Nernst effect and Critical Field in Cuprate Superconductors

0.26 MB·English

by  D. N. Dias

Magnetic Properties Calculations of Cuprate Superconductors based on a Phase
Separation Theory
D. N. Dias1, E. S. Caixeiro2, and E. V. L. de Mello1
1Instituto de F´ısica, Universidade Federal Fluminense,
Niter´oi, RJ 24210-340, Brazil
7
and
0
2Centro Brasileiro de Pesquisas F´ısicas,
0
Rio de Janeiro, RJ 22290-180 Brazil
2

n
(Dated: February 6, 2008)
a
J There is a great debate concerning the hole of the inhomogeneities in high critical temperature

9 superconductors (HTS). In this context, there are many experiments and proposals related with
1 a possible electronic phase separation (PS). However there is not a method to quantify how such
transitionoccursandhowitdevelops. TheCahn-Hilliard(CH)theoryofphaseseparation provides

] a way which we can trace the phase separation process as a function of temperature in agreement
n
withsomeexperiments. Herewecoupledthesecalculations, withparametersthatyieldastripelike
o
pattern,totheBogoliubov-deGennes(BdG)approachtoaninhomogeneoussuperconductorinorder
c
- toderivemanyHTSpropertiesoftheLa2−xSrCuO4 (LSCO)system. Takingtheupperpseudogap
r asthePStransition line, wecan show that;theonset ofsuperconductivityfollows close theNernst
p
signal, theleading edge shift is close to thezero temperature average gap and the superconducting
u
phaseis achievedbypercolation orJosephson coupling. Ourapproach isalso suitable toreproduce
s
. theexperimental measurements of the Hc2 field and explain why it does not vanish above Tc.
t
a
m PACSnumbers: 74.72.-h,74.80.-g,74.20.De,02.70.Bf
-
d
I. INTRODUCTION metallic phase and an oxygen-poor antiferromagnetic
n
(AF) phaseaboveT=220K.Thereis alsoevidence ofion
o
c diffusion at room temperature in micro crystals of the
The measurement of the pseudogap and many non-
[ Bi2212 superconductors at a very slow rate[17].
conventionalpropertiesinthe normalphase[1,2]ofHTS

2 has become a long standing puzzle. It is a general con- Recent angle resolved photoemission (ARPES) ex-
v sensus that understanding these properties is crucial to periments with improved energy and momentum
5 comprehendthenatureofthesuperconductingtransition resolution[18, 19, 20, 21, 22] have distinguished two
7 and the fundamental interaction in these materials. It is electronic components in ~k-space associated with the
0
7 quite interesting that while low critical temperature su- La2−xSrxCuO4 (LSCO) system: a metallic quasi par-
0 perconductors have a well characterized normal phase, ticle spectral weight at the (π/2,π/2) nodal direction
6 the nature of the normal or pseudogap phase of HTS is which increases with hole doping and an insulator like
0 an open problem which remains a matter of intense ex- spectral weight at the end of the Brillouin zone straight
t/ perimental and theoretical study[3, 4]. segments in the (π,0) and (0,π) antinodal regions which
a are almost insensitive to the doping level.
m One of the possible reason why this problem remains
unsolvedafter20yearsofintenseresearchmaybe dueto New STM data with great resolution have also re-
-
d the uncontrolled intrinsic inhomogeneities in some HTS, vealedstronginhomogeneitiesintheformofapatchwork
n which as some of us have argued[5], may also depend of (nanoscale) local spatial variations in the density of
o on the sample preparation method. Since the theoreti- states which, at low temperature, is related to the lo-
c calprediction[6]andthedetectionofstripes[7],itisclear calsuperconducting gap[23, 24, 25, 26]. More recently it
:
v thatatleastsomefamily ofcompounds exhibitsome de- waspossibletodistinguishtwodistinctbehavior: wellde-
i gree of non-uniformities. In fact, the role of the inhomo- finedcoherentandill-definedincoherentpeaksdepending
X
geneities is an open question: it seems not important in on the exactly spectra location at a Bi2Sr2CaCu2O8+δ
r
a some experiments[8, 9], but on the other hand, the de- (Bi2212) surface[27, 28]. STM experiments have also
tection and the effect of inhomogeneities have rendered detected a regular low energy checkerboard order in
many articles and books[10, 11], as we will discuss in the electronic structure of the Bi2212 family at low
detail in this paper. temperature[29],abovethesuperconductingcriticaltem-
These unusual features of cuprates, like the stripes, perature (Tc)[30] and in the NaxCa2−xCuO2Cl2[31].
