Pauli Paramagnetic Effect Research Articles

Robust quantum Griffiths singularity above 1.5 K in nitride thin films

Quantum Griffiths singularity (QGS), which is closely correlated with the quenched disorder, is characterized by the divergence of the dynamical critical exponent and the presence of activated scaling behavior. Typically such a quantum phenomenon is rather rare and only observed in extremely low temperatures. Here we report the experimental observation of a robust QGS in nitride thin films, NbN, which survives in a rather high temperature range. The electrical transport propertied were measured under the magnetic field up to 12 T. The field induced superconductor-metal transitions were observed with the continuously changed transition points with the decrease of temperature. The dynamical critical exponent based on the conventional power-law scaling reveals a divergent trend when approaching the the low temperature limit. Moreover, the temperature and field dependence of sheet resistance can be described by the activated scaling analysis in the temperature region up to 1.6 K $\leq T \leq$ 4.0 K. We argue that the robustness of QGS in the present system originates from the Pauli paramagnetic effect due to the strong coupling between the magnetic field and the spin degree of freedom.

Investigation of the Pauli paramagnetic effect in systematically tuned NbN thin films

Superconductivity and the normal-state properties of NbN films can be tuned in a wide range, supplying a suitable platform to investigate the systematical evolution of the superconducting performances. Herein, we report the upper critical field of NbN films in both the vertical (B⊥ film) and parallel (B∥ film) orientations over a wide temperature range. Eight samples with the superconducting critical temperature Tc ranging from 2.5 K to 9.8 K are studied. It is found that the magnitudes of the upper critical field in the parallel direction (Bc2∥) exceeds the paramagnetic limiting field Bp in the zero-temperature limit, while the perpendicular one (Bc2⊥) is closely approaching Bp. The temperature dependent Bc2⊥ can be described by the extended Werthamer–Helfand–Hohenberg (WHH) model considering the Pauli spin paramagnetism. Meanwhile, the Bc2∥−T data shows the feature of two-dimensional superconductivity in the temperature range near Tc. Our results show that the Pauli paramagnetic effect is greatly enhanced with the decrease of Tc. The evolution of the obtained Maki parameter with other parameters are discussed to reveal the characteristics of the Pauli paramagnetic effect in this system.

Deviations from the extended London model at high magnetic fields in YBa2Cu3O7

We report on the evolution with the magnetic field and the temperature of the vortex lattice (VL) in fully oxygenated ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7}$ as studied by time-of-flight small-angle neutron scattering. Using the High Field Magnet/Extreme Environment Diractometer beamline at Helmholtz-Zentrum Berlin, we have obtained data up to 25.9 T---much higher than data available previously. Our VL structure results are consistent with the progressive suppression by the field of the superconductivity along the crystallographic $\mathbf{b}$ (CuO chain) direction and an accompanying shift of the nodal directions as the field is increased. The intensity of the diffracted signal reveals the spatial variation of magnetization caused by the VL (the ``form factor''). Instead of a rapid falloff with the field, as seen in superconductors with smaller upper critical fields, we find that the form factor is almost constant with the field above $\ensuremath{\sim}12$ T. We speculate that this is due to Pauli paramagnetic effects.

Observations of the effect of strong Pauli paramagnetism on the vortex lattice in superconducting CeCu2Si2

We present the results of a study of the vortex lattice in the heavy fermion superconductor ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$, using small-angle neutron scattering (SANS). In this material at temperatures well below ${T}_{\mathrm{c}}\ensuremath{\sim}0.6$ K, the value of the upper critical field ${B}_{\mathrm{c}2}\ensuremath{\sim}2.2$ T is strongly limited by the Pauli paramagnetism of the heavy fermions. In this temperature region, our SANS data show an increase in the magnetization of the flux line cores with field, followed by a rapid fall near ${B}_{\mathrm{c}2}$. This behavior is the effect of Pauli paramagnetism and we present a theory-based model, which can be used to describe this effect in a range of materials. The pairing in ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$ appears to arise from the effect of magnetic fluctuations, but the evidence for a $d$-wave order parameter is rather weak. We find that the vortex lattice structure in ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$ is close to regular hexagonal. There are no phase transitions to square or rhombic structures; such transitions are expected for $d$-wave superconductors and observed in ${\mathrm{CeCoIn}}_{5}$; however, the temperature dependence of the SANS intensity indicates that both large and small gap values are present, most likely due to multiband $s$-wave superconductivity, rather than a nodal gap structure.

