Scripta Materialia 50 (2004) 1407–1411 www.actamat-journals.com
Electrical resistivity of fully-relaxed grain boundaries in nanocrystalline Cu L.H. Qian a, Q.H. Lu a, W.J. Kong b, K. Lu
a,*
a
b
Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China Laboratory of Extreme Conditions Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China Received 27 October 2003; received in revised form 16 February 2004; accepted 19 February 2004
Abstract Electrical resistivity of grain boundaries (GBs) was determined in nanocrystalline (nc) Cu specimens prepared by magnetosputtering and subsequent annealing. Extrapolating the microstrain dependence of GB resistivity, we derived electrical resistivity of GBs in a fully-relaxed state in Cu, being 2.04 · 1016 X m2 . 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Electrical resistivity; Nanocrystalline Cu; Relaxation of grain boundary and magnetron sputtering
1. Introduction Experimental determination of electrical resistivity due to grain boundaries (GBs) in metals is of significance for understanding the nature of GBs and benefiting to technological applications. Following the first attempt to measure the GB resistivity in Cu by Andrews in 1965 [1], many systems have been measured [2–7]. However, rather scattered data have been obtained. For example, electrical resistivity of GBs in pure Cu reported by different authors ranges from 3.1–5.1 · 1016 X m2 [2–5]. Such scattered results were mainly attributed to several microstructural variables of the measured sample, including impurities, presence of lattice dislocations, point defects and misorientations of GBs. Segregation of impurities at GBs may significantly alter the scattering of electrons at GBs and therefore change the GB resistivity. Existence of lattice dislocations and point defects will elevate the resistivity due to scattering of electrons at the defects. Minimizing the contributions of lattice dislocations and point defects to electrical resistivity of the polycrystalline metals is essential for deriving the GB resistivity. Hence, the
*
Corresponding author. Tel.: +86-24-2390-6826; fax: +86-24-23998660. E-mail address:
[email protected] (K. Lu).
polycrystalline samples for GB resistivity measurement should have ultrafine grains with sizes smaller than mean free path (mfp) of electrons and the critical distance of dislocation pile-up. And the measurement is preferably performed at low temperatures as the mfp increases with a decreasing temperature. An obvious variation of the GB resistivity was noticed with a change of the GB misorientations, from 2.5–5.5 · 1016 X m2 for low and high angle GBs in Al [6,7]. An extremely small value of 1.5 · 1017 X m2 was obtained for R3 twin boundary in Al [7]. Although these variables affect the GB resistivity, the microstructure of GB is obviously a variable influencing the GB resistivity. Most measurements of GB resistivity were performed in polycrystalline samples with fine grains (with sizes of several microns to several tens of nanometers) and a large number of GBs that constitute a considerable volume fraction. Accompanied with a substantial grain refinement especially in the nanometer regime, GB structures become far from the equilibrium state characterized by a high density of GB dislocations. These dislocations may induce lattice distortion in the vicinity of GBs. The mean microstrain can be as high as a few percents as detected by X-ray diffraction [8], and it increases with a reduction of grain size. The microstructure of GBs in nc metals is obviously a variable to be considered in determining electrical resistivity of GBs. In this work, we studied the
1359-6462/$ - see front matter 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2004.02.026
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microstrain effect on the GB resistivity and extrapolated electrical resistivity of GBs in the fully relaxed state in Cu.
2. Experimental Nc Cu samples were prepared by direct current magnetron sputtering using a Cu target with a purity of 99.999% and a substrate of silicon (1 1 1). The background pressure in the deposition chamber was above 7.0 · 105 Pa, and the working pressure of pure argon gas was about 1 Pa. The temperature of substrate was about 350 K because no cooling medium was used in the course of sputtering, and the thickness of Cu samples was 4 lm. After being peeled from the Si substrate, the as-deposited nc Cu sample (sample A) was annealed in vacuum (better than 1 · 103 Pa) at different temperatures to relax grain boundary, 373 K for 10 min (sample B), 373 K for 30 min (sample C) and 383 K for 30 min (sample D), respectively. The as-deposited sample annealed at 773 K for 4 h was selected as the reference sample (sample E). The nc Cu samples were characterized by means of X-ray diffraction (XRD) and transmission electron
microscopy (TEM). XRD measurements of the nc Cu samples were carried out on a Rigaku DMAX/2400 X-ray diffractometer. TEM experiments were conducted on a Philips EM420 microscope. The content of impurity in the as-deposited sample was measured by means of Auger electron spectroscopy (AES) and less than 200 ppm (wt%) oxygen was detected. A conventional four-probe technique was used to measure the dc electrical resistance at low temperatures (16–290 K). The samples are 1 mm in length and 0.1 mm in width determined by optical microscopy; the thickness of the samples (4 lm) was determined by means of SEM.
