Multi-gap superconductivity in single crystals of Ba 0.65 Na 0.35 Fe 2 As 2 : A calorimetric investigation
A. K. Pramanik, M. Abdel-Hafiez, S. Aswartham, A. U. B. Wolter, S. Wurmehl, V. Kataev, B. Büchner
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Multi-gap superconductivity in single crystals of Ba . Na . Fe As : A calorimetricinvestigation A. K. Pramanik, ∗ M. Abdel-Hafiez, S. Aswartham, A. U. B. Wolter, S. Wurmehl, V. Kataev, and B. B¨uchner
Institute for Solid State Research, IFW Dresden, D-01171 Dresden, Germany (Dated: November 29, 2018)We investigate the electronic properties and the superconducting gap characteristics of a singlecrystal of hole-doped 122 Fe-pnictide Ba . Na . Fe As by means of specific heat measurements.The specific heat exhibits a pronounced anomaly around the superconducting transition temperature T c = 29.4 K, and a small residual part at low temperature. In a magnetic field of 90 kOe, thetransition is broadened and T c is lowered insignificantly by an amount ∼ − K , being consistent with hole-doped 122 compounds. The temperature-dependent superconducting electronic specific heat cannotbe described with single-gap BCS theory under weak coupling approach. Instead, our analysisimplies a presence of two s -wave like gaps with magnitudes ∆ (0) /k B T c = 1.06 and ∆ (0) /k B T c =2.08 with their respective weights of 48% and 52%. While our results have qualitative similaritieswith other hole-doped 122 materials, the gap’s magnitude and their ratio are quite different. PACS numbers: 74.70.Xa, 74.25.Bt, 65.40.Ba, 74.20.Rp
I. INTRODUCTION
The recent discovery of superconductivity (SC) in Fe-based pnictides has lead to wide research activities inboth experimental and theoretical frontiers of solid statephysics. This is primarily because Fe-pnictides exhibita high transition temperature ( T c ), a layered structure,and a proximity between magnetism and SC in its phasediagram - a scenario reminiscent of cuprates. However,Fe-pnictides are multiband metals where all five Fe-3 d or-bitals contribute to the electronic structure in the vicin-ity of the Fermi surface (FS) having a stark contrast withcuprates which are single band Mott-insulators. Detailedband structure calculations, indeed, show that the FS inFe-pnictides is characterized by two electron-like cylin-ders around the M point, and two hole-like cylinders plusa 3D heavy hole-like pocket around the Γ point, there-after implying it a possible multiband superconductor. As for superconductors in general, the central issueremains to understand the superconducting gap symme-try and the mechanism for Cooper pairing, which arestill under debate in case of Fe-pnictides. With a weakelectron-phonon coupling in this class of materials, theoretical calculations predict unconventional SC me-diated by antiferromagnetic (AFM) spin fluctuations,and an s ± type superconducting gap symmetry wherethe order parameter requires a sign change between dif-ferent sheets of the FS. Experimental findings yetexhibit no consensus on the gap symmetry. For in-stance, nearly isotropic two full gaps with different mag-nitudes are evidenced in angle resolved photoemissionspectroscopy (ARPES) experiments for both electron-and hole-doped 122 Fe-pnictides.
