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At least 73 records · Page 4

Exchange coupling in $\mathrm{Bi_2}$ $\mathrm{Se_3}$ / $\mathrm{EuSe}$ heterostructures and evidence of interfacial antiferromagnetic order formation

Spatial confinement of electronic topological surface states (TSSs) in topological insulators poses a formidable challenge because TSSs are protected by time-reversal symmetry. In previous works formation of a gap in the electronic spectrum of TSSs has been successfully demonstrated in topological insulator/magnetic material heterostructures, where ferromagnetic exchange interactions locally lift the time-reversal symmetry. Here we report experimental evidence of exchange interaction between a topological insulator $\mathrm{Bi_2}$ $\mathrm{Se_3}$ and a magnetic insulator EuSe. Spin-polarized neutron reflectometry reveals a reduction of the in-plane magnetic susceptibility within a 2 nm interfacial layer of EuSe, and the combination of superconducting quantum interference device (SQUID) magnetometry and Hall measurements points to the formation of an interfacial layer with a suppressed net magnetic moment. This suppressed magnetization survives up to temperatures five times higher than the Néel temperature of EuSe. Its origin is attributed to the formation of an interfacial antiferromagnetic state. Abrupt resistance changes observed in high magnetic fields are consistent with antiferromagnetic domain reconstruction affecting transport in a TSS via exchange coupling. Importantly, the high-temperature local control of TSSs with zero net magnetization unlocks new opportunities for the design of electronic, spintronic, and quantum computation devices, ranging from quantization of Hall conductance in zero fields to spatial localization of non-Abelian excitations in superconducting topological qubits.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Isotropic nature of the metallic Kagome ferromagnet Fe 3 Sn 2 at high temperatures.

Anisotropy and competing exchange interactions have emerged as two central ingredients needed for centrosymmetric materials to exhibit topological spin textures. Fe 3 Sn 2 is thought to have these ingredients as well, as it has recently been discovered to host room temperature skyrmionic bubbles with an accompanying topological Hall effect. We present small-angle inelastic neutron scattering measurements that unambiguously show that Fe3Sn2 is an isotropic ferromagnet below TC ≈ 660 K to at least 480 K – the lower temperature threshold of our experimental configuration. Fe 3 Sn 2 is known to have competing magnetic exchange interactions, correlated electron behavior, weak magnetocrystalline anisotropy, and lattice anisotropy; all of these features are thought to play a role in stabilizing skyrmions in centrosymmetric systems. Our results reveal that at elevated temperatures, there is an absence of significant magnetocrystalline anisotropy and that the system behaves as a nearly ideal isotropic exchange interaction ferromagnet, with a spin stiffness D(T = 480 K) = 168 meV Å2, which extrapolates to a ground state spin stiffness D(T = 0 K) = 231 meV Å2.

36 MATERIALS SCIENCE↗

Influence of the vertex region on spin dynamics in artificial Kagome spin ice.

We present experimental and theoretical studies of spin-wave mode dynamics in artificial kagome spin ice vertices made of three identical 15-nm thick elongated Ni80Fe20 nanoislands (macrospins). We consider several possible configurations, from completely disjointed macrospins (full dipolar interelement interac-tions) to fully jointed macrospins (full dipolar-exchange interactions). Using angular-resolved magnetic field dependent broadband ferromagnetic resonance (FMR), we demonstrate the occurrence of a mode localized in the vertex region as indicated by the distinct behavior of the FMR spectra at different angles and configurations. Theoretical calculations using micromagnetic simulations support the existence, ori-gin, and behavior of this mode by interpreting it as a localized, quasi-uniform Kittel mode. Our findings pave the way for designing the most appropriate network consisting of ferromagnetic nanomagnets for specific application purposes in magnonics.

