We have performed electrical resistivity measurements on single crystal BaFe2As2 under high pressure P up to 16 GPa with a cubic anvil apparatus, and up to 3 GPa with a modified Bridgman anvil cell. The samples were obtained from the same batch, which was grown with a self-flux method. A cubic anvil apparatus provides highly hydrostatic pressure, and a modified Bridgman anvil cell, which contains liquid pressure transmitting medium, provides quasi hydrostatic pressure. For highly hydrostatic pressure, the crystal phase and magnetic transition temperature decreases robustly with P and disappears at around 10 GPa. The superconducting phase appears adjacent to magnetic phase in narrow pressure region between 11 and 14 GPa. The tiny difference of hydrostaticity between the cubic anvil apparatus and modified Bridgman anvil cell induces a drastic effect on the phase diagram of BaFe2As2. This result indicates that small uniaxial stress along c-axis strongly suppresses the structural/antiferromagnetic ordering and stabilizes superconductivity at much lower pressure.
The effect of uniaxial pressure on superconductivity was examined for a high-Tc cuprate La1.64Eu0.2Sr0.16CuO4, which is located at the boundary between the superconducting and stripe phases. We found remarkably large anisotropy of the uniaxial pressure effect not only between the in-plane and out-of-plane pressures but also within the CuO2-plane. When the pressure is applied along the tetragonal [110] direction, we found the largest pressure effect ever observed in cuprates, dTc/dP - 2.5 K/kbar, while the change of Tc was not appreciable when applied along [100]. This substantial in-plane anis…
Uniaxial stress, as well as hydrostatic pressure are often used to tune material properties in condensed matter physics. Here, we present a setup which allows for the study of the combined effects of quasi-uniaxial stress and hydrostatic pressure. Following earlier designs for measurements under finite stress at ambient pressure (e.g., Chu et al., Science 337, 710 (2012)), the present setup utilizes a piezoelectric actuator to change stress in situ inside the piston-cylinder pressure cell. We show that the actuator can be operated over the full temperature (from 30K up to 260 K) and pressure r…
Starting from the orthorhombic magnetically ordered phase, we investigate the effects of uniaxial tensile and compressive stresses along a, b, and the diagonal a+b directions in BaFe2As2 and CaFe2As2 in the framework of ab initio density functional theory (DFT) and a phenomonological Ginzburg-Landau model. While -contrary to the application of hydrostatic or c-axis uniaxial pressure- both systems remain in the orthorhombic phase with a pressure-dependent nonzero magnetic moment, we observe a sign-changing jump in the orthorhombicity at a critical uniaxial pressure, accompanied by a reversal of…
Uniaxial pressure applied in the b crystallographic direction perpendicular to spontaneous magnetization in heavy fermion ferromagnet URhGe strongly stimulates superconductivity in this compound. The phenomenological approach allows point out two mechanisms of superconducting temperature raising. They originates from stimulation by the uniaxial stress both intraband and interband amplitudes of triplet Cooper pairing. The phenomenon of reentrant superconductivity under magnetic field along b-axis is also strongly sensitive to the uniaxial stress in the same direction. The uniaxial stress accele…
Among unconventional superconductors, Sr$_2$RuO$_4$ has become a benchmark for experimentation and theoretical analysis because its normal-state electronic structure is known with exceptional precision, and because of experimental evidence that its superconductivity has, very unusually, a spontaneous angular momentum, i.e. a chiral state. This hypothesis of chirality is however difficult to reconcile with recent evidence on the spin part of the order parameter. Measurements under uniaxial stress offer an ideal way to test for chirality, because under uniaxial stress the superconducting and chi…
The aim of this study is to compare the magnetic behavior of URu2Si2 under uniaxial stress along the a-axis with the behavior under hydrostatic pressure. Both are very similar, but uniaxial stress presents a critical stress σxa smaller (0.33(5)GPa) than the hydrostatic critical pressure px =0.5 GPa where the ground state switches from HO (hidden order) to AF (antiferromagnetic) ground state. From these critical values and from Larmor neutron diffraction (LND), we conclude that the magnetic properties are governed by the shortest U-U distance in the plane (a lattice parameter). Under stress, th…
