Frustration Index Research Articles

Synthesis, Crystal Structure, and Magnetic Properties of Li3Mg2OsO6, a Geometrically Frustrated Osmium(V) Oxide with an Ordered Rock Salt Structure: Comparison with Isostructural Li3Mg2RuO6

The novel osmium-based oxide Li(3)Mg(2)OsO(6) was synthesized in polycrystalline form by reducing Li(5)OsO(6) by osmium metal and osmium(IV) oxide in the presence of stoichiometric amounts of magnesium oxide. The crystal structure was refined using powder X-ray diffraction data in the orthorhombic Fddd space group with a = 5.88982(5) Å, b = 8.46873(6) Å, and c = 17.6825(2) Å. This compound is isostructural and isoelectronic with the ruthenium-based system Li(3)Mg(2)RuO(6). The magnetic ion sublattice Os(5+) (S = 3/2) consists of chains of interconnected corner- and edge-shared triangles, which brings about the potential for geometric magnetic frustration. The Curie-Weiss law holds over the range 80-300 K with C = 1.42(3) emu·K/mol [μ(eff) = 3.37(2) μ(B)] and θ(C) = -105.8(2) K. Below 80 K, there are three anomalies at 75, 30, and 8 K. Those at 75 and 30 K are suggestive of short-range antiferromagnetic correlations, while that at 8 K is a somewhat sharper maximum showing a zero-field-cooled/field-cooled divergence suggestive of perhaps spin freezing. The absence of magnetic Bragg peaks at 3.9 K in the neutron diffraction pattern supports this characterization, as does the absence of a sharp peak in the heat capacity, which instead shows only a very broad maximum at ∼12 K. A frustration index of f = 106/8 = 13 indicates a high degree of frustration. The magnetic properties of the osmium phase differ markedly from those of the isostructural ruthenium material, which shows long-range antiferromagnetic order below 17 K, f = 6, and no unusual features at higher temperatures. Estimates of the magnetic exchange interactions at the level of spin-dimer analysis for both the ruthenium and osmium materials support a more frustrated picture for the latter. Errors in the calculation and assignment of the exchange pathways in the previous report on Li(3)Mg(2)RuO(6) are identified and corrected.

Study of the physical properties of La2 − xErxTi2O7(0 ≤ x ≤ 0.075) compounds

New complex magnetic frustrated materials La 2 − x Er x Ti 2 O 7 (0 ≤ x ≤ 0.075) have been synthesized by solid-solid method. The crystallographic and magnetic properties were studied as a function of substitution of the La 3+ ion by the Er 3+ one. All samples are found to be single phase and crystallize in the monoclinic structure with P 21 space group. Magnetic measurements have revealed the presence of dominant antiferromagnetic interactions and the absence of magnetic ordering until a temperature of 2 K. The magnetic study suggests that the Curie-Weiss temperature ( θ CW ) is negative, as expected in the frustrated lanthanide sublattice in the Ln 2 – x Ln’ x M 2 O 7 structure. From the frustration index f = - θ CW / T N , we have deduced the presence of a strong frustration phenomenon. The critical properties of the antiferromagnetic behavior, for x = 0.025, 0.05 and 0.075 samples, have been investigated. It was found that the 2D-self-avoiding walk (SAW) model is the best one to describe the critical phenomena.

Six signed Petersen graphs, and their automorphisms

Up to switching isomorphism, there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups, chromatic numbers, and numbers of proper 1-colorations, thereby illustrating some of the ideas and methods of signed graph theory. We also calculate automorphism groups and clusterability indices, which are not invariant under switching. In the process, we develop new properties of signed graphs, especially of their switching automorphism groups.