led to theoretical proposals that PS is essential to un- Bulk sensitive experiments like nuclear magnetic and
derstand their physics[6, 12, 13, 14]. Indeed PS has quadrupolar resonance (NMR and NQR) have also pro-
been observed on the La2CuO4+δ by x-ray and trans- vided ample evidence for spatial charge inhomogeneity
port measurements[15, 16], which have detected a spin- in the CuO2 of some HTS planes[32, 33, 34]. Singer
odal phase segregationinto an oxygen-rich(or hole-rich) atal[33]measuredadistribution ofT1 overthe CuNQR
2
spectruminbulkLSCOwhichcanbeattributedtoadis- clearlydetectedbymanyspectroscopyexperimentsstart-
∗
tributionofholespwithahalfwidthof∆p/p≈0.5. The ing at a temperature that we call T , the lower pseudo-
increasingof ∆p/p asthe temperature decreasesis likely gap temperature, which, depending on the compound,
the strongest indication of a PS transition at tempera- can vary from T∗ ≈ T to roughly 100K above T [4, 43,
c c
turesabove600KinHTS.Morerecently,NMRresultson 44, 45]. On the other hand, Nernst effect experiments
La1.8−xEu0.2SrCuO4 were interpreted as evidence for a which measure a voltage transverse to a thermal gradi-
spatially inhomogeneous charge distribution in a system ent in the presence of a magnetic field perpendicular to
whichthe spinfluctuations aresuppressed[35]. This new the superconducting film are specially sensitive to the
result is also a clear indication that the charge disorder existence and the drift of vortices[37, 38, 39]. Conse-
may be due to a phase separation transition. quently a large Nernst signal above T , which starts at
c
∗
In this paper we work out in detail the scenarioto the Tonset ≈T ,wastakenasastrongindicationofavortex-
physics of HTS based on a PS transition at the (upper) like behavior and the presence of a large region of fluc-
pseudogap curve given in the review of Tallon at al[2], tuating superconductivity[37, 38, 39, 40]. Thus the im-
whichstartsatveryhightemperatures(≈1000K)inthe portantquestioniswhetherthisfluctuatingregionisdue
low underdopedregionandfalls to zeronear the average tothethermallyinducedvorticesofthephasedisordered
doping level ρ = 0.2. This PS transition is treated by scenarioorto someothermechanism,likea non-uniform
m
theCHtheory[36],originallyproposedtodescribethePS density.
transition in alloys, and yields two equilibria densities;
one lowandotherwith highvalueswhichgrowsapartas
the temperature is lowered, exactly as seen in the NQR
Recently, in order to address this question and to
experiments[33] or, indirectly by the stripes structures.
study the pseudogap region, a series of combined mea-
Applying the BdG theoryof superconductivity to sucha
surements on the Nernst effect, the upper critical field
disordered medium we see that, as the temperature de-
creases, the superconductivity appears in nanoscale re- Hc2 and magnetization above the Tc on different com-
pounds were performed by Wang et al[41]. The re-
gions inside the high density phase. With this approach,
sults wereinterpretedas providingstrongsupportto the
we can show that:
phase-disordered theory[41]. On the other hand, Nernst
i-thezerotemperatureaveragesuperconductinggapand
effect studies carried on the presence of induced disor-
the amplitude of pairing |∆| as function of the average
dered demonstrated that T remained basically the
doping level ρ is in reasonable agreement with the su- onset
m
same but T decreasedconsiderably with the presence of
perconducting state gap[18] and the leading edge shift c
controlleddefects[42]andtheyhavedetectedfluctuations
measured by ARPES[22].
only near T , contrary to the phase disordered scenario.
ii-The onset temperature of superconductivity for each c
A possible interpretation is that the additional induced
compound is close the lower pseudogap temperature[1,
disorder hinders the percolation threshold, but does not
4,5]andalsotherecentmeasurementsontheonsettem-
affect the onset temperature (T ) of superconductiv-
perature of Nernst signal[37, 38, 39], providing a simple onset
ity in the nanoscale islands.This result confirms a wide
interpretation to these experiments.
variety of experimental data and theoretical investiga-
iii-ThecalculationofHc2(T)forasystemofnon-uniform
tions which have demonstrated that spatially inhomo-
densityregionswithdifferentlocalsuperconductingT (i)
c
geneities strongly affects the HTS properties[46, 47].
