JAST
2012 March;3(1):1-41.
Published online 2012 February 08.
doi:http://dx.doi.org/10.5355/JAST.2012.1
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| Copyright ⓒ 2010 Journal of Analytical Science & Technology |
| Spin-glass properties of quasicrystals and complex metallic alloys |
| Janez Dolinšek1,2,3*, Zvonko Jagličić4,5 |
1J. Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
2Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
3EN–FIST Centre of Excellence, Dunajska 156, SI-1000 Ljubljana, Slovenia
4Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia
5Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, SI-1000 Ljubljana, Slovenia |
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Corresponding Author:
Janez Dolinšek ,Tel: +386 1 4773 740, Fax: +386 1 4773 191, Email: jani.dolinsek@ijs.si |
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ABSTRACT |
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| Spin-glass properties of magnetic quasicrystals with spins placed on a quasiperiodic lattice, and complex metallic alloys, characterized by giant unit cells, are reviewed. The systems exhibit rich variety of broken-ergodicity phenomena that share properties with site-disordered canonical spin glasses and site-ordered geometrically frustrated antiferromagnets. Magnetic frustration provides basis of a novel concept of digital data storage, where a byte of digital information can be stored into the material by pure thermal manipulation, in the absence of electric, magnetic or electromagnetic field. |
| Keywords: Spin glasses, Quasicrystals, Complex metallic alloys, Magnetic properties |
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Introduction |
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The out-of-equilibrium dynamics of magnetically frustrated systems is one of the challenging fields in condensed matter physics, both theoretically [1,2] and experimentally [3-5]. The dynamics of these systems is characterized by a continuous slowing-down of spin fluctuations (reorientations) upon cooling and frustration implies that there exists a broad spectrum of reorientation times, ranging from microscopic up to the age of the universe. At the spin freezing temperature Tf, the system undergoes an ergodic?nonergodic phase transition, where below Tf, spin reorientations can no longer maintain thermal equilibrium on the experimentally accessible time scales. Ergodicity of the spin system is consequently broken and the out-of-equilibrium dynamics is associated with slow approach towards thermodynamic equilibrium, which can never be reached due to macroscopic equilibration times. Typical broken-ergodicity phenomena observed in magnetically frustrated systems are (i) a large difference between field-cooled (fc) and zero-field-cooled (zfc) magnetic susceptibilities below Tf in small magnetic fields; (ii) the zfc susceptibility exhibits a frequency-dependent cusp associated with a frequency-dependent freezing temperature, Tf(v); (iii) there exists an ergodicity-breaking line in the magnetic field?temperature (H-T) phase diagram (the de Almeida?Thouless line) and (iv) the third-order nonlinear susceptibility x3 shows sharp anomaly in the vicinity of Tf. The ultra-slow approach towards thermal equilibrium in experiments involving isothermal aging periods, where the spin system is let to partially equilibrate during a finite "waiting" (or "aging") time tw under steady external conditions (temperature, magnetic field), yields additional broken-ergodicity phenomena: (v) a logarithmically slow time decay of the thermoremanent dc magnetization (TRM), (vi) the memory effect (ME), where a state of the spin system reached upon isothermal aging can be retrieved after a negative temperature cycle and (vii) "rejuvenation", where small positive temperature cycle within the nonergodic phase erases the effect of previous aging, so that the spin system becomes "young" (unaged) again. The ME is experimentally manifested as a "thermal-imprint" in the zfc electronic magnetization Mzfc at the temperature of aging, which shows depletion (a dip) as compared to Mzfc of the no-aging case. The ME and the rejuvenation are the most spectacular manifestations of the out-of-equilibrium dynamics of a nonergodic frustrated spin system, but are currently incompletely understood (a comprehensive review on the subject can be found in [5]).
The most studied examples of magnetically frustrated systems are spin glasses (SGs) [6], denoting site-disordered spin systems that possess (a) frustration (the interaction between spins is such that no configuration can simultaneously satisfy all the bonds and minimize the energy at the same time) and (b) randomness (the spins are positioned randomly in the material). These two properties lead to highly degenerate free-energy landscapes with a distribution of barriers between different metastable states, resulting in broken ergodicity below Tf. Prototype SGs are canonical spin glasses (dilute magnetic alloys of noble metal host (Cu, Ag, Au) and a magnetic impurity (Fe, Mn)). The interaction between spins in canonical SGs is the conduction-electrons mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction. This interaction oscillates in space and can be either ferromagnetic of antiferromagnetic, depending on the distance between spins. Combined with randomness, the RKKY interaction results in frustration. At Tf , the spin correlation length ξ becomes infinite (ξ3 is the volume within which the spins develop correlations), so that all the spins participate in the collective SG state.
