FFLO State Research Articles

Possible Fulde-Ferrell-Larkin-Ovchinnikov inhomogeneous superconducting state in CeCoIn5

We present specific heat and thermal conductivity of the heavy fermion superconductor CeCoIn5 in the vicinity of the superconducting critical fieldH c2, measured with magnetic field in the plane of this quasi-2D compound and at temperatures down to 50 mK. The superconducting phase diagram and the first order nature of the superconducting phase transition at high fields close to a critical fieldH c2 indicate the importance of the Pauli limiting effect in CeCoIn5. In the same range of magnetic field we observe a second specific heat anomaly within the superconducting state, and interpret it as a signature of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) inhomogeneous superconducting state. In addition, the thermal conductivity data as a function of field display a kink at a fieldH k below the superconducting critical field, which closely coincides with the low temperature anomaly in specific heat tentatively identified with the appearance of the FFLO superconducting state. The enhancement of thermal conductivity within the FFLO state calls for further theoretical investigations of the real space structure of the order parameter (and in particular, the structure of vortices) and of the thermal transport within the inhomogeneous FFLO state.

Comment on “Fluctuation-Driven First-Order Transition in Pauli-Limitedd-Wave Superconductors”

This is a Comment on a recent Letter (Phys.Rev.Lett.93, 247002 (2004)) written by Dalidovich and Yang. It is pointed out that any Pauli-limited approach neglecting the vortices is inapplicable to describing the FFLO state in CeCoIn5.

Penetration depth studies of organic and heavy fermion superconductors in the Pauli paramagnetic limit

Abstract We report penetration depth and resistivity measurements on a number of organic and heavy fermion superconductors in the Pauli paramagnetic limit using a self resonant tank circuit based on a tunnel diode oscillator (TDO). We call a superconductor Pauli limited when the interaction of the applied magnetic field with the electron spin limits the superconducting state, in contrast to orbital limiting, the traditional effect where vortices eventually destroy the superconducting state. In the Pauli limit, Hc2 can change from a second order to a first order phase transition at low temperatures, or in a very clean sample a new superconducting state with a spatially varying order parameter has been predicted, which we refer to as the FFLO state. We have done extensive experiments that show a range of Pauli limited behavior in anisotropic superconductors. Our results for α-(ET)2NH4Hg(SCN)4, κ-(ET)2Cu(NCS)2, and CeCoIn5 show that the nature of the Pauli limiting state is sensitive to the mean freepath in the sample. In CeCoIn5 we have observed clear evidence of the FFLO state using our TDO method, and our data agrees with specific heat and magnetization measurements. We also show that the application of pressure will suppress the Pauli limiting in κ-(ET)2Cu(NCS)2.

Realization and Detection of Fulde-Ferrell-Larkin-Ovchinnikov Superfluid Phases in Trapped Atomic Fermion Systems

In a very interesting recent Letter\cite{machida}, the authors suggested the possibility of realizing the spatially modulated, or Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid state in trapped atomic fermion systems. The authors this Letter used a 1D mean field solution as guidance to estimate the parameter range for the existence of the FFLO phase, and also discussed the possibility of its detection by imaging the atomic density of the system. In this comment I wish to make two points. (i) In 1D there exists an exact solution based on bosonization, which fully takes into account the important quantum fluctuation effects\cite{yang01}; the exact solution suggests a wider parameter range for the FFLO state than that obtained from the mean-field solution of the present Letter. (ii) One can detect the FFLO pairing (in which Cooper pairs carry finite momenta) more directly by extending the methods used to detect BCS pairing\cite{regal,altman,greiner}.

Inhomogeneous superconducting states and umklapp processes in ferromagnet/superconductor nanostructures

We derive a new boundary-value problem (BVP) for the ferromagnetic metal/superconductor (FM/S) nanostructures, where superconductivity is a superposition of the BCS pairing with zero total momentum in the S layers and the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) pairing with nonzero 3D coherent momentum k in the FM layers. It is shown that the processes of mutual transformation between BCS and FFLO pairs at the FM/S boundary are the umklapp processes during which coherent pair momentum k becomes confined in the FM layer. We originally investigate the interplay between the BCS and 3D FFLO states in the pure FM/S bilayers. This competition leads to a partial compensation of the exchange field paramagnetic effect.

