Electrical, magnetic, thermal and thermoelectric properties of the Bergman phase Mg 32 (Al,Zn) 49 complex metallic alloy

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1 Journal of Alloys and Compounds 430 (2007) Electrical, magnetic, thermal and thermoelectric properties of the Bergman phase Mg 32 (Al,Zn) 49 complex metallic alloy A. Smontara a, I. Smiljanić a, A. Bilušić a,b, Z. Jagličić c, M. Klanjšek d, S. Roitsch e, J. Dolinšek d,, M. Feuerbacher e a Institute of Physics, Bijenička 46, P.O. Box 304, HR Zagreb, Croatia b Faculty of Natural Sciences, Mathematics and Education, Teslina 12, HR Split, Croatia c Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia d J. Stefan Institute, University of Ljubljana, Jamova 39, SI-1000 Ljubljana, Slovenia e Institut für Festkörperforschung, Forschungszentrum Jülich, Jülich D-52425, Germany Received 6 April 2006; accepted 6 May 2006 Available online 12 June 2006 Abstract The Mg Al Zn system of intermetallics contains an exceptional crystalline phase Mg 32 (Al,Zn) 49, named the Bergman phase, whose crystal structure is based on a periodic arrangement of icosahedral Bergman clusters within the giant-unit-cell, so that periodic and quasiperiodic atomic orders compete in determining the physical properties of the material. We have investigated electrical, magnetic, thermal and thermoelectric properties of a monocrystalline Bergman phase sample of composition Mg 29.4 (Al,Zn) 51.6, grown by the Bridgman technique. Electrical resistivity is in the range ρ 40 cm and exhibits positive-temperature-coefficient with T 2 dependence at low temperatures and T at higher temperatures, resembling non-magnetic amorphous alloys. Magnetic susceptibility χ measurements revealed that the sample is a Pauli paramagnet with a significant Landau diamagnetic orbital contribution. The susceptibility exhibits a weak increase towards higher temperature. Combined analysis of the ρ(t) and χ(t), together with the independent determination of the Pauli susceptibility via the NMR Knight shift suggests that the observed temperature dependence originates from the mean-free-path effect on the orbital susceptibility. The electronic density of states (DOS) at the Fermi energy E F was estimated by NMR and was found to amount 72% of the DOS of the fcc Al metal, with no evidence on the existence of a pseudogap. Thermal conductivity contains electronic, Debye and hopping of localized vibrations terms, whereas thermopower is small and negative. High structural complexity of the Bergman phase does not result in high complexity of its electronic structure Elsevier B.V. All rights reserved. Keywords: Complex intermetallic alloys; Physical properties 1. Introduction Among the vast number of known intermetallic phases, the term complex metallic alloys (CMAs) denotes a class of exceptional phases with giant-unit-cells containing some hundreds up to several thousand atoms [1]. Examples of CMAs are cubic NaCd 2 with 1152 atoms/unit-cell [2,3], the Bergman phase Mg 32 (Al,Zn) 49 (162 atoms/u.c.) [4], orthorhombic -Al 74 Pd 22 Mn 4 (320 atoms/u.c.) [5 7] and the related phase (about 1500 atoms/u.c.) [8], cubic -Al 3 Mg 2 (1168 atoms/u.c.) [9,10], λ-al 4 Mn (586 atoms/u.c.) [11], c 2 - Corresponding author. Tel.: ; fax: address: jani.dolinsek@ijs.si (J. Dolinšek). Al 39 Fe 2 Pd 21 (248 atoms/u.c.) [12] and the heavy-fermion compound YbCu 4.5, comprising as many as 7448 atoms in the supercell [13]. These giant-unit-cells contrast with elementary metals and simple intermetallics whose unit-cells in general comprise from single up to a few tens atoms only. The giant-unit-cells with lattice parameters of several nanometers provide translational periodicity of the CMA crystalline lattice on the scale of many interatomic distances, whereas on the atomic scale, the atoms are arranged in clusters with polytetrahedral order, where icosahedrally coordinated environments play the prominent role. The structures of CMAs thus show duality; on the scale of several nanometers, CMAs are periodic crystals, whereas on the atomic scale, they resemble quasicrystals (QCs) [14]. A large group of CMAs is based on the 55-atom Mackay icosahedral cluster [1], which is also the building block of icosahedral Al Pd Mn-type /$ see front matter 2006 Elsevier B.V. All rights reserved. doi: /j.jallcom

2 30 A. Smontara et al. / Journal of Alloys and Compounds 430 (2007) QCs [14]. Another group is based on the 105-atom Bergman cluster [1,4], found also in icosahedral Mg Al Zn-type QCs [15]. While in the QC structure these clusters are distributed quasiperiodically, they are arranged on a periodic lattice in the CMAs. The high structural complexity of CMAs together with two competing physical length scales one defined by the unit-cell parameters and the other by the cluster substructure may have a significant impact on the physical properties of these materials, such as the electronic structure and lattice dynamics. On this basis, CMA materials are expected to exhibit novel transport properties, like a combination of metallic electrical conductivity with low thermal conductivity, and electrical and thermal resistances tunable by varying the composition, as recently observed in a series of Al-transition-metal CMAs [16]. Polytetrahedral order is also at the origin of an enhanced hydrogen-storage capability of the alloy, so that CMA materials appear promising candidates for the energy storage [17]. A particular property of CMA structures is the possibility of additional disorder in the giant-unit-cells apart from that due to atomic substitution, interstitials or creation of vacancies, which are also present in simple intermetallics. In CMAs there exist also: (i) split occupation, where two sites are alternatively occupied because they are too close in space to be occupied simultaneously and (ii) configurational disorder resulting from statistically varying orientations of a particular subcluster inside a given cage of atoms. The structures of the giant-unit-cell CMA compounds can be described with reference to a six-dimensional hypercubic lattices in the framework of the cut-and-projection formalism [18] originally developed to describe QCs. The only difference is that a suitable rational cut is employed rather than the irrational one used for the QC lattice. For that reason the giant-unit-cell intermetallics are frequently referred to in the literature as rational approximants to QCs. The structure determination of the giant-unit-cell intermetallics has attracted attention of crystallographers since long time ago (the pioneering work of Pauling on NaCd 2 dates back to 1923 [2] and -Al 3 Mg 2 was