We study theoretically the shift of the superconducting transition temperature (Tc) under uniaxial compression in beta-type organic superconductors, beta-(BEDT-TTF)2I3 and beta-(BDA-TTP)2X[X=SbF6,AsF6], in order to clarify the electron correlation, the spin frustration and the effect of dimerization. The transfer integrals are calculated by the extended Huckel method assuming the uniaxial strain and the superconducting state mediated by the spin fluctuation is solved using Eliashberg's equation with the fluctuation-exchange approximation. The calculation is carried out on both the dimerized (o…
The La-based '214' cuprates host several symmetry breaking phases including superconductivity, charge and spin order in the form of stripes, and a structural othorhombic-to-tetragonal phase transition. Therefore, these materials are an ideal system to study the effects of uniaxial stress onto the various correlations that pervade the cuprate phase diagram. We report resonant x-ray scattering experiments on $\textrm{La}_{1.475}\textrm{Nd}_{0.4}\textrm{Sr}_{0.125}\textrm{Cu}\textrm{O}_{4}$ (LNSCO-125) that reveal a significant response of charge stripes to uniaxial tensile-stress of $\sim$ 0.1 G…
Unidirectional spin and charge density wave order in the cuprates is known to compete with superconductivity. In the stripe order (La,M)$_2$CuO$_4$ family of cuprates, spin and charge order occur as unidirectional order that can be stabilized by symmetry breaking structural distortions, such as the low temperature tetragonal (LTT) phase. Here we examine the interplay between structure and the formation of charge density wave (CDW) order in the LTT phase of La$_{1.475}$Nd$_{0.4}$Sr$_{0.125}$CuO$_4$ by applying uniaxial stress to distort the structure and influence the formation of CDW order. Us…
Muon spin rotation/relaxation ($μ$SR) and polar Kerr effect measurements provide evidence for a time-reversal symmetry breaking (TRSB) superconducting state in Sr$_2$RuO$_4$. However, the absence of a cusp in the superconducting transition temperature ($T_{\rm c}$) vs. stress and the absence of a resolvable specific heat anomaly at TRSB transition temperature ($T_{\rm TRSB}$) under uniaxial stress challenge a hypothesis of TRSB superconductivity. Recent $μ$SR studies under pressure and with disorder indicate that the splitting between $T_{\rm c}$ and $T_{\rm TRSB}$ occurs only when the structu…
CaFe2As2 single crystals under uniaxial pressure applied along the c axis exhibit the coexistence of several structural phases at low temperatures. We show that the room-temperature tetragonal phase is stabilized at low temperatures for pressures above 0.06 GPa, and its weight fraction attains a maximum in the region where superconductivity is observed under applied uniaxial pressure. Simultaneous resistivity measurements strongly suggest that this phase is responsible for the superconductivity in CaFe2As2 found below 10 K in samples subjected to nonhydrostatic pressure conditions.
We report that the pressure-temperature phase diagram of single-crystalline SrFe$_2$As$_2$ is easily affected by the hydrostaticity of a pressure-transmitting medium. For all of the three mediums we used, superconductivity with zero resistance appears, accompanied by the suppression of an antiferromagnetic (orthorhombic) phase, but the critical pressure $P_c$ was found to depend on the type of medium. $P_c$ was estimated to be 4.4 GPa under almost hydrostatic condition, but it decreased to $3.4-3.7$ GPa with the use of the medium already solidified at room temperature. The uniaxial stress alon…
A key question regarding the unconventional superconductivity of Sr$_2$RuO$_4$ remains whether the order parameter is single- or two-component. Under a hypothesis of two-component superconductivity, uniaxial pressure is expected to lift their degeneracy, resulting in a split transition. The most direct and fundamental probe of a split transition is heat capacity. Here, we report development of new high-frequency methodology for measurement of heat capacity of samples subject to large and highly homogeneous uniaxial pressure. We place an upper limit on the heat capacity signature of any second …
Quantum Oscillations of $\mathrm{Sr}_2\mathrm{RuO}_4$ under c-Axis Uniaxial Stress
Max Planck Institute for Chemical Physics of Solids（马克斯·普朗克固体化学物理研究所）；Scottish Universities Physics Alliance (SUPA), School of Physics and Astronomy, University of St Andrews（圣安德鲁斯大学物理与天文学学院苏格兰大学物理联盟）；National Institute for Materials Science（国立材料研究所）；Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院量子材料与技术研究所）
We study single-crystal CeCoIn5 with uniaxial pressure up to 3.97 kbar applied along the c-axis. We find a non-linear dependence of the superconducting transition temperature Tc on pressure, with a maximum close to 2 kbar. The transition also broadens significantly as pressure increases. We discuss the temperature dependence in terms of the general trend that Tc decreases in anisotropic heavy-fermion compounds as they move towards three-dimensional behavior.
