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Studies of icosahedral AlPdRe quasicrystals across the metal–insulator transition
Journal of Non-Crystalline Solids 334&335 (2004) 356–362 www.elsevier.com/locate/jnoncrysol
Studies of icosahedral AlPdRe quasicrystals across the metal–insulator transition € Rapp O.
a,*
, V. Srinivas b, P. Nordblad c, S.J. Poon
d
a b
Department of Solid State Physics, IMIT, KTH-Electrum 229, 164 40 Stockholm-Kista, Sweden Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur, 721 302 India c Department of Materials Science, Uppsala Universitet, Box 534, SE 751 21 Uppsala, Sweden d Department of Physics, University of Virginia, Charlottesville, VA 22901, USA
Abstract The metal–insulator transition (MIT) in icosahedral AlPdRe quasicrystals is investigated by studying changes in sample properties across the transition. The magnetoresistance (MR) the conductivity rðT Þ, and the magnetic susceptibility vðT Þ have been studied. The MR results show that an MIT occurs for resistance ratios R [ ¼ q(4.2 K)/q(295 K)] of about 20–30. r(0 K) is finite, and decreases over four orders of magnitude with increasing R. Besides the MR, the most prominent change in a studied property across the MIT is a change of slope of log rð0Þ vs R. In contrast, no features in the resistivity from 1.5 to 300 K reveals an MIT. New results for vðT ; RÞ are presented. The diamagnetic v decreases from the metallic to the insulating side, while on the insulating side vðT ; RÞ is almost independent of R. Although the MR indicates a disorder driven MIT, there is no support for this picture from the results for v. 2004 Elsevier B.V. All rights reserved. PACS: 71.30.+h; 72.15.)v; 72.20.)i; 75.20.Ck; 75.20.En
1. Problems and outline The existence of a metal–insulator transition (MIT) in icosahedral (i) AlPdRe has been inferred from some recent studies of electronic transport properties. In particular, insulating behavior in the form of Efros– Shklovskii variable range hopping [1], has been found in the magnetoresistance (MR) of samples with large resistance ratios R [ ¼ q(4 K)/q(295 K)] [2], and an MIT has also been suggested from studies on the metallic side of the transition of the MR [3], and the conductivity, r, above 400 mK [4]. These three different methods all gave estimates that an MIT as a function of increasing R should occur in the region of R 20–30. Normally one would expect the low temperature rðT Þ to give the most direct evidence for an MIT. However, this has not been the case for i-AlPdRe, where published results of rðT Þ instead present a manifold of interpretations, and partially contradicting views [5–16]. These differences concern both procedures and results for
*
Corresponding author. Tel.: +46-8 790 4190; fax: +46-8 752 7850. € Rapp). E-mail addresses: [email protected], [email protected] (O.
0022-3093/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2003.12.002
extrapolating rðT Þ to r ¼ 0 or to T ¼ 0 K, as well as methods and results from analyses of the low temperature rðT Þ of assumed insulating samples in terms of Efros–Shklovskii or Mott variable range hopping, VRH. The varying temperature regions often used for such analyses, and the different R-values of samples studied contribute to these discrepancies. An additional difficulty is the finite zero temperature conductivity rð0Þ [17], by itself incompatible with VRH behavior, which may ruin the analyses if neglected, and tends to make them overflexible when taken into account. The MR appears to offer a more powerful tool to study an MIT in i-AlPdRe. First, the MR of quasicrystals is not sensitive to impurity phases. Possible minor metallic impurities precipitating in grain boundaries and not detectable in standard X-ray diffraction, have been assumed to be one reason for a finite rð0Þ. However, the much smaller resistivity of such precipitates would give a negligible contribution to the MR [18]. Second, the vast phase space of magnetic field and temperature is controlled in experiments, and numerical convergence in fitting theoretical expressions is usually satisfactory. In contrast, almost any theory with some fitting parameter(s) can produce an apparent description
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of the smooth rðT Þ. Nevertheless, the interpretation of the MR of i-AlPdRe is not unproblematic. For instance, the ES-VRH model cannot be applied to data at T < 1 K in a straightforward way, and the reason for this failure is not understood [17]. The variation of the magnetic susceptibility, v, with T and R would also be expected to reflect an MIT. Enhanced magnetic interactions in the vicinity of a transition are anticipated in different scenarios, such as the Brinkman–Rice treatment of interacting electrons in a Hubbard model without disorder [19], and scaling theories of interaction and localization in disordered media [20,21]. Such effects have been well verified in disordered systems. In doped semiconductors e.g., enhanced paramagnetic susceptibility and (or) the appearance of local moments have been observed for decreasing temperature or doping concentration [22–24]. In amorphous metals enhanced interactions were observed by monitoring electronic disorder with high energy neutron irradiation [25]. One would expect that an MIT in quasicrystals, though atomically well ordered, belongs to this class of electronically disordered systems, as indicated e.g. from the MR [3]. The MIT in i-AlPdRe is unusual and poorly understood. Recently it was found, however, that the MIT can be monitored by varying R [26]. This finding provides a tool for detailed studies of the transition. In particular, the variation of a physical property across the MIT can be accessed by the variation of R, and may give important clues to the nature of this MIT. In the present paper we exploit this approach, and report on results across the MIT for the MR in Section 2, the conductivity (Section 3), and the magnetic susceptibility (Section 4). Recent results will be reviewed, and new results, in particular for vðT ; RÞ, will be presented. The paper ends with a brief summary.
