Anomalous scaling of ΔC versus Tc in the Fe-based superconductors: the S ± ?> -wave pairing state model

The strong power law behavior of the specific heat jump Δ C ?> versus Tc ( Δ C / T c ∼ T c &agr; , &agr; ≈ 2 ) ?> , first observed by Bud’ko et al (2009 Phys. Rev. B 79 220516), has been confirmed with several families of the Fe-based superconducting compounds with various dopings. We have tested a minimal two band BCS model to understand this anomalous behavior and showed that this non-BCS relation between Δ C ?> versus Tc is a generic property of the multiband superconducting state paired by a dominant interband interaction ( V inter > V intra ?> ) reflecting the relation Δ h Δ e ∼ N e N h ?> near Tc, as in the S ± ?> -wave pairing state. We also found that this Δ C ?> versus Tc power law can continuously change from the ideal BNC scaling to a considerable deviation by a moderate variation of the impurity scattering rate Γ 0 ?> (non-pair-breaking). As a result, our model provides a consistent explanation why the electron-doped Fe-based superconductors follow the ideal BNC scaling very well while the hole-doped systems often show varying degree of deviations.

We also found that this C D versus T c power law can continuously change from the ideal BNC scaling to a considerable deviation by a moderate variation of the impurity scattering rate 0 G (non-pairbreaking). As a result, our model provides a consistent explanation why the electron-doped Fe-based superconductors follow the ideal BNC scaling very well while the hole-doped systems often show varying degree of deviations.

Introduction
The specific heat (SH) jump C D 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 » for a family of doped Ba(Fe x 1-TM x ) 2 As 2 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][3][4][5][6][7][8][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 x 1-K x Fe 2 As 2 for x 0.7 1  < [10] is contrasted to the Na-doped Ba x 1-Na x Fe 2 As 2 ( x 0.1 0.9   ) [8] which displays an excellent BNC scaling. Furthermore, in a more recent measurement [11] on Na-doped K x ( ) D~G p . However, we believe that this result is the consequence of an inconsistent approximation 5 . 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 C D , hence develops a steep variation of C D versus T c . However this theory did not reveal any generic mechanism as to why C T c D follows the BNC scaling T c 2 . Finally, Zannen [16] attributed the origin of C T c 3 D µ to the anomalous temperature dependence of the normal state electronic SH with the scaling form C T n elec 3 µ due to the critical fluctuations near the quantum critical point (QCP). A problem of this theory is that there is no evidence of C T n elec 3 µ 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.