Magnetic properties of frustrated pyrochlore fluorides Cd2M2F6S (M=Fe, Co, and Ni)

Magnetic properties of pyrochlore fluorides ${\mathrm{Cd}}_{2}{M}_{2}{\mathrm{F}}_{6}{\mathrm{S}}_{}$ (${M}_{}=\text{Fe}$, Co, and Ni) were investigated. We found that the $3d$ transition metal ${M}_{}$ is partially substituted by Cd, and the chemical formula is represented by ${\mathrm{Cd}}_{2+2x}{M}_{2\ensuremath{-}2x}{\mathrm{F}}_{6}{\mathrm{S}}_{}$. For all three systems, negative Curie-Weiss temperatures ${\ensuremath{\Theta}}_{\mathrm{CW}}$ indicate dominant antiferromagnetic interactions between the nearest-neighbor magnetic ions. Magnetic orderings are observed much below $|{\ensuremath{\Theta}}_{\mathrm{CW}}|$ for the ${\mathrm{Fe}}_{}$ and ${\mathrm{Co}}_{}$ compounds, and the ordering temperatures are reduced by increasing $x$. The frustration index of the ${\mathrm{Fe}}_{}$ compound is about 10, and that of the Co compound is about 45, indicating strong frustration. No explicit ordering is observed in the ${\mathrm{Ni}}_{}$ compound, which is likely due to large Cd substitution $x$. In addition, field-induced magnetic transitions of the ${\mathrm{Fe}}_{}$ compound are observed. These transitions are likely to correspond to the boundary of a half-magnetization-plateau state found in chromium spinels, which suggests the important role of orbital degrees of freedom to exhibit the field-induced transitions.

Absence of long-range magnetic ordering in the pyrochlore compound Er2Sn2O7

The low temperature behaviour of powder Er2Sn2O7 samples has been studied by magnetic susceptibility, heat capacity, and neutron scattering experiments. We report here the absence of magnetic ordering down to 100 mK. Anomalies in the heat capacity can be accounted for through an analysis of the crystal field spectrum observed by inelastic neutron scattering spectroscopy. These new measurements on Er2Sn2O7 suggest a new lower bound for the frustration index of f = |ΘCW|/TN = 14/0.1 = 140, placing this compound into a highly frustrated regime.

Comparison study of magnetic ordering for Fe-free and Fe-doped LiMn2O4 spinel oxide

The structural and physical properties of LiMn2-xFexO4 (x = 0 to 0.5) were investigated. Contrasting LiMn2O4 with LiMn1.5Fe0.5O4, we observed a remarkable difference in the magnetic frustration. The LiMn2O4 sample has a magnetic ordering transition at 61 K while Fe-doped LiMn1.5Fe0.5O4 has an antiferromagnetic ordering transition with a Neel temperature at TN = 34 K. Our result showed a larger value of the frustration index (f=| w|/TC) for pure LiMn2O4 than for LiMn1.5Fe0.5O4. Fe-doping results in a decrease in the degree of frustration. Fe dopants occupying positions of Mn ion can break the original moment equilibrium so as to suppress the magnetic frustration in LiMn2O4.

Representation and rebellion: the Rockefeller plan at the Colorado Fuel and Iron Company, 1914-1942

Memories of a Massacre Student & Teacher Between Two Extremes Divisions in the Ranks The Rockefeller Plan in Action: The Mines The Rockefeller Plan in Action: The Mill New Union, Same Struggle Depression, Frustration & Real Competition Conclusion Index.

Frustrated and Unfrustrated Magnetic Orders in the 10H Perovskite Ba5Sb1−xMn4+xO15−δ

A 10-layer hexagonal perovskite Ba5Sb1−xMn4+xO15−δ solid solution (0.24 ≤ x ≤ 0.36; space group P63/mmc, a = 5.7095(1) A, and c = 23.4866(3) A for x = 0.36) has been synthesized by solid state reaction and studied using powder X-ray and neutron diffraction and magnetization measurements. This 10H polytype structure contains one corner-sharing (Sb,Mn)O6 octahedron and a tetramer of four face-sharing MnO6 octahedra per formula unit. Two magnetic transitions are observed at 129−167 K (TM) and 13−22 K (TF). The frustration index |θ|/TM, where θ is the Weiss temperature, is very close to unity, but considerable magnetic frustration is evidenced through the long-range order of a small ∼0.5 μB antiferromagnetic moment below TM, and the formation of ferro- or ferrimagnetic clusters. These freeze at TF with the |θ|/TF ≈ 7−10 index representing high magnetic frustration. TF increases directly with the excess of Mn, x, in Ba5Sb1−xMn4+xO15−δ rather than with the overall Mn content, showing that the occupation of the ...

