Anomalous scaling of ΔC versus T_c in the Fe-based superconductors: the ${S}_{\pm }$-wave pairing state model

The specific heat (SH) jump ${\rm{\Delta }}C$ is the most well known thermodynamic signature of the second order phase transition and hence contains the generic information of the transition as well as the material specific information. For example, the BCS theory of superconductivity predicts the universal ratio ${\rm{\Delta }}C/{C}_{\mathrm{el}}{| }_{T={T}_{{\rm{c}}}}=1.43$, hence ${\rm{\Delta }}C/{T}_{{\rm{c}}}=1.43\;\gamma$ is a temperature independent constant and tells us the material specific quantity γ, the Sommerfeld coefficient of the normal state $\gamma =\frac{2{\pi }^{2}}{3}N(0)$ (N(0) density of states (DOSs) at Fermi level). In view of this BCS prediction of ${\rm{\Delta }}C/{T}_{{\rm{c}}}=\mathrm{const}.$, the experimental observation by Bud'ko, Ni, and Canfield (BNC) [1], ${\rm{\Delta }}C/{T}_{{\rm{c}}}\approx {T}_{{\rm{c}}}^{2}$ for a family of doped Ba(Fe${}_{1-x}$TM_x)₂As₂ compounds with TM = Co, Ni is a very intriguing behavior and stimulated active investigations both experimentally and theoretically. After the work of [1], this so-called BNC scaling relation was expanded with an increasing list of the iron pnictide and iron chalcogenide (FePn/Ch) superconducting (SC) compounds [2–9], hence strengthens the speculation that some generic mechanism must exist behind this unusual scaling behavior.

However, recently a few cases—all hole-doped systems—were also reported that they do not follow the ideal BNC scaling behavior. For example, the observation of a strong deviation from the BNC scaling in the K-doped Ba${}_{1-x}$K_xFe₂As₂ for $0.7\lt x\leqslant 1$ [10] is contrasted to the Na-doped Ba${}_{1-x}$Na_xFe₂As₂ ($0.1\leqslant x\leqslant 0.9$) [8] which displays an excellent BNC scaling. Furthermore, in a more recent measurement [11] on Na-doped ${{\rm{K}}}_{1-x}$ Na_xFe₂As₂ the authors claimed that ${\rm{\Delta }}C\sim {T}_{{\rm{c}}}^{2}$ fits the data instead of $\sim {T}_{{\rm{c}}}^{3}$ although data of this compound is limited to a very narrow range of T_c variation. Summarizing the experimental situation, it is fair to say that the large majority of Fe-pnictide superconductors fits the BNC scaling ${\rm{\Delta }}C\sim {T}_{{\rm{c}}}^{3}$ but there exists a small subset of Fe-pnictide systems showing some deviation of varying degrees. In any case, all reported data of the Fe-pnictide superconductors up to now universally exhibit anomalously strong non-BCS dependence⁴ between ${\rm{\Delta }}C\;$ versus T_c and it deserves a theoretical understanding.

For the theoretical investigations, there are three attempted explanations. Kogan [13] argued that strong pair-breaking can cause ${\rm{\Delta }}C/{T}_{{\rm{c}}}\propto {T}_{{\rm{c}}}^{2}$. The essence of this theory is a dimensional counting. The free energy difference near T_c, ${\rm{\Delta }}F={F}_{s}-{F}_{n}$, can be expanded in powers of ${{\rm{\Delta }}}^{2}$ (Δ: the SC order parameter (OP)). In the BCS theory, ${\rm{\Delta }}F\propto -N(0)\frac{{{\rm{\Delta }}}^{4}}{{T}_{{\rm{c}}}^{2}}$ [14]. Using the BCS result of ${{\rm{\Delta }}}^{2}(T)\sim {T}_{{\rm{c}}}^{2}\left(1-\frac{T}{{T}_{{\rm{c}}}}\right)$, we get ${\rm{\Delta }}C/{T}_{{\rm{c}}}\propto \frac{{\partial }^{2}{\rm{\Delta }}F}{\partial {T}^{2}}\sim N(0)$, the well known BCS prediction. In the case of the strong pair-breaking limit, ${{\rm{\Gamma }}}_{\pi }\gg {T}_{{\rm{c}}}$ (${{\rm{\Gamma }}}_{\pi }\;=\;$ pair-breaking rate), considered by Kogan, ${\rm{\Delta }}F\propto -N(0)\frac{{{\rm{\Delta }}}^{4}}{{{\rm{\Gamma }}}_{\pi }^{2}}$ by a dimensional counting. Substituting the same BCS behavior of ${{\rm{\Delta }}}^{2}(T)\propto {T}_{{\rm{c}}}^{2}\left(1-\frac{T}{{T}_{{\rm{c}}}}\right)$, we immediately recover Kogan's result ${\rm{\Delta }}C{/T}_{c}\sim N(0)\frac{{T}_{{\rm{c}}}^{2}}{{{\rm{\Gamma }}}_{\pi }^{2}}\sim {T}_{{\rm{c}}}^{2}$. However, we believe that this result is the consequence of an inconsistent approximation⁵ . The theory of Vavilov et al [15] mainly studied the coexistence region with magnetic order M and SC order Δ. It is a plausible theory that the coexisting magnetic order over the SC order can substantially reduce ${\rm{\Delta }}C$, hence develops a steep variation of ${\rm{\Delta }}C\;$ versus T_c. However this theory did not reveal any generic mechanism as to why ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ follows the BNC scaling $\sim {T}_{{\rm{c}}}^{2}$. Finally, Zannen [16] attributed the origin of ${\rm{\Delta }}C\propto {T}_{{\rm{c}}}^{3}$ to the anomalous temperature dependence of the normal state electronic SH with the scaling form ${C}_{\mathrm{elec}}^{n}\propto {T}^{3}$ due to the critical fluctuations near the quantum critical point (QCP). A problem of this theory is that there is no evidence of ${C}_{\mathrm{elec}}^{n}\propto {T}^{3}$ for a wide doping range of the FePn/Ch superconductors. Moreover all three theories mentioned above are single band theories and do not particularly utilize the unique properties of the FePn/Ch superconductors. In this paper, we propose a theory in which the multi-band nature of the FePn/Ch superconductors is the root cause for producing the BNC scaling behavior.

