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Nb multilayers
Thin Solid Films 520 (2011) 818–823
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Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t s f
Microstructure and ultrahigh strength of nanoscale Cu/Nb multilayers X.Y. Zhu a,b,⁎, J.T. Luo a, F. Zeng a, F. Pan a,⁎ a b
Laboratory of Advanced Materials, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People's Republic of China National Key Lab for Remanufacturing, Academy of Armored Forces Engineering, Beijing 100072, People's Republic of China
a r t i c l e
i n f o
Available online 25 January 2011 Keywords: Cu/Nb multilayers Nanoindentation Microstructure Strength Strain hardening
a b s t r a c t The microstructure and mechanical properties of Cu/Nb multilayers were investigated by X-ray diffraction, transmission electron microscopy, scanning electron microscopy and nanoindentation. Ultrahigh strength of 3.27 GPa is achieved at the smallest layer thickness of 2.5 nm, which agrees well with the theoretical prediction based on the deformation mechanism of crossing of dislocations across interfaces. After that, the strength decreases with the increasing layer thickness and the transition of the deformation mechanism to confined layer slip occurs at the layer thickness of 6.5 nm. Additionally, strength of the Cu/Nb multilayers increases with increasing loading strain rate because of enhanced strain hardening. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Metallic multilayers have attracted extensive attention because of their promising mechanical properties and significant theoretical interest over the past couple of decades [1–3]. It is generally accepted that for layer thickness (h) above approximately 5 nm, the flow strength of multilayers is dependent on the individual layer thickness and several deformation mechanisms have been proposed to describe the plastic deformation in this size scale [4]. While for layer thickness below 5 nm, where the deformation mechanism is thought to be transmission of single glide dislocations across interfaces, the flow strength is independent on layer thickness and the maximum value is determined by interfacial properties [5]. Hoagland et al. have proved that for different interface types, such as coherent interfaces and incoherent interfaces, the factors that affect the resistance to dislocation slip transmission are quite different [6]. For coherent interfaces, such as interfaces in Cu/Ni system, the most important effect on strength derives from the coherency strains and the maximum attainable flow strength should be equal to the coherency stress [7], which has been confirmed by experimental data later [8]. On the other hand, for incoherent interfaces, such as interfaces in Cu/ Nb system, the case is more complex and atomistic simulations are applied to understand the properties of the interfaces and estimate the maximum attainable flow strength. Recently, Wang et al. have investigated the interaction of glide dislocations with Cu/Nb interfaces and found that a single mixed dislocation, from either Cu or Nb, cannot cross the interface until the resolved shear stress (RSS) is
⁎ Corresponding authors at: Laboratory of Advanced Materials, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People's Republic of China. Tel.: +86 10 62772907; fax: +86 10 62771160. E-mail addresses: [email protected] (X.Y. Zhu), [email protected] (F. Pan). 0040-6090/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2010.12.251
increased to extremely high values in excess of 1.1 GPa [9]. This result indicates that the attainable strength maximum in Cu/Nb system should exceed 3.4 GPa. However, such maximum strength or hardness has not yet been observed experimentally, although a lot of researches have been made on the room temperature strength of the Cu/Nb system using nanoindentation or micropillar compression [4,10–13]. The discrepancy between the model prediction and experimental data may be due to the sensitivity of the strength to strain rate. As revealed by atomic modeling, dislocation climb is involved in the process of slip transmission even at room temperature, and reactions between interfacial dislocations assisted by climb could lead to annihilation of dislocation content, resulting in less strain hardening [14]. The room temperature strength values of Cu/Nb multilayers that have been reported were frequently tested at strain rate on the order of 10−4 s−1, quite smaller than the strain rate that usually used in atomistic simulations. This lower strain rate may promote the dislocation climb process, release strain hardening, and then reduce the strength value. In fact, the dependence of yielding strength on the strain rate has been observed in several nanocrystalline materials [15,16]. Although there is no clear understanding of the deformation mechanism, it is postulated that the confined nanoscale grain structure plays an important role in this peculiar phenomenon. Herein we investigate the room temperature strength of Cu/Nb system using nanoindentation with a constant loading strain rate (LSR) of 0.05 s−1 and discuss the length-scale deformation mechanism at such strain rate. Moreover, varying loading strain rate tests are carried out to further investigate the dependence of the strength on strain rate. The experiment results confirm that ultrahigh strength over 3 GPa can be achieved in Cu/Nb multilayers with small layer thickness at relative high strain rate and room temperature dislocation climb is involved in the plastic deformation process. The mechanical properties of Cu/Nb mulitlayers have been intensively studied because of their remarkably high strength, fatigue resistance
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[17] and thermal stability [18]. Due to higher He solubility than either Cu or Nb and ability to prevent the degradation of material properties by the growth of bubbles, Cu/Nb multilayers are markedly more resistant to radiation damage, and thus can be applied to extend the operating limits and lifetimes of nuclear reactors [19–21]. 2. Experimental details Cu/Nb multilayers consisting of alternating equivalent Cu and Nb individual layers with h ranging from 2.5 to 20 nm were prepared at room temperature by alternate electron beam evaporation deposition using an ultrahigh vacuum (UHV) chamber. The base vacuum of the chamber was 5 × 10−6 Pa. The deposition rates were 0.8 Å/s for both Cu and Nb. The nominal thickness of the individual Cu and Nb layers was monitored by an in situ quartz oscillator. All the multilayers were deposited onto Si (100) wafers with native oxide. The deposition began with Cu layer and ended with Nb layer. The total thickness of multilayers was approximately 400 nm. Monolithic Cu film and Nb film of 1 μm were also deposited onto Si (100) in the same condition for comparison. The X-ray diffraction (XRD) experiment was carried out using Rigaku D/max-RB X-ray diffractometer with Cu Kα radiation. Microstructure observation was performed by a JEM2011 transmission electron microscopy (TEM) with 200 kV accelerating voltages. The mechanical properties of the multilayers were investigated by a Nano Indenter XP (MTS Systems Corp.) with a displacement resolution of 0.01 nm and a loading resolution of 50 nN. A Berkovich indenter, a three-sided pyramid with the same area-to-depth function as that of a Vickers indenter, was used in all the experiments. The hardness and modulus of the multilayers were measured by a continuous-stiffness measurement (CSM) technique with a constant LSR of 0.05 s−1. Varying LSR nanoindentation tests are carried out on the multilayer with h = 10 nm with LSR ranging from 0.01 to 0.5 s−1. A frequency of 45 Hz was used to avoid the sensitivity to thermal drift. The maximum indentation depth was 400 nm. More details about this technique and analysis method can be found in references [22–24]. Ten indents were performed on each specimen and an average value is reported with error bars indicating the range. For reference, the mechanical properties of the Cu film and Nb film were also examined under the same experimental condition. After indentation, the indent images of hardness measurement at LRS = 0.05 s−1 were characterized by field-emission scanning electron microscopy (SEM).
