## Introduction to Inhomogeneous Superconductivity - The FFLO State

#### Chuck Agosta - Clark University

In most cases superconductivity is destroyed in an external magnetic field by the action of vortices - non-superconducting regions containing a magnetic field line shielded by circulating Cooper pairs - which increase in density as the magnetic field is increased and ultimately displace the superconducting phase. Clogston 1 and independently Chandrasekhar 2 , were the first to recognize that if the formation of vortices could be suppressed, superconductivity could persist only to a magnetic field limit determined by the Pauli paramagnetism of the electrons denoted as $H_P$, and properly called the Chandrasekhar-Clogston Pauli paramagnetic limit3. We will refer to this level of magnetic field as the paramagnetic limit for short. At this magnetic field, the energy to flip an electron spin between the up and down states, the Zeeman energy $\mu_bH$, exceeds the binding energy of the Cooper pairs, destroying the pairs. Soon after the Chandrasekhar and Clogston papers, it was found that the upper critical field of a superconductor in the paramagnetic limit changes from a continuous to a first-order transition below $t = T/T_c \sim 0.5$ 4, 5. After further study, it was proposed that if Pauli paramagnetism was the dominant cause for limiting superconductivity and the superconductor was clean (r > 1, where $r = \ell/\xi$ and $\ell$ is the mean free path of the quasiparticles and $\xi$ is the superconducting coherence length) a new kind of superconducting state could be stabilized at a magnetic field above $H_P$. This new superconducting state is the FFLO state and is characterized by Cooper pairs with non-zero momentum, and a spatially modulated order parameter and is commonly called the FFLO state after the theorists (Fulde, Ferrell, Larkin, Ovchinnikov) who predicted it.6,7

During the fifty years since the introduction of this theory, physicists have looked for a superconductor in which the vortex effects are suppressed, so that the spin interaction or paramagnetic limiting dominates the high field physics. Two properties of superconductors can suppress the vortex destruction of superconductivity. If the material is layered, that is, quasi-one- or two-dimensional (Q1D or Q2D), the carriers in certain orientations can not freely move from one layer to another, and the vortices can not form as easily as in an isotropic conductor. In particular, if the effective distance between the conducting layers is large enough (larger than the superconducting coherence length, the nominal size of the vortices), the vortices can fit in the less conducting layers of a Q2D superconductor and only weakly affect the superconducting state. (A similar idea works for a single layer too.8) In highly anisotropic superconductors such as the crystallines or cuprates, the superconducting layers are only coupled by Josephson tunneling, forming weak elongated Josephson vortices. Another way to subdue the orbital motion of the quasiparticles is to increase their effective mass. This effect is well known by many physicists who have measured the slope of the critical field verses temperature in the superconducting phase diagram. This slope is determined by the effective mass of the quasiparticles, and the larger the effective mass, the greater the slope, and the higher the critical fields for a given $T_c$. Given these two suggestions, the search for Pauli limited superconductors has focused on two types of materials; the highly anisotropic organic superconductors and the heavy fermion superconductors. The cuprates, although highly anisotropic, are intrinsically dirty, and therefore are not good candidates.

To illustrate the extent to which vorticies can fit between the layers in an anisotropic crystalline organic superconductor below is an example. The high anisotropy allows vortices to hide between the most conducting layers if a magnetic field is aligned precisely parallel to the layers. In this orientation, the vortices have a diminished influence on the superconductivity [59,60]. As explained above, the vortices become Josephson vortices, weak interlayer vortices that can slide in and out of the material in the spaces between the most conducting layers. The figure shows a vivid illustration of the disappearance of the influence of the vortices. A number of features indicated in the figure such as the critical field, $H_{c2}$, the vortex melting transition ($H_m$)—the kink below $H_{c2}$, the irreversibility transition ($H_{irr}$)—the end of the hysteresis, and, at the lowest temperatures flux jumps, can be seen in most of the traces. The last three features, which all depend directly on vortices, are completely absent when the sample layers are parallel (black trace) to the magnetic field. In this orientation, when Jopsephson vortices are confined to the least conducting layers, $H^o_{0rb}$ is essentially infinite and superconductivity is destroyed by Pauli paramagnetism.

H-T phase diagrams of organic and heavy fermion materials (some from the previous page) with the applied field parallel to the conducting planes. The axis are scaled by Tc and HP so that the different materials can now be compared. Notice how some of the superconductor's critical fields never break through the Pauli limit, while others continue to rise down to the lowest temperatures.

On the next page we will show a cartoon of an inhomogeneous superconductor, and a phase diagram of a possible FFLO state in an organic superconductor.
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1 A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962).
2 B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962).
3 R. A. Klemm, A. Luther, and M. R. Beasley, Phys. Rev. B 12, 877 (1975).
4 Saint-James, D.; Sarma, G.; Thomas, E.J. Type II Superconductivity; Saint-James, D., Sarma, G., Thomas, E.J., Eds.; International Series of Monographs on Natural Philosophy; Pergamon Press: Oxford, NY, USA, Volume 17 (1969).
5 Casalbuoni, R.; Nardulli, G. Inhomogeneous superconductivity in condensed matter and QCD. Rev. Mod. Phys. , 76, 263 (2004).
6 P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964).
7 A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964); or Sov. Phys. JETP 20, 762 (1965).
8 P. M. Tedrow and R. Meservey, Phys. Rev. B 8, 5098 (1973).
Last updated 11 Dec 2018
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