PHYSICAL REVIEW B 85, 024205 (2012)

Intrinsic anisotropic magnetic, electrical, and thermal transport properties ofd-Al-Co-Ni decagonal quasicrystals

M. Bobnar,1 P. Jeglic,1,2 M. Klanjsek,1,2 Z. Jaglicic,3 M. Wencka,4 P. Popcevic,5 J. Ivkov,5 D. Stanic,5 A. Smontara,5

P. Gille,6 and J. Dolinsek1,2,*

1Jozef Stefan Institute & University of Ljubljana, Faculty of Mathematics and Physics, Jamova 39, SI-1000 Ljubljana, Slovenia2EN-FIST Centre of Excellence, Dunajska 156, SI-1000 Ljubljana, Slovenia

3Institute of Mathematics, Physics and Mechanics & University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2,SI-1000 Ljubljana, Slovenia

4Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan, Poland5Institute of Physics, Laboratory for the Study of Transport Problems, Bijenicka 46, POB 304, HR-10001 Zagreb, Croatia

6Ludwig-Maximilians-Universitat Munchen, Department of Earth and Environmental Sciences, Crystallography Section, Theresienstrasse 41,D-80333 Munchen, Germany

(Received 4 October 2011; revised manuscript received 26 November 2011; published 20 January 2012)

To address the questions on the anisotropy of bulk physical properties of decagonal quasicrystals and theintrinsic physical properties of the d-Al-Co-Ni phase, we investigated the anisotropic magnetic susceptibility, theelectrical resistivity, the thermoelectric power, the Hall coefficient, and the thermal conductivity of a d-Al-Co-Nisingle crystal of exceptional structural quality. Superior structural order on the local scale of atomic clusters wasconfirmed by 27Al nuclear magnetic resonance spectroscopy. The measurements were performed in the 10-foldperiodic direction of the structure and in three specific crystallographic directions within the quasiperiodic plane,corresponding to the 2 and 2′ twofold symmetry directions and their bisector. The specific heat, being a scalarquantity, was determined as well. The measurements of the second-rank bulk tensorial properties confirm thetheoretical prediction that a solid of decagonal point group symmetry should exhibit isotropic physical propertieswithin the quasiperiodic plane and anisotropy between the in-plane and the 10-fold directions. d-Al-Co-Ni isan anisotropic diamagnet with stronger diamagnetism for the magnetic field in the 10-fold direction. Electricaland thermal transport is strongly metallic in the 10-fold direction but largely suppressed within the quasiperiodicplane, the main reason being the lack of translational periodicity that hinders the propagation of electrons andphonons in a nonperiodic lattice. The third-rank Hall-coefficient tensor shows sign-reversal anisotropy related tothe direction of the magnetic field when applied in the 10-fold direction or within the quasiperiodic plane. Theobserved anisotropy is not a peculiarity of quasicrystals but should be a general feature of solids with brokentranslational periodicity in two dimensions.

DOI: 10.1103/PhysRevB.85.024205 PACS number(s): 61.44.Br, 71.23.Ft, 76.60.−k

I. INTRODUCTION

One of the basic open questions in the physics of qua-sicrystals (QCs) is whether the quasiperiodicity of the structureinfluences the physical properties of a solid in a fundamentalway by introducing qualitatively new phenomena or whetherthe unusual properties are a consequence of complex localatomic order with no direct relationship to the quasiperiodicity.To elucidate this question, decagonal quasicrystals (d-QCs) areof particular importance because their structure can be viewedas a periodic stacking of quasiperiodic atomic planes. Thus,d-QCs are two-dimensional QCs, whereas they are periodiccrystals in a direction perpendicular to the quasiperiodicplanes. Consequently, the physical properties of the d-QCscan be studied in the quasiperiodic (Q) and periodic (10-fold)crystallographic directions on the same sample. Structuralanisotropy of d-QCs results in their anisotropic magnetic,electrical, and thermal transport properties.

The most studied d-QCs are those from the ternary Al-Co-Ni and Al-Co-Cu systems, which form thermodynamicallystable phases and for which conventional crystal growthtechniques can be used to grow large single crystals.1,2 At aqualitative level, their anisotropic physical properties (the mag-netic susceptibility, the electrical resistivity, the thermoelectric

power, the Hall coefficient, and the thermal conductivity)show some common behavior, like the anisotropy of electricalresistivity between the Q and the 10-fold directions in therange ρQ/ρ10 = 4–103–7 and the crystallographic-direction-dependent sign of the Hall coefficient, being electronlike(RH < 0) for the magnetic field application in the 10-folddirection and holelike (RH > 0) for the Q (in-plane) direction.8

However, important quantitative differences were found forsamples of nominally the same or slightly different chemicalcompositions and perhaps structural qualities. A striking ex-ample is electrical resistivity, for which some studies3,4 reportthat both ρ10 and ρQ exhibit a metallic positive temperaturecoefficient (PTC), whereas other studies5–7 report that onlyρ10 is metallic, whereas ρQ exhibits a nonmetallic negativetemperature coefficient (NTC). The nonmetallic behavior ofρQ was suggested to originate from the phonon-assistedtunneling of charge carriers in extended states5 or from anenhanced electron–phonon and phonon–phonon coupling inthe quasiperiodic plane.6 Because phonons cannot propagateeasily in a quasiperiodic structure that lacks translationalperiodicity, the preceding hypotheses deserve further consider-ation. The anisotropic magnetism of d-QCs brings up anothercontroversy. Markert et al.4 reported that d-Al70Co15Ni15 is aCurie paramagnet for the magnetic field application in both the

024205-11098-0121/2012/85(2)/024205(11) ©2012 American Physical Society

M. BOBNAR et al. PHYSICAL REVIEW B 85, 024205 (2012)

10-fold and the Q directions with 10% anisotropy in magneticsusceptibility and containing a weak ferromagnetic (FM)component. In contrast, Yamada et al.8 reported qualitativelydifferent behavior, in which magnetization as a function ofthe magnetic field initially increased with an increasing fieldup to the maximum at about 5 kOe, whereas for higher fieldsit decreased linearly and the negative slope was larger forthe field application in the 10-fold direction. The authorsconcluded that d-Al-Co-Ni is an anisotropic diamagnet, whichis more diamagnetic for the field application in the 10-folddirection, whereas the maximum in M(H ) originates from asmall FM spin fraction present in the samples.

In all preceding studies, the Q direction was not specified inmore detail, lying somewhere arbitrarily in the quasiperiodicplane. Because the quasiperiodic plane of the d-Al-Co-Ni(Cu)structure contains 20 twofold directions,9 it is desirable toperform measurements within this plane in a more controlledmanner in well-defined crystallographic directions. Moreover,frequent contradicting reports on the same physical property,when measured on samples with nominally the same orsimilar chemical compositions but with different structuralperfections, also raise questions about the true intrinsicproperties of the d-Al-Co-Ni phase. In this paper, we reporton the anisotropic physical properties of a large d-Al-Co-Nisingle crystal of exceptional structural quality, so we believethat we report on the true intrinsic properties of this phase.Using 27Al nuclear magnetic resonance (NMR) spectroscopy,we first demonstrate superior structural order on the local scaleof atomic clusters and then present the anisotropic magnetic,electrical, and thermal transport properties (the magnetic sus-ceptibility, the electrical resistivity, the thermoelectric power,the Hall coefficient, and the thermal conductivity) measuredin the 10-fold direction and three crystallographic directionswithin the quasiperiodic plane. The specific heat, being a scalarquantity, was determined as well. Performing analysis of themeasurements in the 10-fold and Q directions, we address thequestion about the influence of quasiperiodicity on the physicalproperties of a solid. This work complements and rounds upour recent investigations of anisotropic physical properties ofgiant-unit-cell decagonal approximant phases that possess astacked-layer structure but contain different numbers of atomiclayers within one periodic unit in the stacking direction—thetwo-layer Y-phase Al-Co-Ni;10,11 the four-layer compoundsAl4(Cr,Fe),12,13 Al13Co4,14 Al13Fe4, and Al13(Fe,Ni)4

15; andthe six-layer Taylor phase Al3(Mn,Fe)16—as compared to thetwo-layer d-Al-Co-Ni QC.