(T (i) will be defined below) is in excellent agreement
c
with the measurements in the LSCO system and also
provides an explanation why the Hc2(T) field does not
vanish at T .
c Thus, taking the spatial disorder as an intrinsic phe-
Our proposal is an alternative scenario to the phase- nomenaassociatedwithaPStransition,wereproducein
disordered theory[13, 14] which has gained increased this paper the main phase diagram boundaries, the up-
attention[37, 38, 39, 40, 41, 42]. Contrary to the famil- perandlowerpseudogapandthesuperconductingphase.
iar BCS theory in which the complex superconducting We also show that the results of Wang et al[41] are also
order parameter Ψ = |∆|eiθ develops with the phase θ compatible with a disordered materialwith non-uniform
essentially locked, in their calculations[13, 14] thermally doping level which can take many forms, from random
generatedvorticesdestroylongrangephasecoherenceat patches to stripes. For this purpose, this paper is orga-
temperatures close to the superconducting critical tem- nized as follows: In section II, we described briefly how
perature Tc. The temperature Tθ which phase rigidity the CH PS calculations are made (the details are in pre-
is lost was estimated to be very low in the underdoped vious paper[48, 49, 50]). In section III, we perform the
region and to increase continuously with doping[13, 14]. BdG superconducting calculation on a system which re-
Therefore, in this scenario,the pseudogapphase is char- sults from the PS. In section IV, we show the results of
acterizedbythepresenceofCooperpairswithnonvanish- both methods combined. In section V, we generalize a
ing pairing amplitude |∆| but, due to thermally excited method to calculate the Hc2 field in a non-uniform sys-
vortices (in zero field), without phase rigidity. tem, again applied to the CH PS results. We finish with
The presence of the pairing amplitude |∆| has been the conclusions.
3
II. THE PHASE SEPARATION ature T. On cooling down the system, the ∆(i,T) start
at temperatures T (i) and increase as the temperature
c
tends to zero. We find that the ∆(i,T) have basically
At least two clearly distinct energy scales are associ-
the same value in small regions with the same charge
ated with the pseudogap[4, 5, 51] and there are several
density. This defines what we call a local superconduct-
indications that the upper pseudogap, which starts at
ing temperature T (i) on a site ”i” or in a small clus-
very low doping and ends near ρ ≈ 0.2[2], may be a c
m
ter. Following the CH results, particularly those which
line of PS(part ofthis PS line is representedin Fig.(2)).
∗ yield stripe patterns, we have kept fixed the input local
Thevaluesofthe upperpseudogapT ,measuredbysus-
chargedensities,whichisanapproachdifferentthanpre-
ceptibility, heat capacity, ARPES, NMR and resistivity,
viousBdGcalculations[53,54,55,56]. Thedetailscanbe
as presented in the review by Tallon and Loram[2], or
found elsewhere[57], but just for completeness, the BdG
the crossover line shown in Timusk and Sttat[1], seem
equations are,
to be independent of the superconductivity phase[2]. In
fact it was verified recently that this line falls inside the
ξ ∆ u (r ) u (r )
sguappetrecmonpdeuracttiunrgesdoinmteh[5e2u].ndTehredovpereydhriegghiounplpeedraplsseouLdeoe- (cid:18)∆∗ −ξ∗(cid:19)(cid:18)vnn(rii)(cid:19)=En(cid:18)vnn(rii)(cid:19), (1)
et al[3] to argue that it is very unlike any relation to the
where E ≥0 are the quasiparticles energies, and
n
superconducting pair formation or to the Nernst onset
temperature T . As mentioned, a strongevidence to-
onset
wardsaspatialPSisthemeasurementsofT1 overtheCu ξun(ri) = − ti,i+δun(ri+δ)−µiun(ri),
NQR spectrum of under and optimally average doping Xδ
ρm samples of bulk LSCO by Singer at al[33], whichwas ∆u (r ) = ∆ (r )+ ∆ (r )u (re +δ), (2)
attributedtodistributionsoflocalholesρ(i)withwidths n i U i δ i n i
of ∆(ρ(i))/ρ ≈ 0.5. They have also showed that the Xδ
m
hole segregation increases as the temperature decreases thet’sarethehoppingparametersandµ thelocalchem-
i
whichmaybecome closetoa bimodalwith twobranches icalpotential. Therearealsosimilarequationsforv (r ).
