Superfluid response in electron-doped cuprate superconductors

0.17 MB·English

by  H. G. Luo

Superfluid response in electron-doped cuprate superconductors
H. G. Luo and T. Xiang
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Institute of Theoretical Physics and Interdisciplinary Center of Theoretical Studies,
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Chinese Academy of Sciences, P. O. Box 2735, Beijing 100080, China
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2
We propose a weakly coupled two-band model with dx2−y2 pairing symmetry to account for the
n anomalous temperature dependence of superfluid density ρs in electron-doped cuprate supercon-
a ductors. This model gives a unified explanation to the presence of an upward curvature in ρs near
J Tc and a weak temperature dependence of ρs in low temperatures. Our work resolves a discrep-
5 ancy in the interpretation of different experimental measurements and suggests that the pairing in
2 electron-doped cuprates has predominately dx2−y2 symmetry in thewhole dopingrange.

]
n
Identification of pairing symmetry has been an im- effectively described as a two-band system [15, 16]. This
o
c portant issue in the investigation of high-Tc supercon- two-bandscenarioisconsistentwiththeconjecturemade
- ductivity. For hole-doped cuprate superconductors, it is byanumberofgroups[17,18,19]ontheexistenceoftwo
r
p commonlyacceptedthatthepairingorderparameterhas kinds of charge carriers in electron-doped materials.
u dx2−y2-wave symmetry [1]. However, for electron-doped Theinterplaybetweentheabovementionedtwobands
s
cuprate superconductors,no consensus has been reached can affect significantly the behavior of superconducting
.
t
a on the pairing symmetry. A number of experiments, in- quasiparticles. Agenericfeatureofaweaklycoupledtwo-
m cludingtheangleresolvedphotoemission(ARPES)[2,3], bands system, as first pointed out by Xiang and Wheat-
theRamanspectroscopy[4],andthephase-sensitivemea- ley [20], is the presence of an upward curvature in the
-
d surements [5, 6], suggested that the electron-doped su- temperaturedependenceofsuperfluiddensityρ nearT .
s c
n
o perconductors also have dx2−y2 -wave symmetry. How- Thisintrinsic upwardcurvatureinthe superfluiddensity
ever, the results revealed by other experiments are con- has indeed been observedin electron-dopedmaterialsby
c
[ troversial [7, 8, 9, 10, 11]. In particular, the magnetic a number of experimental groups [11, 21, 22, 23]. Not