3. Results and discussions 3.1. Microstructural characterization Plane-view TEM observations as shown in Fig. 1(a) showed that the as-deposited sample consists of roughly equiaxed ultra-fine crystals. The volume fraction is the quantity obtained by dividing the volume of certain grains with the same size by the total volumes of all the grains. And the volumes of certain grains with the same
Fig. 1. Bright-field in-plane-view TEM images and the statistical grain size distributions for sample A (a) and D (b), respectively. The inset (a) shows the corresponding selected area electron diffraction (SAED) pattern of sample A.
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Table 1 A list of measured results for various nc Cu samples including the average grain size (D) determined from XRD and TEM observations, the mean microstrain (he2 i1=2 ), the residual resistivity (q0 ), the k in Eq. (1) Sample A B C D E Ref.
D (nm) XRD
TEM
10 ± 5 10 ± 4 10 ± 3 12 ± 4 – –
19 21 22 22 350 1000
he2 i1=2 (%)
q0 (X m)
k
0.242 ± 0.006 0.206 ± 0.007 0.163 ± 0.005 0.141 ± 0.006 0 –
1.15 · 107 9.79 · 108 8.41 · 108 7.18 · 108 2.49 · 109 2.9 · 109
2.00 1.99 1.99 1.99 3.78 –
Results from the literature for the coarse-grained Cu are also included.
grain growth significantly. In addition, the high level of microstrain (which may originate from a high density of dislocations at GBs) also play a role in stabilizing the nanometer-sized crystallites. 3.2. Electrical resistance 3.2.1. Residual electrical resistance of nc Cu Fig. 2 displays temperature dependence of electrical resistivity (q) of the specimens (A–E) with different microstrains. It can be clearly seen that for each sample the resistivity decreases linearly with the temperature down to about 70 K. Below 70 K a non-linear temperature dependence of q is detected. Electrical resistivity of the nc Cu specimens (A–D) is about one order of magnitude larger than that of the coarse-grained sample (E). For example, electrical resistivity at room temperature (q290 ) of the nc Cu specimens ranges from 1.87 · 107 to 1.33 · 107 X m, compared to 2.08 · 108 X m for sample E which is close to the literature value (1.67 · 108 X m) [9]. With reducing microstrain, an evident drop in electrical resistivity is noticed for the nc Cu samples. q290 decreases from 1.87 · 107 to 1.33 · 107 X m when microstrain is reduced from 0.24 to 0.14%. 2.2x10-7 2.0x10-7
Sample A Sample B Sample C Sample D Sample E
1.8x10-7
Electrical resistivity (Ω m)
size are the product of pD2TEM /4, DXRD and the number of the grains with the same size (DTEM and DXRD are grain sizes determined by TEM observations and XRD results). The mean grain size of sample A is 19 nm, and the selected area electron diffraction (SAED) pattern indicates random orientations. XRD analysis in terms of Scherrer–Wilson equation indicates an averaged grain size of about 10 nm, which is smaller than plane-view size. A Æ1 1 1æ texture was detected in the Cu film with an X-ray diffraction peak intensity ratio of Ið1 1 1Þ =Ið2 0 0Þ of 3.1 (with respect to 2.2 for Cu without preferred orientation), as usually observed in the sputtered thin film. The estimated ratio of grain size along the Æ1 1 1æ orientation to the averaged value with random orientations is about 2. The microstrain of the as-deposited sample was determined by XRD, being about 0.24%. Such a large microstrain in the as-deposited sample is a signature of a high level of dislocation density at GBs, which induces a remarkable lattice distortion in the vicinity of GBs. In order to obtain samples with different GBs, a relaxation treatment (annealing) to the as-deposited sample was carried out. As listed in Table 1, the mean grain sizes of sample B, C and D determined by XRD are about 10 ± 4, 10 ± 3 and 12 ± 4 nm respectively, indicating no obvious grain growth during annealing. An obvious decrease in microstrain in these samples is noticed with respect to sample A. It indicates the occurrence of grain boundary relaxation during thermal annealing. Fig. 1(b) indicates a typical plane-view bright-field TEM image for sample D and its corresponding grain size distribution, and the average grain size is 22 nm. The grain size dimension perpendicular to the sample surface is determined by XRD results. Therefore, from the measured results of the averaged grain size (XRD) and planar-view grain size (TEM), as listed in Table 1, one may find that grain growth in the present samples is uniform in all three dimensions. The above analysis shows that rapid abnormal grain growth has not been observed during annealing. This phenomenon implies that some oxygen exists in the sample (e.g., at GBs or triple junctions) that retards
