Calculation of Magnetic Penetration Depth Length $λ(T)$ in High Tc Superconductors

4 Pages·0.14 MB·English

by  Jae-Weon Lee

Calculation of Magnetic Penetration Depth Length λ(T) in High Tc Superconductors
Jae-Weon Lee1, In-Ho Lee2, and Sang Boo Nam1∗
1 Superconductivity Group, Korea Research Institute of Standards and Science,
Doryong-dong 1, Yusung-ku, Taejeon 305-600 Korea
2Korea Institute for Advanced Study, Cheong Ryang Ri, DongdaeMoon Ku, Seoul 130-012, Korea.
1
The notion of a finitepairing interaction energy range via Nam, results in the incomplete conden-
0
sation in which not all states participate in pairings. The states not participating in pairings are
0
2 showntoyieldthelowenergystatesresponsibleforthelinearTdependenceofsuperelectrondensity
at low T in a s-wave superconductor. We present extensive quantitative calculations of λ(T) for
n
all T ranges, in good agreements with experiments. It is not necessary to have nodes in the order
a
parameter, to account for thelinear T dependenceof λ(T) at low T in high Tc superconductors.
J

2
PACS: 74.72.-h, 74.25.Ha, 74.25.Nf

]
One of crucial parameters in a superconductor is the Butnofourfoldisobservedinthesamesystem[18]. Per-
n
o magnetic penetration depth length λ(T) which reflects haps, the observation of [17] may be a reflection of the
c the condensation carrier density, superelectron density Fermi surface.
-
ρ (T), in the London model as Itis highly desirableto carryoutquantitativecalcula-
r s
p tions ofλ(T)forallT rangestosee the accountabilityof
u 2
s ρs(T)/ρs(0)=[λ(0)/λ(T)] . (1) finite Td picture for λ(T) data of HTS. In this letter, we
. presentextensivequantitativecalculationsofλ(T)forall
t The ρ (T) plays an important role for understanding
a s T rangesingoodagreementswithdataofHTS.Forthis,
m the nature of condensation. In the Gorter and Casimir it is worthy to recapitulate the pertinent results for the
two fluid model (GC), ρ (T) varies as 1−(T/T )4. But
- s c notion of a finite Td [8].
d the BCS-ρ (T) has an activation form at low T via the
s To see the phase transition, the transition tempera-
n
orderparameter∆whichindicates the excitationenergy ture T should be a finite value, that is, neither zero nor
o c
gap. The measurements [1–6] of λ(T) at low T in high
c infinite. To have a finite value of Tc, the pairing interac-
[ Tc superconductors (HTS) are compatible with neither tion energy range T should be finite, since T is scaled
d c
the BCS result nor the GC picture. Data indicate the withT withinthepairingtheory[8]. Inotherwords,the
1 d
linearT dependence ofρ (T)atlowT. ThislinearT de-
v s order parameter ∆(k,ω) may be written as [8,9]
1 pendence of λ(T) in fact is taken as providing evidence
1 that the order parameter has nodes, suggesting the d- ∆ for |ǫ |<T
∆(k,ω)= k d (2)
0 wavepairingstate [7]. Onthe other hand,oneofus [8,9] (cid:26)0 for |ǫk|>Td (cid:27)
1
has shown that the notion of a finite pairing interaction
0 for all frequencies ω. Here ǫ is the usual normal state
energy range T results in the incomplete condensation k
1 d
excitation energy with the momentum k, measured with
0 andthelowenergystatesresponsibleforthelinearT de-
respect to the Fermi level.
/ pendence of ρ (T) at low T in a s-wave superconductor
t s
a [10]. Moreover, the incomplete condensation yields the
m
multi-connectedsuperconductors(MS)[11]whichcanac-
- countfortheπ-phaseshiftinPb-YBCOSQUID[12]and
d
1/2 fluxoid quantum in the YBCO ring with three grain
n
o boundary junctions [13].
c Recently, the oxygen isotope effect [14], Tc ∝ M−α
: with α=0.4∼0.49in LSCO single crystal,suggeststhe
v
i electron-phononinteractionwouldplayanimportantrole
X
for understanding superconductivity in cuprate materi-
r als. And the BSCCO bicrystal c-asxis twist Josephson
a
junctionexperiment[15]indicatesthedominantorderpa-
rameter contains the s-wave and not d-wave component.
