Jump in specific heat in the presence of a spin-density wave at the superconducting transition in iron pnictides

Many experiments reveal that in iron-based superconductors the jump of the specific heat $\Delta C$ at the superconducting $T_c$ is not proportional to $T_c$, as expected in BCS theory. Rather, $\Delta C/T_c$ varies with $T_c$, and has a peak near optimal doping and decreases at smaller and larger dopings. We show that this behavior can be naturally explained by the interplay between superconductivity and antiferromagnetism. We demonstrate on general grounds that $\Delta C/T_c$ peaks at the doping where the coexistence phase with antiferromagnetism develops, and decreases at deviations from this doping in both directions. Our results are in quantitative agreement with the experiments.

Figure 1

(a) The specific heat $C\left(T\right)$ and SC and SDW order parameters $\Delta$ and $M$ as functions of $T$. We consider the jump of $C\left(T\right)$ at the onset of SC. (b) The behavior of $\Delta C/\left(\gamma T_c\right)$ as a function of ${\delta }_0$, which scales with doping. In a mean-field theory, $\Delta C/T_c$ is discontinuous at the end point of the coexistence state ($P$) and jumps back to the BCS value $\Delta C/T_c\simeq 1.43\gamma$ at larger dopings (dashed horizontal line). Beyond the mean field, paramagnetic fluctuations smear the discontinuity of $\Delta C/T_c$ and transform it into a maximum, as schematically shown by the solid line.

Figure 2

Top: The phase diagram in the $T$-${\delta }_0$ plane for $T_{m,0}/T_{c,0}=2$ and ${\delta }_2/\left(2\pi T_{c,0}\right)=0.4$, 0.28, 0.26, corresponding to wide, medium, and narrow doping ranges of the coexistence phase. The SDW, SC, SDW+SC and normal (N) phases meet at the tetracritical point $P$. Bottom: The behavior of $\Delta C/\gamma T_c$ vs $T_c/T_{c,0}$ in the coexistence region for these ${\delta }_2$. The arrows indicate $T_c$, below which the whole FS is gapped by SDW. As $T_c$ is lowered through this value, $\Delta C/T_c$ decreases as $T_c^2$ at intermediate $T_c$ and exponentially at lower $T$. Near the tetracritical point, $\Delta C/T_c$ may well exceed the BCS value $1.43\gamma$.

Figure 3

Left: The coexistence region (unshaded) in the ${\delta }_2$-${\delta }_0$ plane. Each point corresponds to a particular ratio $T_{m,0}/T_{c,0}$, and this ratio increases monotonically as ${\delta }_0$ grows at fixed ${\delta }_2$. Right: The value of $\Delta C/\gamma T_c$ at the end point of the coexistence region for ${\delta }_2/2\pi T_{c,0}=0.4,0.28,0.2$. The thin solid line is the BCS value $\Delta C/T_c=1.43\gamma$. Diamonds represent the values $\Delta C/\gamma T_c$ at $T_c=T_{c,0}-0$ for the curves for ${\delta }_2/2\pi T_{c,0}=0.4$ and $0.28$ in Fig. 2.

Figure 4

A comparison between the calculated $\Delta C/T_c$ and the experimental data from Ref. 4 for Ba(Fe${}_{0.925}$Co${}_{0.075}$)${}_2$As${}_2$ for dopings below the optimal one. We used $T_{m,0}/T_{c,0}=1.45$ and ${\delta }_2/2\pi T_{c,0}=0.174$.