# Gap structure in the electron-doped Iron-Arsenide Superconductor Ba(Fe0.92Co0.08)2As2: low-temperature specific heat study

**ABSTRACT** We report the field and temperature dependence of the low-temperature specific heat down to 400 mK and in magnetic fields up to 9 T of the electron-doped Ba(Fe0.92Co0.08)2As2 superconductor. Using the phonon specific heat obtained from pure BaFe2As2 we find the normal state Sommerfeld coefficient to be 18 mJ/mol.K^2 and a condensation energy of 1.27 J/mol. The temperature dependence of the electronic specific heat clearly indicate the presence of the low-energy excitations in the system. The magnetic field variation of field-induced specific heat cannot be described by single clean s- or d-wave models. Rather, the data require an anisotropic gap scenario which may or may not have nodes. We discuss the implications of these results. Comment: New Journal of Physics in press, 10 pages, 5 figures

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**ABSTRACT:**Measurements of London penetration depth are a sensitive tool to study multi-band superconductivity and it has provided several important insights to the behavior of Fe-based superconductors. We first briefly review the "experimentalist - friendly" self-consistent Eilenberger model that relates the measurable superfluid density and structure of superconducting gaps. Then we focus on the \BaFe -derived materials, for which the results are consistent with 1) two distinct superconducting gaps; 2) development of strong in-plane gap anisotropy with the departure from the optimal doping; 3) appearance of gap nodes along the $c-$direction in a highly overdoped regime; 4) significant pair-breaking, presumably due to charge doping; 5) fully gapped (exponential) intrinsic behavior at the optimal doping if scattering is removed (probed in the "self-doped" stoichiometric LiFeAs); 6) competition between magnetically ordered state and superconductivity, which do coexist in the underdoped compounds. Overall, it appears that while there are common trends in the behavior of Fe-based superconductors, the gap structure is non-universal and is sensitive to the doping level. It is plausible that the rich variety of possible gap structures within the general $s_{\pm}$ framework is responsible for the observed behavior.Reports on Progress in Physics 07/2011; 74(12). · 13.23 Impact Factor

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arXiv:1001.4564v1 [cond-mat.supr-con] 25 Jan 2010

Gap structure in the electron-doped Iron-Arsenide

Superconductor Ba(Fe0.92Co0.08)2As2:

low-temperature specific heat study

K. Gofryk1, A. S. Sefat2, E. D. Bauer1, M. A. McGuire2, B. C.

Sales2, D. Mandrus2, J. D. Thompson1and F. Ronning1

1Condensed Matter and Thermal Physics, Los Alamos National Laboratory, Los

Alamos, New Mexico 87545, USA

2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak

Ridge, Tennessee 37831, USA

E-mail: gofryk@lanl.gov

Abstract.

specific heat down to 400 mK and in magnetic fields up to 9 T of the electron-doped

Ba(Fe0.92Co0.08)2As2superconductor. Using the phonon specific heat obtained from

pure BaFe2As2we find the normal state Sommerfeld coefficient to be 18 mJ/mol K2and

a condensation energy of 1.27 J/mol. The temperature dependence of the electronic

specific heat clearly indicate the presence of the low-energy excitations in the system.

The magnetic field variation of field-induced specific heat cannot be described by single

clean s- or d-wave models. Rather, the data require an anisotropic gap scenario which

may or may not have nodes. We discuss the implications of these results.

We report the field and temperature dependence of the low-temperature

PACS numbers: 71.20.Eh, 71.55.Ak, 72.15.Eb, 72.15.Jf, 75.50.Pp

Submitted to: New J. Phys.

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Gap structure in the electron-doped Iron-Arsenide Superconductor

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1. Introduction

The recent discovery of superconductivity in Fe-based pnictides RFeAsO[1, 2] (R-rare

earth) has created a new era in superconductivity research and stimulated a great

interest in these compounds[3]. Subsequently, other types of superconducting materials

containing FeAs layers were discovered such as binary chalocgenides Fe1+xSe[4, 5], so

called ”111” compounds[6, 7] LiFeAs or NaFeAs and 122-systems AFe2As2where A is an

alkaline earth[8, 9, 10]. BaFe2As2, prototypical member of the latter family, crystalizes

with the tetragonal ThCr2Si2-structure type and at ambient pressure exhibits structural

and spin-density-wave (SDW) transitions at about 140 K[11]. Suppression of the SDW

state by either applied pressure[12] or chemical doping[8, 10] results in superconductivity.