n i
whose densities are ρm±∆(ρ(i)). When these BdG equations are solveed, they give the
In order to harmonize these observations with theo- eigen-energies E and the local amplitudes u (r ) and
n n i
retical calculations,we need a model to furnish the local v (r ). Throughtheseamplitudesandthepositiveeigen-
n i
densitiesindopeddisorderedsystems. TheCHapproach energiesitis possible to calculatethe pairingamplitudes
is very convenient because, in general, it describes how
the charges tend to a bimodal distribution below a crit- ∆ (r )=−U u (r )v∗(r )tanh En , (3)
ical temperature Tps(ρm), similar to what was observed U i Xn n i n i 2KBT
by the NQR measurements[33]. Thus we assume that
T (ρ ) is close to the upper pseudogap line mentioned
ps m
above[2,52]. Aswehavediscussedpreviously[48,49,50], ∆ (r ) = −V [u (r )v∗(r +δ)
thePSprocesscanleadtodifferentsegregationpatterns, δ i 2 n i n i
Xn
dependingontheGinzburg-Landau(GL)freeenergyini-
E
tialcoefficients. Sincetheseparametersandthe mobility +v∗(r )u (r +δ)]tanh n (4)
n i n i 2K T
(which is relatedto the time scaleof the PS process)are B
not well known, we adopt values which leads to charge
and the local hole density is given by
stripes formation,in orderto reproducethe observations
inLSCO.Thepossibilitiesofotherpatterns,likedroplets ρ(r )=1−2 [|u (r )|2f +|v (r )|2(1−f )], (5)
i n i n n i n
or patchwork was studied elsewhere[48]. Thus, depend-
Xn
ing on the initial parameters of the GL free energy, the
phase segregation process can form patterns similar to where fn is the Fermi function. The BdG equations (1)
patchwork[24,26]orstripe[7]whichareobservedinHTS are solved self-consistently together with the equations
and in others high correlated electron systems[46, 47]. forthe pairingamplitudes(Eq.(4))andforthe localhole
In what follows, we will take these PS results as the ini- density(Eq.(5))whichiskeptfixedthroughoutthe entire
tialnon-uniforminputchargedistributiononclustersfol- calculation. For calculations using d-wave symmetry we
lowed by the BdG local superconducting calculation. have V <0 and U >0 and Eq.(4) can be written as[54]
1
∆d(ri)= [∆xb(ri)+∆−xb(ri)−∆yb(ri)−∆−yb(ri)]. (6)
III. THE LOCAL GAP CALCULATIONS 4
In the calculations we have used the hopping parame-
At temperatures below the PS transition, we can ob- ters up to third neighbors, close to those derived from
serve the possibility of the local superconducting pair- the ARPES data for YBCO[58], that is, t = 0.23eV
ing amplitude ∆(i,T) formationat a location ri inside a (first neighbors), t2 = −0.61t and t3 = 0.2t. These val-
givencluster[5, 53, 54, 55, 56] as function of the temper- ues are slightly different from our other work[57], which
4
we used hopping values up to 5th neighbors following FollowingthechargepatternsderivedfromtheCHcal-
the ARPES results[58]. In fact we made various studies culations, the 7 stripes at the left are characterized by
aroundtheARPESdata,andtheyallgivethesamequal- local doping ρ(i) ≈ 0 and the 7 at the right side have
itative results,andtherefore,we presentthe calculations ρ(i)≈2ρ . This is the scheme of the three upper panel
m
∗
thatyieldedthebestquantitativeagreementwiththeex- of Fig.(1). The high values of the T ≈T implies that
ps
periments. The potentials are U = 1.1t and V = −0.6t lightly doped compounds like the ρ = 0.05 has ρ(i)
m
which are also close to previous calculations[56]. The strictly zero in the low doping region. At low temper-
value of t=0.23eV was chosenin orderto reproduce[57] atures, the superconducting regions for this compound,
themeasurementsofthezerotemperaturesuperconduct- those with a finite value of ∆(i,T), develops only in the
ing gap by Harris et al[18] and the ARPES leading edge high density region and all together, they never reach
shift by Ino at al[22]. Furthermore this value of t is in more than 50% of the total sites. Consequently, the su-
the range of several experiments on HTS[59]. perconducting sites never percolate and there is no su-
perconducting phase for this compound. For doping of
ρ > 0.06 the low doping sites have some residual fluc-
m
IV. RESULTS tuation(of the phase separation process) ρ(i) > 0 which
grows with ρ , from ≈ 0.03 to 0.05, what changes the
m
properties of a sample from a disordered insulator into
a disorderedmetal, with a superconducting phase atlow

temperatures. Thus, for compounds with 0.06 < ρ <
m
0.012
10K m=0.05 0.20, at low temperatures, one can see in Fig.(1), that
20K
0.008 50K ∆(i,T)developsalsoatthe verylowdopingregions,i.e.,
60K
in the left region of the clusters represented in Fig.(1).
