark, C.P. Frick, P.A. Gruber, E. Arzt, Mater. Sci. Eng. A 508 (2009) 241. [13] A.S. Schneider, D. Kaufmann, B.G. Clark, C.P. Frick, P.A. Gruber, R. Mo¨nig, O. Kraft, E. Arzt, Phys. Rev. Lett. 103 (10) (2009) 105501. [14] T. Suzuki, H. Koizumi, H.O.K. Kirchner, Acta Metall. Mater. 43 (6) (1995) 2177. [15] A. Seeger, U. Holzwarth, Philos. Mag. 86 (2006) 3861. [16] L. Hollang, M. Hommel, A. Seeger, Phys. Status Solidi A 160 (1997) 329–354. [17] J.W. Christian, B.C. Masters, Private communication/Rept. Entitled “Thermally-activated Flow in Body-Centered Cubic Metals” Res. Contract 7/Exptl/729, November 1961.
Scripta Materialia 63 (2010) 1153–1156 www.elsevier.com/locate/scriptamat
Size effects on strength and plasticity of vanadium nanopillars Seung Min Han,a,⇑ Tara Bozorg-Grayeli,b James R. Grovesb and William D. Nixb a
Graduate School of EEWS, Korea Advanced Institute of Science and Technology, 373-1 Guseong Dong, Yuseong Gu, Daejeon 305-701, Republic of Korea b Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, USA Received 23 June 2010; revised 5 August 2010; accepted 6 August 2010 Available online 11 August 2010
A size effect study was conducted on [0 0 1] oriented vanadium nanopillars that were synthesized from both a thin film and a bulk crystal. The results indicate that a size-dependent deformation behavior exists for vanadium; the smaller nanopillars displayed discrete strain bursts and higher stresses during deformation. The size effect exponent is found to be 0.79, and the results are compared with previous reports on other body-centered cubic (bcc) metals: Nb, Ta, Mo and W. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Bcc metal; Size-dependent plasticity; Microcompression; Nanopillar
Mechanical properties of small scale structures are of technological importance in various applications, including microelectronic devices, and MEMS/NEMS devices. Since the microcompression methodology was introduced by Uchic et al. [1], there has been an explosion of research on strength and plasticity of metallic nanopillars made out of face-centered cubic (fcc) single crystals [1–3]. Through extensive study of fcc nanopillar compression, there is wide agreement on the basic results: the pillars display discrete flow behavior with frequent strain bursts and an increase in the yield strength (rYS ) with decrease in the sample diameter (d) according to rYS / d n . The size effect exponent n ranges between 0.6 and 1.0 for fcc metals. However, interpretation of the observed behavior is still under debate. There are two dominant competing theories that aim to explain the observed size-dependent properties in fcc nanopillars. One is the single-armed source theory, as proposed by Parthasarathy et al. [4], and the other is the dislocation starvation/nucleation theory, as proposed by Greer et al. [2]. In both single-armed source and starvation theories, it is important to understand the dislocation multiplication process responsible for the observed plasticity. The multiplication process is achieved by continued operation of a truncated source or by surface nucleation in the single-armed source and starvation theories, respectively. In this regard, studying body-centered cubic (bcc) metals and comparing the results to those for fcc metals might provide insight into
⇑ Corresponding author. E-mail: [email protected]
the operative multiplication processes. BCC metals have higher Peierls stresses and exhibit easier cross-slip of the slower screw dislocations compared to fcc metals. These factors may result in more dislocation interactions that, in turn, may cause more dislocations to be retained in the nanopillar during deformation. In a recent study, Weinberger and Cai showed by dislocation dynamics (DD) simulation that the slower screw components of dislocations are retained during deformation and can cross-slip to form a loop that eventually divides into two separate dislocations [5]. Their work indicated that the multiplication of dislocations in bcc pillars may occur through this mechanism. Weinberger and Cai reported that this mechanism occurs above a critical applied compressive stress, and that this critical stress is pillar size dependent; the larger the pillar dimension, the smaller the critical stress for dislocation multiplication. More recently, a significant body of results for bcc nanopillars has been