s-s*-d-wave superconductor on a square lattice and its BCs phase diagram

4 Pages·0.14 MB·English

by  J. Ferrer

s-s*-d-wave superconductor on a square lattice and its BCS phase diagram
J. Ferrer, M. A. Gonz´alez-Alvarez
Departamento de F´ısica, Facultad de Ciencias, Universidad de Oviedo, E-33007 Oviedo, Spain
J. Sa´nchez-Can˜izares
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Departamento de F´ısica Te´orica de la Materia Condensada, C-V, Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain
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We study an extended Hubbard model with on-site repulsion and nearest neighbors attraction which tries
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to mimic some of the experimental features of doped cuprates in the superconducting state. We draw and
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discuss the phase diagram as a function of the effective interactions among electrons for a wide range of
a
doping concentrations. We locate the region which is relevant for the cuprates setting some constraints on
J
theparameterswhichmaybeusedinthiskindofeffectivemodels. Wealsostudytheeffectsoftemperature
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andorthorrombicityonthesymmetryandmagnitudeofthegapfunction,andmapthemodelontoasimpler
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linearized Hamiltonian, which produces similar phasediagrams.

(February 1, 2008)
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n
o
c Accurate results using a variety of experimental relationsoftheintermediatecouplingregime.16 Itshould
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r techniques like, for instance, ARPES,1,2 Josephson alsobe ausefultoolforstudies oftransport,magneticor
p tunneling,3 penetration depth,4 or thermal conductivity optic properties of the cuprates, where bulk or surface
u
measurements5 provide evidence on the d-wave symme- impurities andotherinhomogeneitiesmustbe takeninto
s
. try of both the gap and order parameter functions of account.17–19
t
a optimally doped and underdoped cuprates. Penetration We present in this article a detailed study of an effec-
m depthmeasurementsinslightlyunderdopedsamples6 de- tive pairing model, where the kinetic energy term comes
- termined that their critical behavior fall on the 3D XY from a tight-binding fit to the ARPES band structure
nd universality class (see also Refs. 7,8). This fact, which of Bi2Sr2CaCu2O8+δ (Bi2212),20 and we use either an
is not consistent with BCS weak coupling theory, indi- attractive or repulsive on-site interaction plus a nearest-
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cates that the phase transition corresponds to the Bose- neighbors attraction term. In other words: we view the
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[ Einsteincondensationofasingle,complexorderparame- ARPESbandstructureofRef.20asthelow-energypair-