Despite a large theoretical and experimental effort (see Ref.[13, 14]) to understand the

nature of the superconductivity in these materials there are still many open questions

that have not been resolved such as the pairing mechanism and the symmetry of

the order parameter.Moreover, the experimental results reported so far are often

contradictory, ranging from nodal to fully gapped isotropic superconductivity. Even

within the Co doped BaFe2As2family the situation is unclear. While surface sensitive

measurements such as ARPES[15] and STM[16] claim fairly isotropic gap values,

penetration depth[17], µSR[18], NMR[19], thermal conductivity[20, 21, 22], specific

heat[23], and Raman scattering[24] argue that an anisotropic gap is necessary, although

the details vary between these measurements as well. Often, the superconducting gap

structure is discussed in terms of the s± model, with a sign reversal of the order

parameter between different Fermi surface sheets[25, 26, 27]. However, the large sample,

family, and doping dependence may favor scenarios where the low energy excitations,

possibly nodal, depend strongly on the particular sample being studied and the probe

used to investigate them (e.g. Ref.[28, 29, 30]).

In this paper we present results of our detailed studies of the specific heat of the

electron-doped Ba(Fe0.92Co0.08)2As2superconductor. By subtracting the lattice contri-

bution (obtained from measurements on non-superconducting samples), we can extract

the full electronic T-dependence. The temperature and magnetic field dependent data

imply an anisotropic gap structure.

2. Experimental details

The large single crystals of Ba(Fe0.92Co0.08)2As2have been grown out of FeAs flux with

the typical size of about 2×1.5×0.2 mm3. The samples crystalize as well-formed plates

with the [001] direction perpendicular to the plane of the crystals. The doping level

was determined by microprobe analysis. More details about the synthesis and char-

acterization of the samples may be found in Ref.[9]. Based on electrical resistivity

measurements, Tcwas established to be 20 K (zero resistance), in agreement with the

heat capacity results presented here. The heat capacity was measured down to 400 mK

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Gap structure in the electron-doped Iron-Arsenide Superconductor

3

and in magnetic fields up to 9 T using a thermal relaxation method implemented in a

Quantum Design PPMS-9 device. All data measured in field were field cooled.

3. Results and discussion

The temperature dependence of the specific heat of Ba(Fe0.92Co0.08)2As2 is shown in

Fig. 1. As can be seen from the figure, a pronounced anomaly of specific heat is observed

at Tc. A magnetic field of 9 T strongly suppresses the anomaly and moves it to lower

temperatures. In general, the total specific heat of any system is the sum of several

different excitations:

Ctot(T) = Cel(T) + Cph(T) + Cmag(T) + ...(1)

where Cel(T), Cph(T) and Cmag(T) describe electronic, lattice and magnetic

contributions to the total specific heat, respectively.

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Figure 1. (Color online) The heat capacity of Ba(Fe0.92Co0.08)2As2 measured in 0

T (circles) and 9 T (squares). The solid line describes the normal state specific heat

(see text). The inset shows the difference between measured C/T and the phonon

contribution. The Tcobtained by entropy balance construction is 20 K.

In order to estimate the phonon contribution in our system, we assume that the

phonon part of the specific heat is independent of doping. Thus, we determine the

phonon specific heat from the parent compound BaFe2As2. BaFe2As2exhibits a spin-

density-wave transition at about 140 K so one may expect the presence of a magnetic

contribution to the low-temperatures specific heat in the system.

inelastic neutron scattering experiments show that, in the ordered state, spin-wave

However, recent

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Gap structure in the electron-doped Iron-Arsenide Superconductor

4

excitations have a gap of about 10 meV (∆ ≈ 116 K)[31]. Consequently, at temperatures

below 40 K, Cmagis negligible, and we separate the contributions to the specific heat of

the parent compound as C = γT + Cph.

Thus, to describe the experimental data of Ba(Fe0.92Co0.08)2As2above Tc= 20 K,

we adjust gamma to obtain the best agreement between the data and C = γT + Cph.

There are four points which give us confidence that our determination of γ and Cphfor

the doped compound is reasonable. (i) the good agreement of C = γT + Cphabove

20 K (see Fig.1). (ii) γ which we obtain in this procedure provides accurate entropy

balance for the electronic specific heat in the superconducting state below Tc. (iii)

the condensation energy which we obtain from the resulting analysis is quantitatively

consistent with other measurements (see below). (iv) Calculations and inelastic x-ray

scattering measurements indicate that the phonons below 10 meV are independent of

dopping[32, 33].