1.6x10-7 1.4x10-7 1.2x10-7 1.0x10-7 8.0x10
-8
6.0x10-8 4.0x10-8 2.0x10-8 0.0 0
50
100
150
200
250
300
Temperature (K)
Fig. 2. The temperature dependence of electrical resistivity (q) of samples A–E at a temperature range of 16–290 K.
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120
Sample A
118
Electrical resistivity (10-9Ω m)
116 76
Sample D
74
72
Sample E 3.2
2.8
2.4 15
20
25
30
35
40
45
50
55
Temperature (K) Fig. 3. The measured temperature dependence of q and the corresponding fitting lines for sample A, D and E at the temperature range of 16–50 K.
Electrical resistivity of a pure metal at low temperature can be represented by q ¼ q0 þ JT k
ð1Þ
where q is the measured electrical resistivity, q0 is the residual resistivity, J and k are constants [10,11]. For obtaining q0 and k, a least-mean-squares fitting of the data in the low temperature region (16–50 K) was carried out by using Eq. (1) as shown in Fig. 3. The fitting lines are consistent with the measured data for each sample, and the values for q0 and k are listed in Table 1. It is interesting to find that the residual resistivity is of the nc Cu samples ranging from 1.15 · 107 to 7.18 · 108 X m are 1–2 orders magnitude larger than that for sample E (2.49 · 109 X m, in comparison to the literature value of 2.8 · 109 X m [12]). The obtained k for sample A–D are almost the same within the error. The k for sample E is 3.78, which is slightly lower than the value (4.40–4.84) in the literature [11]. 3.2.2. Estimation of GB resistivity Usually, the value of q0 is dependent only on the amount of impurity and defect concentration in pure metals. In the present work, the effect of oxygen on q0 can be ruled out as no change is induced during thermal annealing. Otherwise, under high magnification TEM observations, no void has been detected even in the
vicinities of triple junctions and GBs. By using nanoindentation tests [13], the elastic modulus of the films (120 ± 10 GPa) was determined quantitatively, which is very close to the bulk modulus value (128 GPa). This is indicative of a high density in the present film sample, and micro-voids can be excluded, which implies the concentration and or coalescence of trapped argon are negligible during the deposition without substrate bias [14]. Therefore, the effect of impurity on electrical resistivity is ruled out. Consequently, the variation of q0 for the nc Cu mainly originates from the change of defects in the samples. According to the calculation of the lower bound repulsive force per unit length between two edge dislocations, the equilibrium distance between the two dislocations in lattice lC is about 19.3 nm for nc Cu [15]. In the other words, when the grain size (D) is comparable with lC , lattice dislocations and their pile-ups are hardly available, and most dislocations (if any) may locate themselves on GBs. In fact, high resolution TEM observations of the nc samples confirm that no dislocation is present in the crystallites. The average grain size of about 19–22 nm is comparable with lC and smaller than mfp in the lattice (mfp >39 nm below room temperature [16]). Therefore, the effect of lattice defects and electron-phonon scattering on q0 can be neglected in the present case. The variation of q0 after relaxation can be mainly attributed to the change in the concentration of the defects in GBs. Although some twin boundaries are visible in the TEM micrographs, the fraction of twin boundaries is only a small fraction of the total GBs. In addition, electrical resistivity of twin boundaries (R3) is about one order of magnitude less than that of conventional high angle GBs. Therefore, the contribution of twin boundaries to the total resistivity is extremely small and the change in electrical resistivity is related to microstructural variations originating in high angle boundaries. The