Fermi-liquid effects in the Fulde-Ferrell-Larkin-Ovchinnikov state of two-dimensional $d$-wave superconductors

We study the effects of Fermi-liquid interactions on quasi-two-dimensional $d$-wave superconductors in a magnetic field. The phase diagram of the superconducting state, including the periodic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in high magnetic fields, is discussed for different strengths of quasiparticle many-body interactions within Landau’s theory of Fermi liquids. Decreasing the Fermi-liquid parameter $F_0^a$ causes the magnetic spin susceptibility of itinerant electrons to increase, which in turn leads to a reduction of the FFLO phase. It is shown that a negative $F_0^a$ results in a first-order phase transition from the normal to the uniform superconducting state in a finite temperature interval. Finally, we discuss the thermodynamic implications of a first-order phase transition for $\mathrm{Ce}\mathrm{Co}{\mathrm{In}}_5$.

Figure 1

(Color online) The phase diagram of a 2D $d_{x^2-y^2}$-wave superconductor for $F_0^a=0.5$ (left-hand panel) and $F_0^a=0.0$ (right-hand panel). For positive $F_0^a$ the LO state is stabilized over a wider range of fields. Note that the energetically unphysical Pauli-limited transition is first order (dotted-dashed black line) at low temperatures, $0<T<T_P<T_{\mathrm{FFLO}}\approx 0.56T_c$. Above $T_P\approx 0.4T_c$ the instability would become second order (dotted line). Without Fermi liquid effects $T_P=T_{\mathrm{FFLO}}$. At the lower critical field $B_{c1}$ a LO state with modulation $\mathbf{q}$ along nodal directions (110) is stablized (solid magenta line). At even higher fields a LO state with $\mathbf{q}\Vert \left(100\right)$ becomes stable (dashed green line).

Figure 2

(Color online) The phase diagram of a 2D $d$-wave superconductor for $F_0^a=-0.5$ with $T_{\mathrm{FFLO}}⪡T_P\approx 0.75T_c$. Note the first-order LO-N transition between $T_{*}<T<T_{\mathrm{FFLO}}$. Inset: sketch of evolution of the order parameter near $B_{c1}$ transition from uniform (USC) to periodic (PER) LO solution at $T∕T_c=0.15$. In a narrow wedge of magnetic fields single (SDW) and double (DDW) domain wall solutions are favored over either USC or PER states.

Figure 3

(Color online) Left-hand panel, the Pauli-limited upper critical field for Fermi-liquid parameters $F_0^a=\{0.5,0.0,-0.5\}$. Right-hand panel, normalized uniform magnetization vs field for temperatures $T∕T_c=\{0.1,0.3,0.5,0.7,0.9\}$ as labeled in the bottom window.