We report upper critical field $B_{c2}(T)$ measurements on a single-crystalline sample of the ferromagnetic superconductor UCoGe. $B_{c2}(0)$ obtained for fields applied along the orthorhombic axes exceeds the Pauli limit for $B \parallel a,b$ and shows a strong anisotropy $B_{c2}^{a} \simeq B_{c2}^{b} \gg B_{c2} ^{c}$. This provide evidence for an equal spin pairing state and a superconducting gap function of axial symmetry with point nodes along the c axis, which is also the direction of the uniaxial ferromagnetic moment $m_0 = 0.07 μ_{B}$. An unusual curvature or kink is observed in the temperature variation of $B_{c2}$, which possibly indicates UCoGe is a two-band ferromagnetic superconductor.
The temperature-pressure phase diagram of ferromagnetic superconductor UCoGe includes four phase transitions. They are between the paramagnetic and the ferromagnetic states with the subsequent transition in the superconducting ferromagnetic state and between the normal and the superconducting states after which has to occur the transition to the superconducting ferromagnetic state. Here we have developed the Landau theory description of the phase diagram and established the specific ordering arising at each type of transition. The phase transitions to the ferromagnetic superconducting state ar…
We study the ferromagnetic superconductor UCoGe at ambient pressure under $ab$-plane magnetic fields $\vec{H}$ which are perpendicular to the ferromagnetic easy axis. It is shown that, by taking into account the Dyaloshinskii-Moriya interaction arising from the zigzag chain crystal structure of UCoGe, we can qualitatively explain the experimentally observed in-plane anisotropy for critical magnetic fields of the paramagnetic transition. Because of this strong dependence on the magnetic field direction, upper critical fields of superconductivity, which is mediated by ferromagnetic spin fluctuat…
The superconductor UCoGe is analyzed with electronic structure calculations using Linearized Augmented Plane Wave method based on Density Functional Theory. Ferromagnetic and antiferromagnetic calculations with and without correlations (via LDA+U) were done. In this compound the Fermi level is situated in a region where the main contribution to DOS comes from the U-5f orbital. The magnetic moment is mainly due to the Co-3d orbital with a small contribution from the U-5f orbital. The possibility of fully non-collinear magnetism in this compound seems to be ruled out. These results are compared …
We report thermal expansion measurements on a single crystal of the superconducting ferromagnet UCoGe for magnetic fields applied along the main orthorhombic axes. The thermal expansion cell was mounted on a piezo-electric rotator in order to fine-tune the magnetic field angle. The superconducting and magnetic phase diagram has been determined. With our bulk technique we confirm the $S$-shape of the upper-critical field, $B_{c2}$, for $B \parallel b$ and reinforcement of superconductivity above 6 T. At the same time the Curie point shifts towards lower temperatures on increasing the field alon…
Magnetic-field-angle-resolved specific heat and magnetization measurements were conducted on a ferromagnetic superconductor UCoGe with remarkable anisotropic upper critical field $H_{\rm c2}$. Although $H_{\rm c2}$ reaches a high magnetic field ($\sim 20$~T) along the $b$ axis, it is small ($\sim~0.6$~T) when a magnetic field is applied along the magnetic easy $c$-axis. This study indicates that the specific heat is abruptly suppressed when the magnetic field is applied toward the $c$ axis from the $a$ and $b$ axes in the ferromagnetic state. The field response of density of states (DOS) is an…
We report neutron scattering measurements of critical magnetic excitations in the weakly ferromagnetic superconductor UCoGe. The strong non-Landau damping of the excitations we observe, although unusual has been found in another related ferromagnet, UGe2 at zero pressure. However, we also find there is a significant anisotropy of the magnetic correlation length in UCoGe that contrasts with an almost isotropic length for UGe2. The values of the magnetic correlation length and damping are found to be compatible with superconductivity on small Fermi surface pockets. The anisotropy may be importan…
The upper critical field for field along the b-axis of the p-wave ferromagnetic superconductor UCoGe has a particular S-shape, akin to the re-entrant superconducting phase of URhGe. To explore the evolution of the superconducting phase under this transverse magnetic field, we report the thermal conductivity and resistivity measurements, revealing a possible field-induced vortex liquid phase, and supporting a field-induced change of the superconducting order parameter.