A similar situa-tion is observed in other studies, like, point contact An-dreev reflection spectroscopy (PCARS), and penetra-tion depth measurements. On the other hand, possi-ble existence of nodes is revealed in the temperature ( T )dependence of penetration depth and nuclear magnetic resonance (NMR) measurements for both the 122 and1111 series. It can be mentioned that most of theseinvestigations (except NMR) are surface sensitive, there-fore, sample impurity or inhomogeneity at the surfacemay cause such contradicting results.In this situation, specific heat ( C ) rather provides a keysource of information regarding the bulk thermodynamicproperties, exploring the electronic- and gap-structure inmaterials. In addition, it probes the system in equilib-rium and low energy state. Recent specific heat studiesin different families of Fe-pnictides have explored dissim-ilar gap properties with a single to double gaps and eventhe presence of nodes, and this variation appears relatedto the nature as well as the level of doping. In this contribution we investigate the electronic prop-erties and superconducting gap characteristics in a hole-doped 122 compound, Ba . Na . Fe As ( T c = 29.4 K),by means of specific heat measurements. Although, suchinvestigations have been performed in great details forits K-doped analogues Ba − x K x Fe As , such studies arelacking in the Na-doped 122 family. It is, however, nec-essary to scrutinize how these properties are sensitive todifferent dopant species with unlike sizes and chemistry,considering the fact that the gap properties significantlymodify with the nature of the doping elements. To beprecise, ARPES experiments reveal that for hole-dopedBa . K . Fe As , the average gap for the inner hole-likeand two electron-like cylinders is similar with a largevalue ∼
12 meV, while the outer hole cylinder is hav-ing a lower value ∼ In contrast, for electron-doped BaFe . Co . As , the study shows that the innerhole pocket disappears and the average gaps of compara-ble sizes (6.6 and 5 meV) are observed in the outer holeand two electron cylinders. These observations under-line that detailed investigations are necessary in differentcompositional materials for a generalized understandingof these issues.Our results show a pronounced specific heat anomalyat T c where the jump height is consistent with the T c value according to recent results on Fe-pnictides. Thistransition is minimally suppressed in magnetic fields of90 kOe. Our estimated electronic coefficient in the nor-mal state is high in agreement with other hole-doped 122compounds. Our analysis further shows that the super-conducting electronic specific heat cannot be describedwith the single-band weak-coupling BCS scheme, ratherit implies the presence of two s -wave like gaps with dif-ferent magnitudes and contributions. These results havequalitative similarities with K-doped materials. How-ever, the quantitative difference in the gap ratio mayindicate the different density of states (DOS) in respec-tive bands and the different interband interaction in thesematerials, hence highlighting the element specific role ofthe dopant. II. EXPERIMENTAL DETAILS
Single crystals of Ba . Na . Fe As (BNFA) andBaFe As (BFA) used in the present study have beengrown using a self-flux method. The details ofsample preparation and characterization are describedelsewhere. The parent compound, i.e., BaFe As hasbeen used to estimate the lattice specific heat contri-bution. The crystals have been characterized with x-ray diffraction (XRD) which implies the absence of anychemical impurity phase within the experimental accu-racy. The mentioned chemical compositions of this ma-terial have been determined by energy dispersive anal-ysis of x-ray (EDAX) spectroscopy performed at differ-ent places of the sample. For the sample BNFA, theNa variation in the used piece is found within the in-strumental error limit. A recent study on polycrys-talline Ba − x Na x Fe As shows an unstable crystallo-graphic phase in Na-rich compositions where the mate-rial is susceptible to chemical impurity phases owing tothe large mismatch in the size of Ba and Na ions. In view of this, our results regarding the crystal homo-geneity are remarkable. The BNFA crystal used in thisstudy is about 2.47 × × . The magnetiza-tion ( M ) data have been collected using a SQUID-VSMmagnetometer made by Quantum Design. The heat ca-pacity is measured along the crystallographic c axis witha Physical Property Measurement System (Quantum De-sign) using a thermal relaxation technique down to 1.8 Kand magnetic fields up to 90 kOe. III. RESULTS AND DISCUSSIONS