Bang, Wonbae↗

Ferromagnetic Exchange and Slow Magnetic Relaxation in Cobalt Bis(1,2-dithiolene)-Bridged Dilanthanide Complexes

The construction of multinuclear lanthanide-based molecules with significant magnetic exchange interactions represents a key challenge in the realization of single-molecule magnets with high operating temperatures. Here, we report the synthesis and magnetic characterization of two series of heterobimetallic compounds, (Cp* 2 Ln) 2 (μ-Co(pdt) 2 ) (Ln = Y 3+ , Gd 3+ , Dy 3+ ; pdt 2– = 1,2-diphenylethylenedithiolate) and [K(18-crown-6)][(Cp* 2 Ln) 2 (μ-Co(pdt) 2 )] (Ln = Y 3+ , Gd 3+ ), featuring two lanthanide centers bridged by a cobalt bis(1,2-dithiolene) complex. Dc magnetic susceptibility data collected for the Gd congeners indicate significant Gd–Co ferromagnetic exchange interactions with fits affording J = +11.5 and +7.33 cm –1 , respectively. Magnetization decay and ac magnetic susceptibility measurements carried out on the single-molecule magnet (Cp* 2 Dy) 2 (μ-Co(pdt) 2 ) reveal full suppression of quantum tunneling and open-loop hysteresis persisting up to 3.5 K. These results, along with those of high-field EPR spectroscopy, suggest that transition metalloligands can enforce strong exchange interactions with adjacent lanthanide centers while maintaining a geometry that preserves molecular anisotropy. Furthermore, the magnetic properties of [K(18-crown-6)][(Cp* 2 Gd) 2 (μ-Co(pdt) 2 )] show that increasing the spin of the ground state of the bridging complex may be a viable alternative to increasing J in obtaining well-isolated, strongly coupled magnetic ground states.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Spin wave theory for the triaxial magnetic anisotropy 2D van der Waals antiferromagnet CrSBr

The magnetic properties of two-dimensional (2D) materials have been attracting increasing attention in recent years due to their unique behavior and possible applications in new devices. One material of great interest is the 2D van der Waals (vdW) crystal CrSBr, which exhibits antiferromagnetic (AF) order at low temperatures due to an interlayer AF exchange interaction. Here, we present a full quantum spin-wave theory for vdW crystals considering one interlayer and three intralayer exchange interactions, and triaxial magnetic anisotropy. The fits of the theoretical results to antiferromagnetic resonance measurements and inelastic neutron scattering data in CrSBr yield reliable values for the seven interaction parameters that can be used to calculate other properties of this interesting material.

2D materials↗

A resonant valence bond spin liquid in the dilute limit of doped frustrated Mott insulators

Abstract Ideas about resonant valence bond liquids and spin–charge separation have led to key concepts in physics such as quantum spin liquids, emergent gauge symmetries, topological order and fractionalization. Despite extensive efforts to demonstrate the existence of a resonant valence bond phase in the Hubbard model that originally motivated the concept, a definitive realization has yet to be achieved. Here we present a solution to this long-standing problem by uncovering a resonant valence bond phase exhibiting spin–charge separation in realistic Hamiltonians. We show analytically that this ground state emerges in the dilute-doping limit of a half-filled Mott insulator on corner-sharing tetrahedral lattices with frustrated hopping, in the absence of exchange interactions. We confirm numerically that the results extend to finite exchange interactions, finite-sized systems and finite dopant density. Although much attention has been devoted to the emergence of unconventional states from geometrically frustrated interactions, our work demonstrates that kinetic energy frustration in doped Mott insulators may be essential for stabilizing robust, topologically ordered states in real materials.

Physics↗

Spin-dependent electron transfer in electrochemically transparent van der Waals heterostructures for oxygen evolution reaction

Spin selective catalysis is an emerging approach for improving the thermodynamics and kinetics of reactions. The role of electron spins has been scarcely studied in catalytic reactions. One exception is the oxygen evolution reaction (OER) where strongly correlated metals and oxides are used as catalysts. In OER, spin alignment facilitates the transition of singlet state of the reactant to the triplet state of O 2 . However, the influence of strong correlations on spin exchange mechanism and spin selective thermodynamics of most catalytic reactions remain unclear. Here we decouple the strongly correlated catalyst from the electrolyte to study spin exchange in two-dimensional (2D) magnetic iron germanium telluride (FGT) heterostructure. We demonstrate that transmission of spin and electrochemical information between the catalyst and the reactant can occur through quantum exchange interaction despite the catalyst of FGT being completely encapsulated by graphene or hexagonal boron nitride (hBN). The strong correlations in FGT that lead to enhanced spin exchange in OER are observed in graphene or hBN layers with thicknesses of up to 6 nm. We demonstrate that spin alignment in FGT leads to a lowering of thermodynamic barrier for adsorption of hydroxide ion and electron transfer to the catalyst. This results in up to fivefold enhancement in OER performance and improved kinetics. Our results provide clear evidence that transmission of both quantum mechanical and electrochemical information through quantum spin exchange interaction in FGT leads to an enhancement in catalytic performance.