The SH jump ${\rm{\Delta }}C$ in any 2nd order phase transition is an indication of the entropy change through the phase transition at T_c. For SC transition, regardless of pairing mechanism, ${\rm{\Delta }}C$ is due to the opening of the SC gap on Fermi surface and its magnitude is proportional to how rapidly the gap opens⁶ . We also note that the Fe-based superconductors are multi-band superconductors, hence the total SH jump ${\rm{\Delta }}C$ should be a summation of the SH jump contribution of each band. Therefore, the standard single band formula [17] should be generalized for many bands as

$$\begin{eqnarray}&&{\rm{\Delta }}C=\displaystyle \sum _{i={\rm{h}},{\rm{e}}}{N}_{i}(0){\left.\left(\displaystyle \frac{-{\rm{d}}{{\rm{\Delta }}}_{i}^{2}}{{\rm{d}}T}\right)\right|}_{{T}_{{\rm{c}}}},\end{eqnarray} \tag{ 1 }$$

where the band index 'i' counts the different bands, and N_i are the DOSs and ${{\rm{\Delta }}}_{i}$ are the SC OPs of each band. For the minimal two band model, which will be used in this paper, 'i' counts the hole and electron bands typical in the Fe-based superconductors.

In the one band BCS superconductor, using ${{\rm{\Delta }}}^{2}(T)\sim {T}_{{\rm{c}}}^{2}\left(1-\frac{T}{{T}_{{\rm{c}}}}\right)$, the above equation yields the BCS result ${\rm{\Delta }}C/{T}_{{\rm{c}}}\propto N(0)=\mathrm{const}$. However, in the case of a multiband superconductor, with the same BCS pairing mechanism but generalized to multiband, the standard single band BCS result can strongly deviate for the following reasons. First, each ${{\rm{\Delta }}}_{i}^{2}\sim {a}_{i}{T}_{{\rm{c}}}^{2}$ has non-universal and strongly band dependent coefficient a_i because a_i is determined by the combination of N_i of several bands and the inter- and intra-band pairing interactions V_ij among them. Second, the size of ${{\rm{\Delta }}}_{i}$ and N_i is inversely correlated when the interband pairing is dominant [18], hence equation (1) is not a simple summation of the BCS behavior of many bands. Therefore, the behavior of equation (1) for the multi-band superconductors can reveal more information about the pairing interactions as well as the pairing state.

At present the most widely accepted pairing state in the Fe-based superconductors is the sign-changing S-wave state (S${}_{\pm }$-wave) mediated by a dominant interband repulsive interaction (${V}_{\mathrm{inter}}\gt {V}_{\mathrm{intra}}$) between the hole band(s) around Γ point and the electron band(s) around M point [19]. In real compounds, there exist multiple hole and multiple electron bands with their corresponding SC OPs ${{\rm{\Delta }}}_{i}$ [20] and there should exist a multitude of inter- and intra-band pairing interactions V_ij among them. However, we notice that the SH and T_c, studied in this paper, are the thermodynamically averaged quantities and these two quantities do not sensitively depend on the details of the multiple hole bands and multiple electron bands. Instead, the total SH and T_c are determined by the interaction between the averaged hole bands and the averaged electron bands in the following way: ${N}_{{\rm{h}}}={N}_{{\rm{h}}1}+{N}_{{\rm{h}}2}+\ldots \;$ and ${{\rm{\Delta }}}_{{\rm{h}}}^{2}=[{N}_{{\rm{h}}1}{{\rm{\Delta }}}_{{\rm{h}}1}^{2}+{N}_{{\rm{h}}2}{{\rm{\Delta }}}_{{\rm{h}}2}^{2}+\ldots ]/{N}_{{\rm{h}}}$ (when $T\to 0$) and a similar definition for the electron bands. Therefore, as far as we are interested in the relation between the total SH jump ${\rm{\Delta }}C$ and T_c, the minimal two band model is sufficient and the values of ${N}_{{\rm{h}}({\rm{e}})}$ and ${{\rm{\Delta }}}_{{\rm{h}}({\rm{e}})}$ in our two band model should be understood as the thermodynamically averaged quantities as above⁷ .

Having justified the two band model for the study of the SH jump ${\rm{\Delta }}C\;$ versus T_c, we again emphasize that our purpose of this paper is not to prove any pairing mechanism or model for the Fe-based superconductors but to test a minimal model to see how much we can understand the BNC scaling with it. The essential physics of the two band ${S}_{\pm }$-wave state can be studied with the two coupled gap equations [21]

$$\begin{eqnarray}{{\rm{\Delta }}}_{{\rm{h}}} & = & -[{V}_{\mathrm{hh}}{N}_{{\rm{h}}}\chi ]{{\rm{\Delta }}}_{{\rm{h}}}-[{V}_{\mathrm{he}}{N}_{{\rm{e}}}\chi ]{{\rm{\Delta }}}_{{\rm{e}}},\\ {{\rm{\Delta }}}_{{\rm{e}}} & = & -[{V}_{\mathrm{ee}}{N}_{{\rm{e}}}\chi ]{{\rm{\Delta }}}_{{\rm{e}}}-[{V}_{\mathrm{eh}}{N}_{{\rm{h}}}\chi ]{{\rm{\Delta }}}_{{\rm{h}}},\end{eqnarray} \tag{ 2 }$$