We have carried out direction-dependent ^{59}Co NMR experiments on a single crystal sample of the ferromagnetic superconductor UCoGe in order to study the magnetic properties in the normal state. The Knight shift and nuclear spin-lattice relaxation rate measurements provide microscopic evidence that both static and dynamic susceptibilities are ferromagnetic with strong Ising anisotropy. We discuss that superconductivity induced by these magnetic fluctuations prefers spin-triplet pairing state.
We study the strong-coupling superconductivity near ferromagnetic quantum critical points, mainly focusing on the upper critical fields $H_{c2}$. Based on our simple model calculations, we discuss experimentally observed unusual behaviors of $H_{c2}$ in a recently discovered ferromagnetic superconductor UCoGe. Especially, the large anisotropy between $H_{c2}\parallel a$-axis and $H_{c2}\parallel c$-axis, and the strong-coupling behaviors in $H_{c2}^{\parallel a}$ are investigated. We also examine effects of non-analytic corrections in the spin susceptibility on the superconductivity, which can…
From detailed angle-resolved NMR and Meissner measurements on a ferromagnetic (FM) superconductor UCoGe (T_Curie ~ 2.5 K and T_SC ~ 0.6 K), we show that superconductivity in UCoGe is tightly coupled with longitudinal FM spin fluctuations along the c axis. We found that magnetic fields along the c axis (H || c) strongly suppress the FM fluctuations and that the superconductivity is observed in the limited magnetic field region where the longitudinal FM spin fluctuations are active. These results combined with model calculations strongly suggest that the longitudinal FM spin fluctuations tuned b…
We report a high-pressure single crystal study of the superconducting ferromagnet UCoGe. Ac-susceptibility and resistivity measurements under pressures up to 2.2 GPa show ferromagnetism is smoothly depressed and vanishes at a critical pressure $p_c = 1.4$ GPa. Near the ferromagnetic critical point superconductivity is enhanced. Upper-critical field measurements under pressure show $B_{c2}(0)$ attains remarkably large values, which provides solid evidence for spin-triplet superconductivity over the whole pressure range. The obtained $p-T$ phase diagram reveals superconductivity is closely conne…
We investigate the effect of substituting Si for Ge in the ferromagnetic superconductor UCoGe. Dc-magnetization, ac-susceptibility and electrical resistivity measurements on polycrystalline UCoGe$_{1-x}$Si$_x$ samples show that ferromagnetic order and superconductivity are progressively depressed with increasing Si content and simultaneously vanish at a critical concentration $x_{cr} \simeq 0.12$. The non-Fermi liquid temperature variation in the electrical resistivity near $x_{cr}$ and the smooth depression of the ordered moment point to a continuous ferromagnetic quantum phase transition. Su…
We succeeded in growing high quality single crystals of the ferromagnetic superconductor UCoGe and measured the magnetoresistance at fields up to 34T. The Shubnikov-de Haas signal was observed for the first time in a U-111 system (UTGe, UTSi, T: transition metal). A small pocket Fermi surface (F~1kT) with large cyclotron effective mass 25m0 was detected at high fields above 22T, implying that UCoGe is a low carrier system accompanyed with heavy quasi-particles. The observed frequency decreases with increasing fields, indicating that the volume of detected Fermi surface changes nonlinearly with…
We review our recent studies on ferromagnetic superconductors, UGe2, URhGe and UCoGe, where the spin-triplet state with the so-called equal spin pairing is realized. We focus on experimental results of URhGe and UCoGe in which the superconductivity occurs already at ambient pressure. The huge upper critical field Hc2 on UCoGe for the field along the hard magnetization axis (b-axis) is confirmed by the AC susceptibility measurements by the fine tuning of field angle. Contrary to the huge Hc2 along the hard-magnetization axis, Hc2 along the easy-magnetization axis (c-axis) is relatively small in…
We performed thermal conductivity measurements on a single crystal of the ferromagnetic superconductorUCoGe under magnetic field. Two different temperature dependencies of the thermal conductivity are observed, for H//b linear at low magnetic field and quadratic for magnetic field larger than 1 Tesla. At the same field value, a plateau appears in the field dependency of the residual term of thermal conductivity. Such observations suggest a multigap superconductivity with a line of nodes in the superconducting gap.
