Table Of ContentThe Electronic Specific Heat of Ba K Fe As from 2K to 380K.
1−x x 2 2
J.G. Storey1, J.W. Loram1, J.R. Cooper1, Z. Bukowski2 and J. Karpinski2
1Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, U.K. and
2Laboratory for Solid State Physics, ETH Zurich, Zurich, Switzerland
(Dated: January 4, 2010)
Using a differential technique, we have measured the specific heats of polycrystalline
Ba K Fe As samples with x = 0, 0.1 and 0.3, between 2K and 380K and in magnetic fields
1−x x 2 2
0to13Tesla. Fromthisdatawehavedeterminedtheelectronicspecificheatcoefficientγ(≡C /T)
el
over the entire range for the three samples. The most heavily doped sample (x = 0.3) exhibits a
largesuperconductinganomaly∆γ(T )∼48mJ/molK2 atT =35K,andwedeterminetheenergy
c c
gap, condensationenergy, superfluiddensityandcoherence length. Inthenormal stateforthe x=
0
1 0.3sample,γ ∼47mJ/molK2 isconstantfromTc to380K.Intheparentcompound(x=0)there
0 isalargealmostfirstorderanomalyatthespindensitywave(SDW)transitionatTo =136K.This
2 anomaly is smaller and broader for x = 0.1. At low T, γ is strongly reduced by the SDW gap for
n both x = 0 and 0.1, but above To, γ for all three samples are similar.
a
PACSnumbers: 74.25.Bt,74.70.-b
J
4
The electronic specific heat contains a wealth of quan- mina crucibles sealed in evacuated quartz ampoules. Fi-
]
n titative information on the electronic spectrum over an nally, samples of Ba1−xKxFe2As2 with x = 0.1 and 0.3
o energy region ±100meV about the Fermi level, cru- were prepared from appropriate amounts of single-phase
c cial to the understanding of high temperature supercon- BaFe As and KFe As . The components were mixed,
2 2 2 2
-
r ductivity. Measurements of the electronic specific heat pressed into pellets, placed into alumina crucibles and
p
played an important role in revealing key properties of sealed in evacuated quartz tubes. The samples were an-
u
the copper-oxide based ‘cuprate’ high-temperature su- nealed for 50 h at 700◦C with one intermittent grind-
s
. perconductors (HTSCs). Some examples include the ing, and were characterized by room temperature pow-
t
a normal-state “pseudogap”[1–4], the bulk sample inho- derX-raydiffractionusingCuK radiation. Thediffrac-
α
m
mogeneity length scale[5], and more recently the degree tionpatterns were indexedon thebasis ofthe tetragonal
- to which the superconducting transition temperature is ThCr Si typestructure(spacegroupI4/mmm). Lattice
d 2 2
suppressed due to superconducting fluctuations[6]. In parameters calculated by a least-squares method agree
n
o this work we extend such measurements to the recently wellwiththosereportedbyChenetal[8]. TracesofFeAs
c discovered iron-arsenide based ‘pnictide’ HTSCs. Here as an impurity were detected for compositions with x =
[ we present results obtained from polycrystalline samples 0.1 and 0.3. The samples for heat capacity measurement
1 of Ba1−xKxFe2As2 (x = 0, 0.1 and 0.3) using a high- weighed ∼ 0.8g.
v resolution differential technique[7]. The total specific heats of the three samples are plot-
4 Withthistechniquewedirectlymeasurethedifference ted as γtot ≡ Ctot/T in Fig. 1. In the x = 0 sample
7
in the specific heats of a doped sample and the undoped there is a sharp and almost first order anomaly at the
4
reference sample (BaFe As ). This eliminates most of magneto-structural transition at T = 136K, in agree-
0 2 2 0
. the large phonon term from the raw data and yields a ment with published single crystal data[9]. For the x =
1
curve which is dominated by the difference in electronic 0.1 sample the corresponding anomaly at T = 135K is
0 0
0 termsbetweenthesampleandreference. Aftermakinga broaderandconsiderablyreducedinmagnitude,inagree-
1 smallcorrectionforanyresidualphononterm,thisdiffer- ment with the data of Rotter et al.[10] who have tracked
: ence in electronic specific heats can be determined with T in the specific heat down to 105K in a sample with x
v 0
i a resolution of ∼ 0.1 mJ/mol K2 at temperatures from = 0.2. The magnetic field dependence of this anomaly is
X
1.8K to 380K and magnetic fields from 0 to 13T. During extremelyweakforbothsamples. Weseenoevidencefor
r a measurement run the total specific heats of the sample a structural transition in the x = 0.3 sample (the very
a
and reference are also measured. weak anomaly at 67K in the differential data (Fig. 2(a))
The polycrystalline samples of Ba K Fe As were is probably due to an FeAs impurity phase[11]). The su-
1−x x 2 2
preparedbyasolidstatereactionmethodsimilartothat perconducting transition in this sample is clearly visible
reported by Chen et al[8]. First, Fe As, BaAs, and in γtot at 35K, and the field dependence is shown in the
2
KAs were prepared from high purity As (99.999%), Fe inset to Fig. 1.