Phase diagram and spectroscopy of Fulde-Ferrell-Larkin-Ovchinnikov states of two-dimensionald-wave superconductors

Experimental evidence suggests that the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state may be realized in the unconventional, heavy-fermion superconductor ${\mathrm{CeCoIn}}_{5}$. We present a self-consistent calculation of the field versus temperature phase diagram and order parameter structures for the FFLO states of quasi-two-dimensional $d$-wave superconductors. We calculate the spatially nonuniform order parameter, free energy density, and local density of states for magnetic fields parallel to the superconducting planes. We predict that the lower critical magnetic field transition between the spatially uniform and nonuniform FFLO state is second order. We discuss the signatures of the nonuniform FFLO state which should be observable in scanning tunneling microscopy measurements of the local density of states.

Topological Structure of a Vortex in the Fulde-Ferrell-Larkin-Ovchinnikov State

We find theoretically that the vortex core in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is quite different from the ordinary core for a simple topological reason. The intersection point of a vortex and nodal plane of the FFLO state empties the excess spins. This leads to observable consequences in the spatial structure of the spontaneous magnetization. We analyze this topological structure based on the low lying excitation spectrum by solving a microscopic Bogoliubov-de Gennes equation to clarify its physical origin.

Fulde–Ferrell–Larkin–Ovchinnikov states in a superfluid Fermi gas

Recently, both the experimental and theoretical studies to realize superfluid states in atomic gases have been performed by using a 50–50 mixture of atoms in two hyperfine spin states. We discuss favorable superfluid states in an unequal mixture. Such situation is analogous to a superconducting state in the presence of a magnetic field acting on the electron spins and then the natural candidate for the ground state is known as the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state which is a spatially inhomogeneous superfluid state. We examine the possibility of the FFLO state in two species gas of Fermions confined in a realistic three-dimensional harmonic trap. We propose a clear experimental way to create and directly detect the spatially modulated FFLO state from the macroscopic signature.

Fermiology and Fulde-Ferrell-Larkin-Ovchinnikov Phase Formation

The Bogoliubov–de Gennes equations for singlet superconductivity in an exchange field are analyzed with real materials having complex Fermi surfaces in mind. The resulting gap equation is reformulated in terms of a velocity spectrum on the Fermi surface in which the surface geometry is built in. The resulting analysis can readily be used for arbitrary dispersion relations. Fulde-Ferrell-Larkin-Ovchinnikov FFLO phases are studied in the temperature-field plane, with results providing a physically clear interpretation of why certain directions of pair momentum q are energetically favored. We present clarifying results for models the two-dimensional square Fermi surface, one-, two-, and three-dimensional isotropic surfaces and provide an application to the weak ferromagnetic ZrZn2 showing it is not a favorable case for an FFLO phase.

New high field state of flux line lattice in CeCoIn 5

We have measured the ultrasound velocity of quasi-two-dimensional superconductor CeCoIn 5 with extremely large Pauli paramagnetic susceptibility. The results indicate that the new high field superconducting phase, which is revealed by the recent heat capacity measurements, is characterized by the unusual softening of flux line lattice. The softening is most likely due to the collapse of the flux line lattice tilt modulus and transition to quasi-two-dimensional vortex state. These results provide a strong evidence that the high field phase is the Fulde–Ferrell–Larkin–Ovchinnikov phase, in which the order parameter is spatially modulated and has planar nodes aligned perpendicularly to the vortices.

Transition to Fulde-Ferrell-Larkin-Ovchinnikov phases in three dimensions: A quasiclassical investigation at low temperature with Fourier expansion