total area of GBs per unit volume can be estimated by s ¼ 2=DTEM þ 1=DXRD supposing grains are cubic in shape (a similar relation should hold for the grains with complicated shape). One may calculate the electrical resistivity per unit GB area (qgb ) in terms of q0 =s. The relationship between the qgb and microstrain for the nc Cu samples as shown in Fig. 4 indicates that qgb decreases monotonously from 5.6 · 1016 to 4.12 · 1016 X m2 with a reduction of microstrain from 0.24% to 0.14%. It is clear that these qgb for the nc Cu samples are consistent with the literature data for GB resistivity (3.1–5.1 · 1016 X m2 ) [2–5], which were supposed to originate from different misorientation angles of the GBs. However, grains in nc Cu samples possess random orientations, and relaxation of the as-deposited sample cannot modify the average misorientation angle of the crystallites. The obvious decrease in qgb values should be
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[8,17–20]. The remarkable change in the GB structure has a pronounced effect in altering physical properties of the nc samples, such as an evident drop in thermal expansion coefficient and Debye–Waller parameter with a smaller microstrain [20]. The observed variation of GB resistivity with microstrain is another indication that physical properties of nc materials are not only dependent upon grain size, but closely related to the defects in GBs.
7
ρ gb (10-16Ω m2)
6 5 4 3 2
2.04 x 10-16 Ω m2
1 0.00
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4. Conclusion 0.05
0.10
0.15
0.20
0.25
0.30
Mean microstrain (%) Fig. 4. The relationship between the qgb value and the mean microstrain. The least-mean-squares fitting of the measured data is indicated by the dotted line.
attributed to the change in concentration of defects in GBs during annealing. 3.2.3. Extrapolation of GB resistivity in fully-relaxed state In this work, because of a large microstrain originating from a high density of GB dislocations in the asdeposited nc Cu sample, a large qgb value is obtained. Annealing leads to a relaxation of the GB structure and the dislocation density at GBs decreases. Because of the reduced scattering of electrons at GBs the obvious change of qgb is observed, indicating that electrical resistivity of GBs is sensitive to the density of dislocations at GBs. Consequently, it is reasonable to interpret the microstrain dependence of qgb in terms of varied density of GB dislocations. Therefore, it is significant to derive the GB resistivity when the GB structure is in fully-relaxed state (i.e., without GB dislocation). Extrapolating microstrain dependence of GB resistivity to zero microstrain, one may calculate the electrical resistivity of GBs in a fullyrelaxed state in Cu, being about 2.04 · 1016 X m2 . It is the lowest value of electrical resistivity of GBs with random misorientations in Cu. Actually, this values is much smaller than the GB resistivity values reported in the literature (3.1–5.1 · 1016 X m2 ) [2–5]. In this sense, the fully-relaxed state without GB dislocations (when microstrain tends to zero) serves as a ground state for GBs, of which the electrical resistivity is a fundamental physical parameter, identifying the intrinsic structure characteristics of GBs. It is well known that microstructures of GBs are sensitive to the thermal and/or mechanical history during the processing of the sample. Very different states of GBs may be obtained in samples with different synthesis or processing routes, manifested by different microstrains that can be varied by 1–2 orders of magnitude
In conclusion, we found that electrical resistivity of GBs in nc Cu is sensitive to the density of GB dislocations, besides impurity and misorientation angles of GBs. By extrapolating microstrain dependence of GB resistivity, we obtained electrical resistivity of GBs in a fully-relaxed state, being 2.04 · 1016 X m2 , which is much smaller than the GB resistivity values reported in the literature.
Acknowledgements We acknowledge the financial support from the National Science Foundation of China (No. 50021101 and 50125103), the Ministry of Science and Technology of China (No. 1999064505) and Max-Planck Society of Germany.
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