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Effect of disorder on the electrical resistivity in the partially crystalline Zr76Ni24 metallic glasses
Journal of Alloys and Compounds 421 (2006) 12–18
Effect of disorder on the electrical resistivity in the partially crystalline Zr76Ni24 metallic glasses I. Kokanovi´c ∗ Department of Physics, Faculty of Science, University of Zagreb, P.O. Box 331, Zagreb, Croatia Received 17 October 2005; received in revised form 4 November 2005; accepted 5 November 2005 Available online 20 December 2005
Abstract The electrical resistivity has been measured for annealed amorphous and partially crystalline Zr76 Ni24 metallic glasses in the temperature range from 2 to 290 K. The effect of disorder on the electrical resistivity in the amorphous and partially crystalline Zr76 Ni24 metallic glasses has been studied. Quenching rate and thermal annealing have been used to modify the disorder of the samples. The electrical resistivity of the annealed partially crystalline samples up to an annealing temperature slightly below the first crystallization exotherm increases with decreasing heating rates, whereas the annealed amorphous samples show an opposite effect. After annealing the samples slightly above the first crystallization exotherm the electrical resistivity of the annealed samples decreases drastically. The temperature dependence of the electrical resistivity of the samples has been interpreted in terms of the weak localisation of electrons and contribution by electron–phonon scattering. The contribution by electron–phonon scattering increases with increasing the fraction of crystalline phases in the annealed samples. The amorphous and partially crystalline Zr76 Ni24 metallic glasses become superconducting at temperatures below 3.5 K. © 2005 Elsevier B.V. All rights reserved. Keywords: Metallic glasses; Crystallisation; Electrical transport; Superconductors
1. Introduction In this work the amorphous and partially crystalline Zr76 Ni24 metallic glasses were prepared by rapid solidification. Rapid solidification from the melt is a method widely used to obtain metallic glasses. On the other hand, by varying the quenching rate during melt spinning or the crystallization of some metallic glasses it is possible to produce partially crystalline alloys [1,2]. Their structure consists of a crystalline phase, of a few nanometer grain size, embedded in the remaining amorphous matrix. This crystalline phase is often in a non-equilibrium state and grows with a highly faulted morphology [3,4]. Thus, the metastable state of partially crystalline metallic glasses can relax structurally to a more stable state whenever the atoms attain an appreciable mobility upon annealing. Associated with the structural relaxation, many physical properties, such as electrical transport, change drastically [5–8]. It is well known that the temperature coefficient of the electrical resistivity, TCR = (1/ρ)dρ/dT,
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is extremely sensitive to the number of quenched-in crystallites within the glassy matrix. Thus, the TCR provides a rather sensitive tool for probing the microscopic state of partially crystalline metallic glasses. Also, electrical resistivity measurements can help in understanding various phenomena in partially crystalline metallic glasses. It is an easy technique for studying phase transformations, defects and other structural changes in crystalline and amorphous alloys. On the other hand, the motion of electrons in disordered systems is one of the most fundamental problems in solid state physics. Both theoretical [9–12] and experimental [13–16] studies of the low temperature behaviour of electrical resistivity of disordered electronic systems have led to quantum corrections to the Boltzmann contribution. It has been shown that the diffusive motion of electrons in two-dimensional as well as in three-dimensional disordered systems entails quantum corrections to the electrical resistivity and the magnetoresistivity. There are two principal sources of quantum corrections, arising from weak localisation [9,10] and the Coulomb interaction [11,12]. Both of these corrections are important when the mean free path becomes short so that electron propagation between scattering events is no longer free-electron-like but diffusive. At low enough temperatures, where the elastic scattering time is
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a few orders of magnitude shorter than the inelastic scattering time, the quantum corrections arising from the interference of the electronic partial waves are very important. It has been shown that constructive interference of the electronic waves can only be expected in a back-scattering geometry. This quantum interference effect will produce an increase of the sample resistivity. The magnitude of this additional contribution at given temperature is reduced by the presence of the inelastic electron–phonon, spin–orbit or spin–flip scattering, since they destroy the constructive interference. Recently results showed that, according to X-ray diffraction (XRD) results, two sides of quenched ribbons of the partially crystalline Zr76 Ni24 metallic glass have revealed different structures [17] (that in contact with the wheel—the dark side of the ribbon is called DSR and that in contact with the inert atmosphere—shiny side of the ribbon is called SSR). The diffraction pattern of the DSR shows a broad maximum centred at 2α = 36.1◦ of β = 4.2◦ corresponding to an amorphous phase. On the other hand, the XRD pattern of the SSR showed to consist of a set of sharp lines. We were able to fit the peaks to the orthorhombic Zr3 Ni phase. The lattice constants are obtained as follows: a = (1.098 ± 0.002), b = (0.880 ± 0.001) and c = (0.333 ± 0.002) nm. The SSR crystalline layer was relatively thin. After the SSR crystalline layer was abraded by approximately 5 m and examined with XRD only a broad maximum was present in the diffraction pattern, which means that the thickness of the crystalline layer was about 5 m. This crystalline layer was built of the orthorhombic Zr3 Ni nanocrystalline grains. The XRD patterns of both side of the partially crystalline Zr76 Ni24 metallic glass are shown in Ref. [17] and Figs. 1 and 2. Also, a thin interphase region was between amorphous and nanocrystalline phases. This interphase region or two-phase region was composed of the orthorhombic Zr3 Ni nanocrystalline grains embedded in the amorphous matrix. The average crystallite sizes of the orthorhombic Zr3 Ni nanocrystalline phase were from 10 to 30 nm and the crystallite sizes decrease with increasing annealing temperature up to slightly below the first crystallisation exotherm [17]. The purpose of this experiment was to study the effect of disorder on the electrical resistivity and to gain insight into the amorphous structure of the glass and the metastable nanocrystaline structure of the intermediate crystallisation products in the partially crystalline Zr76 Ni24 metallic glass. Different quenching rate and thermal annealing have been used to modify the disorder of the samples. All samples, including those, which are completely amorphous, are characterised by high room-temperature resistivity, they are paramagnetic and become superconducting at temperatures below 3.5 K [18].
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then cut from the ribbon. The thermal stability of the samples was measured by means of a calibrated Perkin-Elmer DSC-4 differential scanning calorimeter using an atmosphere of purified argon gas. The samples (typically 5 mg) were packed tightly in sealed aluminium cans and an empty aluminium can was used as the reference. The temperature and heat flow axes were calibrated using Zn solid–liquid transition recorded with the same heating rates. The structural changes after heating the samples with heating rates of 10 and 60 K/min up to temperatures between 593 and 663 K were investigated by X-ray diffraction using Cu K␣ radiation. Particle induced X-ray emission spectroscopy (PIXE) analysis indicated that the samples had a composition Zr(76.0 ± 0.1) Ni(24.0 ± 0.1) . The electrical resistance was measured by a low-frequency (23.2 Hz) fourprobe ac method in the temperature range of 2–290 K; the precision extended to a few parts in 106 . The temperature measurement was performed in a 4 He cryostat. A calibrated cernox thermometer was used to monitor the temperature of the sample. The critical magnetic field, Hc2 (T), measurements were performed at temperatures down to 2 K in magnetic fields up to 1 T oriented parallel to the sample.
3. Results and discussion The values of heat flow, cp , determined from the DSC measurements of the partially crystalline Zr76 Ni24 metallic glass at heating rates of 10 and 60 K/min are shown in Fig. 1. The DSC trace of the partially crystalline Zr76 Ni24 metallic glass shows one sharp exothermic peak. In all cases, the cooling curves obtained after measuring up to 823 K did not show any thermal effects. This demonstrates the irreversible nature of the processes that give rise to the exothermal peaks during heating. The crystallization peak temperature Tpx corresponding to the maximum of the first exotherm is designated Tp1 . The values of Tp1 observed with the heating rates of s = 10 and 60 K/min were 620.4 ± 0.5 and 642.2 ± 0.5 K, respectively. In order to study the influence of quenched-in crystallites of the orthorhombic Zr3 Ni crystalline phase on the thermal stability of the residual amorphous phase of the as-quenched sample, the thermal stability of the Zr76 Ni24 metallic glass was also studied. The DSC trace of the Zr76 Ni24 metallic glass exhibited the similar DSC crystallization characteristics as did the partially crystalline Zr76 Ni24 metallic glass (as-quenched sample) and the position of the maximum of exotherm was unchanged for heating rates of s = 10 and 60 K/min.
2. Experimental techniques Ribbons of the amorphous and partially crystalline Zr76 Ni24 metallic glasses were prepared by rapid solidification of the melt on a single-roll spinning copper wheel in an argon atmosphere. By adjusting the melt-spinning parameters, primarily the wheel velocity, it is possible to reduce the cooling rate and produce partially crystalline metallic glasses. So, we were able to obtain Zr76 Ni24 ribbons containing approximately 90.4% of an amorphous phase and 9.6% of a crystalline phase. The samples, 5 mm long, 3.4 mm wide and 50 m thick were
Fig. 1. The heat flow cp of the partially crystalline Zr76 Ni24 metallic glass in the temperature range of 300–700 K at the heating rates, s, () s = 60 K/min and () s = 10 K/min.
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Fig. 2. The change in the temperature-dependent part of the electrical resistivity relative to its value at 290 K (ρ(T) − ρ(290 K))/ρ(290 K) of the Zr76 Ni24 metallic glass [as-quenched (), the annealed up to 593 K with s = 60 K/min (), the annealed up to 593 K with s = 10 K/min (), the annealed up to 635 K with s = 10 K/min () and the annealed up to 663 K with s = 60 K/min) (♦)].