2D materials↗

Origin of Ferromagnetic Exchange Coupling in Donor–Acceptor Biradical Analogues of Charge-Separated Excited States

A new donor–acceptor biradical complex, Tp Cum,Me Zn(SQ-VD) (Tp Cum,Me Zn + = zinc(II) hydro-tris(3-cumenyl-5-methylpyrazolyl)borate complex cation; SQ = orthosemiquinone; VD = oxoverdazyl), which is a ground-state analogue of a charge-separated excited state, has been synthesized and structurally characterized. Here, the magnetic exchange interaction between the S = 1/2 SQ and the S = 1/2 VD within the SQ-VD biradical ligand is observed to be ferromagnetic, with $J_{SQ-VD}$ = +77 cm –1 (H = -2$J_{SQ-VD}\bar{S}_{SQ}·\bar{S}_{VD}$) determined from an analysis of the variable-temperature magnetic susceptibility data. The pairwise biradical exchange interaction in Tp Cum,Me Zn(SQ-VD) can be compared with that of the related donor–acceptor biradical complex Tp Cum,Me Zn(SQ-NN) (NN = nitronyl nitroxide, S = 1/2), where $J_{SQ-NN}$ ≅ +550 cm –1 . This represents a dramatic reduction in the biradical exchange by a factor of ~7, despite the isolobal nature of the VD and NN acceptor radical SOMOs. Computations assessing the magnitude of the exchange were performed using a broken-symmetry density functional theory (DFT) approach. These computations are in good agreement with those computed at the CASSCF NEVPT2 level, which also reveals an S = 1 triplet ground state as observed in the magnetic susceptibility measurements. A combination of electronic absorption spectroscopy and CASSCF computations has been used to elucidate the electronic origin of the large difference in the magnitude of the biradical exchange coupling between Tp Cum,Me Zn(SQ-VD) and Tp Cum,Me Zn(SQ-NN). A Valence Bond Configuration Interaction (VBCI) model was previously employed to highlight the importance of mixing an SQ SOMO → NNLUMO charge transfer configuration into the electronic ground state to facilitate the stabilization of the high-spin triplet (S = 1) ground state in Tp Cum,Me Zn(SQ-NN). Here, CASSCF computations confirm the importance of mixing the pendant radical (e.g., VD, NN) LUMO (VD LUMO and NN LUMO ) with the SOMO of the SQ radical (SQ SOMO ) for stabilizing the triplet, in addition to spin polarization and charge transfer contributions to the exchange. An important electronic structure difference between Tp Cum,Me Zn(SQ-VD) and Tp Cum,Me Zn(SQ-NN), which leads to their different exchange couplings, is the reduced admixture of excited states that promote ferromagnetic exchange into the Tp Cum,Me Zn(SQ-VD) ground state, and the intrinsically weaker mixing between the VD LUMO and the SQ SOMO compared to that observed for Tp Cum,Me Zn(SQ-NN), where this orbital mixing is significant. The results of this comparative study contribute to a greater understanding of biradical exchange interactions, which are important to our understanding of excited-state singlet–triplet energy gaps, electron delocalization, and the generation of electron spin polarization in both the ground and excited states of (bpy)Pt(CAT-radical) complexes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Programming correlated magnetic states with gate-controlled moiré geometry

The ability to control the underlying lattice geometry of a system may enable transitions between emergent quantum ground states. We report in situ gate switching between honeycomb and triangular lattice geometries of an electron many-body Hamiltonian in rhombohedral (R)–stacked molybdenum ditelluride (MoTe 2 ) moiré bilayers, resulting in switchable magnetic exchange interactions. At zero electric field, we observed a correlated ferromagnetic insulator near one hole per moiré unit cell with a widely tunable Curie temperature up to 14 K. Applying an electric field switched the system into a half-filled triangular lattice with antiferromagnetic interactions; further doping this layer-polarized superlattice tuned the antiferromagnetic exchange interaction back to ferromagnetic. Our work demonstrates R-stacked MoTe 2 moirés to be a laboratory for engineering correlated states with nontrivial topology.