Evidence for the Fulde\u2013Ferrell\u2013Larkin\u2013Ovchinnikov state in bulk NbS2

We present measurements of the magnetic torque, specific heat and thermal expansion of the bulk transition metal dichalcogenide (TMD) superconductor NbS2 in high magnetic fields, with its layer structure aligned strictly parallel to the field using a piezo rotary positioner. The upper critical field of superconducting TMDs in the 2D form is known to be dramatically enhanced by a special form of Ising spin orbit coupling. This Ising superconductivity is very robust to the Pauli paramagnetic effect and can therefore exist beyond the Pauli limit for superconductivity. We find that superconductivity beyond the Pauli limit still exists in bulk single crystals of NbS2 for a precisely parallel field alignment. However, the comparison of our upper critical field transition line with numerical simulations rather points to the development of a Fulde-Ferrell-Larkin-Ovchinnikov state above the Pauli limit as a cause. This is also consistent with the observation of a magnetic field driven phase transition in the thermodynamic quantities within the superconducting state near the Pauli limit.

Interplay of charge density wave and multiband superconductivity in layered quasi-two-dimensional materials: The case of 2H−NbS2 and 2H−NbSe2

Despite intense efforts on all known quasi-two-dimensional superconductors, the origin and exact boundary of the electronic orderings, particularly charge density waves and superconductivity, are still attractive problems with several open questions. Here, in order to reveal how the superconducting gap evolves, we report on high quality complementary measurements of magneto-optical imaging, specific heat, magnetic susceptibility, resistivity measurements, Andreev spectroscopy, and London penetration depth ${\ensuremath{\lambda}}_{ab}(T)$ measurements supplemented with theoretical calculations for $2\mathrm{H}\text{\ensuremath{-}}\mathrm{Nb}{\mathrm{Se}}_{2}$ and $2\mathrm{H}\text{\ensuremath{-}}\mathrm{Nb}{\mathrm{S}}_{2}$ single crystals. The temperature dependence of ${\ensuremath{\lambda}}_{ab}(T)$ calculated from the lower critical field and Andreev spectroscopy can be well described by using a two-band model with $s$-wave-like gaps. The effect of pressure on the superconducting gap of both systems illustrates that both bands are practically affected. Upon compression, the Fermi surfaces do not change significantly, and the nesting remains almost unaffected compared to that at ambient condition. However, a strong bending in the upper critical fields (${H}_{\mathrm{c}2}$) curves is obtained under pressure and support the presence of a strong Pauli paramagnetic effect. In $\mathrm{Nb}{\mathrm{Se}}_{2}$, using a two-band model with $s$-wave-like gaps, the temperature dependence ${H}_{\mathrm{c}2}(T)$ can be properly described. In contrast to that, the behavior of ${H}_{\mathrm{c}2}$ for $\mathrm{Nb}{\mathrm{S}}_{2}$ is ruled by the spin paramagnetic effect. The estimated values of the penetration depth at $T=0\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ confirm that $\mathrm{Nb}{\mathrm{Se}}_{2}$ and $\mathrm{Nb}{\mathrm{S}}_{2}$ superconductors depart from a Uemura-style relationship between ${T}_{\mathrm{c}}$ with ${\mathbit{\ensuremath{\lambda}}}_{ab}^{\ensuremath{-}2}(T)$, the in-plane superconducting penetration depth.