II. STRUCTURAL CONSIDERATIONS AND SAMPLEPREPARATION

The quasiperiodic plane of the d-Al-Co-Ni structure isperpendicular to the [00001] 10-fold direction and contains20 directions (including the opposite ones) that correspondto the positions of 10 twofold axes.9 These twofold axesbelong to two sets, sometimes called 2 and 2′ (denoted crys-tallographically as sets 〈10000〉 and 〈10100〉, respectively),which are not equivalent to each other but rather are rotatedby 18◦ with respect to the other set. The angle between twoneighboring twofold directions of a given set amounts to 36◦.The quasiperiodic plane with the 20 twofold directions is

FIG. 1. (Color online) (a) The quasiperiodic plane of the d-Al-Co-Ni structure with the 20 twofold directions, where the directionsof set 2 are given by solid lines and those of set 2′ by dashed lines.The particular directions [01000] of set 2 and [10100] of set 2′, whichare orthogonal to each other and were employed as the measurementdirections, are shown in bold. (b) An idealized polyhedron of thed-Al-Co-Ni QC of point group 10/mmm, using the concept of netplanes (Ref. 17). Each of the 20 side facets of the polyhedron isnormal to one of the twofold directions shown in panel (a).

shown in Fig. 1(a), where the directions of set 2 are given bysolid lines and those of set 2′ by dashed lines. The particulardirections [01000] of set 2 and [10100] of set 2′, which areperpendicular to each other, are shown in bold. An idealizedpolyhedron of the d-Al-Co-Ni QC of point group 10/mmm,using the concept of net planes17 (being analogous to latticeplanes for periodic crystals), is shown in Fig. 1(b). Each ofthe 20 side facets of the polyhedron is normal to one of thetwofold directions shown in Fig. 1(a).

Our centimeter-size single crystal was grown bythe Czochralski method. Electron probe microanalysis(EPMA) measurements yielded the average composition ofAl69.7Co10.0Ni20.3 with a standard deviation of 0.2 at.% foreach component. This composition is close to the Ni-rich limitof the QC stability region. Details of the sample preparationand characterization by x-ray transmission topography andother techniques are published elsewhere,18 demonstrating thehigh structural quality of the material. The radial elementaldistribution was found to be absolutely homogeneous withinthe error of EPMA. The crystal exhibited facets on as-grownsurfaces, corresponding to those of the polyhedron shown inFig. 1(b).

To perform crystallographic-direction-dependent measure-ments, we cut from the parent crystal four rectangular barsof the dimensions 10 × 2 × 2 mm3. Three of the bars hadthe edges of the parallelepipeds directed in three orthogonalcrystallographic directions, the [00001] 10-fold direction, the[01000] 2 direction, and the [10100] 2′ direction (the last twoare shown in bold in Fig. 1(a)), where each bar had its long axisin a different direction of this set. The fourth bar was cut withits long axis in the in-plane direction [20100] that is a bisectorof two neighboring 2 and 2′ directions, lying at an angle of 9◦to each of the former ones (in the following text, this directionis referred to as the bis direction), whereas the other two edgeswere in the [00001] 10-fold direction and the [13110] in-planedirection. A sketch of the crystallographic directions of the fourbars is shown in Fig. 2(a). Our samples were thus oriented with

024205-2

INTRINSIC ANISOTROPIC MAGNETIC, ELECTRICAL, . . . PHYSICAL REVIEW B 85, 024205 (2012)

FIG. 2. (a) The crystallographic directions of the four bar-shapedd-Al-Co-Ni samples used in the measurements of the anisotropicphysical properties. Here, [00001] is a 10-fold direction, [01000] isa 2 twofold direction, [10100] is a 2′ twofold direction, [20100] is abisector of two neighboring 2 and 2′ directions lying at an angle of9◦ to each of them (denoted as the bis direction), and [13110] is anin-plane direction perpendicular to the 10-fold and the bis directions.(b) The choice of axes of a Cartesian x,y,z crystal-fixed coordinatesystem for symmetry analysis of a general second-rank tensor ofdecagonal point group 10/mmm symmetry.

their long axes in the 10-fold direction and three directions inthe quasiperiodic plane, the 2, the 2′, and the bis directions; theorientations of the other two orthogonal directions were knownfor each sample as well. The thus-prepared samples enabledus to determine the physical properties of the d-Al-Co-Ni QCin well-defined directions of the decagonal symmetry group.

III. SYMMETRY ANALYSIS OF THE BULK TENSORIALPHYSICAL PROPERTIES

In an anisotropic crystal, bulk physical properties like themagnetic susceptibility χ , the electrical conductivity σ (theinverse resistivity σ = ρ−1), the thermoelectric power S, andthe thermal conductivity κ are symmetric (and diagonalizable)second-rank tensors. For example, the electrical conductivity

tensor σij relates the current density→j to the electrical field

→E via the relation ji = ∑

j σijEj , where i,j = x,y,z denotecrystallographic directions in a Cartesian coordinate system.The Hall coefficient R

ijk

H = Ej/jiBk is a third-rank tensor,with i,j ,k denoting crystallographic directions of the currentji , the Hall electrical field Ej , and the magnetic field Bk .The tensorial ellipsoids exhibit the same symmetry axes asthe crystallographic structure. The invariance of the tensors onthe symmetry operations of a given point group determinesthe form of the tensors. Using the decagonal point groupsymmetry, we determine the general form of a second-ranktensor of a d-QC.

Because the [00001] 10-fold, the [01000] 2, and the [10100]2′ crystallographic directions are mutually perpendicular, theyrepresent a convenient choice for the axes of a Cartesian x,y,zcrystal-fixed coordinate system. We assume that x‖2′, y‖2, andz‖10 in the preceding set of directions [Fig. 2(b)]. Because ofthe twofold axes in the x and y directions, the Txz and Tyz

off-diagonal elements of an arbitrary symmetric second-ranktensor Tij are zero by symmetry. The 10-fold axis in the z

direction causes the Txy off-diagonal element to vanish andthe Txx and Tyy diagonal elements to be equal. A second-rank tensor Tij of decagonal symmetry is thus diagonal in thepreceding Cartesian system with the general form

Tij =

∣∣∣∣∣∣∣

a 0 0

0 a 0

0 0 b

∣∣∣∣∣∣∣. (1)

Therefore, the material of ideal decagonal symmetry shouldbe isotropic in the quasiperiodic plane regarding the bulkphysical properties but should show anisotropy between thequasiperiodic plane and the periodic 10-fold direction.