UCoGe exhibits superconductivity in the presence of ferromagnetism. When a field is applied along the b axis (perpendicular to the easy axis), ferromagnetism is weakened and superconductivity is enhanced. This enhancement has been attributed to an increase in coupling as observed in the enhanced effective mass produced by the critical fluctuations as the ferromagnetic transition is strongly suppressed. However it is also important to know if and how the Fermi surface changes near the critical point. Here we report measurements of the thermoelectricity of UCoGe which reveal a low carrier densit…
We have performed 59Co NMR experiments on the ferromagnetic superconductor UCoGe under magnetic fields (H) along the a- and b- axes to investigate the relationship between ferromagnetic properties and superconductivity. The ferromagnetic ordering temperature TCurie is suppressed and the nuclear spin-lattice relaxation rate 1/T1 at 2 K is enhanced in H || b, although TCurie and 1/T1 are unchanged in H || a, indicating that the ferromagnetic criticality is induced only when H is applied along the b axis. We show the close relationship between the magnetic anisotropies and the superconducting one…
UCoGe is reported as a weak ferromagnetic (FM) two-band superconductor (SC) with the critical temperature, $T_{\mathrm{SC}}\sim$ 0.7 K and the Curie temperature, $T_{\mathrm{C}}\sim$ 3 K at ambient pressure. We report exotic attributes of the SC and FM state in moderate magnetic fields. The observed phenomena clearly demonstrate that the SC regime is superior to the paramagnetic state in the vicinity of $T_{\mathrm{SC}}$. Above $T_{\mathrm{SC}}$, the zero-field state is characterized by a regime with strong FM spin fluctuations, suggesting proximity of the FM quantum critical point (FM-QCP). I…
In order to determine the superconducting paring state in the ferromagnetic superconductor UCoGe, ^{59}Co NMR Knight shift, which is directly related to the microscopic spin susceptibility, was measured in the superconducting state under magnetic fields perpendicular to spontaneous magnetization axis: ^{59}K^{a, b}. ^{59}K^{a, b} shows to be constant, but does not decrease below a superconducting transition. These behaviors as well as the invariance of the internal field at the Co site in the superconducting state exclude the spin-singlet pairing, and can be interpreted with the equal-spin pai…
The phase separation of the ferromagnetic (FM) and paramagnetic (PM) phases in the superconducting (SC) state of UCoGe at the FM critical region was investigated using $^{59}$Co nuclear quadrupole resonance (NQR) technique by taking advantage of its site-selective feature. The NQR measurements revealed that the first-order quantum phase transition occurs between the FM and the PM states. The nuclear spin-lattice relaxation rate $1/T_1$ exhibited a clear drop at the SC state in the PM phase, whereas it was not detected in the FM phase, which indicates that the superconductivity in the FM phase …
We report the coexistence of ferromagnetic order and superconductivity in UCoGe at ambient pressure. Magnetization measurements show that UCoGe is a weak ferromagnet with a Curie temperature T_{C}= 3 K and a small ordered moment $m_{0}$= 0.03 $μ_B$. Superconductivity is observed with a resistive transition temperature T_{s} = 0.8 K for the best sample. Thermal-expansion and specific-heat measurements provide solid evidence for bulk magnetism and superconductivity. The proximity to a ferromagnetic instability, the defect sensitivity of T_{s}, and the absence of Pauli limiting, suggest triplet s…