Fig. 1. The cross-section transmission electron micrographs taken in bright field of the multilayer with h = 20 nm.
superlattice structure can be produced in ultrathin layers of two dissimilar multilayers [25]. In fact, X-ray pole figure measurements have confirmed that epitaxial relationship of Kurdjumov–Sachs orientation Cu{111}//Nb{110} and Cu b 110N//Nb b 111 N exits in the Cu/Nb interfaces in ultra thin layer thickness [26]. Due to the large lattice mismatch of ~11%, misfit dislocations will be generated in the interfaces to accommodate the elastic strain, resulting in a compressive contribution to the interface stress [27,28]. Owing to the Poisson effect, the spacing perpendicular to the interfaces will expand, resulting in the shift of the diffraction peaks of Cu (111) and Nb (110) to a low-angle. 3.2. Mechanical properties The nanoindentation data were measured continuously during the loading of the indenter by the CSM method, and then they were analyzed by the Oliver and Pharr method [22]. Fig. 3(a) and (b) show the variations in hardness and modulus with indentation depth of the Cu/Nb multilayer with h = 2.5 nm for ten different continuous stiffness indentation tests, respectively. Both the curves approach
3. Results and discussion 3.1. Microstructure Cross-section transmission electron microscopy was performed to confirm the layered structure of the Cu/Nb multilayers. Fig. 1 shows the bright field image of the multilayer with h = 20 nm. Discrete layered structure with sharp planar interfaces between the two phases is clearly visible. Meanwhile, we can see that the multilayer show columnar structure with the columns mostly extending through the whole thickness of the film. The average width of the column grains is around 130 nm, much larger than the layer thickness. Fig. 2 shows the high-angle symmetrical X-ray spectra for Cu/Nb multilayers in the 2θ range of 30–65°. One can see that only the peaks of Nb (110) and Cu (111) are rather strong, and the other diffraction peaks are quite weak. The results indicate that all the multilayers are polycrystalline with strong Nb (110) and Cu (111) out of plane texture. As the layer thickness decreases to 4 nm, some additional satellite peaks are observed to flank the principal reflections and their position agrees well with the periodicity predicted by the deposition rate calibrations. Meanwhile, the peaks of the Cu (111) and Nb (110) begin to shift to a low-angle. The appearance of satellite peaks is a sign of the formation of superlattice structure and it has been proved that
Fig. 2. The XRD patterns for all the Cu/Nb multilayers.
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plateau values before the indentation depth exceeds 200 nm and no apparent variation is observed after that. As the thickness of Cu/Nb multilayers is only about 400 nm, the substrate effect should be considered firstly. Generally speaking, for soft films on hard substrates systems, such as Cu/Nb multilayers deposited on silicon substrate, because the plastic deformation is expected to occur only in the film when the penetration depth is less than the film thickness, substrate should have insignificant effects on the hardness measurement [29]. Nanoindentation with different penetration depths of 100, 150, 200, 250, 300, 350, 400 nm were carried out on the multilayer with h = 2.5 nm to further clarify the substrate effects. The corresponding maximum load (P) and indentation size (d) were recorded to make a Meyer plot [30], that is, a log–log plot of indentation size vs. applied load, as shown in Fig. 4. It is well known that a plot of log P vs. log d for measurements on a bulk material generally yields a straight line, and a deviation from the straight line occurs at the indentation depth where the substrate begins to influence the indentation process [30]. It can be seen in Fig. 4 that logP and logd yield a straight line relationship before the indentation depth exceeds 400 nm and the slope keeps constant with the increasing load. It verifies that the plastic deformation during the penetration is confined to the film itself until the indenter reaches the film/substrate interface, thus the substrate effects can be ruled out in the hardness measurement. Therefore, the plateau value of 8.84 GPa in the hardness curves can be regarded as the intrinsic hardness value for the Cu/Nb multilayer with h = 2.5 nm. As for the modulus measurement, it is more complex
Fig. 3. (a) Hardness and (b) modulus as a function of indentation depth of the Cu/Nb multilayer with h = 2.5 nm for ten different continuous stiffness indentation tests. The plateau values present the intrinsic hardness or modulus of this specimen.
Fig. 4. The Meyer plot for nanoindentation of the multilayer with h = 2.5 nm at indentation depth from 100 to 400 nm.
because the elastic deformation will extend into the whole content including the film and the substrate. However, as shown in Fig. 3(b), the modulus values almost keep constant with the increasing indentation depth, indicating that the subtract effect on the modulus measurement is negligible. Thereby we can take the plateau value of 152 GPa as the intrinsic modulus for this specimen. The modulus values of the other Cu/Nb multilayers with different layer thickness do not vary significant. For comparison, the hardness and modulus of monolithic Cu film and Nb film were also measured by nanoindentation at LSR of 0.05 s−1, and they are 2.86 GPa and 126 GPa for Cu film and 5.57 GPa and 113 GPa for Nb film. Fig. 5 shows the strength of Cu/Nb multilayers plotted as a function of layer thickness. The strength is estimated as nanoindentation hardness divided by a factor of 2.7. The strength (also estimated as H/ 2.7) of the Cu/Nb multilayers reported by Misra et al. [4] is also shown in Fig. 5 for comparison. The discrepancy will be discussed later. An obvious increase in the strength with decreasing layer thickness can be observed and peak flow stress of 3.27 GPa is achieved at the smallest layer thickness of 2.5 nm. All multilayers exhibit flow stress greater than the rule of mixture value σROM = (σCu + σNb)/2 = 1.56 GPa as shown on the intercept in Fig. 5. It is accepted that when the layer thickness decreases down to nanometer, the plastic deformation is controlled by single dislocations slip in confined layer and the
Fig. 5. The flow strength as a function of the layer thickness. The curve calculated by the CLS model gives a good fit to the data at 20 nm ≤ h ≤ 6.5 nm. The flow strength reported by Misra et al. is also plotted for comparison.