where the pair susceptibility χ at T_c is defined as

$$\begin{eqnarray}&&\chi ({T}_{{\rm{c}}})={T}_{{\rm{c}}}\displaystyle \sum _{n}{\int }_{-{{\rm{\Lambda }}}_{0}}^{{{\rm{\Lambda }}}_{0}}{\rm{d}}\xi \displaystyle \frac{1}{{\omega }_{n}^{2}+{\xi }^{2}}\approx \mathrm{ln}\left[\displaystyle \frac{1.14{{\rm{\Lambda }}}_{0}}{{T}_{{\rm{c}}}}\right],\end{eqnarray} \tag{ 3 }$$

where ${\omega }_{n}=\pi {T}_{{\rm{c}}}(2n+1)$ and ${{\rm{\Lambda }}}_{0}$ is a pairing energy cut-off. The pairing potentials V_ab ($a,b={\rm{h}},{\rm{e}}$) are all positive and further simplified in this paper as ${V}_{\mathrm{he}}={V}_{\mathrm{eh}}={V}_{\mathrm{inter}}$ and ${V}_{\mathrm{hh}}={V}_{\mathrm{ee}}={V}_{\mathrm{intra}}$ without loss of generality.

In the limit ${V}_{\mathrm{intra}}/{V}_{\mathrm{inter}}\to 0$, equation (2) can be analytically solved and provides the interesting kinematic constraint relation [18]

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Delta }}}_{{\rm{h}}}}{{{\rm{\Delta }}}_{{\rm{e}}}}\sim \sqrt{\displaystyle \frac{{N}_{{\rm{e}}}}{{N}_{{\rm{h}}}}}\;\;\;\;\text{as}T\to {T}_{{\rm{c}}},\end{eqnarray} \tag{ 4 }$$

and the critical temperature is given by

$$\begin{eqnarray}&&{T}_{{\rm{c}}}\approx 1.14{{\rm{\Lambda }}}_{0}\mathrm{exp}[-1/({V}_{\mathrm{inter}}\sqrt{{N}_{{\rm{e}}}{N}_{{\rm{h}}}})].\end{eqnarray} \tag{ 5 }$$

To calculate the experimental data of ${\rm{\Delta }}C$ versus T_c for a Fe-122 compound with a series of doping, we need a modeling of doping. First, we notice that the undoped parent compound such as BaFe₂As₂ is a compensated metal, hence has the same number of electrons and holes, i.e. n_h = n_e. Therefore it is a reasonable approximation to take N_h = N_e at no doping and then the doping of holes (K, Na, etc) or electrons (Co, Ni, etc) is simulated by varying N_h and N_e while keeping ${N}_{{\rm{e}}}+{N}_{{\rm{h}}}={N}_{\mathrm{tot}}=\mathrm{const}.$ We have several remarks on this modeling of doping: (1) the assumption ${N}_{\mathrm{tot}}=\mathrm{const}.$ is only for convenience. The sensitive parameters of our model are the relative values between N_e and N_h but not the total DOS N_tot⁸ . Therefore, we use normalized parameters such as ${\tilde{N}}_{{\rm{h}}({\rm{e}})}={N}_{{\rm{h}}({\rm{e}})}/{N}_{\mathrm{tot}}$ and dimensionless coupling constants ${\tilde{V}}_{\mathrm{inter}/\mathrm{intra}}={N}_{\mathrm{tot}}\cdot {V}_{\mathrm{inter}/\mathrm{intra}}$ in our calculations. (2) We do not literally mean ${n}_{{\rm{h}}({\rm{e}})}={N}_{{\rm{h}}({\rm{e}})}$. The actual change of DOS ${N}_{{\rm{h}}({\rm{e}})}$ with doping '$x$' (holes or electrons) should be complicated and also nonlinear. What our model assumes is: ${N}_{{\rm{h}}}\approx {N}_{{\rm{e}}}$ at no doping and the difference of DOSs $| {N}_{{\rm{h}}}-{N}_{{\rm{e}}}|$ increases with doping⁹ . This assumption is consistent with the angle-resolved-photo-emission-spectroscopy measurements of (Ba${}_{1-x}$K_x)Fe₂As₂[22] and Ba(Fe${}_{1-x}$Co_x)₂As₂ [23] which show the systematic changes of hole (electron) FS sizes with dopings. For the rest of this paper, it is convenient to use the normalized DOSs as ${\bar{N}}_{{\rm{h}},{\rm{e}}}={N}_{{\rm{h}},{\rm{e}}}/{N}_{\mathrm{tot}}$ and define the dimensionless coupling constants as ${\bar{V}}_{\mathrm{intra}/\mathrm{inter}}={N}_{\mathrm{tot}}\cdot {V}_{\mathrm{intra}/\mathrm{inter}}$.