first investigated in 1936 [19]), whereas studies of their physical properties remain scarce due to the problem of materials preparation in efficient quantities and single-crystalline form. While for the crystallographic structure determination single-grain samples of a few hundred micrometers in size are sufficient, centimeter-sized single crystals of high structural perfection are generally required for physical property measurements. The intense research on QCs in the past two decades mostly ternary and quaternary intermetallic phases has also brought about dramatic technical advances in singlecrystal growth procedures for other intermetallic compounds [8]. Novel procedures were developed for the treatment of complex alloys containing incongruent phases and inconvenient elements, possessing, e.g., high vapor pressure. Consequently, fairly high-quality large single-crystalline samples of several CMAs, grown by the Bridgman, Czochralski and flux growth techniques, are now available. Recently, centimeter-size single crystals of the Mg 32 (Al,Zn) 49 Bergman phase were successfully grown by the Bridgman technique. Here, we present the investigation of their physical properties: the electrical resistivity, the magnetism, the thermal conductivity, the thermoelectric power and the determination of the electronic density of states at the Fermi level via the NMR Knight shift and the nuclear spin-lattice relaxation. This work complements our recent studies of physical properties of the giant-unit-cell CMA phases in several other intermetallic systems: (i) the orthorhombic O 1,O 2 and -brass phases of Al Cr Fe [20], (ii) the orthorhombic and phases of Al Pd Mn [7] and (iii) the cubic -Al 3 Mg 2 phase [21]. 2. Structural details and sample characterization The structure of the Bergman phase Mg 32 (Al,Zn) 49 was first solved in 1957 by Bergman et al. [4], whereas a refinement of the structure was reported recently [22]. The unit cell is cubic with space group Im 3, and contains 162 atoms. The Bergman phase can exist in a wide compositional region in the Mg Al Zn phase diagram [23], in contrast to the thermally stable icosahedral QCs that can form only in a narrow compositional range around Mg 44 Al 15 Zn 41 [24]. Within the homogeneity range of the Bergman phase, the lattice constant was reported [22] to vary between a = and Å for increasing Zn content between 14 and 51 at.%. The basic element of the structure is the Bergman icosahedral cluster consisting of six successive shells (Fig. 1a). From the center these shells are: (1) an icosahedron, (2) a pentagonal dodecahedron, (3) a larger icosahedron, (4) a truncated icosahedron (a soccer ball structure), (5) a larger pentagonal dodecahedron and (6) an even larger icosahedron. The final structure is obtained by a body-centered packing of these clusters (Fig. 1b) by sharing a hexagonal plane of the fourth shell (the soccer ball ). Eight crystallographically different kinds of atomic positions were identified in the unit cell. The coordination polyhedra around these sites have ligancy 12 (icosahedron), 14, 15 and 16 (Friauf polyhedron). All of the 98 zinc and aluminum atoms in the unit cell have icosahedral coordination. The only difference between the original structure by Bergman et al. [4], and the refined one by Sun et al. [22], is that the refined structure does not have an atom in the center of the cluster. All sites of the Bergman cluster are fully occupied by atoms (occupancy 1), so that no fractional occupation is present in the Bergman phase. The sample material was grown by the Bridgman technique and the details of growth and structural characterization will be reported elsewhere [25]. The sample for measurements was cut from a Bridgman monocrystalline ingot and was shaped in the form of a rectangular prism with dimensions 2 mm 2mm 7 mm and the long axis parallel to the [1 0 0] crystal direction. This was also the direction of the current and heat flow in the transport measurements. Due to the cubic symmetry, however, no orientation-dependence of the transport coefficients is expected. The chemical composition was determined using energy-dispersive X-ray spectroscopy (EDXS) to be Mg 36.3 Al 32.0 Zn 31.7 in at.% (corresponding to Mg 29.4 (Al,Zn) 51.6, which is within the homogeneity range of the Bergman phase [23]), with the possible error up to 1%. To check for the presence of secondary phases, scanning electron microscope (SEM) investigations were carried out with a JEOL JSM 5800 microscope. The backscattered electron (BSE) image

3 A. Smontara et al. / Journal of Alloys and Compounds 430 (2007) Fig. 1. (a) Successive atomic shells of the six-shell Bergman cluster. (b) Body-centered packing of the Bergman clusters, sharing a hexagonal face of the fourth shell produce the structure of the Mg 32 (Al,Zn) 49 Bergman phase (the model is drawn by using atomic coordinates of [22]). showed featureless, homogeneous pattern, confirming that the sample consists of a single phase with no inclusion of secondary phases. 3. Electrical resistivity The electrical resistivity ρ(t) was determined in the temperature interval between 300 and 4 K using the standard four-terminal technique and the data are displayed in Fig. 2. ρ(t) exhibits positive-temperature-coefficient (PTC), indicating that lattice vibrations play an important role in the temperature dependence of the resistivity. The residual resistivity at 4 K amounts ρ 4K = 40.6 cm and the room-temperature (RT) resistivity is ρ 300 K = 46.8 cm, so that the total increase from 4 to 300 K is by a factor R =(ρ 300 K ρ 4K )/ρ 4K = 15%. The ρ(t) dependence of Fig. 2 closely resembles that of nonmagnetic amorphous alloys with Pauli paramagnetism of conduction electrons [26]. This similarity is not surprising; the non-magnetic state with Pauli paramagnetism of our sample is demonstrated in Fig. 3 and discussed in the next chapter, whereas structural similarity between amorphous alloys and the Bergman phase on the local scale is also straightforward. While icosahedral coordination provides basis of the Bergman phase structure [4,22] (recall that all of the zinc and aluminum atoms in the unit cell have icosahedral coordination), icosahedrally coordinated local environments are considered to dominate small-scale atomic order of amorphous alloys as well [27]. The resistivity of amorphous alloys can be conveniently treated within the Baym Meisel Cote theory [26] that considers the inelas- Fig. 2. Temperature-dependent electrical resistivity ρ(t) of the Mg (Al,Zn) 51.6 Bergman phase monocrystalline sample. Solid curves show the lowtemperature ρ T 2 and the high-temperature ρ T behaviour.