In Fig. 1a we present the temperature dependenceof the volume susceptibility ( χ vol ) measured followingthe zero-field-cooled (ZFC) and field-cooled (FC) pro-tocols for BNFA. χ vol has been deduced from the dc-magnetization data measured in a field of 20 Oe appliedparallel to the c axis. Care has been taken to correct vo l T (K)
H (||c)= 20 Oe ZFC FC (a) vo l T (K)
H(||ab) = 5 Oe ZFC FC (b) M / H ( e m u cc - O e - ) T (K)
H(||c) = 20 Oe ZFC FC
FIG. 1: (a) The volume susceptibility χ vol after demagnetiza-tion correction has been plotted as function of temperature.The χ vol has been deduced from the dc magnetization mea-sured with H || c = 20 Oe following ZFC and FC protocols forBa . Na . Fe As . The inset shows the similarly deduced χ vol with H || ab plane ( H = 5 Oe). (b) The same data inmain panel of (a) have been plotted without demagnetizationcorrection. the magnetization data for demagnetization effect wherethe demagnetization factor has been estimated from anellipsoidal approximation based on the dimensions of thecrystal. The material exhibits bulk SC which is evidentfrom the diamagnetic signal in the M ZF C curve at lowtemperatures. Although, the ZFC and FC magnetiza-tion already start to bifurcate around 34 K, our materialshows a sharp superconducting transition (width ∼ dM ZF C /dT . However, similarly deduced χ vol with H || ab plane ( H = 5 Oe) shows a clear bifurcation between theZFC and FC magnetization data around 29.5 K, as evi-dent from the inset of Fig. 1a. This difference in onsetof bifurcation between our ZFC and FC magnetizationdata for fields applied along different crystallographic di-rections will be studied in more detail in the future. No-tably, χ vol exhibits an almost full diamagnetic shieldingat low temperatures with fields parallel to both c axisand ab plane. These results are in support of the goodquality of our single crystal. The fact, that M F C > H || c axis within the superconducting state seems tobe an artifact in the data, probably arising from fluxtrapping during the FC process which is likely for thisfield geometry due to layered structure in this material.Fig. 1b shows the same data presented in main panel ofFig. 1a without demagnetization correction, demonstrat-ing anomalies are not significantly evident. We wouldlike to mention that the onset of negative magnetizationat temperatures higher than the sharp transition in Fig.1 with H || c axis is not visible in the specific heat datawhich exhibit a sharp jump around 29.4 K observed inthe M ZF C curve (shown below).The temperature dependence of the specific heat inthe form
C/T vs T is shown in Fig. 2 for BFA in 0 Oeand for BNFA in 0 and 90 kOe. For our further analy-sis, the electronic contribution to the specific heat ( C el )is required for the material under study (BNFA). SinceBNFA is nonmagnetic, the subtraction of the lattice spe- C / T ( m J m o l e - K - ) T (K)
BNFA, H = 0 kOe BNFA, H = 90 kOe BFA, = H = 0 kOe C / T ( m J m o l e - K - ) T (K ) BNFA BFA
FIG. 2: (Color online) Temperature dependence of the specificheat
C/T measured in 0 and 90 kOe for Ba . Na . Fe As and BaFe As . The inset shows the plot C/T vs T . Thestraight lines represent linear fits to C/T = γ + βT (seetext). cific heat ( C ph ) from the total specific heat ( C tot ) willsimply serve our purpose. Conventionally, C ph is esti-mated by suppressing the superconducting transition inhigh magnetic fields. However, the upper critical field( H c ) is significantly high in this class of superconduc-tors. Thus, we have estimated C ph from its parent com-pound BFA, which is not superconducting throughoutthe temperature range, as evident from Fig. 2 where C/T does not exhibit any anomalous behavior as func-tion of temperature. On cooling from room temperature,BFA exhibits a long-range magnetic order of AFM-typepaired with spin density wave (SDW) formation around140 K, implying a likely magnetic contribution toits specific heat. In fact, our specific heat data showa sharp peak around this AFM-SDW transition in BFA(not shown). However, a recent neutron scattering mea-surement has revealed that the energy gap for low-energyspin-wave excitations in the magnetically ordered state isabout 9.8 meV ( ≡