Science & Technology - Other Topics↗

Time resolved electrical, optical, and thermal probes of topological spin textures in magnetic nanostructures

This project is a fundamental study of magnetization dynamics in nanoscale spin textures with topological properties. One example is magnetic skyrmions that form in chiral ferromagnetic material systems that lack inversion symmetry. We have concentrated on the scalable chiral magnetic metal B20 FeGe thin film that has an asymmetric exchange interaction due to non-centrosymmetric cubic lattice. While the asymmetric exchange interaction can be found at the interface of a ferromagnet and a heavy metal, manipulation of skyrmions at these interfaces is non-trivial due to pinning and it requires higher critical current. In contrast, B20 materials such as FeGe have superior material quality to offer pinning-free and low-current skyrmion motion.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Solving the Bethe-Salpeter equation on a subspace: Approximations and consequences for low-dimensional materials

It is well known that the ambient environment can dramatically renormalize the quasiparticle gap and exciton binding energies in low-dimensional materials, but the effect of the environment on the energy splitting of the spin-singlet and spin-triplet exciton states is less understood. A prominent effect is the renormalization of the exciton binding energy and optical strength (and hence the optical spectrum) through additional screening of the direct Coulomb term describing the attractive electron-hole interaction in the kernel of the Bethe-Salpeter equation. The repulsive exchange interaction responsible for the singlet-triplet splitting, on the other hand, is unscreened within formal many-body perturbation theory. However, Benedict argued that in practical calculations restricted to a subspace of the full Hilbert space, the exchange interaction should be appropriately screened by states outside of the subspace, the so-called S approximation [L. X. Benedict, Phys. Rev. B 66, 193105 (2002)PRBMDO0163-182910.1103/PhysRevB.66.193105]. Here, we systematically explore the accuracy of the S approximation for different confined systems, including a molecule and heterostructures of semiconducting and metallic layered materials. We show that the S approximation is actually exact in the limit of small exciton binding energies (i.e., small direct term) and can be used to significantly accelerate convergence of the exciton energies with respect to the number of empty states, provided that a particular effective screening consistent with the conventional Tamm-Dancoff approximation is employed. We further find that the singlet-triplet splitting in the energy of the excitons is largely unaffected by the external dielectric environment for most quasi-two-dimensional materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

A graphene edge-mediated quantum gate

We propose a quantum gate architecture that allows for the systematic control of the effective exchange interactions between magnetic impurities embedded in nanoscale graphene flakes connected by a gated bridge. The entanglement between the magnetic moment and the edge states of the fragments is used to electrostatically tune the exchange interaction from ferro to antiferromagnetic by merely changing the bridge's carrier density. By characterizing the effects of size and coupling parameters, we explore different operation regimes of this device by means of exact calculations with the density matrix renormalization group. We analyze the results utilizing a simplified model that accounts for the main many-body mechanisms. Finally, we discuss how to use arrays of these devices to build quantum simulators for quantum many-body Hamiltonians.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Magnetic coupling between Fe(NO) spin probe ligands through diamagnetic Ni II , Pd II and Pt II tetrathiolate bridges

