This article was downloaded by: [Smontara, Ana] On: 8 September 2010 Access details: Access Details: [subscription number 926714732] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Philosophical Magazine Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713695589 Anisotropic Hall effect in Al 13 TM 4 approximants J. Ivkov a ; P. Popčević a ; D. Stanić a ; B. Bauer b ; P. Gille b ; J. Dolinšek c ; A. Smontara a a Laboratory for the Physics of Transport Phenomena, Institute of Physics, Zagreb Croatia b Department of Earth and Environmental Sciences, Crystallography Section, Ludwig-Maximilians- Universität München, München, Germany c J. Stefan Institute, University of Ljubljana, Ljubljana, Slovenia First published on: 08 September 2010 To cite this Article Ivkov, J., Popčević, P., Stanić, D., Bauer, B., Gille, P., Dolinšek, J. and Smontara, A.(2010) 'Anisotropic Hall effect in Al 13 TM 4 approximants', Philosophical Magazine,, First published on: 08 September 2010 (ifirst) To link to this Article: DOI: 10.1080/14786435.2010.509053 URL: http://dx.doi.org/10.1080/14786435.2010.509053 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
Philosophical Magazine 2010, 1 7, ifirst Anisotropic Hall effect in Al 13 TM 4 approximants J. Ivkov a, P. Popcˇevic a, D. Stanic a, B. Bauer b, P. Gille b, J. Dolinsˇek c and A. Smontara a * a Laboratory for the Physics of Transport Phenomena, Institute of Physics, Zagreb Croatia; b Department of Earth and Environmental Sciences, Crystallography Section, Ludwig-Maximilians-Universita t Mu nchen, Mu nchen, Germany; c J. Stefan Institute, University of Ljubljana, Ljubljana, Slovenia (Received 28 May 2010; final version received 12 July 2010) The Hall coefficient, R H, in monoclinic Y Al Ni Co, orthorhombic Al 13 Co 4 and monoclinic Al 13 Fe 4 and Al 13 (Fe,Ni) 4 single crystals was investigated for all combinations of the electrical current and magnetic field directions and in the temperature interval from 90 to 370 K. In all three intermetallics, which belong to the Al 13 TM 4 (TM ¼ transition metal) class of approximants to the decagonal quasicrystals, the Hall coefficient exhibits well-defined anisotropy. R H is positive hole-like or zero for the magnetic field parallel to the plane that corresponds to the quasiperiodic plane in decagonal (d-) quasicrystals, and is negative electron-like or zero for the magnetic field perpendicular to this plane. The only exception is R H in Al 13 Fe 4 for the field parallel to the stacking direction, which changes its sign from positive to negative value with an increase of temperature. The results for the anisotropy of R H are correlated to the anisotropy of R H in d-al Ni Co and d-al Cu Co quasicrystals and a brief overview of the theoretical results is presented. Keywords: complex metallic alloy; Hall effect anisotropy; approximant 1. Introduction It is well understood that decagonal quasicrystals (d-qcs) are formed by a periodic stack of quasiperiodic atomic planes. On the other hand, their approximant phases are characterised by large unit cells that periodically repeat in space, and by one set of the atomic planes (named atomic layers) that correspond to the quasiperiodic atomic planes of d-qcs in the sense that they show locally similar patterns. This means that their structure, on the scale of near-neighbour atoms, closely resemble each other. Further, the periodicity lengths along the stacking direction of these layers in the approximant phases are almost identical to those along the periodic direction of d-qcs. Therefore, decagonal approximants offer valid comparison to the d-qcs. Here, it is important that the translational periodicity of decagonal approximants may enable the straightforward theoretical simulation of the physical properties. *Corresponding author. Email: ana@ifs.hr ISSN 1478 6435 print/issn 1478 6443 online ß 2010 Taylor & Francis DOI: 10.1080/14786435.2010.509053 http://www.informaworld.com
2 J. Ivkov et al. A consequence of the anisotropic and layered structure of both d-qcs and their approximants is distinct anisotropy of electrical and thermal transport properties (electrical resistivity, thermoelectric power, Hall coefficient, thermal conductivity) when measured along different crystalline directions [1 4]. The anisotropy of the Hall coefficient, R H, of d-qcs [5,6] is especially intriguing, being positive hole-like (R H 4 0) for the magnetic field lying in the quasiperiodic plane, whereas it changes sign to negative (R H 50) for the field along the periodic direction, thus becoming electron-like. This R H anisotropy has been reported for the d-al Ni Co, d-al Cu Co and d-al Si Cu Co and is considered to be a universal feature of d-qcs. In this paper, a brief overview is presented