Microscopic Eilenberger theory of Fulde-Ferrell-Larkin-Ovchinnikov states in the presence of vortices

We theoretically investigate the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state by using the microscopic quasi-classical Eilenberger equation. The Pauli paramagnetic effects and the orbital depairing effects due to vortices are treated in an equal footing for three dimensional spherical Fermi surface model and $s$-wave pairing. The field evolution of the LO state is studied in detail, such as the $H$-$T$ phase diagram, spatial structures of the order parameter, the paramagnetic moment, and the internal filed. Field-dependences of various thermodynamic quantities: the paramagnetic moment, entropy, and the zero-energy density of states are calculated. Those quantities are shown to start quickly growing upon entering the LO state. We also evaluate the wave length of the LO modulation, the flux line lattice form factors for small angle neutron scattering, and the NMR spectra to facilitate the identification of the LO state. Two cases of strong and intermediate Pauli paramagnetic effect are studied comparatively. The possibility of the LO phase in Sr$_2$RuO$_4$, CeCoIn$_5$, CeCu$_2$Si$_2$, and the organic superconductors is critically examined and crucial experiments to identify it are proposed.

Competing pairing interactions responsible for the large upper critical field in a stoichiometric iron-based superconductor CaKFe4As4

The upper critical field of multiband superconductors is an important quantity that can reveal the details about the nature of the superconducting pairing. Here we experimentally map out the complete upper critical field phase diagram of a stoichiometric superconductor, CaKFe$_4$As$_4$, up to 90T for different orientations of the magnetic field and at temperatures down to 4.2K. The upper critical fields are extremely large, reaching values close to ~3$T_c$ at the lowest temperature, and the anisotropy decreases dramatically with temperature leading to essentially isotropic superconductivity at 4.2K. We find that the temperature dependence of the upper critical field can be well described by a two-band model in the clean limit with band coupling parameters favouring intraband over interband interactions. The large Pauli paramagnetic effects together with the presence of the shallow bands is consistent with the stabilization of an FFLO state at low temperatures in this clean superconductor.

Strong Pauli paramagnetic effect in the upper critical field of KCa2Fe4As4F2

Recently, 12442 system of Fe-based superconductors has attracted considerable attention owing to its unique double-FeAs-layer structure. A steep increase in the in-plane upper critical field with cooling has been observed near the superconducting transition temperature, $T_c$, in KCa$_2$Fe$_4$As$_4$F$_2$ single crystals. Herein, we report a high-field investigation on upper critical field of this material over a wide temperature range, and both out-of-plane ($H\|c$, $H_{c2}^{c}$) and in-plane ($H\|ab$, $H_{c2}^{ab}$) directions have been measured. A sublinear temperature-dependent behavior is observed for the out-of-plane $H_{c2}^{c}$, whereas strong convex curvature with cooling is observed for the in-plane $H_{c2}^{ab}$. Such behaviors could not be described by the conventional Werthamer--Helfand--Hohenberg (WHH) model. The data analysis based on the WHH model by considering the spin aspects reveals a large Maki parameter $\alpha=9$, indicating that the in-plane upper critical field is affected by a very strong Pauli paramagnetic effect.

Effect of Hydrostatic Pressure on Superconductivity of FeSe Thin Films

The electronic transport in superconducting epitaxial c-axis-oriented FeSe thin films grown on (001)-oriented MgO substrates was investigated. To this end, the in-plane resistivity was measured in dependence on temperature, pressure, and magnetic field. The temperature ranged from 1.2 to 35 K; static magnetic fields with strengths up to 14 T were applied normal to the film surface, i.e. parallel to the FeSe c-axis; and hydrostatic pressure was applied from 0 to 2.7 GPa. Concerning the role of the MgO substrate in the pressure experiments, it is suggested that the substrate mainly reduces the in-plane compressibility of the film in comparison to bulk. The transition to superconductivity shifted to higher temperatures with increasing pressure. The onset critical temperature raised from 11.5 K at zero applied pressure with an initial rate of 2.5 K/GPa to 18.2 K at 2.7 GPa. The pressure-induced increase of the critical temperature was accompanied by a twofold broadening of the transition width. As a counterpart of pressure, the magnetic field shifted the superconducting transition to lower temperature. In addition to pressure, the field also induced a noticeable broadening of the superconductive transition rather than a parallel shift. The positive magnetoresistance at 20 K increased with enhanced pressure and reached 24% at the highest pressure and field. For each applied pressure, the magnetoresistance could be fitted by a Lorentzian function, i.e. it originates from classical Lorentz scattering. The resulting charge carrier mobility increased under pressure suggesting a decreasing collision rate. The upper critical field raised with higher pressure. Its temperature dependence could be fitted by conventional Werthamer-Hohenberg-Helfand theory under the assumption of the Pauli paramagnetic effect that became more pronounced under pressure. The anomalous behaviour of the normalized negative slope of the upper critical field at the critical temperature suggested a change of the Fermi surface above a critical pressure of 2 GPa.