114 K) for this material. Therefore,magnetic contributions to the specific heat will be neg-ligible in the range of our working temperatures ( < C/T around 29.4 K (Fig. 2), which ismarked by the superconducting transition. The temper-ature where this anomaly appears is consistent with thesharp superconducting transition in magnetization mea-surements (see Fig. 1). The jump in the specific heatis reasonably pronounced with a ∆
C/T c value around 84mJ mol − K − which is comparable to other hole (potas-sium) doped 122 compounds with a quantity around 100mJ mol − K − . It is worth to mention that the ob-tained ∆
C/T c for the present material scales well with its T c in perspective of recent results of Fe-pnictides. The T C = 29.4 K r = 3.3 mJ mole -1 K C e l / T ( m J m o l e - K - ) T (K) n = 57.5 mJ mole -1 K T c T (K) n TS e S ( m J m o l e - K - ) FIG. 3: The electronic specific heat C el /T as function of tem-perature for the sample Ba . Na . Fe As . γ n and γ r rep-resent the normal state and residual electronic coefficient ofthe specific heat. The inset shows the entropy in the normaland superconducting state as a function of temperature. specific heat measured in a magnetic field of 90 kOe (Fig.2) shows that the superconducting transition is broad-ened and insignificantly shifted ( ∼ H c estimated tobe above 100 Tesla. In the inset of Fig. 2, specific heat data have beenplotted in the form
C/T vs T for the compounds BFAand BNFA. At low temperature, the data can be linearlyfitted to C/T = γ + βT , where γ and β are the electronicand lattice coefficients of the specific heat. For BFA, weobtain γ = 6.13(8) mJ mol − K − and β = 0.369(7) mJmol − K − . From the obtained β -value, we calculatethe Debye temperature θ D following the relation θ D =[(12 π Rn )/(5 β )] / , where R is the molar gas constant,and n is the number of atoms per formula unit. Thisgives θ D = 297 K for BFA. The extracted value of γ in our crystal is consistent with other studies on singlecrystals of BFA ( ∼ − K − ), and close totheoretically predicted values, i.e., 5.68 mJ mol − K − (Ref. 42) or 7.22 mJ mol − K − (Ref. 43). The fact thatthe specific heat data C/T versus T for BNFA exhibit alinear behavior at low temperatures without any upturndiscards the possibility of Schottky-like contributions inour sample under study.The phononic contribution C ph to the specific heat ofBFA has been determined following the relation C BF Aph = C BF Atot - C BF Ael , where the C BF Ael is γ BF A · T . We findthat the specific heat is dominated by phonons in this re-gion, i.e., around T c , C el is only about 10% of C ph . Using C BNF Ael /T = C BNF Atot /T - f · C BF Aph /T , we can calculate C BNF Ael . The scaling factor f has been introduced dueto slightly different atomic compositions between BNFAand BFA. To determine the value of f , we have used acriterion that normal- and superconducting-state entropyare equal at T c , i.e., R T c ( C el /T ) dT = γ n T c , where γ n isthe normal-state electronic specific heat coefficient. We TABLE I: The superconducting transition temperature T c , the jump height of the electronic specific heat ∆ C el /T c , the normalstate electronic specific heat coefficient γ n , and the superconducting gap properties α i and γ i /γ n extracted from specific heatmeasurements for Ba . Na . Fe As along with other hole- and electron-doped 122 Fe-pnictides. The α i and γ i /γ n representthe zero temperature gap ratio and its weight in the i -th band, respectively.Compounds Ref. T c ∆ C el /T c γ n α i = ∆ i (0) /k B T c , γ i /γ n ( K ) (mJ mol − K − ) (mJ mol − K − )Ba . Na . Fe As a This work 29.4 72.5 57.5 α =1.06, γ /γ n = 0.48 α =2.08, γ /γ n = 0.52Ba . K . Fe As a
22 36.5 98.1 63.3 α =1.945, γ /γ n = 1Ba . K . Fe As b