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strength values in this scale region can be predicted by a confined layer slip (CLS) model proposed by Misra et al. [4]. The CLS stress is express as: σ cls = M
μb 4−ν αh′ f μb ln − + h ð1−νÞλ b 8πh′ 1−ν
ð1Þ
The first term on the right-hand side of the equation is modified Orowan stress, representing a structure-induced layer thickness dependent hardening, where M is Taylor factor of 3.1, μ is the shear modulus, b is the length of the Burgers vector, ν is the Poisson ratio, α is the core cut-off parameter and h′ is the layer thickness measured parallel to the glide plane. The second term f/h presents interface stress which stems from elastic deformation of the interfacial region and is defined as the gradient of the interfaces energy with respect to strain [27]. As indicated by the XRD patterns, significant compressive stress only exists at low layer thickness. It means that at this range the interfacial region is effectively pre-stressed and would assist the applied stress to cause yielding via the CLS mechanism. The last term presents strain hardening which, at room temperature, can be estimated as C = μb/(1 − ν). Here we use Eq. (1) to calculate the CLS stress for comparison with experimentally measured strength. According to the extent of core spreading revealed in atomistic modeling [4], we take the core cut-off parameter α as 0.2 for Cu/Nb multilayers that approximates to a core cut-off dimension of five times the Burgers vector magnitude. Interface stress (f) is estimated by a typical value f ≈ 2 J/m2 [4]. The spacing of the interface dislocation array (λ) is unknown and is used as a fitting parameter. As the flow strength of all the multilayers is larger than that of Cu film or Nb film, yielding in Cu/Nb multilayers is expected to occur in both Cu and Nb layers. Thus, substituting μ = 43 GPa, b = 0.28 nm, ν = 0.37 that derived from a rule-ofmixtures calculation for Cu and Nb, we obtain the fitting results shown in Fig. 5 with λ = 13.5 nm. That is to say, dislocation content at the interface can be built up by CLS with an average spacing of 13.5 nm as each gliding loop deposits one dislocation segment at the two interfaces, which contributes to strain hardening. While the CLS model fit the experimental data quite well, we notice that at layer thickness down to 4 nm, the CLS stress tends to decrease with decreasing layer thickness. This results from the fact that the equation used to calculate the dislocation self-energy is not valid when the layer thickness is close to the core cut-off dimension [4]. Since Eq. (1) cannot be used to calculate strength values in this range, we extrapolate the model calculation at larger h to smaller h (shown with a dashed line in Fig. 5) and find that the CLS model overestimates the strength and turns out to be invalid in this scale region. In fact, a number of researches have confirmed that as layer thickness decreases below approximately 5 nm, CLS stress for confined layer slip exceeds the stress for single dislocation transmission across the interface and crossing of dislocations across interfaces becomes the dominant deformation mechanism [4,31,32]. At the same time, the strength reaches a plateau, which is determined by the strengthening of interface barrier and can be predicted by atomistic simulations. Wang et al. have predicted that incoherent interfaces in the Cu/Nb system are very strong barriers and a single mixed dislocation, from either Cu or Nb, cannot cross the interface until the RSS is increased in excess of 1.1 GPa [9]. Multiplied by the Taylor factor of 3.1, the RSS leads to ultrahigh yield strength above 3.41 GPa in Cu/Nb multilayers. The maximum flow strength of 3.27 we obtain at h = 2.5 nm agrees quite well with the atomic modeling prediction and proves directly in the experiment that extremely high strength value exceeding 3 GPa can be achieved at small length scale in such system. The indent morphology may also throw light on the understanding of the length-scale-dependent deformation mechanisms in Cu/Nb multilayers. Fig. 6 is the typical SEM images of the indents taken after hardness measurement at LSR of 0.05 s−1. Well-shaped circular shear
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bands (SBs) preferentially occurred both around the indent and on the indent surface in the multilayer with h = 2.5 nm, as shown in Fig. 6(a). Only a few SBs occurred around the indent in the multilayer with h = 4 nm, as shown in Fig. 6(b). In the multilayers with h ≥ 6.5 nm, no SBs are observed around the indent or on the indent surface, as indicated by a typical indent of the multilayer with h = 6.5 nm in Fig. 6(c). The formation of shear bands is usually related with grain boundary mediated deformation. However, because grain boundary sliding is always accompanied with a drop in strength, the continuous increase in the flow strength till h = 2.5 nm denies the grain boundary mediated deformation mechanism. Here, the formation of shear bands at small layer thickness is attributed to the different deformation mechanisms controlling in the deformation process. As we discussed earlier, at h ≥ 6.5 nm, deformation is controlled by single dislocations slip in confined layers, during which dislocations bow out leaving segments on the interfaces. Hence, both layers can deform homogeneously and no SBs form. While at h b 6.5 nm, crossing of single dislocations across interfaces operates in deformation process. Because only one set of slip planes in Cu/Nb multilayers with Cu{111}∥Nb{110} has a common trace of intersection in the interface plane and the other slip planes are discontinuity, dislocation lines are not parallel when single dislocations enter the interfaces from Cu layers or Nb layers except for the particular slip system [9]. As a result, the crossing of single dislocations is difficult except in the particular slip system. Two aspects may cause the formation of shear bands. First, when the indent tip penetrates the multilayers, the stress concentration can rotate the layers and make the particular slip system turn to an easy shear direction after the adequate amount of rotation has taken place. As a result, the shear instability will suddenly occur when the RSS is greater than or equal to the critical RSS required for the transmission in that appropriate plane. Second, as predicted by Wang et al. through atomic simulations, lattice dislocations can enter the interfaces, including those with discontinuity slip planes, spread and rearrange by climb process [14], and finally dislocation debris scattered in interfaces can reassemble into lattice glide dislocations, facilitating slip transmission. As a result, dislocation production and storage occurs predominantly at the interfaces, and shear transfer across the interfaces unsteadily. We notice that the amount of SBs increases as the layer thickness decreases from 4 to 2.5 nm. It implies that at h = 4 nm, the mechanism of dislocations slip in confined layers may still operate in the deformation process, although crossing of dislocations across interfaces has already become an important deformation mechanism. While at h = 2.5 nm, slip transmission through interfaces has dominated the deformation process thoroughly. Additionally, comparing our results with the strength of Cu/Nb multilayers reported by Misra et al., we find that the flow strength we attain here is almost 30% larger than the values obtained by Misra et al. at a constant displacement rate of 2 nm/s [4], as shown in Fig. 5. The results reported by Misra et al. were also explained well by the CLS model and the fitting parameters used in their experiment are quite similar to ours except for the spacing λ of interfacial dislocations that was introduced by CLS. A fitting parameter λ of 15 nm was given by Misra et al., larger than that we get in our fitting. This means that a stronger strain hardening term is obtained in our experiment, which may be ascribed to higher LSR we used in the tests. For depth-sensing self-similar indentation, the effective strain rate in our tests can be calculated as ˙εeff ≈˙h = h = 0:5˙P = P = 2:5 × 10−2 s−1 . Although the strain rate used by Misra et al. was not reported, another micropillar compression test on 5 nm Cu/Nb multilayer performed by Mara et al. at a constant displacement rate of 1.7 nm/s gave an initial strain rate of 2 × 10−4 s−1 [11]. The strength that obtained in the micropillar compression testing reached 2.4 GPa, agreeing well with the value reported by Misra et al. Thus, we presume that the strain rate that Misra et al. used in the nanoindentation tests must on the order of 10−4, two order magnitude lower than what we used in our measurement. As the strain hardening process is attributed to interfacial dislocation– dislocation interactions during CLS, where dislocation climb is involved
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Fig. 6. SEM images of the indents taken after CSM hardness measurement for the multilayers (a) h = 2.5 nm, (b) h = 4 nm and (c) h = 6.5 nm.