Theoretical studies for identifying horizontal line nodes via angle-resolved density-of-states measurements: Application to Sr2RuO4

On the basis of the microscopic quasi-classical Eilenberger theory, we analyze the recent angle-resolved specific heat experiment carried out at low temperature for Sr$_2$RuO$_{4}$ to identify the superconducting gap symmetry, comprising either horizontal or vertical line nodes relative to the tetragonal crystal symmetry. Several characteristics, in particular, the landscape of the in-plane oscillation amplitude $A_4(B, T)$ with a definite sign for almost the entire $B$-$T$ plane are best explained by the horizontal line node symmetry, especially when the multiband effect and Pauli paramagnetic effect are taken into account. The present analysis of $A_4(B,T)$ with definite sign points to the presence of an anomalous field region at a lower temperature in the experimental data, whose origin is investigated. Our theory demonstrates the application and uniqueness of the field-rotating thermodynamic measurements in uncovering the precise gap structure for target materials.

Searching for Gap Zeros in Sr2RuO4 via Field-Angle-Dependent Specific-Heat Measurement

The gap structure of Sr$_2$RuO$_4$, which is a longstanding candidate for a chiral p-wave superconductor, has been investigated from the perspective of the dependence of its specific heat on magnetic field angles at temperatures as low as 0.06 K ($\sim 0.04T_{\rm c}$). Except near $H_{\rm c2}$, its fourfold specific-heat oscillation under an in-plane rotating magnetic field is unlikely to change its sign down to the lowest temperature of 0.06 K. This feature is qualitatively different from nodal quasiparticle excitations of a quasi-two-dimensional superconductor possessing vertical lines of gap minima. The overall specific-heat behavior of Sr$_2$RuO$_4$ can be explained by Doppler-shifted quasiparticles around horizontal line nodes on the Fermi surface, whose in-plane Fermi velocity is highly anisotropic, along with the occurrence of the Pauli-paramagnetic effect. These findings, in particular, the presence of horizontal line nodes in the gap, call for a reconsideration of the order parameter of Sr$_2$RuO$_4$.

Unusual evolution of Bc2 and Tc with inclined fields in restacked TaS2 nanosheets

Recently we reported an enhanced superconductivity in restacked monolayer TaS2 nanosheets compared with the bulk TaS2, pointing to the exotic physical properties of low dimensional systems. Here we tune the superconducting properties of this system with magnetic field along different directions, where a strong Pauli paramagnetic spin-splitting effect is found in this system. Importantly, an unusual enhancement as high as 3.8 times of the upper critical field Bc2, as compered with the Ginzburg-Landau (GL) model and Tinkham model, is observed under the inclined external magnetic field. Moreover, with the out-of-plane field fixed, we find that the superconducting transition temperature Tc can be enhanced by increasing the in-plane field and forms a dome-shaped phase diagram. An extended GL model considering the special microstructure with wrinkles was proposed to describe the results. The restacked crystal structure without inversion center along with the strong spin–orbit coupling may also play an important role for our observations.