(99.9%),Ba(99.9%)andK(99.95%)inevacuatedquartz Inthetemperaturerange2−8K,γtot =γ(0)+β(0)T2
ampoules at 800, 650 and 500◦C respectively. Next, atzerofieldwithγ(0)=5.1,9.3and1.8mJ/molK2 and
the terminal compounds BaFe As and KFe As were β(0) = 0.45, 0.37 and 0.57 mJ/mol K4 for the x = 0, 0.1
2 2 2 2
synthesized at 950 and 700◦C respectively, from stoi- and 0.3 samples respectively. On the assumption that
chiometric amounts of BaAs or KAs and Fe As in alu- β(0)isentirelyduetophononsweobtainDebyetemper-
2
2
to its saturation value 3nR (where n is the number of
SC anomaly atoms/formula unit) shown by the dotted line in Fig. 1.
1250 650 0T
) Toextendtheusefulrangeofthehightemperatureregion
-2 K 1000 600 13T we exploit the fact that the Debye temperature θD(T)
-1 ol 550 deduced from Cv is generally only weakly T-dependent
m forT >0.5θD(∞)(or∼180Kforthepresentmaterials).
750
J 28 30 32 34 36 Withthisconstraintweobtainlimitinghightemperature
m
Debye temperatures θ (∞) = 368, 360 and 356K and
γ tot ( 500 xx == 00.1 high-T values γ(300K)D= 55, 47 and 47 mJ/mol K2 for
x = 0.3 the 0, 0.1 and 0.3 samples.
250 3nR/T Differentialmeasurements(seeFig.2(a))betweeneach
sample and the x = 0 reference give ∆γtot =∆γ+∆γph
0 (assuming ∆γan = 0). It is clear from Figs. 1 and
0 50 100 150 200 250 300 350 400 2(a) that the phonon terms for the three samples are
Temperature (K) very similar. The broad negative peak at 35K seen in
FIG. 1: (Color online) γtot ≡ Ctot/T of Ba K Fe As for ∆γtot(0.1,0)=γtot(x=0.1)−γtot(x=0),andwhichcan
1−x x 2 2
x = 0, 0.1 and 0.3. The dotted line shows the high temper- also be inferred in the data for ∆γtot(0.3,0) in the same
ature limiting value of the lattice contribution, 3nR/T. The temperature region, is compatible with fractional in-
inset shows the suppression of the superconducting anomaly creasesof0.018and0.030intheacousticphononfrequen-
in magnetic fields from 1 to 13T in 1T increments. cies for the 0.1 and 0.3 samples relative to the x = 0 ref-
erence. Sinceweexpectthedifferenceofelectronicterms
∆γ(0.1,0) for the two non-superconducting samples to
atures θD(0)=289, 298and 257Kforthethree samples. be only weakly T-dependent (at least below 100K), we
We cannot, however, ignore the possibility that for the obtain ∆γph(0.1,0) = ∆γtot(0.1,0) − ∆γ(0.1,0) using
superconducting 0.3 sample β(0) is enhanced by an elec- the low temperature value ∆γ(0.1,0) = −4.2 mJ/mol
tronic component ∝ T2, leading to too low a value of K2. To ensure that the resulting ∆γph(0.1,0) has a
θD(0). This would explain the reduced value of θD(0) T-dependence compatible with that of a phonon spec-
forthissamplewhichcontradictsthetrendseenin∆γtot trum it was modelled with a histogram for the difference
(Fig.2(a))thatacousticphononfrequencies,whichdom- phonon spectrum, as discussed previously[15]. Making
inatethephononspecificheatinthistemperatureregion, this phonon correction removes the broad negative peak
increase with doping. If the 0 and 0.1 samples have an in ∆γtot at 35K for this sample (Fig. 2(b)). As shown in
initial electronic T2 term in γ(T) we would expect this the same figure, subtracting a phonon term ∆γph(0.3,0)
to have a magnitude ∼ γnT2/To2 ∼ 0.003 mJ/mol K2 = 1.68∆γph(0.1,0) also removes the negative peak in
if it is controlled by the same energy scale as T0. This ∆γtot(0.3,0). Following this correction, the electronic
contribution to β would be negligible compared with the ∆γ(0.3,0)hasanadditionalnegativeT-dependencegiven
measured β(0). by ∼ −20×10−6T3 mJ/mol K2 in the range 40 - 110K.