penetration depth data measured by Kokales et al. [8] only does it lend further support to the two-band pic-
3
and by Prozorov et al. [9, 10] showed that the low ture,butalsoshedslightontheunderstandingofvarious
v
6 temperature superfluid density of electron-doped super- controversialexperimental observations.
9 conductors varies quadratically with temperature in the In this paper, we propose to use a two-band BCS-like
4 whole range of doping, in agreement with the theoreti- model with dx2−y2-wave pairing symmetry to account
7
calpredictionforad-wavesuperconductorwithimpurity forthe lowenergyelectromagneticresponseofsupercon-
0
scattering. However,theexperimentaldatapublishedby ducting quasiparticles in electron-doped materials. This
4
0 Kimet al. [11]suggestedthatthereisad-toanisotropic model, as will be shown later, captures the main fea-
/ s-wavetransitionacrosstheoptimaldoping. Foroptimal tures of quasiparticle excitations in the superconducting
t
a and overdoped samples, they found that the low tem- state and gives a unified account for the experimental
m
perature superfluid density exhibits an exponential tem- data. Ourresultsuggeststhatthesuperconductingpair-
- perature dependence, in favor of an anisotropic s-wave ing in electron-doped materials is governed by the same
d
pairing state. mechanism as in hole-doped ones, although their phase
n
o diagrams look asymmetric.
The above discrepancy indicates that low lying quasi-
c The two-band model we study is defined by
: particle excitations in electron-doped cuprates behave
v
i quite differently than in hole doped ones. To resolve the H = ξikc†ikσcikσ + Vikk′c†ik′↑c†i−k′↓ci−k↑cik↓
X discrepancy, a thorough understanding of the electronic Xikσ Xikk′
ar srterguacrtdu,rtehoefdeolepcitnrgone-vdoolpuetidonmoaftetrhiaelsFiesrmdeissiruerdfa.cIen(tFhSis) + V3kk′c†1k′↑c†1−k′↓c2−k↑c2k↓+h.c. , (1)
Xkk′ (cid:16) (cid:17)
revealedbythe ARPESofNd2−xCexCuO4 (NCCO)[12]
is of great interest. At low doping, a small FS pocket where i = 1,2 represents the band around (π,0) and
first appears around (π,0), in contrast to the hole dop- that around (π/2,π/2), respectively. c1kσ and c2kσ are
ing case where the low-lying states are centered around thecorrespondingelectronoperators. V1kk′ andV2kk′ are
(π/2,π/2)[13]. Byfurtherdoping,anotherpocketbegins the reduced pairing potentials for the two bands. V3kk′
to form around (π/2,π/2). The presence of the two sep- is the interband pair interaction. This model has also
arateFS pockets mayresult fromthe band folding effect been used to describe superconducting properties of the
induced by the antiferromagnetic correlations[14, 15]. It two-bandsuperconductorMgB2[24]. InMgB2,the inter-
mayalsobeamanifestationofthelowerandupperHub- band coupling is weak since the two relevant bands have
bard bands[16]. At the mean field level, the theoretical different parity symmetry [25]. In the present case, the
calculations indicate that these two FS pockets can be interband coupling is also weak since the strong antifer-
2
romagneticfluctuationsdonotcouplethefirstbandwith functions. This will suppress, for example, the tempera-
the second one in electron-doped cuprates. turedependenceofthenormalizedsuperfluiddensityand
Inelectron-dopedmaterials,the superconductivityoc- aggrandize the experimental difficulty in identifying the
curs at much higher doping than in hole doped ones. expected power law behavior for a d-wave superconduc-
However, as shown by the ARPES experiments, the ap- tor.
pearanceofthesuperconductingphasecoincideswiththe The superfluid density is inversely proportionalto the
appearance of the second band at the Fermi level. This squareofthe magneticpenetrationdepth,i.e.,ρ ∝λ−2.
s
reveals a close resemblance between electron- and hole- Underthe BCSmean-fielddecomposition,the superfluid
doped materials. It suggests that it is the interaction densityofthe systemissimplyasumofthe contribution
driving the second band to superconduct that leads the from each band and can be expressed as
whole system to superconduct in electron-doped mate-
rials, and that the pairing potential V2kk′ has predomi- ρs(T)=ρs,1(T)+ρs,2(T), (2)
nantlydx2−y2 symmetry,resemblingtheholedopedcase.
where ρ is the superfluid density of the i’th band. It
V1kk′ can in principle be different to V2kk′. However, s,i
can be evaluated with the formula given in Ref. [26]. In
if pairing in the first band is originated from the same
lowtemperatures,since thereis afinite gapinthe quasi-
mechanismas the second bandor induced by the second
bandbytheproximityeffect,V1kk′ shouldmostprobably particle excitations of the first band, ρs,1(T) is expected
to be given by
have dx2−y2 symmetry.
1,2In,3t)hceacnalaclullbateiofancstobreilzoewd,: wVe1kaks′su=mge1γthkγakt′V,ikVk2′kk(i′ == ρs,1(T)∼ρs,1(0)(cid:16)1−ae−∆′1/kBT(cid:17), (3)
g2γkγk′ and V3kk′ = g3γkγk′, where g1, g2, and g3 are
the corresponding coupling constants, γ1k = γ2k = γk = where∆′1 istheminimumvalueof∆1γk ontheFSofthe
coskx−cosky is the dx2−y2-wavepairing function. Here first band and a is a constant. There are gap nodes in
we have implicitly assumed that the first band has the the second band, therefore ρs,2 should behave similarly
same pairingsymmetryas the secondone. This assump- as in a pure d-wave superconductor and show a linear T
tion can in fact be relaxed. The qualitative conclusion dependence in low temperatures due to the low energy
draw below does not depend much on the detailed form linear density of states:
ofthe pairingfunctionforthe firstbandnear(π,0), pro-
T
vided there are no gap nodes on the FS of this band. ρs,2(T)∼ρs,2(0) 1− . (4)
(cid:18) T (cid:19)
Taking the BCS mean field approximation, the inter- c
actionbetweenthetwobandsisdecoupled. Itisstraight-
Thus, in the limit T ≪ T , the normalized total super-
c
forward to show that the quasiparticle eigenspectrum of
fluid density is approximately given by
the i’th band is given by the following expression E =
ik
ξi2k+∆2iγk2, where ∆i is the gap amplitude of the i’th ρs(T) ≈1− ρs,2(0) T − ρs,1(0)ae−∆′1/kBT (5)
bpand. Theyaredeterminedbythe followingcoupledgap ρ (0) ρ (0) T ρ (0)
s s c s
equations ∆1 = kγk(g1hc1−k↓c1k↑i + g3hc2−k↓c2k↑i)
and ∆2 = kγk(Pg2hc2−k↓c2k↑i+g3hc1−k↓c1k↑i), where where ρs(0)=ρs,1(0)+ρs,2(0).
h···i denotePs thermal average. For a pure d-wave superconductor, as shown by Eq.
The above expression of E indicates that there are (4), the slope of the linear T term in the normalized su-
ik
gap nodes in the quasiparticle excitations of the second perfluid density is proportional to 1/T . However, for
c
band. However, there is a finite excitation gap in the the coupled two-band system considered here, this lin-
first band since the nodal lines of γk do not intersect ear slope is normalized by a factor ρs,2(0)/ρs(0). The
with the FS of that band if the system is not heavily zero temperature superfluid density ρ (0) is a measure
s,i
overdoped. Therefore, as far as thermal excitations are of the diamagnetic response in the i’th band. It is ap-
concerned, the first band behaves as in a s-wave super- proximatelyproportionaltotheratiobetweenthecharge
conductor, although the pairing is of dx2−y2 symmetry. carrierconcentrationandtheeffectivemassinthatband,
Thisindicatesthatthesuperconductingstateofelectron- i.e., ρ (0)∝n /m∗. It is difficult to estimate this ratio
s,i i i
doped cuprates is actually a mixture of d-wave and s- for each individual band. However, as the FS pocket of
wave-like pairing states. Apparently, the low tempera- thefirstbandappearsimmediatelyafterdopingandthat
ture/energy behavior of quasiparticle excitations is gov- ofthesecondbandappearsonlyafterthelongrangeanti-
ernedbythesecondbandsincethefirstbandisthermally ferromagnetic order is completely suppressed, one would
activated. This would naturally explain why the typical expect ρs,2(0) to be much smaller than ρs,1(0). This
d-wave behaviors were observed in quite many experi- means that ρs,2(0)/ρs(0) ≪ 1 and the linear T term in
ments [2, 3, 4, 5, 6]. However, the presence of the first ρ (T) is greatly suppressed. Thus the low temperature
s
band will change the relative contribution of the second curve of the normalized superfluid density looks much
band to the superfluid as well as other thermodynamic flatter than in a pure d-wave system, although ρ (T) is
s
3
behave similarly as in a conventional d-wave supercon-