It has been discovered later that very similar broken-ergodicity properties develop also in pure (i.e. site-ordered) systems without quenched disorder [7-11]. These are geometrically frustrated antiferromagnets (AFMs) with kagomé and pyrochlore lattices, where triangular or tetrahedral distribution of nearest-neighbor AFM-coupled spins frustrates an ordered periodic system. Such systems are known as "topological" or "geometrically-frustrated" SGs. Many of their properties (the zfc?fc magnetization splitting, the frequency-dependent Tf(v), the de Almeida-Thouless line, the x3 anomaly and the slow relaxation of magnetization below Tf) closely resemble the situation in site-disordered SGs. However, the important difference is the short correlation length ξ usually encountered in geometrically frustrated SGs, where ξ is non-zero already at relatively high temperatures (compared to Tf) and does not increase significantly with decreasing temperature. What really changes upon cooling is the characteristic time scale of the fluctuating moments, which slow down and exhibit a dramatic spin freezing below Tf. The short ξ demonstrates that the spins form magnetic clusters. The appropriateness of describing these systems as SGs depends on the coupling between clusters. In the case of interacting clusters, such a system may be viewed as a usual SG with renormalized magnetic moments, whereas the system of noninteracting clusters would be just a superparamagnet. Superparamagnetic clusters below the blocking temperature TB exhibit very similar features as SGs, i.e. their ergodicity is broken on the experimental time scale. Due to some anisotropy energy, the reorientation of a cluster by the magnetic field may be "blocked" over a macroscopic time scale. It is, therefore, many times difficult to discriminate between a true SG and a superparamagnet.
Another class of magnetically frustrated materials are magnetic quasicrystals (QCs), where spins are placed on a quasiperiodic lattice that exhibits nonperiodic long-range order with crystallographically forbidden symmetries like 5-, 8-, 10- or 12-fold rotation axes. SG phenomena were observed in two kinds of magnetic QCs. The first are the Al-based icosahedral i-Al-Pd-Mn and i-Al-Cu-Fe families, where the d electrons of the transition metal (TM) atoms represent the basic reorientable magnetic dipoles. In these systems only a small amount of Mn (typically 1% of all Mn atoms) [12-14] and Fe (typically 10-4 of all Fe atoms) [15] carry magnetic moments, the rest being nonmagnetic. It is sometimes difficult to classify these d moments as localized or itinerant unambiguously. The second kind is the rare-earth-containing QCs, where the f magnetic moments of the rare-earth (RE) atoms are well localized and sizable. Currently there are two members of this class, the icosahedral i-RE-Mg-Zn [16-18] and i-RE-Mg-Cd [18-20] families. The RE concentration in these samples is large, about 10 at. %, and all the RE atoms are magnetic. Together with good localization of the f moments, this makes these systems ideally suitable to study the behavior of spins in a quasiperiodic structure. The QC structures contain many lattice sites with partial, split or mixed occupancy, leading to substitutional and topological disorder in the lattice. A certain degree of randomness is thus present in the QC structures. The spins placed on a quasiperiodic lattice may also exhibit geometrical frustration. For instance, a QC structure of icosahedral symmetry is usually built of icosahedral atomic clusters. An icosahedron is composed of twenty equilateral triangles, so that the AFM-coupled spins placed on a perfect icosahedron should exhibit geometrical frustration in very much the same manner as the spins on a triangular or tetrahedral kagomé and pyrochlore lattices. The classification of the SG phase in magnetic QCs as site-disordered or geometrically frustrated remains an open question; perhaps it is a combination of both.
Recently, magnetic frustration was reported for another class of materials, the complex metallic alloys (CMAs), characterized by giant unit cells that comprise from hundreds to thousands of atoms. Typical broken-ergodicity phenomena were reported for the orthorhombic Taylor phase T-Al3Mn and its ternary extensions T-Al3(Mn,Pd) and T-Al3(Mn,Fe) [21,22], all comprising 156 atoms in the unit cell. The Taylor phase is recognized as an approximant phase to the d-Al-Mn-Pd decagonal quasicrystal (d-QC), showing structural similarity to the d-QC on an intermediate scale of atomic clusters. Due to this structural similarity, and being periodic solids in three dimensions, QC approximant phases are suitable to investigate and model the physical properties of the true QC phases without translational periodicity, where the lack of periodicity prevents any quantitative theoretical analysis by standard theoretical approaches. The study of approximant phases may thus elucidate the origin of magnetic frustration in QCs (i.e., site disorder versus geometrical frustration). In the case of the Taylor phase, most lattice sites in the structure show either fractional occupation (the sites are too close in space to be occupied simultaneously) or mixed TM/Al occupancy, so that there exists substitutional and topological disorder on the lattice. Randomness is thus present in the Taylor phase, suggesting site-disordered origin of the SG phase. Geometric frustration in the Taylor phase has not been confirmed, though it may be present as well. Magnetic frustration and SG ordering were observed also in the μ-Al4Mn complex intermetallic phase with 563 atoms in the giant unit cell [23], which is an approximant phase to the Al?Mn quasicrystal. The μ-Al4Mn structure, although highly complex, shows a very low amount of disorder with only two Al sites being partially occupied. In the absence of site disorder, magnetic frustration in the μ-Al4Mn phase is of purely geometrical origin due to triangular spin clusters in the lattice. SG phases originating from randomness and geometric frustration were reported also for the FeAl2 and Fe2Al5 complex intermetallic phases [24].