2. Magnetoresistance across the MIT The MR gives the most clear evidence for an MIT in i-AlPdRe, since it can be quantitatively described in terms of well-established theories on both the metallic and the insulating side of the MIT. In fact, such a successful description of an electronic transport property is exceptional in quasicrystals. To illustrate an MIT in the MR, it is not necessary to resort to detailed numerical analyses. Qualitative studies of the field dependence are adequate since the MR displays a sequence of characteristic B-dependences for increasing magnetic field B, which are distinctly different on the metallic and insulating sides. Fig. 1 illustrates this qualitative argument for an MIT in between two i-Al70:5 Pd21 Re8:5 samples with R ¼ 11, well into the metallic side [3], and R ¼ 160, which is far into the insulating side [2]. For metallic samples the MR
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Fig. 1. The magnetoresistance, ½qðB; T Þ qð0; T Þ=qð0; T Þ at T ¼ 4:2 K vs. magnetic field for two samples; R ¼ 11 well into the metallic side of the MIT, and R ¼ 160 far into the insulating side. Characteristic field dependences of the metallic and insulating MR are shown by the curve segments. (Data from [2,3].)
is well described by weak localization and electron– electron interaction theories [3]. These theories, which have been developed in detail [27,28], describe quantum corrections to transport properties in weakly (electronically) disordered systems. All contributions to the MR increase as B2 for small B (in alloys with not too weak spin–orbit scattering), and as B1=2 at intermediate fields. In Efros–Shklovskii theory [1] for insulators, MR decreases at small B as )B, due to interference between different trajectories contributing to the hopping process, followed at larger B by an increase, MR +B2 , due to the shrinking of electronic wave functions in field. At still larger fields the continued increase of the MR is characterized by a series of different exponents, where the first one is MR B2=3 . All these feature of metallic and insulating MR are born out by experiments (Fig. 1). One well-known, remarkable property of many quasicrystals is that increased atomic disorder leads to decreased resistivity [29]. This has long been expected also for i-AlPdRe, and was shown recently by structural and transport studies of samples irradiated with high energy neutrons [26]. In this way an insulator–metal transition can be monitored by an increasing irradiation dose, from an insulating state with large R, to metallic
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samples with small R. It is a great advantage of this method that a single sample can be studied across the transition, thus focussing on intrinsic properties, largely circumventing worries on the role of sample quality and impurities, ubiquitous in experimental work on quasicrystals. Fig. 2 illustrates some results. Three sets of data are shown: (i) a sample with R ¼ 67 in the original state, exposed to a series of irradiation doses, D, up to 6.8 · 1019 cm2 (circles) [26]; (ii) a sample with R ¼ 27, irradiated with D up to 3.7 · 1019 cm2 (rhomboids) [26]; and (iii) a series of as made arc-quenched foil samples, where R varied from 45 to 2 [3]. MR vs. R is found to follow qualitatively the same function for these three sets of samples, with an MR which first increases with increasing R, goes through a maximum and decreases to negative values. Assigning the sign change of the MR to an MIT, Fig. 2 suggests that the MIT in i-AlPdRe occurs in the region R 20–30, in agreement with the estimates quoted above [2–4]. The MR is thus a characteristic function of R only, independent of sample history and the details of impurities or defects in the icosahedral state. The MR thus provides a fairly perspicuous picture of the MIT, yet an incomplete one. For instance, in the vicinity of the transition one would like to understand the development
Fig. 2. MR vs. R at T ¼ 4:2 K, B ¼ 2 T for three sets of samples; circles and rhomboids: two series of samples which were neutron irradiated (data from [26]), down triangles: as-made samples, [3]. Open symbols are as-made samples, filled symbols are irradiated samples. The MR is a similar function MRðRÞ for these different samples.
of the localization length and the hopping length, which govern the variable range hopping. It has not yet been possible to find reliable estimates of these parameters.