ductor. This picture for the doping dependence of low
40
40 temperatureρs(T)/ρs(0)agreesqualitativelywithallex-
x = 0.152 K) perimental observations.
30 D(’12300
2 )
- mm x = 0.131 10 1.0
-2l(T) ( 20 x = 0.124 00.11 0.12 0D.1o3pin0g.1 x4 0.15 0) 0.8 r rs
( s, 1
s
r
10 Fits T) / 0.6
(s0.4
r
0
0 5 10 15 20 25 r
0.2 s, 2
T (K)
FIG. 1: Fitting curves of Eq. (7) to the low temperature
0.0
superfluid density data published in Ref. [11] for x = 0.124,
0.131 and 0.152. The inset shows the doping dependence of 0.0 0.2 0.4 0.6 0.8 1.0
thefitting parameter ∆′1. T/Tc
FIG. 2: Illustration of the contributions from the two bands
to thesuperfluid density in electron-doped cuprates.
stillgovernedbyapowerlawT dependenceatsufficiently
low temperatures.
Close to T , a positive curvature will appear in ρ (T).
c s
In real materials, the low temperature dependence of
This is a simple but universal property of a weakly cou-
ρ (T)/ρ (0) will be further suppressed by impurity scat-
s s pled two-band system[20]. To understand this, let us
tering and the linear term will be replaced by a T2 term
firstconsiderthecaseg3 =0. Inthiscase,thetwobands
in the limit T ≪Γ0 [27] aredecoupledandwillbecomesuperconductingindepen-
k2T2 dently. Let us denote their transition temperatures by
ρs,2(T)∼ρs,2(0)(cid:18)1− 6πBΓ0∆2(cid:19), (6) Tg3c0,1tahnedsuTpc02eracnonddauscstuinmgetTrac0n1s<itioTnc02w. iFlloorcficunriteatbautcrsimticaalll
whereΓ0 is the scatteringrate. Inthis case,ρs(T)/ρs(0) temperature closeto Tc02, i.e. Tc ∼Tc02 (Fig. 2). Justbe-
becomes low Tc, ρs is mainly contributed from the second band.
However,whenT dropsbelowT0,theintrinsicsupercon-
ρs(T) ≈1− ρs,2(0) kB2T2 − ρs,1(0)ae−∆′1/kBT. (7) ducting correlation of the first bc1and will appear in addi-
ρs(0) ρs(0) 6πΓ0∆2 ρs(0) tion to the induced one, and the contribution to ρs(T)
fromthisbandwillriserapidlywithdecreasingtempera-
We believe this formula capturesthe mainfeature oflow
ture. Consequently,aclearupturnwillshowupinρ (T)
temperaturesuperfluiddensity. Indeed,byfittingtheex- s
around T0. The appearance of a positive curvature in
perimental data with the aboveequation, we find that it c1
the experimental data of ρ (T), as already mentioned, is
does give a good accountfor the low temperature super- s
a strong support to the two-band picture.
fluid in the whole doping range. This can be seen from
To calculate explicitly the temperature dependence of
Fig. 1 where the fitting curves of Eq. (7) to the mea-
ρ in the whole temperature range, one needs to know
surementdata published inRef. [11] areshownfor three s
the band dispersion ξ . For this purpose, we adopt
representative doping cases in the under-, optimal and ik
the expressions first proposed by Kusko et al. [16]
over-doping regimes, respectively.
Inelectron-dopedmaterials,dopingwillreducethedis- ξik =± εi,k+εi,k+Q± (εi,k−εi,k+Q)2+4δ2 /2−µi
tance between the FS of the first band and the nodal where ±(cid:16)corresponds topthe first/second band(cid:17), ε =
ik
lines of γ . At low doping, the contribution from the −2t (cosk + cosk ) − 4t′cosk cosk − 4t′′(cos2k +
k i x y i x y i x
exponential term is small and the T2 term is dominant. cos2k −1), t′ = −0.25t and t′′ = 0.2t . Q = (π,π)
y i i i i
By further doping, ∆′ begins to drop (the inset of Fig. is the antiferromagnetic wave vector and here δ is taken
1
1), the contribution from the exponential term becomes as a constant. µ is the chemical potential determined
i
comparable with the T2 term in certain low tempera- by the occupation number for each band. It was shown
ture regime. In this case, the T2 dependence of ρ (T) thattheFScontoursdeterminedfromthisformulaagree
s
would become difficult to be identified if the exponen- qualitativelywiththeARPESdata[15,16]. Followingthe
tial term is not clearly separated. In heavily overdoped suggestion of Ref. [17, 18], we assume that the second
regime,theFSofthefirstbandwillstrideoverthenodal band is hole-like. The doping concentration is therefore
lines of γ . In this case, ∆′ =0 and ρ (T)/ρ (0) should given by the difference x = n −n , where n and n
k 1 s s e h e h
4
further detect the gap structure in electron-doped ma-