The value of the normal state electronic specific heat may be compared with the

density of states at the Fermi level calculated for pure BaFe2As2. Within a single band

model approximation, γ = 18 mJ/mol K2gives N(EF) to be 7.64 eV−1/f.u. Using LDA

approximation together with GGA-PBE[34] or general potential in LAPW method[35]

the calculated values of N(EF), for the parent BaFe2As2, are 3.93 and 3.06 eV−1/f.u.,

respectively. Thus the mass renormalization is roughly a factor of 2, consistent with

mass renormalization determined by optics[36] and ARPES[37].

At 20 K the specific heat exhibits a jump ∆C/Tc= 24 mJ/mol K2(see the inset

in Fig.1) being consistent with previous reports (Ref.[9, 23, 39, 38]). It is also similar

to ∆C/Tc= 28 mJ/mol K2obtained for Ba0.6K0.4Fe2As2[23, 40]. The size of the jump

at Tcdepends on the details of the superconducting state. Taking γ = 18 mJ/mol K2

the ratio ∆C/Tcγ = 1.33 is very close to, albeit smaller than, the weak-coupling BCS

value 1.43.

Using the normal state specific heat, we can extract the condensation energy

and relate it to the thermodynamic critical field. This quantity can be obtained by

integrating the entropy difference:

U =

?

[Sn(T) − Ss(T)]dT(2)

where Sn and Ss denote entropy at the normal and superconducting state

respectively. In the case of Ba(Fe0.92Co0.08)2As2this analysis give U = 1.27 J/mol and

the thermodynamic critical field Hc= 0.23 T. Using a penetration depth λ = 325 nm

obtained from MFM[41] at a slightly different doping level and consistent with µSR

results[43] and coherence length ξ = 27.6 obtained by STS[16] gives a Ginzburg-Landau

parameter K =

obtain Hc2= 38 T, from the expression Hc2=√2KHc, in reasonable agreement with

the published value of about 40 T[44] and the value of 39 T obtained from the slope of

the upper critical field measured by specific heat and the expression Hc2=0.69dHc2

This provides additional confidence in our phonon substraction

λ

ξ= 118. Using this value and our thermodynamic critical field Hcwe

dTcTc[42].

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Gap structure in the electron-doped Iron-Arsenide Superconductor

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Figure 2.

Ba(Fe0.92Co0.08)2As2. The dashed and solid lines are theoretical curves based on BCS

theory (Eq.3) with a one and two s-wave gaps, respectively (see text).

(Color online) Temperature dependence of the electronic specific of

We begin our investigation on the possible symmetry of the superconducting gap by

examining the temperature dependence of the electronic specific heat. Fig.2 displays the

non-lattice part of the specific heat of Ba(Fe0.92Co0.08)2As2obtained by subtracting the

phonon contribution together with a small Schottky contribution below 1 K (see below).

At low temperatures, a sizeable residual specific heat coefficient γ0= 3.7 mJ/mol K2

is observed in this system. Similar behavior has also been reported in Ba0.6K0.4Fe2As2

(γ0= 7.7 mJ/mol K2)[45] and Ba(Fe1−xCox)2As2(γ0≈ 3 mJ/mol K2for optimal doped

samples)[23]. Interestingly, the sizeable value of the residual specific heat coefficient

also has been observed in superconducting cuprates[46, 47]. For Ba(Fe0.92Co0.08)2As2,

γ0 = 3.7 mJ/mol K2amounts to 20 % of γ. We rule out that γ0 originates from

a non-superconducting portion of the sample based on the fact that x-ray analysis

limits impurity phases to less than 5%, and we observe full diamagnetic shielding from

magnetization measurements. Alternative explanations for the origin of the residual γ0

include pair breaking effects of an unconventional superconductor[56], crystallographic

defects or spin glass behavior.

To fit the Cel/T data in figure 2 we use the BCS expression for specific heat:

CBCS= td

dt

?∞

0

dy

?

−6γ∆0

kBπ

?

[flnf + (1 − f)ln(1 − f)](3)

where t =

T

Tc, f is the Fermi function f =

1

e

E

kBT+1, E =

√ǫ2+ ∆2and y =

ǫ

∆

(see Ref.[48]).

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Gap structure in the electron-doped Iron-Arsenide Superconductor

6

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Figure 3. (Color online) The heat capacity of Ba(Fe0.92Co0.08)2As2plotted as C/T

vs T2with magnetic field applied along the c-axis.

