Small anisotropy of the lower critical field and the s±-wave two-gap feature in single-crystal LiFeAs
ABSTRACT The in- and out-of-plane lower critical fields and magnetic penetration depths of LiFeAs were examined. The anisotropy ratio ΓHc1(0) was smaller than the expected theoretical value, and increased slightly with increasing temperature from 0.6Tc to Tc. The small degree of anisotropy was numerically confirmed by considering the electron correlation effect. The temperature dependence of the penetration depths followed a power law (~Tn) below 0.3Tc, with n>3.5 for both λab and λc. Based on theoretical studies of iron-based superconductors, these results suggest that the superconductivity of LiFeAs can be represented by an extended s±-wave due to the weak impurity scattering effect. And the magnitudes of the two gaps were also evaluated by fitting the superfluid density for both the in- and out-of-plane directions to the two-gap model. The estimated values for the two gaps are consistent with the results from angle-resolved photoemission spectroscopy and specific-heat experiments.
arXiv:1007.4906v1 [cond-mat.supr-con] 28 Jul 2010
Small anisotropy of the lower critical fields and two-gap features in single crystal
Yoo Jang Song1, Jin Soo Ghim1, Jae Hyun Yoon1, Kyu Joon Lee2, Myung
Hwa Jung2, Hyo-Seok Ji3, Ji Hoon Shim3, and Yong Seung Kwon1∗
1Department of Physics, Sungkyunkwan University, Suwon 440-746, South Korea
2Initiative Center for Superconductivity, Department of Physics,Sogang University, Seoul 121-741, South Korea
3Department of Chemistry, Pohang University of Science and Technology, Pohang 790-784, South Korea
(Dated: July 29, 2010)
The in- and out-of-plane lower critical fields and magnetic penetration depths for LiFeAs were
examined. The anisotropy ratio, γHc1(0)∼1.2 is smaller than the previous theoretical expected value,
and slightly increase with increasing temperature from 0.6Tc to Tc. This small degree of anisotropy
was numerically confirmed by considering electron correlation. The penetration depths of LiFeAs
follows a power law(∼Tn) below 0.3Tc, with n>3.5 for both λab and λc. These results imply that
the superconductivity of LiFeAs could be represented by an extended s±-wave due to weak impurity
scattering effect. And two superconducting gaps for in- and out-of-plane were found by fitting the
superfluid density of LiFeAs, respectively. The estimated values for the two gaps are consistent with
the results of angle resolved photoemission spectroscopy.
PACS numbers: 74.20.Rp, 74.25.Op, 74.70.Dd
Since the discovery of Fe-based superconductors, much
effort has been devoted to identifying a higher transi-
tion temperature (Tc) and to investigating the mecha-
nism of the superconducting (SC) behavior. As a result,
the number of reports related to the symmetry of the su-
perconducting order-parameter (OP) has increased. The
SC gap symmetries for 1111-type and 122-type SCs have
been studied mostly by estimating the magnetic pen-
etration depth, one of the most useful probes of OP
symmetry, using the tunneling diode resonator (TDR)
technique [1–3], the microwave method [4–6], or scan-
ning SQUID susceptometry [7, 8]. Previous studies on
the magnetic penetration depth in SmFeAsO1.8F0.2 ,
PrFeAsO1−y , and Ba1−xKxFe2As2  showed expo-
nential temperature dependence supporting the existence
of a s-wave-type SC gap, while those for LaFePO  ex-
hibited a near linear temperature dependence support-
ing a nodal gap. Recent reports on Fe-based SCs, how-
ever, revealed power law behaviors with an exponent of
2≤n≤2.5 at low temperature [2, 9]. Nevertheless, the
various symmetric properties of the SC gaps such as
an extended s-wave , an s±-wave [10–12], a nodal d-
wave [7, 13], and a point nodal gap  have been argued.
Although the question of OP symmetry for Fe-pnictide
SCs still remains, the s±-wave model is more favorable.