3. Conductivity across the MIT The search for a vanishing conductivity at the MIT has frequently been used as a method to locate the MIT. In i-AlPdRe alloys these efforts, have however, given rather different results. In extrapolations as a function of T 1=2 , it was found that a finite rðT Þ vanished at about R ¼ 20 [11] or 12.8 ± 0.5 [14]. Using the logarithmic derivative of the measured resistance with respect to in T , it was instead concluded that thin film i-AlPdRe samples were metallic at R ¼ 3 and insulating at R ¼ 9 [15]. What function to extrapolate against is a question of debate. Different results obtained in extrapolations vs. different functions of T were recently illustrated [4]. In addition a different conduction mechanism may dominate at low temperatures, below those from which extrapolations were made. A finite zero temperature conductivity, rð0Þ, also in insulating i-AlPdRe, is one mechanism which invalidates extrapolations. Results in this area have been controversial [10–16]. When measurements for high R samples were taken to below 20 mK, it was found that saturation of rðT Þ appears to be a general property of i-AlPdRe [17]. In Fig. 3 the results of [17] have been extended, in particular by including also tabulated results at 1.5 K for polygrained samples with small R, ðR 6 4Þ [3], and a datum at 4.2 K for single grain i-AlPdRe with R ¼ 1:8 [30]. For low R samples, with small q(4.2 K) and jdq=dT j, this difference in evaluation temperatures is insignificant. rð0Þ decreases continuously over 4 orders of magnitude with increasing R. The range of rð0Þ, and the different samples used, strongly suggest that a finite rð0Þ is an intrinsic property of i-AlPdRe also for insulating samples. At the location of the MIT, as obtained from the MR, the relation between rð0Þ and R changes slope in a log r vs. T diagram. In the metallic phase there is a precipitous drop of rð0Þ with increasing R, while in the insulating state, rð0Þ decreases exponentially with R at a slower rate. The reason for a finite rð0Þ is not known. The observations hint at two conduction channels in quasicrystals. This possibility was suggested early [31,32]. In insulating i-AlPdRe we propose that these two channels could be variable range hopping and quantum tunneling between residual critical states, electron states which are neither localized as in insulators, nor extended as in metals, but instead fall off as a power law [33]. Monitoring R by neutron irradiation provides a convenient way to search for characteristic features in qðT Þ when traversing the MIT. Such studies of qðT Þ
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Fig. 3. Estimated zero temperature conductivity, rð0Þ vs. R. The filled square is a single grain sample [30], open triangles: melt-spun and annealed samples from [3,10]. Circles: estimates from measurements to below 15 mK. The open circle at R ¼ 20 is from [4], the other circles are data from [17]. Melt-spun samples, (open circles) and ingot samples (filled circles) were further described in [17].
have so far only been made in the range 1.5–300 K [26]. Nevertheless, as described above, an MIT in the magnetoresistance is clearly observable in this temperature range, and a comparison is interesting. Fig. 4 shows qðT Þ vs. T on double logarithmic scales for a series of irradiated samples with doses, D, from 5 · 1017 to 6.8 · 1019 n/cm2 . In this process R decreased from 57 to 1.2. Although q and jdq=dT j increase with decreasing dose at all temperatures, no particular feature can be observed which would distinguish an MIT anywhere on these curves. This conclusion remains valid if data are plotted vs. a function of temperature chosen to reveal some special, characteristic behavior. Variable range hopping conductivity, a likely mechanism on the insulating side, is of the form rðT Þ ¼ r0 exp½ðT0 =T Þm ;
ð1Þ
with m ¼ 1=4 or 1/2, in Mott [34] or Efros–Shklovski [1] VRH, respectively. Fig. 5 illustrates VRH according to these theories for the two samples in Fig. 4, which encompass an MIT, as determined from the MR. Data in Fig. 5 cover the range 1.5–4.2 K, where VRHbehavior could be expected. However, the results are inconclusive. First, a description in terms of a VRH theory is of comparable quality both for the insulating R ¼ 57 sample, and for the R ¼ 13 metallic sample.
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Fig. 4. qðT Þ vs. T across the MIT for a neutron irradiated sample. R, indicated in figure, decreases with increasing dose from 5 · 1017 (top curve) to 6.8 · 1019 n/cm2 (bottom curve). Data from [26].
Second, there is no apparent difference between Mott and Efros–Shklovskii theories, with VRH fits showing comparable discrepancies for both samples. Data in both panels are slightly concave. This may be due to the finite rð0Þ. Subtracting rð0Þ from the observed rðT Þ before taking the logarithm, has a stronger effect at low temperatures, where rðT Þ is smaller and may thus improve such fits. Another possibility is a temperature dependent prefactor in Eq. (1), affecting the analyses.