Moreover,no node in the order parameter is observedin
theangulardependenceofthenon-lineartransversemag-
neticmomentofYBCOintheMeissnerstate[16]. Onthe
other hand, the scanning tunneling microscope imaging
the effects of individual zinc impurity atoms on super-
conductivityinBSCCO[17]showsthefourfoldsymmet- FIG. 1. Schematic diagram showing the order parameter
ricquasiparticlecloud,indicatingthed-wavecomponent. ∆ in thefinite pairing interaction energy ranges Td view.
1
Here the natural units of h¯ = c = k = 1 are used.
B
Laterweuse∆(T)withT for∆aswell. Notethatwhas (cid:13)
1.0(cid:13)
no constraint and that in the case of pairings of carriers
via the electron-phonon interaction, the T corresponds
d
to the Debye temperature. However, the nature of T in Nam(cid:13)
d
HTS is still unknown. Our results are not depending on
the nature of Td. The order parameter ∆ is the solution d(cid:13)
of the BCS like equation [8] c / T 0.5(cid:13) BCS(cid:13) (cid:13)
T
1 Td dǫ β
= tanh E, (3)
g Z0 E 2
where E = (ǫ2+∆2)1/2,β = 1/T, and g corresponds to
0.0(cid:13)
the BCScoupling parameterN(0)V . The solutionof
BCS 0(cid:13) 1(cid:13) 2(cid:13)
∆ for g are shown in the unit of T in Fig. 2.
d g(cid:13)
FIG.3. Tc versusg forBCSandNam’smodel[8],respec-
tively.
The BCS parameter ∆(0)/T can be easily calculated
c
from Eq. (3) as [8]
∆(0)/T =T /[T sinh(1/g)]. (5)
c d c
FIG.2. The order parameter ∆ as a function of g and T.
The equation for T from Eq. (3) via ∆(T ) = 0 may
c c
be written as [8,9]
−1
1/g=(2/π) (2/j)tan (y/j), (4)
Xj
where y = T /πT , and sum is over the positive odd
d c
integers j. The factor of arctangent function makes the
sum converge. For large y, Eq. (3) yields the BCS result
T (BCS). The quantitative calculations of T are given
incFig. 3, together with the BCS T (BCS). cUnlike the FIG.4. The BCS parameter ∆(0)/Tc versus g.
c
BCS result of T (BCS), the T from Eq.(4) does not
c c As is shown in Fig. 4, ∆(0)/T is a function of g or
have any upper limit. The fact is that for large g > c
T /T , and increases with increasing g or T /T . In the
2.32, T increases with increasing g as T = gT /2. One c d c d
c c d rangeofg <0.2orT /T <0.0076,ithasaconstantBCS
interesting value of g = 0.657 yields T = 100 K with T c d
c d value. In fact, this range corresponds to the case of low
= 400 K which is of the order of the Debye temperature
T superconductors(LTS). In the range of 0.5 < g < 1.5
in HTS. This value of g may be realized in YBCO by c
it increases almost in a linear of g, and has a saturated
considering the electron-phonon interaction of the order
value of 2 for large g.
of λ =1.3∼2.3 [19].
p In the sprit of Bardeen [20], the normal fluid density
ρ (T) = ρ−ρ (T), within the pairing theory, may be
n s
written as [7]
2
∞
ρ (T)/ρ=2 d(ω/T)n(ω)f(ω/T)[1−f(ω/T)], (6)
n
Z0
wheref(x)istheusualFermifunction1/[1+exp(x)]and
the density of states n(ω)=N(ω)/N(0) is given by [8]
n(ω)=q(ω/Td)+nBCS(ω)r(ω/Td), (7)
−1
q(ω/T )=(2/π)tan (ω/T ), (8)
d d
−1
r(ω/Td)=(2/π)tan [nBCS(ω)Td/ω], (9)
nBCS(ω)=Re{ω/(ω2−∆2)1/2}. (10)
Physically,ρ (T)wouldberesultedfromthesingleparti-
n
cle excitation not pairs. Thus, the factor f(x) in Eq. (6)
is the occupation probability of the state |k ↑> and the
factor [1−f(x)] is the unoccupation probability of the FIG. 5. The temperature dependence of [λ(0)/λ(T)]2
partnerstate, say,|−k ↓>, andvice versa,respectively. (solidlines)comparedwiththeexperimentaldataforHBCCO
The factor 2 comes from the spin sum. The states of [4], BSCCO [3], LSCO [5] and Sr214 [6].