Expanding the gap equation (2) near T_c and using equation (4), we obtain ${{\rm{\Delta }}}_{{\rm{h}},{\rm{e}}}(T)$ near T_c as

$$\begin{eqnarray}{{\rm{\Delta }}}_{{\rm{h}}}^{2}(T) & \approx & \displaystyle \frac{2}{1+{N}_{{\rm{h}}}/{N}_{{\rm{e}}}}{{\rm{\Delta }}}_{\mathrm{BCS}}^{2}(T),\\ {{\rm{\Delta }}}_{{\rm{e}}}^{2}(T) & \approx & \displaystyle \frac{2}{1+{N}_{{\rm{e}}}/{N}_{{\rm{h}}}}{{\rm{\Delta }}}_{\mathrm{BCS}}^{2}(T)\end{eqnarray} \tag{ 6 }$$

with ${{\rm{\Delta }}}_{\mathrm{BCS}}^{2}(T)={\pi }^{2}\frac{8}{7\zeta (3)}{T}_{{\rm{c}}}^{2}(1-T/{T}_{{\rm{c}}})$. Combining the results of equations (4) and (6), equation (1) provides

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}C}{{T}_{{\rm{c}}}}\approx 4\times {(3.06)}^{2}{N}_{\mathrm{tot}}\cdot ({\bar{N}}_{{\rm{h}}}{\bar{N}}_{{\rm{e}}}).\end{eqnarray} \tag{ 7 }$$

This is our key result of this paper. In contrast to the one band BCS superconductor, equation (7) clearly shows that ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ can have a strong T_c dependence through ${\bar{N}}_{{\rm{h}}}{\bar{N}}_{{\rm{e}}}$ even with a constant N_tot (see equation (5)). With doping in a given FePn/Ch compound, ${\bar{N}}_{{\rm{h}}}$ and ${\bar{N}}_{{\rm{e}}}(=1-{\bar{N}}_{{\rm{h}}})$ varies over the range of $[0,1]$ (see footnote 8). As such if $({\bar{N}}_{{\rm{h}}}{\bar{N}}_{{\rm{e}}})\sim {T}_{{\rm{c}}}^{2}$ at least for some region of ${\bar{N}}_{{\rm{h}},{\rm{e}}}$, we would obtain the BNC scaling for that region.

Having analyzed the simple case (${V}_{\mathrm{intra}}=0$), in the next section we numerically study more realistic cases with ${V}_{\mathrm{intra}}\ne 0$ and including the impurity scatterings. We solve the coupled gap equation (2) for ${{\rm{\Delta }}}_{{\rm{h}},{\rm{e}}}(T)$ near T_c and directly calculate ${\rm{\Delta }}C$ using equation (1). We find that the non-pair-breaking impurity scattering plays a crucial role in order to explain the ideal BNC scaling ${\rm{\Delta }}C/{T}_{{\rm{c}}}\propto {T}_{{\rm{c}}}^{2}$ in Ba(Fe${}_{1-x}$TM_x)₂As₂ (TM = Co, Ni) as well as a strong deviation in Ba${}_{1-x}$K_xFe₂As₂[10].

In figure 1(A), we calculated T_c versus ${\bar{N}}_{{\rm{h}}}$ of the two band model equation (2) for ${\bar{V}}_{\mathrm{inter}}=1.0$ and 2.0, respectively, with ${\bar{V}}_{\mathrm{intra}}=0$ for both cases. Indeed, the calculated T_c shows a strong dependence on ${\bar{N}}_{{\rm{h}}}$, symmetric with respect to ${\bar{N}}_{{\rm{h}}}=0.5$. We plot the same data as ${\bar{N}}_{{\rm{h}}}\cdot {\bar{N}}_{{\rm{e}}}$ versus T_c in figure 1(B). In the case of ${\bar{V}}_{\mathrm{inter}}=2.0$, we find ${\bar{N}}_{{\rm{h}}}\cdot {\bar{N}}_{{\rm{e}}}\sim {T}_{{\rm{c}}}^{2}$ near the maximum T_c region which is the necessary condition for the BNC scaling from equation (7). It also shows that the overall power of the relation ${\bar{N}}_{{\rm{h}}}\cdot {\bar{N}}_{{\rm{e}}}\sim {T}_{{\rm{c}}}^{\alpha }$ becomes weaker with the weaker pairing potential ${\bar{V}}_{\mathrm{inter}}$. Now we calculate ${\rm{\Delta }}C({\bar{N}}_{{\rm{h}}})$ and ${T}_{{\rm{c}}}({\bar{N}}_{{\rm{h}}})$ from equations (1) and (2). ${\rm{\Delta }}C({\bar{N}}_{{\rm{h}}})$ and ${T}_{{\rm{c}}}({\bar{N}}_{{\rm{h}}})$ are implicitly related through ${\bar{N}}_{{\rm{h}}}\in [0,1]$. In figure 1(C), we plot ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ versus T_c in log–log scale, for different pairing potentials ${\bar{V}}_{\mathrm{inter}}=1.0,1.5,2.0,3.0$ and 5.0, respectively, with ${\bar{V}}_{\mathrm{intra}}=0.0$ for all cases. First, as seen in figure 1(C), we can see the general trend that the region where the BNC scaling ${\rm{\Delta }}C/{T}_{{\rm{c}}}\sim {T}_{{\rm{c}}}^{2}$ holds becomes widened near the maximum T_c region with increasing the pairing potential strength ${\bar{V}}_{\mathrm{inter}}$. With extensive numerical experiments, we found: (1) ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ becomes $\sim {T}_{{\rm{c}}}^{2}$ for the whole region if ${\bar{V}}_{\mathrm{inter}}\gt 5.0$, however, this strength of pairing potential might be unrealistically strong. (2) Including ${V}_{\mathrm{intra}}\ne 0.0$ does not change the general behavior shown in figure 1(C) as long as ${\bar{V}}_{\mathrm{intra}}\lt {\bar{V}}_{\mathrm{inter}}/2$. (3) Finally, as T_c approaches its maximum value (as ${\bar{N}}_{{\rm{h}},{\rm{e}}}\to 0.5$), ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ approaches the BCS limit ($9.36{N}_{\mathrm{tot}}$) for weak coupling (${\bar{V}}_{\mathrm{inter}}\lt 1.0$) and exceeds it for strong coupling (${\bar{V}}_{\mathrm{inter}}\gt 1.0$). This shows that our results—although our model is a mean field theory—partially capture a strong coupling effect by full numerical calculations of ${\rm{\Delta }}(T){| }_{{T}_{{\rm{c}}}}$.