Despite intense experimental efforts, the nature of the unconventional superconducting order parameter of UTe$_2$ remains elusive. This puzzle stems from different reported numbers of superconducting transitions at ambient pressure, as well as a complex pressure-temperature phase diagram. To bring new insights into the superconducting properties of UTe$_2$, we measured the heat capacity and electrical resistivity of single crystals under compressive uniaxial stress $σ$ applied along different crystallographic directions. We find that the critical temperature $T_{\rm c}$ of the single observed …
We measure the effect of uniaxial pressure on the superconducting transition temperature Tc in CeIrIn5. We find a linear change in Tc with both a-axis and c-axis pressure, with slopes of 56 mK/kbar and -66 mK/kbar, respectively. By comparing results from doping studies and different types of pressure measurements, we separate the influences of hybridization and dimensionality on Tc. We find the true geometric influence, for constant hybridization, is dTc/d(c/a)=44 K.
Raman scattering can detect spontaneous point-group symmetry breaking without resorting to single-domain samples. Here we use this technique to study $\mathrm{BaFe_2As_2}$, the parent compound of the "122" Fe-based superconductors. We show that an applied compression along the Fe-Fe direction, which is commonly used to produce untwinned orthorhombic samples, changes the structural phase transition at temperature $T_{\mathrm{s}}$ into a crossover that spans a considerable temperature range above $T_{\mathrm{s}}$. Even in crystals that are not subject to any applied force, a distribution of subs…
Capitalizing on the two-dimensional nature of superconductivity in field-effect doped C60, we show that it should be possible to increase the transition temperature Tc by applying uniaxial stress perpendicular to the gate electrode. This method not only holds the promise of substantially enhancing Tc (by about 30 K per GPa), but also provides a sensitive check of the current understanding of superconductivity in the doped Fullerenes.
The recent discovery of superconductivity in heavy Fermion compound UTe2, a candidate topological and triplet-paired superconductor, has aroused widespread interest. However, to date, there is no consensus on whether the stoichiometric sample of UTe2 is superconducting or not due to lack of reliable evidence to distinguish the difference between the nominal and real compositions of samples. Here, we are the first to clarify that the stoichiometric UT2 is non-superconducting at ambient pressure and under hydrostatic pressure up to 6 GPa, however we find that it can be compressed into supercondu…
While the superconducting transition temperature Tc of Sr2RuO4 is 1.5 K, its onset Tc is enhanced as high as 3 K in the Sr2RuO4-Ru eutectic system, which is often referred to as the 3-K phase. We have investigated effects of uniaxial pressure on the non-bulk superconductivity in the 3-K phase. While Tc of pure Sr2RuO4 is known to be suppressed by hydrostatic pressure, a large enhancement of the superconducting volume fraction of the 3-K phase was observed for both out-of-plane and in-plane uniaxial pressures. Especially, under the in-plane pressure, the shielding fraction at 1.8 K of only less…
Superconductivity of transition metal dichalcogenide $1T$-TiTe$_2$ under high pressure was investigated by the first-principles calculations. Our results show that the superconductivity of $1T$-TiTe$_2$ exhibits very different behavior under the hydrostatic and uniaxial pressure. The hydrostatic pressure is harmful to the superconductivity, while the uniaxial pressure is beneficial to the superconductivity. Superconducting transition temperature $T_C$ at ambient pressure is 0.73 K, and it reduces monotonously under the hydrostatic pressure to 0.32 K at 30 GPa. While the $T_C$ increases dramati…
Pressure represents a clean tuning parameter for traversing the complex phase diagrams of interacting electron systems and as such has proved of key importance in the study of quantum materials. Application of controlled uniaxial pressure has recently been shown to more than double the transition temperature of the unconventional superconductor Sr$_{2}$RuO$_{4}$ for example, leading to a pronounced peak in $T_\mathrm{c}$ vs. strain whose origin is still under active debate. Here, we develop a simple and compact method to apply large uniaxial pressures passively in restricted sample environment…