Upper critical fields in Ba2Ti2Fe2As4O single crystals: Evidence for dominant Pauli paramagnetic effect

We report on magneto-optical imaging and the temperature dependency of the upper critical fields ${H}_{c2}^{c}(T)$ parallel to the $c$ axis and ${H}_{c2}^{ab}(T)$ parallel to the $ab$ plane in ${\mathrm{Ba}}_{2}{\mathrm{Ti}}_{2}{\mathrm{Fe}}_{2}{\mathrm{As}}_{4}\mathrm{O}$ single crystals. These data were inferred from the measurements of the temperature-dependent resistance in static magnetic fields up to 14 T and magnetoresistance in pulsed fields up to 60 T. ${H}_{c2}$ values are found to be 52 and 50 T for $H\ensuremath{\parallel}ab$ and $H\ensuremath{\parallel}c$, respectively. These values are 1.2--1.35 times larger than the weak-coupling Pauli paramagnetic limit $({H}_{p}\ensuremath{\sim}1.84{T}_{c})$, indicating that enhanced paramagnetic limiting is essential and this superconductor is unconventional. Our observations of strong bending in the ${H}_{c2}^{ab}(T)$ curves and a nearly isotropic maximum upper critical field ${H}_{c2}^{ab}(0)\phantom{\rule{0.16em}{0ex}}\ensuremath{\approx}\phantom{\rule{0.16em}{0ex}}{H}_{c2}^{c}(0)$ support the presence of a strong Pauli paramagnetic effect. We show that the Werthamer-Helfand-Hohenberg (WHH) formula that includes the spin-orbit scattering can effectively describe the ${H}_{c2}^{ab}(T)$ curve, whereas ${H}_{c2}$ deviates from the conventional WHH theoretical model without considering the spin paramagnetic effect for the $H\ensuremath{\parallel}c$ and $H\ensuremath{\parallel}ab$ directions. For $H\ensuremath{\parallel}c$, a two-band model is required to fully reproduce the behavior of ${H}_{c2}$, while for $H\ensuremath{\parallel}ab$ the spin paramagnetic effect is responsible for the behavior of ${H}_{c2}$. The anisotropy of ${H}_{c2}$ is close to 3 near ${T}_{c}$ and decreases rapidly at lower temperatures.

Gap structure of FeSe determined by angle-resolved specific heat measurements in applied rotating magnetic field

Quasiparticle excitations in FeSe were studied by means of specific heat ($C$) measurements on a high-quality single crystal under rotating magnetic fields. The field dependence of $C$ shows three-stage behavior with different slopes, indicating the existence of three gaps ($\Delta_1$, $\Delta_2$, and $\Delta_3$). In the low-temperature and low-field region, the azimuthal-angle ($\phi$) dependence of $C$ shows a four-fold symmetric oscillation with sign change. On the other hand, the polar-angle ($\theta$) dependence manifests as an anisotropy-inverted two-fold symmetry with unusual shoulder behavior. Combining the angle-resolved results and the theoretical calculation, the smaller gap $\Delta_1$ is proved to have two vertical-line nodes or gap minima along the $k_z$ direction, and is determined to reside on the electron-type $\varepsilon$ band. $\Delta_2$ is found to be related to the electron-type $\delta$ band, and is isotropic in the $ab$-plane but largely anisotropic out of the plane. $\Delta_3$ residing on the hole-type $\alpha$ band shows a small out-of-plane anisotropy with a strong Pauli-paramagnetic effect.

Anisotropy and multiband superconductivity in Sr2RuO4 determined by small-angle neutron scattering studies of the vortex lattice

Despite numerous studies the exact nature of the order parameter in superconducting Sr2RuO4 remains unresolved. We have extended previous small-angle neutron scattering studies of the vortex lattice in this material to a wider field range, higher temperatures, and with the field applied close to both the <100> and <110> basal plane directions. Measurements at high field were made possible by the use of both spin polarization and analysis to improve the signal-to-noise ratio. Rotating the field towards the basal plane causes a distortion of the square vortex lattice observed for H // <001>, and also a symmetry change to a distorted triangular symmetry for fields close to <100>. The vortex lattice distortion allows us to determine the intrinsic superconducting anisotropy between the c-axis and the Ru-O basal plane, yielding a value of ~60 at low temperature and low to intermediate fields. This greatly exceeds the upper critical field anisotropy of ~20 at low temperature, reminiscent of Pauli limiting. Indirect evidence for Pauli paramagnetic effects on the unpaired quasiparticles in the vortex cores are observed, but a direct detection lies below the measurement sensitivity. The superconducting anisotropy is found to be independent of temperature but increases for fields > 1 T, indicating multiband superconductvity in Sr2RuO4. Finally, the temperature dependence of the scattered intensity provides further support for gap nodes or deep minima in the superconducting gap.