0.004 70K
At the temperature Tc(ρm), when the ∆(i,T) arise at
0.000
theselowdopingsites,thesystembecomessuperconduc-
1200KK Tc(m) m=0.08 0.015 tor either by the percolation of the many local super-
70K 0.010 conducting regions (the regions where ∆(i,T) are non-
80K
90K 0.005 vanishing) or by Josephson coupling[60, 61]. The values
(i,T) 0.000 of Tc(ρm) are shown in each panel of Fig.(1). Conse-

0.015 1200KK m=0.15 quently, below Tc(ρm), superconducting critical temper-
0.010 35K Tc(m) ature, the system can hold a dissipation-less current, as-
60K suming, as usual in mean field (BCS) theories, that all
0.005 80K
90K these superconducting regions form a conventional su-
0.000
0.012 perconducting phase with a unique phase θ which is es-
10K m=0.22
20K Tc(m) 0.008 sential to percolation and Josephson coupling. Above
30K T (ρ ) the compounds form a mixture of superconduct-
c m
0.004 ing,insulatorandnormaldomainsandabovethepairing
50 100 150 0.000 formation temperature Tonset(ρm), they are disordered
metals with mixtures of normal (ρ(i)≥0.03−0.05) and
i=N/a
insulator (ρ(i)≤0.03−0.05) regions.
FIG.1: (coloronline)Temperatureevolutionofthelocalpair-
Fromtheseresults,weidentify T (ρ )asthe high-
ing amplitude ∆(i,T),in unitsof t=0.23eV at each site ”i” onset m
est temperature to induce a ∆(i,T) in any region of a
of a 14×14 cluster with 196 sites. Because the CH phase
separation for ρm ≤ 0.20 and the stripe pattern, the sites in given compound which is easily seen from the panels of
the left have ρ(i) ≈ 0 and the ones in the right ρ(i) ≈ 2ρm. Fig.(1)(andfromsimilarstudiesonothersvaluesofρm).
In contrast, ρm =0.22 which is beyond the phase separation Thus Tonset(ρm) is identified with the onset of Nernst
limit Tps, has just a Gaussian charge disorder and its profile signal because the rising of superconducting regions in
is very different. a metallic matrix increases the vortices drift. The val-
ues of T (ρ ) and T (ρ ) are shown in Fig.(2). The
onset m c m
maximumpairingamplitudesforeachρ atlowtemper-
Lets apply our method to the LSCO system. We then m
perform calculations with stripe disordered clusters as ature (∆0(pm)) is in good agreement with the ARPES
zero temperature leading edge shift[57] or the maximum
derived from a set of parameters in the CH approach
magnitude of the superconducting gap[18, 22].
and keep this initial doping configuration fixed in the
temperature range of our calculations. As we have al- The zero temperature gap can be also obtained by
ready mentioned, the T (ρ ) ends near ρ ≥ 0.20[2], a different procedure through the study of the local
ps m m
and for higher density values there is not any PS, only density of states (LDOS), which is given by Ni(E) =
smallfluctuations aroundρ . Fig.(1)showsthe temper- [|u (x )|2f′(E −E )+|v (x )|2f′(E +E )], where
m n n i n n n i n n
ature evolution of the superconducting local pair ampli- tPhe primeisthe derivativewithrespecttothe argument.
