Specific heat in strongly hole-doped iron-based superconductors

We compute specific heat $C\left(T\right)$ in a strongly hole-doped Fe-based superconductor, like ${\mathrm{KFe}}_2{\mathrm{As}}_2$, which has only hole pockets. We model the electronic structure by a three-orbital/three-pocket model with two smaller hole pockets made out of $d_{xz}$ and $d_{yz}$ orbitals and a larger pocket made out of $d_{xy}$ orbital. We use as an input the experimental fact that the mass of $d_{xy}$ fermion is several times heavier than that of $d_{xz}/d_{yz}$ fermions. We argue that the heavy $d_{xy}$ band gives the largest contribution to the specific heat in the normal state, but the superconducting gap on the $d_{xy}$ pocket is much smaller than that on $d_{xz}/d_{yz}$ pockets. We argue that in this situation the jump of $C\left(T\right)$ at $T_c$ is determined by $d_{xz}/d_{yz}$ fermions, and the ratio $(C_s-C_n)/C_n$ is a fraction of that in a one-band BCS superconductor. At $T<T_c,C\left(T\right)$ remains relatively flat down to some $T^{*}$, below which it rapidly drops. This behavior is consistent with the data for ${\mathrm{KFe}}_2{\mathrm{As}}_2$ and related materials. We use one-parameter model for the interactions and fix this only parameter by matching the experimental ratio of the gaps on the two $d_{xz}/d_{yz}$ pockets. We argue that the resulting parameter-free model reproduces quantitatively the data on $C\left(T\right)$ for ${\mathrm{KFe}}_2{\mathrm{As}}_2$. We further argue that the very existence of a finite $T^{*}<T_c$ favors $s^{+-}$ gap structure over $d$-wave, because in the latter case $T^{*}$ would almost vanish.

Figure 1

Four different scenarios for the behavior of $C\left(T\right)/T$ in the three-pocket model with light $d_{xz}/d_{yz}$ bands and a heavy $d_{xy}$ band. (a) The specific heat both above and below $T_c$ is determined by the $d_{xy}$ band. (b) The specific heat jump is defined by the gap opening on $d_{xz}/d_{yz}$ bands, but $T$ dependence of $C\left(T\right)$ below $T_c$ is still determined by the $d_{xy}$ band. (c) The specific heat jump at $T_c$ and the behavior at $T_{xy}<T<T_c$ is determined by $d_{xz}/d_{yz}$ bands, while the contribution to $C\left(T\right)$ from the $d_{xy}$ band remains the same as in the normal state (the dashed line). Below $T_{xy}$, the gap on the $d_{xy}$ band becomes larger than $T$, and $C\left(T\right)/T$ rapidly drops. (d) The case when $T_{xy}=0$.

Figure 2

Diagrammatic expressions for Gorkov's gap equations. Triangles with different filling represent SC vertexes on different bands. Solid, dashed, and dotted lines represent $c,d$, and $f$ fermions, respectively. Wavy lines represent interactions between fermions.

Figure 3

The result of the numerical evaluation of the specific heat coefficient $C\left(T\right)/\left(\gamma T\right)$ within our model $\mathrm{[}C\left(T\right)=\gamma T$ above $T_c]$. The dashed line shows $C\left(T\right)/\left(\gamma T\right)$ for $d_{xy}$ orbital in the normal state. The magnitude of the jump of $C\left(T\right)$ at $T_c$ and the overall behavior of $C\left(T\right)/\left(\gamma T\right)$ below $T_c$ agrees well with the experimental data from [25, 28, 29, 30, 43, 44, 45].