Reaction of the nitrosylated-iron metallodithiolate ligand, paramagnetic (NO)Fe(N 2 S 2 ), with [M(CH 3 CN) n ][BF 4 ] 2 salts (M = Ni II , Pd II , and Pt II ; n = 4 or 6) affords di-radical tri-metallic complexes in a stairstep type arrangement ([FeMFe] 2+ , M = Ni, Pd, and Pt), with the central group 10 metal held in a MS 4 square plane. These isostructural compounds have nearly identical ν(NO) stretching values, isomer shifts, and electrochemical properties, but vary in their magnetic properties. Despite the intramolecular Fe∙∙∙Fe distances of ca. 6 Å, antiferromagnetic coupling is observed between {Fe(NO)} 7 units as established by magnetic susceptibility, EPR, and DFT studies. The superexchange interaction through the thiolate sulfur and central metal atoms is on the order of Ni II < Pd II $\ll$ Pt II with exchange coupling constants (J) of -3, -23, and -124 cm -1 , consistent with increased covalency of the M–S bonds (3d < 4d < 5d). This trend is reproduced by DFT calculations with molecular orbital analysis providing insight into the origin of the enhancement in the exchange interaction. Specifically, the magnitude of the exchange interaction correlates surprisingly well with the energy difference between the HOMO and HOMO-1 orbitals of the triplet states, which is reflected in the central metal's contribution to these orbitals. These results demonstrate the ability of sulfur-dense metallodithiolate ligands to engender strong magnetic communication by virtue of their enhanced covalency and polarizability.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Effects of pressure on the electronic and magnetic properties of bulk NiI 2

Transition metal dihalides have recently garnered interest in the context of two-dimensional van der Waals magnets as their underlying geometrically frustrated triangular lattice leads to interesting competing exchange interactions. In particular, NiI 2 is a magnetic semiconductor that has been long known for its exotic helimagnetism in the bulk. Recent experiments have shown that the helimagnetic state survives down to the monolayer limit with a layer-dependent magnetic transition temperature that suggests a relevant role of the interlayer coupling. Here, we explore the effects of hydrostatic pressure as a means to enhance this interlayer exchange and ultimately tune the electronic and magnetic response of NiI 2 . We study first the evolution of the structural parameters as a function of external pressure using first-principles calculations combined with x-ray diffraction measurements. We then examine the evolution of the electronic structure and magnetic exchange interactions via first-principles calculations and Monte Carlo simulations. Further, we find that the leading interlayer coupling is an antiferromagnetic second-nearest-neighbor interaction that increases monotonically with pressure. The ratio between isotropic third- and first-nearest-neighbor intralayer exchanges, which controls the magnetic frustration and determines the magnetic propagation vector q of the helimagnetic ground state, is also enhanced by pressure. As a consequence, our Monte Carlo simulations show a monotonic increase in the magnetic transition temperature, indicating that pressure is an effective means to tune the magnetic response of NiI 2 .

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Site Occupancy Preference and Magnetic Properties in Nd 2 (Fe,Co) 14 B

Partial replacement of Fe by Co is an effective method to increase Curie temperature (T C ) , which improves the thermal stability of magnetic properties in Nd 2 Fe 14 B-based permanent magnets. The correlation between Fe substitution and magnetic properties has been studied in Nd 2 (Fe,Co) 14 B via a first-principles calculation. The calculated Fe substitution energies indicate that the Co atoms avoid the 8 j 2 site, which agrees with the experiments. The Co atoms are ferromagnetically coupled with Fe sublattice and show magnetic moments of about 1.2 to 1.7 μ B at different crystallographic sites, less than that of Fe (2.1–2.7 μ B ), resulting in the decrease in total magnetization at ground state (0 K) with increasing Co content. The effective exchange interaction parameter, derived from the energy difference between varied magnetic structures, increases from 7.8 meV to 17.0 meV with increasing Co content from x = 0 to x = 14 in Nd 2 Fe 14–x Co x B. This change in the effective exchange interaction parameter is responsible for the enhancement of T C in Nd 2 (Fe,Co) 14 B. The total magnetization at 300 K, derived from mean-field theory, shows a peak maximum value at x = 1 in Nd 2 Fe 14–x Co x B. The phenomenon results from the interplay between the reduction of the magnetic moment in the Fe(Co) sublattice and the enhancement of T C with increasing Co content.

