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resents the residual zero-temperature electronic specific heat and the second term the lattice specific heat. The fitting pa- rameters for zero field are γ (0) = 1 . 2(2) mJ/mol K 2 and β =0.496(1) mJ/mol K $^{4}$. For magnetic field H = 9 T we used the same β and obtained γ (0) = 6 . 4(2) mJ/mol K $^{2}$. These residual specific heat values are the lowest reported for FeAs- based superconductors so far, indicating the superior quality and high purity of our samples. The ratio of the residual elec- tronic specific heat to its normal-state counterpart ( γ N $_{K}$, see below) yields an estimate of the non-superconducting phase fraction γ (0) /γ N K ≈ 1 . 2 / 50 = 2 . 4 %. 

Figure 1b highlights the superconductivity-induced jump anomaly of C$_{p}$ , which exhibits the largest magnitude (125 mJ/mol K $^{2}$) and smallest width ( ≤ 0 . 4 K) reported to-date [2– 4, 10, 11]. An external magnetic field applied perpendicular to the ab -plane gradually suppresses the jump and reduces the transition temperature by 1.4 K for H = 9 T. The transition temperature T$_{c}$ determined as the maximum of the derivative of C$_{p}$ ( T ) decreases linearly with magnetic field with a slope of δH$_{c}$$_{2}$ ( T ) /δT |$_{T}$$_{c}$ = - 6.7 T/K (Fig. 2c), that is consistent with the v alue reported in Ref. 10. This is a factor of two larger than the value obtained from resistivity measurements on samples from the same batch [12]. This difference may be due to flux-flo w effects. 

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In order to reliably extract the electronic contribution, C$_{el}$ ( T ) , from the total measured specific heat, the contri- bution of lattice excitations, C$_{latt}$ ( T ) , has to be accurately determined. The lattice specific heat of the parent com- pound BaFe$_{2}$As$_{2}$ can not be accurately obtained because of the magnetic and structural phase transitions at ∼ 140 K. We found that substitution of 12% Fe by Mn suppresses the spin-density-wave state and does not induce superconductiv- ity. The specific heat of Ba(Fe$_{0}$$_{.}$$_{88}$Mn$_{0}$$_{.}$$_{12}$)$_{2}$As$_{2}$ (red cir- cles in Fig. 1a) was found to be independent of magnetic field. The electronic and lattice specific heat terms extracted from a fit to low-temperature data on this compound (inset in Fig. 1a) are γ (0) = γ N Mn = 14.9(2) mJ/mol K 2 and β$_{Mn}$ = 0.420(4) mJ/mol K $^{4}$, respectively. The lattice specific heat of Ba(Fe$_{0}$$_{.}$$_{88}$Mn$_{0}$$_{.}$$_{12}$)$_{2}$As$_{2}$ , C Mn $_{latt}$( T ) , was obtained by subtract- ing γ N $_{Mn}$T from its total specific heat, C Mn tot ( T ) . The esti- mated Debye temperature is 306 K, somewhat higher than the corresponding value for superconducting Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ (277 K). Since the difference of the lattice parameters of both compounds does not exceed 1.5%, we assume that the phonon contributions to their specific heat and to the entropy obey a law of corresponding states. The normal-state specific heat of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ can then be obtained from the commonly used corresponding states approximation [13]: 

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where γ N K is the Sommerfeld constant in the normal state of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ , and A and B are close to unity. From a least-squares fit of our data to Eq. (1) within the temper- ature range 40 - 150 K (more than 300 data points) under the constraint of entropy conservation, we obtained γ N $_{K}$= 50 mJ/mol K $^{2}$, with A = 0 . 95 and B = 1 . 03 . Figure 2 shows the difference, Δ C$_{el}$ ( T ) /T , between the measured specific heat C$_{p}$ ( T ) /T of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ (Fig. 1a) and its normal- state counterpart C K $_{tot}$( T ) /T estimated from Eq. (1). The 