of our experimental results for the anisotropy of the Hall coefficient in monoclinic Y Al Ni Co, orthorhombic Al 13 Co 4 and monoclinic Al 13 Fe 4 and Al 13 (Fe,Ni) 4 complex metallic alloys. These phases belong to the Al 13 TM 4 group of complex metallic compounds and are approximants to the decagonal quasicrystals. The monoclinic Y Al Ni Co phase has two atomic layers within one periodic unit [7]. The orthorhombic (o-) Al 13 Co 4 and monoclinic (m-) Al 13 Fe 4 complex metallic compounds have similar structures [8] with four atomic layers within one periodic unit. Al 13 (Fe,Ni) 4 is a ternary extension of the Al 13 Fe 4 compound [9]. In addition to the experimental results the possibilities and some shortcomings of the direct theoretical calculations of the R H (T) are discussed. For completeness, the results for the anisotropic electrical resistivity,,ofal 13 Fe 4 and Al 13 (Fe,Ni) 4 are also presented. 2. Experimental The large single crystals used in our study were grown by the Czochralski technique [10]. The structures of Y Al Co Ni (Al 76 Ni 2 Co 22 ) and o-al 13 Co 4 are given in [7]. The structure of Al 13 Fe 4 matches well the monoclinic unit cell of the Grin et al. model [11] with a ¼ 1.5492 nm, b ¼ 0.8078, c ¼ 1.2471 nm and ¼ 107.69. The parameters of Al 13 (Fe,Ni) 4, with the exact chemical composition Al 76.5 Fe 21.3 Ni 2.2, were practically the same as those of the Al 13 Fe 4 compound. In order to perform crystalline-direction-dependent studies of the Hall effect, as well as of the other electrical and thermal transport properties, we have cut from the ingots of each alloy three bar-shaped samples of dimensions 1 1 7mm 3, with their long axes along three orthogonal directions. For each sample the long sides of the bar were perpendicular to other two orthogonal directions. Two of the three orthogonal directions are equal to b and c crystallographic directions for all the three alloys. In orthorhombic Al 13 Co 4 the third direction is the a direction. For monoclinic Y Al Co Ni, Al 13 Fe 4 and Al 13 (Fe,Ni) 4 the third orthogonal direction denoted as a* lies in the monoclinic plane at an angle with respect to a (26 in Y Al Co Ni, and 17 in Al 13 Fe 4 and Al 13 (Fe,Ni) 4, respectively), and is perpendicular to c and b. Note that for these monoclinic samples the stacking direction corresponds to the b direction, whereas it is a direction for o-al 13 Co 4. For each of the samples the Hall coefficient was measured with the current direction along the long axis, and for magnetic field directed along each of the other
Philosophical Magazine 3 two orthogonal directions, making six experiments altogether per phase. The measurements were performed by the five-point method using standard ac technique in magnetic fields up to 1 T with the experimental uncertainty of 0.1 10 10 m 3 C 1, and in the temperature interval from 90 to 370 K. Electrical resistivity was measured between 300 and 2 K using the standard four-terminal technique. 3. Results and discussions The Hall coefficients, R H ¼ E y /j x B z (where B z is magnetic field perpendicular to the current of the density j x, and E y is transverse Hall field), of Y Al Ni Co and o-al 13 Co 4, for all combinations of the current and magnetic field directions, are displayed in Figures 1a and 1b, respectively. The superscript on R H denotes the direction of the field with respect to the crystallographic directions: a or a*, b and c. The anisotropic electrical resistivity of these alloys was reported in [7]. Corresponding R H (T) and (T ) results for Al 13 Fe 4 and Al 13 (Fe,Ni) 4, are displayed in Figures 2a 2d. The anisotropic values of R H at 100 and 300 K are collected in Table 1. For each alloy, the six sets of R H data form three groups of two practically identical R H curves, where the magnetic field in a given crystallographic direction yields, in accordance with the Onsager relations [12], the same R H for the current along the other two crystallographic directions in the perpendicular plane. The R H values are typically metallic in the range 10 9 10 10 m 3 C 1. For each alloy, R H and exhibit a pronounced anisotropy. Except for B z k b and B z k a* in Al 13 Fe 4 and for B z k b in Al 13 (Fe,Ni) 4, R H (T ) exhibits a rather weak temperature dependence that tends to disappear at higher temperatures, as is typical for metallic systems. The rather large temperature dependence of R H for B z k b in Al 13 Fe 4 is due neither to the anomalous Hall effect (a) (b) Figure 1. The Hall coefficient, R H ¼ E y /j x B z, of monoclinic Y Al Ni Co (a) and o-al 13 Co 4 (b) for different combinations of current j x and magnetic field B z. Note that the stacking directions for Y Al Ni Co and o-al 13 Co 4 are the b and a directions, respectively.