UCoGe is one of the few compounds showing the coexistence of ferromagnetism and superconductivity at ambient pressure. With T_Curie = 3 K and T_SC = 0.6 K it is near a quantum phase transition; the pressure needed to suppress the magnetism is slightly higher than 1 GPa. We report simultaneous resistivity and ac-susceptibility measurements under pressure on a polycrystal with very large single-crystalline domains and a resistivity ratio of about 6. Both methods confirm the phase diagram established before by resistivity measurements on a polycrystal. The ferromagnetic phase is suppressed for P …
Magnetoresistivity measurements with fine tuning of the field direction on high quality single crystals of the ferromagnetic superconductor UCoGe show anomalous anisotropy of the upper critical field H_c2. H_c2 for H // b-axis (H_c2^b) in the orthorhombic crystal structure is strongly enhanced with decreasing temperature with an S-shape and reaches nearly 20 T at 0 K. The temperature dependence of H_c2^a shows upward curvature with a low temperature value exceeding 30 T, while H_c2^c at 0 K is very small (~ 0.6 T). Contrary to conventional ferromagnets, the decrease of the Curie temperature wi…
A $^{59}$Co nuclear quadrupole resonance (NQR) was performed on a single-crystalline ferromagnetic (FM) superconductor UCoGe under pressure. The FM phase vanished at a critical pressure $P_c$, and the NQR spectrum just below $P_c$ showed phase separation of the FM and paramagnetic (PM) phases below Curie temperature $T_{\textrm{Curie}}$, suggesting first-order FM quantum phase transition (QPT). We found that the internal field was absent above $P_c$, but the superconductivity is almost unchanged. This result suggests the existence of the nonunitary to unitary transition of the superconductivit…
Co nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) studies were performed in the recently discovered UCoGe, in which the ferromagnetic and superconducting (SC) transitions were reported to occur at $T_{\rm Curie} \sim 3$ K and $T_S \sim 0.8$ K (N. T. Huy {\it et al.}, Phys. Rev. Lett. {\bf 99} (2007) 067006), in order to investigate the coexistence of ferromagnetism and superconductivity as well as the normal-state and SC properties from a microscopic point of view. From the nuclear spin-lattice relaxation rate $1/T_1$ and Knight-shift measurements, we confirmed that fe…
$^{73}$Ge and $^{59}$Co nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) measurements have been performed on a $^{73}$Ge-enriched single-crystalline sample of the ferromagnetic superconductor UCoGe in the paramagnetic state. The $^{73}$Ge NQR parameters deduced from NQR and NMR are close to those of another isostructural ferromagnetic superconductor URhGe. The Knight shifts of the Ge and Co sites are well scaled to each other when the magnetic field is parallel to the $b$ or $c$ axis. The hyperfine coupling constants of Ge are estimated to be close to those of Co. The la…
Recently, coexistence of ferromagnetism (T_Curie = 2.8K) and superconductivity (T_sc = 0.8K) has been reported in UCoGe, a compound close to a ferromagnetic instability at ambient pressure P. Here we present resistivity measurements under pressure on a UCoGe polycrystal. The phase diagram obtained from resistivity measurements on a polycrystalline sample is found to be qualitatively different to those of all other ferromagnetic superconductors. By applying high pressure, ferromagnetism is suppressed at a rate of 1.4 K/GPa. No indication of ferromagnetic order has been observed above P ~ 1GPa. …
Single-crystals of the new ferromagnetic superconductor UCoGe have been grown. The quality of as-grown samples can be significantly improved by a heat-treatment procedure, which increases the residual resistance ratio (RRR) from ~5 to ~30. Magnetization and resistivity measurements show the annealed samples have a sharp ferromagnetic transition with a Curie temperature T_C is 2.8 K. The ordered moment of 0.06 mu_B is directed along the orthorhombic c-axis. Superconductivity is found below a resistive transition temperature T_s = 0.65 K.