The crystallisation processes of the partially crystalline and amorphous Zr76 Ni24 metallic glass were studied in more detail. The XRD patterns for different heat-treated samples are given in Ref. [17]. The samples were subjected to heating rates of 60 and 10 K/min in separate runs from 298 K to slightly below and to slightly above the first crystallization exotherm; then the heating was stopped and the sample cooled quickly to 298 K. XRD measurements performed at room temperature show no significant changes on the XRD pattern upon annealing of the partially crystalline and amorphous Zr76 Ni24 metallic glasses up to 593 K with the heating rate of 10 K/min. On the other hand, the samples that underwent the heating rate of 60 K/min up to 663 K show changes in the XRD patterns [17]. The XRD pattern consists of two sets of lines. One set of lines was successfully indexed by the Zr2 Ni tetragonal structure, space group I4/mcm with lattice parameters a = (0.648 ± 0.002) and c = (0.522 ± 0.002) nm. The other set of lines belongs the -Zr nanocrystalline phase. The crystalline peaks of the -Zr nanocrystalline phase were fitted to the hexagonal structure space group P6/mmm with lattice parameters: a = (0.506 ± 0.001) and c = (0.314 ± 0.001) nm. The first maximum peak of such XRD pattern is sharper and shifted to smaller scattering angle 2θ than one of the partially crystalline Zr76 Ni24 metallic glass. The maximum peak is centred at 2θ = 35.5◦ with a full width at half-maximum of β = 0.8◦ . By contrast the XRD pattern of the SSR of this annealed sample showed additional crystalline lines corresponding to the orthorhombic Zr3 Ni crystalline phase already described above. The XRD patterns of the DSR of the annealed partially crystalline Zr76 Ni24 metallic glass and annealed amorphous sample subjected to heating at 60 K/min up to 823 K consist of two sets of sharp lines, which correspond to ␣-Zr crystalline phase and Zr2 Ni crystalline phase. The crystalline peaks of the ␣-Zr crystalline phase were fitted to the hexagonal structure space group P63 /mmc with lattice parameters: a = (0.323 ± 0.002) and c = (0.516 ± 0.005) nm. The first maxi-
mum of the sample annealed up to 823 K could be resolved in two peaks with maxima at 2θ 1 = 35.35◦ (Zr2 Ni) and 2θ 2 = 36.43◦ (␣-Zr) with full widths at half-maximum of β1 = 0.52◦ and β2 = 0.36◦ , respectively. It should be point out that only a small fraction of the metastable cubic Zr2 Ni crystalline phase with a lattice parameter a = 1.227 nm was also identified. The XRD pattern of the SSR of the annealed partially crystalline Zr76 Ni24 metallic glass subjected to the heating rate 60 K/min up to 823 K shows an additional set of the sharp diffraction lines corresponding to the orthorhombic Zr3 Ni crystalline phase already described above. Thus, we conclude that the existence of the orthorhombic Zr3 Ni crystalline phase does not change the crystallization transformations of the Zr76 Ni24 amorphous phase and remains unchanged during annealing of the samples. On the basis of the DSC and XRD data obtained it was possible to identify the phases which occur in the heat treatment of the partially crystalline and amorphous Zr76 Ni24 metallic glass up to 823 K. The first stage of crystallization of the amorphous phase shows the evolution of -Zr and Zr2 Ni nanocrystalline phases. The second stage of crystallization of the samples shows the transformation of -Zr phase to a stable ␣-Zr nanocrystalline phase. The measured values of room-temperature electrical resistivity of the partially crystalline Zr76 Ni24 metallic glass, the Zr76 Ni24 metallic glass and their annealed samples are listed in Tables 1 and 2, respectively. The as-quenched samples were subjected to the heating rates of 60 and 10 K/min in separate runs from 298 K up to the annealing temperature, Ta (slightly below and to slightly above the crystallisation exothem), then the heating was stopped and sample cooled quickly to 298 K. The feature in all these electrical resistivity data is the rather high residual resistivity, reflecting the high atomic disorder in the samples. It can be seen from Tables 1 and 2 that the value of room-temperature electrical resistivity of the partially crystalline Zr76 Ni24 metallic glass (ρ(290K) = 136 cm) is lower than the value of the resistivity of the Zr76 Ni24 metallic glass (ρ(290 K) = 163 cm). According to the XRD results [17], it is explained as a consequence that the crystalline orthorhombic Zr3 Ni phase of about 5 m of thickness, embedded in the remaining amorphous matrix of the partially crystalline Zr76 Ni24 metallic glass involves the Zr3 Ni nanocrystalline layer of relatively low electrical resistivity immersed in a glassy matrix of high electrical resistivity since ordering of crystallographic structure increases electron mobility and leads to a resistivity reductions. Thus, the electrical resistivity of the partially crystalline Zr76 Ni24 metallic glass is composed of the structural components of both fractions (amorphous and Zr3 Ni nanocrystalline layer). The structural components of the amorphous phase and Zr3 Ni nanocrystalline layer in the electrical resistivity of the partially crystalline Zr76 Ni24 metallic glass were numerically separated and the electrical resistivity of the Zr3 Ni nanocrystalline layer of the partially crystalline Zr76 Ni24 metallic glass was estimated using following model. Suppose that the partially crystalline Zr76 Ni24 metallic glass, according to the XRD measurements, is composed of a parallel circuit made from the amorphous phase and Zr3 Ni nanocrystalline layer. Thus, the
I. Kokanovi´c / Journal of Alloys and Compounds 421 (2006) 12–18
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Table 1 Effect of heat treatment on various physical properties Ta ± 1 (K)
s (K/min)
ρ(290 K) ± 5 ( cm)
(1/ρ)∂ρ/∂T ± 0.01 (×10−4 K−1 )
Tc ± 0.01 (K)
Tc ± 0.005 (K)
(∂Hc2 /∂T ) ± 0.1 (T/K)
D ± 0.1 (×10−5 m2 s−1 )
ξ 0 ± 5 (×10−10 m)
295 593 593 663
0 60 10 60
136 140 148 72
0.84 × 10−4 0.55 × 10−4 0.02 × 10−4 9.25 × 10−4
3.49 3.46 3.42 2.88
0.03 0.03 0.03 0.05
−2.7 −2.5 −2.5 −1.0
4.1 4.4 4.4 11.0
59 62 62 107
Ta is the maximum annealing temperature of partially crystalline Zr76 Ni24 metallic glass, s the heating rate at what Ta is reached, ρ(290 K) the electrical resistivity, (1/ρ)∂ρ/∂τ the temperature coefficient of the electrical resistivity, Tc the superconducting transition temperature, Tc the 10–90% superconducting transition width in the electrical resistivity, (∂Hc2 /∂T )Tc the shape of the upper critical field at Tc (0), D the electron diffusion constant and ξ 0 is the zero-temperature coherence length.
measured electrical resistivity for the parallel circuit, ρp , is given by; ρp =
ρ c ρa (1 − c)ρc + cρa
(1)
where ρc and ρa are the resistivities of the Zr3 Ni nanocrystalline layer and amorphous phase, respectively, and c is the volume fraction of the Zr3 Ni nanocrystalline layer. In order to estimate the volume fraction of the Zr3 Ni nanocrystalline layer of the partially crystalline Zr76 Ni24 metallic glass the total integrated crystallisation heat, H, was calculated. The H of the sample was determined from the area of the crystallisation exotherm (Fig. 1). The total crystallisation heats were H1 = (63.8 ± 0.5) J/g for the partially crystalline Zr76 Ni24 metallic glass, and H2 = (69.9 ± 0.5) J/g for the Zr76 Ni24 metallic glass. Analysis of the H2 compared with that of the H1 gives a direct estimate of the volume fractions of the two phases. Thus, the estimated volume fraction of the Zr3 Ni nanocrystalline layer was c = (9.6 ± 0.2)%, which is in good agreement with the estimated thickness of the Zr3 Ni nanocrystalline layer of about 5 m using the XRD measurements. The electrical resistivity of the Zr3 Ni nanocrystalline layer is obtained using Eq. (1) and the experimental values for the ρp (290 K) = 136 and ρa (290 K) = 163 gives ρc (290 K) = (53 ± 5) cm. Furthermore, the electrical resistivity data indicates an increase in the room-temperature electrical resistivity of the annealed partially crystalline Zr76 Ni24 metallic glass upon decreasing heating rate at an annealing temperature, Ta = 593 K, slightly below the crystallization exotherm (Table 1). These changes of the electrical resistivity of the annealed samples are irreversible. The increase in the electrical resistivity is related to
the rearrangement in atomic position in the interphase or twophase region of the amorphous and Zr3 Ni nanocrystaline phases described above. Also, the atomic rearrangements in the twophase region destroy the Zr3 Ni nanocrystals and decrease their nanocrystalline sizes [17]. Thus, the resistivity upturn is additionally enhanced by low grain sizes and a high density of the grain boundaries present in the two-phase region of the annealed partially crystalline Zr76 Ni24 metallic glasses. On the other hand, the decrease in the room-temperature electrical resistivity of the annealed Zr76 Ni24 metallic glass up to a temperature slightly below the crystallisation exotherm was ascribed with relaxation voids and strain in these annealed samples (Table 2). If the as-quenched samples are heated at 60 K/min up to 663 K, slightly above the crystallisation exotherm, the room-temperature electrical resistivity decreases drastically (Tables 1 and 2). This tendency is also in good agreement with the XRD measurements which showed that during the first stage of crystallisation, the amorphous phase transforms in the -Zr and Zr2 Ni nanocrystalline phases with nanocrystallite size of D = 10 nm [17]. Thus, further an ordering of crystallographic structure increases electron mobility and leads to resistivity reductions. The change in the temperature-dependent electrical resistivity, relative to its value at 290 K, ρ/ρ(290 K), of the Zr76 Ni24 metallic glass, the partially crystalline Zr76 Ni24 metallic glass and their annealed samples for the temperature range of 5–290 K is shown in Figs. 2 and 3, respectively. The error bars (not shown) are smaller than the size of the symbols. It can be seen from Fig. 2 that the temperature coefficients of the electrical resistivity, TCR, of the Zr76 Ni24 metallic glass is negative in the temperature range 10–295 K. The TCR of the Zr76 Ni24 metallic glass annealed in the annealing temperature range of 298–593 K
Table 2 Effect of heat treatment on various physical properties Ta ± 1 (K)
s (K/min)
ρ(290 K) ± 5 ( cm)
(1/ρ)∂ρ/∂T ± 0.01 (×10−4 K−1 )
Tc ± 0.01 (K)
Tc ± 0.005 (K)
(∂Hc2 /∂T ) ± 0.1 (T/K)
D ± 0.1 (×10−5 m2 s−1 )
ξ0 ± 5 (×10−10 m)
295 593 593 635 663
0 60 10 10 60
163 160 156 125 107
−1.30 × 10−4 −1.24 × 10−4 −0.47 × 10−4 6.64 × 10−4 7.31 × 10−4
3.52 3.48 3.45 2.95 2.90
0.020 0.015 0.018 0.020 0.020
−3.5 −3.4 −3.2 −3.0 −2.4
3.1 3.2 3.4 3.6 4.5
52 53 54 61 69
Ta is the maximum annealing temperature of Zr76 Ni24 metallic glass, s the heating rate at what Ta is reached, ρ(290 K) the electrical resistivity, (1/ρ)∂ρ/∂τ the temperature coefficient of the electrical resistivity, Tc the superconducting transition temperature, Tc the 10–90% superconducting transition width in the electrical resistivity, (∂Hc2 /∂T )Tc the shape of the upper critical field at Tc (0), D the electron diffusion constant and ξ 0 is the zero-temperature coherence length.