4 32 A. Smontara et al. / Journal of Alloys and Compounds 430 (2007) in samples of increasingly larger ρ 0, the PTC resistivity gradually transforms into the negative-temperature-coefficient (NTC) one [26]. For amorphous alloys with residual resistivitiy lower than about cm (where our Bergman phase sample also classifies), the term ρ(t) dominates [26], promoting a T 2 dependence at low temperatures and T at higher temperatures, whereas the Debye Waller factor can provide small correction to this behaviour at elevated temperatures. The ρ(t) data shown in Fig. 2 are in qualitative agreement with this prediction. The analysis of the low- and high-temperature regimes (solid curves in Fig. 2) shows that the T 2 dependence holds between 4 and 120 K, whereas the linear T regime is observed above 200 K. No significant influence of the Debye Waller factor on the resistivity is observed. The similarity of ρ(t) of the investigated Mg 29.4 (Al,Zn) 51.6 Bergman phase sample to the low-resistivity non-magnetic amorphous alloys is, therefore, evident. 4. Magnetic and NMR measurements Fig. 3. (a) Temperature-dependent magnetic susceptibility χ in a field H = 10 koe. Solid line is a polynomial fit with Eq. (2), whereas dashed line represents the best fit with Eq. (3) (plus the Curie term) of the temperaturedependent Pauli paramagnetism (equivalent to using Eq. (2) without the A 1 T and A 3 T 3 terms). The inset shows the same data on the full vertical scale. (b) Magnetization M as a function of the magnetic field at 300 and 5 K. Solid lines are χ = M/H fits and f.u. denotes formula unit (i.e. one Mg 36.3 Al 32.0 Zn 31.7 unit). On both graphs, the Larmor diamagnetic core susceptibility has already been subtracted. tic electron phonon interaction to be of prime importance for the electron transport at temperatures below the Debye temperature θ D. Using the Baym equation, the ρ(t) of non-magnetic amorphous alloys was shown to be well approximated by [28] ρ = [ρ 0 + ρ(t )] exp[ 2W(T )]. (1) Here, ρ 0 is the residual resistivity at low temperatures, ρ(t) the term arising from the inelastic electron phonon interaction and the exponential term exp[ 2W(T)] represents the Debye- Waller factor. The term ρ(t) exhibits T 2 dependence at low temperatures (below few tens of degrees K) and T at higher temperatures. This is to be contrasted with the Bloch Grüneisen law for regular crystalline metals, where T 5 dependence holds at low temperatures and T at higher temperatures. The Debye Waller factor yields (1 αt 2 ) dependence at low temperatures and (1 βt) above several tens of degrees K, so that the overall ρ(t) temperature dependence is a result of interplay between ρ(t) exp[ 2W(T)] and ρ 0 exp[ 2W(T)] terms. Consequently, The magnetization as a function of the magnetic field, M(H), and the temperature-dependent magnetic susceptibility, χ(t), were measured with a Quantum Design SQUID magnetometer, equipped with a 5 T magnet. The susceptibility χ was investigated in the temperature interval between 300 and 2 K in a magnetic field H = 10 koe applied along the [1 0 0] direction (Fig. 3a). As the M(H) dependence is linear in the whole investigated field range up to 5 T (Fig. 3b), we analyse χ = M/H in the following. On graphs, the Larmor diamagnetic susceptibility due to closed atomic shells, χ Larmor = emu/mol, calculated from literature data [29], has already been subtracted. The slopes of the M(H) lines in Fig. 3b yield the susceptibility values χ 300 K = emu/mol and χ 5K = emu/mol. These values are of the same order (but of opposite sign) as the Larmor diamagnetic susceptibility, indicating that this is the Pauli paramagnetic susceptibility of conduction electrons. Due to the low electrical resistivity of the investigated material, a significant Landau diamagnetic contribution due to orbital electronic motion induced by the field may be present also, so that χ = χ Pauli + χ Landau. The susceptibility χ(t) exhibits slight increase towards higher temperatures, where the smallness of this increase is evident from the inset in Fig. 3a, where χ(t) is displayed on the full vertical scale. The susceptibility exhibits a Curie upturn below 20 K that may be attributed to dilute localized paramagnetic impurities of extrinsic origin. The χ(t) data in Fig. 3a were fitted with the following polynomial function that reproduces well the experimental points in the whole investigated temperature interval χ = χ 0 + A 1 T + A 2 T 2 + A 3 T 3 + A 4 T 4 + C T. (2) Here, χ 0 is a temperature-independent term, the terms A i T i are the temperature-dependent corrections and the last term is the Curie susceptibility of the paramagnetic impurities. The resulting fit parameter values yield the solid line in Fig. 3a and are collected in Table 1. Neglecting the trivial Curie upturn below 20 K, the temperature dependence of χ(t) can be analysed by assum-