We next discuss the separation of electronic and We conclude below that γ (T) for the 0.3 sample is T-
n
phonon contributions from the total specific heat over independent, and attribute this negative T-dependence
a wider temperature range. The total specific heat co- to a corresponding positive term in the electronic spe-
efficient is given by γtot = γ + γph + γan where the cific heat coefficient γ(x = 0) of the undoped sample in
electronic term γ = C /T, the harmonic phonon term the SDW phase.
el
γph =C /T and the anharmonic phonon (dilation) term Aftercorrectingfor∆γph(0.3,0)wefindthatjustabove
v
γan = (C − C )/T[12]. C and C are the lattice T ,γ(x=0.3)∼47mJ/molK2. Thisisveryclosetothe
p v p v c
heat capacities at constant pressure and volume respec- value S/T(T ), where S = (cid:82)T γ(T)dT is the electronic
c 0
tively. Theanharmonictermisgivenbyγan =VBβ2[12] entropy. By definition, S/T(T(cid:48)) is the average value of γ
where V is the molar volume, B is the bulk modu- below T(cid:48). From conservation of entropy S/T(T ) is also
c
lus and β is the volume expansion coefficient. We as- the average value of the underlying normal state γ (T)
n
sume that γan is doping independent and use V = 61 below T . Therefore the result above is consistent with
c
cm3/mol[10], B ∼ 0.80×108 mJ/cm3[13], and β(300K) a T-independent γ (T) for T < T . In addition, this
n c
∼ 50×10−6 /K[14] to obtain a room temperature value value for γ (T) below T is close to its high temperature
n c
γan(300K) ∼ 12 mJ/mol K2 for each sample. To a value estimated above, and it is reasonable to assume
very good approximation β(T) ∝ Cv(T)[12] and thus that for this sample γn(T) is T-independent over the en-
γan(T) = [Cv(T)/Cv(300K)]2 · γan(300K). A plot of tire temperature range. Subtracting this constant term
γan(T) is shown in the inset to Fig. 2(a). from γtot(x = 0.3) gives γph(x = 0.3), which we then fit
After correcting for γan(T), the electronic term γ can withanine-termhistogramforthephononspectrum[15].
be determined at high temperatures where C is close Finally, subtracting ∆γph(0.3,0) and ∆γph(0.3,0.1) from
v
3
(a-12)mol K) 5705 12060 100 200γ an300 V) 1250 d wave Density of States
mJ 25 me
∆γ tot ( 0 ∆ ( 10 0 1 E/∆ 2
s + id wave
0.3 - 0 0T 5
0.3 - 0 13T
-25
0.1 - 0 0T s wave
(b) 150 100 0
0 10 20 30 40
-2 K) 125 50 Temperature (K)
-1 ol 100 00 50 100 150 200 250 300 350 FIG.3: (Coloronline)Temperaturedependenceofthesuper-
mJ m 75 xx == 00..31 cdoentedrumctininegdSga/pT ewxittrhacthteadtcbaylccuolamtepdarfirnogmtmheodeexlpseorfimtheentdaelnly-
γ ( x = 0 sity of states (shown in the inset) given by Eqns. 1-3.