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ρ ∼ −T 1/2 ,
Fig. 3. The change in the temperature-dependent part of the electrical resistivity relative to its value at 290 K (ρ(T) − ρ(290 K))/ρ(290 K) of the partially crystalline Zr76 Ni24 metallic glass [as-quenched (), the annealed up to 593 K with s = 60 K/min (), the annealed up to 593 K with s = 10 K/min () and the annealed up to 663 K with s = 60 K/min) (♦)].
is also negative and its absolute value decreases with decreasing heating rates. If the Zr76 Ni24 metallic glass is heated up to slightly above the crystallisation exotherm, the TCR changes sign and become positive. On the other hand, it is apparent from Fig. 3 that the electrical resistivity of the partially crystalline samples follows a decreasing trend with decreasing temperature and saturates in the temperature region from 10 to 40 K. The values of the TCR of the partially crystalline Zr76 Ni24 metallic glass and its annealed samples in the annealing temperature range of 298–663 K are positive. A positive TCR is often a sign of the presence quenched-in crystalline within glassy matrix. The values of TCR at room temperature of the annealed samples are listed in Table 1. The TCR of the annealed partially crystalline Zr76 Ni24 metallic glasses decreases as the heating rates decrease for annealing temperatures slightly below the crystallisation exotherm. These observations are consistent with well-known Mooij correlation for disordered systems containing d electrons [19]. By confining his attention to those systems that had a relatively small TCR, he found that resistivity tends to be larger in systems which have a smaller value of TCR (Table 1). In strongly disordered amorphous samples when the electron mean path becomes comparable with interatomic distance an incipient weak localisation shows up and the correction to the Boltzmann conductivity [20,21] is given by σ =
2e2 πhLi (T )
(2)
where Li (T) is the inelastic diffusion length and Li (T) = (Dτ i )1/2 where D is the electronic diffusion constant and τ i is the inelastic scattering time. This was shown to lead to temperature dependence of electrical resistivity in strongly disordered alloys of the form: ρ ∼ −T,
Tθ Tθ
(3)
Tθ < T < Tθ 3
(4)
where Tθ is Debye temperature. On the other hand, at T > 20 K, in the samples with a nanocrystalline fraction and positive TCR, the temperature behaviour of the electrical resistivity has been explained with help of Ziman theory where the electrical resistivity depends on the Debye–Waller factor [22,23]. Several temperature regions may be distinguished in the temperature variation of the electrical resistivity of the Zr76 Ni24 metallic glass (Fig. 2). In the first one, from 5 up to 15 K the electrical resistivity changes insignificantly, in the second range from 20 to 90 K the electrical resistivity varies proportionally to temperature and in the third temperature region from 110 up to 250 K, the electrical resistivity is proportional to T1/2 . The best fit yields ρ/ρ(290 K) = (6.03 ± 0.01) × 10−2 − (3.27 ± 0.01) × 10−4 × T in the second temperature region. This temperature behaviour of the electrical resistivity can be treated within the framework of the weak localisation of electrons where the temperature dependence of ρ is determined by temperature dependence of the inelastic scattering time τ i which is assumed to be equal to βT−2 where β is the inelastic scattering parameter. In the third temperature region, the best fit yields ρ/ρ(290 K) = (6.84 ± 0.01) × 10−2 − (4.03 ± 0.01) × 10−3 × T1/2 . The root temperature dependence presumably comes from the weak localisation effect which changes over to ρ(T) ∼ −T1/2 due to the fact that at high temperatures, the τ i is given by simply form, τ i ∼ T−1 . However, for the annealed Zr76 Ni24 metallic glass the quality of fitting was considerably improved when an additional T2 or T dependent term was added in the second and third temperature regions, respectively. Moreover, it was found that the temperature dependence of the structural contribution of the nanocrystalline phase of the annealed samples subjected to the heating rate 60 K/min at the annealing temperature above the crystallisation exotherm (Ta = 663 K) fits to the Ziman theory [22,23]. According to XRD results [17], in the temperature region from 20 to 70 K, the best fit yields ρ/ρ(290 K) = (−2.42 ± 0.02) × 10−1 + (8.85 ± 0.08) × 10−6 × T2 whereas in the temperature region from 200 to 300 K, ρ/ρ(290 K) = (−2.10 ± 0.02) × 10−1 + (7.65 ± 0.08) × 10−4 × T. Thus, the temperature dependence of the electrical resistivity for the annealed Zr76 Ni24 metallic glass suggests that contribution by electron–phonon scattering increases with increasing a fraction of crystalline phase in the annealed samples. In contrast to the metallic glass, the temperature dependence of the electrical resistivity for the partially crystalline samples shows very peculiar electrical resistivity behaviour (Fig. 3). It consists of a plateau region in the temperature region from 10 up to 45 K. The plateau in the electrical resistivity plot originates from the competing effects of the two different scattering mechanisms due to simultaneous presence of the amorphous and orthorhombic Zr3 Ni crystalline phases, both of which are pronounced in these samples. The XRD patterns of these samples support this explanation [17]. The electrical resistivity data for the partially crystalline sample are well fitted to a function
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Fig. 4. The temperature-dependent electrical resistivity relative to its value at 4.2 K, ρ(T)/ρ(4.2 K), vs. temperature below 4.5 K of the partially crystalline Zr76 Ni24 metallic glass [as-quenched (), the annealed up to 593 K with s = 60 K/min (), the annealed up to 593 K with s = 10 K/min () and the annealed up to 663 K with s = 60 K/min (♦)].