Zoom In Zoom Out Reset image size

Although it is not completely successful to produce the ideal BNC scaling behavior, the results of the generic two band model in figure 1(C) is remarkably non-BCS and very encouraging in that it shows that ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ has changed more than one order of magnitude with ${N}_{\mathrm{tot}}=\mathrm{const}.$ for all calculations. However, the BNC scaling behavior is still limited in a region near the higher T_c region (where ${\rm{\Delta }}N=| {\bar{N}}_{{\rm{e}}}-{\bar{N}}_{{\rm{h}}}|$ is small) and we need an extra mechanism to extend the power law scaling behavior at lower temperatures. We remind the fact that the jump ${\rm{\Delta }}C$ occurs at T_c (a finite temperature) and the impurity scattering effect will be increasingly effective at lower temperatures. In particular, if the impurity scattering should efficiently reduce ${\rm{\Delta }}C$ but not so much affect T_c, then the slope of ${\rm{\Delta }}C/{T}_{{\rm{c}}}\;$ versus T_c would become steeper in this region and the region of the BNC scaling would be widened even with a moderate strength of ${\bar{V}}_{\mathrm{inter}}$.

Phenomenologically we can consider two kind of impurity scattering in the two band model: ${{\rm{\Gamma }}}_{0}$ (intra-band scattering) and ${{\rm{\Gamma }}}_{\pi }$ (inter-band scattering)¹⁰ . As we assumed the S_± -wave state, ${{\rm{\Gamma }}}_{\pi }$ causes strong pair-breaking effect (e.g. suppression of T_c and reduction of ${{\rm{\Delta }}}_{{\rm{h}},{\rm{e}}}$), while ${{\rm{\Gamma }}}_{0}$ does not affect the superconductivity itself [24]. For example, the pair-breaking scattering ${{\rm{\Gamma }}}_{\pi }$ enters the pair-susceptibility ${\chi }_{{\rm{h}},{\rm{e}}}({T}_{{\rm{c}}})={T}_{{\rm{c}}}{\sum }_{n}{\int }_{-{{\rm{\Lambda }}}_{{\rm{h}}i}}^{{{\rm{\Lambda }}}_{{\rm{h}}i}}{\rm{d}}\xi \frac{1}{{\tilde{\omega }}_{n}^{2}+{\xi }^{2}}$ of equation (3) replacing as ${\omega }_{n}\to {\tilde{\omega }}_{n}={\omega }_{n}+{{\rm{\Gamma }}}_{\pi }$. As a result, ${{\rm{\Gamma }}}_{\pi }$ directly affects both T_c and $\left(\frac{-{\rm{d}}{{\rm{\Delta }}}_{i}^{2}}{{\rm{d}}T}\right)$. However, we have already discussed in the Introduction that increasing ${{\rm{\Gamma }}}_{\pi }$ does not help to produce the BNC scaling (see footnote 4). On the other hand, there is another effect of the impurity scattering which does not directly affect the superconductivity, namely the quasiparticle broadening. This quasiparticle broadening is determined by the total scattering rate ${{\rm{\Gamma }}}_{\mathrm{tot}}={{\rm{\Gamma }}}_{0}+{{\rm{\Gamma }}}_{\pi }$ and the SH jump of ${\rm{\Delta }}C$ of equation (1) should take into account of this quasiparticle broadening as follows [25],

$$\begin{eqnarray}&&{\rm{\Delta }}C=\displaystyle \sum _{i={\rm{h}},{\rm{e}}}{N}_{i}{\left.\left(\displaystyle \frac{-{\rm{d}}{{\rm{\Delta }}}_{i}^{2}}{{\rm{d}}T}\right)\right|}_{{T}_{{\rm{c}}}}{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{\rm{d}}x}{2}\left[\displaystyle \frac{1}{{\mathrm{cosh}}^{2}\left(\frac{x}{2}\right)}\right]\frac{{x}^{2}}{{x}^{2}+{(\displaystyle \frac{{{\rm{\Gamma }}}_{\mathrm{tot}}}{{T}_{{\rm{c}}}})}^{2}},\end{eqnarray} \tag{ 8 }$$

where $x=\omega /{T}_{{\rm{c}}}$. Compared to equation (1), the additional integration part in the above equation contains the thermal average effect including the quasiparticle broadening ${{\rm{\Gamma }}}_{\mathrm{tot}}$. And it shows that the non-pair-breaking scattering rate ${{\rm{\Gamma }}}_{0}$ entering the thermal average part can strongly reduce ${\rm{\Delta }}C$ without affecting T_c.