In addition, recent advances in theoretical studies [14, 15]
which consider an impurity scattering effect in the Eliash-
berg equation have provided convincing evidence that
the s±-wave SC state is the most promising candidate
for the true pairing state of Fe-pnictide SC. Moreover,
experimental measurements on Fe-based SCs have re-
vealed small anisotropy ratios due to relatively strong
interplane coupling [16–18] in contrast to cuprates with
a large anisotropy. According to first principles calcu-
lations for LiFeAs compounds, the anisotropy ratios of
γλ(0) and γρ(0), estimated using the magnetic penetra-
tion depth and the electrical resistivity are approximately
3 and 9, respectively . Furthermore, some theoretical
works have implied the importance of electron correla-
tions  and their effects on the unconventional super-
conductivity [21, 22] of Fe pnictide SC.
In this letter, we report that the magnetic anisotropy
ratioof single crystalLiFeAs
smaller than the previous theoretical expected value of
∼3 . The small anisotropy ratio was confirmed by
considering the electron correlation effect on the band
structure using the local density approximation (LDA)
and dynamical mean field theory (DMFT) approach. We
also show that both magnetic penetration depths, λab
for H?c and λc for H?ab, follow a power law (∼Tn)
with n>3.5 below 0.3Tc. These results imply that the
superconductivity of LiFeAs could be represented by an
extended s±-wave model due to a weak impurity scat-
tering effect.Through fittings to superfluid density
for the in- and out-of-plane directions (as in Ref. 
for Ba1−xKxFe2As2), it was revealed that LiFeAs pos-
sesses two SC gaps. Recently, the s-wave two-gap fea-
ture was examined using angle resolved photoemission
spectroscopy (ARPES) , specific heat , and lower
critical field  experiments.
group  reported lower critical field, anisotropy, and
two-gap features of LiFeAs using a method similar to
that in Ref. . In our study, however, the temperature
dependence of γHc1due to the multiband effect, the SC
gap symmetry and the isotropic properties of the gaps
were novel, and the measured small anisotropy ratio was
also confirmed using the LDA+DMFT approach consid-
ering the electron correlation effect.
In particular, the Chu
The single crystal growth of LiFeAs with Tzero
FIG. 1: (Color online) The initial region of the magnetiza-
tion curve M(H) of single crystal LiFeAs for H?ab at various
temperatures. Inset : The same in-field magnetization data
subtracted from the meissner line. The dashed line in the
inset represents 3×10−4emu.
K is detailed Ref. . The single crystal used in this
study was obtained from another batch of the same ingot,
referred to in Ref. . The magnetic field dependence
of the magnetization, M(H), was scanned from -70 to
70 kOe from 2 to 7 K at 0.5 K intervals and from 7 to
Tcat 1 K intervals for H?ab and H?c, using a vibrating
sample magnetometer (VSM SQUID, Quantum Design).
The curves of magnetization, M(H), for H?ab be-
low Tc are shown in Figure 1.
shows a similar behavior(data not shown). The value
c1is evaluated from the magnetic field deviat-
ing from the Meissner line on the initial slope of the
M(H) curve.The inset of Figure 1 reveals how one
may determine H∗
emu at different temperatures.
Hc1 is evaluated by excluding the demagnetizing effect
that arises from orbital motions.
ple, Hc1 is given by Hc1=H∗
a and b are the width and thickness of the sample,
respectively .Here, a=2.8 mm and b=1.3 mm
for H?c, while a=1.3 mm and b=2.8 mm for H?ab.
These values yield H?c
respectively.To estimate the penetration depth, we
used the London formulas,
quantum and Ginzberg-Landau parameter κc=λab/ξab,
κab=?λabλc/ξabξc. Figure 2 shows the extracted Hc1as
a function of T of LiFeAs for H?ab and H?c. Hc1(0) is
evaluated via the extrapolation of Hc1(T) in the low tem-
perature regions. As shown in Fig. 2, the flat behavior
below 0.2Tcand the existence of an inflection point near
0.5Tc suggest that LiFeAs is a superconductor with an
s-wave symmetry and a two-gap structure. The inset of
Fig. 2 shows that the anisotropy of Hc1, γHc1=H?c
The M(H) for H?c
c1using the criterion of ∆M=3×10−4
The absolute value of
For a platelet sam-
c1=(Φ0/4πλabλc)lnκab, where Φ0 is the flux
FIG. 2: (Color online). The extracted Hc1 of LiFeAs as a
function of T for H?ab and H?c. The solid lines are a guide
to the eye. The inset shows the temperature dependence of
the anisotropy of Hc1, γHc1=H?c
FIG. 3: (Color online) The temperature dependence of the
magnetic penetration depths, λab and λc, of LiFeAs. Inset :
A Uemura plot of the superconducting transition temperature
Tc versus λ−2
had a small value between 1.2 and 2, and γHc1slightly
increased with increasing temperature above 0.6Tc. The
temperature dependence of γHc1was similar to those ob-
served for MgB2 and PrFeAsO1−y[4, 5], indicating
multiband superconductivity. Such behavior may be due
to the existence of Fermi surfaces (FSs) with different
sizes and different anisotropies .