a + bT + cT2 in the temperature region from 45 up to 80 K, while at higher temperature to a function a + bT1/2 + cT. This can be treated within the framework of the weak localisation and the Ziman theory [22,23]. Also, it was found that the temperature dependence of the structural contribution of the nanocrystalline phases of the annealed partially crystalline Zr76 Ni24 metallic glass subjected to 60 K/min at the annealing temperature above the exotherm (Ta = 663 K) fits to the Ziman theory [22,23]. Thus, in the temperature region from 15 to 65 K, the best fit yields ρ/ρ(290 K) = (−3.29 ± 0.02) × 10−1 + (1.59 ± 0.08) × 10−5 × T2 whereas in the temperature region from 210 to 300 K, ρ/ρ(290 K) = (−2.51 ± 0.02) × 10−1 + (8.67 ± 0.08) × 10−4 × T. The numerical analysis of the electrical resistivity shows that weak localisation contribution to scattering does not exceed 6% of the measured electrical resistivity of the Zr76 Ni24 metallic glass and diminishes with increasing annealing temperature and decreasing heating rates in its annealed samples. Furthermore, it was found that the temperature dependence of electrical resistivity of the Zr3 Ni nanocrystalline phase fits well to the Ziman theory and dominates over structural contribution of the amorphous phase of the partially crystalline Zr76 Ni24 metallic glass at temperatures higher than 45 K. Moreover, for samples possessing a room-temperature resistivity below 150 cm for which the TCR is positive, the Ziman theory should be applied to explain the temperature dependence of the electrical resistivity of the annealed samples. However, as the electrical resistivity increases above 150 cm and the mean free path of electron propagation approaches the value of the interatomic distance, the TCR becomes negative and the theory of weak localisation has been invoked for a suitable explanation of the electrical resistivity results. The temperature-dependent electrical resistivity relative to its value at 4.2 K, ρ(T)/ρ(4.2 K), of the partially crystalline Zr76 Ni24 metallic glass and its annealed samples in the vicinity of Tc is shown in Fig. 4. The Tc was determined as the midway point
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on the resistivity transition versus temperature transition. The values of Tc are listed in Tables 1 and 2. It can be seen from Tables 1 and 2 and Fig. 4 that all the superconducting transitions are very sharp and the temperature difference between the 90% and 10% points of the resistivity change is typically less than 50 mK. The Tc of the samples heated at an annealing temperature below the crystallisation exotherm changes slightly with decreasing heating rates. In all the annealed samples, the Tc decreases with decreasing heating rates and increasing annealing temperatures. This result is in contrast to the annealed Zr80 Fe20 and Zr80 Co20 metallic glasses [8,24]. The increase in the Tc upon annealing up to an annealing temperature below the first crystallisation exotherm in the Zr80 Fe20 and Zr80 Co20 metallic glasses has been related to the decrease in the spin-fluctuation mass enhancement parameter, λsp . It is well known that the influence of spin fluctuations on superconductivity in Zr-3d metallic glasses decreases with increasing Zr concentration whereas at fixed Zr concentration it decreases as one move from Fe towards Ni [13,24–26]. Thus, the effect of the λsp on superconductivity in the investigated samples can be negligible. Using the modified form of the McMillan equation [27] it can be shown that this change in Tc upon annealing is related to a decrease in the electron–phonon coupling constant. According to the results of the analysis XRD data in Ref. [17], the decrease in λph created by a hardening of phonon modes as a result of relaxation of the quenched-in strains and redistribution of the defects decreases the Tc . Thus, we can conclude that for the heating rates of 60 and 10 K/min and annealing temperature slightly below the crystallisation exoterm the annealing reduces λph , hence the Tc decreases in the annealed samples. Furthermore, according to XRD data [17], the modification in the chemical short-range order due to heating above the crystallisation exotherm resulting in evolution of the -Zr phase and Zr2 Ni nanocrystalline phase plays an important role in determining the Tc of the annealed samples subjected to different annealing temperatures. The values of the Tc of these annealed samples are very close with the value of the Tc of the Zr2 Ni metallic glass (Tc = 2.78 K) [13]. It is well-known that the Tc –s decrease approximately linearly with decreasing Zr content in amorphous Zr-3d (3d = Ni, Co and Fe) alloys [26]. The decrease of the Zr content reduces the electronic density of states at the Fermi level, N(EF ) and this decrease of the N(EF ) is related with the decrease of λph . The superconducting properties of the partially crystalline Zr76 Ni24 metallic glass up to slightly above the crystallisation exotherm are characterised by a somewhat less sharp electrical resistive transition than observed in the as-quenched sample (Table 1). Also, below 3.5 K down to superconducting transition small anomaly of the order 5 % in the electrical resistivity of the same annealed sample is observed (Fig. 4). It seems that the anomaly is due to small inhomogeneities, which are still present in this sample. The value of the (∂Hc2 /∂T )Tc was determined from the slope of the measured Hc2 versus Tc curve at Tc (0) and listed in Tables 1 and 2. The applied magnetic field of 1 T shifts the Tc of the Zr76 Ni24 metallic glass down by approximately 0.29 K, which correspond to a slope of −3.5 T/K for the upper critical magnetic field near Tc . The values of the (∂Hc2 /∂T )Tc depend
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very much on the microstructural defect density in the sample. For instance, the slope of the partially crystalline Zr76 Ni24 metallic glass is (∂Hc2 /∂T )Tc = −2.7 T/K whereas for the annealed partially crystalline Zr76 Ni24 metallic glass subjected to the heating rate of 60 K/min at the annealing temperature of 663 K, (∂Hc2 /∂T )Tc = −1 T/K. The absolute value of the (∂Hc2 /∂T )Tc decreases with decreasing heating rates and increasing annealing temperatures or increasing the nanocrystalline fraction of the annealed samples (Tables 1 and 2). Furthermore, from the Tc , (∂Hc2 /∂T )Tc and by using wellknown expression for the zero-temperature coherence length, ξ 0 = 1.81 × 10−8 × (Tc × ∂Hc2 (T)/∂T)−1/2 [28], the value of the ξ 0 was estimated. The values of the ξ 0 are given in Tables 1 and 2. It can be seen from Tables 1 and 2 that the ξ 0 increases significantly with annealing at annealing temperature above the crystallisation exotherm. For instance, the value of the ξ 0 of the partially crystalline Zr76 Ni24 metallic glass is ξ 0 = 5.9 nm whereas for its annealed sample at the annealing temperature of 663 K, ξ 0 = 10.7 nm. The increase of the ξ 0 is larger for the annealed partially crystalline Zr76 Ni24 metallic glass than for the annealed Zr76 Ni24 metallic glass (Tables 1 and 2). The homogeneity of the annealed samples is judged to be high as evidenced by a small superconducting transition width and sharp electrical resistive transition. This suggests that the homogeneity is on a spatial scale of less than the ξ 0 . The values of the diffusion coefficient were also obtained, using the relation 4kB D= πe
∂H −1 c2 ∂T Tc
(5)
where kB is the Boltzmann constant and e is the electronic charge. The values of D are given in Tables 1 and 2. It can be seen from Tables 1 and 2 that the values of D increase with increasing annealing temperatures and decreasing heating rates. In the case of strong scattering, the electron motion becomes diffusive so that the electrical resistivity is expressed as ρ = (e2 N(EF )D)−1 . Thus, we can conclude that the decrease of the electrical resistivity of the annealed amorphous samples with the decreasing heating rates and increasing annealing temperatures is caused mostly by the increase of the D in the annealed samples. 4. Conclusion The Zr76 Ni24 metallic glass and partially crystalline Zr76 Ni24 metallic glass were prepared by melt spinning. The effect of annealing on the resistivity in the Zr76 Ni24 metallic glass and partially crystalline Zr76 Ni24 metallic glass has been investigated. The resistivity of the annealed partially crystalline samples up to a temperature slightly below the crystallization exotherm increases with decreasing heating rates, whereas after annealing the samples slightly above crystallization exotherm the resistivity decreases drastically. The temperature dependence of the electrical resistivity of the samples has been interpreted in terms of the weak localisation of electrons and
contribution by electron–phonon scattering. The contribution by electron–phonon scattering increases with increasing the fraction of crystalline phase in the annealed samples. The asquenched samples become superconducting at temperatures below 3.52 K. The superconducting transition temperature, Tc , of the amorphous and partially crystalline Zr76 Ni24 metallic glass subjected to different heating rates up to slightly below and above the crystallisation exotherm decreases with decreasing heating rates and increasing annealing temperatures. The homogeneity of the annealed samples is judged to be high as evidenced by a small superconducting transition width and sharp electrical resistive transition. This suggests that the homogeneity is on a spatial scale of less than the zero-temperature coherence length ξ 0 . The resistivity decrease of the annealed Zr76 Ni24 metallic glass subjected to the heating rates of 60 and 10 K/min at temperatures slightly below or above the crystallization exotherm is caused mostly by the increase of the D. References [1] Y. Yoshizawa, S. Oguma, K. Yamauchi, J. Appl. Phys. 64 (1988) 6044. [2] A. Inoue, K. Nakazato, Y. Kawamura, T. Masumoto, Mater. Sci. Eng. A 179–180 (1994) 654. [3] Y.D. Dong, G. Gregan, M.G. Scott, J. Non-Cryst. Solids 43 (1981) 403. [4] U. Koster, U. Herold, in: H.J. G¨untherodt, H. Beck (Eds.), Glassy Metalls I, Springer Verlag, Heidelberg, 1981, p. 225. [5] C.C. Koch, D.M. Kroeger, J.S. Lin, J.O. Scarbrough, W.L. Johnson, A.C. Anderson, Phys. Rev. B 27 (1983) 1586. [6] O. Laborde, A. Ravex, J.C. Lasjaunias, O. B´ethoux, J. Low Temp. Phys. 56 (1984) 461. [7] S.J. Poon, Phys. Rev. B 27 (1982) 5519. [8] M. Sabouri-Ghomi, Z. Altounian, J. Non-Cryst. Solids 205–207 (1996) 692. [9] H. Fukuyama, T. Hoshino, J. Phys. Soc. Jpn. 50 (1981) 2131. [10] B.L. Al’tshuler, A.G. Aronov, A.I. Larkin, D.E. Khmel’nitzkii, Sov. Phys. JETP 54 (1981) 411. [11] P.A. Lee, T.V. Ramakrishnan, Phys. Rev. B 26 (1982) 4009. [12] B.L. Al’tshuler, A.G. Aronov, in: A.L. Efros, M. Pollak (Eds.), Electron–Electron Interaction in Disordered Systems, North-Holland, Amsterdam, 1985, pp. 1 and 109. [13] I. Kokanovi´c, B. Leonti´c, J. Lukatela, Phys. Rev. B 41 (1990) 958. [14] A.K. Meikap, Y.Y. Chen, J.J. Lin, Phys. Rev. B 69 (2004) 212202. [15] I. Kokanovi´c, B. Leonti´c, J. Lukatela, Phys. Status Solidi (b) 241 (2004) 908. [16] A.K. Majumdar, J. Magn. Magn. Mater. 263 (2003) 26. [17] I. Kokanovi´c, A. Tonejc, Mater. Sci. Eng. A 373 (2004) 26. [18] I. Kokanovi´c, B. Leonti´c, J. Lukatela, J. Ivkov, Phys. Rev. B 42 (1990) 11587. [19] H. Mooij, Phys. Status Solidi A 17 (1973) 521. [20] E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [21] M.A. Howson, J. Phys. F 14 (1984) L25. [22] J.M. Ziman, Principles of the Theory of Solids, Cambridge University Press, Cambridge, 1972, p. 225. [23] S.R. Nagel, Phys. Rev. B 16 (1977) 1694. [24] I. Kokanovi´c, B. Leonti´c, J. Lukatela, A. Tonejc, Mater. Sci. Eng. A 375–377 (2004) 688. [25] I. Kokanovi´c, B. Leonti´c, J. Lukatela, Phys. Rev. B 60 (1999) 7440. [26] Z. Altounian, J.O. Strom-Olsen, Phys. Rev. B 27 (1983) 4149. [27] J.M. Daams, B. Mitrovic, J.P. Carbotte, Phys. Rev. Lett. 46 (1981) 65. [28] P.H. Kes, C.C. Tsuei, Phys Rev. B1 28 (1983) 5126.