Minimal two band model for the SH jump C D
The SH jump C D 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, C D is due to the opening of the SC gap on Fermi surface and its magnitude is proportional to how rapidly the gap opens 6 . We also note that the Fe-based superconductors are multi-band superconductors, hence the total SH jump C D 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 where the band index 'i' counts the different bands, and N i are the DOSs and i D 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 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 D~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 i D 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. 4 According to the standard single band BCS theory, C T N 0 c t o t ( ) D~. If we adopt this single band BCS formula to understand the typical BNC scaling data, it would require that the total DOS N 0 tot ( ) of Fe-pnictides should change by two orders of magnitude by doping. Although there is a large uncertainty in the experimental data, the variation of N 0 tot ( ) by doping is less than a factor 3 (within the hole and electron doping regions, respectively) [12]. 5 In the strong pair-breaking limit ( Then the SH jump in this limit becomes , quite opposite to the BNC scaling. 6 One assumption for this statement is that fermionic quasiparticles are defined as low energy excitations both at normal state and superconducting state, of course, with different contents for each phase. Then, the general form of entropy can always be written as . This statement is more general than the BCS theory and holds regardless of pairing mechanism. The SH jump C D is derived by taking a differentiation as At present the most widely accepted pairing state in the Fe-based superconductors is the sign-changing Swave state (S  -wave) mediated by a dominant interband repulsive interaction (V V inter 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 i D [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: Having justified the two band model for the study of the SH jump C D 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  -wave state can be studied with the two coupled gap equations [21] where the pair susceptibility χ at T c is defined as ) are all positive and further simplified in this paper as without loss of generality.
In the limit V V 0 intra inter  , equation (2) can be analytically solved and provides the interesting kinematic constraint relation [18]  To calculate the experimental data of C D 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 BaFe 2 As 2 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 N N const. increases with doping 9 . This assumption is consistent with the angle-resolved-photo-emission-spectroscopy measurements of (Ba x 1-K x )Fe 2 As 2 [22] and Ba (Fe x 1-Co x ) 2 As 2 [23]   T c  ). 8 As we mentioned in footnote 4, N tot can change by a factor 3 with dopings in real compounds. In our two band model, changing the value N tot would change the results of C D and T c . However, when we plot C D versus T c , the effect of varying N tot is mostly cancelled in the relation of C D versus T c . 9 For the undoped parent compound BaFe 2 As 2 , our model assumes For the hole doping cases, electron pockets(s) disappears Expanding the gap equation (2) near T c and using equation (4), we obtain Combining the results of equations (4) and (6), equation (1) provides his is our key result of this paper. In contrast to the one band BCS superconductor, equation (7) clearly shows that C T c D can have a strong T c dependence through N N h ē¯e ven with a constant N tot (see equation (5)). (¯¯)~at least for some region of N h,ē , we would obtain the BNC scaling for that region. Having analyzed the simple case (V 0 intra = ), in the next section we numerically study more realistic cases with V 0 intra ¹ and including the impurity scatterings. We solve the coupled gap equation (2) for near T c and directly calculate C D using equation (1). We find that the non-pair-breaking impurity scattering plays a crucial role in order to explain the ideal BNC scaling C T T c c 2 D µ in Ba(Fe x 1-TM x ) 2 As 2 (TM=Co, Ni) as well as a strong deviation in Ba x 1-K x Fe 2 As 2 [10].
. In figure 1(C), we plot C T c D versus T c in log- ). This shows that our results-although our model is a mean field theory-partially capture a strong coupling effect by full numerical calculations of T T c ( )| D . 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 C T c D has changed more than one order of magnitude with N const. tot = for all calculations. However, the BNC scaling behavior is still limited in a region near the higher T c region (where N N N e h |¯¯| D = 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 C D 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 C D but not so much affect T c , then the slope of C T c D 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 V inter .
Phenomenologically we can consider two kind of impurity scattering in the two band model: 0 G (intra-band scattering) and G p (inter-band scattering) 10 . As we assumed the S ± -wave state, G p causes strong pair-breaking effect (e.g. suppression of T c and reduction of h,e D ), while 0 G does not affect the superconductivity itself [24]. . Compared to equation (1), the additional integration part in the above equation contains the thermal average effect including the quasiparticle broadening tot G . And it shows that the non-pair-breaking scattering rate 0 G entering the thermal average part can strongly reduce C D without affecting T c .
In figure 2, we show the numerical results of C T c D versus T c in log-log scale with a choice of a moderate strength of the pairing potentials, V 2.0 inter = and V 0.5 intrā = 11 and varied the impurity scattering rates 0 G , and G p . 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 '´' symbols, 0 shows the T c 2 scaling only for the limited region near the maximum T c and it quickly becomes flattened and slower than T c . Interestingly, this behavior looks very similar to the experimental data of Ba x 1-K x Fe 2 As 2 [10]. Therefore, we speculate that Ba x 1-K x Fe 2 As 2 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 D~, a super-strong scaling. 10 In general, the impurity potential is a momentum dependent function q ( ) G . For the S  -wave pairing state, a large momentum (q Q , ( ) p p = ) scattering part acts as a pair-breaking component G p and the small momentum (q Q  ) scattering part acts as a nonpair-breaking component 0 G . 11 As can be seen in figure 1(c), any choice of values in between V 1. 5 3.0 inter < < would yield qualitatively similar results. Also the dimensionless interaction V V N inter inter tot · = defined in our paper should be compared to a more conventional dimensionless interaction, defined in the T c -equation (see equation (5) . This means that the typical dimensionless coupling strength can be 0.75 Although it is not our main interest in this paper, we have a remark on the absolute magnitude of C T c D which is traditionally compared to the Sommerfeld coefficient normal is a common criterion to indicate the strong coupling superconductors (if R 1.43 > ).
However, reliable measurement of normal g is difficult in most of cases and that was the original reason why Canfield and coworkers [1] 12 , it is notable that the Co-doped Ba(Fe x 1-Co x ) 2 As 2 compounds have uniformly smaller value of R 1.43 < while the K-doped Ba x 1-K x Fe 2 As 2 compounds have values R from slightly to moderately larger than R 1.43 BCS = . As discussed in the next section, this different trend can be understood from the fact that the the K-doping in Ba x 1-K x Fe 2 As 2 does not introduce strong impurity scattering while the Co-doping in Ba(Fe x 1-Co x ) 2 As 2 does introduce strong impurity scattering. And in general the magnitude of C T c D increases with increasing the strength of the pairing interaction (V inter ) due to a strong coupling effect as shown in figure 1(C).

BNC scaling and its deviation
The main message of the numerical calculations in figures 1 and 2 is: (1) the strong non-BCS relation (see D~a with a power α larger than 1 a = (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,    [34,35] 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][2][3][4][5][6][7][8][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 x 1-TM x ) 2 As 2 (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-pairbreaking (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 x 1-K x )Fe 2 As 2 [10], (K x 1-Na x )Fe 2 As 2 [8,11], and (Ba x 1-Na x )Fe 2 As 2 [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 x 1-K x )Fe 2 As 2 [10] shows a very strong deviation from the BNC scaling and the case of (K x 1-Na x )Fe 2 As 2 [8,11] shows a moderate deviation as C T c 2 D~(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 x 1-Na x )Fe 2 As 2 [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 Na 1+ (period 3) and Ba 2+ (period 6) is the largest compared to the other cases where the combinations are between K 1+ (period 4) and Ba 2+ (period 6), and between Na 1+ (period 3) and K 1+ (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 C D 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 C D versus T c into a BNC power law C T c 3 D~. When the impurity scattering is not sufficiently strong (mostly, hole doped systems), a varying degrees of deviation from the ideal BNC scaling would occur.

Coexistence region with the magnetic and SC orders
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 N 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 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 M h,e ( ) c and cuts out some part of DOS from N tot 0 , 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 C D versus T c including the fluctuation effects is beyond the scope of our theory.

Summary and conclusions
We have shown that the puzzling BNC scaling relation C T T  In this sense, the BNC plotting of C D versus T c was an ingenious idea-although it was originally done reluctantly in that way instead of the standard plotting of C C N ele D versus C T N ele 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 N N h ē¯w hich governs both C D and T c (see equations (5) and (7)) and the kinematic constraint near T c . Then a consideration of the non-pairbreaking impurity effect which broadens the quasiparticle spectra near T c explains the systematic evolution from the ideal BNC scaling [1][2][3][4][5][6][7][8][9] to its deviations as found in Ba x 1-K x Fe 2 As 2 [10] and K x 1-Na x Fe 2 As 2 [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.