50
25
theelectronictermcomesfromtheobservationofsimilar
0
0 20 40 60 80 100 anomalouscurvaturebetween15and20Kinthefieldde-
pendence ∆γ(H)=γ(H,T)−γ(0,T) shown in the inset
Temperature (K)
to Fig. 4(a). Since the phonon term is H-independent,
FIG. 2: (Color online) (a) The directly measured raw differ-
this anomalous curvature in ∆γ(H) can only be of elec-
enceinspecificheatsbetweenthedopedsamplesandundoped
reference in 0 and 13T fields. The inset shows the anhar- tronic origin, and may signal the presence of a second
monic phonon term γan. (b) The electronic specific heat of energy gap as inferred by Mu et al[17].
Ba1−xKxFe2As2 for x = 0, 0.1 and 0.3. To determine the magnitude and temperature depen-
dence of the superconducting gap ∆ for x = 0.3 (shown
inFig.3)wefirstsubtractfromγ thesmallresidualγ(0)
γph(x=0.3) gives γph(x=0) and γph(x=0.1). = 1.8 mJ/mol K2 and then match the corrected S/T
The electronic specific heats of the three samples are with that calculated assuming the following models for
shown in Figure 2(b). Above 150K, γ is at most only the density of states (see Fig. 3 inset):
weakly T-dependent and almost independent of doping. s-wave
At low temperatures γ for the x = 0 and 0.1 samples
(cid:112)
is heavily reduced by factors of ∼ 11 and 5 respectively N(E)=E/ E2−∆2 (1)
duetotheopeningofaspindensitywave(SDW)gapbe-
low T ∼ 136K, and increases as ∼ 20×10−6T3 mJ/mol d-wave
0
K2 in the range 40 − 110K. Band splitting and signs of 1 (cid:88) (cid:112)
partial gapping of the Fermi surface have been observed N(E)= E/ E2−∆2cos22θ (2)
N
θ
intheSDWstateofBaFe As byangle-resolvedphotoe- θ
2 2
mission spectroscopy[16]. γ for the x = 0.3 sample is
and s+id-wave
dominated by the large superconducting anomaly with
∆γ(Tc) ≈ 48 mJ/mol K2, a normal-state γN(0) of about N(E)= 1 (cid:88)E/(cid:112)E2−0.98∆2cos22θ−0.02∆2 (3)
47 mJ/mol K2 and γ(0) = 1.8 mJ/mol K2 . Measure- N
θ
θ
ments on a more overdoped crystal with x = 0.4 by Mu
et al.[17] show an even larger ∆γ(Tc) ≈ 100 mJ/mol K2 where the summations over θ are from 0 to 45◦ and Nθ
and γN(0) ≈ 63 mJ/mol K2, implying a further growth is the number of θ values. Although such modelling is
in the density of states with doping. For our x =0.3 too crude to pin down the exact nature of the gap, the
samplethesuperconductingcondensationenergyU(0)= rapid increase in ∆ below 15K in the case of the as-
10.6 J/mol, as determined from (cid:82)Tc(S −S )dT, where sumed d-wave gap function, shows that nodes on the
0 n s
S and S are the normal and superconducting state en- Fermi surface are incompatible with the T-dependence
n s
tropies respectively. In Fig. 2(b), the curvature in γ for of γ observed at low temperature, even in the presence
x =0.3 between 10 and 20K is reduced compared to the ofweakpairbreaking. Thes+id-wavegapfunctionpro-
raw ∆γtot data (Fig. 2(a)) by the phonon correction de- ducesamorereasonable∆(T). Thisfunctioncompletely
scribed above. Evidence that the remaining curvature gaps the Fermi surface but is strongly anisotropic. Such
in γ may reflect a genuine anomalous T-dependence of anisotropycouldarisefromthepresenceofmultiplegaps
4
-1-2 ol K) 150 13T (a)T) 112400 Hc1 x 104 235000
γγ(H) - (0) (mJ m --11-0550 25..50 0T H, H, H (c1cc2 12468000000 κHHcc2 x 102 51120050000κ
0.0 (a)
-20 0 10 20 30 0 0
(b) 16
-1 K) 100 13T 40 14
-1 mol A) 30 λ 12 -2 m)
0) (mJ 5705 ξλ, ( 20 ξ/100 1/λ2 6810 λµ2 1/ (
S( 10 4
H) - 25 2
S( (b) 0 0
0 5 10 15 20 25 30 35 40
0 1T Temperature (K)
2. 0
-1 mol) 2.0 13T 11..05 2.5K FBfrIeaGe0..3eKn5:e0r.3PgFyoel2l∆yAcFsr2(yHsdt,aeTlrlii)vn:eed(aav)freCormargiteitdchaeml fiifixeeelldddssdHtaept,eenHpdaern,acmHeetoe,frastnhodef
0) (J 1.5 00..05 Gλ,inssubpuerrgfl-uLiadnddeanusiptayra∝m1e/tλer2κa;n(db)suLpoenrdcoonndpcuecnteitncrg1atcioonhc2edreepntche
H) - F( 1.0 0 2 4 H6 (T8)1012 length ξ.