Effect of disorder on the electrical resistivity in the partially crystalline Zr76Ni24 metallic glasses I. Kokanovi´c ∗ Department of Physics, Faculty of Science, University of Zagreb, P.O. Box 331, Zagreb, Croatia Received 17 October 2005; received in revised form 4 November 2005; accepted 5 November 2005 Available online 20 December 2005
Abstract The electrical resistivity has been measured for annealed amorphous and partially crystalline Zr76 Ni24 metallic glasses in the temperature range from 2 to 290 K. The effect of disorder on the electrical resistivity in the amorphous and partially crystalline Zr76 Ni24 metallic glasses has been studied. Quenching rate and thermal annealing have been used to modify the disorder of the samples. The electrical resistivity of the annealed partially crystalline samples up to an annealing temperature slightly below the first crystallization exotherm increases with decreasing heating rates, whereas the annealed amorphous samples show an opposite effect. After annealing the samples slightly above the first crystallization exotherm the electrical resistivity of the annealed samples decreases drastically. The temperature dependence of the electrical resistivity of the samples has been interpreted in terms of the weak localisation of electrons and contribution by electron–phonon scattering. The contribution by electron–phonon scattering increases with increasing the fraction of crystalline phases in the annealed samples. The amorphous and partially crystalline Zr76 Ni24 metallic glasses become superconducting at temperatures below 3.5 K. © 2005 Elsevier B.V. All rights reserved. Keywords: Metallic glasses; Crystallisation; Electrical transport; Superconductors
1. Introduction In this work the amorphous and partially crystalline Zr76 Ni24 metallic glasses were prepared by rapid solidification. Rapid solidification from the melt is a method widely used to obtain metallic glasses. On the other hand, by varying the quenching rate during melt spinning or the crystallization of some metallic glasses it is possible to produce partially crystalline alloys [1,2]. Their structure consists of a crystalline phase, of a few nanometer grain size, embedded in the remaining amorphous matrix. This crystalline phase is often in a non-equilibrium state and grows with a highly faulted morphology [3,4]. Thus, the metastable state of partially crystalline metallic glasses can relax structurally to a more stable state whenever the atoms attain an appreciable mobility upon annealing. Associated with the structural relaxation, many physical properties, such as electrical transport, change drastically [5–8]. It is well known that the temperature coefficient of the electrical resistivity, TCR = (1/ρ)dρ/dT,
∗
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is extremely sensitive to the number of quenched-in crystallites within the glassy matrix. Thus, the TCR provides a rather sensitive tool for probing the microscopic state of partially crystalline metallic glasses. Also, electrical resistivity measurements can help in understanding various phenomena in partially crystalline metallic glasses. It is an easy technique for studying phase transformations, defects and other structural changes in crystalline and amorphous alloys. On the other hand, the motion of electrons in disordered systems is one of the most fundamental problems in solid state physics. Both theoretical [9–12] and experimental [13–16] studies of the low temperature behaviour of electrical resistivity of disordered electronic systems have led to quantum corrections to the Boltzmann contribution. It has been shown that the diffusive motion of electrons in two-dimensional as well as in three-dimensional disordered systems entails quantum corrections to the electrical resistivity and the magnetoresistivity. There are two principal sources of quantum corrections, arising from weak localisation [9,10] and the Coulomb interaction [11,12]. Both of these corrections are important when the mean free path becomes short so that electron propagation between scattering events is no longer free-electron-like but diffusive. At low enough temperatures, where the elastic scattering time is
I. Kokanovi´c / Journal of Alloys and Compounds 421 (2006) 12–18
a few orders of magnitude shorter than the inelastic scattering time, the quantum corrections arising from the interference of the electronic partial waves are very important. It has been shown that constructive interference of the electronic waves can only be expected in a back-scattering geometry. This quantum interference effect will produce an increase of the sample resistivity. The magnitude of this additional contribution at given temperature is reduced by the presence of the inelastic electron–phonon, spin–orbit or spin–flip scattering, since they destroy the constructive interference. Recently results showed that, according to X-ray diffraction (XRD) results, two sides of quenched ribbons of the partially crystalline Zr76 Ni24 metallic glass have revealed different structures [17] (that in contact with the wheel—the dark side of the ribbon is called DSR and that in contact with the inert atmosphere—shiny side of the ribbon is called SSR). The diffraction pattern of the DSR shows a broad maximum centred at 2α = 36.1◦ of β = 4.2◦ corresponding to an amorphous phase. On the other hand, the XRD pattern of the SSR showed to consist of a set of sharp lines. We were able to fit the peaks to the orthorhombic Zr3 Ni phase. The lattice constants are obtained as follows: a = (1.098 ± 0.002), b = (0.880 ± 0.001) and c = (0.333 ± 0.002) nm. The SSR crystalline layer was relatively thin. After the SSR crystalline layer was abraded by approximately 5 m and examined with XRD only a broad maximum was present in the diffraction pattern, which means that the thickness of the crystalline layer was about 5 m. This crystalline layer was built of the orthorhombic Zr3 Ni nanocrystalline grains. The XRD patterns of both side of the partially crystalline Zr76 Ni24 metallic glass are shown in Ref. [17] and Figs. 1 and 2. Also, a thin interphase region was between amorphous and nanocrystalline phases. This interphase region or two-phase region was composed of the orthorhombic Zr3 Ni nanocrystalline grains embedded in the amorphous matrix. The average crystallite sizes of the orthorhombic Zr3 Ni nanocrystalline phase were from 10 to 30 nm and the crystallite sizes decrease with increasing annealing temperature up to slightly below the first crystallisation exotherm [17]. The purpose of this experiment was to study the effect of disorder on the electrical resistivity and to gain insight into the amorphous structure of the glass and the metastable nanocrystaline structure of the intermediate crystallisation products in the partially crystalline Zr76 Ni24 metallic glass. Different quenching rate and thermal annealing have been used to modify the disorder of the samples. All samples, including those, which are completely amorphous, are characterised by high room-temperature resistivity, they are paramagnetic and become superconducting at temperatures below 3.5 K [18].
13
then cut from the ribbon. The thermal stability of the samples was measured by means of a calibrated Perkin-Elmer DSC-4 differential scanning calorimeter using an atmosphere of purified argon gas. The samples (typically 5 mg) were packed tightly in sealed aluminium cans and an empty aluminium can was used as the reference. The temperature and heat flow axes were calibrated using Zn solid–liquid transition recorded with the same heating rates. The structural changes after heating the samples with heating rates of 10 and 60 K/min up to temperatures between 593 and 663 K were investigated by X-ray diffraction using Cu K␣ radiation. Particle induced X-ray emission spectroscopy (PIXE) analysis indicated that the samples had a composition Zr(76.0 ± 0.1) Ni(24.0 ± 0.1) . The electrical resistance was measured by a low-frequency (23.2 Hz) fourprobe ac method in the temperature range of 2–290 K; the precision extended to a few parts in 106 . The temperature measurement was performed in a 4 He cryostat. A calibrated cernox thermometer was used to monitor the temperature of the sample. The critical magnetic field, Hc2 (T), measurements were performed at temperatures down to 2 K in magnetic fields up to 1 T oriented parallel to the sample.
3. Results and discussion The values of heat flow, cp , determined from the DSC measurements of the partially crystalline Zr76 Ni24 metallic glass at heating rates of 10 and 60 K/min are shown in Fig. 1. The DSC trace of the partially crystalline Zr76 Ni24 metallic glass shows one sharp exothermic peak. In all cases, the cooling curves obtained after measuring up to 823 K did not show any thermal effects. This demonstrates the irreversible nature of the processes that give rise to the exothermal peaks during heating. The crystallization peak temperature Tpx corresponding to the maximum of the first exotherm is designated Tp1 . The values of Tp1 observed with the heating rates of s = 10 and 60 K/min were 620.4 ± 0.5 and 642.2 ± 0.5 K, respectively. In order to study the influence of quenched-in crystallites of the orthorhombic Zr3 Ni crystalline phase on the thermal stability of the residual amorphous phase of the as-quenched sample, the thermal stability of the Zr76 Ni24 metallic glass was also studied. The DSC trace of the Zr76 Ni24 metallic glass exhibited the similar DSC crystallization characteristics as did the partially crystalline Zr76 Ni24 metallic glass (as-quenched sample) and the position of the maximum of exotherm was unchanged for heating rates of s = 10 and 60 K/min.
2. Experimental techniques Ribbons of the amorphous and partially crystalline Zr76 Ni24 metallic glasses were prepared by rapid solidification of the melt on a single-roll spinning copper wheel in an argon atmosphere. By adjusting the melt-spinning parameters, primarily the wheel velocity, it is possible to reduce the cooling rate and produce partially crystalline metallic glasses. So, we were able to obtain Zr76 Ni24 ribbons containing approximately 90.4% of an amorphous phase and 9.6% of a crystalline phase. The samples, 5 mm long, 3.4 mm wide and 50 m thick were
Fig. 1. The heat flow cp of the partially crystalline Zr76 Ni24 metallic glass in the temperature range of 300–700 K at the heating rates, s, () s = 60 K/min and () s = 10 K/min.
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I. Kokanovi´c / Journal of Alloys and Compounds 421 (2006) 12–18
Fig. 2. The change in the temperature-dependent part of the electrical resistivity relative to its value at 290 K (ρ(T) − ρ(290 K))/ρ(290 K) of the Zr76 Ni24 metallic glass [as-quenched (), the annealed up to 593 K with s = 60 K/min (), the annealed up to 593 K with s = 10 K/min (), the annealed up to 635 K with s = 10 K/min () and the annealed up to 663 K with s = 60 K/min) (♦)].