Dynamics of coupled adaptive elements: Bursting and intermittent oscillations generated by frustration in networks

Adaptation to environmental change is a common property of biological systems. Cells initially respond to external changes in the environment, but after some time, they regain their original state. By considering an element consisting of two variables that show such adaptation dynamics, we studied a coupled dynamical system containing such elements to examine the diverse dynamics in the system and classified the behaviors on the basis of the network structure that determined the interaction among elements. For a system with two elements, two types of behaviors, perfect adaptation and simple oscillation, were observed. For a system with three elements, in addition to these two types of dynamics, rapid-burst-type oscillation and a slow cycle were discovered; depending on the initial conditions, these two types of dynamics coexisted. These behaviors are a result of the characteristic dynamics of each element, i.e., fast response and slow adaptation processes. The behaviors depend on the network structure (in specific, a combination of positive or negative feedback among elements). Cooperativity among elements due to a positive feedback loop leads to simple oscillation, whereas frustration involving alternating positive and negative interactions among elements leads to the coexistence of rapid bursting oscillation and a slow cycle. These behaviors are classified on the basis of the frustration indices defined by the network structure. The period of the slow cycle is much longer than the original adaptation time scale, while the burst-type oscillation is a continued response that does not involve any adaptation. We briefly discuss the universal applicability of our results to a network of a larger number of elements and their possible relevance to biological systems.

Spin glass transitions in the absence of chemical disorder for the pyrochlores A2Sb 2O 7 ( A=Mn, Co, Ni)

The pyrochlores in the series A 2Sb 2O 7 have been synthesized and characterized as exhibiting spin glass transitions at T SG=41, 4.5, and 2.6 K (for A=Mn 2+ S = 5 2 , Co 2+ S = 3 2 and Ni 2+ S=1, respectively) despite the lack of chemical disorder. Since the Curie–Weiss temperature remains essentially constant for all members in the series ( θ ∼ − 40 K ), the frustration index for these materials increases significantly as the moment size is reduced from f = | θ | / T SG = 1.1 (Mn 2+), to 9.3 (Co 2+) to 14.6 (Ni 2+). There is also a corresponding change in the spin dynamics measured by the shift in the AC susceptibility signal as a function of frequency. These new materials provide an avenue to investigate the effect of quantum fluctuations on the Heisenberg pyrochlore lattice in the low spin limit, and show there is a dramatic change in the spin dynamics as the quantum regime is approached.

Magnetic properties of theS=32geometrically frustrated double perovskitesLa2LiRuO6andBa2YRuO6

Two $B$-site ordered double perovskites, ${\text{Ba}}_{2}{\text{YRuO}}_{6}$ and ${\text{La}}_{2}{\text{LiRuO}}_{6}$, have been reinvestigated as part of a systematic study of geometric magnetic frustration in this class of oxide materials. Both involve ${\text{Ru}}^{5+}(4{d}^{3},S=3/2)$ as the magnetic ion residing on a face-centered-cubic lattice---one of the canonical frustrated lattices. Results from dc susceptibility, neutron-diffraction, heat-capacity, $^{7}\text{L}\text{i}$ and $^{89}\text{Y}$ NMR studies are presented. ${\text{La}}_{2}{\text{LiRuO}}_{6}$ $(P{2}_{1}/n)$ shows long-range antiferromagnetic order below 24 K from $^{7}\text{L}\text{i}$ NMR, heat-capacity and magnetic-susceptibility (Fisher's heat-capacity) data which is well below the susceptibility maximum at 30 K. Analysis of the entropy loss and the $^{7}\text{L}\text{i}$ data indicates the importance of short-range spin correlations at higher temperatures, consistent with a frustrated system. ${\text{Ba}}_{2}{\text{YRuO}}_{6}$ retains $Fm3m$ symmetry found at room temperature down to 2.8 K with cell constants, $a=8.33559(9)$ and $a=8.3239(5)\text{ }\text{\AA{}}$, respectively. However, $^{89}\text{Y}$ magic-angle-spinning NMR detects a very low $\ensuremath{\sim}1%$ site mixing between Y and Ru ions. Magnetic-susceptibility data are more complex than reported previously with two broad peaks around 37 and 47 K. The transition temperature is 36 K from heat capacity and variable-temperature neutron-diffraction data. The Weiss temperatures and frustration indices, $|\ensuremath{\theta}|/{T}_{\text{N}}$, for ${\text{Ba}}_{2}{\text{YRuO}}_{6}$ are $\ensuremath{-}522$ K and 16 while much smaller values are observed for ${\text{La}}_{2}{\text{LiRuO}}_{6}$, $\ensuremath{-}184$ K and 8, which can be attributed to the monoclinic structural distortion in the latter which weakens the superexchange interactions.