in at room temperature [14,33,34], it is temperature and strain-rate dependent and can be enhanced by higher strain rate or lower temperature. In order to further confirm the dependence of flow strength on strain rate for Cu/Nb multilayers, we measure the hardness of the multilayer with h=10 nm at different LSR from 0.01 s−1 to 0.5 s−1, and present the plots of the flow stress vs. LSR in Fig. 7(a). As we expect, the flow stress rises apparently with increasing LSR. When LSR increases from 0.01 s−1 to 0.5 s−1, the flow stress augments from 2.64 GPa to 3.27 GPa. Using atomistic simulations, Wang et al. have revealed that multiple dislocations that deposited at the interfaces can dissociate into interfacial dislocations
through dislocation climb process [14]. As a consequence, reactions between interfacial dislocations assisted by climb could lead to the annihilation of dislocation content (recovery). That is to say, when layers deform at faster LSR, this process provides insufficient time for the dislocation annihilation and lead to larger strain hardening. At slower LSR, in contrast, dislocation climb process is easier to operate, resulting in less strain hardening. Therefore, it is confirmed that the ultrahigh strength that we obtain in the Cu/Nb multilayers is partly due to larger strain hardening that was caused by high strain rate. Elevating temperature can make the same effect as reducing LRS because thermal activation also enables increased dislocation climb. The high-temperature mechanical behavior of Cu/Nb multilayer with h=60 nm has verified that the contribution of strain hardening to strength value increases with decreasing of the temperatures tested [35]. The strain-rate sensitivity (SRS) of the flow stress, which is defined as m = ∂ lnσ = ∂ ln˙ε [36], is calculated to be 0.056 from the typical double logarithmic curve of flow strength vs. effective strain rate, as shown in Fig. 7(b). For comparison, the dependence of flow stress on strain rate for Cu and Nb thin films is also measured and presented in Fig. 7(b). The SRS is 0.091 for Cu film (the out-of plane grain size determined by XRD analysis is 35 nm) and 0.057 for Nb film (the out-of plane grain size determined by XRD analysis is 9 nm), the value for Cu film with nanoscale grain size is consistent well with the magnitude and general trend of mCu that is reported in Ref. [37,38]. It is interesting to notice that the SRS of the Cu/Nb multilayer with h=10 nm almost equals to that of the Nb film. It may be rationalized as follows: The diffusion of the Cu atoms in the interfaces is much faster than that of the Nb atoms, thus the rate of climb in the Cu layers will be much larger than that in the Nb layers [13,14,39]. Therefore, the resistive force that is caused by increasing strain is easier to be released by dislocation annihilation process assisted by climb in the Cu layers, although the yielding in the multilayer is supposed to be controlled by dislocation glide in both the Cu and Nb layers. As a result, the majority of the extra load that required overcoming strain hardening is taken up by the Nb layers, and the rate-controlling deformation in the Cu/Nb multilayer is mostly controlled by the Nb layers and exhibits similar characteristic as the Nb film. However, when the layer thickness decreases to 6.5 nm where the deformation mechanism will change to crossing of single dislocations across interfaces, as dislocation climb process dominates the accommodation process of slip transmission, variation in SRS with the decreasing layer thickness is another interesting issue and will be explored experimentally in our future work. 4. Conclusions
Fig. 7. (a) Variation in flow strength of the multilayer with h = 10 nm with varying LSR. (b) Flow stress–effective strain rate log–log plots for the multilayer with h = 10 nm.