The crystallisation processes of the partially crystalline and amorphous Zr76 Ni24 metallic glass were studied in more detail. The XRD patterns for different heat-treated samples are given in Ref. [17]. The samples were subjected to heating rates of 60 and 10 K/min in separate runs from 298 K to slightly below and to slightly above the first crystallization exotherm; then the heating was stopped and the sample cooled quickly to 298 K. XRD measurements performed at room temperature show no significant changes on the XRD pattern upon annealing of the partially crystalline and amorphous Zr76 Ni24 metallic glasses up to 593 K with the heating rate of 10 K/min. On the other hand, the samples that underwent the heating rate of 60 K/min up to 663 K show changes in the XRD patterns [17]. The XRD pattern consists of two sets of lines. One set of lines was successfully indexed by the Zr2 Ni tetragonal structure, space group I4/mcm with lattice parameters a = (0.648 ± 0.002) and c = (0.522 ± 0.002) nm. The other set of lines belongs the -Zr nanocrystalline phase. The crystalline peaks of the -Zr nanocrystalline phase were fitted to the hexagonal structure space group P6/mmm with lattice parameters: a = (0.506 ± 0.001) and c = (0.314 ± 0.001) nm. The first maximum peak of such XRD pattern is sharper and shifted to smaller scattering angle 2θ than one of the partially crystalline Zr76 Ni24 metallic glass. The maximum peak is centred at 2θ = 35.5◦ with a full width at half-maximum of β = 0.8◦ . By contrast the XRD pattern of the SSR of this annealed sample showed additional crystalline lines corresponding to the orthorhombic Zr3 Ni crystalline phase already described above. The XRD patterns of the DSR of the annealed partially crystalline Zr76 Ni24 metallic glass and annealed amorphous sample subjected to heating at 60 K/min up to 823 K consist of two sets of sharp lines, which correspond to ␣-Zr crystalline phase and Zr2 Ni crystalline phase. The crystalline peaks of the ␣-Zr crystalline phase were fitted to the hexagonal structure space group P63 /mmc with lattice parameters: a = (0.323 ± 0.002) and c = (0.516 ± 0.005) nm. The first maxi-
mum of the sample annealed up to 823 K could be resolved in two peaks with maxima at 2θ 1 = 35.35◦ (Zr2 Ni) and 2θ 2 = 36.43◦ (␣-Zr) with full widths at half-maximum of β1 = 0.52◦ and β2 = 0.36◦ , respectively. It should be point out that only a small fraction of the metastable cubic Zr2 Ni crystalline phase with a lattice parameter a = 1.227 nm was also identified. The XRD pattern of the SSR of the annealed partially crystalline Zr76 Ni24 metallic glass subjected to the heating rate 60 K/min up to 823 K shows an additional set of the sharp diffraction lines corresponding to the orthorhombic Zr3 Ni crystalline phase already described above. Thus, we conclude that the existence of the orthorhombic Zr3 Ni crystalline phase does not change the crystallization transformations of the Zr76 Ni24 amorphous phase and remains unchanged during annealing of the samples. On the basis of the DSC and XRD data obtained it was possible to identify the phases which occur in the heat treatment of the partially crystalline and amorphous Zr76 Ni24 metallic glass up to 823 K. The first stage of crystallization of the amorphous phase shows the evolution of -Zr and Zr2 Ni nanocrystalline phases. The second stage of crystallization of the samples shows the transformation of -Zr phase to a stable ␣-Zr nanocrystalline phase. The measured values of room-temperature electrical resistivity of the partially crystalline Zr76 Ni24 metallic glass, the Zr76 Ni24 metallic glass and their annealed samples are listed in Tables 1 and 2, respectively. The as-quenched samples were subjected to the heating rates of 60 and 10 K/min in separate runs from 298 K up to the annealing temperature, Ta (slightly below and to slightly above the crystallisation exothem), then the heating was stopped and sample cooled quickly to 298 K. The feature in all these electrical resistivity data is the rather high residual resistivity, reflecting the high atomic disorder in the samples. It can be seen from Tables 1 and 2 that the value of room-temperature electrical resistivity of the partially crystalline Zr76 Ni24 metallic glass (ρ(290K) = 136 cm) is lower than the value of the resistivity of the Zr76 Ni24 metallic glass (ρ(290 K) = 163 cm). According to the XRD results [17], it is explained as a consequence that the crystalline orthorhombic Zr3 Ni phase of about 5 m of thickness, embedded in the remaining amorphous matrix of the partially crystalline Zr76 Ni24 metallic glass involves the Zr3 Ni nanocrystalline layer of relatively low electrical resistivity immersed in a glassy matrix of high electrical resistivity since ordering of crystallographic structure increases electron mobility and leads to a resistivity reductions. Thus, the electrical resistivity of the partially crystalline Zr76 Ni24 metallic glass is composed of the structural components of both fractions (amorphous and Zr3 Ni nanocrystalline layer). The structural components of the amorphous phase and Zr3 Ni nanocrystalline layer in the electrical resistivity of the partially crystalline Zr76 Ni24 metallic glass were numerically separated and the electrical resistivity of the Zr3 Ni nanocrystalline layer of the partially crystalline Zr76 Ni24 metallic glass was estimated using following model. Suppose that the partially crystalline Zr76 Ni24 metallic glass, according to the XRD measurements, is composed of a parallel circuit made from the amorphous phase and Zr3 Ni nanocrystalline layer. Thus, the
I. Kokanovi´c / Journal of Alloys and Compounds 421 (2006) 12–18
15
Table 1 Effect of heat treatment on various physical properties Ta ± 1 (K)
s (K/min)
ρ(290 K) ± 5 ( cm)
(1/ρ)∂ρ/∂T ± 0.01 (×10−4 K−1 )
Tc ± 0.01 (K)
Tc ± 0.005 (K)
(∂Hc2 /∂T ) ± 0.1 (T/K)
D ± 0.1 (×10−5 m2 s−1 )
ξ 0 ± 5 (×10−10 m)
295 593 593 663
0 60 10 60
136 140 148 72
0.84 × 10−4 0.55 × 10−4 0.02 × 10−4 9.25 × 10−4
3.49 3.46 3.42 2.88
0.03 0.03 0.03 0.05
−2.7 −2.5 −2.5 −1.0
4.1 4.4 4.4 11.0
59 62 62 107
Ta is the maximum annealing temperature of partially crystalline Zr76 Ni24 metallic glass, s the heating rate at what Ta is reached, ρ(290 K) the electrical resistivity, (1/ρ)∂ρ/∂τ the temperature coefficient of the electrical resistivity, Tc the superconducting transition temperature, Tc the 10–90% superconducting transition width in the electrical resistivity, (∂Hc2 /∂T )Tc the shape of the upper critical field at Tc (0), D the electron diffusion constant and ξ 0 is the zero-temperature coherence length.
measured electrical resistivity for the parallel circuit, ρp , is given by; ρp =
ρ c ρa (1 − c)ρc + cρa
(1)
where ρc and ρa are the resistivities of the Zr3 Ni nanocrystalline layer and amorphous phase, respectively, and c is the volume fraction of the Zr3 Ni nanocrystalline layer. In order to estimate the volume fraction of the Zr3 Ni nanocrystalline layer of the partially crystalline Zr76 Ni24 metallic glass the total integrated crystallisation heat, H, was calculated. The H of the sample was determined from the area of the crystallisation exotherm (Fig. 1). The total crystallisation heats were H1 = (63.8 ± 0.5) J/g for the partially crystalline Zr76 Ni24 metallic glass, and H2 = (69.9 ± 0.5) J/g for the Zr76 Ni24 metallic glass. Analysis of the H2 compared with that of the H1 gives a direct estimate of the volume fractions of the two phases. Thus, the estimated volume fraction of the Zr3 Ni nanocrystalline layer was c = (9.6 ± 0.2)%, which is in good agreement with the estimated thickness of the Zr3 Ni nanocrystalline layer of about 5 m using the XRD measurements. The electrical resistivity of the Zr3 Ni nanocrystalline layer is obtained using Eq. (1) and the experimental values for the ρp (290 K) = 136 and ρa (290 K) = 163 gives ρc (290 K) = (53 ± 5) cm. Furthermore, the electrical resistivity data indicates an increase in the room-temperature electrical resistivity of the annealed partially crystalline Zr76 Ni24 metallic glass upon decreasing heating rate at an annealing temperature, Ta = 593 K, slightly below the crystallization exotherm (Table 1). These changes of the electrical resistivity of the annealed samples are irreversible. The increase in the electrical resistivity is related to
the rearrangement in atomic position in the interphase or twophase region of the amorphous and Zr3 Ni nanocrystaline phases described above. Also, the atomic rearrangements in the twophase region destroy the Zr3 Ni nanocrystals and decrease their nanocrystalline sizes [17]. Thus, the resistivity upturn is additionally enhanced by low grain sizes and a high density of the grain boundaries present in the two-phase region of the annealed partially crystalline Zr76 Ni24 metallic glasses. On the other hand, the decrease in the room-temperature electrical resistivity of the annealed Zr76 Ni24 metallic glass up to a temperature slightly below the crystallisation exotherm was ascribed with relaxation voids and strain in these annealed samples (Table 2). If the as-quenched samples are heated at 60 K/min up to 663 K, slightly above the crystallisation exotherm, the room-temperature electrical resistivity decreases drastically (Tables 1 and 2). This tendency is also in good agreement with the XRD measurements which showed that during the first stage of crystallisation, the amorphous phase transforms in the -Zr and Zr2 Ni nanocrystalline phases with nanocrystallite size of D = 10 nm [17]. Thus, further an ordering of crystallographic structure increases electron mobility and leads to resistivity reductions. The change in the temperature-dependent electrical resistivity, relative to its value at 290 K, ρ/ρ(290 K), of the Zr76 Ni24 metallic glass, the partially crystalline Zr76 Ni24 metallic glass and their annealed samples for the temperature range of 5–290 K is shown in Figs. 2 and 3, respectively. The error bars (not shown) are smaller than the size of the symbols. It can be seen from Fig. 2 that the temperature coefficients of the electrical resistivity, TCR, of the Zr76 Ni24 metallic glass is negative in the temperature range 10–295 K. The TCR of the Zr76 Ni24 metallic glass annealed in the annealing temperature range of 298–593 K
Table 2 Effect of heat treatment on various physical properties Ta ± 1 (K)
s (K/min)
ρ(290 K) ± 5 ( cm)
(1/ρ)∂ρ/∂T ± 0.01 (×10−4 K−1 )
Tc ± 0.01 (K)
Tc ± 0.005 (K)
(∂Hc2 /∂T ) ± 0.1 (T/K)
D ± 0.1 (×10−5 m2 s−1 )
ξ0 ± 5 (×10−10 m)
295 593 593 635 663
0 60 10 10 60
163 160 156 125 107
−1.30 × 10−4 −1.24 × 10−4 −0.47 × 10−4 6.64 × 10−4 7.31 × 10−4
3.52 3.48 3.45 2.95 2.90
0.020 0.015 0.018 0.020 0.020
−3.5 −3.4 −3.2 −3.0 −2.4
3.1 3.2 3.4 3.6 4.5
52 53 54 61 69
Ta is the maximum annealing temperature of Zr76 Ni24 metallic glass, s the heating rate at what Ta is reached, ρ(290 K) the electrical resistivity, (1/ρ)∂ρ/∂τ the temperature coefficient of the electrical resistivity, Tc the superconducting transition temperature, Tc the 10–90% superconducting transition width in the electrical resistivity, (∂Hc2 /∂T )Tc the shape of the upper critical field at Tc (0), D the electron diffusion constant and ξ 0 is the zero-temperature coherence length.