IV. LOCAL-SCALE SYMMETRY ANALYSIS BY 27Al NMR

While x-ray crystallography is an indispensable tool forthe structure determination of QCs, it yields an averagestructure over the crystal and cannot give local detailssuch as local deviations from the average structure. NMRspectroscopy offers complementary structural informationon the local atomic scale of several interatomic distances.Orientation-dependent NMR spectra of quadrupolar nucleisuch as 27Al in a magnetic field provide information on(1) the distribution of electrical field gradient (EFG) and themagnetic-shielding tensors and the associated distribution oflocal atomic environments around the resonant nuclei, (2)the number of crystallographically inequivalent lattice sites,(3) the local symmetry of the crystalline lattice, and (4) theconfigurational disorder of a particular atomic cluster in thestructure. 27Al NMR is thus suitable to assess the structuralquality of our d-Al-Co-Ni samples on the scale of atomicclusters. The details of the method to determine the symmetryof local chemical environments in Al-containing complexintermetallics were published elsewhere.19,20 The method isbased on the measurement of rotation patterns of 27Al NMRspectra for rotations about particular symmetry directionsof the crystallographic structure. The orientation-dependentintensity of absorption in a given frequency window on thesatellite part of the 27Al spectrum then shows the samesymmetry as the local chemical environments around the 27Alresonant nuclei within a spherical-like volume of diameterof several interatomic distances. The local character of theinformation originates from the EFG at the 27Al sites beingproduced predominantly by the electrical charges from thenear-neighbor atomic coordination shells and the EFG fallingoff with the distance from the electrical charge as 1/r3.

Our 27Al NMR experiments were conducted in a magneticfield B0 = 9.39 T at the temperature T = 80 K. Thebroad absorption spectra extend over the frequency intervalof 5 MHz and were recorded by a frequency-sweep technique.The shape of the strongly inhomogeneously broadened 27AlNMR spectrum of d-Al-Co-Ni is predominantly determinedby the electrical quadrupole interaction.20 A 27Al (spinI = 5/2) spectrum of our Al69.7Co10.0Ni20.3 for the magneticfield parallel to a 2′ twofold direction (B0‖2′) is shown inFig. 3. The spectrum shows the structure of a broad, low-intensity “background” line, corresponding to the first-orderquadrupole-perturbed ±5/2 ↔ ±3/2 and ±3/2 ↔ ±1/2satellite transitions and a narrow, high-intensity central line

024205-3

M. BOBNAR et al. PHYSICAL REVIEW B 85, 024205 (2012)

FIG. 3. (Color online) The 27Al frequency-swept NMR spectrumof d-Al-Co-Ni in a magnetic field B0 = 9.39 T at the temperatureT = 80 K for the field parallel to a 2′ twofold direction (B0‖2′). Thereference frequency on the horizontal scale corresponds to the 27AlLarmor frequency νref = 104.101 MHz. The dashed box enclosespart of the satellite spectrum used to determine the intensity variationupon rotating the crystal (see the text).

(1/2 ↔ −1/2 transition) in the middle of the spectrum thatis quadrupole perturbed in the second order. By rotating thesamples in the magnetic field, the position and shape of thecentral line did not change considerably, whereas significantintensity variations were observed on the satellite part of thespectrum.

To record the rotation patterns of the intensity variationover the spectrum, the samples were rotated about an axisperpendicular to the magnetic field. The rotation axis waschosen to coincide with the 10-fold, 2, and 2′ directions.In Fig. 4(a), part of the satellite spectrum (the part that isenclosed in a dashed box in Fig. 3) for the rotation aboutthe 2 twofold direction is shown on an expanded scale forthree particular angles, α = 0◦, 45◦, and 90◦, between the10-fold axis and the magnetic field. The orientation-dependentintensity variation in the interval α = [0◦, 180◦] recorded atthe frequency ν1 (marked by a dashed arrow in Fig. 4(a)),is shown in Fig. 4(b). The pattern shows the symmetry of atwofold axis, where the angle α = 90◦ represents a plane ofmirror symmetry. A twofold symmetric pattern is also obtainedfor rotation about the 2′ direction, as shown in Figs. 4(c)and 4(d). The orientation-dependent spectra and the intensitypatterns for rotation about the 10-fold direction are shown inFigs. 5(a)–5(c), where α now denotes the angle between the2 twofold axis and the magnetic field. The intensity patternswere recorded at two frequencies, ν1 and ν2 [marked by dashedarrows in Fig. 5(a)], and are displayed in the interval α = [0◦,36◦] in Figs. 5(b) and 5(c). The patterns repeat themselves

FIG. 4. (Color online) (a) An expanded part of the 27Al NMRsatellite spectrum (the part enclosed in a dashed box in Fig. 3) forrotation about the 2 twofold direction for three particular angles,α = 0◦, 45◦, and 90◦, between the 10-fold axis and the magnetic field.(b) The orientation-dependent intensity variation in the interval α =[0◦, 180◦] recorded at the frequency ν1 (marked by a dashed arrow inpanel (a)). (c) and (d) The corresponding spectrum and the intensityvariation for the rotation about the 2′ twofold direction.

exactly after 36◦, consistent with the 10-fold symmetry of thestructure. The patterns repeat approximately, but not exactly,after 18◦ [the minimum at 18◦ in Fig. 5(b) is deeper than theminima at 0◦ and 36◦, whereas in Fig. 5(c) the maximum at18◦ has a different shape than the maxima at 0◦ and 36◦]. Theangle α = 0◦ corresponds to the orientation B0‖2 and α = 18◦corresponds to B0‖2′, which confirms the nonequivalence ofthe 2 and 2′ sets of twofold axes. The angle α = 9◦ correspondsto the orientation at which a bisector of the 2 and 2′ directions,i.e., the in-plane bis direction [20100], is parallel to themagnetic field. The repetition of the intensity patterns after 36◦

FIG. 5. (Color online) (a) An expanded part of the 27Al NMRsatellite spectrum (the part enclosed in a dashed box in Fig. 3) for rota-tion about the 10-fold direction for three particular angles, α = 0◦, 18◦,and 24◦, between the 2 twofold axis and the magnetic field. (b) and(c) The orientation-dependent intensity variation in the interval α =[0◦, 36◦] recorded at the frequencies ν1 and ν2 (marked by dashedarrows in panel (a)).

024205-4

INTRINSIC ANISOTROPIC MAGNETIC, ELECTRICAL, . . . PHYSICAL REVIEW B 85, 024205 (2012)

thus clearly demonstrates the crystallographic inequivalence ofthe 2, 2′, and bis directions.

The 27Al NMR-intensity rotation patterns of Figs. 4 and 5demonstrate well-developed symmetries of local chemical en-vironments in our d-Al-Co-Ni material that are consistent withthe elements of the decagonal point group. The macroscopicaverage decagonal symmetry, as determined by x-rays, isthus preserved on the local scale of atomic clusters as well,confirming that our quasicrystalline material is locally wellordered.

