Theoretical Study of Interplay of Superconductivity and Ferromagnetism in Hybrid Rutheno–Cuprate Superconductors RuSr2RCu2O8

We consider the extended two-band s–f model with additional terms, describing intersite Cooper pairs’ interaction between 4f (5f) and conduction electrons. Following Green’s function technique and equation of motion method, self-consistent equations for superconducting order parameter (Δ) and magnetic order parameter (m _f ) are derived. The expressions for specific heat, density of states, and free energy are also derived. The theory has been applied to explain the coexistence of superconductivity and ferromagnetism in hybrid rutheno-cuprate superconductors RuSr₂RECu₂O₈ (RE = Gd, Eu). The theory shows that it is possible to become superconducting if the system is already ferromagnetic. A study of specific heat, density of states and free energy is also presented. The agreement between theory and experimental observations is quite satisfactory.      

Synthesis and Superconducting Properties of Layered Ruthenocuprates

Superconducting properties of layered ruthenocuprates, RuSr ₂ RECu ₂ O ₈ and RuSr ₂ (RE _1-x Ce _x ) ₂ Cu ₂ O ₁₀ (RE = Sm, Eu, Gd), are investigated. The differences between superconductivity of these two phases are considered to be due to, their structural differences ; The RE/Sr substitution is more feasible in RuSr ₂ RECu ₂ O ₈ than in RuSr ₂ (RE _1-x Ce _x ) ₂ Cu ₂ O ₁₀ , while the latter phase shows larger oxygen nonstoichiometry. The superconducting RuSr ₂ Eu ₂ Cu ₂ O ₈ (T _{c, onset} = 25K, T _{c, zero} = 7K) phase was successfully synthesized for the first time, and coexistence of superconductivity and ferromagnetism in these systems was reconfirmed by using single-phase samples.     

Theoretical Study of Specific Heat, Density of States and Free Energy of Itinerant Ferromagnetic Superconductor URhGe

Following the equation of motion method and Green’s function technique, the coexistence of itinerant ferromagnetism (FM) and superconductivity (SC) is investigated in a single band homogeneous system. Self-consistent equations for superconducting order parameter (Δ) and magnetic order parameter (Δ_FM) are derived. It is shown that there generally exists a coexistent (Δ≠0 and Δ_FM≠0) solutions to the coupled equations of the order parameter in the temperature range 0<T<min (T _C,T _FM) where T _C and T _FM are respectively the superconducting and ferromagnetic transition temperatures. Expressions for the specific heat, density of states and free energy are derived. The specific heat has a linear temperature dependence at low temperatures as opposed to the exponential decrease in the BCS theory. The density of states for a finite Δ_FM increases as opposed to that of a standard ferromagnetic metal. The free energy shows that the superconducting ferromagnetic state has lower energy than the normal ferromagnetic state and therefore is realized at low enough temperature. The theory is applied to explain the observations of URhGe. The agreement between theory and experimental results is quite satisfactory.      

Coexistence of Superconductivity and Ferromagnetism in Electron and Nuclear Magnets

A phenomenological model uniting the London model of superconductivity and the micromagnetic approach to ferromagnetism is applied to new examples of coexistence of ferromagnetism and superconductivity revealed in recent experiments: (i) Type-I superconductivity in a nuclear ferromagnet AuIn ₂ ; (ii) Type-II high-Tc superconductivity in a compound R _1.5 Ce _0.5 RuSr ₂ Cu ₂ O ₁₀ which is weakly ferromagnetic.     

The Unusual Phase-Transitions in Ruthenate–Cuprate Superconducting Ferromagnets

We have systematically investigated the magnetic, electrical, and structural properties of the high-purity superconducting ferromagnet RuSr₂GdCu₂O₈, in which superconductivity and significant ferromagnetism have been reported to coexist. Our results show that there may exist in this compound a novel superconducting state which is characterized by the absence of a bulk Meissner effect, the presence of negligible superconducting condensation energy, and the appearance of an unusually large effective penetration depth. Such a state appears to have a nonuniform filamentary structure existing in the less magnetic walls between the ferromagnetic domains or a fine-grain structure beset by ferromagnetic walls between antiferromagnetic domains, similar to the case of NdNi₂B₂C. These results are presented and contrasted with the recently reported Meissner effect in RuSr₂GdCu₂O₈ in terms of the various phase transition sequences predicted.     

Cooperative phenomena in ternary superconductors

A microscopic theory of ferromagnetic superconductors is developed from first principles. Self-consistent equations for the superconducting order parameter Δ and spontaneous magnetizations 〈S ^z 〉 are derived using a Green's function technique and considering thef-d exchange effect up to the second order. The theory is applied to explain the experimental results in the reentrant superconducting ternary system ErRh₄B₄. The present model explains reentrant behavior, predicts the coexistence of superconductivity and ferromagnetism in a very small range of temperature, the suppression of superconductivity by ferromagnetism, and vice versa. These results are in excellent agreement with the experimental data and predictions of other models. The behavior of the spontaneous magnetization, the superconducting order parameter, the specific heat, and the density of states is also studied.     