Frustrated Orders in the Perovskite (Bi0.5Sr0.5)CrO3

A new perovskite, (Bi0.5Sr0.5)CrO3, has been prepared at 1000 °C and ambient pressure, although BiCrO3 and SrCrO3 are both high pressure phases. The crystal structure is rhombohedral (R3̅c, a = 5.4652(5) and c = 13.3502(2) Å at 300 K). Well-defined local Bi3+ displacements are evident despite an absence of long-range Bi/Sr cation order, but the degree of ferroelectric order is unclear. Long range G-type antiferromagnetic order occurs below 30 K; however, the low value of the ordered magnetic moment (1.2 μB), a high frustration index, and low temperature divergence of zero field and field cooled susceptibilities show that a substantial glassy component is also present.

Ba3NbFe3Si2O14: A New Multiferroic with a 2D Triangular Fe3+ Motif

Single crystals of the new multiferroic Ba3NbFe3Si2O14 have been synthesized and characterized via X-ray diffraction, magnetic susceptibility, NMR, and electric polarization measurements. The underlying topology of the Fe 3+ spins is composed of isolated triangular units stacked in 2D layers along the c-direction of the trigonal lattice. Magnetic susceptibility experiments verify the spin state of the iron moments (S = 5/2), and large antiferromagnetic interactions with θ = −190 K. A cusp in the susceptibility and lambda-like feature in the specific heat indicate a magnetic phase transition at TN = 26 K, resulting in a frustration index of f = 7. A significant amount of spin entropy is released at higher temperatures, suggestive of short-ranged magnetic correlations developing at 50 K (as evidenced by features in the specific heat and thermal conductivity). The development of a spontaneous electric polarization below 26 K verifies that Ba3NbFe3Si2O14 is a multiferroic.

Long-Range Antiferromagnetic Ordering in the Novel Magnetically Frustrated Rock Salt Oxide System: Li3Mg2RuO6

The new ruthenate, Li3Mg2RuO6, was synthesized in polycrystalline form by a solid-state method. The crystal structure of Li3Mg2RuO6 was refined from powder neutron diffraction data in the orthorhombic Fddd space group in an ordered NaCl structure type with a = 5.8759 (2) Å, b = 8.4206 (1) Å, and c = 17.6455 (5) Å. The Ru5+ (S = 3/2) sublattice shows a unique topology, which consists of chains of edge-sharing triangles interconnected by corner sharing and thus there exists the potential for geometric magnetic frustration. This is the first time that magnetic behavior of a material with such sublattice has been investigated. The isostructural, nonmagnetic Li3Mg2NbO6 was prepared as the lattice match for the heat capacity analysis. The temperature-dependent magnetic susceptibility, heat capacity, and neutron diffraction data reveal that the system undergoes a long-range AFM ordering below 17 K but also provide evidence for short-range spin correlations well above TN. The magnetic frustration index, f ≈ 109/17 = 6.4 suggests that the system is geometrically frustrated. Spin dimer analysis of the magnetic exchange interactions supports this characterization. Comparisons are made with other materials containing the Ru5+ ion on geometrically frustrated lattices.

Superconducting transitions in frustrated Josephson-junction arrays on a dice lattice

Transport measurements were carried out on dice Josephson-junction arrays with the frustration index f = 1/3 and 1/2 which possess, within the limit of the XY model, an accidental degeneracy of the ground states as a consequence of the formation of zero-energy domain walls. The experimental results are in apparent contradiction with the theoretical expectation that frustration effects in the f = 1/3 system are particularly strong enough to suppress a vortex-ordering transition down to near zero temperature. The data for f = 1/2 are more consistent with theoretical evaluations. The agreement between the experiments and the Monte Carlo simulations of a XY model for f = 1/3 suggests that the order-from-disorder mechanism for the removal of an accidental degeneracy may still be effective in the f = 1/3 system.