V. EXPERIMENTAL RESULTS

Magnetic measurements were conducted by a QuantumDesign MPMS XL-5 superconducting quantum interferencedevice magnetometer equipped with a 50-kOe magnet, oper-ating in the temperature range 1.9–400 K. The measurementsof the electrical resistivity, the thermoelectric power, the Hallcoefficient, the thermal conductivity, and the specific heatwere conducted by a Quantum Design physical propertymeasurement system (PPMS 9T) equipped with a 90-kOemagnet and operating in the temperature range 2–400 K.Electrical resistivity was measured by a standard four-terminaltechnique. The thermoelectric power and the thermal con-ductivity were measured simultaneously by monitoring boththe temperature and the voltage drop across the sample asa square-wave heat pulse is applied to one of its ends. TheHall-coefficient measurements were performed by the five-point method using the standard alternating current techniquein magnetic fields up to 10 kOe. Specific heat was measuredby a thermal-relaxation calorimeter.

A. Magnetization and magnetic susceptibility

In the first set of measurements, the magnetization versusthe magnetic field curves M(H ) were determined at T = 5 Kfor the magnetic field sweep of ±50 kOe applied in the 10, 2,2′, and bis crystallographic directions [Fig. 6(a)]. All curvesshow linear, negative-slope, diamagnetic behavior. While thereis no anisotropy among the three in-plane directions (2, 2′, andbis), anisotropy exists to the 10-fold direction, which has alarger negative slope, i.e., the material is more diamagnetic inthis direction.

The magnetic susceptibility χ = M/H , determined in thetemperature range 1.9–300 K in the magnetic field H =1 kOe applied in the 10, 2, 2′, and bis directions, is shown inFig. 6(b). The results are consistent with the M(H ) curves ofFig. 6(a), with all susceptibilities being negative diamagnetic.There is no anisotropy among the three in-plane directions,whereas anisotropy exists to the 10-fold direction, whichis more diamagnetic. The zero-field-cooled and field-cooledsusceptibilities show small splitting below 10 K, which can beattributed to a tiny FM component, likely of extrinsic origin,to the d-Al-Co-Ni phase (e.g., FM surface oxides and/orCo- or Ni-rich magnetic clusters in the vicinity of defects).The FM component is, however, very small, amounting to asmall fraction of the diamagnetic susceptibility, and is muchsmaller than the FM fractions reported in other investigationsof magnetic properties of d-Al-Co-Ni.4,8

FIG. 6. (Color online) (a) The magnetization versus the magneticfield curves M(H ) of d-Al-Co-Ni determined at T = 5 K forthe magnetic field sweep of ±50 kOe applied in the 10, 2, 2′,and bis crystallographic directions. (b) The magnetic susceptibilityχ = M/H determined in the temperature range 1.9–300 K in themagnetic field H = 1 kOe. The index Q is conveniently used for thethree investigated in-plane directions (2, 2′, and bis).

The Larmor diamagnetic susceptibility of closed atomicshells for the Al69.7Co10.0Ni20.3 composition was calcu-lated from literature tables21 to amount to χdia = −5 ×10−6 emu/mol, which is very close to the room-temperature(RT) values of the three isotropic in-plane susceptibilitiesshown in Fig. 6(b) (amounting to χ2 ≈ χ2′ ≈ χbis =−7 × 10−6

emu/mol). This demonstrates that the in-plane susceptibilitycan be practically fully accounted for by the Larmor dia-magnetism. Because the contribution of the FM spin fractionto the susceptibility is a small fraction of |χdia|, this alsodemonstrates that negligible fractions of Co and/or Ni spinsare involved in the FM-ordered clusters. The susceptibilityin the 10-fold direction is about two times more diamag-netic than the in-plane susceptibilities, with the RT valueχ10 = −14 × 10−6 emu/mol. This can be explained by anadditional Landau diamagnetic contribution present for thefield application in this crystallographic direction because ofthe conduction-electron orbital circulation in a plane perpen-dicular to the 10-fold direction, thus in the quasiperiodic plane.The Landau diamagnetic contribution to the susceptibility istheoretically of the same order of magnitude as that due to theLarmor diamagnetism. The presence of Landau diamagneticcontribution for the field application in the 10-fold directionand its absence for the in-plane application can be understoodfrom the structural considerations of the d-Al-Co-Ni phase.According to the structural model by Burkov,22 and observedexperimentally,23 the d-Al-Co-Ni structure can be viewedas a set of overlapping decagonal columnar clusters of a

024205-5

M. BOBNAR et al. PHYSICAL REVIEW B 85, 024205 (2012)

10.3-A radius with the long axis along the 10-fold axis,where decagonal atomic rings are located in the perpendicularquasiperiodic planes. These atomic rings represent circularelectrical current loops when the magnetic field is appliedin the 10-fold direction and provide Landau diamagneticcontribution to the magnetic susceptibility χ10. Because thedecagonal atomic rings and the associated circular currentloops exist only in the quasiperiodic planes, no Landaudiamagnetism is present for the magnetic field direction otherthan along the 10-fold axis, and χ10 is consequently morediamagnetic than the in-plane susceptibilities.

The d-Al-Co-Ni QC can thus be classified as an anisotropicdiamagnet with no in-plane anisotropy and slight anisotropyto the 10-fold direction, where the magnetic anisotropyoriginates from the additional Landau orbital diamagnetic termpresent for the field application along the 10-fold axis dueto the conduction-electron circulation within the decagonalatomic rings located in the quasiperiodic atomic planes. Thediamagnetism of the d-Al-Co-Ni material also demonstratesthat the transition metals Co and Ni are in a nonmagneticstate with their d-bands fully occupied (no unpaired d-typeelectrons are present).

B. Electrical resistivity

The electrical resistivity ρ(T ) data, measured between 340and 2 K in the 10, 2, 2′, and bis directions, are shown inFig. 7(a), whereas the resistivities normalized to their 340 Kvalues, ρ/ρ340K, are shown in Fig. 7(b). The resistivitiesshow no in-plane anisotropy, whereas there is considerable

FIG. 7. (Color online) (a) Temperature-dependent electrical re-sistivity ρ (T ) measured in the 10, 2, 2′, and bis directions. (b) Theresistivities normalized to their 340 K values, ρ/ρ340K. The index Qis conveniently used for the three investigated in-plane directions (2,2′, and bis).

anisotropy between the in-plane direction and the 10-folddirection. At 340 K, ρ2 ≈ ρ2′ ≈ ρbis = 303 ± 4 μ�cm, wherethe scatter of the three in-plane resistivity values is by 3%,whereas ρ10 = 36 μ�cm. This yields the anisotropy factorρQ/ρ10 = 8.4 (where the index Q is conveniently used forall three in-plane directions). The 3% scatter in the absolutevalues of the in-plane resistivities is within the experimentalerror due to uncertainty in the determination of the samples’geometrical parameters (the length and the cross section). Thisis confirmed because the normalized in-plane resistivities inFig. 7(b) perfectly overlap, demonstrating that they exhibitidentical temperature dependence, and the tiny differences intheir absolute values can be accounted for by constant scalingfactors. The isotropy of the in-plane resistivity can thus beclaimed unambiguously.