In this review paper, we present spin glass properties and out-of-equilibrium dynamics of magnetically frustrated QCs [18] (the icosahedral RE-containing class) and CMAs [21], by pointing out their similarities as well as differences to canonical SGs and geometrically frustrated AFMs. We also show that magnetic frustration provides basis of a novel concept of digital data storage [25], where a byte of digital information is stored into the system of frustrated spins by pure thermal manipulation, in the absence of electric, magnetic or electromagnetic field. |
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Spin-glass properties of rare- |
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2.1. Origin of magnetic frustration in rare-earths-containing quasicrystals
Being electrically conducting solids, the basic interaction between spins in the RE-containing class of icosahedral QCs is the RKKY exchange interaction. The aperiodic distribution of RE?RE distances frustrates the spin system [18]. Here it is important noticing that the RE spins are not positioned randomly in the i-RE-Mg-Zn(Cd) structure, but occupy well-defined lattice sites [26], so that frustration is of geometrical origin. In real samples, structural and chemical disorders on the lattice reintroduce some degree of randomness. However, it was reported [27] that real i-RE-Mg-Zn samples could be grown to the best structural perfection, containing small amount of disorder only, so that these systems can be considered as a physical realization of a topological glass. From this point of view, the situation is similar to the geometrically frustrated AFMs, but with an important difference. The interaction between spins in the above AFMs is the nearest-neighbor direct exchange, whereas in the i-RE-Mg-Zn(Cd), it is the RKKY interaction.
Another similarity of the i-RE-Mg-Zn(Cd) QCs to the geometrically frustrated AFMs is the cluster structure of the spins. There are several features that indicate the presence of magnetic clusters in the i-RE-Mg-Zn(Cd). The most direct observation was reported for the i-Ho-Mg-Zn, where neutron scattering has detected novel short-range spin correlations [28], described by a six-dimensional modulation vector. The remarkable issue is the fact that the spin correlations terminate with a short correlation distance, 1 nm. An almost identical short-range spin cluster structure was observed also in the i-Tb-Mg-Cd [29], even though its atomic structure belongs to a fundamentally different type of icosahedral lattice (primitive icosahedral or P type), as compared to the i-Ho-Mg-Zn (face-centered icosahedral or F type). An indirect observation of magnetic clusters was also provided by the Vogel-Fulcher-law analysis of the frequency-dependent ac magnetic susceptibility in the i-Tb-Mg-Zn [30], where the Vogel-Fulcher temperature (amounting T0= 4.8 K, as compared to the freezing temperature T(f) = 5.8 K) is considered as a measure of the interaction strengths between clusters in a spin glass. Similar features (short AFM correlations and Vogel-Fulcher-law ac susceptibility) were reported also for the geometrically frustrated kagomé AFM (H3O)Fe3(SO4)2(OH)6 [11] (where ξ≈ 1.9 nm) and pyrochlore Tb2Mo2O7 [7].
Magnetic clustering properties of a purely geometrically frustrated quasiperiodic system were also predicted theoretically for some prototypical QC lattices. A study of Ising spins on a one-dimensional Fibonacci chain [31] found a ground state with a hierarchical structure of nested clusters, which underwent gradual paramagnetization in an external field with increasing temperature. A complicated spin structure was also found for a two-dimensional Penrose lattice [32]. Magnetic clustering thus seems to be an intrinsic feature of the geometrically frustrated spin systems, either periodic or quasiperiodic.