F( 0.5
(c) tronic entropy with field (see Fig. 4(b)).
1T
0.0 (cid:90) T
0 10 20 30 40 50
∆S(H,T)= ∆γ(H,T)dT (5)
Temperature (K) 0
FIG. 4: (Color online) (a) Magnetic field dependent change The change in free energy is obtained by integrating the
in electronic specific heat γ(H)−γ(0) vs temperature, in 1T
entropy over temperature (see Fig. 4(c)).
increments, from the data in the inset of Fig. 1. (Inset) En-
largement of the 0 to 30K region. (b) Field dependence of (cid:90) T
theelectronicentropy,S(H)−S(0),calculatedfrom(a)using ∆F(H,T)=− ∆S(H,T)dT +∆F(H,0) (6)
Eqn.5. (c)Fielddependenceofthefreeenergy,F(H)−F(0), 0
calculatedfrom(b)usingEqn.6. (Inset)AfittoF(H)−F(0)
∆F(H,0) is determined from the condition that the su-
at 2.5K using Eqn. 8.
perconducting contribution to ∆F(H,T) tends to zero
for T (cid:29) T . At each temperature, the field dependence
c
of the free energy is fitted to a theoretical expression de-
on different sheets of the Fermi surface, or from an in- rived from the model of Hao and Clem[18] for an s-wave
trinsically anisotropic gap(s). superconductor.
Weturnnowtotheinformationcontainedinthemag- (cid:18) (cid:19)
aφ eβH
netic field dependence of γ. The inset to Fig. 1 shows ∆Fs,rev(H,T) = 32π20λ2Hln Hc2 (7)
the suppression of the superconducting anomaly in mag-
netic fields from 1 to 13T in 1T increments. Because the
= a Hln(a H) (8)
1 2
phononspecificheatisindependentofmagneticfield,the
changeinelectronicspecificheatwithfieldisobtainedby The coefficients a and β are weakly field dependent and
subtracting the zero field data from the data measured are given by a ∼ 0.77 and β ∼ 1.44 in the range 0.02 <
in a field (see Fig. 4(a)). H/H < 0.3. A fit to the free energy at 2.5K is shown
c2
in the inset to Fig. 4(c).
The penetration depth, λ, and the upper critical field,
∆γ(H,T)=γ(H,T)−γ(0,T) (4)
H , are determined directly from the fit parameters a
c2 1
and a . Then using the following Ginsburg-Landau rela-
2
Integrating over temperature gives the change in elec- tions we extract the: critical field H2 = (φ /4πλ2)H ,
c 0 c2
5
√
Ginsburg-Landau parameter κ = H /H 2, lower crit- [3] J.W.Loram,J.L.Luo,J.R.Cooper,W.Y.Liang,and
c2 c
ical field H = (φ lnκ)/(4πλ2), and superconducting J. L. Tallon, Physica C 341–348, 831 (2000).
c1 0
coherence length ξ = (φ /2πH )1/2. The temperature [4] J. W. Loram, J. Luo, J. R. Cooper, W. Y. Liang, and
0 c2
J. L. Tallon, J. Phys. Chem. Solids. 62, 59 (2001).
dependenciesofthesequantitiesisplottedinFig.5. The
[5] J. W. Loram, J.L. Tallon, and W.Y. Liang, Phys. Rev.
valuesshownarepolycrystallineaverages. Thelargeκ≈
B 69, 060502(R) (2004).
130 indicates the strong type II nature of this material.