5 Table 1 Parameter values obtained from the fit of the χ(t) data displayed in Fig. 3a using Eq. (2) χ 0 (10 4 emu/mol) 6.6 ± 0.2 A 1 (10 7 emu/mol K) 1.6 ± 0.2 A 2 (10 9 emu/mol K 2 ) 6.1 ± 0.4 A 3 (10 11 emu/mol K 3 ) 2.7 ± 0.3 A 4 (10 14 emu/mol K 4 ) 3.3 ± 0.8 C (10 4 emu K/mol) 1.8 ± 0.2 A. Smontara et al. / Journal of Alloys and Compounds 430 (2007) ing temperature-dependent Pauli paramagnetism. Within the independent-electron approximation, the spin part of the electronic magnetization can be written as [30] χ Pauli = χ 0 + A 2 T 2 + A 4 T 4 (3) Here, χ 0 = μ 2 B g(e F) is the temperature-independent term (where μ B is the Bohr magneton and g(e F ) is the conductionelectron density of states (DOS) at the Fermi level E F ) and A 2 T 2 and A 4 T 4 are the two lowest-order temperature corrections emerging from the temperature dependence of the Fermi energy and the variation of the DOS with energy in the vicinity of E F. Assuming that the DOS around E F can be expanded in Taylor series, we have A 2 = χ 0 (π 2 kb 2 /6){(g /g) (g /g) 2 }, where g = g(e F ), g = ( g/ E) EF and g = ( 2 g/ E 2 ) EF. The explicit form of A 4 can be found in ref. [30] and depends on the DOS derivatives at E F up to the fourth order. The fit of the susceptibility with the temperature-dependent Pauli susceptibility of Eq. (3) (plus the Curie term) does not, however, reproduce the data satisfactorily (dashed curve in Fig. 3a). Instead, Eq. (2) that contains additional linear and cubic terms, A 1 T and A 3 T 3, gives excellent fit (solid curve in Fig. 3a) in the whole investigated temperature interval. Considering the magnitudes of the various A i T i terms calculated from the values in Table 1, we find at 100 K A 1 T:A 2 T 2 : A 3 T 3 :A 4 T 4 = 1:3.8:1.7:0.2, so that the linear and cubic terms are of comparable size to the quadratic and quartic terms. At 300 K, these ratios become 1:11.4:15:5.6, where the cubic term is now the largest. The linear and cubic terms cannot be derived under the assumptions that lead to the expression for χ Pauli given by Eq. (3), so that they must originate from other terms in the susceptibility. An independent check of this consideration can be made by measuring Knight shift [31] of the NMR line, which provides a straightforward way to distinguish experimentally the electron spin paramagnetic contribution χ Pauli to the susceptibility of a metal from other sources of magnetization. This distinction relies on the fact that the magnetic moments of the resonant nuclei couple much more strongly to the spin magnetic moment of s- or sp-type conduction electrons (these electrons extend into the nucleus) than to the magnetic fields arising from electronic translational motion. The Knight shift K of the NMR resonance line is directly proportional to the Pauli paramagnetic susceptibility [31] K = ν m ν 0 = ( ) 8π φ EF (0) 2 χ Pauli, (4) 3 Fig. 4. (a) 27 Al NMR spectra of the Mg 29.4 (Al,Zn) 51.6 Bergman phase sample at 80 K in two magnetic fields: 2.35 T (solid line) and 4.7 T (dashed line). The origin of the frequency scale is taken at the 27 Al resonance frequency of the AlCl 3 aqueous solution. The narrow, high-intensity line in the center of each spectrum represents the central transition (1/2 1/2) of the 27 Al (spin I = 5/2) nucleus, whereas the broad background intensity in the wings corresponds to the satellite transitions. (b) Temperature dependence of the frequency shift of the 27 Al central line relative to ν AlCl3 in B = 4.7 T (left scale) and the 27 Al nuclear spin-lattice relaxation rate in a (T 1 T) 1 vs. T plot (right scale). where φ EF (0) 2 is the density of conduction electrons at the nucleus averaged over the Fermi surface, ν 0 the nuclear Larmor frequency and ν m is the magnetic shift relative to ν 0, so that the temperature-dependent Pauli susceptibility when observed in the static susceptibility experiment should be observed in the Knight shift as well. Our NMR Knight shift measurements were performed on the central line (the narrow, high-intensity line in Fig. 4a) of the 27 Al spectrum (spin I = 5/2). The details of the 27 Al spectral shape and the Knight shift determination by measuring the shift of the peak of the central line from the nuclear Larmor frequency in two magnetic fields are given elsewhere [32].Asthe 27 Al nucleus possesses nuclear electric quadrupole moment, the total NMR frequency shift of the central line is generally a sum of the second-order quadrupolar shift that scales inversely with the magnetic field B, ν Q = a/b, and the magnetic shift that is directly proportional to the field, ν m = bb. Our experiments (Fig. 4a) were performed at 80 K in two magnetic fields of strength 2.35 T (where ν 0 ( 27 Al) = MHz) and 4.7 T (ν 0 ( 27 Al) = MHz) and the total shift ν = a/b + bb, measured relative to the AlCl 3 aqueous solution, amounted ν 2.35 T = 27 khz and ν 4.7 T = 59 khz. Decomposition of ν