In figure 2, we show the numerical results of ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ versus T_c in log–log scale with a choice of a moderate strength of the pairing potentials, ${\bar{V}}_{\mathrm{inter}}=2.0$ and ${\bar{V}}_{\mathrm{intra}}=0.5$ ¹¹ and varied the impurity scattering rates ${{\rm{\Gamma }}}_{0}$, and ${{\rm{\Gamma }}}_{\pi }$. First, all cases, with and without impurity scattering, show a strong deviation from the standard BCS limit (the horizontal dashed line). This behavior is the generic feature of the multiband superconductors as we explained in the previous section. The case without impurity scattering (red '$\times$' symbols, ${{\rm{\Gamma }}}_{0}$ = ${{\rm{\Gamma }}}_{\pi }=0$), ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ shows the T_c² scaling only for the limited region near the maximum T_c and it quickly becomes flattened and slower than $\sim {T}_{{\rm{c}}}$. Interestingly, this behavior looks very similar to the experimental data of Ba${}_{1-x}$K_xFe₂As₂[10]. Therefore, we speculate that Ba${}_{1-x}$K_xFe₂As₂ system belongs to the clean limit superconductor. This is understandable since K and Ba ions are out of the Fe–As planes and hence the doped K ions would introduce much weaker impurity potentials for the superconductivity. Next, only a small increase of impurity scattering (green '$+$' symbols, ${{\rm{\Gamma }}}_{0}={{\rm{\Gamma }}}_{\pi }=0.02$ in unit of ${{\rm{\Lambda }}}_{{\rm{h}}i}$) immediately changes ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ versus T_c steeper such as ${\rm{\Delta }}C/{T}_{{\rm{c}}}\sim {T}_{{\rm{c}}}^{1.5}$ over the whole T_c range. Further increasing the impurity scattering rate the case with ${{\rm{\Gamma }}}_{0}=0.1$ and ${{\rm{\Gamma }}}_{\pi }=0.05$ (pink '$\blacklozenge$' symbols) displays an ideal BNC scaling ${\rm{\Delta }}C/{T}_{{\rm{c}}}\sim {T}_{{\rm{c}}}^{2}$ for the entire range of T_c. For the demonstration purposes, we also calculated the case with unrealistically large impurity scattering rates, ${{\rm{\Gamma }}}_{0}=0.5$ and ${{\rm{\Gamma }}}_{\pi }=0.05$ (dark yellow '$\bullet$' symbols), which displays ${\rm{\Delta }}C\sim {T}_{{\rm{c}}}^{4}$, a super-strong scaling.

Zoom In Zoom Out Reset image size

Although it is not our main interest in this paper, we have a remark on the absolute magnitude of ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ which is traditionally compared to the Sommerfeld coefficient ${\gamma }_{\mathrm{normal}}$ ($=\frac{2{\pi }^{2}}{3}N(0)$). The BCS value $R=\frac{{\rm{\Delta }}C}{{T}_{{\rm{c}}}}/{\gamma }_{\mathrm{normal}}=1.43$ is a common criterion to indicate the strong coupling superconductors (if $R\gt 1.43$). However, reliable measurement of ${\gamma }_{\mathrm{normal}}$ is difficult in most of cases and that was the original reason why Canfield and coworkers [1] plotted ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ versus T_c instead of the conventional plotting of ${\rm{\Delta }}C/{T}_{{\rm{c}}}\;$ versus ${\gamma }_{\mathrm{normal}}$. In figure 2, the values of ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ in clean limit (${{\rm{\Gamma }}}_{0}=0,{{\rm{\Gamma }}}_{\pi }=0$) have a maximum value slightly above the BCS limit ($1.43\;{\gamma }_{n}=9.36\;{N}_{\mathrm{tot}}$) and other cases with finite impurity scatterings have the reduced values of ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ because the SH jump becomes broadened by the impurity scatterings—both by ${{\rm{\Gamma }}}_{0}$ and ${{\rm{\Gamma }}}_{\pi }$. These values and the trend are not inconsistent with most of experiments. Among the data collected in¹² , it is notable that the Co-doped Ba(Fe${}_{1-x}$Co_x)₂As₂ compounds have uniformly smaller value of $R\lt 1.43$ while the K-doped Ba${}_{1-x}$K_xFe₂As₂ compounds have values R from slightly to moderately larger than ${R}_{\mathrm{BCS}}=1.43$. As discussed in the next section, this different trend can be understood from the fact that the the K-doping in Ba${}_{1-x}$K_xFe₂As₂ does not introduce strong impurity scattering while the Co-doping in Ba(Fe${}_{1-x}$Co_x)₂As₂ does introduce strong impurity scattering. And in general the magnitude of ${\rm{\Delta }}C/{T}_{{\rm{c}}}$ increases with increasing the strength of the pairing interaction (V_inter) due to a strong coupling effect as shown in figure 1(C).

The main message of the numerical calculations in figures 1 and 2 is: (1) the strong non-BCS relation (see footnote 3) of ${\rm{\Delta }}C\sim {T}_{{\rm{c}}}^{\alpha }$ with a power α larger than $\alpha =1$(BCS limit) is a generic feature of the multiband superconductors, and this is due to a pure kinematic constraint of equation (4); (2) however, the system needs some amount of impurity scattering (for example, ${{\rm{\Gamma }}}_{0}/{T}_{{\rm{c}}}^{0}\approx 1/3$, the pink '$\blacklozenge$' symbols in figure 2) to have the observed BNC scaling. If the impurity scattering strength is weaker, a deviation from the BNC scaling becomes stronger, in particular in the low T_c region. These theoretical features can consistently explain why a majority of the Fe-pnictide systems [1–9] follow the BNC scaling but some systems [10, 11] show a deviation from it.