In summary, Cu/Nb multilayers with a good compositionally modulated structure were prepared by evaporation deposition. Their microstructure and mechanical properties were investigated using XRD, TEM, SEM and nanoindentation. The results show that the strength values obey the CLS model as the layer thickness decreases
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from 20 to 6.5 nm. After that, ultrahigh strength value of 3.27 GPa is attained at h = 2.5 nm, which is consistent well with the prediction made by atomistic simulations based on the crossing of dislocations across interfaces mechanism in an incoherent system. There are SBs formed around the indents for the multilayers with h ≤ 6.5 nm, which is ascribed to shear instability that occurs when the mechanism of crossing of dislocations across interfaces operates in the deformation process. These results imply that there is a transition of the deformation mechanism at h = 6.5 nm. Furthermore, the varying LSR hardness measurement confirms that the hardness values of the multilayers increase with increasing LSR, which is ascribed to the enhanced strain hardening caused by higher LSR. Acknowledgements The authors are grateful for the financial support from the National Natural Science Foundation of China (Grant Nos. 50871060 and 50772055) and National ‘863’ High-tech project of China (Grant No. 2007AA03Z426). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t s f
Microstructure and ultrahigh strength of nanoscale Cu/Nb multilayers X.Y. Zhu a,b,⁎, J.T. Luo a, F. Zeng a, F. Pan a,⁎ a b
Laboratory of Advanced Materials, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People's Republic of China National Key Lab for Remanufacturing, Academy of Armored Forces Engineering, Beijing 100072, People's Republic of China
a r t i c l e
i n f o
Available online 25 January 2011 Keywords: Cu/Nb multilayers Nanoindentation Microstructure Strength Strain hardening
a b s t r a c t The microstructure and mechanical properties of Cu/Nb multilayers were investigated by X-ray diffraction, transmission electron microscopy, scanning electron microscopy and nanoindentation. Ultrahigh strength of 3.27 GPa is achieved at the smallest layer thickness of 2.5 nm, which agrees well with the theoretical prediction based on the deformation mechanism of crossing of dislocations across interfaces. After that, the strength decreases with the increasing layer thickness and the transition of the deformation mechanism to confined layer slip occurs at the layer thickness of 6.5 nm. Additionally, strength of the Cu/Nb multilayers increases with increasing loading strain rate because of enhanced strain hardening. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Metallic multilayers have attracted extensive attention because of their promising mechanical properties and significant theoretical interest over the past couple of decades [1–3]. It is generally accepted that for layer thickness (h) above approximately 5 nm, the flow strength of multilayers is dependent on the individual layer thickness and several deformation mechanisms have been proposed to describe the plastic deformation in this size scale [4]. While for layer thickness below 5 nm, where the deformation mechanism is thought to be transmission of single glide dislocations across interfaces, the flow strength is independent on layer thickness and the maximum value is determined by interfacial properties [5]. Hoagland et al. have proved that for different interface types, such as coherent interfaces and incoherent interfaces, the factors that affect the resistance to dislocation slip transmission are quite different [6]. For coherent interfaces, such as interfaces in Cu/Ni system, the most important effect on strength derives from the coherency strains and the maximum attainable flow strength should be equal to the coherency stress [7], which has been confirmed by experimental data later [8]. On the other hand, for incoherent interfaces, such as interfaces in Cu/ Nb system, the case is more complex and atomistic simulations are applied to understand the properties of the interfaces and estimate the maximum attainable flow strength. Recently, Wang et al. have investigated the interaction of glide dislocations with Cu/Nb interfaces and found that a single mixed dislocation, from either Cu or Nb, cannot cross the interface until the resolved shear stress (RSS) is
⁎ Corresponding authors at: Laboratory of Advanced Materials, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People's Republic of China. Tel.: +86 10 62772907; fax: +86 10 62771160. E-mail addresses: [email protected] (X.Y. Zhu), [email protected] (F. Pan). 0040-6090/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2010.12.251
increased to extremely high values in excess of 1.1 GPa [9]. This result indicates that the attainable strength maximum in Cu/Nb system should exceed 3.4 GPa. However, such maximum strength or hardness has not yet been observed experimentally, although a lot of researches have been made on the room temperature strength of the Cu/Nb system using nanoindentation or micropillar compression [4,10–13]. The discrepancy between the model prediction and experimental data may be due to the sensitivity of the strength to strain rate. As revealed by atomic modeling, dislocation climb is involved in the process of slip transmission even at room temperature, and reactions between interfacial dislocations assisted by climb could lead to annihilation of dislocation content, resulting in less strain hardening [14]. The room temperature strength values of Cu/Nb multilayers that have been reported were frequently tested at strain rate on the order of 10−4 s−1, quite smaller than the strain rate that usually used in atomistic simulations. This lower strain rate may promote the dislocation climb process, release strain hardening, and then reduce the strength value. In fact, the dependence of yielding strength on the strain rate has been observed in several nanocrystalline materials [15,16]. Although there is no clear understanding of the deformation mechanism, it is postulated that the confined nanoscale grain structure plays an important role in this peculiar phenomenon. Herein we investigate the room temperature strength of Cu/Nb system using nanoindentation with a constant loading strain rate (LSR) of 0.05 s−1 and discuss the length-scale deformation mechanism at such strain rate. Moreover, varying loading strain rate tests are carried out to further investigate the dependence of the strength on strain rate. The experiment results confirm that ultrahigh strength over 3 GPa can be achieved in Cu/Nb multilayers with small layer thickness at relative high strain rate and room temperature dislocation climb is involved in the plastic deformation process. The mechanical properties of Cu/Nb mulitlayers have been intensively studied because of their remarkably high strength, fatigue resistance
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[17] and thermal stability [18]. Due to higher He solubility than either Cu or Nb and ability to prevent the degradation of material properties by the growth of bubbles, Cu/Nb multilayers are markedly more resistant to radiation damage, and thus can be applied to extend the operating limits and lifetimes of nuclear reactors [19–21]. 2. Experimental details Cu/Nb multilayers consisting of alternating equivalent Cu and Nb individual layers with h ranging from 2.5 to 20 nm were prepared at room temperature by alternate electron beam evaporation deposition using an ultrahigh vacuum (UHV) chamber. The base vacuum of the chamber was 5 × 10−6 Pa. The deposition rates were 0.8 Å/s for both Cu and Nb. The nominal thickness of the individual Cu and Nb layers was monitored by an in situ quartz oscillator. All the multilayers were deposited onto Si (100) wafers with native oxide. The deposition began with Cu layer and ended with Nb layer. The total thickness of multilayers was approximately 400 nm. Monolithic Cu film and Nb film of 1 μm were also deposited onto Si (100) in the same condition for comparison. The X-ray diffraction (XRD) experiment was carried out using Rigaku D/max-RB X-ray diffractometer with Cu Kα radiation. Microstructure observation was performed by a JEM2011 transmission electron microscopy (TEM) with 200 kV accelerating voltages. The mechanical properties of the multilayers were investigated by a Nano Indenter XP (MTS Systems Corp.) with a displacement resolution of 0.01 nm and a loading resolution of 50 nN. A Berkovich indenter, a three-sided pyramid with the same area-to-depth function as that of a Vickers indenter, was used in all the experiments. The hardness and modulus of the multilayers were measured by a continuous-stiffness measurement (CSM) technique with a constant LSR of 0.05 s−1. Varying LSR nanoindentation tests are carried out on the multilayer with h = 10 nm with LSR ranging from 0.01 to 0.5 s−1. A frequency of 45 Hz was used to avoid the sensitivity to thermal drift. The maximum indentation depth was 400 nm. More details about this technique and analysis method can be found in references [22–24]. Ten indents were performed on each specimen and an average value is reported with error bars indicating the range. For reference, the mechanical properties of the Cu film and Nb film were also examined under the same experimental condition. After indentation, the indent images of hardness measurement at LRS = 0.05 s−1 were characterized by field-emission scanning electron microscopy (SEM).