4. Magnetic susceptibility across the MIT Previous studies have shown that the magnetic susceptibility of i-AlPdRe is diamagnetic and weakly temperature dependent above 100 K [35–37]. Only few measurements have been made on samples of different resistivities with R in the range 7–47 [35], and about 2–9 [36], respectively [38]. Irregular trends of v as a function of R were observed. However, some considerations discussed below, which are important for i-AlPdRe, were not, or may not have been taken into account in [35,36]. When looking for a relation between v and R of iAlPdRe, two possible sources of errors must be controlled: (i) R-values may vary between different portions of a larger ingot or between different pieces of samples from the same ingot in a melt-spinning process. Measurements of v must therefore be made on the same
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samples were selected for the present studies with Rvalues of 3, 31, 71, and 160. Magnetic measurements were made in a Quantum Design MPMS SQUID magnetometer. The samples were investigated by magnetization vs. magnetic field measurements on increasing and decreasing magnetic field in the range 0–4 T at several different temperatures between 2 and 300 K. The resistivity measurements were made in standard four point arrangement with contacts of silver paint on the same sample pieces. The measured magnetization M vs. applied magnetic field H did not extrapolate to the origin. This was ascribed to small ferromagnetic clusters of weight fraction x and saturation magnetization r. The measured magnetization M was therefore assumed to be of the form M ¼ vm H ¼ vH þ xr:
ð2Þ
vm is the measured apparent magnetic susceptibility, and v the bulk value. The results were analyzed in the form M=H vs. 1=H . Above about 0.3 T these data were well described by straight lines from which v and xr were determined. The general form of vðT Þ in Fig. 6 is in agreement with previous results [35–37]. The small increase of v at low temperatures, similar in all samples, may be unrelated to the icosahedral phase and can be due e.g. to small clusters which behave paramagnetically at low T , or to unsaturated ferromagnetic clusters. xr in Fig. 7 is nearly constant for each sample, suggesting that the Curie temperatures of such clusters are above the present measurement range. The irregular variation of xr between samples could be due to extrinsic impurities or to slightly varying Fe distribution in the samples due to varying annealing conditions. Taking a simplified view that r in Eq. (2) is due to saturated Fe of 200 erg/Oe g, gives x in the range 8–20 ppm for the present samples. The nominal Fe concentration is about 700 ppm. The results thus suggest that most Fe atoms substitute on Fig. 5. Test of VRH behavior between 1.5 and 4.2 K: (a) the R ¼ 57, and (b) the R ¼ 13 samples. (Filled circles) top scales, ES-VRH and (open circles) bottom scales, Mott VRH.
small sample piece from which the R-value is determined. (ii) Minor magnetic impurities in the starting elements, or spurious magnetic contaminations from sample handling and measurement procedures, can change a weak diamagnetic signal. The samples had nominal composition Al70:5 Pd21 Re8:5 . Starting materials were powders of the constituting elements. The dominating magnetic impurity was 0.1 wt% Fe in the Al powder, with negligible magnetic contributions from the other elements. Ingot samples were prepared as previously described in some detail [3]. Pieces for measurements had masses of 10–20 mg. Four
Fig. 6. v vs. T for four samples. The R-values are: () 3; (n) 31; (m) 71 and (.) 160. Bars indicate the largest estimated errors.
€ Rapp et al. / Journal of Non-Crystalline Solids 334&335 (2004) 356–362 O.
Fig. 7. xr vs. T . The R-values of the samples are: () 3; (n) 31; (m) 71 and (.) 160. Some of the largest estimated errors have been indicated.
non-magnetic sites, in analogy with Mn in i-AlPdMn [39]. No enhancement is observed in magnetic interactions with increasing R, neither in a paramagnetic component, nor in an increased fraction of spontaneous magnetic moments. This conclusion does not depend on the reasons for an enhanced v at low T , or the scatter in values of xr between different samples. In fact the only observation of a clear R-dependence that can be made from these data is that the diamagnetic susceptibility for the metallic sample ðR ¼ 3Þ is comparatively weak, (Fig. 6), while for R P 31, it is stronger and within measurement accuracy roughly independent of R. Thus any change in v ¼ vðRÞ across the MIT is opposite to that expected from conventional results in electronically disordered systems [20–26]. A possible explanation for the drop in v from R ¼ 3 to 31 is a reduction of the small metallic Pauli contribution at the MIT. A comparison with the density of states, DðeF Þ, from the electronic specific heat, c, is complicated by the expected but unknown contribution to c from two level tunneling states [37,40]. In a rough order of magnitude calculation we assumed that the Pauli contribution to v is lost at the MIT. For c ¼ 0:2 mJ/mol K2 , as for a metallic sample of i-AlPdRe with R ¼ 4 [41] close to our R ¼ 3 sample, and neglecting tunneling contributions and electron–phonon interaction, Dv ðeF Þ was evaluated from c. Similarly, any Stoner enhancement was neglected when estimating Dv ðeF Þ from the decrease, Dv 8 108 cm3 /g, of v from R ¼ 3 to the other samples in Fig. 6. We then found Dv ðeF Þ and Dc ðeF Þ to be of the same order of magnitude, ½Dv ðeF Þ=Dc ðeF Þ 1:5 in qualitative support of this conjecture. 5. Conclusions An MIT can be identified from the magnetoresistance of icosahedral AlPdRe to occur as a function of increasing resistance ratio R in the region of R 20–30.
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This result allows monitoring an MIT as a function of R. We have pursued the idea to select samples over a suitable range of R-values to look for and study the variation of physical properties across the MIT. MRðRÞ, rðRÞ, and vðRÞ were investigated. The most clear case, besides the MR, where a distinctly different behavior can be observed across the transition is the Rdependence of the finite zero temperature conductivity, r(0 K, R), which shows a much slower exponential decrease for increasing R on the insulating side. In contrast, the finite temperature rðT ; RÞ does not display any characteristic feature from which an MIT could be identified. vðT ; RÞ exhibits a roughly temperature independent drop across the MIT, possibly associated with loss of the Pauli contribution, while on the insulating side vðT ; RÞ appears to be independent of R. The absence of enhanced magnetic interactions in v at an MIT, which from the MR appears to be driven by electronic disorder, is puzzling, and adds to the already remarkable accumulation of the anomalous properties of quasicrystals. Acknowledgements Financial support from the Swedish Science Agency Vetenskapsr adet and the US National Science Foundation, NSF Grant No DMR-9700584 are gratefully acknowledged.