The resistivities exhibit metallic PTC for all four investi-gated directions. The PTC is large for the 10-fold direction,giving fractional increase between 2 and 340 K by R10 =(ρ340K

10 − ρ2K10 )/ρ340K

10 = 26%. For the three in-plane direc-tions, the fractional increase is much smaller, amounting toRQ = (ρ340K

Q − ρ2KQ )/ρ340K

Q ≈ 2%. Another marked differencebetween the in-plane and the periodic resistivities is theirtemperature dependence. While ρ10 exhibits linearlike increaseupon heating at temperatures above the low-temperaturesaturated region, the in-plane resistivities show a tendencyto exhibit maximum or at least leveling off at temperaturesclose to RT in their much weaker temperature dependence.

C. Thermoelectric power

The thermoelectric power data (the Seebeck coefficientS), measured between 310 and 2 K in the 10, 2, 2′, andbis directions are displayed in Fig. 8. There is again noanisotropy among the three in-plane directions, S2 = S2′ =Sbis, but there is significant anisotropy to S10 in the 10-folddirection. The thermopower values for all directions are small,up to a few microvolts per Kelvin at most and show rathercomplicated temperature dependence. S10 is negative in thelow-temperature region but exhibits a minimum at 60 Kwith the value −4 μV/K where its slope is reversed so thatS10 changes sign to positive at 250 K. The three in-plane

FIG. 8. (Color online) Temperature-dependent thermoelectricpower (S), measured in the 10, 2, 2′, and bis directions. The index Qis conveniently used for the three investigated in-plane directions (2,2′, and bis).

024205-6

INTRINSIC ANISOTROPIC MAGNETIC, ELECTRICAL, . . . PHYSICAL REVIEW B 85, 024205 (2012)

FIG. 9. (Color online) Temperature-dependent Hall coefficientRH = Ey/jxBz. Eight sets of experimental data were collected bydirecting the current jx along the long axes of the samples, thus inthe 10, 2, 2′, and bis directions, whereas the magnetic field Bz wasdirected in the other two orthogonal directions of each sample (detailsof the current and field directions are given in the legend). The eightRH data sets form two groups of identical Hall coefficients. The firstgroup (denoted as R

Q

H ) contains five data sets, all corresponding tothe application of the magnetic field in the quasiperiodic plane, thusin the 2, 2′, and bis directions. The second group (denoted as R10

H )contains three sets in which the magnetic field was parallel to the10-fold direction.

thermopowers (conveniently denoted as SQ) are positive inmost of the investigated temperature range, except at about40 K, where a minimum with a slightly negative value of −0.2μV/K is observed.

D. Hall coefficient

The temperature-dependent Hall coefficient RH =Ey/jxBz was determined in the temperature interval between360 and 10 K. Eight sets of experimental data were collected(Fig. 9) by directing the current jx along the long axes of thesamples, thus in the 10, 2, 2′, and bis directions, whereas themagnetic field Bz was directed in the other two orthogonaldirections of each sample. The eight RH data sets form twogroups of Hall coefficients with the following regularity. Thefirst group contains five identical data sets, all corresponding tothe application of the magnetic field in the quasiperiodic plane,thus in the 2, 2′, and bis directions. These data sets show noanisotropy, regardless of the current direction (that was setparallel to the 10, 2, 2′, and bis directions) and give a positiveholelike and almost temperature-independent value of the Hallcoefficient R

QH ≈ 1.6 × 10−10 m3C−1 (where the superscript

Q denotes the in-plane direction of the magnetic field) with theexperimental uncertainty of ±0.2 × 10−10 m3C−1. The secondgroup contains three sets of identical Hall coefficients in whichthe magnetic field was always parallel to the 10-fold direction,whereas the current was in the three in-plane directions 2,2′, and bis. The application of the field along the 10-foldaxis yields a negative electronlike Hall coefficient with thevalue at 340 K of R10

H ≈ −6.2 × 10−10 m3C−1 and moderatetemperature dependence. The values of both R

QH and R10

H aretypical metallic. The Hall coefficient thus shows anisotropy,which is related to the direction of the magnetic field and is

holelike for the in-plane field application (RQH > 0), whereas it

changes sign and becomes electronlike (R10H < 0) for the field

application along the 10-fold axis. This kind of Hall-coefficientanisotropy is in agreement with the previously published resulton d-Al-Co-Ni(Cu) QCs.24

E. Specific heat and the electronic density of states at εF

Though the specific heat is a scalar quantity and hence doesnot give information on the anisotropic physical propertiesof d-Al-Co-Ni, the low-temperature specific heat C(T ) is aconvenient quantity to estimate the value of the electronicdensity of states (DOS) at the Fermi energy εF and theDebye temperature θD . Because our d-Al-Co-Ni material isdiamagnetic, the total specific heat is a sum of the electronicand lattice specific heats. The electronic specific heat dependslinearly on temperature, Cel(T ) = γ T , with the electronicspecific heat coefficient γ = (π2/3)k2

Bg(εF ), where g(εF ) isthe DOS at εF . At low temperatures below ∼10 K, the latticespecific heat can usually be well approximated by the Debyemodel and is expressed as a function of temperature in theform Clatt(T ) = αT 3. The lattice specific heat coefficient α

is related to the Debye temperature via the relation θD =(12π4R/5α)1/3, where R is the gas constant. The total specificheat at low temperatures can then be written as

C(T ) = γ T + αT 3. (2)

The specific heat measurements were performed in thetemperature range between 2 and 300 K. The low-temperaturemolar specific heat is displayed in Fig. 10 in a C/T versus T 2

plot; the specific heat in the entire investigated temperaturerange is displayed in the inset. The analysis yielded thevalues γ = 0.52 mJ/mol·K2 and θD = 545 K. The referenceelectronic specific heat coefficient of the Al metal is25 γAl =1.348 mJ/mol·K2, which allows estimation of the value of theDOS g at εF of d-Al-Co-Ni relative to the Al metal. We obtainγ /γAl = g/gAl = 0.39, so the DOS at εF of d-Al-Co-Ni isreduced to 39% of the DOS of the Al metal. The reduced DOSat εF is one of the reasons for the moderately high electricalresistivity of d-Al-Co-Ni.

FIG. 10. (Color online) Low-temperature molar specific heat ina C/T versus T 2 plot. The solid line is the fit with Eq. (2). Thespecific heat in the entire investigated temperature range (2–300 K)is displayed in the inset.

024205-7

M. BOBNAR et al. PHYSICAL REVIEW B 85, 024205 (2012)

FIG. 11. (Color online) Thermal conductivity κ in the 10, 2, 2′,and bis directions. The index Q is conveniently used for the threeinvestigated in-plane directions (2, 2′, and bis).

F. Thermal conductivity

The thermal conductivity κ in the 10, 2, 2′, and bis directionsis displayed in Fig. 11. There is no anisotropy among the threein-plane directions 2, 2′, and bis but strong anisotropy to the10-fold direction, where the thermal conductivity in the 10-folddirection κ10 is considerably higher than the in-plane thermalconductivity κQ. Both κ10 and κQ show an initial fast increaseat low temperatures up to ∼30 K. Above that temperature, thegrowth of κ10 becomes slower but continues to increase almostlinearly up to RT with the 300 K value κ10 = 22 W/mK. Incontrast, κQ exhibits a plateau above 30 K and grows onlyinsignificantly upon further heating, reaching the RT value ofκQ = 5 W/mK.