emerging. Bei et al. have shown that their alloyed Mo pillars, uniquely synthesized from directional solidification, deform at close to the theoretical strength and do not display any size-dependent strength [6]. On the other hand, Nb, Ta, Mo, and W nanopillars synthesized using focused ion beam milling show a size effect that differs for the different metals [7–13]. Molybdenum nanopillars showed size-dependent strength with n varying between 0.3 and 0.5 [7–9,12]. However, the size effect exponent for Nb was observed to be higher (n = 0.48– 1.06) than that for Mo nanopillars [10,11,13]. Kim et al. and Schneider et al. proposed that the difference is due to the different Peierls stresses for these metals at room
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.08.011
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S. M. Han et al. / Scripta Materialia 63 (2010) 1153–1156
temperature [10,13]. It is well known that low temperature deformation in bcc metals is strongly dependent on temperature, and there is a critical temperature above which the screw dislocations can glide easily due to thermal activation over the Peierls barriers [14,15]. This critical temperature for Nb is 350 K, whereas for Mo the critical temperature is much higher, 480 K [15,16]. Schneider et al. and Kim et al. explored two additional bcc metals (Ta, W in addition to Nb, Mo), and they both reported that the size-dependent strength depends on the critical temperatures for each of these materials [11,13]. Schneider et al. reported that the exponent scales well with Tc, while Kim et al. reported that the size exponents are approaching those of the fcc metals when tested at or above Tc but do not scale with Tc. In the present study, we report on the size-dependent strength of vanadium single crystal nanopillars. This study of vanadium provides useful insights into the deformation mechanisms of bcc pillars in two respects. First, vanadium offers a unique opportunity to explore in situ TEM compression testing to reveal the dislocation mechanisms that are responsible for the observed size effects in bcc crystals. Vanadium is a good choice for this purpose because it is relatively electron transparent due to its low atomic number, compared to other bcc metals such as Mo, Nb, and Ta. Although in situ TEM testing of nanopillars is a powerful technique, a major disadvantage of the technique is that a wide range of pillar sizes cannot be studied because the sample dimensions must be limited to achieve electron transparency. Therefore, we have conducted ex situ microcompression tests on V nanopillars to obtain a more complete picture of size effects for V. Another objective of this work is to compare the size effect study of the V, which has Tc = 380 K [17], to those for other bcc metals with different Tc to potentially shed light on to the Tc dependent size-dependent plasticity. Two vanadium samples were used for the microcompression tests. The first sample was a bulk single crystal of vanadium with a (1 0 0) surface that had been electro-polished and well-annealed to remove the surface defects arising from the polishing process. The second sample was a vanadium thin film that had been grown epitaxially onto a (1 0 0) MgO substrate using e-beam evaporation. The (1 0 0) MgO substrate was initially subjected to a light ion etch (50 V for 1 min with Ar ions) to remove any surface contaminants. Vanadium was then evaporated onto the substrate at 650 °C at a deposition ˚ s1. The growth process was monitored using rate of 1 A reflection high energy electron diffraction (RHEED) to ensure epitaxy, and X-ray diffraction was used to confirm the epitaxy after the film had been deposited. The total thickness of the vanadium film was 1.2 lm. Nanopillars were then machined using the focused ion beam (FIB) from both of these samples to produce a wide range of sample diameters, all of which were kept close to a length to diameter ratio of 3:1. Thin film pillars were limited in length by the film thickness, but they have the advantage that the length of the pillar can be determined precisely. The bulk sample, on the other hand, was useful for producing a wide range of sample sizes. The FIB synthesized pillars were then deformed using two different nanoindentation systems, the Agilent NanoXP and the Hysitron Ti-750, both fitted with a diamond flat punch tip.