Eq. (8) are reflections of states being not participated in
pairings. A word of caution is in order. The ω is the However, as shown in Fig. 6, we have obtained poor
dynamical energy which has the kinetic as well as in- agreement near Tc between calculation and data of
teraction parts. Physically, the sum of spectral weights YBCO by Hardy et al[1] andanisotropicdata of YBCO
outside T < |ǫ |, result in the states of Eq. (8). Thus, by Kamal et al [2]. The YBCO b case is good.
d k
the low energy states are realized. In other words, carri-
ers which do not participate in pairings yield the linear
T dependence of ρ (T) at low T. In fact, these states
n
results in the linear T dependence of λ(T) at low T. To
seethis,byinsertingEq.(8)intoEq.(6),onecangetthe
variation of λ(T) at low T as [9,10]
1
∆λ/λ(0)= ρ (T)/ρ=(T/T )(T /T )(2/π)ln2, (11)
n c c d
2
similar to the result by d-wave picture [7],
[∆λ/λ(0)]d =(T/Tc)(Tc/∆0)ln2 (12)
viand(ω)=ω/∆0,where∆0isthemaximumvalue(anti-
node) of the order parameter.
For the quantitative calculations of λ(T), we have de-
termined T /T or g [Eq. (4)] via Eq. (11), by taking
c d
the slope of [λ(0)/λ(T)]2 near zero temperature. Once g
or T /T is set, no adjustable parameter is used in our
c d
2
calculations of Eq. (6). FIG.6. [λ(0)/λ(T)] (solid line) compared with data for
As is shown in Fig. 5, we have obtained good agree- bulk YBCO [1], and a- and b-axes,respectively [2].
ments between calculations and data of BSCCO by Lee
et al [3], HBCCO by Panagopoulos et al [4] and LSCO ThenotionofafiniteTd resultsinthe incompletecon-
byPanagopoulosetal[5],andSr214byBonaldeetal[6], densation at zero temperature. By considering the sum
respectively. We picked up not all of data points in the rule,wecancalculatethe fractionofstates,R,beingnot
papers for clarity. Bonalde et al [6] reported that their participated in pairings as [8]
dataatlowT varyasT2 whichareresultedfromscatter-
∞
ings by impurities or defects. In a finite Td picture, the R(z)= [nBCS(ω)−n(w)]dω/∆ (13)
impurity scatteringsmakesome states atthe Fermilevel Z∆
not participate in pairings, and result in the T2 term in ∆
= n(ω)dω/∆
λ(T) at low T [21]. Z0
3
−1 2
=(2/π)tan (z)−(1/zπ)ln(1+z ),
where z =∆/T =[∆(0)/T ](T /T ) which is a function
d c c d
of g or T /T .
c d
∗
Correspondence address: [email protected]
(cid:13) [1] W. N. Hardy et al., Phys. Rev. Lett. 60, 3999 (1993).
0.50(cid:13) [2] S. Kamal et al., Phys. Rev. B 58, R8933 (1998).
[3] S. Lee et al., Phys. Rev. Lett. 77, 735 (1996).
0.0050(cid:13) (cid:13) [4] C. Panagopoulos et al., Phys. Rev. B 53, R2999 (1996).
[5] C. Panagopoulos et al., Phys. Rev. B 60, 14617 (1999).
(cid:13)0.00 (cid:13)25(cid:13) (cid:13) [6] I. Bonalde et al., Phys. Rev. Lett. 85, 4775 (2000).
0.00000(cid:13).0(cid:13) 0.1(cid:13) 0.2(cid:13) [7] D. J. Scalapino, Phys. Rep. 250, 329 (1995).
R(cid:13) 0.25(cid:13) (cid:13) [8] S.B. Nam,Phys. Lett.A193, 111 (1994), ibid(E)A197,
(cid:13)
458(1995).ThedensityofstatesN(ω)isobtainedbythe
sum over k (the integral with respect to ǫk) of spectral
weights (the imaginary parts of the Green’s function).