Figure 3 exhibits the temperature dependence of the
magnetic penetration depths, λab for H?c and λc for
H?ab, of LiFeAs. The magnetic penetration depths at
T=0, λab(0)≈198.4nm and λc(0)≈250nm, were obtained
TABLE I: The calculated renormalization factors of each Fe
3d orbital and the calculated specific heat coefficient, γ, at-
tained from the LAD+DMFT approach.
Orbitaldz2 dx2−y2 dxz,yz dxy γ (mJ/mol K2)
1/z=m∗/m 2.09 2.082.89 3.86 29.1
by extrapolating λ(T) in the low temperature regions. In
the inset of Figure 3, the relationship between Tc and
Uemura plot (red-dotted line), consistent with the re-
sults using the transverse-field muon-spin rotation (TF-
µSR) of two polycrystalline LiFeAs compounds . Sim-
ilar to γHc1, the anisotropy of the penetration depth,
γλ=λc/λab, exhibited small values of 1.26 at 0 K and
1.8 near Tc.
To confirm the small anisotropy of the LiFeAs com-
pound, we investigated the electron correlation effect on
the band structure of LiFeAs using the LDA+DMFT ap-
proach. The LDA calculations are performed using the
full-potential linearized augmented plane-wave method
in the Wien2k code, and DMFT has been implemented
based on the LDA Hamiltonian method. The impurity
problem within the DMFT self-consistency equation was
solved using the continuous time quantum Monte Carlo
calculation.We used the parameters of Coulomb in-
teraction U=5.0 eV and the Hund’s rule coupling con-
stant J=0.54 eV, which reproduce the ARPES experi-
ments  very well. Also, the specific heat coefficient,
γ, was calculated as 29.1 mJ/mol·K2, in good agree-
ment with the experimental value of 35 mJ/mol·K2for
our sample. Due to the electron correlation effect, the
ARPES of LiFeAs showed strong band renormalization
by a factor of three compared with that of the LDA
bands. Because each Fe 3d orbital has a different strength
of hybridization each also has a unique renormalization
factors m∗/m (m∗is an effective mass). The calculated
renormalization factors of each Fe 3d orbital are shown
in Table 1. The dz2 and dx2−y2 orbitals have the small-
est renormalization factors, while the dxyorbital has the
largest renormalization factor. Due to the different size of
the renormalization factor, the band renormalization can
be varied depending on the orbital character in momen-
tum space. Because the electrical anisotropy is strongly
dependent on the anisotropy of the band dispersion, the
different band renormalizations can result in large mod-
ification in the electrical anisotropy compared to that of
the LDA calculations. Indeed, the LDA+DMFT calcu-
lation showed ρc/ρab=2.19, a much smaller value than
the ρc/ρab=9 result obtained via LDA . This result
attained from the LDA+DMFT approach for LiFeAs cor-
responded with the electrical anisotropy ratio estimated
from the experimental results, γρ∼3.3.
The ∆λ(T)s estimated from the superfluid densities,
out-of-plane directions are shown in Figure 4; note
ab(0)=25.4 µm−2is represented by the electron-doped
of the in- and
∆λ(T)/∆λ(0) for the penetration depths λab and λc. The
solid line(blue) is ∆λ ∼ (T/Tc)n.
that ∆λ(T)=λ(T)-λ(0). In the low-temperature region
below 0.3Tc, the power law ∆λ(T)∼(T/Tc)n(blue solid
line) was a suitable fit for both ∆λab(T) and ∆λc(T),
with n=3.8±0.1 for H?c and n=3.6±0.1 for H?ab. In
previous reports on doped Fe-based SCs, the majority
of the penetration depth behavior can be modeled using
the power law, with an exponent value of 2≤n≤2.5 in
the low temperature region [2, 9].