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Studies of icosahedral AlPdRe quasicrystals across the metal–insulator transition € Rapp O.
a,*
, V. Srinivas b, P. Nordblad c, S.J. Poon
d
a b
Department of Solid State Physics, IMIT, KTH-Electrum 229, 164 40 Stockholm-Kista, Sweden Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur, 721 302 India c Department of Materials Science, Uppsala Universitet, Box 534, SE 751 21 Uppsala, Sweden d Department of Physics, University of Virginia, Charlottesville, VA 22901, USA
Abstract The metal–insulator transition (MIT) in icosahedral AlPdRe quasicrystals is investigated by studying changes in sample properties across the transition. The magnetoresistance (MR) the conductivity rðT Þ, and the magnetic susceptibility vðT Þ have been studied. The MR results show that an MIT occurs for resistance ratios R [ ¼ q(4.2 K)/q(295 K)] of about 20–30. r(0 K) is finite, and decreases over four orders of magnitude with increasing R. Besides the MR, the most prominent change in a studied property across the MIT is a change of slope of log rð0Þ vs R. In contrast, no features in the resistivity from 1.5 to 300 K reveals an MIT. New results for vðT ; RÞ are presented. The diamagnetic v decreases from the metallic to the insulating side, while on the insulating side vðT ; RÞ is almost independent of R. Although the MR indicates a disorder driven MIT, there is no support for this picture from the results for v. 2004 Elsevier B.V. All rights reserved. PACS: 71.30.+h; 72.15.)v; 72.20.)i; 75.20.Ck; 75.20.En
1. Problems and outline The existence of a metal–insulator transition (MIT) in icosahedral (i) AlPdRe has been inferred from some recent studies of electronic transport properties. In particular, insulating behavior in the form of Efros– Shklovskii variable range hopping [1], has been found in the magnetoresistance (MR) of samples with large resistance ratios R [ ¼ q(4 K)/q(295 K)] [2], and an MIT has also been suggested from studies on the metallic side of the transition of the MR [3], and the conductivity, r, above 400 mK [4]. These three different methods all gave estimates that an MIT as a function of increasing R should occur in the region of R 20–30. Normally one would expect the low temperature rðT Þ to give the most direct evidence for an MIT. However, this has not been the case for i-AlPdRe, where published results of rðT Þ instead present a manifold of interpretations, and partially contradicting views [5–16]. These differences concern both procedures and results for
*
Corresponding author. Tel.: +46-8 790 4190; fax: +46-8 752 7850. € Rapp). E-mail addresses: [email protected], [email protected] (O.
0022-3093/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2003.12.002
extrapolating rðT Þ to r ¼ 0 or to T ¼ 0 K, as well as methods and results from analyses of the low temperature rðT Þ of assumed insulating samples in terms of Efros–Shklovskii or Mott variable range hopping, VRH. The varying temperature regions often used for such analyses, and the different R-values of samples studied contribute to these discrepancies. An additional difficulty is the finite zero temperature conductivity rð0Þ [17], by itself incompatible with VRH behavior, which may ruin the analyses if neglected, and tends to make them overflexible when taken into account. The MR appears to offer a more powerful tool to study an MIT in i-AlPdRe. First, the MR of quasicrystals is not sensitive to impurity phases. Possible minor metallic impurities precipitating in grain boundaries and not detectable in standard X-ray diffraction, have been assumed to be one reason for a finite rð0Þ. However, the much smaller resistivity of such precipitates would give a negligible contribution to the MR [18]. Second, the vast phase space of magnetic field and temperature is controlled in experiments, and numerical convergence in fitting theoretical expressions is usually satisfactory. In contrast, almost any theory with some fitting parameter(s) can produce an apparent description
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of the smooth rðT Þ. Nevertheless, the interpretation of the MR of i-AlPdRe is not unproblematic. For instance, the ES-VRH model cannot be applied to data at T < 1 K in a straightforward way, and the reason for this failure is not understood [17]. The variation of the magnetic susceptibility, v, with T and R would also be expected to reflect an MIT. Enhanced magnetic interactions in the vicinity of a transition are anticipated in different scenarios, such as the Brinkman–Rice treatment of interacting electrons in a Hubbard model without disorder [19], and scaling theories of interaction and localization in disordered media [20,21]. Such effects have been well verified in disordered systems. In doped semiconductors e.g., enhanced paramagnetic susceptibility and (or) the appearance of local moments have been observed for decreasing temperature or doping concentration [22–24]. In amorphous metals enhanced interactions were observed by monitoring electronic disorder with high energy neutron irradiation [25]. One would expect that an MIT in quasicrystals, though atomically well ordered, belongs to this class of electronically disordered systems, as indicated e.g. from the MR [3]. The MIT in i-AlPdRe is unusual and poorly understood. Recently it was found, however, that the MIT can be monitored by varying R [26]. This finding provides a tool for detailed studies of the transition. In particular, the variation of a physical property across the MIT can be accessed by the variation of R, and may give important clues to the nature of this MIT. In the present paper we exploit this approach, and report on results across the MIT for the MR in Section 2, the conductivity (Section 3), and the magnetic susceptibility (Section 4). Recent results will be reviewed, and new results, in particular for vðT ; RÞ, will be presented. The paper ends with a brief summary.