23 37.3 ∼
100 49 α =2.07, γ /γ n = 1Ba . K . Fe As a
24 38.5 ∼
120 50 α =1.1, γ /γ n = 0.5 α =3.3, γ /γ n = 0.5KFe As b
25 3.5 ∼
21 69.1 α =0.3, γ /γ n = 0.55 α =2.4, γ /γ n = 0.45Ba(Fe . Co . ) As a α =0.95, γ /γ n = 0.33 α =2.2, γ /γ n = 0.67Ba(Fe . Co . ) As a
28 20 ∼
22 18 α =0.957, γ /γ n = 0.38 α =2.175, γ /γ n = 0.62 a Single crystal b Polycrystal started with f = 1 but the entropy conservation criterionis satisfied for f = 0.95 (inset of Fig. 3). This practiceyields T c = 29.4 K. The resulting C el /T for BNFA ispresented in the main panel of Fig. 3. It is obviousfrom the figure that the superconducting transition at T c is reasonably sharp, yielding a jump in C el /T at T c around 72.5 mJ mol − K − . From our determined γ n =57.5 mJ mol − K − , we estimate the universal parame-ter C el /γ n T c = 1.26. This value, however, is lower thanthe weak-coupling BCS prediction of 1.43. Followingthe fact that the superconducting anomaly at T c is rea-sonably sharp in BNFA, therefore a distribution in T c orthe presence of an impurity phase is an unlikely expla-nation for such a reduced value of C el /γ n T c . Instead,we believe that the presence of multiple SC gaps pos-sibly render a low C el /γ n T c in BNFA, as evidenced inother 122 Fe-pnictides. Moreover, the signature ofa multi-gap scenario in BNFA is evidenced by a signif-icant hump around 12 K in our C el /T vs T data (Fig.3), which will be discussed below. Note that C e /T al-most saturates at low temperature, however, it does notextrapolate to zero, yielding a residual electronic specificheat value γ r = 3.3 mJ mol − K − . We mention that thepresence of a finite γ r is common in both electron- andhole-doped 122 crystals, and that the value of γ r in our present case is remarkably low, showing the goodquality of our single crystal. The origin of γ r in BNFAis not clear, however, it may arise due to an incompletetransition to the superconducting state or broken pairs inthe superconducting condensate. Nonetheless, as-suming a superconducting volume fraction ( γ n - γ r )/ γ n ≈ γ n for BNFA is consistentwith other members in the hole-doped 122 series whereasfor electron-doped 122 compounds γ n is much lower (see Table I). Utilizing our value for γ n , we can obtain infor-mation about the normal state electronic properties, i.e.,the DOS at the Fermi energy N ( ǫ F ) of BNFA using therelation: γ n = γ (1 + λ ) , (1) γ = π k B N ( ǫ F ) , (2)where λ is the electron-phonon coupling constant and k B is the Boltzmann constant. Since in the case of Fe-pnictides λ is not significant, we can set γ n ≡ γ . There-fore, γ n is mainly contributed by N ( ǫ F ) which implies ahigher N ( ǫ F ) in hole-doped compounds than in electron-doped ones. From our γ n , we calculate N ( ǫ F ) = 24.37states eV − f.u. − . It is worth to mention that the gen-eral high values of γ n or N ( ǫ F ) for hole-doped 122 Fe-pnictides remains controversial with theoretical calcula-tions yielding γ n = 13.03 mJ mol − K − and N ( ǫ F ) =5.526 states eV − f.u. − for Ba . K . Fe As , where therelated values are only around 20% higher than the par-ent compound. However, another calculation clarifiesthat upon including the band parameters from exper-imental ARPES data as well as mass renormalizationeffects, the calculated γ n is close to the experimentalvalues. Nonetheless, this controversy calls for furtherrigorous theoretical investigations adopting possible rec-onciliations within the experimental findings.After exploring the electronic specific heat and elec-tronic structure, we now examine the superconductinggap properties in BNFA. In many cases, specific heatmeasurements have already been proved to be an effectivetool in understanding the superconducting gap structure C e l - s / n T t (=T/T c ) Experimental data Weak coupling BCS theory = 1.06, / n = 0.48 = 2.08, / n = 0.52 Sum FIG. 4: The normalized superconducting electronic specificheat ( C el − s /γ n T ) of Ba . Na . Fe As as a function of re-duced temperature t = T /T c . The dashed line represents thetheoretical curve based on single-band weak coupling BCStheory with the s -wave gap ∆(0) /k B T c = 1.76 following Eq.4 and 5. The solid lines represent the