We note that most of data which best fit the BNC scaling are from the electron doped compounds Ba(Fe${}_{1-x}$TM_x)₂As₂ (TM transition metals such as Co, Ni). Electron doping ions, replacing Fe sites, directly enters onto the Fe–As plane so that they introduce more efficient scattering centers (pair-breaking and non-pair-breaking (see footnote 9)). Therefore, the electron doped systems will have a sufficient strength of impurity scattering to produce an ideal BNC scaling as shown in figure 2. On the other hand, in the case of the hole doped systems such as (Ba${}_{1-x}$K_x)Fe₂As₂[10], (K${}_{1-x}$Na_x)Fe₂As₂[8, 11], and (Ba${}_{1-x}$Na_x)Fe₂As₂[8], etc the doped ions enters in between the Fe–As planes so that they will introduce relatively weak scattering centers. Therefore, the hole doped systems would have more chance to be in the weak impurity scattering limit, hence would have more chance to deviate from the ideal BNC scaling. In fact, the case of (Ba${}_{1-x}$K_x)Fe₂As₂[10] shows a very strong deviation from the BNC scaling and the case of (K${}_{1-x}$Na_x)Fe₂As₂[8, 11] shows a moderate deviation as ${\rm{\Delta }}C\sim {T}_{{\rm{c}}}^{2}$ (although this power law has a large uncertainty because it has a very limited data distribution). On the other hand, another hole doped system (Ba${}_{1-x}$Na_x)Fe₂As₂ [8] displays a perfect BNC scaling. According to our theory, we should conclude that Na ions substituting Ba ions should create relatively strong scattering centers although they enter in between the Fe–As planes. This speculation is quite reasonable because the difference of the ion sizes between Na¹⁺ (period 3) and Ba²⁺ (period 6) is the largest compared to the other cases where the combinations are between ${{\rm{K}}}^{1+}$ (period 4) and Ba²⁺ (period 6), and between Na¹⁺ (period 3) and K¹⁺ (period 4).

All these behaviors tell us that the BNC scaling is indeed a reflection of the specific features of the Fe-based superconductors. Namely, the slope of ${\rm{\Delta }}C\;$ versus T_c in the Fe-based superconductors is universally much steeper than the BCS limit due to the generic feature of the multiband superconductors. Then doping inevitably provides a sufficient strength of impurity scattering (in particular, the electron doped cases) which makes the already steep slope of ${\rm{\Delta }}C\;$ versus T_c into a BNC power law ${\rm{\Delta }}C\sim {T}_{{\rm{c}}}^{3}$. When the impurity scattering is not sufficiently strong (mostly, hole doped systems), a varying degrees of deviation from the ideal BNC scaling would occur.

Experiments showed that the BNC scaling continues to be valid even when the spin density wave (SDW) order coexists with the SC order in the underdoped regime [5]. Therefore we would like to extend our model including the magnetic order in the underdoped regime. We consider only the hole doped region ${\bar{N}}_{{\rm{h}}}\in [0.5,1]$, because our model is symmetric with respect to the hole or electron dopings. We took a simple phenomenological approach ignoring the self-consistent calculation of the magnetic and the SC orders. We arbitrarily chose the coexistence region for $0.5\leqslant {\bar{N}}_{{\rm{h}}}\lt 0.7$ (this is the underdoped regime; ${\bar{N}}_{{\rm{h}}}=0.5$ means no doping in our model), just for the sake of demonstration, and introduced the magnetic order $M({\bar{N}}_{{\rm{h}}})$ for this region by hand. This negligence of the self-consistency is a posteriori well justified by the result we found that the BNC scaling relation ${\rm{\Delta }}C/{T}_{{\rm{c}}}\sim {T}_{{\rm{c}}}^{2}$ is insensitive to the details of $M({\bar{N}}_{{\rm{h}}})$.

We assumed that the magnetic OP $M({\bar{N}}_{{\rm{h}}})$ linearly grows from zero at ${\bar{N}}_{{\rm{h}}}=0.7$ to a maximum value M_max at ${\bar{N}}_{{\rm{h}}}=0.5$ as shown in figure 3(A). When a finite M exists, it affects the superconductivity in two important ways: (1) it weakens the SC pair susceptibility and we take the simplest approximation as ${\chi }_{{\rm{h}},{\rm{e}}}(M)={T}_{{\rm{c}}}{\sum }_{n}{\int }_{0}^{{{\rm{\Lambda }}}_{{\rm{h}}i}}{\rm{d}}\xi \frac{2}{{\tilde{\omega }}_{n}^{2}+{\xi }^{2}\quad +\quad {M}^{2}}$ [15]; (2) the presence of SDW order M also removes a part of the FSs. Phenomenologically, we take this effect into account by linearly reducing the total DOS N_tot starting from ${\bar{N}}_{{\rm{h}}}=0.7$ to a maximum reduction at ${\bar{N}}_{{\rm{h}}}=0.5$ as ${N}_{\mathrm{tot}}({\bar{N}}_{{\rm{h}}})={N}_{\mathrm{tot}}^{0}\left[1-a\frac{M({\bar{N}}_{{\rm{h}}})}{{{\rm{\Lambda }}}_{{\rm{h}}i}}\right]$ (a = 0.5 was chosen for calculations in figure 3; for example, N_tot reduced to $\frac{1}{2}{N}_{\mathrm{tot}}^{0}$ at ${\bar{N}}_{{\rm{h}}}=0.5$ with $\frac{{M}_{\mathrm{max}}({\bar{N}}_{{\rm{h}}}=0.5)}{{{\rm{\Lambda }}}_{{\rm{h}}i}}=1.0$). With these phenomenological ansätze, we solved the T_c-equations from equation (2) with fixed pairing interactions and impurity scattering rates (${\bar{V}}_{\mathrm{inter}}=2.0,{\bar{V}}_{\mathrm{intra}}=0.5;$ and ${{\rm{\Gamma }}}_{0}=0.10,{{\rm{\Gamma }}}_{\pi }=0.05$) for three different strengths of $M({\bar{N}}_{{\rm{h}}})$ in figure 3(A). The results qualitatively simulate the experimental phase diagram: T_c starts decreasing when $M({\bar{N}}_{{\rm{h}}})$ starts growing from ${\bar{N}}_{{\rm{h}}}=0.7$ to a maximum value M_max at ${\bar{N}}_{{\rm{h}}}=0.5$ and the reduction of T_c is faster with a stronger magnetic order. For the region where M = 0 for ${\bar{N}}_{{\rm{h}}}\gt 0.7$, the calculations of T_c and ${\rm{\Delta }}C$ are the same as the results in figure 2.