The results of our analysis using a single s-wave gap[49] (dashed blue line) and

two separate s-wave gaps (solid red line) are shown in Fig.2. In an s-wave model a

residual linear term must be an extrinsic contribution to the specific heat, and hence

we subtracted γ0from γ for the purposes of these fits. The normal state Sommerfeld

coefficient γ = (18 - 3.7) mJ/mol K2and Tc= 20 K were held fixed during the fits.

The gap value obtained from the single gap fit is 3 meV and may be compared with

∆0 = 6 meV derived for hole-doped Ba0.6K0.4Fe2As2 with Tc

∆0 = 3 meV and Tc = 20 K gives ∆0/(kBTc) = 1.74, close to the weak coupling

value of 1.76. However, as can be seen from Fig.2 the single gap fit does not describe

the data sufficiently and clearly indicates the presence of low energy excitations in

the system below 8 K. A much better description is obtained by fitting the data to

C = (1-A)CBCS(∆1) + ACBCS(∆2) which gives the solid red line and the parameters

∆1= 1.65 meV, ∆2= 3.75 meV and A = 0.62. While this fit provides a reasonable

description of the data, we emphasize that there are multiple anistropic gap descriptions

that could provide a similarly good fit. From this data alone we cannot determine

whether or not nodes exist.

The low-temperature specific heat of Ba(Fe0.92Co0.08)2As2 measured in several

magnetic fields is presented in Fig.3. Below 1 K an upturn in C/T is observed. Such

behavior has been already observed in Ba(Fe0.92Co0.08)2As2samples grown by In flux as

well as in Ba0.6K0.4Fe2As2and FeSe single crystals[50, 45, 51]. The origin of the anomaly

is most probably related to the presence of a small amount of magnetic impurities. With

increasing magnetic field the upturn shifts to higher temperatures and transforms into

≈ 37 K[45]. Taking

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Gap structure in the electron-doped Iron-Arsenide Superconductor

7

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Figure 4. (Color online) Field-induced change in low temperature specific heat of

Ba(Fe0.92Co0.08)2As2, obtained at 0 K by extrapolating the experimental data of Fig.3

to zero temperatures (see text). The green dotted line and red dashed line represent

field dependencies expected for s-wave and d-wave descriptions, respectively. The blue

solid line is a theoretical curve for anisotropic s-wave superconductors (see text).

a maximum, which gradually broadens and diminishes in magnitude with further rising

field. Such behavior indeed resembles that expected for a degenerate ground state split

due to the Zeeman effect in internal and external magnetic fields. Thus, to avoid the

effects of this magnetic contribution, we rely on a linear extrapolation of the data from

above 1.5 K. The red solid lines in Fig.3 display the linear tendency of low-temperature

specific heat displayed as C/T(T2). We have used it to determine the electronic specific

at 0 K by extrapolating the low-temperature data to zero temperature.

The so obtained ∆γ =

T

derived at 0 K, as described above, as well as at

several other temperatures is presented in Fig.4. Similar data were obtained in Ref.[23].

The data have been presented in the form ∆γ/γ vs H/Hc2with γ = 18 mJ/mol K2and

Hc2= 39 T. In fully gapped superconductors the localized quasiparticle states in vortex

cores result in ∆γ proportional to H/Hc2since the number of vortices is proportional

to the magnetic field. As can be seen from the figure, the specific heat rises much faster

with field than expected for a simple s-wave gap. This could indicate an anisotropic gap

or point to a field dependent coherence length[52, 53]. The latter scenario, however, is

unlikely to be a sole explanation as the temperature dependence indicates low-energy

excitations in the system (see Fig.2). Thus, we further explore the expectation of an

anisotropic gap for an explanation of the ∆γ(H) dependence.

[C(H)−C(0)]

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Gap structure in the electron-doped Iron-Arsenide Superconductor

8

It was shown theoretically[55] by using microscopic quasiclassical theory that

an anisotropic gap structure can display a significant field dependence in ∆γ(H) .

Taking into account this model the experimental data may be well described at low

magnetic field (see the blue solid line in Fig.4) using the gap anisotropy ratio α = 0.5

(∆min/∆max = 0.5), which is in resonable agreement with the two gap fit of the

temperature dependence in Fig.2.

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Figure 5. (Color online) Normalized field-induced change in low temperature specific

heat of Ba(Fe0.92Co0.08)2As2, obtained at 0 K, versus magnetic field. The red solid line

is a fit of Eq.4 to the experimental data. Inset: the same data plotted versus a

with a = 0.5 and Hc2= 40 T.

?