Compression tests were performed at a constant nominal displacement rate of 2 nm s1. Since bcc materials are known to be strain-rate-dependent, a few tests at nominally constant strain rates were also performed. A series of scanning electron microscopy (SEM) images of the V pillars made from both the bulk single crystal and the thin film, before and after deformation, are shown in Figure 1. These images clearly indicate that there are multiple slip events that lead to inhomogeneous deformation behavior. This behavior is more apparent for the sub-micrometer scale pillars compared to the larger pillars. The SEM image in Figure 1b shows a thin film nanopillar after compression. From this image, the angle between the top surface of the pillar and the slip plane was calculated to be 40.4°. This is in approximate agreement with {1 1 0} slip of a (0 0 1) orientated bcc pillar. The conventional constant volume, homogeneous deformation model was used to calculate the true stress–true strain curves. It should be noted that the computed true stresses and true strains are only approximate since the deformation geometry is not uniform. However, we minimize the error by working with stress vs. strain up to 5% strain where inhomogeneous deformation is minimized. First, assuming that the pillar deforms homogeneously during compression such that the pillar becomes shorter and wider uniformly, the instantaneous cross-sectional area at the top during compression was estimated using ð1Þ A0 L0 ¼ Ap Lp ! Ap ¼ A0 L0 =Lp where A0, L0 are initial top cross-sectional area and length of the sample, and Ap, Lp are the top area and length of the sample during deformation. To determine Lp, the plastic displacement was first defined by subtracting the compliances associated with Sneddon sinkin and compression of the substrate pedestal: ð2Þ up ¼ utot uSneddon uSubs where uSneddon ¼ uSubs ¼
P k Smed
P k Subs
¼
P v2Subs
2ESubs = 1 P ¼ ESubs Abottom =LSubs
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Abottom =p ð3Þ
The Sneddon compliance arises due to the elastic punching effect of the pillar (pedestal), and the substrate compliance refers to the MgO pedestal’s finite compliance for the pillars made from the thin film. For the case of the
Figure 1. Before and after deformation SEM images of V pillars. (a)– (c) are from V thin film and (d)–(e) are from bulk V.
S. M. Han et al. / Scripta Materialia 63 (2010) 1153–1156
True Stress (MPa)
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Figure 3. Log–log plot of 2% flow stress plotted against pillar diameter. The slope of the plot or the size effect exponent was calculated to be 0.79. a Flow Stress (MPa)
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Vanadium Size Effect Plot
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Size Effect Plot of BCC Pillars 10000
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Figure 2. True stress vs. true strain of bulk and vanadium pillars tested with (a) Hysitron Ti-750 and (b) Agilent NanoXP.
Size Effect Plot of BCC Pillars
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Flow Stress (MPa)
Representative true stress vs. true strain plots for V pillars with different diameters are reported in Figure 2. Figure 2a shows results of pillars made from both thin film and bulk V tested using the Hysitron Ti-750 and Figure 2b shows results of V bulk pillars tested with the Agilent NanoXP. For smaller pillars with diameters <1 lm, discrete strain bursts, similar to those for fcc pillars, are observed. Pillars tested using the Hysitron Ti750 nanoindenter, Figure 2a, show unloads associated with discrete bursts indicating that the testing system is able to quickly remove the load to keep up with the specified constant displacement rate. Pillars tested using the Agilent NanoXP, Figure 2b, show displacement bursts at constant loads, indicating that this system is not able to remove the load quickly to maintain constant displacement rate. However, the overall deformation behavior, excluding the response during the burst events, is similar for pillars tested with the two systems. We note that the initial loading slopes of the stress vs. strain plot vary between different sized pillars, and the values also do not correspond to the V modulus, which is 128 GPa. There are two reasons why the loading slopes may differ from the modulus. First of all, any tapering that will cause an increase in the cross-sectional areas during compression results in an overestimation of the stresses since we are using the initial top diameters for the stress calculations. Smaller pillars tend to have more tapering of the side walls, and therefore the loading slopes appear bigger than those of larger pillars. The second reason why the loading slopes are not the same as the V modulus is that the punch is not perfectly aligned with the top surface of the pillar, and this results in gradual contact and reduced initial slopes. However, if we take the slope from the unloads that occur during the strain bursts, then the modulus is closer to the actual modulus of the sample. For example, the slope taken from the unload at a strain of 0.015 from the 1103 nm bulk V pillar from Figure 2a is 128 GPa, which is the expected modulus for V. As shown in Figure 2, there clearly exists a size dependence in the deformation behavior, with the smaller diameter pillars have the higher yield and flow stresses. We also note that pillars with diameter less than 1 lm experience obvious discrete strain bursts, whereas
pillars with diameters of 2–3 lm show more traditional, continuous plastic flow with strain hardening. For all pillar compression results, flow stresses taken at 2% plastic strain were determined and plotted against their diameters as shown in Figure 3. This log–log plot quantifies the size effect; the slope of the line is 0.79. Our results for V can be compared with microcompression studies for other bcc metals. The 2% flow stresses from our V study are first compared with the 8% flow stresses of Nb, Ta, Mo, W from Kim et al. [11] in Figure 4a and with the 5% flow stresses for these same bcc metals reported by Schneider et al. [13] in Figure 4b. The resulting size exponents from these studies are summarized in Table 1 together with the respective critical temperatures (Tc). Collected results for different bcc metal nanopillars are analyzed with respect to their Tc to determine whether the Tc dependent size effect theory proposed by Schneider et al. and Kim et al. [10,13] is consistent with these results. The Tc dependent size effect theory argues that a bcc metal would have a higher Peierls stress below Tc, but that it would behave more like an fcc metal above Tc, at which point the dislocations would glide more easily, resulting in fewer interactions and less multiplications. According to this picture, above the Tc, the mechanism for dislocation multiplication in nanopillars, as described by Weinberger and Cai [5], may not occur, and the nanopillars may approach a starvation state instead. Based on this Tc theory, we would then expect the V to behave more like the fcc metals since the Tc of 380 K is closer to the testing temperature of 298 K (room temperature). Thus, the size effect exponent might be expected to be close to the values for fcc metals, but may still be slightly smaller than that for bcc Nb, which has a smaller Tc than the V. As can be seen in Figure 4a, the present results are in good agreement with the data reported by Kim et al.