We report a magnetoresistance study of the superconducting ferromagnet UCoGe. The data, taken on single-crystalline samples, show a pronounced structure at $B^* = 8.5$~T for a field applied along the ordered moment $m_0$. Angle dependent measurements reveal this field-induced phenomenon has an uniaxial anisotropy. Magnetoresistance measurements under pressure show a rapid increase of $B^*$ to 12.8~T at 1.0~GPa. We discuss $B^*$ in terms of a field induced polarization change. Upper critical field measurements corroborate the unusual S-shaped $B_{c2}(T)$-curve for a field along the $b$-axis of …
Identification of pairing mechanisms leading to the unconventional superconductivity realized in copper-oxide, heavy-fermions, and organic compounds is one of the most challenging issues in condensed-matter physics. Clear evidence for an electron-phonon mechanism in conventional superconductors is seen by the isotope effect on the superconducting transition temperatures $T_{\rm SC}$, since isotopic substitution varies the phonon frequency without affecting the electronic states. In unconventional superconductors, magnetic fluctuations have been proposed to mediate superconductivity, and consid…
We report measurements of the coefficient of linear thermal expansion, $α(T)$, of the superconducting ferromagnet UCoGe. The data taken on a single-crystalline sample along the orthorhombic crystal axes reveal a pronounced anisotropy, with the largest length changes along the $b$ axis. The large values of the step sizes $Δα$ at the magnetic and superconducting phase transitions provide solid evidence for bulk magnetism and superconductivity. Specific-heat measurements corroborate bulk superconductivity. Thermal-expansion measurements in magnetic fields $B \parallel a,b$ show $Δα$ at $T_C$ grow…
A variety of heavy fermion superconductors, such as UCoGe, UGe$_2$, and URhGe exhibit a striking coexistence of bulk ferromagnetism and superconductivity. In these systems, the magnetic moment decreases with pressure, and vanishes at a ferromagnetic quantum critical point (qcp). Remarkably, the superconductivity in UCoGe varies smoothly with pressure across the qcp and exists in both the ferromagnetic and paramagnetic regimes. We argue that in UCoGe, spin-orbit interactions stabilize a time-reversal invariant odd-parity superconductor in the high pressure paramagnetic regime. Based on a simple…
The pressure-temperature phase diagram of the orthorhombic ferromagnetic superconductor UCoGe was determined by resistivity measurements up to $10.5~$GPa. The Curie temperature $T_C$ is suppressed with pressure and vanishes at the critical pressure $p_c\approx1~$GPa. Superconductivity is observed in both the ferromagnetic state at low pressure, and in the paramagnetic state above $p_c$ up to about 4$~$GPa. Non-Fermi liquid behavior appears in a large pressure range. The resistivity varies linearly with temperature around $p_c$ and evolves continuously with pressure to a $T^2$ Fermi-liquid beha…
The discovery in 2000 that the ferromagnetic (FM) compound UGe2 (T_Curie = 52 K at ambient pressure) becomes superconducting under a pressure of P = 1.1 GPa until it enters the paramagnetic (PM) phase above Pc = 1.6 GPa was a surprise. Successive searches for new materials (URhGe and UCoGe) led to the discovery of the coexistence of superconductivity (SC) and ferromagnetism at ambient pressure. Furthermore in UCoGe, it was found that SC survives in the PM regime from P_c = 1.1 to 4 GPa. Focus has been on low-temperature experiments under extreme conditions of magnetic field (H), pressure, and …
We present the group-thoretical classification of gap functions in superconductors coexisting with some magnetic order in non-symmorphic magnetic space groups. Based on the weak-coupling BCS theory, we show that UCoGe-type ferromagnetic superconductors must have horizontal line nodes on either $k_z=0$ or $\pmπ/c$ plane. Moreover, it is likely that additional Weyl point nodes exist at the axial point. On the other hand, in UPd$_2$Al$_3$-type antiferromagnetic superconductors, gap functions with $A_g$ symmetry possess horizontal line nodes in antiferromagnetic Brillouin zone boundary perpendicul…