A key question regarding the unconventional superconductivity of Sr$_2$RuO$_4$ remains whether the order parameter is single- or two-component. Under a hypothesis of two-component superconductivity, uniaxial pressure is expected to lift their degeneracy, resulting in a split transition. The most direct and fundamental probe of a split transition is heat capacity. Here, we report measurement of heat capacity of samples subject to large and highly homogeneous uniaxial pressure. We place an upper limit on the heat-capacity signature of any second transition of a few per cent of that of the primar…
Static stripe order is detrimental to superconductivity. Yet, it has been proposed that transverse stripe fluctuations may enhance the inter-stripe Josephson coupling and thus promote superconductivity. Direct experimental studies of stripe dynamics, however, remain difficult. From a strong-coupling perspective, transverse stripe fluctuations are realized in the form of dynamic "kinks" -- sideways shifting stripe sections. Here, we show how modest uniaxial pressure tuning reorganizes directional kink alignment. Our starting point is La$_{1.88}$Sr$_{0.12}$CuO$_4$, where transverse kink ordering…
We have employed uniaxial stress along the principal axes of the quasi-two dimensional organic superconductor $κ$--(BEDT-TTF)$_{2}$Cu(SCN)$_{2}$. The lattice anisotropy is thereby altered, with corresponding changes in the intermolecular transfer energies. The effect of uniaxial stress on the superconducting transition temperature $T_{c}$ and critical field $B_{c2}$ is found to be anisotropic.There is an indication of an increase in $T_{c}$ and $B_{c2}$ for in-plane stress, but both parameters decrease rapidly for transverse (inter-plane) stress. Magnetotransport studies reveal stress-induced …
Nematic states are characterized by rotational symmetry breaking without translational ordering. Recently, nematic superconductivity, in which the superconducting gap spontaneously lifts the rotational symmetry of the lattice, has been discovered. However the pairing mechanism and the mechanism determining the nematic orientation remain unresolved. A first step is to demonstrate control of the nematicity, through application of an external symmetry-breaking field, to determine the sign and strength of coupling to the lattice. Here, we report for the first time control of the nematic orientatio…
We report muon spin rotation and magnetic susceptibility experiments on in-plane stress effects on the static spin-stripe order and superconductivity in the cuprate system La2-xBaxCuO4 with x = 0.115. An extremely low uniaxial stress of 0.1 GPa induces a substantial decrease in the magnetic volume fraction and a dramatic rise in the onset of 3D superconductivity, from 10 to 32 K; however, the onset of at-least-2D superconductivity is much less sensitive to stress. These results show not only that large-volume-fraction spin-stripe order is anti-correlated with 3D superconducting (SC) coherence,…
In this paper we report the impact of uniaxial strain $\varepsilon$ applied along the crystalline $a$ axis on the newly discovered kagome superconductor CsV$_3$Sb$_5$. At ambient conditions, CsV$_3$Sb$_5$ shows a charge-density wave (CDW) transition at $T_{\rm CDW}=94.5$ K and superconducts below $T_C = 3.34$ K. In our study, when the uniaxial strain $\varepsilon$ is varied from $-0.90\%$ to $0.90\%$, $T_C$ monotonically increases by $\sim 33\%$ from 3.0 K to 4.0 K, giving rise to the empirical relation $T_C (\varepsilon)=3.4+0.56\varepsilon+0.12\varepsilon^2$. On the other hand, for $\varepsi…
We explore the effect of lattice anisotropy on the charge-ordered superconductor 2H-NbSe$_{2}$. Using a novel strain apparatus, we measure the superconducting transition temperature $T_{sc}$ as a function of uniaxial strain. It is found that $T_{sc}$ is independent of tensile(compressive) strain below a threshold of 0.2\% (0.1\%), but decreases strongly with larger strains with an average rate of $1.3\,$K/\% ($2.5\,$K/\%). Transport signatures of charge order are largely unaffected as a function of strain. Theoretical considerations show that the change in the behavior of $T_{sc}$ with strain …
Spin-orbit coupled semiconductor nanowires with Zeeman splitting in proximity contact with bulk $s$-wave superconductivity have recently been proposed as a promising platform for realizing Majorana fermions. However, in this setup the chemical potential of the nanowire is generally pinned by the Fermi surface of the superconductor. This makes the tuning of the chemical potential by external electrical gates, a crucial requirement for unambiguous detection of Majorana fermions, very challenging in experiments. Here we show that tunable topological superconducting regime supporting Majorana ferm…