6 34 A. Smontara et al. / Journal of Alloys and Compounds 430 (2007) into the quadrupolar and magnetic contributions yielded a = 7.8 khz T and b = 12.9 khz/t, so that νq 2.35 T = 3.3 khz and νm 2.35 T = 30.3 khz, whereas νq 4.7T = 1.6 khz and νm 4.7T = 60.6 khz. The magnetic shift is thus by far dominant over the quadrupolar shift and yields the Knight shift K = ν m /ν 0 = , which amounts to about 2/3 of the value in fcc Al (free-electronlike) metal where K Al = The temperature dependence of the Knight shift was determined between 300 and 4 K in a magnetic field 4.7 T and is displayed in Fig. 4b. We observe that the shift and consequently also χ Pauli is constant from 4 to 100 K, whereas it exhibits a tiny increase by about 3% (from 59 khz at 100 K to 61 khz at 300 K) at higher temperatures. This is to be contrasted with the continuous temperature-dependent increase of the static susceptibility χ by 18% in the interval from 20 to 300 K (Fig. 3a). Therefore, the temperature dependence of the susceptibility χ(t) from Fig. 3a cannot be associated with the temperature-dependent Pauli paramagnetism. On the other hand, the tiny increase of the frequency shift by 3% at elevated temperatures may still be due to the temperature-dependent Pauli paramagnetism or due to orbital contribution to the Knight shift (that was neglected in Eq. (4)). Eliminating χ Pauli to be the source of temperature dependence of χ, the observed χ(t) can be qualitatively explained by the following consideration. According to Mizutani [26], the increase of the electrical resistivity in non-magnetic amorphous alloys of residual resistivity less than cm is entirely due to shortening of the electron mean-free-path. This should also affect the electron orbital motion and, consequently, the orbital susceptibility χ Landau. A temperature-dependent increase of the resistivity upon heating will decrease the diamagnetic (negative) χ Landau, so that the sum χ = χ Pauli + χ Landau will be an increasing function of temperature in very much the same manner as the resistivity ρ(t) itself. A comparison of ρ(t) from Fig. 2 and χ(t) from Fig. 3a shows that these two quantities exhibit qualitatively similar temperature behaviour and relative increase (the resistivity increase of 15% compares well to the 18% increase of χ in the investigated temperature interval). The terms A 2 T 2 and A 4 T 4 in Eq. (2) should, therefore, not be considered as the two lowest-order temperature corrections to the Pauli susceptibility, but the entire phenomenological Eq. (2) (excluding the Curie term) should be considered merely as a polynomial fit to the susceptibility, where the temperature dependence of χ results from the mean-free-path effect on the Landau diamagnetic susceptibility. Here, one should stress that, for a free-electron metal, χ Landau is also temperature-independent and related to χ Pauli by a fixed ratio, so that the temperature-dependent χ(t) from Fig. 3a demonstrates dissimilarity of the Mg 29.4 (Al,Zn) 51.6 Bergman phase CMA from free-electron systems. 5. NMR determination of the electronic density of states at E F In Fig. 4b, it is shown that the 27 Al Knight shift of the Bergman phase sample is constant below 100 K, whereas it becomes only marginally temperature-dependent at higher temperatures. Since, according to Eq. (4), K χ Pauli, this means that the temperature-dependent correction terms to χ Pauli in Eq. (3) may be to a good approximation neglected and we have χ Pauli = μ 2 B g(e F). Adopting the approximation that φ EF (0) 2 in Eq. (4) of the investigated Bergman phase sample is not much different from that of pure fcc Al metal, we can estimate the electronic DOS g(e F ) of the Bergman phase relative to the DOS of the Al metal g Al (E F ) from the relation g(e F )/g Al (E F )=K/K Al = This reduction is not large, suggesting that the significantly larger electrical resistivity of the Bergman phase as compared to the Al metal (recall that at RT ρ Mg29.4 (Al,Zn) cm, whereas ρ Al 2.7 cm) is mainly a consequence of smaller electronic diffusion constant of conduction electrons (shorter mean-free path) in the giant-unit-cell Bergman phase CMA. NMR spectroscopy offers another independent criterion on how free-electronlike a metallic compound may be. The method relies on the simultaneous measurement of the Knight shift K and the NMR spin-lattice relaxation rate T1 1. For a free-electron gas, a simple equation relating these two quantities, called the Korringa relation [31], holds ( ) K 2 h 2 γe =, (5) 4πk B TT 1 γ n where γ e and γ n are the electronic and nuclear gyromagnetic ratios, respectively. The degree to which this relation is fulfilled, by inserting the experimental values of K and T1 1, is a rough measure of the free-electronlike nature of the compound. In metals, the nuclear spins are relaxed by coupling to the spins of conduction electrons and the relaxation rate is directly proportional to the temperature [31] 1 T 1 = β s g 2 (E F )k B T, (6) where β s = (64/9)π 3 h 3 γ 2 e γ2 n φ E F (0) 4. Eq. (6) yields the relation (T 1 T) 1 = const, which is known as the Korringa law, characteristic of regular metals. Our spin-lattice relaxation experiments were performed in the field 4.7 T by using an inversionrecovery pulse sequence. The 27 Al relaxation rate T 1 1 was determined on the central line of the 27 Al spectrum and was extracted from the magnetization-recovery curves by the short-saturation relaxation model of Narath [33]. The data are displayed in Fig. 4bina(T 1 T) 1 versus T plot. We observe that T 1 T =3.67sK is constant in the whole investigated temperature range from RT to 6 K, suggesting regular metallic nature of the Bergman phase material. Testing the Korringa relation of Eq. (5), weget K 2 = and hγ 2 e /(4πk BT 1 Tγ 2 n ) = , which may be considered as good matching, comparable to that found typically in regular metals and alloys (recall that for the fcc Al free-electronlike metal these figures are K 2 = and hγ 2 e /(4πk BT 1 Tγ 2 n ) = , by using T 1 T Al =1.88sK [34]). The NMR results are thus in favor of regular metallic nature of the Bergman phase CMA, which is not much different from simple metals and alloys. Since Eq. (6) yields T 1 1 g 2 (E F ), we may make an independent estimate of the reduction of the DOS at E F of the Bergman phase relative to the Al metal by adopting the same approximation as in the case of Knight shift ( φ EF (0) 2 is taken the same for