We study the interplay of the ferromagnetic (FM) state and the p-wave superconducting (SC) state observed in several materials such as UCoGe and URhGe in a totally nonperturbative manner. To this end, we introduce a lattice Ginzburg-Landau model that is a genuine generalization of the phenomenological Ginzburg-Landau theory proposed previously in the continuum and also a counterpart of the lattice gauge-Higgs model for the s-wave SC transition, and study it numerically by Monte-Carlo simulations. The obtained phase diagram has qualitatively the same structure as that of UCoGe in the region whe…
We report on theoretical studies on ferromagnetic superconductors, URhGe, and UCoGe and identifies the pairing state as a non-unitary spin-triplet one, analogous to superfluid $^3$He-A phase. A recently found superconductor UTe$_2$ with almost ferromagnet is analyzed by the same manner. Through investigating their peculiar upper critical field $H_{\rm c2}$ shapes, it is shown that the pairing symmetry realized in all three compounds can be tuned by their magnetization curves under applied fields. This leads to the reentrant $H_{\rm c2}$ in URhGe, an S-shaped in UCoGe and an L-shaped $H_{\rm c2…
We extended the Scharnberg-Klemm theory of $H_{c2}(T)$ in p-wave superconductors with broken symmetry to cases of partially broken symmetry in an orthorhombic crystal, as is appropriate for the more exotic ferromagnetic superconductor UCoGe in strong magnetic fields. For some partially broken symmetry cases, $H_{c2}(T)$ can mimic upward curvature in all three crystal axis directions, and reasonably good fits to some of the UCoGe data are obtained.
The correlated metal UCoGe is a weak itinerant ferromagnet with a Curie temperature T_C = 3 K and a superconductor with a transition temperature T_s = 0.6 K. We review its basic thermal, magnetic - on the macro and microscopic scale - and transport properties, as well as the response to high pressure. The data unambiguously show that superconductivity and ferromagnetism coexist below T_s = 0.6 K and are carried by the same 5f electrons. We present evidence that UCoGe is a p-wave superconductor and argue that superconductivity is mediated by critical ferromagnetic spin fluctuations.
Recent advances on ferromagnetic superconductors, UGe2, URhGe and UCoGe are presented. The superconductivity (SC) peacefully coexists with the ferromagnetism (FM), forming the spin-triplet state of Cooper pairs. The striking new phenomena, such as SC reinforced by the magnetic field, are associated with Ising-type ferromagnetic fluctuations. A variety of ferromagnetic ordered moments between UGe2, URhGe and UCoGe affords to understand the relation between FM, tricriticality and SC.
We report zero-field muon spin rotation and relaxation measurements on the superconducting ferromagnet UCoGe. Weak itinerant ferromagnetic order is detected by a spontaneous muon spin precession frequency below the Curie temperature $T_C = 3$ K. The $μ^+$ precession frequency persists below the bulk superconducting transition temperature $T_{sc} = 0.5$ K, where it measures a local magnetic field $B_{loc} = 0.015$ T. The amplitude of the $μ$SR signal provides unambiguous proof for ferromagnetism present in the whole sample volume. We conclude ferromagnetism coexists with superconductivity on th…
We study phenomenologically by using the previously derived Landau free energy, the role of anisotropy in ferromagnetic superconductors UGe2, URhGe, and UCoGe. The three compounds are separately discussed with the special stress on UGe2. The main effect comes from the strong uniaxial anisotropy of magnetization while the anisotropy of Cooper pairs and crystal anisotropy only slightly change the phase diagram in the vicinity of Curie temperature. The limitations of this approach are also discussed.