[6] J. L. Tallon, J. G. Storey, and J. L. Loram,
Our Hc2 values are very similar to those obtained from arXiv:0908.4428 (2009).
radio frequency penetration depth measurements on a [7] J. W. Loram, J. Phys. E 16, 367 (1983).
single crystal with x = 0.45[19], and the value we ob- [8] H. Chen, Y. Ren, Y. Qiu, W. Bao, R. H. Liu, G. Wu,
tain for λ(0) ≈ 260nm is similar to those measured by T. Wu, Y. L. Xie, X. F. Wang, Q. Huang, et al., EPL
85, 17006 (2009).
infrared spectroscopy[20] and tunnel diode resonator[21]
[9] J. K. Dong, L. Ding, H. Wang, X. F. Wang, T. Wu,
techniques.
G. Wu, X. H. Chen, and S. Y. Li, New J. Phys. 10,
In summary, above 150K the electronic specific heats
123031 (2008).
for the three samples are large and almost identical. For [10] M.Rotter,M.Tegel,I.Schellenberg,F.M.Schappacher,
the x = 0 sample there is a large and almost first order R. P¨ottgen, J. Deisenhofer, A. Gu¨nther, F. Schrettle,
anomalyattheSDWtransitionatT =136K,whilstthe A. Loidl, and D. Johrendt, New J. Phys. 11, 025014
0
corresponding anomaly for the x = 0.1 sample at T = (2009).
0
135K is smaller and broader and more closely resembles [11] K. Selte, A. Kjekshus, and A. F. Andresen, Acta Chem.
Scand. 26, 3101 (1972).
a second order phase transition. At low temperatures, γ
[12] N.W.AschcroftandN.D.Mermin,Solid State Physics.
for these two samples is strongly reduced by the SDW
(Holt Saunders, 1976).
gapbyfactorsof11and5forx=0and0.1respectively.
[13] S.A.J.Kimber,A.Kreyssig,Y.-Z.Zhang,H.O.Jeschke,
Inthex=0.3samplethelargesuperconductinganomaly R. Valenti, F. Yokaichiya, E. Colombier, J. Yan, T. C.
and associated value of S/T(T ) shows that the under- Hansen, T. Chatterji, et al., Nature Mat. 8, 471 (2009).
c
lying normal state γ at low temperature is close to its [14] S.L.Bud’ko,N.Ni,S.Nandi,G.M.Schmiedeshoff,and
hightemperaturevalue. Thissuggeststhatasxreduces, P. C. Canfield, Phys. Rev. B 79, 054525 (2009).
[15] J. W. Loram, K. A. Mirza, J. R. Cooper, and W. Y.
the growth of the SDW gap and consequent reduction in
Liang, Phys. Rev. Lett. 71, 1740 (1993).
the density of states is responsible for the disappearance
[16] L.X.Yang,Y.Zhang,H.W.Ou,J.F.Zhao,D.W.Shen,
of superconductivity. The temperature dependence of γ
B. Zhou, J. Wei, F. Chen, M. Xu, C. He, et al., Phys.
(x = 0.3) supports a nodeless superconducting gap func- Rev. Lett. 102, 107002 (2009).
tionthatisstronglyanisotropicabouttheFermisurface, [17] G. Mu, H. Luo, Z. Wang, L. Shan, C. Ren, and H. H.
possibly due to the presence of multiple gaps. Wen, Phys. Rev. B 79, 174501 (2009).
WegratefullyacknowledgefundingfromtheEngineer- [18] Z.HaoandJ.R.Clem,Phys.Rev.Lett.67,2371(1991).
[19] M. M. Altarawneh, K. Collar, C. H. Mielke, N. Ni, S. L.
ing and Physical Sciences Research Council (U.K.) and
Bud’ko,andP.C.Canfield,Phys.Rev.B78,220505(R)
the Swiss National Science Foundation pool MaNEP.
(2008).
[20] G. Li, W. Z. Hu, J. Dong, Z. Li, P. Zheng, G. F. Chen,
J.L.Luo,andN.L.Wang,Phys.Rev.Lett.101,107004
(2008).
[21] R. Prozorov, M. A. Tanatar, R. T. Gordon, C. Martin,
[1] J. Loram, K. A. Mirza, J. R. Cooper, W. Y. Liang, and H.Kim,V.G.Kogan,N.Ni,M.E.Tillman,S.L.Bud’ko,
J. M. Wade, J. Supercon. 7, 243 (1994). and P. C. Canfield, Physica C 469, 582 (2009).
[2] J.W.Loram,K.A.Mirza,J.R.Cooper,andJ.L.Tallon,
J. Phys. Chem. Solids 59, 2091 (1998).