We investigate unconventional superconductivity in three-dimensional electronic systems with the chemical potential close to a quadratic band touching point in the band dispersion. Short-range interactions can lead to d-wave superconductivity, described by a complex tensor order parameter. We elucidate the general structure of the corresponding Ginzburg--Landau free energy and apply these concepts to the case of an isotropic band touching point. For vanishing chemical potential, the ground state of the system is given by the superconductor analogue of the uniaxial nematic state, which features…
Superconducting strontium titanate (STO) is in the spotlight as a low carrier concentration semiconductor in proximity to polar order. Its superconducting pairing mechanism poses an open fundamental challenge, which may be resolved by controlling how strongly the superconducting and polar orders interact. Here, we show for the first time that non-monotonic superconducting transition temperature behavior in uniaxially compressed STO can occur, revealing a reduction followed by a peak in the superconducting transition temperature. The corresponding displacements in the lattice are extremely smal…
We investigate the evolution of the electrical resistivity of BaFe2As2 single crystals with pressure. The samples used were from the same batch grown from self flux and showed properties that were highly reproducible. Samples were pressurised using three different pressure media: pentane-isopentane (in a piston cylinder cell), Daphne oil (in an alumina anvil cell) and steatite (in a Bridgman cell). Each pressure medium has its own intrinsic level of hydrostaticity, which dramatically affects the phase diagram. An increasing uniaxial pressure component in this system quickly reduces spin densit…
We have investigated uniaxial pressure effect on superconductivity of pure Sr2RuO4, whose intrinsic superconducting transition temperature Tc is 1.5 K. It was revealed that a very low uniaxial pressure along the c axis, only 0.2 GPa, induces superconductivity with the onset Tc above 3 K. The present results indicate that pure Sr2RuO4 has two superconducting phases with Tc = 1.5 K and with varying Tc up to 3.2 K. The latter phase exhibits unusual features and is attributable to anisotropic crystal distortions beyond the elastic limit.
Applying in-plane uniaxial pressure to strongly correlated low-dimensional systems has been shown to tune the electronic structure dramatically. For example, the unconventional superconductor Sr$_2$RuO$_4$ can be tuned through a single Van Hove singularity which results in a strong enhancement of both $T_\text{c}$ and $H_\text{c2}$. Out-of-plane ($c$ axis) uniaxial pressure is expected to tune the quasi-two-dimensional structure even more strongly, causing it to approach two Van Hove singularities simultaneously. Here we achieve a record value of $3.2$ GPa compression along the $c$ axis of Sr$…
We use neutron diffraction and muon spin relaxation to study the effect of in-plane uniaxial pressure on the antiferromagnetic (AF) orthorhombic phase in BaFe$_2$As$_2$ and its Co- and Ni-substituted members near optimal superconductivity. In the low temperature AF ordered state, uniaxial pressure necessary to detwin the orthorhombic crystals also increases the magnetic ordered moment, reaching an 11$\%$ increase under 40 MPa for BaFe$_{1.9}$Co$_{0.1}$As$_2$, and a 15$\%$ increase for BaFe$_{1.915}$Ni$_{0.085}$As$_2$. We also observe an increase of the AF ordering temperature ($T_N$) of about …
Iron-based superconductors are well-known for their intriguing phase diagrams, which manifest a complex interplay of electronic, magnetic and structural degrees of freedom. Among the phase transitions observed are superconducting, magnetic, and several types of structural transitions, including a tetragonal-to-orthorhombic and a collapsed-tetragonal transition. In particular, the widely-observed tetragonal-to-orthorhombic transition is believed to be a result of an electronic order that is coupled to the crystalline lattice and is, thus, referred to as nematic transition. Nematicity is therefo…
Correlation between crystal structure and superconducting properties of the BiS2-based superconductor LaO0.5F0.5BiS2 was investigated. We have prepared LaO0.5F0.5BiS2 polycrystalline samples with various lattice constants. It was found that the annealing the sample under high pressure generated uniaxial strain along the c axis. Further, the highly-strained sample showed higher superconducting properties. We concluded that the uniaxial strain along the c axis was positively linked with the enhancement of superconductivity in the LaO1-xFxBiS2 system.