behavior of the penetration depth at low temperature,
many authors of theoretical articles argue that most
pnictides are s±-wave SCs due to the impurity scattering
effect [14, 15]. According to Ref. , the low tempera-
ture power-law behavior of the undoped superconductor
LiFeAs, with an exponent of n>3.5, implies that a
LiFeAs SC could be represented by an extended s±-wave
model due to a weak impurity scattering effect.
following expressions were used to study the two gap
feature of LiFeAs:
For the power-law
˜ ρs(T)=x ˜ ρs(T,△1(0))+(1 − x) ˜ ρs(T,△2(0))(1)
0.00.2 0.4 0.60.81.0
FIG. 5: (Color online) Temperature dependence of the super-
fluid density λ2(0)/λ2(T) for the in-plane and out-of-plane in
LiFeAs. The solid lines (black) are the results that best fit
the s±-wave two-gap model(Eq.(1)).
f(E)=[1+exp(E/kBT)]−1is the Fermi function.
temperature dependence of the gap, △(T), can be ex-
pressed as △(0)tanh[1.82[1.018(Tc/T-1)]0.51] .
As shown in Figure 5(a) and (b), the fitting re-
sults with two gaps (black solid line) are more consis-
tent with the experimental data than are those with
one gap (blue dash-dot line).
the model were evaluated as follows: △ab
H?ab. These results are in good agreement with those
attained using ARPES , specific heat , and lower
critical field . In summary, we illustrated that the
small anisotropy ratio of the lower critical fields and
penetration depth in LiFeAs is confirmed by both mag-
netic measurements and numerical calculation using the
LDA+DMFT approach, considering the electron corre-
lation effect. The low temperature power-law behavior
(n>3.5) of the penetration depth implies that a LiFeAs
superconductor could be represented using an extended
s±-wave model due to a weak impurity scattering effect.
In addition, the sizes of the two SC gaps in LiFeAs for in-
x (0≤x≤1) is the weighting factor that
The parameters in
2(0)=1.4±0.1 meV, x=0.55 for H?c, and
1(0)=2.9±0.2 meV, △c
2(0)=1.2±0.1 meV, x=0.51 for
and out-of-plane were obtained. The results were found
to be consistent with those of previous reports.
This research was supported by the Basic Science
Research Program (2010-0007487), the Mid-career Re-
searcher Program (No.
the Nuclear R&D Programs (2006-2002165 and 2009-
0078025) through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education, Sci-
ence and Technology.
∗Electronic address: firstname.lastname@example.org
 C.Martin, M.E. Tillman, H.Kim, M.A.Tanatar, S.K.Kim,
and R. Prozorov, Phys. Rev. Lett. 102, 247002 (2009)
 R.T. Gordon, H. Kim, M.A. Tanatar, R. Prozorov, and
V.G Kogan, Phys. Rev. B 81, 180501(R) (2010)
 L. Malone, J. D. Fletcher, A. Serafin, and A. Carring-
ton, N. D. Zhigadlo, Z. Bukowski, S. Katrych, and J.
Karpinski, Phys. Rev. B 79, 140501(R) (2009)
 T. Shibauchi, K. Hashimoto, R. Okazaki, Y. Matsuda,
Physica C 469 (2009) 590-598
 K. Hashimoto, T. Shibauchi, T. Kato, K. Ikada, R.
Okazaki, H. Shishido, M. Ishikado, H. Kito, A. Iyo, H.
Eisaki, S. Shamoto, and Y. Matsuda, Phys. Rev. Lett.
102, 017002 (2009)
 K. Hashimoto, T. Shibauchi, S. Kasahara, K. Ikada, S.
Tonegawa, T. Kato, R. Okazaki, C. J. van der Beek, M.
Konczykowski, H. Takeya, K. Hirata, T. Terashima, and
Y. Matsuda, Phys. Rev. Lett. 102, 207001 (2009)
 C. W. Hicks, T. M. Lippman, M. E. Huber, J. G. Ana-
lytis, J. H. Chu, A. S. Erickson, I. R. Fisher, and K. A.