Coexistence of Superconductivity and Itinerant Ferromagnetism in Ferromagnetic Metals UCoGe and UIr

A microscopic coexistence of itinerant ferromagnetism (FM) and superconductivity (SC) is studied in a single band homogenous system, following an equation of motion method and Green’s function technique. Self-consistent equations for superconducting order parameter (Δ) and magnetization parameter (M) are derived. It is shown that there generally exists a coexistent (Δ≠0, M≠0) solution to the coupled equations of the order parameters in the temperature range 0<T<min (T _C,T _FM), where T _C and T _FM are respectively the superconducting and ferromagnetic transition temperatures. The expressions for electronic specific heat (C/T), density of states, free energy, transition probabilities, ultrasonic attenuation, and nuclear relaxation are also derived. The theory is applied to explain the observations in UCoGe and UIr. The specific heat capacity at low temperature shows linear temperature dependence as opposed to the activated behavior. Density of states increases as opposed to the case of a standard ferromagnetic metal. Free energy study reveals that the superconducting ferromagnetic state has lower energy than the normal ferromagnetic state and, therefore, coexistence of FM and SC realized at a low enough temperature. The agreement between theory and experimental results for UCoGe and UIr is quite encouraging.     

Multiple Magnetic Ordering Temperatures in RuSr<Subscript>2</Subscript>Eu<Subscript>1.5</Subscript>Ce<Subscript>0.5</Subscript>Cu<Subscript>2</Subscript>O<Subscript>10−<Emphasis Type="Italic">δ</Emphasis></Subscript> System

We report synthesis and magnetic characterization of variously processed magneto-superconducting (Rutheno-cuprates) RuSr₂Eu_1.5Ce_0.5Cu₂O_10−δ . The compound crystallizes in I4/mmm tetragonal structure. Magnetization measurements showed bulk magnetic ordering and superconductivity at around 100 K and 30 K, respectively. Further, the careful examination of the low field magnetic susceptibility reveals two minor magnetic transitions at around 135 K and 200 K, in addition to the major transition at 100 K. When the samples are processed in different environments of air, O₂ and slightly pressurized O₂, the nature of magnetic transitions and the superconductivity changes dramatically. The highest superconducting transition is achieved for the high pressure O₂ annealed samples and the lowest for the air annealed one. On the other hand, the minor magnetic transitions are more prominent in air-annealed samples. Though the minor magnetic transitions are intrinsic to all variously processed samples, they are not clearly seen for higher O₂ content samples. Basically, the Ru spins order antiferro magnetically at around 200 K (first minor transition), and reorients themselves at 125 K (second minor transition) before finally ordering in canted ferromagnetic state or a spin glass structure. These results can be explained on the basis of fluctuating valance of Ru⁴⁺/Ru⁵⁺.      

Coexistence of Itinerant Ferromagnetism and Superconductivity in ZrZn<Subscript>2</Subscript>

A microscopic coexistence of itinerant ferromagnetism and superconductivity is studied in a single- band homogeneous system, using an equation of motion method and Green’s function technique. Self-consistent equations for the superconducting order parameter Δ and magnetization m are derived. It is shown that there generally exists coexistent (Δ≠0 and m≠0) solutions to the coupled equations of the order parameters in the temperature range 0<T<min (T _c ,T _FM ), where T _c and T _FM are respectively the superconducting and ferromagnetic transition temperatures. The expressions for specific heat, density of states, free energy, and critical field are also derived. The theory is applied to explain the observations in ZrZn₂. The agreement between the theory and experimental observations is quite satisfactory.      

Magnetic and Superconducting Properties of RuSr2Sm1.4Ce0.6Cu2O10−δ Samples

Samples with nominal composition RuSr₂Sm_1.4Ce_0.6Cu₂O_10−δ were synthesized at different conditions and their magnetic and superconducting properties were investigated. It was established that the magnetic order is of a long range type. It was shown that the increase of the oxygen content and the presence of larger microcrystals enhance the temperature of the onset of diamagnetism and the diamagnetic volume fraction of the samples, but do not significantly affect the temperatures of the intragrain superconducting and ferromagnetic transitions.     