VI. DISCUSSION

Our measurements of the anisotropic physical propertiesof a d-Al-Co-Ni single crystal in the 10-fold direction andthree specific, crystallographically nonequivalent directionswithin the quasiperiodic plane confirm that a QC of decagonalsymmetry is an isotropic solid in the quasiperiodic plane re-garding its bulk physical properties, but there exists anisotropybetween the quasiperiodic plane and the periodic direction.The theoretical symmetry analysis of a general second-ranktensor that is invariant under the symmetry elements of adecagonal point group is in agreement with the experimentalresults. In a Cartesian crystal-fixed frame with the z axis inthe 10-fold direction and the x and y axes in the 2′ and 2directions, respectively, the tensor is diagonal and containsonly two different diagonal elements, as expressed by Eq. (1),so we determined the complete tensors by specifying thevalues of a given physical property in the in-plane and the10-fold directions, e.g., the resistivities ρQ and ρ10 or thethermopowers SQ and S10. Though previous studies of theanisotropic physical properties of d-QCs did not distinguishamong different directions within the quasiperiodic plane(i.e., the measurement direction in the quasiperiodic planewas not specified), the in-plane to the 10-fold directionanisotropy was still correctly determined. However, differentdegrees of structural perfection of the previously investigatedd-Al-Co-Ni(Cu) samples of nominally the same composition,and the associated scatter of their physical properties, leave

the question about the true intrinsic properties of the d-Al-Co-Ni(Cu) phase open, in particular the question about theinfluence of quasiperiodicity on the physical properties of asolid. For our d-Al-Co-Ni material, the high structural qualityof the single-crystalline material was confirmed by 27Al NMRsymmetry analysis on the local scale of atomic clusters, aswell as by structural assessment by other techniques publishedpreviously.18 The high structural quality of our material makesus believe that we are reporting on the true intrinsic physicalproperties of the d-Al-Co-Ni phase. In the following, wediscuss the investigated physical properties in view of thepreceding open questions.

A. Magnetic properties

The ground state of d-Al-Co-Ni is diamagnetic, wherethe transition metals Co and Ni are in a nonmagnetic statewith their d-bands fully occupied. Regarding the anisotropyof magnetization and magnetic susceptibility, d-Al-Co-Ni QCcan be classified as an anisotropic diamagnet with no in-planeanisotropy and moderate anisotropy to the 10-fold direction,which is about two times more diamagnetic direction. Strongerdiamagnetism in the 10-fold direction originates from theadditional Landau orbital diamagnetic term present for the fieldapplication in this direction due to the magnetic force–inducedconduction-electron circulation within the decagonal atomicrings located in the quasiperiodic atomic planes. However,the in-plane magnetic susceptibility can be practically fullyaccounted for by the Larmor diamagnetism of closed atomiccore shells. Similar anisotropic magnetism with no in-planeanisotropy and stronger diamagnetism in the stacking (pseudo-10-fold) direction was also found in the Y-phase Al-Co-Ni two-layer decagonal approximant,10 so long-range quasiperiodicityof the structure plays no role in the (dia)magnetism ofd-Al-Co-Ni.

B. Electrical resistivity and thermal conductivity

The values of the in-plane and the periodic electricalresistivities and their ratio ρ10/ρQ = 8.4 of our d-Al-Co-Niare in the range found in previous studies.3–7 The muchlarger in-plane resistivity can be attributed to the lack oftranslational periodicity within the quasiperiodic plane and tothe chemical and phason disorder that are at the origin of largeresidual T → 0 resistivity. Large resistivity in nonperiodicsystems is a general effect, also found in, e.g., amorphousmetals, so it is not a peculiarity of QCs. The very differenttemperature dependences of ρ10 and ρQ can be understoodas follows. The periodic ρ10 shows temperature dependencetypical of regular metals. Away from the low-temperaturesaturated region, where the plateau in the resistivity is due toelastic electron scattering by quenched defects, the resistivityincreases linearly with the temperature and exhibits quitestrong PTC, which can be attributed to inelastic electron–phonon scattering. This suggests a ballistic-type motion ofconduction electrons in the 10-fold direction (where the meanfree path l between scattering events is much larger thanthe extension of the conduction-electron wave packet Lwp)and confirms that phonons are excited in this direction, asexpected for a periodic solid. The very weak PTC temperature

024205-8

INTRINSIC ANISOTROPIC MAGNETIC, ELECTRICAL, . . . PHYSICAL REVIEW B 85, 024205 (2012)

dependence of ρQ can be, on the other hand, understood inthe frame of the theory of slow charge carriers,26 applicable tosystems with weak dispersion of the electronic band energies

En(→k ), where n is the band index and

→k is the wave vector. In

such a situation, the electron velocity→v= (1/h)∂En(

→k )/∂

→k

is small, and the traveling distance between two scatteringevents becomes comparable to the extension of the electronicwave packet (l ≈ Lwp); thus, the motion of electrons is nolonger ballistic but becomes diffusive. Moreover, becauseonly long-wavelength acoustic phonons can propagate withinthe quasiperiodic plane (for which the solid is an elasticcontinuum), inelastic electron–phonon scattering is largelysuppressed and the phonons no longer give the dominantcontribution to the temperature dependence of the in-planeresistivity. Weak dispersion of the electronic bands in the Qdirections of a model approximant of the d-Al-Cu-Co phasewas demonstrated theoretically by Trambly de Laissardiereand Fujiwara.27 The model of slow charge carriers predicts thetransition from ballistic to nonballistic electrical conductivityas a function of the relaxation time τ between scattering eventsthat enters the electronic mean free path l = vτ . The electricalconductivity is σ = e2g(εF )D, where D is the electronic diffu-sion constant that can be written by simple kinetic arguments asD = v2τ/3 = l2/3τ , with v2 being the mean square electronicspeed. For long τ , usually realized at low temperatures,we obtain in the free-electron limit the Drude conductivityσ = ne2τ/m. Because τ shortens with rising temperature, thisyields a PTC resistivity in the low-temperature ballistic regime.At higher temperatures, τ becomes short enough to drive thesystem into the l ≈ Lwp limit of a nonballistic (diffusive)type of motion. The diffusion constant then becomes inverselyproportional to the relaxation time, D = L2/τ , where L is aconstant proportional to the elementary diffusion step. Thisyields the conductivity of the type σ ∝ 1/τ that yields anonmetallic NTC resistivity in the high-temperature regime.26

At the transition from the ballistic to the diffusive type ofmotion, the resistivity exhibits a maximum and a crossoverfrom the low-T PTC to the high-T NTC resistivity. Thetemperature of the resistivity maximum depends sensitively onthe electronic velocity v (hence on the electronic structure andband dispersion), the temperature-dependent relaxation timeτ (T ), and the concentration of quenched defects, so differenttemperature dependences of the resistivity can be obtained forsamples of nominally the same chemical composition but dif-ferent structural perfections and defect concentrations. Variouspossible temperature-dependent electrical resistivities withinthe slow-charge-carriers model (PTC, NTC, and mixed PTC-NTC with a maximum), expressed by the formula of the form

ρ−1 = Aτ + B/τ, (3)

are elaborated on theoretically in detail in our previouspublication (see Fig. 7 of Ref. 12). The theory was successfullyapplied to the complex resistivities of the Al4(Cr,Fe) decagonalapproximant12 and the giant-unit-cell heavy-fermion inter-metallic compound YbCu4.25.28 In Fig. 12(a), we show the fit ofthe temperature-dependent normalized resistivity ρQ/ρQ,340 K

of our d-Al-Co-Ni from Fig. 7(b) by using Eq. (3). Therelaxation time was taken in the form τ−1 = τ−1