Zoom In Zoom Out Reset image size

In figure 3(B), ${\rm{\Delta }}C/{T}_{{\rm{c}}}\;$ versus T_c is calculated for the corresponding three cases of figure 3(A). The data with dark yellow '$\diamond$' symbols, displaying the ${\rm{\Delta }}C\sim {T}_{{\rm{c}}}^{3}$ BNC scaling, is the same calculation as in figure 2 with ${{\rm{\Gamma }}}_{0}=0.10$ and ${{\rm{\Gamma }}}_{\pi }=0.05$ but only over the region ${\bar{N}}_{{\rm{h}}}\in [0.7,1]$ where M = 0. Then the three solid symbols are the calculation results for the region of ${\bar{N}}_{{\rm{h}}}\in [0.5,0.7]$ with three different strengths of magnetic order $M({\bar{N}}_{{\rm{h}}})$ of figure 3(A). The results of figure 3(B) reveal an interesting behavior. Namely, while it is more natural to expect that the relation between ${\rm{\Delta }}C$ and T_c should dramatically change with and without a magnetic order [15], the calculations of figure 3(B) with the coexisting magnetic and SC orders show that the BNC scaling is robustly obeyed even with widely different strengths of M. We trace the origin of this surprising result to the fact that the underdoped region (i.e. where ${\rm{\Delta }}\bar{N}=| {\bar{N}}_{{\rm{h}}}-{\bar{N}}_{{\rm{e}}}|$ is small and T_c is maximum) is the region where the BNC scaling is best obeyed due to the kinematic constraint of the multiband superconductor when the magnetic order was not considered (see figures 1(c) and 2). Therefore, even if the magnetic order modifies the pair susceptibility ${\chi }_{{\rm{h}},{\rm{e}}}(M)$ and cuts out some part of DOS from N_tot⁰, the generic kinematic constraint (equation (4)) of the multiband superconductor is still operative to enforce the BNC scaling relation.

Finally, we emphasize that our approach here is only a mean field theory and did not include the fluctuation effect of the SDW order which would enhance the SH jump [36]. However, this fluctuations would also affect T_c and fully self-consistent calculations of the scaling relation of ${\rm{\Delta }}C\;$ versus T_c including the fluctuation effects is beyond the scope of our theory.

We have shown that the puzzling BNC scaling relation ${\rm{\Delta }}C/{T}_{{\rm{c}}}\sim {T}_{{\rm{c}}}^{2}$ observed in a wide range of the FePn/Ch SC compounds [1–9] is a manifestation of the generic property of the multiband superconductor paired by a dominant inter-band pairing potential ${V}_{\mathrm{inter}}\gt {V}_{\mathrm{intra}}$. While each of ${\rm{\Delta }}C({N}_{i},{V}_{\alpha },M)$ and ${T}_{{\rm{c}}}({N}_{i},{V}_{\alpha },M)$ have non-trivial and non-universal dependencies on the material specific parameters such as the DOSs (N_i), pairing potentials (${V}_{\alpha }$), competition with SDW (M), etc, these non-universal dependencies are largely cancelled in the plot of ${\rm{\Delta }}C\;$ versus T_c because ${N}_{i},{V}_{\alpha },M$, etc are implicit variables in this plot. As such, the ${\rm{\Delta }}C\;$ versus T_c plot filters out these material specific details and only reveals the universal and generic features of the Fe-based superconductors.

In this sense, the BNC plotting of ${\rm{\Delta }}C\;$ versus T_c was an ingenious idea—although it was originally done reluctantly in that way instead of the standard plotting of ${\rm{\Delta }}C/{C}_{N}^{\mathrm{ele}}\;$ versus ${C}_{N}^{\mathrm{ele}}/{T}_{{\rm{c}}}$ because of the lack of the reliable data of C^ele_N (the normal state electronic SH). In a similar sprit, but in theory, our minimal two band model is a maximally simplified model only to capture the origin of this universal behavior by filtering out (in fact ignoring) the numerous details of the real materials. Using this minimal model, we were able to identify the origin of this universal behavior: the presence of one single parameter ${\bar{N}}_{{\rm{h}}}{\bar{N}}_{{\rm{e}}}$ which governs both ${\rm{\Delta }}C$ and T_c (see equations (5) and (7)) and the kinematic constraint $\frac{{{\rm{\Delta }}}_{{\rm{h}}}}{{{\rm{\Delta }}}_{{\rm{e}}}}\sim \sqrt{\frac{{N}_{{\rm{e}}}}{{N}_{{\rm{h}}}}}$ near T_c. Then a consideration of the non-pair-breaking impurity effect which broadens the quasiparticle spectra near T_c explains the systematic evolution from the ideal BNC scaling [1–9] to its deviations as found in Ba${}_{1-x}$K_xFe₂As₂[10] and K${}_{1-x}$Na_xFe₂As₂[11]. In essence, the BNC scaling and its varying degree of deviations reconfirm the fact that the Fe-based superconductors are the multiband superconductors paired by a dominant interband interaction.

YB was supported by Chonnam National University Grant 2014 and Grant No. 2013-R1A1A2-057535 of the National Research Foundation of Korea. GRS was supported by the US Department of Energy, Office of Basic Energy Sciences, Contract No. DE-FG02-86ER45268.