4 J. Ivkov et al. (a) (b) (c) (d) Figure 2. The Hall coefficient, R H ¼ E y /j x B z, of monoclinic Al 13 Fe 4 (a) and Al 13 (Fe,Ni) 4 (b) for different combinations of current j x and magnetic field B z ; (c,d) the electrical resistivity of the same alloys along three orthogonal crystallographic directions, a*, b, and c. Note that the stacking direction for these two alloys is the b direction. nor to the variable number of the current carriers. Measurements of the magnetic properties have shown that Al 13 Fe 4 has too low a magnetic susceptibility (510 5 )to yield anomalous contribution to R H of the order of 10 10 m 3 C 1. Next, from the metallic behaviour of resistivity it is evident that no change of the number of charge carriers is taking place. As to the sign of R H it can be seen that, except for B z k b in Al 13 Fe 4 for which the field direction R H (T) changes the sign with temperature, in Al 13 TM 4 alloys R H is positive or close to zero for magnetic field parallel to the atomic layers, and is negative or close to zero for the magnetic field parallel to the stacking direction. Therefore, it can be stated that there is reasonable analogy in the anisotropy of R H in Al 13 TM 4 compounds and the anisotropy of R H in d QCs. As to the electrical resistivity, (T), it is observed that for all of the four alloys (T ) is the lowest in the stacking direction so there is a complete analogy of the resistivity anisotropy of d QCs [13 15] and their Al 13 TM 4 approximants. Here it is worth emphasising the strong increase of the residual resistivity upon the addition of 2 at. % Ni to the pure binary Al 13 Fe 4 alloy. At the same time there is the change of
Philosophical Magazine 5 Table 1. Anisotropic Hall coefficient, R H ¼ E y /j x B z,of Y Al Ni Co, o-al 13 Co 4, m-al 13 Fe 4 and m Al 13 (Fe,Ni) 4 for various directions of the current j x and magnetic field B z at 100 and 300 K. Average values for a particular field and two current directions are given. R H (10 10 m 3 C 1 ) Alloy B z j x 100 K 300 K Y Al Ni Co a* b, c 10.3 8.5 b a*, c 0.4 0 c a*, b 5.7 4.5 o Al 13 Co 4 a b, c 8.8 6.5 b a, c 3.6 3.5 c a, b 0.7 0.6 m Al 13 Fe 4 a* b, c 17 4 b a*, c 68 8 c a*, b 16 10 m Al 13 (Fe,Ni) 4 a* b, c 0 1 b a*, c 46 30 c a*, b 5 4 R b H from large positive values at lower temperatures in Al 13Fe 4 to large negative values in Al 13 (FeNi) 4. The large sensitivity of the electrical properties of these alloys with respect to the Ni content indicates that it is worth undertaking a detailed study on this particular system, in order to investigate the effects of disorder on the properties of complex intermetallic compounds in general. Now we briefly report on the results of the theoretical analysis of the Hall effect in these alloys performed up to now. Quantitative theoretical analysis is difficult and requires knowledge of the Fermi surface. The anisotropic R H reflects the complicated structure of the anisotropic Fermi surface that contains electron-like and hole-like contributions. Depending on the combination of the crystallographic directions, electron-like (R H 50) or hole-like (R H 4 0) contributions dominate, or the two contributions compensate each other (R H 0). Ab initio calculations of the electronic band structure have been performed for Y Al Co Ni [2], o-al 13 Co 4 [3] and m-al 13 Fe 4 [4], yielding, respectively, eleven, eight and six bands that cross the Fermi energy. In all of the three compounds, the Fermi surface is highly anisotropic, and this is the origin of the experimentally observed anisotropy of electron transport properties of these