curves of the two s -wavegap model (see text). and its distributions. For our sample, however, C el first needs to be corrected due to a finite γ r . At low tem-perature C el is assumed to be contributed by the super-conducting ( C el − s ) as well as the non-superconductingnormal ( C n ) parts of the specific heat. While the normalelectrons will have specific heat contributions linear intemperature ( γ r T ) the superconducting electronic con-tribution will be scaled by 1 - γ r /γ n . On this basis, thesum of the individual contributions to C el allows to ex-tract C el − s from the following relation: C el − s /T = γ n γ n − γ r ( C el /T − γ r ) . (3)In Fig. 4 we present the normalized data C el − s /γ n T asa function of reduced temperature t (= T /T c ) for BNFA.As mentioned earlier, C el − s /γ n T exhibits a broad humparound t = 0.4, which implies the presence of multi-ple SC gaps in this compound. We have analyzed ourspecific heat data utilizing the α -model which was origi-nally proposed to account for the thermodynamic proper-ties of a strongly coupled single-gap superconductor un-der semiempirical approach. This model, however, laterhad been generalized to explain the specific heat behaviorin multi-band, multi-gap superconductors, i.e., MgB . Following this model, the thermodynamic properties likethe entropy ( S ) and C can be calculated for a system ofindependent quasiparticles as: Sγ n T c = − π ∆(0) k B T c Z ∞ [ f ln f + (1 − f ) ln(1 − f )] dy, (4) Cγ n T c = t d ( S/γ n T C ) dt , (5)where f = [ exp ( βE ) + 1] − and β = ( k B T ) − . The en-ergy of the quasiparticles is given by E = p [ ǫ + ∆ ( t )],where ǫ is the energy of the normal electrons relative tothe Fermi surface. In Eq. 4, the integration variable y = ǫ/ ∆(0), where ∆(0) is the zero temperature gap magni-tude and the scaled gap α = ∆(0) /k B T c is the only ad-justable fitting parameter. The temperature dependenceof the gap is determined as ∆( t ) = ∆(0) δ ( t ), with δ ( t )being obtained from the table in Ref. 51. In the case oftwo gaps, the thermodynamic properties are determinedas the sum of contributions from the two gaps, i.e., α (= ∆ (0) /k B T c ) and α (= ∆ (0) /k B T c ) with their re-spective weights γ /γ n and γ /γ n respectively, where γ + γ = γ n .Using Eqs. 4 and 5, we first calculate the specificheat C el − s /γ n T as a function of t with α = 1.76 for thesingle-band weak coupling BCS theory. As evident fromFig. 4, the calculations disagree significantly with ourexperimental data, where the former is characterized bya higher jump anomaly at T c . Moreover, an oppositecurvature and different magnitude below and above t ≈ C el − s /γ n T introducing two gaps and their appropriate weights. Ap-parently, values α = 1.06 [∆ (0) = 2.68 meV], γ /γ n =0.48 and α = 2.08 [∆ (0) = 5.27 meV], γ /γ n = 0.52,yield the closest matching with our experimental data(see Fig. 4). The Fig. 4 also shows the C el − s /γ n T vs t plot for an individual α and its weights.For the sake of comparison, we have summarized thevalues of scaled gaps α and α and their respectiveweights, T c , ∆ C el /T c and γ n for Ba . Na . Fe As along with other hole- and electron-doped 122 materialsin Table I. For BNFA, the larger gap α has a higher valuethan the weak-coupling BCS gap value (1.76) while thesmaller one α has a lower value. Although the gap mag-nitudes are scattered for different compounds within theBa-122 family, their relative weights exhibit a consistenttrend. Upon electron doping the smaller gap has around(30 - 40)% contribution to the electronic specific heat,whereas for hole doped compounds both the bands con-tribute almost equally. While our obtained gap structurefor Ba . Na . Fe As has qualitative similarity withother hole doped materials, such as Ba . K . Fe As ,the gap ratio ∆ /∆ differs significantly (Table I).In the scenario of an interband pairing model forFe-pnictides, the gap ratio is predicted as ∆ /∆ = p N /N , where N and N are the Fermi-level DOSin the respective bands, and ∆ /∆ is shown to evolvewith the effective coupling among the bands. Therefore,one can speculate that the DOS in different bands aswell as their coupling modify with K and Na dop-ing. Indeed, K