2. Magnetoresistance across the MIT The MR gives the most clear evidence for an MIT in i-AlPdRe, since it can be quantitatively described in terms of well-established theories on both the metallic and the insulating side of the MIT. In fact, such a successful description of an electronic transport property is exceptional in quasicrystals. To illustrate an MIT in the MR, it is not necessary to resort to detailed numerical analyses. Qualitative studies of the field dependence are adequate since the MR displays a sequence of characteristic B-dependences for increasing magnetic field B, which are distinctly different on the metallic and insulating sides. Fig. 1 illustrates this qualitative argument for an MIT in between two i-Al70:5 Pd21 Re8:5 samples with R ¼ 11, well into the metallic side [3], and R ¼ 160, which is far into the insulating side [2]. For metallic samples the MR
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Fig. 1. The magnetoresistance, ½qðB; T Þ qð0; T Þ=qð0; T Þ at T ¼ 4:2 K vs. magnetic field for two samples; R ¼ 11 well into the metallic side of the MIT, and R ¼ 160 far into the insulating side. Characteristic field dependences of the metallic and insulating MR are shown by the curve segments. (Data from [2,3].)
is well described by weak localization and electron– electron interaction theories [3]. These theories, which have been developed in detail [27,28], describe quantum corrections to transport properties in weakly (electronically) disordered systems. All contributions to the MR increase as B2 for small B (in alloys with not too weak spin–orbit scattering), and as B1=2 at intermediate fields. In Efros–Shklovskii theory [1] for insulators, MR decreases at small B as )B, due to interference between different trajectories contributing to the hopping process, followed at larger B by an increase, MR +B2 , due to the shrinking of electronic wave functions in field. At still larger fields the continued increase of the MR is characterized by a series of different exponents, where the first one is MR B2=3 . All these feature of metallic and insulating MR are born out by experiments (Fig. 1). One well-known, remarkable property of many quasicrystals is that increased atomic disorder leads to decreased resistivity [29]. This has long been expected also for i-AlPdRe, and was shown recently by structural and transport studies of samples irradiated with high energy neutrons [26]. In this way an insulator–metal transition can be monitored by an increasing irradiation dose, from an insulating state with large R, to metallic
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samples with small R. It is a great advantage of this method that a single sample can be studied across the transition, thus focussing on intrinsic properties, largely circumventing worries on the role of sample quality and impurities, ubiquitous in experimental work on quasicrystals. Fig. 2 illustrates some results. Three sets of data are shown: (i) a sample with R ¼ 67 in the original state, exposed to a series of irradiation doses, D, up to 6.8 · 1019 cm2 (circles) [26]; (ii) a sample with R ¼ 27, irradiated with D up to 3.7 · 1019 cm2 (rhomboids) [26]; and (iii) a series of as made arc-quenched foil samples, where R varied from 45 to 2 [3]. MR vs. R is found to follow qualitatively the same function for these three sets of samples, with an MR which first increases with increasing R, goes through a maximum and decreases to negative values. Assigning the sign change of the MR to an MIT, Fig. 2 suggests that the MIT in i-AlPdRe occurs in the region R 20–30, in agreement with the estimates quoted above [2–4]. The MR is thus a characteristic function of R only, independent of sample history and the details of impurities or defects in the icosahedral state. The MR thus provides a fairly perspicuous picture of the MIT, yet an incomplete one. For instance, in the vicinity of the transition one would like to understand the development
Fig. 2. MR vs. R at T ¼ 4:2 K, B ¼ 2 T for three sets of samples; circles and rhomboids: two series of samples which were neutron irradiated (data from [26]), down triangles: as-made samples, [3]. Open symbols are as-made samples, filled symbols are irradiated samples. The MR is a similar function MRðRÞ for these different samples.
of the localization length and the hopping length, which govern the variable range hopping. It has not yet been possible to find reliable estimates of these parameters.
