Enhancement of $T_c$ by disorder in underdoped iron pnictide superconductors

We analyze how disorder affects the transition temperature $T_c$ of the $s^{+-}$ superconducting state in the iron pnictides. The conventional wisdom is that $T_c$ should rapidly decrease with increasing inter-band nonmagnetic impurity scattering, but we show that this behavior holds only in the overdoped region of the phase diagram. In the underdoped regime, where superconductivity emerges from a pre-existing magnetic state, disorder gives rise to two competing effects: breaking of the Cooper pairs, which tends to reduce $T_c$, and suppression of the itinerant magnetic order, which tends to bring $T_c$ up. We show that for a wide range of parameters the second effect wins; i.e., in the coexistence state $T_c$ can increase with disorder. Our results provide an explanation for several recent experimental findings and lend additional support to $s^{+-}$ pairing in the iron pnictides.

Figure 1

Temperatures of normal (N)-to-SDW, SDW-to-SC, and N-to-SC transitions as functions of doping for the clean and dirty cases (dashed and solid lines, respectively) for two sets of system parameters. In the underdoped region, where SC emerges from a pre-existing SDW phase, $T_c$ for $s^{+-}$ pairing increases with the concentration of nonmagnetic impurities for one set of parameters (a) and weakly decreases for the other set (b). These two behaviors are consistent with the data in Refs. 9, 11, and 13, respectively. Temperatures and ${\delta }_0$ are measured in units of $T_{c,0}$, which is the SC transition temperature at perfect nesting and without SDW (for pure SDW, the corresponding temperature is $T_{N,0}$). We used in (a) $T_{N,0}/T_{c,0}=2$, ${\delta }_2/\left(2\pi T_{c,0}\right)=0.4$, and impurity-scattering amplitudes ${\Gamma }_0={\Gamma }_{\pi }=0.006\left(2\pi T_{c,0}\right)$; and in (b) $T_{N,0}/T_{c,0}=4$, ${\delta }_2/\left(2\pi T_{c,0}\right)=0.8$, and ${\Gamma }_0={\Gamma }_{\pi }=0.012\left(2\pi T_{c,0}\right)$.

Figure 2

$\Delta T_c=T_c^{\mathrm{dirty}}-T_c^{\mathrm{clean}}$ (in units of $T_c^{\mathrm{clean}}$) as function of the ratio between intraband and interband impurity scattering amplitudes (${\Gamma }_0$ and ${\Gamma }_{\pi }$, respectively). Line SDW${}_{\mathrm{a}}$ is for the parameters of Fig. 1a at a fixed ${\delta }_0/\left(2\pi T_{c,0}\right)=0.2$ and line SDW${}_{\mathrm{b}}$ is for the parameters of Fig. 1b, at a fixed ${\delta }_0/\left(2\pi T_{c,0}\right)=0.6$. For both curves the system is in the coexistence region and $({\Gamma }_0+{\Gamma }_{\pi })/\left(2\pi T_c^{\mathrm{clean}}\right)=0.01$. The lower curve is for a pure superconductor, $M=0$. In the coexistence region, $\Delta T_c$ is definitely positive when ${\Gamma }_{\pi }$ is small. For ${\Gamma }_0\sim {\Gamma }_{\pi }$, the behavior of $\Delta T_c$ is a result of the competition between the direct pair-breaking effect of impurities, which tends to reduce $T_c$, and the suppression of the SDW order parameter, which tends to increase $T_c$.

Figure 3

(a) SDW transition temperature (in units of $T_{N,0}$) as function of doping ${\delta }_0$ (in units of $2\pi T_{N,0}$) for the clean ($\gamma =0$) and dirty cases ($\gamma /2\pi T_{N,0}=0.06\mathrm{and}0.1$). We set ${\delta }_2=0$, but the behavior at a finite ${\delta }_2$ is quite similar. The dashed lines denote metastable solutions of the SDW gap equations, characteristic of the first-order character of the transition. In the dirty case, the second-order SDW transition extends to $T=0$ above a certain threshold of scattering amplitude. (b) Magnetization $M$ (in units of $2\pi T_{N,0}$) as function of temperature for ${\delta }_0/\left(2\pi T_{N,0}\right)=0.1$ and ${\delta }_2/\left(2\pi T_{N,0}\right)=0.2$ [same as in Fig. 1a]. The dashed and solid lines are for the clean and dirty cases, respectively [in the dirty case $\gamma /\left(2\pi T_{N,0}\right)=0.006$].