The above considerations suggest that the i-RE-Mg-Zn(Cd) QCs are closely related to other geometrically frustrated spin systems, whereas they are less similar to site-disordered spin glasses. It, therefore, depends on the coupling strength between clusters in order to classify them as SGs or superparamagnets (or a combination of both). The experimental observations of their magnetic properties [30,33] show typical broken-ergodicity phenomena, like the zfc-fc magnetization splitting, the frequency-dependent Tf(v), the ergodicity-breaking line in the TH−diagram, the negative anomaly in χ3 and aging effects in the dc magnetization. The minimum in χ3 is usually considered as a particularly strong evidence of a thermodynamic phase transition to a SG state. Its standard interpretation is that the spin fluctuations freeze at Tfcritically with a power law X3 ≈ (T-Tf)-r and the correlation length ξ tends to infinity. However, in view of the short correlation length observed in the i-Ho-Mg-Zn and i-Tb-Mg-Cd, this interpretation needs further consideration in the case of the i-RE-Mg-Zn(Cd) QCs. Another argument suggesting superparamagnetic nature of spin freezing in the RE-containing QCs is the recently reported [17] linear (superparamagnetic) relation between the thermoremanent magnetization (TRM) and the field-cooling magnetic field, MTRM ∝ Hfc, in the i-Tb-Mg-Zn. This behavior is just opposite to that expected for site-disordered SGs, where the TRM amplitude (normalized to its field-cooled magnetization value Mfc) should decrease for an increasing field [34]. The above experimental results therefore do not allow making an unambiguous classification of the i-RE-Mg-Zn(Cd) systems as SGs or superparamagnets. It is important to stress that a superparamagnetic component in the magnetization was found quite commonly in the geometrically frustrated AFMs. A nice example is the kagomé AFM (H3O)Fe3(SO4)2(OH)6 [11], where the progressive freezing of superparamagnetic entities over a wide temperature range is manifested in a continuous growth of the fc magnetization also below the zfc–fc splitting temperature Tf. A related situation was observed in the i-RE-Mg-Cd QCs [33]. There, the zfc-fc magnetization splitting occurs at the temperature Tf1(Fig. 1a) [18], whereas short-range spin correlations (spin clusters) appear already slightly above Tf1, as shown by the neutron experiment [29]. However, the fc magnetization continues to grow also below Tf1 down to the temperature of another anomaly at Tf2, below which it becomes temperature-independent. The origin of the second anomaly at Tf2 (that is almost H–independent [33]) is not clear at present, but it is straightforward to associate the growth of the fc magnetization between Tf1 and Tf2 with a superparamagnetic component in the magnetization. In contrast, the fc magnetization below Tf was found temperature-independent in the i-RE-Mg-Zn family [30] (Fig. 1b), as typical for site-disordered SGs. Strong superparamagnetic components were found also in other geometrically frustrated jarosite samples [35]. These samples also showed a divergence of the nonlinear susceptibility χ3, normally associated with a SG transition, which could not be explained properly.
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Concluding Remarks |
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| The out-of-equilibrium dynamics of magnetically frustrated systems remains one of the challenging fields in condensed matter physics. At the spin freezing temperature, the system of coupled spins in a frustrated configuration undergoes an ergodic–nonergodic phase transition, where spin reorientations can no longer maintain thermal equilibrium on the experimentally accessible time scales. Ergodicity of the spin system is consequently broken and the out-of-equilibrium dynamics is associated with slow approach towards thermodynamic equilibrium, which can never be reached due to macroscopic equilibration times. Magnetic quasicrystals with spins placed on a quasiperiodic lattice that exhibits nonperiodic long-range order with crystallographically forbidden symmetries, and complex metallic alloys, characterized by giant unit cells, exhibit variety of broken-ergodicity phenomena that share properties with the site-disordered canonical spin glasses and the site-ordered geometrically frustrated antiferromagnets. They can be best classified as a combination of both. Magnetic frustration provides basis of a novel concept of digital data storage, where a byte of digital information can be stored into the system of frustrated spins by pure thermal manipulation, in the absence of electric, magnetic or electromagnetic field. The concept of a thermal memory cell was successfully demonstrated on the magnetically frustrated complex intermetallic phase T-Al3(Mn,Fe), the decagonal quasicrystal d-Al-Mn-Fe and the canonical spin glass Cu-Mn as the storage media. |
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Acknowledgement |
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| We thank Marko Jagodič for his help in the experimental measurements and Michael Feuerbacher and Marc Heggen for provision of the T-Al3(Mn,Pd,Fe) samples. |
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In Eq. (2) we assume a stretched-exponential time-decay of the TRM. In another study (Dasgupta, C.; Ma, S.K.; Hu, C.K. Dynamic properties of a spin-glass model at low temperatures. Phys. Rev. B 1979, 20, 3837-3849), a theoretical TRM decay in a superparamagnet due to flipping clusters was derived as a power-law, aTTRMtM?∝. This is the same law as that derived for a spin glass 45., but with an explicit temperature dependence of the coefficient ()()THaHTa?=,. |
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