Superconductivity and Magnetism in Pure and Substituted RuSr2GdCu2O8

We describe results of several macroscopic and local measurements of magnetic and superconducting properties for pure and Ce and Cu substituted RuSr₂GdCu₂O₈. From various experiments the physical properties phase diagram is derived as a function of the hole doping. We show that the large contribution of polarized Gd³⁺ magnetic moments to magnetization causes apparent absence of the Meissner state and augments the weak Ru ferromagnetism. This weak ferromagnetism in the superconducting state can be significantly enhanced by the external magnetic field. These results indicate that the low-temperature behavior can be qualitatively explained assuming a quasi-two-dimensional character of the superconducting regions.     

Superconductivity and Magnetism in Pure and Substituted RuSr2GdCu2O8

We describe results of several macroscopic and local measurements of magnetic and superconducting properties for pure and Ce and Cu substituted RuSr₂GdCu₂O₈. From various experiments the physical properties phase diagram is derived as a function of the hole doping. We show that the large contribution of polarized Gd³⁺ magnetic moments to magnetization causes apparent absence of the Meissner state and augments the weak Ru ferromagnetism. This weak ferromagnetism in the superconducting state can be significantly enhanced by the external magnetic field. These results indicate that the low-temperature behavior can be qualitatively explained assuming a quasi-two-dimensional character of the superconducting regions.     

Coexistence of Superconductivity and Ferromagnetism

An overview of the problem of the coexistence of superconductivity and ferromagnetism in uranium-based superconductors such as UGe₂, URhGe, UCoGe, UIr and ZrZn₂ will be briefly presented. Starting with a pairing Hamiltonian with equal spin superconducting triplet pairs and using quantum field theory Green function formalism, the coexistent ferromagnetic order with superconductivity will be shown to be a very distinct possibility. The singlet superconductivity coexisting with ferromagnetism is against the experimental observations. It will be argued that the superconductivity can be destroyed at two points as a result of magnetic fields both intrinsic and or extrinsic, and thus establishing reentrant superconductivity as observed most recently in URhGe. In the absence of consensus on the mechanism, the same is not highlighted although the magnetic quantum fluctuations emanating from quantum critical point could provide the pairing. The limitations of the model will also be discussed.     

Interplay of Superconductivity and Ferromagnetism in UGe<Subscript>2</Subscript>

The emergence of pressure induced superconductivity (SC) under the background of ferromagnetic state in 5f-electron based itinerant ferromagnetic superconductor UGe₂ is studied in the single band model by using a mean-field approximation. The solutions to the coupled equations of superconducting gap (Δ) and magnetization (m) are obtained using Green’s function technique and equation of motion method. It is shown that there generally exists a coexistent (Δ≠0, m≠0) solution to the coupled equations of the order parameters in the temperature range 0<T<min (T _C,T _FM), where T _C and T _FM are respectively the superconducting and ferromagnetic transition temperatures. The study of electronic specific heat (C/T), density of states, free energy, etc. are also presented. The specific heat capacity at low temperature shows linear temperature dependence as opposed to the activated behavior. Density of states increases as opposed to the case of a standard ferromagnetic metal. Free energy study reveals that the superconducting ferromagnetic state has lower energy than the normal ferromagnetic state and, therefore, realized at low enough temperature. The agreement between theory and experimental results for UGe₂ is quite satisfactory.      

Magneto-Transport Properties of the Rutheno-Cuprate RuSr2GdCu2O8

RuSr ₂ GdCu ₂ O ₈ (Ru-1212) is a triple perovskite containing both CuO ₂ and RuO ₂ layers. It has attracted a great deal of interest recently because it displays electronic ferromagnetism and superconductivity coexisting on a microscopic scale, with T _Curie 135K and T _c up to 50K. ^{1, 2} We report magnetisation and magnetoresistance (MR) data that exhibit effects due to the interaction between the ferromagnetic ruthenium moments and the conduction electrons. The MR is negative at temperatures above T _Curie , and is proportional to the square of the applied field well above this temperature. Below T _Curie the MR displays a positive peak at fields of around 15 kOe, but at high fields it becomes negative again, and approximately linear. We analyse the high temperature data in terms of spin-disorder scattering theory and extract a value for the exchange interaction between the carriers in the CuO ₂ planes and the Ru spins.     

Superconductivity and Magnetism in RuSr2GdCu2O8

The ruthenocuprate RuSr₂GdCu₂O₈ orders magnetically at 135 K and then becomes superconducting at 45 K. These transitions have been observed by several groups, but the intrinsically complex nature of the compound, as well as the unexpected coexistence of magnetism with high-temperature superconductivity, makes it uncertain what the magnetic, electronic, and structural character of its ground state is. We find that density functional-based evaluations give a magnetic and structural character that is consistent with the latest data: the RuO₆ octahedra rotate (by 7° around the c^ axis) causing a doubling of the cell, and an antiferromagnetic structure that has this same cell doubling is favored over a ferromagnetic alignment of Ru spins. The minority Ru d _xy states are partially occupied, leaving a metallic RuO₂ layer that dopes the CuO₂ bilayer and may lead to canting of the Ru spins.     