Moler, Phys. Rev. Lett. 103, 127003 (2009)
 L. Luan, O. M. Auslaender, T. M. Lippman, C. W. Hicks,
B. Kalisky, J. H. Chu, J. G. Analytis, I. R. Fisher, J. R.
Kirtley, and K. A. Moler, Phys. Rev. B 81, 100501(R)
 R.T. Gordon, C. Martin, H. Kim, N. Ni, M.A. Tanatar,
J. Schmalian, I.I. Mazin, S.L. Bud’ko, P.C. Canfield, and
R. Prozorov, Phys. Rev. B 79, 100506(R) (2009)
 R. Sknepnek, G. Samolyuk, Yong-bin Lee, and Jorg
Schmalian, Phys. Rev. B 79, 054511 (2009)
 M. Machida, Y. Nagai, Y. Ota, N. Naki, H. Nakamura,
N. Hayashi, doi:10.1016/j.physc.2009.10.087
 A. B. Vorontsov, M. G. Vavilov, and A. V. Chubukov,
Phys. Rev. B 79, 140507(R) (2009)
 J. D. Fletcher, A. Serafin, L. Malone, J. G. Analytis, J.-
H. Chu, A. S. Erickson, I. R. Fisher, and A. Carrington,
Phys. Rev. Lett. 102, 147001 (2009)
 Yunkyu Bang, EPL, 86 (2009) 47001
 O V Dolgov, A A Golubov, and D Parker, New Journal
of Physics 11 (2009) 075012
 G.F. Chen, Z. Li, J. Dong, G. Li, W.Z. Hu, X. D. Zhang,
X. H. Song, P. Zheng, N. L. Wang, and J. L. Luo, Phys.
Rev. B 78, 224512 (2008)
 N. Ni, S. L. Bud’ko, A. Kreyssig, S. Nandi, G. E. Rustan,
A. I. Goldman, S. Gupta, J. D. Corbett, A. Kracher, and
P. C. Canfield, Phys. Rev. B 78, 014507(2008)
 Y. Jia, P. Cheng, L. Fang, H. Q. Luo, H. Yang, C. Ren,
L. Shan, C. Z. Gu, and H. H. Wen, Appl. Phys. Lett. 93,
 H. Nakamura, M. Machida, T. Koyama, and N. Hamada,
JPSJ 78, No.12 123712 (2009)
 K. Haule, J. H. Shim, and G. Kotliar, Phys. Rev. Lett.
100, 226402 (2008)
 I.I. Mazin, M.D. Johannes, L. Boeri, K. Koepernik, D.J.
Singh, Phys. Rev. B 78, 085104 (2008)
 X. Dai, Z. Fang, Y. Zhou, F. C. Zhang. Phys. Rev. Lett.
101, 057008 (2008)
 S.V. Borisenko, V.B. Zabolotnyy, D.V. Evtushinsky,
T.K. Kim, I.V. Morozov, A.N. Yaresko, A.A. Ko-
rdyuk, G. Behr, A. Vasiliev, R. Follath, B. Buchner,
 F. Wei, F. Chen, K. Sasmal, B. Lv, Z.J. Tang, Y.Y. Xue,
A.M. Guloy, and C.W. Chu, Phys. Rev. B 81, 134527
 K. Sasmal, B. Lv, Z. Tang, F.Y.Wei, Y.Y. Xue, A.M.
Guloy, and C.W. Chu, Phys. Rev. B 81, 144512 (2010)
 C. Ren, Z. S. Wang, H. Q. Luo, H. Yang, L. Shan, and
H. H. Wen, Phys. Rev. Lett. 101, 257006 (2008)
 Y. J. Song, J. S. Ghim, B. H. Min, Y. S. Kwon, M. H.
Jung, and J. S. Rhyee, arXiv: 1002.2249, Appl. Phys.
Lett. 96, 212508 (2010)
 E. H. Brandt, Phys. Rev. B 60, 11939 (1999)
 L. Lyard, P. Szabo, T. Klein, J. Marcus, C. Marcenat,
K. H. Kim, B. W. Kang, H. S. Lee, and S. I. Lee, Phys.
Rev. Lett. 92, 057001 (2004)
 A. Carrington, F. Manzano, Physica C 385 (2003) 205-
 D.J. Singh, Phys. Rev. B 78, 094511 (2008)
 F. L. Pratt, P.J. Baker, S. J. Blundell, T. Lancaster, H.
J. Lewtas, P. Adamson, M.J. Pitcher, D.R. Parker, and
S.J. Clarke, Phys. Rev. B 79, 052508 (2009)