7 A. Smontara et al. / Journal of Alloys and Compounds 430 (2007) both compounds). We obtain g(e F )/g Al (E F ) = T 1 T Al /T 1 T = 0.72, which is exactly the same factor as also determined from the ratio of the Knight shifts. This matching gives confidence in the experimental values of the K and T1 1 parameters and the conclusions derived from the NMR results. 6. Thermal conductivity The thermal conductivity was measured between 8 and 300 K using an absolute steady-state heat-flow method. The thermal flux was generated by a 1 k RuO 2 chip-resistor, glued to one end of the sample, while the other end was attached to a copper heat sink. The temperature gradient across the sample was monitored by a chromel constantan differential thermocouple. The temperature dependence of the thermal conductivity parameter κ(t) is displayed in Fig. 5. The κ(t) value at 300 K amounts 24 W/mK (recall that the corresponding value for fcc Al metal is about 240 W/mK). The electronic contribution κ el to the thermal conductivity can be calculated using the Wiedemann Franz law and the measured electrical resistivity ρ(t) data, presented in Fig. 2 κ el = L 0T ρ, (7) where L 0 = W K 2 is the Lorenz number. The lattice contribution κ l = κ el is analysed by considering propagation of long-wavelength phonons within the Bergman phase structure and, at elevated temperatures, hopping of localized vibrations. While the long-wavelength phonons may be analysed within the Debye model, the hopping of localized vibrations represents thermally activated motion, which we assume can be described by activation energy E a. Within this model, hopping yields a contribution to the thermal conductivity κ H = κ 0 H exp ( E a k B T ), (8) Fig. 5. Temperature-dependent thermal conductivity κ(t) between 8 and 300 K reproduced theoretically (solid line) by Eq. (11). The three contributions to the total κ(t) are shown separately: the electronic contribution κ el, dotted line; the Debye contribution κ D, dashed line; the hopping contribution κ H, dash dot line. where κh 0 is a constant. The Debye thermal conductivity is written as [35] θd κ D = C D T 3 /T x 4 e x τ(x) (e x dx. (9) 2 1) 0 Here, C D = kb 4 /2π2 v h 3, v is the average sound velocity (defined by 3/ v 3 = 1/v 3 L + 2/v3 T, where v L and v T are the longitudinal and transversal sound velocities, respectively), θ D the Debye temperature, τ the phonon relaxation time and x = ω/k B T, where ω is the phonon energy. The different phonon-scattering processes are incorporated into the relaxation time τ(x) and we assume that Matthiessen s rule is valid, τ 1 = τj 1, where τj 1 is a scattering rate related to the jth-scattering channel. In amorphous solids (to which, according to the ρ(t) analysis, the Bergman phase shows certain resemblance) the two dominant phonon-scattering processes at low temperatures are Casimir scattering at the sample boundaries and scattering on tunnelling states. Because in our experiment, which was performed between 300 and 8 K, we do not really enter the low-temperature regime, these two processes may be neglected. There exist two other processes that dominate thermal conduction in the investigated temperature regime. The first process is scattering of phonons on structural defects of stacking-fault type, for which the scattering rate is given by [36] τsf 1 = 7 a 2 10 v γ2 ω 2 N s. (10) Here, a is a lattice parameter, γ the Grüneisen parameter and N s the linear density of stacking faults. A considerable density of stacking faults, produced by thermal stresses during the material cool-down after preparation, was directly observed by TEM in our Bergman phase samples [25]. For the convenience of the fitting procedure, we rewrite τsf 1 = Ax 2 T 2 (note that, since x 2 T 2, τsf 1 does not show an explicit temperature dependence). The second scattering mechanism is the umklapp processes. In periodic crystals, the acoustic vibrational spectrum exhibit a gap only at the Brillouin zone boundary and the umklapp scattering rate is described by an exponential factor, τ 1 um ω2 T exp ( θ D /βt ), where β is a dimensionless parameter of the order one. In quasicrystals, on the other hand, the vibrational spectrum contains a dense distribution of energy gaps, which implies that umklapp-type phononscattering processes are much more frequent. The model of quasiumklapp phonon scattering in a quasiperiodic lattice yields power-law-type frequency- and temperature dependence of the scattering rate τ 1 um ω2 T 4 [37]. Since the local atomic order in the Bergman phase is quasicrystals-like, one can expect a power-law-type umklapp scattering rate also in this compound. As many different power-law expressions, such as τum 1 ω 3 T, ω 2 T 2,ω 2 T 4, can be found in the literature, we assume phenomenologically τum 1 = Bxα T 4, where the exponent α (yielding a power-law dependence τum 1 ωα T 4 α ) should be determined from the fit.