Fig. 1. The cross-section transmission electron micrographs taken in bright field of the multilayer with h = 20 nm.
superlattice structure can be produced in ultrathin layers of two dissimilar multilayers [25]. In fact, X-ray pole figure measurements have confirmed that epitaxial relationship of Kurdjumov–Sachs orientation Cu{111}//Nb{110} and Cu b 110N//Nb b 111 N exits in the Cu/Nb interfaces in ultra thin layer thickness [26]. Due to the large lattice mismatch of ~11%, misfit dislocations will be generated in the interfaces to accommodate the elastic strain, resulting in a compressive contribution to the interface stress [27,28]. Owing to the Poisson effect, the spacing perpendicular to the interfaces will expand, resulting in the shift of the diffraction peaks of Cu (111) and Nb (110) to a low-angle. 3.2. Mechanical properties The nanoindentation data were measured continuously during the loading of the indenter by the CSM method, and then they were analyzed by the Oliver and Pharr method [22]. Fig. 3(a) and (b) show the variations in hardness and modulus with indentation depth of the Cu/Nb multilayer with h = 2.5 nm for ten different continuous stiffness indentation tests, respectively. Both the curves approach
3. Results and discussion 3.1. Microstructure Cross-section transmission electron microscopy was performed to confirm the layered structure of the Cu/Nb multilayers. Fig. 1 shows the bright field image of the multilayer with h = 20 nm. Discrete layered structure with sharp planar interfaces between the two phases is clearly visible. Meanwhile, we can see that the multilayer show columnar structure with the columns mostly extending through the whole thickness of the film. The average width of the column grains is around 130 nm, much larger than the layer thickness. Fig. 2 shows the high-angle symmetrical X-ray spectra for Cu/Nb multilayers in the 2θ range of 30–65°. One can see that only the peaks of Nb (110) and Cu (111) are rather strong, and the other diffraction peaks are quite weak. The results indicate that all the multilayers are polycrystalline with strong Nb (110) and Cu (111) out of plane texture. As the layer thickness decreases to 4 nm, some additional satellite peaks are observed to flank the principal reflections and their position agrees well with the periodicity predicted by the deposition rate calibrations. Meanwhile, the peaks of the Cu (111) and Nb (110) begin to shift to a low-angle. The appearance of satellite peaks is a sign of the formation of superlattice structure and it has been proved that
Fig. 2. The XRD patterns for all the Cu/Nb multilayers.
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plateau values before the indentation depth exceeds 200 nm and no apparent variation is observed after that. As the thickness of Cu/Nb multilayers is only about 400 nm, the substrate effect should be considered firstly. Generally speaking, for soft films on hard substrates systems, such as Cu/Nb multilayers deposited on silicon substrate, because the plastic deformation is expected to occur only in the film when the penetration depth is less than the film thickness, substrate should have insignificant effects on the hardness measurement [29]. Nanoindentation with different penetration depths of 100, 150, 200, 250, 300, 350, 400 nm were carried out on the multilayer with h = 2.5 nm to further clarify the substrate effects. The corresponding maximum load (P) and indentation size (d) were recorded to make a Meyer plot [30], that is, a log–log plot of indentation size vs. applied load, as shown in Fig. 4. It is well known that a plot of log P vs. log d for measurements on a bulk material generally yields a straight line, and a deviation from the straight line occurs at the indentation depth where the substrate begins to influence the indentation process [30]. It can be seen in Fig. 4 that logP and logd yield a straight line relationship before the indentation depth exceeds 400 nm and the slope keeps constant with the increasing load. It verifies that the plastic deformation during the penetration is confined to the film itself until the indenter reaches the film/substrate interface, thus the substrate effects can be ruled out in the hardness measurement. Therefore, the plateau value of 8.84 GPa in the hardness curves can be regarded as the intrinsic hardness value for the Cu/Nb multilayer with h = 2.5 nm. As for the modulus measurement, it is more complex
Fig. 3. (a) Hardness and (b) modulus as a function of indentation depth of the Cu/Nb multilayer with h = 2.5 nm for ten different continuous stiffness indentation tests. The plateau values present the intrinsic hardness or modulus of this specimen.
Fig. 4. The Meyer plot for nanoindentation of the multilayer with h = 2.5 nm at indentation depth from 100 to 400 nm.
because the elastic deformation will extend into the whole content including the film and the substrate. However, as shown in Fig. 3(b), the modulus values almost keep constant with the increasing indentation depth, indicating that the subtract effect on the modulus measurement is negligible. Thereby we can take the plateau value of 152 GPa as the intrinsic modulus for this specimen. The modulus values of the other Cu/Nb multilayers with different layer thickness do not vary significant. For comparison, the hardness and modulus of monolithic Cu film and Nb film were also measured by nanoindentation at LSR of 0.05 s−1, and they are 2.86 GPa and 126 GPa for Cu film and 5.57 GPa and 113 GPa for Nb film. Fig. 5 shows the strength of Cu/Nb multilayers plotted as a function of layer thickness. The strength is estimated as nanoindentation hardness divided by a factor of 2.7. The strength (also estimated as H/ 2.7) of the Cu/Nb multilayers reported by Misra et al. [4] is also shown in Fig. 5 for comparison. The discrepancy will be discussed later. An obvious increase in the strength with decreasing layer thickness can be observed and peak flow stress of 3.27 GPa is achieved at the smallest layer thickness of 2.5 nm. All multilayers exhibit flow stress greater than the rule of mixture value σROM = (σCu + σNb)/2 = 1.56 GPa as shown on the intercept in Fig. 5. It is accepted that when the layer thickness decreases down to nanometer, the plastic deformation is controlled by single dislocations slip in confined layer and the
Fig. 5. The flow strength as a function of the layer thickness. The curve calculated by the CLS model gives a good fit to the data at 20 nm ≤ h ≤ 6.5 nm. The flow strength reported by Misra et al. is also plotted for comparison.