0 + τ−1p , where

τ−10 is the temperature-independent rate due to scattering by

FIG. 12. (Color online) (a) Theoretical fit of the normalizedρQ(T ) from Fig. 7(b) by using Eq. (4) of the slow-charge-carriersmodel. The fit parameter values are given in the text. (b) TheoreticalρQ (T ) curves normalized to the ρ0 = ρ (T = 0) value for the cases inwhich the transition from a ballistic to a nonballistic (diffusive) regimewould occur at lower temperatures because of stronger disorder inthe material (shorter relaxation time τ ). The original fit (C = 8.63 ×10−4 and α = 1.0) from panel (a) up to the highest temperature of theexperimental measurements (340 K) is shown by the bold curve. Themodified C and α parameter values of the other curves are given inthe legend.

quenched defects, whereas τ−1p contains any temperature-

dependent scattering processes, e.g., scattering by phonons. Inthe simplest case, τp can be phenomenologically written as apower law of temperature τp = β/T α , at least within a limitedtemperature interval. Eq. (3) can then be rewritten in the form

ρ−1 = A

1 + CT α+ B (1 + CT α) , (4)

whereA = e2g(εF )v2τ0, B = e2g (εF ) L2/τ0, and C = τ0/β.The fit [solid curve in Fig. 12(a)] was made using the parametervalues A = 2.19 × 10−3 (μ�cm)−1, B = 1.27 × 10−3

(μ�cm)−1, C = 8.63 × 10−4, and α = 1.0, where the unitsof the coefficient C are chosen such that the temperaturein the expression CT α is dimensionless. Eq. (4) reproduceswell the temperature-dependent resistivity ρQ, except in thelow-temperature limit, where it cannot reproduce the T → 0leveling off. The PTC resistivity with the continuously decreas-ing slope upon heating is a sign that the conduction-electronsystem is in the ballistic regime within our investigatedtemperature range but the temperature-dependent shorteningof τ drives the system toward the nonballistic regime, whichwould be entered at temperatures higher than our highestinvestigated temperature of 340 K. Extrapolation of thetheoretical curve in Fig. 12(a) to higher temperatures indicatesthat the resistivity maximum would appear at T = 360 K.

024205-9

M. BOBNAR et al. PHYSICAL REVIEW B 85, 024205 (2012)

Eq. (4) allows us to study the effect of shortening τp onthe shape of the resistivity curve ρQ (T ) shown in Fig. 12(a),as a consequence of increased dynamic disorder in thelattice. τp shortens faster because of either a decrease of thecoefficient β [thus increasing the coefficient C in Eq. (4)]or an increased exponent α. Comparing d-Al-Co-Ni samplesof nominally the same chemical composition but differentstructural perfections, τp is generally shorter in less perfectsamples. Slight increase of the C and α parameter values fromthose determined in the fit of Fig. 12(a) should thus mimic theresistivity change because of decreased structural perfection.In Fig. 12(b), the original fit from Fig. 12(a) up to the highesttemperature of the experimental measurements (340 K) isdisplayed by the bold curve; the curves with C = 1 × 10−3

and 2 × 10−3 (by keeping the original α value) and the curveswith α = 1, 1.05, and 1.1 (by keeping the original C value) areshown as well. For all combinations of modified parameters,the transition from PTC to NTC resistivity occurs at lowertemperatures, so a slight increase of the disorder eitherintroduces a maximum in the resistivity or changes it fromPTC to NTC within the experimentally observed temperatureregime.

The preceding result offers the answer to the question aboutthe origin of different literature-reported electrical resistivitiesin the Q direction of the d-Al-Co-Ni(Cu) QCs of nominallythe same composition, where different authors have reportedPTC, NTC, or mixed PTC-NTC behavior. The origin of thisdiscrepancy appears to be the different structural perfectionsof the investigated samples. Structurally more perfect samplesgenerally possess a longer relaxation time τ and hence a longerelectronic mean free path l = vτ ; thus, the transition fromthe ballistic PTC to the nonballistic NTC resistivity occurs athigher temperatures, and the PTC resistivity is measured inthe temperature range between the T → 0 limit and, e.g., theRT for more perfect samples.

The transport of heat (the thermal conductivity) in d-Al-Co-Ni is in agreement with the transport of the electrical charge(the electrical conductivity). Both are the highest in the 10-folddirection; thus, the d-Al-Co-Ni material is the best electricaland heat carrier in this direction, where both the electronsand the phonons contribute to the transport. Away from thelow-temperature T → 0 region, both ρ10 and κ10 increase withtemperature in a linearlike manner that is typical for regularmetals and alloys. Within the quasiperiodic plane, the lackof translational periodicity of the lattice largely suppressesthe electronic and phononic degrees of freedom, resulting inlow values of electrical and thermal conductivity ρ−1

Q andκQ and their very weak temperature dependence. The typeof anisotropy in ρ and κ observed in d-Al-Co-Ni is not apeculiarity of d-QCs but is a general feature of solids withbroken translational periodicity in two dimensions, i.e., a solidwith a periodic stacked-layer structure like d-QCs. However,amorphous disorder within the atomic layers is expected toexhibit the same type of anisotropy of electrical and thermalconductivity as the d-QCs.

C. Thermopower and Hall coefficient

Unlike electrical resistivity, which is proportional to thesquare of the electrical charge (ρ−1 ∝ e2) and hence does notdistinguish between the negative electron-type carriers (−e)

and the positive hole-type carriers (+e), the thermopower S

and the Hall coefficient RH distinguish between the electronsand the holes, because the charge in their expressions appearsas e and 1/e, respectively. S and RH are thus more sensitiveto the details of the anisotropic Fermi surface that is at theorigin of the anisotropy of the electronic transport coefficientsand the sign reversal of these coefficients for differentcrystallographic directions. In the absence of a unit cell inthe structural description of d-QCs, an ab initio calculation ofthe Fermi surface is not feasible. In our recent work on theY-phase Al-Co-Ni (Ref. 11) and the orthorhombic Al13Co4

(Ref. 14) decagonal approximants, we showed that the ab initiocalculated Fermi surfaces change quite drastically for differentstructural models (one model is usually a refined version ofan older model) or even just for the relaxation of an existingmodel (atoms are shifted by small fractions of interatomicdistances to energetically more favorable positions). Thus,conclusions on the electronic transport coefficients of d-QCson the basis of Fermi surfaces calculated for the approximantphases should be made with great care. Some qualitativeconclusions can still be reached on the basis of experimentalS10, SQ, R10

H , and RQH parameters, as shown in Figs. 8 and 9.

