Interplay between spin density wave and $\pi$ phase shifted superconductivity in the Fe pnictide superconductors

TL;DR

This study investigates the relationship between spin density wave incommensurability and superconductivity in Fe pnictides, revealing conditions for their coexistence or phase separation and the nature of SDW ordering.

Contribution

It introduces a systematic method combining Ginzburg--Landau analysis and numerical gap equations to map SDW and SC phase boundaries in Fe pnictides.

Findings

IC-SDW and SC coexist for T_c < T^*

SDW is commensurate for T_c > T_M^* and incommensurate otherwise

First order transition occurs between C and IC SDW for large T_M^0

Abstract

We explore if the phase separation or coexistence of the spin density wave (SDW) and superconductivity (SC) states has any relation to the incommensurability of the SDW in the Fe pnictide superconductors. A systematic method of determining the phase separation or coexistence was employed by computing the anisotropy coefficient $\beta$ from the the 4th order terms of the Ginzburg--Landau (GL) expansion of the free energy close to the tricritical/tetracritical point. It was complemented by the self-consistent numerical iterations of the gap equations to map out the boundaries between the phase separation and coexistence of the SDW and SC phases, and between commensurate (C) and incommensurate (IC) SDW in the temperature--doping plane. Our principal results for the sign reversed $s$-wave pairing SC, in terms of the multicritical temperature, $T_c$, the phase separation/coexistence boundary between the SDW and SC, $T^*$, and the boundary between C/IC SDW, $T_M^*$, are: (a) IC-SDW and SC coexist for $T_c < T^*$ and phase separate otherwise, (b) SDW takes the C form for $T_c>T_M^*$ and IC form for $T_c<T_M^*$, and (c) the thermodynamic first order phase transition intervenes in between the C-SDW and IC-SDW boundary for large $T_M^0$, where $T_M^0$ is the SDW transition temperature at zero doping, $T^*=0.35 ~T_M^0$ and $T_M^*=0.56\ T_M^0$. The intervention makes the phase diagram more complicated than previously reported. By contrast no coexistence was found for the equal sign pairing SC. These results will be compared with the experimental reports in the Fe pnictide superconductors.