terials. The scanning tunneling measurement that was
1.0 used for testing the two-band nature of MgB2 from the
vortex core state along the c-axis[28], for example, can
) 0.8 x = 0.131 be used to examine the two-gap picture here. Since the
0
(s interlayer hopping is highly anisotropic [26] and the c-
r
/ 0.6 x = 0.124 axistunnelingcurrentiscontributedmainlyfromthefirst
T) band,this measurementwouldallowus todetermine the
r(s0.4 coherencelengthofthefirstbandfromthespatialexten-
x = 0.152 sionofthe vortexcore. Comparingitwiththe coherence
0.2 lengthofthe secondbandwhichcanbe determinedfrom
the measurement of Hc2, this will provide a direct test
0.0 for our two-band theory.
0.0 0.2 0.4 0.6 0.8 1.0 In conclusion, we showed that the temperature de-
T/Tc pendence of ρs in electron-doped cuprate superconduc-
torscanbe wellexplainedbyaweaklycoupledtwo-band
FIG. 3: Comparison between theoretical calculations (lines)
model. Our work resolves the discrepancy in the inter-
andexperimentaldata(symbols)[11]forthetemperaturede-
pendenceofthenormalized superfluiddensityρs(T)/ρs(0)of pretation of different measurement results. It suggests
PCCO at three different doping levels. thatthe pairingpotentialinelectron-dopedcuprateshas
dx2−y2 symmetry in the whole doping range, same as in
hole-doped materials.
are the carrierconcentrations of the first and the second We are grateful to T. R. Lemberger, D. H. Lee and
bands, respectively. However, it should be emphasized L. Yu for helpful discussions, and M. S. Kim and T. R.
that similar results can also be obtained if both bands Lembergerforkindlyprovidingustheexperimentaldata.
are electron-like. ThisworkwassupportedbytheNationalNaturalScience
Fig. 3 compares the theoretical results of superfluid Foundation of China.
density for a pure system with the corresponding ex-
perimental data (symbols) [11] for x = 0.124, 0.131,
and 0.152. The parameters used are t1 = 5, t2 =
1, (g1,g2,g3,ne,nh) = (1.3,1.082,0.005,0.214,0.09) for
x = 0.124, (1.3,1,0.01,0.231,0.1) for x = 0.131, and [1] C. C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, 969
(2000), and references therein.
(1.3,0.984,0.001,0.261,0.11) for x = 0.152. As can be
[2] T. Sato et al.,Science 291, 1517 (2001).
seen, the overall agreement between theoretical calcu-
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[10] A. Snezhkoet al.,Phys. Rev.Lett. 92, 157005 (2004).
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[24] N.Nakai,M.Ichioka,andK.Machida,J.Phys.Soc.Jpn.,
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5
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