tudes∆(ri,T)orsimply∆(i,T)ateachsiteiinasquare In a typical cluster of our calculations, the opening of
mesh 14×14. this maximum ∆(i,T), occurs at the neighbor of site
5
ρ =0.03 ρ =0.08 ρ =0.12 ρ =0.22
m m m m
140 196
120 Tonset( m) Data
Tonset( m)
100 Tonset( m) LDOS
Tc( m) Data
T[K] 80 TTcP(Sm) Site
60
40
20
0 1
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 −0.05 0.05 −0.05 0 0.05 −0.05 0 0.05−0.02 0 0.02
E/t
m
FIG. 3: The study of the superconducting gap from the first
FIG. 2: The local superconducting onset temperature taken peak of the local density of states (LDOS) at zero tempera-
as the Nernst signal (solid line, from Ref.14) and the super- ture. At ρm =0.8 the superconductinggap is of theorder of
conducting percolation temperature Tc(ρm). The black cir- 20meVanddecreases asρm increases, whichagreeswiththe
cles and squares are the experimental data of Tonset and Tc, measurements made bythe ARPES[18, 22]
respectively. The triangles are the values derived from the
densityofstates. ThephaseseparationlineTps isalsoshown.
we follow along the lines described by Caixeiro et al[62].
The GL upper critical field of a homogeneous supercon-
i=127 (see, for instance, the compound with ρm =0.15 ductor may be written as[62]
inFig.(1))andT (ρ )wasdefinedasthetemperature
onset m
whichsuchlocalsuperconductinggapvanishes. Now, by
tahnedltoackaeltdheenssiutypeorfcostnadtuecs,tiwnge gcaanpaansatlhyezemtihneimloucmalvgaalupes Hc2(T) = 2πξΦa20b(0)(cid:18)TcT−c T(cid:19). (T <Tc) (7)
of E with a non-vanishing spectral weight, as shown in
n LetusnowapplythisexpressiontoaHTSwithstripes
Fig.(3). The corresponding temperature which this su-
inhomogeneities in the charge distribution. As showed
perconductinggapvanishesisalsoshowninFig.(2)andit
in Fig.(1) , when a superconducting amplitude develops
isquitesimilartoT (ρ ),infact,webelievetheywill
onset m in a given region “ri”, it has a local superconducting
be equal for larger clusters. Thus, both ways of defining
temperature T (i) and it will contribute to the critical
theonsettemperatureofsuperconductinggapofthesys- c
field with a locallinear upper criticalfield Hi (T) below
tem are in good agreement with the Nernst signal onset c2
T (i). Therefore, the total contribution of the various
temperature. c
local superconducting regions to the upper critical field
is the sum of all the Hi (T)’s at temperatures below
c2
T (ρ ), which is the temperatures that the system
onset m
V. THE CRITICAL FIELD
starts to develop some superconducting region. In this
way, the Hc2 for a whole sample is
Thepresenceofsuperconductingregionswithdifferent
critical temperature will affect the magnetic response.
W
Furthermore, those regions with the local critical tem- Φ0 1 Tc(i)−T
perature Tc(i) above the superconducting critical tem- Hc2(T)= 2πξa2b(0)W Xi=1(cid:18) Tc(i) (cid:19)
perature T (ρ ) greatly affects the normal state prop-
c m
W
erties. Such effects have been seen by many measure- 1
= Hi (T) (T <T (i)≤T (ρ )) (8)
ments of anomalous magnetic properties of the normal W c2 c onset m
Xi=1
phase[62, 63, 64, 65]. In particular, the anomalous up-
per critical field Hc2 was recently measured by Wang et where W is the number of superconducting regions,(the
al[41] on the LSCO system. high hole densities stripes), each with its local T (i)≤
c
A few years ago some of us have developed a way to T (ρ ). For the LSCO series a coherence length of
onset m
calculate Hc2 of a disordered superconductor[62]. While ξab(0)≈22˚Aisinagreementwiththemeasurements[62].
inthatpaperwe havedealtwitha generaldisorder,here This value of ξab(0) leads to Hc2(0)=Φ/2πξa2b(0)=64T.
we apply this method to the stripe disorder described This value is close with the extrapolated values as it is
above, appropriated to the LSCO system. To explain it, seen in Fig.(4).