We investigate, in three spatial dimensions, the transition from the normal state to the Fulde-Ferrel-Larkin-Ovchinnikov superfluid phases. We make use of a Fourier expansion for the order parameter and the Green's functions to handle the quasiclassical equations in the vicinity of the transition. We show that, below the tricritical point, the transition is always first order. We find that, at the transition, the higher Fourier components in the order parameter are always essentially negligible. Below the tricritical point we have the already known result that the order parameter has a spatial dependence which is essentially $\cos({\bf q}.{\bf r})$. However when the temperature is lowered, the order parameter switches to a sum of two cosines, with equal weigths and wavevector with the same length, but orthogonal directions. Finally by further lowering the temperature, and down to T=0, one finds a another transition toward an order parameter which is the sum of three cosines with again equal weigths and orthogonal directions. Hence the structure of the order parameter gets more complex as the temperature is lowered. On the other hand the resulting critical temperatures are found to be only slightly higher than the ones corresponding to the standard second order FFLO transition. We apply our results to the specific case of ultracold Fermi gases and show that the differences in atomic populations of the two hyperfine states involved in the BCS condensation display sizeable variations when one goes from the normal state to the superfluid FFLO phases, or one FFLO phase to another. Experimentally this should allow to identify clearly the various phase transitions.

Possible Fulde–Ferrell–Larkin–Ovchinnikov inhomogeneous superconducting state in [formula omitted

We report specific heat and thermal conductivity measurements of the heavy fermion superconductor CeCoIn 5 in the vicinity of the superconducting critical field H c 2 , with magnetic field in the plane of this quasi-2D compound and at temperatures down to 50 mK. The superconducting phase transition changes from second to first order for field above 10 T, as evident from a sharp peak in specific heat and a jump in thermal conductivity, indicating the importance of the Pauli limiting effect in CeCoIn 5 . In the same range of magnetic field, we observe a second specific heat anomaly within the superconducting state. We interpret this anomaly as a signature of a Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) inhomogeneous superconducting state. In addition, the thermal conductivity data as a function of field display a kink at a field H k below the superconducting critical field, which closely coincides with the low-temperature anomaly in specific heat, tentatively identified with the appearance of the FFLO superconducting state. Our results indicate that the thermal conductivity is enhanced within the FFLO state, and call for further theoretical investigations of the real-space structure of the order parameter (and in particular, the structure of vortices) and of the thermal transport within the inhomogeneous FFLO state.

Texture in the Superconducting Order Parameter ofCeCoIn5Revealed by Nuclear Magnetic Resonance

We present a ^{115}In NMR study of the quasi two-dimensional heavy-fermion superconductor CeCoIn_5 believed to host a Fulde-Ferrel-Larkin-Ovchinnkov (FFLO) state. In the vicinity of the upper critical field and with a magnetic field applied parallel to the ab-plane, the NMR spectrum exhibits a dramatic change below T*(H) which well coincides with the position of reported anomalies in specific heat and ultrasound velocity. We argue that our results provide the first microscopic evidence for the occurrence of a spatially modulated superconducting order parameter expected in a FFLO state. The NMR spectrum also implies an anomalous electronic structure of vortex cores.

Evidence for the Fulde-Ferrell-Larkin-Ovchinnikov state inCeCoIn5from penetration depth measurements

We report penetration depth and resistivity measurements on the heavy fermion superconductor ${\mathrm{CeCoIn}}_{5}$ using a self-resonant tank circuit based on a tunnel diode oscillator. For magnetic fields applied near parallel to the $ab$ planes and temperatures below 250 mK, two phase transitions were found. The lower field transition, within the superconducting state, is of a second order and we identify it as the transition from the ordinary vortex state to the Fulde, Ferrell, Larkin, Ovchinnikov (FFLO) state. The higher field transition marks the change from the FFLO to the normal state. This higher field transition, ${H}_{c2}$, is of first order up to 900 mK, the highest temperature measured. Our normal-state resistivity measurements at temperatures between 100 and 900 mK suggest that the FFLO state is related to the change of the quasiparticle interaction strength, ${F}_{0}^{a}$.

Current-carrying Andreev bound states in a superconductor-ferromagnet proximity system

We study the ground-state properties of a ferromagnet-superconductor heterostructure on the basis of a quasiclassical theory. We have solved the Eilenberger equations together with Maxwell's equation fully self-consistently and found that due to the proximity effect a Fulde-Ferrel-Larkin-Ovchinnikov (FFLO)-like state is realized in such a system. Moreover, this state has oscillations of the pairing amplitude in either one or two directions, depending on the exchange splitting and thickness of the ferromagnet. In particular, using semiclassical arguments (Bohr-Sommerfeld quantization rule) we show that owing to the presence of the Andreev bound states in the ferromagnet, a spontaneous current in the ground state is generated as a hallmark of the FFLO state in the direction parallel to the interface. We also discuss the effects of the elastic disorder and finite transparency of the interface on the properties of the FFLO state in the system.