H

Hc2

?1

2

An extreme anisotropic gap is that of a nodal superconductor as occurs in the

cuprates. For clean d-wave superconductors, it has been shown by Volovik[54] that

the quasiparticle excitation spectrum is shifted by the superfluid velocity, resulting in

∆γ ∝

from regions far from the vortex cores and close to the nodes. This situation is presented

in Fig.4 by the red dashed line. As can be seen from the figure ∆γ is not increasing so

strong as expected for clean d-wave superconductors.

Finally, we consider an additional alternative.

observed in Ba(Fe0.92Co0.08)2As2 may also be related to impurity scattering effect

as expected for a dirty d-wave superconductor.

Hirschfeld[56] that in the dirty d-wave limit, ∆γ behaves like HlogH at the lowest fields

(Hc1 ≤ H ≪ Hc2). In this approach the field dependence of field-induced specific

heat may be expressed by[57]:

?H/Hc2. The quasiparticles that contribute to the density of states are coming

The finite density of states γ0

It has been shown by K¨ ubert and

∆γ

γn

= A

?H

B

?

log

?B

H

?

(4)

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Gap structure in the electron-doped Iron-Arsenide Superconductor

9

where A = 0.322?∆0

of the gap and impurity scattering rate, respectively. The solid line in Fig.5 is a fit

of Eq.4 to the experimental data, resulting in the parameter

analysis the value of Hc2has been fixed to 40 T. The inset in Fig.5 shows the data for

Ba(Fe0.92Co0.08)2As2together with a curve calculated for a clean d-wave superconduc-

tor. As may be seen the field dependence of γ(H)/γnas well as the magnitude of the

residual specific heat may be consistent with the dirty d-wave scenario. The point of

these two fits (Fig.4 and 5) is to demonstrate that we can not distinguish between these

two different interpretations. The quality of agreement between the two gap analysis in

Fig.4 and the dirty d-wave analysis of Fig.5 is comparable.

Γ

?1

2, B =

πHc2

2a2, a ≈ 0.5 and ∆0 and Γ denote a maximum

∆0

Γ= 78. During this

From the above analysis, we can determine that a single isotropic s-wave gap is in-

capable of describing the specific heat data. The form of the anisotropic gap, however,

cannot be determined by our data alone. We see that two extreme cases of multi-

band s-wave, and a dirty d-wave scenario are each in reasonable agreement with the

data. Results from NMR[19], thermal conductivity[20, 21, 22], ARPES[15], penetration

depth[17], µSR[18] and Raman[24] also conclude that an anisotropic gap is necessary

to describe their data on Co-doped BaFe2As2, although the extent to which varies from

gapless to mild multiband behavior. A nodal gap imposed by symmetry, as in the case

of the cuprates, is ruled out by the vanishingly small residual linear term of the thermal

conductivity. Hence, the dirty d-wave analysis applied above should not be directly

applicable. However, accidental nodes as anticipated in some spin-fluctuation models

of the pnictides (see Ref.[28, 29, 30]), cannot be ruled out by the thermal conductivity

results[58] and are also consistent with our specific heat data. The accidental node sce-

nario has the favorable aspect that the nodes could be lifted by disorder and/or doping

which would help reconcile some of the seemingly contradictory results[59]. Further

measurements as a function of doping and disorder are necessary to help elucidate the

gap structure, not to mention the variations between different families and dopant atoms.

4. Summary

In summary, we have used low-temperature specific heat and its magnetic field re-

sponse to explore details of the symmetry of the superconducting gap in electron-doped

Ba(Fe0.92Co0.08)2As2 superconductor with Tc = 20 K. Using the phonon part of the

specific heat of pure BaFe2As2, we determine the normal state Sommerfeld coefficient

in Ba(Fe0.92Co0.08)2As2 to be 18 mJ/mol K2. The temperature variation of the elec-

tronic specific heat below Tcmay be well described by the presence of two supercon-

ducting gaps, pointing to complex gap structure in the system. The field-induced low-

temperature specific heat can not be explained by simple clean s- or d-wave descriptions.

Its behavior also indicates a strongly anisotropic gapped superconductor.

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Gap structure in the electron-doped Iron-Arsenide Superconductor

10

Note: During completion of this manuscript we became aware of ref.[60] which

used a similar procedure to determine the phonon contribution to the specific heat of a

similar crystal. The resulting temperature dependence was analyzed within a two band

model with results in good agreement with ours.

4.1. Acknowledgments

Work at Los Alamos National Laboratory was performed under the auspices of the U.S.

Department of Energy, Office of Science and supported in part by the Los Alamos LDRD

program. Research at Oak Ridge National Laboratory is sponsored by the Division of

Material Sciences and Engineering Office of Basic Energy Sciences.

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