3. Conductivity across the MIT The search for a vanishing conductivity at the MIT has frequently been used as a method to locate the MIT. In i-AlPdRe alloys these efforts, have however, given rather different results. In extrapolations as a function of T 1=2 , it was found that a finite rðT Þ vanished at about R ¼ 20 [11] or 12.8 ± 0.5 [14]. Using the logarithmic derivative of the measured resistance with respect to in T , it was instead concluded that thin film i-AlPdRe samples were metallic at R ¼ 3 and insulating at R ¼ 9 [15]. What function to extrapolate against is a question of debate. Different results obtained in extrapolations vs. different functions of T were recently illustrated [4]. In addition a different conduction mechanism may dominate at low temperatures, below those from which extrapolations were made. A finite zero temperature conductivity, rð0Þ, also in insulating i-AlPdRe, is one mechanism which invalidates extrapolations. Results in this area have been controversial [10–16]. When measurements for high R samples were taken to below 20 mK, it was found that saturation of rðT Þ appears to be a general property of i-AlPdRe [17]. In Fig. 3 the results of [17] have been extended, in particular by including also tabulated results at 1.5 K for polygrained samples with small R, ðR 6 4Þ [3], and a datum at 4.2 K for single grain i-AlPdRe with R ¼ 1:8 [30]. For low R samples, with small q(4.2 K) and jdq=dT j, this difference in evaluation temperatures is insignificant. rð0Þ decreases continuously over 4 orders of magnitude with increasing R. The range of rð0Þ, and the different samples used, strongly suggest that a finite rð0Þ is an intrinsic property of i-AlPdRe also for insulating samples. At the location of the MIT, as obtained from the MR, the relation between rð0Þ and R changes slope in a log r vs. T diagram. In the metallic phase there is a precipitous drop of rð0Þ with increasing R, while in the insulating state, rð0Þ decreases exponentially with R at a slower rate. The reason for a finite rð0Þ is not known. The observations hint at two conduction channels in quasicrystals. This possibility was suggested early [31,32]. In insulating i-AlPdRe we propose that these two channels could be variable range hopping and quantum tunneling between residual critical states, electron states which are neither localized as in insulators, nor extended as in metals, but instead fall off as a power law [33]. Monitoring R by neutron irradiation provides a convenient way to search for characteristic features in qðT Þ when traversing the MIT. Such studies of qðT Þ
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Fig. 3. Estimated zero temperature conductivity, rð0Þ vs. R. The filled square is a single grain sample [30], open triangles: melt-spun and annealed samples from [3,10]. Circles: estimates from measurements to below 15 mK. The open circle at R ¼ 20 is from [4], the other circles are data from [17]. Melt-spun samples, (open circles) and ingot samples (filled circles) were further described in [17].
have so far only been made in the range 1.5–300 K [26]. Nevertheless, as described above, an MIT in the magnetoresistance is clearly observable in this temperature range, and a comparison is interesting. Fig. 4 shows qðT Þ vs. T on double logarithmic scales for a series of irradiated samples with doses, D, from 5 · 1017 to 6.8 · 1019 n/cm2 . In this process R decreased from 57 to 1.2. Although q and jdq=dT j increase with decreasing dose at all temperatures, no particular feature can be observed which would distinguish an MIT anywhere on these curves. This conclusion remains valid if data are plotted vs. a function of temperature chosen to reveal some special, characteristic behavior. Variable range hopping conductivity, a likely mechanism on the insulating side, is of the form rðT Þ ¼ r0 exp½ðT0 =T Þm ;
ð1Þ
with m ¼ 1=4 or 1/2, in Mott [34] or Efros–Shklovski [1] VRH, respectively. Fig. 5 illustrates VRH according to these theories for the two samples in Fig. 4, which encompass an MIT, as determined from the MR. Data in Fig. 5 cover the range 1.5–4.2 K, where VRHbehavior could be expected. However, the results are inconclusive. First, a description in terms of a VRH theory is of comparable quality both for the insulating R ¼ 57 sample, and for the R ¼ 13 metallic sample.
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Fig. 4. qðT Þ vs. T across the MIT for a neutron irradiated sample. R, indicated in figure, decreases with increasing dose from 5 · 1017 (top curve) to 6.8 · 1019 n/cm2 (bottom curve). Data from [26].
Second, there is no apparent difference between Mott and Efros–Shklovskii theories, with VRH fits showing comparable discrepancies for both samples. Data in both panels are slightly concave. This may be due to the finite rð0Þ. Subtracting rð0Þ from the observed rðT Þ before taking the logarithm, has a stronger effect at low temperatures, where rðT Þ is smaller and may thus improve such fits. Another possibility is a temperature dependent prefactor in Eq. (1), affecting the analyses.
4. Magnetic susceptibility across the MIT Previous studies have shown that the magnetic susceptibility of i-AlPdRe is diamagnetic and weakly temperature dependent above 100 K [35–37]. Only few measurements have been made on samples of different resistivities with R in the range 7–47 [35], and about 2–9 [36], respectively [38]. Irregular trends of v as a function of R were observed. However, some considerations discussed below, which are important for i-AlPdRe, were not, or may not have been taken into account in [35,36]. When looking for a relation between v and R of iAlPdRe, two possible sources of errors must be controlled: (i) R-values may vary between different portions of a larger ingot or between different pieces of samples from the same ingot in a melt-spinning process. Measurements of v must therefore be made on the same
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samples were selected for the present studies with Rvalues of 3, 31, 71, and 160. Magnetic measurements were made in a Quantum Design MPMS SQUID magnetometer. The samples were investigated by magnetization vs. magnetic field measurements on increasing and decreasing magnetic field in the range 0–4 T at several different temperatures between 2 and 300 K. The resistivity measurements were made in standard four point arrangement with contacts of silver paint on the same sample pieces. The measured magnetization M vs. applied magnetic field H did not extrapolate to the origin. This was ascribed to small ferromagnetic clusters of weight fraction x and saturation magnetization r. The measured magnetization M was therefore assumed to be of the form M ¼ vm H ¼ vH þ xr:
ð2Þ
vm is the measured apparent magnetic susceptibility, and v the bulk value. The results were analyzed in the form M=H vs. 1=H . Above about 0.3 T these data were well described by straight lines from which v and xr were determined. The general form of vðT Þ in Fig. 6 is in agreement with previous results [35–37]. The small increase of v at low temperatures, similar in all samples, may be unrelated to the icosahedral phase and can be due e.g. to small clusters which behave paramagnetically at low T , or to unsaturated ferromagnetic clusters. xr in Fig. 7 is nearly constant for each sample, suggesting that the Curie temperatures of such clusters are above the present measurement range. The irregular variation of xr between samples could be due to extrinsic impurities or to slightly varying Fe distribution in the samples due to varying annealing conditions. Taking a simplified view that r in Eq. (2) is due to saturated Fe of 200 erg/Oe g, gives x in the range 8–20 ppm for the present samples. The nominal Fe concentration is about 700 ppm. The results thus suggest that most Fe atoms substitute on Fig. 5. Test of VRH behavior between 1.5 and 4.2 K: (a) the R ¼ 57, and (b) the R ¼ 13 samples. (Filled circles) top scales, ES-VRH and (open circles) bottom scales, Mott VRH.