We have investigated uniaxial and hydrostatic pressure effects on superconductivity in Fe1.07Te0.88S0.12 through magnetic-susceptibility measurements down to 1.8 K. The superconducting transition temperature Tc is enhanced by out-of-plane pressure (uniaxial pressure along the c-axis); the onset temperature of the superconductivity reaches 11.8 K at 0.4 GPa. In contrast, Tc is reduced by in-plane pressure (uniaxial pressure along the ab-plane) and hydrostatic pressure. Taking into account these results, it is inferred that the superconductivity of Fe1+yTe1-xSx is enhanced when the lattice const…
Bending-like stress induced by solder joint under uniaxial tensile testing in 2G-HTS tapes: Impact and optimization approach
School of Electrical Engineering, Shanghai Jiao Tong University（上海交通大学电气工程学院）；State key Laboratory of High-Efficiency special cable technology, Shanghai Jiao Tong University（上海交通大学高效特种电缆技术国家重点实验室）；Key Laboratory of Applied Superconductivity, Institute of Electrical Engineering, Chinese Academy of Sciences（中国科学院电气工程技术研究所超导应用重点实验室）；University of Chinese Academy of Sciences（中国科学院大学）
We report resistivity and calorimetric measurements on two single crystals of CePd2Si2 pressurized up to 7.4 GPa. A weak uniaxial stress induced in the pressure cell demonstrates the sensitivity of the physics to anisotropy. Stress applied along the c-axis extends the whole phase diagram to higher pressures and enhances the superconducting phase emerging around the magnetic instability, with a 40% increase of the maximum superconducting temperature, Tc, and a doubled pressure range. Calorimetric measurements demonstrate the bulk nature of the superconductivity.
Understanding the microscopic origins of electronic phases in high-transition temperature (high-Tc) superconductors is important for elucidating the mechanism of superconductivity. In the paramagnetic tetragonal phase of BaFe2-xTxAs2 (where T is Co or Ni) iron pnictides, an in-plane resistivity anisotropy has been observed. Here we use inelastic neutron scattering to show that low-energy spin excitations in these materials change from four-fold symmetric to two-fold symmetric at temperatures corresponding to the onset of the in-plane resistivity anisotropy. Because resistivity and spin excitat…
The effects of uniaxial compressive stress on the normal state $^{17}$O nuclear magnetic resonance properties of the unconventional superconductor Sr$_{2}$RuO$_{4}$ are reported. The paramagnetic shifts of both planar and apical oxygen sites show pronounced anomalies near the nominal $\mathbf{a}$-axis strain $\varepsilon_{aa}\equiv\varepsilon_v$, that maximizes the superconducting transition temperature, $T_{c}$. The spin susceptibility weakly increases on lowering the temperature below $T$$\simeq$10 K, consistent with an enhanced density of states associated with passing the Fermi energy thro…
The stress caused by vortices in tetragonal superconductors contributes to the intervortex interaction which depends on vortex orientation within the crystal, on elastic moduli, and is attractive within certain angular regions even in fields along the $c$ crystal axis. For sufficiently strong stress dependence of the critical temperature, this contribution may result in distortions of the hexagonal vortex lattice for $\bm H||\bm c$. In small fields it leads to formation of a square vortex lattice with a fixed $H$ independent spacing. This should be seen in the magnetization $M(H)$ as a discont…
The effect of uniaxial tensile stress and the resultant strain on the structural/magnetic transition in the parent compound of the iron arsenide superconductor, BaFe$_2$As$_2$, is characterized by temperature-dependent electrical resistivity, x-ray diffraction and quantitative polarized light imaging. We show that strain induces a measurable uniaxial structural distortion above the first-order magnetic transition and significantly smears the structural transition. This response is different from that found in another parent compound, SrFe$_2$As$_2$, where the coupled structural and magnetic tr…
The Hubbard model on a square lattice is one of the most studied condensed-matter quantum problems.Here we find evidence that for intermediate $U/4t$ values and a hole-concentration range $x\in (x_c,x_*)$ the ground state of the Hubbard model on the square lattice perturbed by weak three-dimensional (3D) uniaxial anisotropy has long-range d-wave superconducting order. Here $t$ is the effective nearest-neighbor transfer integral and $U$ the effective on-site repulsion. The lower critical concentration $x_c$ involves the Ginzburg number Gi and is approximately given by $x_c\approx {\rm Gi}+x_0\a…