The field dependence of the electronic specific heat gamma(H) of URhGe is determined using temperature-dependent magnetization measurements and Maxwell's relation for all three orthorhombic directions. A large (40 %) enhancement of gamma(H) is found at the reorientational transition for fields applied along the b hard axis, which we argue occurs when the field-induced moment is of the same size as the zero-field ordered moment M0. Fields applied along the easy direction, on the other hand, lead to a rapid suppression of gamma(H). We emphasize that this behavior is also applicable to the closel…
The field-angular dependence of Co-NMR spin-lattice relaxation rate 1/T1 has been measured for a 10% Co-doped single crystal of URhGe. The experiment revealed that spin fluctuations in ferromagnetic (FM) state of URhGe are robust against magnetic field below about 4 T, applied along any direction in the bc crystal plane. This is in clear contrast with the sister compound UCoGe, in which FM spin fluctuations are rapidly suppressed by a tiny applied field along the c axis. We show that such a difference in the character of the spin fluctuations is reflected in their two distinct phase diagrams f…
The appearance of magnetization during the application of mechanical stress and the creation of elastic deformation during the application of a magnetic field are two fundamental properties of piezomagnetic materials. The symmetry of superconducting ferromagnets UGe_2, URhGe, UCoGe allows piezomagnetism. In addition to conventional piezomagnetic properties occurring in both normal and superconducting states, superconducting state of these piezomagnets has its own specificity. Unlike conventional superconductors, in uranium ferromagnets, the critical transition temperature to the superconductin…
The mechanism of unconventional superconductivity, such as high-temperature-cuprate, Fe-based, and heavy-fermion superconductors, has been studied as a central issue in condensed-matter physics. Spin fluctuations, instead of phonons, are considered to be responsible for the formation of Cooper pairs, and many efforts have been made to confirm this mechanism experimentally. Although a qualitative consensus seems to have been obtained, experimental confirmation has not yet been achieved. This is owing to a lack of the quantitative comparison between theory and experiments. Here, we show a semi-q…
The spin-triplet state is most likely realized in uranium ferromagnetic superconductors, UGe2, URhGe, UCoGe. The microscopic coexistence of ferromagnetism and superconductivity means that the Cooper pair should be realized under the strong internal field due the ferromagnetism, leading to the spin-triplet state with equal spin pairing. The field-reinforced superconductivity, which is observed in all three materials when the ferromagnetic fluctuations are enhanced, is one of the strong evidences for the spin-triplet superconductivity. We present here the results of a newly discovered spin-tripl…
The symmetry approach to the description of the (P,T) phase diagram of ferromagnet superconductors with triplet pairing is developed. Taking into account the recent experimental observations made on UCoGe it is considered the case of a crystal with orthorhombic structure and strong spin-orbital coupling. It is shown that formation of ferromagnet superconducting state from a superconducting state is inevitably accompanied by the first order type transition.
We review our recent results on ferromagnetic superconductors, URhGe and UCoGe. High quality single crystals of both compounds were successfully grown. The specific heat shows a clear jump related to the superconducting transition in UCoGe. The finite values of C/T at 0K are discussed in terms of the self-induced vortex state and the value of the ordered moment. With increasing fields for H // b-axis in URhGe, the jump of thermal expansion increases and shifts to lower temperature. The re-entrant and S-shaped superconducting phases for URhGe and UCoGe respectively are explained by the unusual …
We have measured magnetization at high pressure in the uranium ferromagnetic superconductor UGe$_2$ and analyzed the magnetic data using Takahashi's spin fluctuation theory. There is a peak in the pressure dependence of the width of the spin fluctuation spectrum in the energy space $T_0$ at $P_x$, the phase boundary of FM1 and FM2 where the superconducting transition temperature $T_{sc}$ is highest. This suggests a clear correlation between the superconductivity and pressure-enhanced magnetic fluctuations developed at $P_x$. The pressure effect on $T_{Curie}/T_0$, where $T_{Curie}$ is the Curi…
We have developed the quasi-classical formalism for a self-induced vortex state in ferromagnetic superconductors. By combining the spatially averaged approximation of the superconductivity with the phenomenological form of the ferromagnetism, we demonstrate the thermodynamic properties of the p-wave triplet equal-spin-pairing state with the spontaneous vortex lattice. The occurrence condition for the self-induced vortex state is discussed within this formalism. The comparison of the calculated quantities with the observed ones indicates that the self-induced vortex state indeed occurs in UCoGe…
We extend the Scharnberg-klemm theory of $H_{c2}$ in p-wave superconductors with broken symmetry to cases of partially broken symmetry in an orthorhombic crystal, as is appropriate for the more exotic ferromagnetic superconductor UCoGe in strong applied magnetic fields. For some partially broken symmetry cases, $H_{c2}$ can mimic upward curvature in all three crystal axis directions, and reasonably good fits to some of the the low-field UCoGe data are obtained.