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I. Kokanovi´c / Journal of Alloys and Compounds 421 (2006) 12–18
ρ ∼ −T 1/2 ,
Fig. 3. The change in the temperature-dependent part of the electrical resistivity relative to its value at 290 K (ρ(T) − ρ(290 K))/ρ(290 K) of the partially crystalline Zr76 Ni24 metallic glass [as-quenched (), the annealed up to 593 K with s = 60 K/min (), the annealed up to 593 K with s = 10 K/min () and the annealed up to 663 K with s = 60 K/min) (♦)].
is also negative and its absolute value decreases with decreasing heating rates. If the Zr76 Ni24 metallic glass is heated up to slightly above the crystallisation exotherm, the TCR changes sign and become positive. On the other hand, it is apparent from Fig. 3 that the electrical resistivity of the partially crystalline samples follows a decreasing trend with decreasing temperature and saturates in the temperature region from 10 to 40 K. The values of the TCR of the partially crystalline Zr76 Ni24 metallic glass and its annealed samples in the annealing temperature range of 298–663 K are positive. A positive TCR is often a sign of the presence quenched-in crystalline within glassy matrix. The values of TCR at room temperature of the annealed samples are listed in Table 1. The TCR of the annealed partially crystalline Zr76 Ni24 metallic glasses decreases as the heating rates decrease for annealing temperatures slightly below the crystallisation exotherm. These observations are consistent with well-known Mooij correlation for disordered systems containing d electrons [19]. By confining his attention to those systems that had a relatively small TCR, he found that resistivity tends to be larger in systems which have a smaller value of TCR (Table 1). In strongly disordered amorphous samples when the electron mean path becomes comparable with interatomic distance an incipient weak localisation shows up and the correction to the Boltzmann conductivity [20,21] is given by σ =
2e2 πhLi (T )
(2)
where Li (T) is the inelastic diffusion length and Li (T) = (Dτ i )1/2 where D is the electronic diffusion constant and τ i is the inelastic scattering time. This was shown to lead to temperature dependence of electrical resistivity in strongly disordered alloys of the form: ρ ∼ −T,
Tθ Tθ
(3)
Tθ < T < Tθ 3
(4)
where Tθ is Debye temperature. On the other hand, at T > 20 K, in the samples with a nanocrystalline fraction and positive TCR, the temperature behaviour of the electrical resistivity has been explained with help of Ziman theory where the electrical resistivity depends on the Debye–Waller factor [22,23]. Several temperature regions may be distinguished in the temperature variation of the electrical resistivity of the Zr76 Ni24 metallic glass (Fig. 2). In the first one, from 5 up to 15 K the electrical resistivity changes insignificantly, in the second range from 20 to 90 K the electrical resistivity varies proportionally to temperature and in the third temperature region from 110 up to 250 K, the electrical resistivity is proportional to T1/2 . The best fit yields ρ/ρ(290 K) = (6.03 ± 0.01) × 10−2 − (3.27 ± 0.01) × 10−4 × T in the second temperature region. This temperature behaviour of the electrical resistivity can be treated within the framework of the weak localisation of electrons where the temperature dependence of ρ is determined by temperature dependence of the inelastic scattering time τ i which is assumed to be equal to βT−2 where β is the inelastic scattering parameter. In the third temperature region, the best fit yields ρ/ρ(290 K) = (6.84 ± 0.01) × 10−2 − (4.03 ± 0.01) × 10−3 × T1/2 . The root temperature dependence presumably comes from the weak localisation effect which changes over to ρ(T) ∼ −T1/2 due to the fact that at high temperatures, the τ i is given by simply form, τ i ∼ T−1 . However, for the annealed Zr76 Ni24 metallic glass the quality of fitting was considerably improved when an additional T2 or T dependent term was added in the second and third temperature regions, respectively. Moreover, it was found that the temperature dependence of the structural contribution of the nanocrystalline phase of the annealed samples subjected to the heating rate 60 K/min at the annealing temperature above the crystallisation exotherm (Ta = 663 K) fits to the Ziman theory [22,23]. According to XRD results [17], in the temperature region from 20 to 70 K, the best fit yields ρ/ρ(290 K) = (−2.42 ± 0.02) × 10−1 + (8.85 ± 0.08) × 10−6 × T2 whereas in the temperature region from 200 to 300 K, ρ/ρ(290 K) = (−2.10 ± 0.02) × 10−1 + (7.65 ± 0.08) × 10−4 × T. Thus, the temperature dependence of the electrical resistivity for the annealed Zr76 Ni24 metallic glass suggests that contribution by electron–phonon scattering increases with increasing a fraction of crystalline phase in the annealed samples. In contrast to the metallic glass, the temperature dependence of the electrical resistivity for the partially crystalline samples shows very peculiar electrical resistivity behaviour (Fig. 3). It consists of a plateau region in the temperature region from 10 up to 45 K. The plateau in the electrical resistivity plot originates from the competing effects of the two different scattering mechanisms due to simultaneous presence of the amorphous and orthorhombic Zr3 Ni crystalline phases, both of which are pronounced in these samples. The XRD patterns of these samples support this explanation [17]. The electrical resistivity data for the partially crystalline sample are well fitted to a function
I. Kokanovi´c / Journal of Alloys and Compounds 421 (2006) 12–18
Fig. 4. The temperature-dependent electrical resistivity relative to its value at 4.2 K, ρ(T)/ρ(4.2 K), vs. temperature below 4.5 K of the partially crystalline Zr76 Ni24 metallic glass [as-quenched (), the annealed up to 593 K with s = 60 K/min (), the annealed up to 593 K with s = 10 K/min () and the annealed up to 663 K with s = 60 K/min (♦)].