8 36 A. Smontara et al. / Journal of Alloys and Compounds 430 (2007) The thermal conductivity data of Fig. 5 were fitted, using Eqs. (7) (9), by the expression κ(t ) = κ el (T ) + κ D (T ) + κ H (T ), (11) where τ 1 = τsf 1 + τum 1. The Debye temperature of our Mg 29.4 (Al,Zn) 51.6 Bergman phase sample is not known as yet from complementary measurements, so that a trial value θ D = 500 K was used, based on the value used for the κ(t) analysis of the related -Al 3 Mg 2 CMA compound [21] and the literature data [38] for several Al-based icosahedral QCs that all exhibit similar icosahedral cluster substructure. Since our κ(t) data are available only up to 300 K, and κ D (T) gives the smallest contribution to κ(t) above 100 K (see Fig. 5), the fit was rather insensitive to a change of this θ D value, which was then used as a fixed parameter in the fit. The Debye constant C D was also not taken as a free parameter, but was instead calculated by using v = 4000 ms 1, a typical value determined for Al-based icosahedral QCs from ultrasonic data [39]. There still remain a number of fit parameters involved in the fitting procedure: κh 0 and E a for the hopping contribution and A, B and α for the Debye contribution. In spite of the excellent fits of the κ(t) data obtained with Eq. (11), to be presented in the following, and of the reasonable values of the fit parameters, the fits should not be considered more than just qualitative. The theoretical fit of κ(t) using Eq. (11) (solid line in Fig. 5) shows excellent agreement with the experimental data over the whole investigated temperature range, and the fit parameters are collected in Table 2. On the graph, the individual contributions κ el (T), κ D (T) and κ H (T) to the total κ(t) are displayed also. The Debye contribution κ D (T) exhibits a maximum at about 40 K. Similar behaviour is commonly found in periodic solids, where the maximum is attributed to the phonon phonon umklapp processes [40]. The hopping contribution κ H becomes significant above 70 K and the hopping activation energy was found to be E a = 15.7 mev. This E a value is very similar to that found in the -Al 3 Mg 2 CMA compound [21] (where E a = 16.9 mev for the monocrystalline sample), as well as to those determined by inelastic neutron (INS) [41 43] and X-ray [44] scattering experiments in Al-based QCs, such as the i-al Pd Mn and i- Al Cu Fe families, where dispersionless vibrational states were identified for energies higher than 12 mev. In QCs, such dispersionless states indicate localized vibrations and are considered to be a consequence of a dense distribution of energy gaps in the phonon excitation spectrum. This prevents extended phonons from propagating through the lattice, whereas localized vibrations may still be excited. Therefore, localized vibrations appear to be present in the giant-unit-cell Bergman phase material as well, where their origin may be attributed to the icosahe- dral cluster substructure, a feature that is characteristic of the icosahedral QCs as well. The parameter A, which is close to 10 7 s 1 K 2, makes it possible to estimate the linear density of stacking faults N s. We take a 1 nm as the characteristic length scale of the cluster substructure in the unit cell and γ 2 for the Grüneisen parameter, wherefrom we get N s = 10A v h 2 /7a 2 γ 2 kb 2 = 0.8 m 1, a typical value for intermetallic systems. The parameters B and α define phonon scattering by umklapp processes in a phenomenological way. The fit yielded α = 2.6, which yields the frequency and temperature dependence of the umklapp rate τum 1 ω2.6 T 1.4. This power-law dependence suggests that a dense distribution of energy gaps could exist in the acoustic phonon branch of our Mg 29.4 (Al,Zn) 51.6 Bergman phase sample. Comparing the electronic and lattice contributions to the thermal conductivity, we find at 300 K κ el = 17 W/mK and κ l = κ κ el = 7 W/mK with the ratio κ el /κ l = 2.4. Electronic conductivity thus dominates over the lattice conductivity at room temperature, but κ el and κ l are still of comparable size. This is different from both simple metals, where the electronic contribution is usually one to two orders of magnitude larger than the lattice contribution, and Al-based QCs, where electrons carry less than 1% of the heat, but is similar to the ratio found in other CMAs (for the monocrystalline -Al 3 Mg 2, κ el /κ l = 1.4 was determined at RT [21], whereas ratios between 0.67 and 1 were found for the and CMA phases of Al Pd Mn [7]). From this point of view, the investigated Bergman phase material, as well as other investigated CMAs, lie somewhere between regular metals and QCs. 7. Thermoelectric power The thermopower measurements were performed at between 300 and 4 K by applying a differential method with two identical thermocouples (chromel gold with 0.07% iron), attached to the sample with silver paint. The thermoelectric power data (the Seebeck coefficient S(T)) are shown in Fig. 6. The negative S(T) value indicates that electrons are the dominant charge carriers. The room-temperature S value is rather small, amounting Table 2 Fit parameters of the thermal conductivity κ(t) from Fig. 5 E a (mev) 15.7 ± 0.7 κh 0 (W/mK) 10.7 ± 0.5 A (10 6 s 1 K 2 ) 5.8 ± 0.9 B (10 3 s 1 K 4 ) 3.7 ± 0.7 α 2.6 ± 0.1 Fig. 6. Thermoelectric power S(T) between 4 and 300 K.