2% Flow Stress (MPa)
bulk sample, uSubs is zero and only the Sneddon compliance displacement was subtracted from total displacement. The plastic length, Lp, was then determined by Lp ¼ L0 up . Finally, Lp was inserted to Eq. (1) to calculate Ap, which was used to calculate the true stress and true strain as stated below: r ¼ P =Ap ; e ¼ eel þ ep ¼ P =ðEAp Þ þ lnðL0 =Lp Þ ð4Þ
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Figure 4. V size effect plot compared with Nb, Mo, Ta, W by (a) Kim et al. [11] and (b) Schneider et al. [13].
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Table 1. Critical temperature and size effect exponent reported for different bcc nanopillars [11,13,17]. Material
Nb V (this work) Ta
Critical temperature (K) 350 380 n (with Kim et al. [11]) 0.93 0.79 n (with Schneider et al. [13]) 0.48 0.79
Mo W
450 480 800 0.43 0.44 0.44 0.41 0.38 0.21
[11], where our size effect exponent (0.79) is comparable to but slightly smaller than that of Nb (0.93), as expected. As also expected, the V exponent is larger than that for Ta (0.43), Mo (0.44), and W (0.44) reported by Kim et al., and therefore is consistent with the Tc theory. Our results are also compared with Schneider et al.’s Nb, Ta, Mo, and W results in Figure 4b [13]. The V size exponent is larger than those from Schneider et al.’s Ta (0.41), Mo (0.38), and W (0.21), and this trend is in accordance with the Tc theory. However, there is a discrepancy between our results and Schneider et al.’s Nb results; our V exponent is larger than their exponent for Nb (0.48), even though V has a higher Tc. The reported flow stresses for the Nb nanopillars of Schneider et al. are generally higher for larger diameters and this resulted in a smaller size effect when compared to the present V results. A possible explanation for higher flow stresses at larger diameters might be the presence of impurities in the bcc bulk crystal, which can be sensitive to impurity concentrations. Another possible explanation might involve different surface defect concentrations introduced during polishing steps. Alternatively, the present experiments might be underestimating the flow stresses at large diameters because testing at a constant nominal displacement rate gives lower strain rates for larger pillars and possibly a lower flow stress due to strain rate effects. We discuss this possibility below. The effect of strain rate on the flow stress of V nanopillars was also explored. Bcc metals are known to display strong strain rate effects, with higher flow stresses at higher strain rates. The strain rate effect on Mo bcc nanopillars was investigated by Schneider et al. and Kim et al.; Schneider et al. reported a 50% increase in flow stresses at 2.5% strain when the strain rate was increased by a factor of 100 [12], and Kim et al. reported a factor of 3 increase in yield strengths when the strain rate was increased by the same amount for 400 nm Mo pillars [12]. As noted above, most of our tests were performed at a fixed displacement rate, which resulted in different strain rates for differently sized pillars. The nominal engineering strain rate can be calculated as _ 0 , where u_ is the nominal displacement rate and e_ ¼ u=L L0 is the initial length of the pillar. Since we keep the aspect ratio, L0/D0, close to 3, then the strain rate is given _ by e_ ¼ u=3D 0 , where D0 is the initial diameter. Our smallest pillar has a diameter of 200 nm and that results in a strain rate of 3.3 103; one of our largest pillars has a diameter of 2000 nm, which results in a strain rate of 3.3 104 when tested at the same constant displacement rate of 2 nm s1. This strain rate ratio is smaller than that used by Schneider et al. in their strain rate sensitivity test, and therefore the difference in resulting flow stresses is expected to be smaller. Nevertheless, we tested three pillars with large diameters at a faster constant strain rate of 2 103, which is comparable to the rate