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strength values in this scale region can be predicted by a confined layer slip (CLS) model proposed by Misra et al. [4]. The CLS stress is express as: σ cls = M
μb 4−ν αh′ f μb ln − + h ð1−νÞλ b 8πh′ 1−ν
ð1Þ
The first term on the right-hand side of the equation is modified Orowan stress, representing a structure-induced layer thickness dependent hardening, where M is Taylor factor of 3.1, μ is the shear modulus, b is the length of the Burgers vector, ν is the Poisson ratio, α is the core cut-off parameter and h′ is the layer thickness measured parallel to the glide plane. The second term f/h presents interface stress which stems from elastic deformation of the interfacial region and is defined as the gradient of the interfaces energy with respect to strain [27]. As indicated by the XRD patterns, significant compressive stress only exists at low layer thickness. It means that at this range the interfacial region is effectively pre-stressed and would assist the applied stress to cause yielding via the CLS mechanism. The last term presents strain hardening which, at room temperature, can be estimated as C = μb/(1 − ν). Here we use Eq. (1) to calculate the CLS stress for comparison with experimentally measured strength. According to the extent of core spreading revealed in atomistic modeling [4], we take the core cut-off parameter α as 0.2 for Cu/Nb multilayers that approximates to a core cut-off dimension of five times the Burgers vector magnitude. Interface stress (f) is estimated by a typical value f ≈ 2 J/m2 [4]. The spacing of the interface dislocation array (λ) is unknown and is used as a fitting parameter. As the flow strength of all the multilayers is larger than that of Cu film or Nb film, yielding in Cu/Nb multilayers is expected to occur in both Cu and Nb layers. Thus, substituting μ = 43 GPa, b = 0.28 nm, ν = 0.37 that derived from a rule-ofmixtures calculation for Cu and Nb, we obtain the fitting results shown in Fig. 5 with λ = 13.5 nm. That is to say, dislocation content at the interface can be built up by CLS with an average spacing of 13.5 nm as each gliding loop deposits one dislocation segment at the two interfaces, which contributes to strain hardening. While the CLS model fit the experimental data quite well, we notice that at layer thickness down to 4 nm, the CLS stress tends to decrease with decreasing layer thickness. This results from the fact that the equation used to calculate the dislocation self-energy is not valid when the layer thickness is close to the core cut-off dimension [4]. Since Eq. (1) cannot be used to calculate strength values in this range, we extrapolate the model calculation at larger h to smaller h (shown with a dashed line in Fig. 5) and find that the CLS model overestimates the strength and turns out to be invalid in this scale region. In fact, a number of researches have confirmed that as layer thickness decreases below approximately 5 nm, CLS stress for confined layer slip exceeds the stress for single dislocation transmission across the interface and crossing of dislocations across interfaces becomes the dominant deformation mechanism [4,31,32]. At the same time, the strength reaches a plateau, which is determined by the strengthening of interface barrier and can be predicted by atomistic simulations. Wang et al. have predicted that incoherent interfaces in the Cu/Nb system are very strong barriers and a single mixed dislocation, from either Cu or Nb, cannot cross the interface until the RSS is increased in excess of 1.1 GPa [9]. Multiplied by the Taylor factor of 3.1, the RSS leads to ultrahigh yield strength above 3.41 GPa in Cu/Nb multilayers. The maximum flow strength of 3.27 we obtain at h = 2.5 nm agrees quite well with the atomic modeling prediction and proves directly in the experiment that extremely high strength value exceeding 3 GPa can be achieved at small length scale in such system. The indent morphology may also throw light on the understanding of the length-scale-dependent deformation mechanisms in Cu/Nb multilayers. Fig. 6 is the typical SEM images of the indents taken after hardness measurement at LSR of 0.05 s−1. Well-shaped circular shear
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bands (SBs) preferentially occurred both around the indent and on the indent surface in the multilayer with h = 2.5 nm, as shown in Fig. 6(a). Only a few SBs occurred around the indent in the multilayer with h = 4 nm, as shown in Fig. 6(b). In the multilayers with h ≥ 6.5 nm, no SBs are observed around the indent or on the indent surface, as indicated by a typical indent of the multilayer with h = 6.5 nm in Fig. 6(c). The formation of shear bands is usually related with grain boundary mediated deformation. However, because grain boundary sliding is always accompanied with a drop in strength, the continuous increase in the flow strength till h = 2.5 nm denies the grain boundary mediated deformation mechanism. Here, the formation of shear bands at small layer thickness is attributed to the different deformation mechanisms controlling in the deformation process. As we discussed earlier, at h ≥ 6.5 nm, deformation is controlled by single dislocations slip in confined layers, during which dislocations bow out leaving segments on the interfaces. Hence, both layers can deform homogeneously and no SBs form. While at h b 6.5 nm, crossing of single dislocations across interfaces operates in deformation process. Because only one set of slip planes in Cu/Nb multilayers with Cu{111}∥Nb{110} has a common trace of intersection in the interface plane and the other slip planes are discontinuity, dislocation lines are not parallel when single dislocations enter the interfaces from Cu layers or Nb layers except for the particular slip system [9]. As a result, the crossing of single dislocations is difficult except in the particular slip system. Two aspects may cause the formation of shear bands. First, when the indent tip penetrates the multilayers, the stress concentration can rotate the layers and make the particular slip system turn to an easy shear direction after the adequate amount of rotation has taken place. As a result, the shear instability will suddenly occur when the RSS is greater than or equal to the critical RSS required for the transmission in that appropriate plane. Second, as predicted by Wang et al. through atomic simulations, lattice dislocations can enter the interfaces, including those with discontinuity slip planes, spread and rearrange by climb process [14], and finally dislocation debris scattered in interfaces can reassemble into lattice glide dislocations, facilitating slip transmission. As a result, dislocation production and storage occurs predominantly at the interfaces, and shear transfer across the interfaces unsteadily. We notice that the amount of SBs increases as the layer thickness decreases from 4 to 2.5 nm. It implies that at h = 4 nm, the mechanism of dislocations slip in confined layers may still operate in the deformation process, although crossing of dislocations across interfaces has already become an important deformation mechanism. While at h = 2.5 nm, slip transmission through interfaces has dominated the deformation process thoroughly. Additionally, comparing our results with the strength of Cu/Nb multilayers reported by Misra et al., we find that the flow strength we attain here is almost 30% larger than the values obtained by Misra et al. at a constant displacement rate of 2 nm/s [4], as shown in Fig. 5. The results reported by Misra et al. were also explained well by the CLS model and the fitting parameters used in their experiment are quite similar to ours except for the spacing λ of interfacial dislocations that was introduced by CLS. A fitting parameter λ of 15 nm was given by Misra et al., larger than that we get in our fitting. This means that a stronger strain hardening term is obtained in our experiment, which may be ascribed to higher LSR we used in the tests. For depth-sensing self-similar indentation, the effective strain rate in our tests can be calculated as ˙εeff ≈˙h = h = 0:5˙P = P = 2:5 × 10−2 s−1 . Although the strain rate used by Misra et al. was not reported, another micropillar compression test on 5 nm Cu/Nb multilayer performed by Mara et al. at a constant displacement rate of 1.7 nm/s gave an initial strain rate of 2 × 10−4 s−1 [11]. The strength that obtained in the micropillar compression testing reached 2.4 GPa, agreeing well with the value reported by Misra et al. Thus, we presume that the strain rate that Misra et al. used in the nanoindentation tests must on the order of 10−4, two order magnitude lower than what we used in our measurement. As the strain hardening process is attributed to interfacial dislocation– dislocation interactions during CLS, where dislocation climb is involved
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Fig. 6. SEM images of the indents taken after CSM hardness measurement for the multilayers (a) h = 2.5 nm, (b) h = 4 nm and (c) h = 6.5 nm.