36 MATERIALS SCIENCE↗

Transforming Thermal Expansion from Positive to Negative: The Case of Cubic Magnetic Compounds of (Zr,Nb)Fe 2

Negative thermal expansion (NTE) is an intriguing property for not only fundamental studies but also technological applications. However, few NTE materials are available compared with the huge amount of positive thermal expansion materials. The discovery of new NTE materials remains challenging. In this paper, we report a chemical modification strategy to transform thermal expansion from positive to negative in cubic magnetic compounds of (Zr,Nb)Fe 2 by tuning the magnetic exchange interaction. Furthermore, an isotropic zero thermal expansion can be established in Zr 0.8 Nb 0.2 Fe 2 (α 1 = 1.4 x 10 -6 K -1 , 3-470 K) over a broad temperature range that is even wider than that of the prototype Invar alloy of Fe 0.64 Ni 0.36 . The NTE of (Zr,Nb)Fe 2 is originated from the weakened magnetic exchange interaction and the increased d electrons of Fe by the Nb chemical substitution, so that the magnetovolume effect overwhelms the contribution of anharmonic lattice vibration.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Spin dynamics of the spin-chain antiferromagnet RbFeS 2

In this work, we report transport and inelastic neutron scattering studies on electronic properties and spin dynamics of the quasi-one-dimensional spin-chain antiferromagnet RbFeS 2 . An antiferromagnetic phase transition at TN≈195 K and dispersive spin waves with a spin gap of 5 meV are observed. By modeling the spin excitation spectra using linear spin wave theory, intra and interchain exchange interactions are found to be SJ 1 =100(5) meV and SJ 3 =0.9(3) meV, respectively, together with a small single-ion anisotropy of SD zz =0.04(1) meV. Comparison with previous results for other materials in the same class of Fe 3+ spin-chain systems reveals that although the magnetic order sizes show significant variation from 1.8 to 3.0μ B within the family of materials, the exchange interactions SJ are nevertheless quite similar, analogous to the iron pnictide superconductors where both localized and delocalized electrons contribute to the spin dynamics.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Quantum spin ice response to a magnetic field in the dipole-octupole pyrochlore Ce 2 Zr 2 O 7

The pyrochlore magnet Ce 2 Zr 2 O 7 has attracted much attention as a quantum spin ice candidate whose novelty derives in part from the dipolar-octupolar nature of the Ce 3+ pseudospin-1/2 degrees of freedom it possesses. We report heat capacity measurements on single crystal samples of Ce 2 Zr 2 O 7 down to T~0.1K in a magnetic field along the [$1,\bar{1},0$] direction. These measurements show that the broad hump in the zero-field heat capacity moves higher in temperature with increasing field strength and is split into two separate humps by the [$1,\bar{1},0$] magnetic field at ~2T. These separate features are due to the decomposition of the pyrochlore lattice into effectively decoupled chains for fields in this direction: One set of chains (α chains) is polarized by the field while the other (β chains) remains free. This situation is similar to that observed in the classical spin ices Ho 2 Ti 2 O 7 and Dy 2 Ti 2 O 7 , but with the twist that here the strong transverse exchange interactions produce substantial quantum effects. Our theoretical modeling suggests that the β chains are close to a critical state, with nearly-gapless excitations. We also report elastic and inelastic neutron scattering measurements on single crystal Ce 2 Zr 2 O 7 in [$1,\bar{1},0$] and [0,0,1] magnetic fields at temperatures down to T=0.03K. The elastic scattering behaves consistently with the formation of independent chains for a [$1,\bar{1},0$] field, while the [0,0,1] field produces a single field-induced elastic magnetic Bragg peak at (0,2,0) and equivalent wavevectors, indicating a polarized spin ice state for fields above ~3T. For both [$1,\bar{1},0$] and [0,0,1] magnetic fields, our inelastic neutron scattering results show an approximately dispersionless continuum of scattering that increases in both energy and intensity with increasing field strength. By modeling the complete set of experimental data using numerical linked cluster and semiclassical molecular dynamics calculations, we demonstrate the dominantly multipolar nature of the exchange interactions in Ce 2 Zr 2 O 7 and the smallness of the parameter θ, which controls the mixing between dipolar and octupolar degrees of freedom. Finally, these results support previous estimates of the microscopic exchange parameters and place strong constraints on the theoretical description of this prominent spin ice candidate.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