Absence of the Pauli-Paramagnetic Limit in a Superconducting U6Co

We performed $^{59}$Co nuclear magnetic resonance (NMR) measurements of single-crystalline U$_6$Co. There is a small decrease in the Knight shift in the superconducting (SC) state, but this change mainly arises from the SC diamagnetic effect. The negligible change of the spin part of the Knight shift, together with the absence of the Pauli-paramagnetic effect in the SC U$_6$Co, is understood as a consequence of the small spin susceptibility. The nuclear spin-lattice relaxation rate $1/T_1$ is also measured in the SC state under the magnetic field, and exhibits a tiny Hebel-Slichter peak just below the SC transition temperature and exponential behavior at lower temperatures. These behaviors are in agreement with the full-gap s-wave pairing in U$_6$Co.

Temperature and angular dependence of the upper critical field in K2Cr3As3

We report measurements of the upper critical field $H_{\mathrm{c2}}$ as functions of temperature $T$, polar angle $\theta$ (of the field direction with respect to the crystallographic $c$ axis), and azimuthal angle $\phi$ (of the field direction relative to the $a$ axis within the $ab$ plane) for the Cr-based superconductor K$_{2}$Cr$_{3}$As$_{3}$ with a quasi-one-dimensional and non-centrosymmetric crystal structure. We confirm that the anisotropy in $H_{\mathrm{c2}}(T)$ becomes inverse with decreasing temperature. At low temperatures, $H_{\mathrm{c2}}(\theta)$ data are featured by two maxima at $\theta$ = 0 ($\mathbf{H}\parallel c$) and $\pi/2$ ($\mathbf{H}\bot c$), which can be quantitatively understood only if uniaxial effective-mass anisotropy and absence of Pauli paramagnetic effect for $\mathbf{H}\bot c$ are taken simultaneously into consideration. The in-plane $H_{\mathrm{c2}}(\phi)$ profile shows a unique threefold modulation especially at low temperatures. Overall, the characteristic of the $H_{\mathrm{c2}}(\theta, \phi, T)$ data mostly resemble those of the heavy-fermion superconductor UPt$_3$, and we argue in favor of a dominant spin-triplet superconductivity with odd parity in K$_{2}$Cr$_{3}$As$_{3}$.

Absence of the O17 Knight-shift changes across the first-order phase transition line in Sr2RuO4

We performed $^{17}$O nuclear magnetic resonance measurements on superconducting (SC) Sr$_2$RuO$_4$ under in-plane magnetic fields. We found that no new signal appears in the SC state and that the $^{17}$O Knight shifts obtained from the double-site measurements remain constant across the first-order phase-transition line, as well as across the second-order phase-transition line as already reported. The present results indicate that the SC spin susceptibility does not decrease in the high-field region, although a magnetization jump in the SC state was reported at low temperatures. Because the spin susceptibility is unchanged in the SC state in Sr$_2$RuO$_4$, we suggest that the first-order phase transition across the upper critical field should be interpreted as a depairing mechanism other than the conventional Pauli-paramagnetic effect.

Anisotropic Upper Critical Field of Iron-Based Superconductors

The upper critical field and its anisotropy are the easiest properties to examine in the research of iron-based superconductors. Based on warped cylindrical Fermi surface models, we investigate the temperature and angle dependence of the upper critical field in detail by employing the quasi-classical formalism of the Werthamer–Helfand–Hohenberg (WHH) theory. Our numerical results reveal the anisotropy of the upper critical field, which may be caused by an anisotropic gap function (e.g., d-wave pairing) or an anisotropic Fermi surface, respectively. Further, according to our analysis, this anisotropy can be modulated by the deformation of the Fermi surface and will be strongly suppressed by the Pauli paramagnetic effect.