in at room temperature [14,33,34], it is temperature and strain-rate dependent and can be enhanced by higher strain rate or lower temperature. In order to further confirm the dependence of flow strength on strain rate for Cu/Nb multilayers, we measure the hardness of the multilayer with h=10 nm at different LSR from 0.01 s−1 to 0.5 s−1, and present the plots of the flow stress vs. LSR in Fig. 7(a). As we expect, the flow stress rises apparently with increasing LSR. When LSR increases from 0.01 s−1 to 0.5 s−1, the flow stress augments from 2.64 GPa to 3.27 GPa. Using atomistic simulations, Wang et al. have revealed that multiple dislocations that deposited at the interfaces can dissociate into interfacial dislocations
through dislocation climb process [14]. As a consequence, reactions between interfacial dislocations assisted by climb could lead to the annihilation of dislocation content (recovery). That is to say, when layers deform at faster LSR, this process provides insufficient time for the dislocation annihilation and lead to larger strain hardening. At slower LSR, in contrast, dislocation climb process is easier to operate, resulting in less strain hardening. Therefore, it is confirmed that the ultrahigh strength that we obtain in the Cu/Nb multilayers is partly due to larger strain hardening that was caused by high strain rate. Elevating temperature can make the same effect as reducing LRS because thermal activation also enables increased dislocation climb. The high-temperature mechanical behavior of Cu/Nb multilayer with h=60 nm has verified that the contribution of strain hardening to strength value increases with decreasing of the temperatures tested [35]. The strain-rate sensitivity (SRS) of the flow stress, which is defined as m = ∂ lnσ = ∂ ln˙ε [36], is calculated to be 0.056 from the typical double logarithmic curve of flow strength vs. effective strain rate, as shown in Fig. 7(b). For comparison, the dependence of flow stress on strain rate for Cu and Nb thin films is also measured and presented in Fig. 7(b). The SRS is 0.091 for Cu film (the out-of plane grain size determined by XRD analysis is 35 nm) and 0.057 for Nb film (the out-of plane grain size determined by XRD analysis is 9 nm), the value for Cu film with nanoscale grain size is consistent well with the magnitude and general trend of mCu that is reported in Ref. [37,38]. It is interesting to notice that the SRS of the Cu/Nb multilayer with h=10 nm almost equals to that of the Nb film. It may be rationalized as follows: The diffusion of the Cu atoms in the interfaces is much faster than that of the Nb atoms, thus the rate of climb in the Cu layers will be much larger than that in the Nb layers [13,14,39]. Therefore, the resistive force that is caused by increasing strain is easier to be released by dislocation annihilation process assisted by climb in the Cu layers, although the yielding in the multilayer is supposed to be controlled by dislocation glide in both the Cu and Nb layers. As a result, the majority of the extra load that required overcoming strain hardening is taken up by the Nb layers, and the rate-controlling deformation in the Cu/Nb multilayer is mostly controlled by the Nb layers and exhibits similar characteristic as the Nb film. However, when the layer thickness decreases to 6.5 nm where the deformation mechanism will change to crossing of single dislocations across interfaces, as dislocation climb process dominates the accommodation process of slip transmission, variation in SRS with the decreasing layer thickness is another interesting issue and will be explored experimentally in our future work. 4. Conclusions
Fig. 7. (a) Variation in flow strength of the multilayer with h = 10 nm with varying LSR. (b) Flow stress–effective strain rate log–log plots for the multilayer with h = 10 nm.
In summary, Cu/Nb multilayers with a good compositionally modulated structure were prepared by evaporation deposition. Their microstructure and mechanical properties were investigated using XRD, TEM, SEM and nanoindentation. The results show that the strength values obey the CLS model as the layer thickness decreases
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from 20 to 6.5 nm. After that, ultrahigh strength value of 3.27 GPa is attained at h = 2.5 nm, which is consistent well with the prediction made by atomistic simulations based on the crossing of dislocations across interfaces mechanism in an incoherent system. There are SBs formed around the indents for the multilayers with h ≤ 6.5 nm, which is ascribed to shear instability that occurs when the mechanism of crossing of dislocations across interfaces operates in the deformation process. These results imply that there is a transition of the deformation mechanism at h = 6.5 nm. Furthermore, the varying LSR hardness measurement confirms that the hardness values of the multilayers increase with increasing LSR, which is ascribed to the enhanced strain hardening caused by higher LSR. Acknowledgements The authors are grateful for the financial support from the National Natural Science Foundation of China (Grant Nos. 50871060 and 50772055) and National ‘863’ High-tech project of China (Grant No. 2007AA03Z426). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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