and Na have different ionic sizesand electronic configurations which may contribute dif-ferently to these issues. However, ∆ /∆ ratio impliesthat N /N in Ba . K . Fe As is surprisingly abouttwice higher than in Ba . Na . Fe As , which seems tobe an unlikely situation with just this kind of dopantvariation (see Table 1). On the other hand, the fact that γ ∼ γ ∼ . γ n for both compounds implies that bothgaps open up at the FS with almost equal DOS irrespec-tive of the dopant species. These observations proba-bly suggest that the theoretical discussions in Ref. 52need to include more than two bands. In fact, ARPESresults strongly hint towards the inclusion of at leastfour bands (two hole-like and two electron-like) openingthe superconducting gaps in Ba − x K x Fe As . Also, theneed for four bands to describe the thermodynamic sig-natures has been pointed out in theoretical calculation. At the same time, one can clearly see in Table 1 thatwithin multi-gap analysis the smaller gap α remains al-most close to 1 for all materials (except for the extremelyhole-doped case KFe As ). However, the larger one α ,which appears in the strongly nested inner hole-like andelectron-like bands, varies with both the doping elementand their concentration, illustrating that the nesting con-dition is modified with the doping in Fe-pnictides, whichis quite intriguing.It is worth to mention here that the employed α -modelfollows a simple semiempirical approach where the su-perconducting gap is assumed to have BCS temperaturedependence and the interband coupling is not taken intoaccount. Despite such simplification this model has beenextensively used to analyze the experimental thermody-namic data for many kinds of materials. One, however,certainly needs to check other self-consistent models tocompare the results. In this scenario, within the frame-work of Eliashberg approach for MgB Dolgov et al. has shown that α -model is sufficiently accurate to findthe superconducting gap values though the gap’s par-tial contribution lacks in full agreement. Another re-cently proposed γ -model by Kogan et al. is also aneffective approach which takes into account the inter-band pairing potential and is successfully tested for twoband superconductors MgB and V Si. Our experimen-tal work calls therefore for a detailed theoretical analysisof our data with these and other appropriate models tofully understand the multigap superconducting nature in Fe-pnictides. In addition, considering the fact that su-perconducting gaps estimated by using different exper-imental techniques like ARPES, PCARS, or muon spinrotation ( µ SR) exhibit a wide distribution of absolutevalues, further studies involving specific heat measure-ments are required on doped 122 Fe-pictides with differ-ent kinds of doping elements as well as doping concen-trations to develop a comprehensive understanding anda generalized view on this matter. IV. CONCLUSIONS
In summary, the electronic properties and supercon-ducting gap structure of hole doped 122 Fe-pnictideBa . Na . Fe As are studied by measuring specificheat. A reasonably pronounced anomaly has been foundaround T c = 29.4 K. In applied magnetic fields, the tran-sition becomes broadened, however, there is only a min-imal decrease in T c of about 1.5 K in 90 kOe. Employ-ing an entropy conservation criterion at T c , we extract γ n = 57.5 mJ mole − K − which agrees well with otherhole doped 122 compounds. It is interesting that thetemperature-dependent superconducting electronic spe-cific heat cannot be explained within single-band weak-coupling BCS theory. From our analysis we find thatthe presence of s -wave like two gaps with magnitudes∆ (0) /k B T c = 1.06 and ∆ (0) /k B T c = 2.08 and respec-tive weights of about γ / γ n = 0.48 and γ / γ n = 0.52matches well with our experimental data. Though theseresults are qualitatively similar to K-doped 122 com-pounds, on a quantitative level their gap magnitudes andtheir ratios are quite different. This calls for further stud-ies on materials with different doping levels to reach a fullunderstanding of the gap structure and related mecha-nisms. V. ACKNOWLEDGMENT
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