The usual method to get N(ω) is taking the residues
at the poles in the Green’s function, via 1/X = P/X −
0.00(cid:13) iπδ(X),(A),wherePstandsfortheprincipalvaule.How-
0(cid:13) 1(cid:13) 2(cid:13)
ever,inthecaseofafiniteTd,withtheorderparameter∆
g(cid:13) of Eq. (2), the (A) is not useful, since the pole ǫp(ω) in
the superconducting Green’s function, has multi-values
for a given ω. To see the inadequacy of the (A), let us
FIG.7. The fraction of states, R, being not participated consider ω = Td. The pole ǫp(ω) via ǫp2(ω) = ω2 − ∆2,
in pairings versus g. has different valuesat ǫk = Td − 0 and ǫk = Td + 0, re-
spectivley.Thus,the(A)isuseless,sincethe(A)isvalid
onlyinthecaseofthepolehavingthesamevalueforthe
InFig.7isshownR(z)asafunctionofg. Intherange
left and right of it, respectively. Oneway to get N(ω) is
of g < 0.2 or T /T < 0.0076, R(z) is negligible. As
c d to carry out the ǫk integral of the imaginary part of the
stated before, this range correspondsto the case of LTS.
Green’s functionon thecontourof twofan (eight) shape
Thus,thelinearT dependenceofλ(T)atlowT ishardly of the real and imaginary axes in the ǫk plane, with the
observed in LTS. causality condtion, as described in this reference, yield-
Insummary,eventhoughthemodelofEq.(2)isideal, ing the states of Eq. (8). On the other hand, the decay
the quantitative calculations account very well for data type of ∆ ouside Td may be imagined, but such a type
for all T ranges without any adjustable parameter, ex- functioncannotbeaself-consistentsolutionoftheorder
ceptforYBCOdatanearT . Perhaps,the Fermisurface parameterequation.The∆ofEq.(2)isasolutionofthe
c
order parameter equation.
effectwouldplayanimportantroleforλ(T)inthecaseof
[9] S. B. Nam, J. Korean Phys. Soc. 31, 426 (1997).
YBCO. Of course, the retardation and non-local effects
[10] S. B. Nam, Prog. in Supercond. 2, 11 (2000).
shouldbetakenintoaccountaswellforimprovement. In
[11] S. B. Nam, Phys. Lett. A198 , 447 (1995).
all, the calculations are quite satisfactory and theoreti- [12] D.A.Wollman et al., Phys. Rev. Lett. 71 ,2134 (1993).
cally sound. We suggest the pairing interaction energy [13] C. C. Tsuei et al., Phys. Rev. Lett. 73 , 593 (1994).
range Td in HTS may be of the order of 1∼ 2 times Tc. [14] J. hofer et al., Phys. Rev. Lett. 84, 4192 (2000).
ThelinearT dependenceofλ(T)atlowT doesnotimply [15] Q. Li et al., Phys. Rev. Lett. 83, 4160 (2000).
nodes in the order parameter,contrary to general belief. [16] A.Bhattacharya et al, Phys. Rev. Lett. 82, 3132 (1999).
InthespiritofafiniteT ,itisrecentlyshown[21]that [17] S. H. Pan et al., Nature 403, 746 (2000).
d
thespinlessimpurityscatteringssuppressT anddestroy [18] A. Yazdaniet al., Phys. Rev. Lett. 83, 176 (1999).
c
superconductivity. Some states at the Fermi level are [19] W. Weber and L. I. Mattheiss, Phys. Rev. B 37, 599
(1988).
shownnottoparticipateinpairingswhentherearescat-
[20] J. Bardeen, Phys. Rev. Lett. 1, 399 (1958).
tering centers suchas impurities, andresultin the linear
[21] S. B. Nam, Bull. Am. Phys. Soc 45, 256 (2000).
T term in the specific heat at low T. The quantitative
[22] I. Lee et al., tobe published.
calculations[22]accountwellforthe reductionofTc[23] [23] Gang Xiao et al, Phys. Rev. Lett. 60 , 1446 (1994).
andthespecificheatdata[24,25]intheZn-dopedYBCO, [24] K. A. Moler et al, Phys. Rev. B 55, 3954 (1997).
respectively. [25] D. L. Sisson et al, Phys. Rev. B 61, 3604 (2000).
WethankKRISSmembersfortheirwarmhospitalities
at KRISS. Specially JWL thanks Dr. Y. H. Lee for his
kindnessandSBNthanksDrsJ.C.ParkandY.K.Park
for various discussions. This work is supported in part
by KOFST.
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