used for the smaller pillars. The resulting flow stresses are plotted in Figure 3, together with other data collected at a constant displacement rate. As can be seen in Figure 3, the flow stresses for the pillars tested at the constant strain rate (higher displacement rate for larger pillars) are close to the overall trend of the rest of data, and the calculated size effect exponent remains at 0.79. In summary, a study of size-dependent plasticity was conducted for bcc vanadium using both a thin film and a bulk crystal with the (1 0 0) orientation. Microcompression tests revealed a clear size-dependent deformation behavior with a size effect exponent of 0.79. Our results are in general agreement with other size effect studies of different bcc metals such as Nb, Ta, W and Mo, and the expected trend based on the critical temperature dependent plasticity theory was observed. Vanadium has a Tc of 380 K which is higher than that for Ta, Mo, and W, and the size effect exponent for V was correspondingly highest. The observed size effect exponent is close to those found for fcc metals, but the mechanism for deformation is still expected to differ from that for fcc metals because of the increased Peierls stress and corresponding easiness for cross-slip for bcc V. In situ TEM deformation of V nanopillars will be critical in developing a full understanding of the deformation mechanism for bcc metals. The authors thank A.S. Schneider and P.A. Gruber for providing the V bulk single crystal and for insightful discussions. This research was supported by the Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under Contract No. DE-FG02-04ER46163. [1] M.D. Uchic, D.M. Dimiduk, J. Florando, W.D. Nix, Science 305 (2004) 986. [2] J.R. Greer, W.C. Oliver, W.D. Nix, Acta Mater. 53 (2005) 1821. [3] C.A. Volkert, E.T. Lilleodeen, Philos. Mag. 86 (2006) 5567. [4] T.A. Parthasarathy, S.I. Rao, D.M. Dimiduk, M.D. Uchic, D.R. Trinkle, Scripta Mater. 56 (2007) 313. [5] C.R. Weinberger, W. Cai, Proc. Natl. Acad. Sci. USA 105 (38) (2008) 14304. [6] H. Bei, S. Shim, E.P. George, M.K. Miller, E.G. Herbert, G.M. Pharr, Scripta Mater. 57 (2007) 397. [7] J.R. Greer, C.R. Weinberger, W. Cai, Proc. Natl. Acad. Sci. USA 493 (2008) 21. [8] J.Y. Kim, J.R. Greer, Appl. Phys. Lett. 93 (10) (2008) 101916. [9] J.Y. Kim, J.R. Greer, Acta Mater. 57 (17) (2009) 5245. [10] J.Y. Kim, D.C. Jang, J.R. Greer, Scripta Mater. 61 (3) (2009) 300. [11] J.Y. Kim, D.C. Jang, J.R. Greer, Acta Mater. 58 (2010) 2355. [12] A.S. Schneider, B.G. Clark, C.P. Frick, P.A. Gruber, E. Arzt, Mater. Sci. Eng. A 508 (2009) 241. [13] A.S. Schneider, D. Kaufmann, B.G. Clark, C.P. Frick, P.A. Gruber, R. Mo¨nig, O. Kraft, E. Arzt, Phys. Rev. Lett. 103 (10) (2009) 105501. [14] T. Suzuki, H. Koizumi, H.O.K. Kirchner, Acta Metall. Mater. 43 (6) (1995) 2177. [15] A. Seeger, U. Holzwarth, Philos. Mag. 86 (2006) 3861. [16] L. Hollang, M. Hommel, A. Seeger, Phys. Status Solidi A 160 (1997) 329–354. [17] J.W. Christian, B.C. Masters, Private communication/Rept. Entitled “Thermally-activated Flow in Body-Centered Cubic Metals” Res. Contract 7/Exptl/729, November 1961.