Long-range antiferromagnetic ordering in theS=12ordered rocksalt oxideLi5OsO6: Comparison with the isoelectronic and isostructural spin glassLi4MgReO6

${\mathrm{Li}}_{5}\mathrm{Os}{\mathrm{O}}_{6}$ and ${\mathrm{Li}}_{5}\mathrm{Re}{\mathrm{O}}_{6}$ polycrystalline samples were synthesized by conventional solid state methods. Employing powder neutron diffraction data, the crystal structure of ${\mathrm{Li}}_{5}\mathrm{Os}{\mathrm{O}}_{6}$ was reinvestigated. ${\mathrm{Li}}_{5}\mathrm{Os}{\mathrm{O}}_{6}$ crystallizes in $C2∕m$ in an ordered NaCl structure type, where $a=5.0472(1)\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$, $b=8.7827(2)\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$, $c=5.0079(1)\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$, $\ensuremath{\beta}=109.777(2)\ifmmode^\circ\else\textdegree\fi{}$ and $V=208.90(1)\phantom{\rule{0.3em}{0ex}}{\mathrm{\AA{}}}^{3}$. Magnetic susceptibility and heat capacity data indicate an antiferromagnetic long-range order below $40\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, although there is evidence for low dimensional short-range order below $80\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. As well, the frustration index, $f=\ensuremath{\mid}\ensuremath{\theta}\ensuremath{\mid}∕{T}_{N}\ensuremath{\sim}1$, in contrast to the isostructural and isoelectronic compound, ${\mathrm{Li}}_{4}\mathrm{Mg}\mathrm{Re}{\mathrm{O}}_{6}$, which is a spin glass below $12\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and has $f\ensuremath{\sim}14$. An attempt was made to rationalize these differences using spin dimer analysis. The key results are that the spin exchange interactions in the Re-based compound are stronger and are consistent with a frustrated triangular lattice model, while a low dimensional short-range order model is better for ${\mathrm{Li}}_{5}\mathrm{Os}{\mathrm{O}}_{6}$. The main reason for this is a strong Jahn-Teller distortion in the $\mathrm{Os}{\mathrm{O}}_{6}$ octahedron material which constrains the unpaired electron to occupy the ${d}_{xy}$ orbital.

Partial Field-Induced Magnetic Order in the Spin-Liquid KagoméNd3Ga5SiO14

The distorted kagomé system Nd3Ga5SiO14 has been investigated with neutron scattering down to 0.046 K with no evidence of magnetic long range order of the Nd3+ moments in a zero field. Substantial diffuse scattering is observed which is in agreement with nearest-neighbor correlations between the fluctuating spins. Upon the application of a field in the c direction, the diffuse scattering is reduced in intensity while the magnetic Bragg peaks grow in intensity to saturate by 1 T to 1/2 of the expected magnetization. These measurements suggest that a unique spin-liquid state develops in Nd3Ga5SiO14 with a frustration index of f approximately |theta|/T_{C}> or =1300.

Frustrated magnetism and the effects of titanium substitution in the mixed-valence oxideBaV10−xTixO15