6
InFig.(4)weplottheresultsforρ =0.1and0.18and VI. CONCLUSION
m
comparedwiththeexperimentalvaluesofWangetal[41]
for overdoped Bi2201(La:04 and 02). We also compare
Taking the upper pseudogap line as a PS transition
ourcalculationstothepreviousexperimentalvaluesfrom
temperature, we have monitored the development of
the same group([39]) on LSCO because our calculations
chargeinhomogeneitiesbytheCHtheorywhichdescribes
aredirectedtoreproducethenon-uniformitiesonthisse-
howchargesegregationincreasesasthetemperaturegoes
ries. The excellent agreementindicates that LSCO com-
down, similarly as seen in the NQR experiments[33]. In
poundshaveindeedadistributionoflocalsuperconduct-
this way we have followed the process of charge segrega-
ing regions Tc(i) close to that derived here(see Fig.(1)). tion in the LSCO system and studied the superconduct-
On the other hand, since the Hc2 curves of Bi2201 are ingpropertiesofthenormalandsuperconductingphases
so flat near T/Tc ≈ 1, according our results, it indi- asfunctionofdoping. Ourapproachisgeneralandcould
catesthatthismaterialishighlydisorderedinT (i),with
c be directed to other patterns of charge disorder. Each
Tonset(ρm)muchlargerthancriticaltemperatureTc(ρm). chargedomain with constantdensity, in general,is char-
Asimilarprocedure,takenintoaccountanon-uniform
acterized by its local superconducting pairing amplitude
system with variations on the local Tc(i), was applied ∆(i,T) which arises at Tc(i). This introduces the new
before to reproduce experimental values of the magne-
concept of a local superconducting temperature, which
tization above Tc on LSCO single crystals[63], and the marks the appearence of the local pairing amplitude .
same results[64] are in good qualitative agreement with
The maximum local value of T (i) of a given compound
c
the measurements of Wang et al[41]. Recently, a distri- with average doping level ρ , as we demonstrated, can
m
bution of local T (i) was also used to reproduce and to
c berelatedtotheNernstonsettemperatureT (ρ )or
onset m
interpret the magnetization data above T [65].
c lower pseudogap. This approach, with different regions
orislandswithlocalsuperconductingtempearturesT (i)
c
provides a clear explanation to the non-vanishing of Hc2
above the critical temperature T (ρ ). It also explains
60 c m
why early tunneling experiments[4, 18] did not see any
LSCO Bi-2201
60 special signal at Tc(ρm).
In conclusion, we have provided the steps to an inter-
50
pretation of the HTS phase diagram where the disorder
plays a key role. We have also shownthat some physical
T) 40 properties, like the experimental data of Wang et al[41]
H(c240 on the Nernst effect and Hc2, which were presented as
0 evidences of the phase disordered scenario, can also be
favorable interpreted as special features of conventional
30 Hc2 LLaa::00..24 20 ssuyspteermcosnadtuchtiigvhiteyrwhhoilcehddeenvseitloiepssrinegsiponatsi.allOyudriscoarldceurlead-
m=0.1 m=0.1 tions furnish also an interpretation to the normal phase
m=0.18 m=0.18 as a disordered metal and to the three most measured
20 0 phase diagramboundaries in HTS: the upper pseudogap
0 10 20 30 0 0.5 1.0 1.5
as a phase separation transition, the lower pseudogap as
T(K) t=T/Tc the onset of islands of superconductivity and the system
FIG. 4: The lines are calculations of Hc2 considering sam- superconducting critical temperature as the Josephson
ples with different local superconducting temperatures Tc(i), coupling or percolation temperature among the various
similar toFig.(1). Left panelare comparison with theexper- superconducting regions.
imental points from Ref.([39]) for LSCO (ρm = 0.2) and the This work has been partially supported by CAPES,
right panel with points from Ref.([41]).
CNPq and CNPq-Faperj Pronex E-26/171.168/2003.
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