We review our recent studies on ferromagnetic superconductors, UGe2, URhGe and UCoGe, together with the ferromagnetic quantum criticality and paramagnetic singularity on the Ising 5f-itinerant system UCoAl. Thanks to the variety of ordered moment in ferromagnetic superconductors from 1.5 muB to 0.05 muB, interesting systematic changes or similarities are clarified. All ferromagnetic superconductors show large upper critical field Hc2, and the field-reentrant (-reinforced) phenomena are observed in the field-temperature phase diagram, when the pressure or field direction is tuned for particular…
A general quasi-phenomenological theory that describes phases and phase transitions of ferromagnetic superconductors with spin-triplet electron Cooper pairing is presented. The theory is based on extended Ginzburg-Landau expansion in powers of superconducting and ferromagnetic order parameters. A simple form for the dependence of theory parameters on the pressure ensures a correct theoretical outline of the temperature-pressure phase diagram where a stable phase of coexistence of p-wave superconductivity and itinerant ferromagnetism appears. This new theory is in an excellent agreement with th…
On the basis of microscopic theory it is demonstrated how the coupling between the electrons by means of magnetization fluctuations in ferromagnetic metal with orthorhombic symmetry gives rise equal spin pairing superconducting state with general form of the order parameter dictated by symmetry. The strong upturn of the upper critical field along $b$-direction above 5 Tesla in UCoGe is explained by the increase of pairing interaction caused by the suppression of the Curie temperature by magnetic field parallel to b-axis. It is proposed that similar phenomenon at much higher field must take pla…
Motivated by the recent theoretical and experimental progress in the heavy fermion system UCoGe, we study ferromagnetic chiral superconductors in the presence of magnetic domains. Within mean field approximations, it is shown that chiral superconducting domains are naturally induced by the ferromagnetic domains. The domain wall current flows in the opposite direction to the naively expected one as in $^3$He-A phase due to contributions from "unpaired electrons". Consequently, the domain wall current flows in the same direction with that of surface currents when the magnetic domain wall lies pa…
We report a dc magnetization study of the critical phenomenon around the ferromagnetic transition temperature T_C in high-quality single crystals of uranium ferromagnetic superconductors UGe2 and URhGe. The critical exponents, beta for the temperature dependence of the magnetization below T_C, gamma for the magnetic susceptibility, and delta for the magnetic isothermal at T_C have been determined with a modified Arrott plot, a Kouvel-Fisher plot, and the scaling analysis. Magnetization in the ferromagnetic state has strong uniaxial magnetic anisotropy in the two compounds. However, the univers…
Ferromagnetic superconductor URhGe has orthorhombic structure and possesses spontaneous magnetisation along the c-axis. Magnetic field directed along the $b$-axis suppresses ferromagnetism in $c$-direction and leads to a metamagnetic transition into polarised paramagnetic state in the $b$-direction. The theory of these phenomena based on the specific magnetic anisotropy of this material in $(b,c)$ plane is given. Line of the first order metamagnetic transition ends at a critical point. The Van der Waals - type description of behaviour of physical properties near this point is developed. The tr…
We model the normal state of ferromagnetic superconductors with two general ellipsoidal Fermi surfaces (FSs), one for each spin projection $σ=\{\uparrow,\downarrow\}$, each with its ferromagnetically split chemical potential $μ_σ$ and its three distinct single particle effective masses, $\{m_{iσ}\}$, the geometric mean of which is $m_σ$. We study this model in the presence of an arbitrarily oriented magnetic induction, ${\bf B}=μ_{0}{\bf H}+{\bf M_{0}}$, where ${\bf M_{0}}$ includes the Ising-like spontaneous ferromagnetic order, which for URhGe is in the $c$-axis direction above the supercond…