The in-plane isotropy of SQ suggests that the Fermi surface ofd-Al-Co-Ni is a rotational body around the 10-fold symmetryaxis. The rotational symmetry of the Fermi surface was alsoproposed to be at the origin of the negative electron-type valueof R10

H for the field application along the 10-fold axis, aselaborated by Yun-ping et al.,24 whereas the positive hole-typeR

QH for the in-plane field application was proposed to reflect

the Fermi surface curvature typical of nearly full bands inthe cross sections perpendicular to the 10-fold symmetrydirection. The anisotropies and the temperature dependencesof the S10, SQ, R10

H , and RQH parameters are sensitive to the

details of the d-Al-Co-Ni-specific Fermi surface and can beaffected by the anisotropic phonon spectrum. In the absenceof knowledge of the Fermi surface, a quantitative analysis ofthe preceding phenomena is not possible. At the qualitativelevel, the considerably less negative SQ value at the minimum,as compared to the minimum value of S10, is compatiblewith the reduced electronic term in the in-plane thermopower,whereas the shift of the SQ minimum to a lower tempera-ture suggests stronger in-plane phononic/structural scatteringthan that of the 10-fold direction. Both features correlatereasonably with the much higher in-plane resistivity ρQ (andthe residual ρQ resistivity value at T = 2 K), as comparedto ρ10.

VII. CONCLUSIONS

We have investigated the anisotropic physical properties ofa d-Al-Co-Ni single crystal of exceptional structural qualitythat was confirmed by both standard materials characterizationmethods and 27Al NMR spectroscopy on the local scale ofatomic clusters. The measurements were performed in the 10-fold (periodic) direction of the structure and in three specificcrystallographic directions within the quasiperiodic plane,corresponding to the 2 and 2′ twofold symmetry directionsand their bisector. The measurements of the second-rank bulktensorial properties (the magnetic susceptibility, the electrical

024205-10

INTRINSIC ANISOTROPIC MAGNETIC, ELECTRICAL, . . . PHYSICAL REVIEW B 85, 024205 (2012)

resistivity, the thermopower, and the thermal conductivity)confirm the theoretical prediction that a solid of decagonalpoint group symmetry should exhibit isotropic physical prop-erties within the quasiperiodic plane and anisotropy betweenthe in-plane and the 10-fold directions. The third-rank Hall-coefficient tensor showed anisotropy related to the directionof the magnetic field, when applied in the 10-fold directionor within the quasiperiodic plane. The electrical and thermaltransport is strongly metallic in the periodic crystalline planebut largely suppressed within the quasiperiodic plane, the mainreason being the lack of translational periodicity that hindersthe propagation of electrons and phonons in a nonperiodiclattice. However, this kind of anisotropy is not a peculiarity of

d-QCs but should be a general feature of solids with brokentranslational periodicity in two dimensions, while retainingthe periodicity in the third dimension. For example, any solidwith a stacked-layer structure like d-QCs but with amorphousdisorder within the atomic layers is expected to exhibit thesame type of anisotropy of the electrical and thermal transportcoefficients as the d-QCs.

ACKNOWLEDGMENTS

This work was partially supported by the Croatian Ministryof Science, Education and Sports Grant No. 035-0352826-2848.

*jani.dolinsek@ijs.si1A. P. Tsai, A. Inoue, and T. Masumoto, Mater. Trans. 30, 463 (1989).2T. Godecke, Z. Metallkd. 88, 557 (1997).3T. Shibuya, T. Hashimoto, and S. Takeuchi, J. Phys. Soc. Jpn. 59,1917 (1990).

4J. T. Markert, J. L. Cobb, W. D. Bruton, A. K. Bhatnagar, D. G.Naugle, and A. R. Kortan, J. Appl. Phys. 76, 6110 (1994).

5S. Martin, A. F. Hebard, A. R. Kortan, and F. A. Thiel, Phys. Rev.Lett. 67, 719 (1991).

6W. Yun-ping and Z. Dian-lin, Phys. Rev. B 49, 13204 (1994).7Q. Guo and S. J. Poon, Phys. Rev. B 54, 6046 (1996).8Y. Yamada, Y. Yokoyama, K. Matono, K. Fukaura, and H. Sunada,Jpn. J. Appl. Phys. 38, 52 (1999).

9W. Steurer and T. Haibach, in Physical Properties of Quasicrystals,edited by Z. M. Stadnik (Springer, Berlin, 1999), pp. 65; P. Gille,G. Meisterernst, and N. Faber, J. Cryst. Growth 275, 224 (2005).

10A. Smontara, I. Smiljanic, J. Ivkov, D. Stanic, O. S. Barisic,Z. Jaglicic, P. Gille, M. Komelj, P. Jeglic, M. Bobnar, andJ. Dolinsek, Phys. Rev. B 78, 104204 (2008).

11M. Komelj, J. Ivkov, A. Smontara, P. Gille, P. Jeglic, and J. Dolinsek,Solid State Comm. 149, 515 (2009).

12J. Dolinsek, P. Jeglic, M. Komelj, S. Vrtnik, A. Smontara,I. Smiljanic, A. Bilusic, J. Ivkov, D. Stanic, E. S. Zijlstra, B. Bauer,and P. Gille, Phys. Rev. B 76, 174207 (2007).

13J. Dolinsek, S. Vrtnik, A. Smontara, M. Jagodic, Z. Jaglicic,B. Bauer, and P. Gille, Phil. Mag. 88, 2145 (2008).

14J. Dolinsek, M. Komelj, P. Jeglic, S. Vrtnik, D. Stanic, P. Popcevic,J. Ivkov, A. Smontara, Z. Jaglicic, P. Gille, and Y. Grin, Phys. Rev.B 79, 184201 (2009).

15P. Popcevic, A. Smontara, J. Ivkov, M. Wencka, M. Komelj,P. Jeglic, S. Vrtnik, M. Bobnar, Z. Jaglicic, B. Bauer, P. Gille,H. Borrmann, U. Burkhardt, Y. Grin, and J. Dolinsek, Phys. Rev. B81, 184203 (2010).

16M. Heggen, M. Feuerbacher, J. Ivkov, P. Popcevic, I. Batistic,A. Smontara, M. Jagodic, Z. Jaglicic, J. Janovec, M. Wencka, andJ. Dolinsek, Phys. Rev. B 81, 184204 (2010).

17W. Steurer and A. Cervellino, Acta Crystallogr. A 57, 333 (2001).18B. Bauer, G. Meisterernst, J. Hartwig, T. Schenk, and P. Gille,

Philos. Mag. 86, 317 (2006).19P. Jeglic, M. Komelj, M. Klanjsek, U. Tkalec, S. Vrtnik,

M. Feuerbacher, and J. Dolinsek, Phys. Rev. B 75, 014202(2007).

20P. Jeglic and J. Dolinsek, Phys. Rev. B 71, 014204 (2005).21P. W. Selwood, Magnetochemistry (Interscience Publishers,

New York, 1956).22S. E. Burkov, Phys. Rev. Lett. 67, 614 (1991).23W. Steurer and K. H. Kuo, Acta Crystallogr. B 46, 703 (1990).24W. Yun-ping, Z. Dian-lin, and L. F. Chen, Phys. Rev. B 48, 10542

(1993).25U. Mizutani, Introduction to the Electron Theory of Metals

(Cambridge University Press, Cambridge, 2001), pp. 43.26G. Trambly de Laissardiere, J.-P. Julien, and D. Mayou, Phys. Rev.

Lett. 97, 026601 (2006).27G. Trambly de Laissardiere, and T. Fujiwara, Phys. Rev. B 50, 9843

(1994).28J. Dolinsek, M. Wencka, M. Jagodic, Z. Jaglicic, S. Gottlieb-

Schonmeyer, F. Ritter, and W. Assmus, Solid State Comm. 150,1629 (2010).

024205-11