compounds. For Al 13 (Fe,Ni) 4 the calculation of the Fermi surface is uncertain due to the chemical disorder of the alloy. Further, the theoretical Hall coefficient was calculated for Y Al Co Ni [2] and o-al 13 Co 4 [3] by means of the Boltzmann semiclassical theory and by applying the calculated band structure. This calculation was simplified with the approximation of the band-independent and isotropic relaxation time, by which the scattering time is cancelled out in the calculation, and which is justified by a weak temperature dependence of R H. Then, the temperature dependence originates from the Fermi Dirac function only. The Hall coefficient calculated for the Y Al Ni Co
6 J. Ivkov et al. phase reproduces well the experimental data for this alloy. For the o-al 13 Co 4 compound the theory gives too large values for all the field directions, but reproduces well the order of appearance of the anisotropic Hall coefficient (R b H 4 Rc H 4 Ra H ) and the crystallographic-direction-dependent change of sign, R b H 4 0 and Ra H 5 0. However, for Al 13 Fe 4, for which R b HðTÞ strongly depends on temperature and even changes its sign, the above assumption about isotropic, which would justify the direct calculations of R H (T), is evidently not applicable. Therefore, there is no satisfactory matching between R H calculated with this approximation and experimental results. For R H in Al 13 (Fe,Ni) 4 the isotropic relaxation time approximation would give some qualitative results, but, as already noted, the calculation of the Fermi surface is uncertain due to the chemical disorder. Therefore a more detailed theoretical analysis for the transport properties of Al 13 Fe 4 and Al 13 (Fe,Ni) 4 still has to be elaborated. 4. Conclusions Anisotropic Hall coefficients have been measured, up to now, in four Al 13 TM 4 compounds, Y Al Co Ni, o-al 13 Co 4, m-al 13 Fe 4 and one composition of Al 13 (Fe,Ni) 4. There is a correlation between the anisotropy of R H in Al 13 TM 4 compounds on one side and d-qcs on the other. R H is negative electron-like or zero for the magnetic field perpendicular to the plane that is, or corresponds to, the quasiperiodic plane, and R H is positive hole-like or zero for the magnetic field parallel to this plane. For all these alloys, and field and current directions, there is only one exception from this correlation. This is R H of Al 13 Fe 4 for the magnetic field perpendicular to the stacking planes, for which R H exhibits a change of sign and unusually large temperature dependence. The origin of the anisotropic Hall coefficient in Al 13 TM 4 compounds is the anisotropic Fermi surface, the anisotropy which originates from the specific stacked-layer structure and the chemical decoration of the lattice. When the chemical order enables the calculation of the Fermi surface, and the weak temperature dependence of R H (T) justifies the approximation of a single and isotropic scattering time, theory reproduces the experimental results for R H either qualitatively, as for o-al 13 Co 4, or even quantitatively, as for Y Al Ni Co. Acknowledgements This work was done within the activities of the 6th Framework EU Network of Excellence Complex Metallic Alloys, and has been supported in part by the Ministry of Science, Education and Sports of the Republic Croatia through the Research Project No. 035-0352826-2848. References [1] A. Smontara, I. Smiljanic, J. Ivkov, D. Stanic, O.S. Barisˇic, Z. Jagličic, P. Gille, M. Komelj, P. Jeglicˇ, M. Bobnar and J. Dolinsˇek, Phys. Rev. B 78 (2008) p.104204.
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