Anisotropy of thermal conductivity and possible signature of the Fulde-Ferrell-Larkin-Ovchinnikov state inCeCoIn5

We have measured the thermal conductivity of the heavy-fermion superconductor CeCoIn_5 in the vicinity of the upper critical field, with the magnetic field perpendicular to the c axis. Thermal conductivity displays a discontinuous jump at the superconducting phase boundary below critical temperature T_0 ~ 1 K, indicating a change from a second to first order transition and confirming the recent results of specific heat measurements on CeCoIn_5. In addition, the thermal conductivity data as a function of field display a kink at a field H_k below the superconducting critical field, which closely coincides with the recently discovered anomaly in specific heat, tentatively identified with the appearance of the spatially inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state. Our results indicate that the thermal conductivity is enhanced within the FFLO state, and call for further theoretical investigations of the order parameter's real space structure (and, in particular, the structure of vortices) and of the thermal transport within the inhomogeneous FFLO state.

Upper critical field and Fulde-Ferrell-Larkin-Ovchinnikov state inCeCoIn5

Angle dependent magnetothermal conductivity experiments on ${\mathrm{CeCoIn}}_{5}$ indicate that this compound is a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductor. In this study, the low-temperature behavior of the upper critical field is measured in a single crystal of ${\mathrm{CeCoIn}}_{5}$ along the directions $\stackrel{\ensuremath{\rightarrow}}{H}\ensuremath{\Vert}\stackrel{\ensuremath{\rightarrow}}{a}$ and $\stackrel{\ensuremath{\rightarrow}}{H}\ensuremath{\Vert}\stackrel{\ensuremath{\rightarrow}}{c}.$ The data are compared with model calculations of the upper critical field in a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductor. It is found that the observed ${H}_{c2}(T)$ along $\stackrel{\ensuremath{\rightarrow}}{H}\ensuremath{\Vert}\stackrel{\ensuremath{\rightarrow}}{a}$ is consistent with a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state at low temperatures, $T<0.7\mathrm{K},$ whereas for $\stackrel{\ensuremath{\rightarrow}}{H}\ensuremath{\Vert}\stackrel{\ensuremath{\rightarrow}}{c}$ the FFLO state appears to be absent in ${\mathrm{CeCoIn}}_{5}.$ Furthermore, it is predicted that the quasiparticle density of states in the FFLO state exhibits a complex peak structure which should be observable by scanning tunneling microscopy.

Fulde–Ferrell–Larkin–Ovchinnikov state in d-wave superconductors

Recent reports of anomalous low-temperature properties at high magnetic fields of κ-(BEDT-TFF) 2Cu(NCS) 2 and λ-(BEDTS) 2GaCl 4 have sparked a renewed interest in possible Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states in d-wave superconductors. In contrast to conventional three-dimensional s-wave superconductors, there is evidence that FFLO states are realizable for quasi-two-dimensional anisotropic superconductors in a magnetic field parallel to the conducting plane. In this paper, the quasiparticle density of states N( E) are studied in two-dimensional d xy -wave and d x 2− y 2 -wave superconductors. The energy dependence of N( E) provides clear signatures of the FFLO state.

Magnetic enhancement of superconductivity

Movshovich et al. suggest that our calorimetric observation of the FFLO superconducting phase boundary1 and its angular dependence are inconsistent with their own data reporting the possible presence of an FFLO phase, and that we have incorrectly used magnetothermal measurements to identify the order of the FFLO transition. They also contest that the correlation of the magnetization steps with the calorimetrically identified FFLO superconducting phase of CeCoIn5 is coincidental. We consider each of these points in turn.

FFLO superconducting state in CeCoIn 5

We present specific heat data for the heavy fermion superconductor CeCoIn 5 close to the upper critical field H c2 for magnetic fields applied along the [1 0 0] crystallographic axis. For fields above 10 T , the superconducting phase transition becomes first-order. In the same field range, we observe a second specific heat anomaly within the superconducting state, which we attribute to a Fulde–Ferrell–Larkin–Ovchinnikov state.