9 A. Smontara et al. / Journal of Alloys and Compounds 430 (2007) V/K, so that the Bergman phase material does not appear to be promising thermoelectric material. The thermopower shows a strong (negative) increase between 4 and 50 K, followed by a weaker increase to higher temperatures. While it is difficult to explain the exact temperature dependence of S(T) from Fig. 6 in the absence of a proper microscopic model of electrical conduction of the Bergman phase, a crude order-of-magnitude estimate of the Fermi energy can be obtained by approximating S(T) by a linear function with zero at 0 K and S 300 K 5 V/K, yielding S/T = V/K 2. Using the expression S = (π 2 k 2 B /2 e E F)T that is valid in the free-electron limit with the electron collision time independent of energy [45], we obtain E F = 2.2 ev. 8. Summary and conclusions We have determined magnetic, electrical, thermal transport and thermoelectric properties of the Mg 29.4 (Al,Zn) 51.6 Bergman phase material comprising 162 atoms in the unit cell, in order to investigate how the exceptional structural complexity of the giant-unit-cell and the coexistence of two competing length scales one defined by the unit-cell parameters and the other by the cluster substructure affect the physical properties of a metallic material. The study was performed on structurally homogeneous monocrystalline material, free of secondary phases and grown by the Bridgman technique. The electrical resistivity of the material between 4 K and RT is in the range ρ cm and exhibits a PTC increase to higher temperatures, where this type of ρ(t) can be attributed to the inelastic electron phonon interaction. ρ(t) of the Bergman phase shows similarity to non-magnetic amorphous alloys, exhibiting T 2 dependence at low temperatures (below 100 K), and T at higher temperatures. This is to be contrasted with the Bloch Grüneisen law for a regular crystal metal, where T 5 dependence holds at low temperatures and T at higher temperatures. Magnetic susceptibility measurements have shown that the sample is a Pauli paramagnet with significant Landau diamagnetic contribution due to orbital motion of conduction electrons. The susceptibility exhibits weak increase towards higher temperature, which can be modeled phenomenologically by a polynomial fit including temperature correction terms up to T 4 order, where all terms in the polynomial expansion are of comparable magnitude. Complementary measurements of the Pauli susceptibility by the NMR Knight shift of the 27 Al resonance have shown that χ Pauli is only marginally temperature-dependent and cannot be the origin of the much stronger temperature dependence of χ. Combined analysis of the electrical resistivity and the magnetic susceptibility suggests that the temperature dependence of the susceptibility originates from the mean-free-path effect of conduction electrons on the orbital susceptibility term χ Landau. The electronic DOS at E F relative to the fcc Al metal was estimated by two independent techniques, the NMR Knight shift and the spin-lattice relaxation. Both techniques yielded the result that the DOS at E F of the investigated Mg 29.4 (Al,Zn) 51.6 Bergman phase sample amounts 72% of the DOS of the Al free-electronlike metal. Good matching of the Korringa relation between the Knight shift K and the relaxation rate T 1 1 also supports similarity of the electronic structure of the Bergman phase CMA to regular metallic alloys and compounds. The thermal conductivity of the Bergman phase contains an electronic contribution κ el that was estimated by the Wiedemann Franz law, and the lattice contribution κ l that can be reproduced by the sum of the Debye term (long-wavelength phonons) and the term due to hopping of localized vibrations. While hopping becomes the dominant lattice heat-carrying channel at temperatures above 70 K, the Debye term dominates at low temperatures and exhibits a maximum at about 40 K. At the lowest measured temperature (8 K), the scattering of phonons on stacking-fault-like defects limits the heat transport, whereas at higher temperatures, umklapp processes become excited with a τum 1 ω2.6 T 1.4 dependence of the umklapp rate. The room-temperature value of the electronic conductivity is a factor 2.4 larger than the lattice conductivity, but κ el and κ l are still of comparable size. This feature is intermediate between simple metals and alloys on the one hand and Al-based QCs on the other hand, but seems to be a general feature of the Al-based giant-unit-cell CMAs. The thermoelectric power of about 5 V/K at room temperature is small and negative, indicating that electrons are the majority charge carriers. The small thermopower suggests that the Bergman phase material is not promising candidate for the thermoelectric application. Though the Bergman phase CMA compound appears to be excellent candidate to investigate the influence of high structural complexity and the competition of two different length scales on the physical properties of a metallic solid, no spectacular differences with respect to simple metals and alloys were found. The dominant icosahedral local atomic coordination in the giant-unit-cell makes the Bergman phase resembling amorphous metals on the local scale, which is evident in the temperature-dependent electrical resistivity. This also shows that the electronic transport properties of the Bergman phase are predominantly determined by the small-scale atomic clustering, whereas the large-scale periodicity of the CMA lattice is of more or less marginal importance. The high structural complexity does not result in high complexity of the electronic structure. The electronic DOS at E F is reduced to 72% of the value in fcc Al metal and the NMR measurements are in support of simple electronic structure of the Bergman phase CMA. We found no experimental evidence on the existence of a pseudogap in the DOS at E F. This is to be contrasted with the theoretical calculations of the electronic structure of a series of higher-order (1/1, 2/1, 3/2 and 5/3) rational approximants to the icosahedral Mg Al Zn QCs [46,47], where the lowest-order (1/1) approximant corresponds to the Bergman phase Mg 32 (Al,Zn) 49, whereas the highest 5/3 approximant contains 12,380 atoms in the unit cell, so that the series converges towards the QC state. Structureinduced pseudogaps in the DOS at E F, practically identical for all studied approximants, were predicted in those calculations. For the Bergman phase, it was shown (see Fig. 5 of ref. [47]) that the partial s, p and d DOSs, as well as the site-resolved Mg-, Zn- and Al DOSs all exhibit a pseudogap at or very near E F. These results lead to the conclusion [46] that the pseudogaps at E F are a truly generic property of the Mg Al Zn approximants, approaching the quasicrystalline structure very closely. Therefore, from the theoretical point of view, no pro-

10 38 A. Smontara et al. / Journal of Alloys and Compounds 430 (2007) nounced differences in the DOS-related physical properties, like the electrical resistivity, are expected between the Mg Al Zn QCs and their approximants, including the Bergman phase. In an early paper on the electrical resistivity determination of icosahedral Mg Al Zn QCs [48], performed on melt-spun ribbons (samples were reported to include holes, cracks and contamination by secondary phases up to 5%), it was reported that the resistivity exhibits a negative-temperature-coefficient (only low-temperature measurements below 10 K are presented in ref. [48]). Anticipating the theoretical prediction on no significant difference in the DOS between the QC and the approximant phases, this hints that the Bergman phase should also exhibit an NTC resistivity. Our ρ(t) measurements on a structurally highquality monocrystalline material do not support this conclusion. 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