small sample piece from which the R-value is determined. (ii) Minor magnetic impurities in the starting elements, or spurious magnetic contaminations from sample handling and measurement procedures, can change a weak diamagnetic signal. The samples had nominal composition Al70:5 Pd21 Re8:5 . Starting materials were powders of the constituting elements. The dominating magnetic impurity was 0.1 wt% Fe in the Al powder, with negligible magnetic contributions from the other elements. Ingot samples were prepared as previously described in some detail [3]. Pieces for measurements had masses of 10–20 mg. Four
Fig. 6. v vs. T for four samples. The R-values are: () 3; (n) 31; (m) 71 and (.) 160. Bars indicate the largest estimated errors.
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Fig. 7. xr vs. T . The R-values of the samples are: () 3; (n) 31; (m) 71 and (.) 160. Some of the largest estimated errors have been indicated.
non-magnetic sites, in analogy with Mn in i-AlPdMn [39]. No enhancement is observed in magnetic interactions with increasing R, neither in a paramagnetic component, nor in an increased fraction of spontaneous magnetic moments. This conclusion does not depend on the reasons for an enhanced v at low T , or the scatter in values of xr between different samples. In fact the only observation of a clear R-dependence that can be made from these data is that the diamagnetic susceptibility for the metallic sample ðR ¼ 3Þ is comparatively weak, (Fig. 6), while for R P 31, it is stronger and within measurement accuracy roughly independent of R. Thus any change in v ¼ vðRÞ across the MIT is opposite to that expected from conventional results in electronically disordered systems [20–26]. A possible explanation for the drop in v from R ¼ 3 to 31 is a reduction of the small metallic Pauli contribution at the MIT. A comparison with the density of states, DðeF Þ, from the electronic specific heat, c, is complicated by the expected but unknown contribution to c from two level tunneling states [37,40]. In a rough order of magnitude calculation we assumed that the Pauli contribution to v is lost at the MIT. For c ¼ 0:2 mJ/mol K2 , as for a metallic sample of i-AlPdRe with R ¼ 4 [41] close to our R ¼ 3 sample, and neglecting tunneling contributions and electron–phonon interaction, Dv ðeF Þ was evaluated from c. Similarly, any Stoner enhancement was neglected when estimating Dv ðeF Þ from the decrease, Dv 8 108 cm3 /g, of v from R ¼ 3 to the other samples in Fig. 6. We then found Dv ðeF Þ and Dc ðeF Þ to be of the same order of magnitude, ½Dv ðeF Þ=Dc ðeF Þ 1:5 in qualitative support of this conjecture. 5. Conclusions An MIT can be identified from the magnetoresistance of icosahedral AlPdRe to occur as a function of increasing resistance ratio R in the region of R 20–30.
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This result allows monitoring an MIT as a function of R. We have pursued the idea to select samples over a suitable range of R-values to look for and study the variation of physical properties across the MIT. MRðRÞ, rðRÞ, and vðRÞ were investigated. The most clear case, besides the MR, where a distinctly different behavior can be observed across the transition is the Rdependence of the finite zero temperature conductivity, r(0 K, R), which shows a much slower exponential decrease for increasing R on the insulating side. In contrast, the finite temperature rðT ; RÞ does not display any characteristic feature from which an MIT could be identified. vðT ; RÞ exhibits a roughly temperature independent drop across the MIT, possibly associated with loss of the Pauli contribution, while on the insulating side vðT ; RÞ appears to be independent of R. The absence of enhanced magnetic interactions in v at an MIT, which from the MR appears to be driven by electronic disorder, is puzzling, and adds to the already remarkable accumulation of the anomalous properties of quasicrystals. Acknowledgements Financial support from the Swedish Science Agency Vetenskapsr adet and the US National Science Foundation, NSF Grant No DMR-9700584 are gratefully acknowledged.
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