Critical Temperatures in Ferromagnetic-Superconducting All-Oxide Superlattices

Ferromagnetic/superconducting superlattices represent a new class of materials with the simultaneous occurrence of superconductivity and ferromagnetism. The mutual interaction of these antagonistic ordering phenomena is of vital fundamental interest and opens novel possibilities for spin-injection devices. Therefore, we systematically studied YBa₂Cu₃O₇-La_2/3Ca_1/3MnO₃ superlattices of different modulation lengths especially with respect to the reduced phase transition temperatures to ferromagnetism and superconductivity, respectively. Conventional models to explain the reduction of T _C and T _Curie fail and novel concepts giving rise to a long-range proximity effect have to be introduced. Furthermore, it is suggested that the pseudogap opening of the YBa₂Cu₃O₇ weakens the innerlayer ferromagnetic coupling of the La_2/3Ca_1/3MnO₃ layers, thus contributing to the reduction of T _Curie.     

Thermodynamic, transport and magnetic properties of the ferromagnetic superconductor RuSr2GdCu2O8 and related compounds

The transport, magnetic and thermodynamic properties of RuSr ₂ GdCu ₂ O ₈ and related compounds are reviewed. These indicate uniform coexistence of superconductivity (T _c =46K) and ferromagnetism (T _M =132K) in which the former arises in the CuO ₂ planes and the latter in the RuO ₂ planes. The earliest data indicated a typical underdoped cuprate with local Ru moment ferromagnetism, the absence of pairbreaking and transport properties dominated by the CuO ₂ planes. However, recent results suggest a more complex scenario of itinerant ferromagnetism and exchange coupling between the CuO ₂ and RuO ₂ layers.     

The Characteristics of the Superconducting and Magnetic Phases in the Polycrystalline Samples of Ruthenocuprates of Nominal Compositions RuSr2GdCu2O8, Ru0.98Sr2GdCu2O8 and Ru0.5Sr2GdCu2.5O8−δ

The temperature dependencies of the resistivity for the superconducting ruthenocuprates of nominal compositions RuSr₂GdCu₂O₈, Ru_0.98Sr₂GdCu₂O₈ and Ru_0.5Sr₂GdCu_2.5O_8?δ were examined for the magnetic field dependent characteristics of the superconducting transitions. The effect of the insignificant diminishing of the Ru/Cu ratio in parent RuSr₂GdCu₂O₈ was confirmed as relevant for the stabilisation of the superconducting phase. Noted differences in the compared characteristics are interpreted for possible inhomogeneous nucleation of the superconducting phase in the parent ruthenocuprate. The phase anisotropy in RuSr₂GdCu₂O₈ and Ru_0.98Sr₂GdCu₂O₈, in presence of the compounds Ru magnetism, appears to be a cause of a significant softening of the H _c2(T) phase line. An anomalous lowering of the magnetoresistivity was observed in the approx. 10 K range above the onset of the superconducting transition, which may suggest the presence of enhanced superconducting fluctuations in the samples. The positive magnetic field shift of the temperatures, which limit the magnetoresistivity and the specific heat signatures of the magnetic ordered state of the Ru sublattice, suggests probing the influence of the ferromagnetic Ru interactions in an effective metallic-like conduction channel present in the samples. Superconducting characteristics of the Ru_0.5Sr₂GdCu_2.5O_8?δ reveal a significant contribution of the Gd paramagnetic signal at low temperatures, interpreted for the presence of a significant anisotropy of the superconducting phase. It is concluded that the Ru–Cu substituted phases of ruthenocuprates may present an opportunity to investigate the effectively anisotropic superconducting phase despite its comparatively high T _c in the compounds related to the 123-type cuprate superconductor.     

Superconductivity in single crystals of the nonperovskite-like cuprate Ba2Cu3O6−x

Spontaneous crystallization from a solution in a melt at a pressure of (0.8–1)×10⁻³ Pa was used to obtain single crystals of a new nonperovskite-like superconducting phase Ba₂Cu₃O_6−x space group P _ccm, a=13.065, b=20.654, and c=11.431 Å). The superconducting properties of the crystals were investigated by modulated microwave absorption. The superconducting transition temperatures were 5 K (sample No. 1), 7 K and 13 K (sample No. 2). The superconductivity in the Ba₂Cu₃O_6−x crystals is attributed to the presence of CuO₂ chains of edge-sharing copper-oxygen squares. Nonperovskite-like Ba₂Cu₃O_6−x crystals are a new class of one-dimensional (1D) superconductors. Pis’ma Zh. Tekh. Fiz. 23, 27–34 (December 26, 1997)