The low temperature magnetic phase transitions in $\mathrm{Ba}{\mathrm{V}}_{10}{\mathrm{O}}_{15}$ at $\ensuremath{\sim}40\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and $\ensuremath{\sim}25\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ are investigated in detail. The topology of the V sublattice, which consists of linked ${\mathrm{V}}_{10}$ clusters formed by four tetrahedra sharing edges and corners, leads to geometric magnetic frustration and a depression of the ordering temperature characterized by the frustration index, $f=\ensuremath{\mid}\ensuremath{\theta}\ensuremath{\mid}∕{T}_{c}\ensuremath{\sim}30$. Specific heat data show the coexistence of both long range and short range magnetic order at low temperature. The position of a $\ensuremath{\lambda}$ peak, identified with ${T}_{c}$ for long range antiferromagnetic order, depends on the ${\mathrm{V}}^{2+}∕{\mathrm{V}}^{3+}$ ratio and varies from $43.0\phantom{\rule{0.3em}{0ex}}\mathrm{K}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}41.6\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ for samples with the highest ${\mathrm{V}}^{2+}$ content but is suppressed to zero for highly oxidized samples. A broad maximum in the specific heat occurs at $\ensuremath{\sim}25\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and is present in all samples regardless of the level of oxidation. For samples with a well defined ${T}_{c}$, estimates of the entropy removal over the range $5\phantom{\rule{0.3em}{0ex}}\mathrm{K}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}55\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ are only 11% of that expected for $8\phantom{\rule{0.3em}{0ex}}{\mathrm{V}}^{3+}(S=1)$ and $2\phantom{\rule{0.3em}{0ex}}{\mathrm{V}}^{2+}(S=3∕2)$ ions per formula unit, suggesting the presence of considerable short-range order. Neutron diffraction data support the presence of long range antiferromagnetic (AF) order with ${T}_{c}\ensuremath{\sim}40\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. The magnetic structure below ${T}_{c}$ is complex: there are five crystallographically independent vanadium ions in the unit cell, leading to 40 magnetic ions in the crystal unit cell and 80 spins per magnetic cell. Furthermore, the magnetic structure is of the multi-$k$ type involving the two unrelated wave vectors $\mathbf{k}=(0\phantom{\rule{0.3em}{0ex}}0\phantom{\rule{0.3em}{0ex}}0)$ and $\mathbf{k}=(1∕2\phantom{\rule{0.3em}{0ex}}0\phantom{\rule{0.3em}{0ex}}0)$. Zero field muon spin relaxation data show spin freezing below $30\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. ac susceptibility results are anomalous. Two peaks, at $\ensuremath{\sim}0\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and $\ensuremath{\sim}25\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ are seen in ${\ensuremath{\chi}}^{\ensuremath{'}}$ but the maxima are frequency independent as is the prominent maximum in ${\ensuremath{\chi}}^{\ensuremath{''}}$ near $25\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, behavior atypical of spin glasses. Ti-doped samples, $\mathrm{Ba}{\mathrm{V}}_{10\ensuremath{-}x}{\mathrm{Ti}}_{x}{\mathrm{O}}_{15}$, show remarkable results as only 5% Ti doping is sufficient to destroy the AF long range order and induce conventional spin glass behavior in the dc and ac susceptibility, with freezing temperatures ranging from $15\phantom{\rule{0.3em}{0ex}}\mathrm{K}--5\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ for doping levels from 5% to 50%. The frustration indices reach enormous values to $f=130$ for $x=3$. The spin glass state is quenched for 70% Ti doping and only paramagnetic behavior is seen.

The investigation of spontaneous magnetization in large-scale two-dimensional hexagonal superconductingπ-ring arrays

We present an analytical investigation on the spontaneous magnetizationof infinitely large two-dimensional identical hexagonal superconductingπ-ring arrays based on an ansatz that each of theπ-rings carries a magnetic flux with the same frustration index. The free energies per ring,Un, in various states characterized by the number of zero-current (‘quiet’) junctionsn in the ring are calculated. A ladder relation of the energies has been demonstrated as, which basically shows that the fewer the quiet junctions of a state, the morefavourable the energy for the state formation. This result is in qualitativeagreement with the recent experimental observations in the large-scale hexagonalπ-loop (triangular half flux quantum) array (Hilgenkamp et al 2003 Nature 422 50–3).

The ordering of spontaneous magnetization moments in large-scale two-dimensional superconductingπ-ring arrays

Large planar arrays of Josephson junctions form two-dimensional systems withmany interesting properties, especially arrays containing superconductingπ-junctions with unconventional superconducting d-wave symmetry. Here we present ananalytical investigation of spontaneous magnetization in two-dimensional superconductingπ-ring arrays, starting from an ansatz that each of the identicalπ-rings carries a magnetic flux with the same frustration index. Our results show that theflux structure with full antiparallel spontaneous magnetization is the most favourable of allpossible arrangements in both square and triangular ring arrays. In a hexagonalπ-ring (triangular flux) array, although there is no full antiparallel flux pattern, theantiparallel bond of nearest-neighbour spontaneous magnetization fluxes is stillenergetically favourable. These results are in agreement with those recently observedexperimentally by Hilgenkamp et al (2003 Nature 422 50–3) in a triangular flux array.