Fluctuation conductivity and vortex state in a superconductor with strong paramagnetic pair breaking

The fluctuation conductivity of a moderately clean type II superconductor with strong Pauli paramagnetic pair breaking (PPB) is studied by focusing on the quantum regime at low temperatures and in high magnetic fields. First, it is pointed out that, as the PPB effect becomes stronger, the quantum superconducting fluctuation is generally enhanced so that the renormalized Aslamasov-Larkin (AL) fluctuation conductivity tends to vanish upon cooling above the irreversibility line. Furthermore, by examining other [the DOS and the Maki-Thompson (MT)] terms of the fluctuation conductivity, the field dependence of the resulting total conductivity is found to depend significantly on the type of the vortex lattice (or, glass) ordered state at low temperatures where the strong PPB plays important roles. By comparing the present results on the fluctuation conductivity with insulating and negative magnetoresistance behaviors seen upon entering a PPB-induced novel SC phase of iron selenide (FeSe), it is argued that the vortex matter states of the superconducting order parameter in the second lowest $(n=1)$ Landau level are realized in FeSe in the parallel field configuration in high fields and at low temperatures.

Figure 1

Schematic high field phase diagrams of a type II superconductor with moderately strong PPB and strong fluctuation, where $h=H/\left[T_{c0}\right|d{H}_{c2}(T=T_{c0})/dT\left|\right]$, and $t=T/T_{c0}$. The blue dashed curve with square symbols denotes the mean field $H_{c2}$ curve of the SC order parameter modes in the lowest ($n=0$) LL, while the red dashed curve with circle symbols is the corresponding one of the second lowest ($n=1$) LL modes. The remaining two dashed curves are the first order $H_{c2}$-transition line (blue) and the FFLO transition line (black) for the $n=0$ LL modes in the mean field approximation. In the present situation with the SC fluctuation, all of these four dashed curves are crossover lines, and the real phase transition lines consist of the solid lines. The blue and red solid lines are the vortex glass (VG) transition lines defined in $n=0$ and $n=1$ LLs, respectively, while the black solid line expresses a structural first order transition line between the ordinary triangular pinned vortex solid (or, the vortex glass) in the $n=0$ LL and an anisotropic $n=1$ LL vortex solid [22, 23] which is believed to be the HFSC phase in FeSe in $\mathbf{H}\perp c$. We stress that all of the solid lines in this figure are just sketches of real phase transition lines. Further details are explained in Sec. 4.

Figure 2

Feynman diagrams describing (a) the free energy of the Gaussian SC fluctuation, (b) $Q_{zz}^{\left(\mathrm{AL}\right)}$, (c) the triangular vertex appearing in $Q_{zz}^{\left(\mathrm{AL}\right)}$, (d) $Q_{zz}^{\left(\mathrm{DOS}\right)}{|}_0$, and (e) the corresponding contribution $Q_{zz}^{\left(\mathrm{MT}\right)}{|}_0$ from the MT diagram. A thick solid curve and a double thin solid one denote the electron Green's function and the fluctuation propagator, Eq. (18), respectively.

Figure 3

One example of the diagrams expressing ${\sigma }_{xx}^{\left(\mathrm{VG}\right)}$. A thick dashed line denotes a square average of the pinning potential $U\left(\mathbf{r}\right)$, a rectangle at the center denotes a VG fluctuation propagator, and the triangular vertices at the left and right edges are defined by Fig. 2.

Figure 4

Diagrams derived from Figs. 2 and 2. A thin dashed line carrying the factor $1/\left[2\pi N\right(0\left)\tau \right]$ and connecting two electron Green's functions denotes a square-averaged impurity potential, and the rectangle in (f) denotes a vertex expressing a mutual interaction between the electrons.

Figure 5

Temperature dependencies of the coefficients defined in Eq. (15) at $H=H_{c2}\left(0\right)$, corresponding to $h=0.1806$ in Fig. 1, which are obtained consistently with the $a_n\left(0\right)=0$ lines in Fig. 1 and in terms of the parameter values listed below Eq. (49).

Figure 6

Examples of the field dependencies of the resulting curves of the normalized resistance, Eq. (45) with Eq. (49), at each value of the normalized temperature $t=T/T_{c0}$. We have used $T_{c0}/E_{\mathrm{F}}=0.312$ besides the parameter values used in obtaining Fig. 5.

Figure 7

Temperature dependencies of the coefficients $b_n$ and $\delta b_n$ at $H=H_{c2}\left(0\right)$ obtained in terms of the parameter values $\tilde{J}=0.94$, $\gamma =1.05$, $\tau T_{c0}=50$, and ${\alpha }_{\mathrm{M}}=3.0$.

Figure 8

Examples of the field dependencies of the resulting curves of the normalized resistance, Eq. (45) with Eq. (50), at the normalized temperatures $t=T/T_{c0}=0.05$ (blue square), 0.01 (black solid curve), and 0.005 (red circle). We have used $T_{c0}/E_{\mathrm{F}}=0.712$ besides the parameter values listed in the text and the caption of Fig. 7.

Figure 9

(a) Diagram expressing ${\sigma }_{zz}^{\left(\mathrm{AL}\right)}+{\sigma }_{zz}^{\left(\mathrm{VG}\right)}$ prior to the pinning potential average. A cross symbol denotes a pinning potential. (b) Diagram correcting ${\sigma }_{zz}^{\left(\mathrm{AL}\right)}$ arising from (a) after the pinning potential average. (c) and (d) Diagrams associated with ${\sigma }_{zz}^{\left(\mathrm{VG}\right)}$ arising from (a) after the pinning potential average. (e) Diagram on the VG fluctuation propagator expressed within the ladder approximation. The thick dashed lines in (b), (d), and (e) denote the pinning potential line carrying the strength ${\mathrm{\Delta }}_{\mathrm{P}}$.