a + bT + cT2 in the temperature region from 45 up to 80 K, while at higher temperature to a function a + bT1/2 + cT. This can be treated within the framework of the weak localisation and the Ziman theory [22,23]. Also, it was found that the temperature dependence of the structural contribution of the nanocrystalline phases of the annealed partially crystalline Zr76 Ni24 metallic glass subjected to 60 K/min at the annealing temperature above the exotherm (Ta = 663 K) fits to the Ziman theory [22,23]. Thus, in the temperature region from 15 to 65 K, the best fit yields ρ/ρ(290 K) = (−3.29 ± 0.02) × 10−1 + (1.59 ± 0.08) × 10−5 × T2 whereas in the temperature region from 210 to 300 K, ρ/ρ(290 K) = (−2.51 ± 0.02) × 10−1 + (8.67 ± 0.08) × 10−4 × T. The numerical analysis of the electrical resistivity shows that weak localisation contribution to scattering does not exceed 6% of the measured electrical resistivity of the Zr76 Ni24 metallic glass and diminishes with increasing annealing temperature and decreasing heating rates in its annealed samples. Furthermore, it was found that the temperature dependence of electrical resistivity of the Zr3 Ni nanocrystalline phase fits well to the Ziman theory and dominates over structural contribution of the amorphous phase of the partially crystalline Zr76 Ni24 metallic glass at temperatures higher than 45 K. Moreover, for samples possessing a room-temperature resistivity below 150 cm for which the TCR is positive, the Ziman theory should be applied to explain the temperature dependence of the electrical resistivity of the annealed samples. However, as the electrical resistivity increases above 150 cm and the mean free path of electron propagation approaches the value of the interatomic distance, the TCR becomes negative and the theory of weak localisation has been invoked for a suitable explanation of the electrical resistivity results. The temperature-dependent electrical resistivity relative to its value at 4.2 K, ρ(T)/ρ(4.2 K), of the partially crystalline Zr76 Ni24 metallic glass and its annealed samples in the vicinity of Tc is shown in Fig. 4. The Tc was determined as the midway point
17
on the resistivity transition versus temperature transition. The values of Tc are listed in Tables 1 and 2. It can be seen from Tables 1 and 2 and Fig. 4 that all the superconducting transitions are very sharp and the temperature difference between the 90% and 10% points of the resistivity change is typically less than 50 mK. The Tc of the samples heated at an annealing temperature below the crystallisation exotherm changes slightly with decreasing heating rates. In all the annealed samples, the Tc decreases with decreasing heating rates and increasing annealing temperatures. This result is in contrast to the annealed Zr80 Fe20 and Zr80 Co20 metallic glasses [8,24]. The increase in the Tc upon annealing up to an annealing temperature below the first crystallisation exotherm in the Zr80 Fe20 and Zr80 Co20 metallic glasses has been related to the decrease in the spin-fluctuation mass enhancement parameter, λsp . It is well known that the influence of spin fluctuations on superconductivity in Zr-3d metallic glasses decreases with increasing Zr concentration whereas at fixed Zr concentration it decreases as one move from Fe towards Ni [13,24–26]. Thus, the effect of the λsp on superconductivity in the investigated samples can be negligible. Using the modified form of the McMillan equation [27] it can be shown that this change in Tc upon annealing is related to a decrease in the electron–phonon coupling constant. According to the results of the analysis XRD data in Ref. [17], the decrease in λph created by a hardening of phonon modes as a result of relaxation of the quenched-in strains and redistribution of the defects decreases the Tc . Thus, we can conclude that for the heating rates of 60 and 10 K/min and annealing temperature slightly below the crystallisation exoterm the annealing reduces λph , hence the Tc decreases in the annealed samples. Furthermore, according to XRD data [17], the modification in the chemical short-range order due to heating above the crystallisation exotherm resulting in evolution of the -Zr phase and Zr2 Ni nanocrystalline phase plays an important role in determining the Tc of the annealed samples subjected to different annealing temperatures. The values of the Tc of these annealed samples are very close with the value of the Tc of the Zr2 Ni metallic glass (Tc = 2.78 K) [13]. It is well-known that the Tc –s decrease approximately linearly with decreasing Zr content in amorphous Zr-3d (3d = Ni, Co and Fe) alloys [26]. The decrease of the Zr content reduces the electronic density of states at the Fermi level, N(EF ) and this decrease of the N(EF ) is related with the decrease of λph . The superconducting properties of the partially crystalline Zr76 Ni24 metallic glass up to slightly above the crystallisation exotherm are characterised by a somewhat less sharp electrical resistive transition than observed in the as-quenched sample (Table 1). Also, below 3.5 K down to superconducting transition small anomaly of the order 5 % in the electrical resistivity of the same annealed sample is observed (Fig. 4). It seems that the anomaly is due to small inhomogeneities, which are still present in this sample. The value of the (∂Hc2 /∂T )Tc was determined from the slope of the measured Hc2 versus Tc curve at Tc (0) and listed in Tables 1 and 2. The applied magnetic field of 1 T shifts the Tc of the Zr76 Ni24 metallic glass down by approximately 0.29 K, which correspond to a slope of −3.5 T/K for the upper critical magnetic field near Tc . The values of the (∂Hc2 /∂T )Tc depend
18
I. Kokanovi´c / Journal of Alloys and Compounds 421 (2006) 12–18
very much on the microstructural defect density in the sample. For instance, the slope of the partially crystalline Zr76 Ni24 metallic glass is (∂Hc2 /∂T )Tc = −2.7 T/K whereas for the annealed partially crystalline Zr76 Ni24 metallic glass subjected to the heating rate of 60 K/min at the annealing temperature of 663 K, (∂Hc2 /∂T )Tc = −1 T/K. The absolute value of the (∂Hc2 /∂T )Tc decreases with decreasing heating rates and increasing annealing temperatures or increasing the nanocrystalline fraction of the annealed samples (Tables 1 and 2). Furthermore, from the Tc , (∂Hc2 /∂T )Tc and by using wellknown expression for the zero-temperature coherence length, ξ 0 = 1.81 × 10−8 × (Tc × ∂Hc2 (T)/∂T)−1/2 [28], the value of the ξ 0 was estimated. The values of the ξ 0 are given in Tables 1 and 2. It can be seen from Tables 1 and 2 that the ξ 0 increases significantly with annealing at annealing temperature above the crystallisation exotherm. For instance, the value of the ξ 0 of the partially crystalline Zr76 Ni24 metallic glass is ξ 0 = 5.9 nm whereas for its annealed sample at the annealing temperature of 663 K, ξ 0 = 10.7 nm. The increase of the ξ 0 is larger for the annealed partially crystalline Zr76 Ni24 metallic glass than for the annealed Zr76 Ni24 metallic glass (Tables 1 and 2). The homogeneity of the annealed samples is judged to be high as evidenced by a small superconducting transition width and sharp electrical resistive transition. This suggests that the homogeneity is on a spatial scale of less than the ξ 0 . The values of the diffusion coefficient were also obtained, using the relation 4kB D= πe
∂H −1 c2 ∂T Tc
(5)
where kB is the Boltzmann constant and e is the electronic charge. The values of D are given in Tables 1 and 2. It can be seen from Tables 1 and 2 that the values of D increase with increasing annealing temperatures and decreasing heating rates. In the case of strong scattering, the electron motion becomes diffusive so that the electrical resistivity is expressed as ρ = (e2 N(EF )D)−1 . Thus, we can conclude that the decrease of the electrical resistivity of the annealed amorphous samples with the decreasing heating rates and increasing annealing temperatures is caused mostly by the increase of the D in the annealed samples. 4. Conclusion The Zr76 Ni24 metallic glass and partially crystalline Zr76 Ni24 metallic glass were prepared by melt spinning. The effect of annealing on the resistivity in the Zr76 Ni24 metallic glass and partially crystalline Zr76 Ni24 metallic glass has been investigated. The resistivity of the annealed partially crystalline samples up to a temperature slightly below the crystallization exotherm increases with decreasing heating rates, whereas after annealing the samples slightly above crystallization exotherm the resistivity decreases drastically. The temperature dependence of the electrical resistivity of the samples has been interpreted in terms of the weak localisation of electrons and
contribution by electron–phonon scattering. The contribution by electron–phonon scattering increases with increasing the fraction of crystalline phase in the annealed samples. The asquenched samples become superconducting at temperatures below 3.52 K. The superconducting transition temperature, Tc , of the amorphous and partially crystalline Zr76 Ni24 metallic glass subjected to different heating rates up to slightly below and above the crystallisation exotherm decreases with decreasing heating rates and increasing annealing temperatures. The homogeneity of the annealed samples is judged to be high as evidenced by a small superconducting transition width and sharp electrical resistive transition. This suggests that the homogeneity is on a spatial scale of less than the zero-temperature coherence length ξ 0 . The resistivity decrease of the annealed Zr76 Ni24 metallic glass subjected to the heating rates of 60 and 10 K/min at temperatures slightly below or above the crystallization exotherm is caused mostly by the increase of the D. References [1] Y. Yoshizawa, S. Oguma, K. Yamauchi, J. Appl. Phys. 64 (1988) 6044. [2] A. Inoue, K. Nakazato, Y. Kawamura, T. Masumoto, Mater. Sci. Eng. A 179–180 (1994) 654. [3] Y.D. Dong, G. Gregan, M.G. Scott, J. Non-Cryst. Solids 43 (1981) 403. [4] U. Koster, U. Herold, in: H.J. G¨untherodt, H. Beck (Eds.), Glassy Metalls I, Springer Verlag, Heidelberg, 1981, p. 225. [5] C.C. Koch, D.M. Kroeger, J.S. Lin, J.O. Scarbrough, W.L. Johnson, A.C. Anderson, Phys. Rev. B 27 (1983) 1586. [6] O. Laborde, A. Ravex, J.C. Lasjaunias, O. B´ethoux, J. Low Temp. Phys. 56 (1984) 461. [7] S.J. Poon, Phys. Rev. B 27 (1982) 5519. [8] M. Sabouri-Ghomi, Z. Altounian, J. Non-Cryst. Solids 205–207 (1996) 692. [9] H. Fukuyama, T. Hoshino, J. Phys. Soc. Jpn. 50 (1981) 2131. [10] B.L. Al’tshuler, A.G. Aronov, A.I. Larkin, D.E. Khmel’nitzkii, Sov. Phys. JETP 54 (1981) 411. [11] P.A. Lee, T.V. Ramakrishnan, Phys. Rev. B 26 (1982) 4009. [12] B.L. Al’tshuler, A.G. Aronov, in: A.L. Efros, M. Pollak (Eds.), Electron–Electron Interaction in Disordered Systems, North-Holland, Amsterdam, 1985, pp. 1 and 109. [13] I. Kokanovi´c, B. Leonti´c, J. Lukatela, Phys. Rev. B 41 (1990) 958. [14] A.K. Meikap, Y.Y. Chen, J.J. Lin, Phys. Rev. B 69 (2004) 212202. [15] I. Kokanovi´c, B. Leonti´c, J. Lukatela, Phys. Status Solidi (b) 241 (2004) 908. [16] A.K. Majumdar, J. Magn. Magn. Mater. 263 (2003) 26. [17] I. Kokanovi´c, A. Tonejc, Mater. Sci. Eng. A 373 (2004) 26. [18] I. Kokanovi´c, B. Leonti´c, J. Lukatela, J. Ivkov, Phys. Rev. B 42 (1990) 11587. [19] H. Mooij, Phys. Status Solidi A 17 (1973) 521. [20] E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [21] M.A. Howson, J. Phys. F 14 (1984) L25. [22] J.M. Ziman, Principles of the Theory of Solids, Cambridge University Press, Cambridge, 1972, p. 225. [23] S.R. Nagel, Phys. Rev. B 16 (1977) 1694. [24] I. Kokanovi´c, B. Leonti´c, J. Lukatela, A. Tonejc, Mater. Sci. Eng. A 375–377 (2004) 688. [25] I. Kokanovi´c, B. Leonti´c, J. Lukatela, Phys. Rev. B 60 (1999) 7440. [26] Z. Altounian, J.O. Strom-Olsen, Phys. Rev. B 27 (1983) 4149. [27] J.M. Daams, B. Mitrovic, J.P. Carbotte, Phys. Rev. Lett. 46 (1981) 65. [28] P.H. Kes, C.C. Tsuei, Phys Rev. B1 28 (1983) 5126.