terfield. Theexistenceofapseudogapwithd-wavesym- ingactionoftheactualHamiltonianwherethestrongin-
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metry above the superconducting state for underdoped teractionsamongelectrons,chargefluctuations,etc,have
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cuprates9–12 provides further confirmation on the non- renormalized the kinetic energy and interaction terms
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0 BCS nature of the superconducting state. giving rise to a set of hopping integrals, an effective on-
1 Thereareseveraltheoreticalschemeswhichseemtofit site repulsion and a nearest-neighbors attraction, while
1 intothisexperimentalstateofaffairs: (a)Inthemagnetic other possible terms are irrelevant. The only channel
0 scenarioforthecuprates,13,14thestrongon-siterepulsion which has not been dealt with in this kind of experi-
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among electrons gives rise to antiferromagnetic collec- mental implementation of the RenormalizationGroup is
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tive excitations. Thesedegreesoffreedomdressthe bare the superconducting one, so we need to perform only a
/
t interaction among the residual electrons, providing the conventional BCS decoupling scheme (the Hartree term
a
m pairing mechanismfor the d-wavesuperconducting state woulddouble-countinteractioneffectsalreadytakeninto
and giving rise to the pseudogap in the normal state. account). This model generates pure s and d, as well
-
d (b) In some spin-charge separation theories,15 the pseu- as mixed s+s*, s+s*+d and s+s*+id superconducting
n dogap is related to the pairing of spinons and the su- statesindifferentregionsofthe phasediagram,wheres*
o perconducting state comes about when holons condense. denotesanextendedsstate. We haveperformedmostof
c
(c) A further line of thought supposes that an undeter- our calculations at zero temperature where it is believed
:
v mined high-energy pairing mechanism (which might be that the BCS approximation provides the gross features
i dressed vertex of case (a)) gives rise to an effective low of the true ground state of the system, at least for small
X
energypairingHamiltonianwherethecouplingconstants enough interactions.22,23
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a set the d-wave superconducting state in an intermediate We assume a tight-binding model on a squared lattice
coupling region.16 with a dispersionrelationwhich keeps hopping terms up
A thorough study of the phase diagram and qualita- to five nearest neighbors
tivefeaturesofasimplemodelwhichretaintheelectronic
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structure found in ARPES experiments is therefore an
ξ(~k)= t η (~k)−µ, (1)
urgenttask,ifsuperconductivityinthecuprateshasany- i i
thingtodowithscenarios(a)or(c). ABCStreatmentof Xi=1
suchamodelshouldserveasastartingpointformoreac- where the hopping integrals t and functions η (~k) are
curatesolutionswhichdealproperlywiththestrongcor- i i
listedintableIofRef.20. Theresultingelectronicstruc-
1
(a)
ture does not possess electron-hole symmetry and has a 400 II II
band width of about 1.1 eV. We vary µ to change the III
IV
occupationnumber n=1−x where x is the hole doping 200
in the sample. The experimental evolution of the Fermi I V
Surface (FS) with doping is not inconsistent with such a
(b)
rigid shift of the band structure even though the ascrip-
tion of a FS to a systemwith a pseudogapis a matter of eV) 400 II II III
controversy.21 We have also performed the calculations V (m2 200 IV
keeping only t1 and t2, and obtained results quantita- I V
tively similar to those explained in this article.
The interaction term has an on-site potential, which (c) s+s*+d
canbeeitherpositiveornegativeplusanearestneighbors 300 II II IV
attraction
III
V
Vint(~kξ)=−V0−V2(cos(kxξ)+cos(kyξ)), (2) 100 I
-400 -200 0 200 400
where ξ is some typical length scale. This potential
resembles the effective pairing interaction found in the V 0 (meV)
magnetic scenario of the cuprates, if we set ξ equal to
FIG. 1. Phase diagram for (a) x = 0.05 (b) x = 0.2 and
the magnetic coherence length (and slightly larger than
(c)x=0.5. LabelsI,IIandVdenotetheregionsofmetallic,
the lattice spacing, see Figure 11 of Ref. 13).
d-wave and s+s* states, while in regions III and IV both d
TheBCSexpressionsforthethermodynamicpotential
ands+s*solutionsarefound. Theboundariesofthes+s*+id
and gap functions are
state are drawn only in figure(b), with dashed lines.
|∆s|2 |∆s∗|2+|∆d|2
Ω(T,µ)=M +M +
V0 V2 We examine first the phase diagram [V0,Vd] for x in
+ ξ −E − 2 ln(1+e−βEk) the range [0.0-0.5] in order to locate the region of pa-
(cid:20) k k β (cid:21) rameter space which can be used to best represent the
Xk
superconducting state of the cuprates. Figure 1 shows
∆(~k)= ∆s+∆s∗Fs∗(~k)+∆dFd(~k), (3) thephasediagramsforx=0.05,0.2and0.5. Thecoupled
BCS equations possess only the trivial normal state so-
where M is the number of sites, Ek = ξk2+|∆k|2 is lution in region I; only a pure d state solution in region
thedispersionrelationforthequasiparticlpes,andFs∗,d = II; only a mixed s+s* solution in region V, and only a
cos(kxξ)±cos(kyξ). pure s solution on the positive V0 axis. There exist both
The gap is a function of the dimensionless variable kξ pure d and mixed s+s* solutions in regions III and IV;
for this particular choice of the interaction term, but we thedstatebeingmorestableinregionIIIandthemixed
believe that such a radial dependence also holds for any s+s*,inregionIV.We alsoobtainamixeds+s*+dsolu-
potential of the form V(~kξ), even for moderate values of tion for rather large values of V2, and a mixed s+s*+id
the coupling constants. In the magnetic scenario of the state. These two states are always the most stable as
cuprates,14ξisthecorrelationlengthand,hence,thefur- long as they exist. The frontiers of stability of the dif-
nished gap function is not commensurate with the Bril- ferent solutions change very slightly in the doping range
louin zone and varies along with x. Plots of ∆(k) in the [0.05-0.4]. When x exceeds 0.4, though, region II be-
ΓM and MY directions for some underdoped and opti- comes increasingly thinner and localized around the V2
mallydopedBi2212samples,1,9ifpossible,mightdemon- axis. The only roleof V0 onthe d-wavestate is to set its
strate if this conjecture is correct. We set nevertheless ξ domain of stability. In particular, the magnitude of ∆
d
equal to the lattice spacing a throughout this paper. does not depend on it.
The saddle point equations can now be easily written Thesizeofthegapsasafunctionofholeconcentration
′
becausethepairinginteractionsV0,2(k−k )areseparable: forpairs(V0,V2)inside regionsII, IVandVis plottedin
Figure 2. The three figures taken as a whole show that
V0 ∆k βEk thedstateismorestableonlyclosetohalf-fillingandfor
∆ = tanh( )
s
M Xk 2Ek 2 largeenoughvalues of the ratio V2/V0, while the s states
exist in the whole range of hole dopings. The shape of
V2 ∆kFs∗,d βEk
∆s∗,d = tanh( ) (4) the curves for ∆d is independent of V0; V2 only changes
M 2E 2
Xk k theirheight,sothatonecanwrite∆d =f(V2)g(x). They
alwaysreachapeakatx=0.2andvanishatx±0.7. The
We compare the free energy per site of the different
form of the curves for the s-state gaps does depend on
superconductingstatesf(T,n)=Ω(T,µ(n))/M+µ(n)n,
both V0 and V2, on the other hand, but its two maxi-
to ascertain their relative stability.
mums always occur at x = 0.2 and 0.8. We find that
2
20
(a)
s
TABLEI. Changes in the gaps dueto orthorrombicity for
10 s* the case V0 = −250 meV, V2 = 160 meV and x=0.2. Gaps
d are measured in meV.
α β ∆d ∆s ∆s∗
0
0 0 22.1 0 0
(b)
0.05 0 20.5 -0.28 0.57
∆ 0 0.05 22.1 -0.23 1.54
10
0.05 0.05 20.6 -0.5 2.01
-0.05 0.05 20.4 0.07 0.86
0.05 -0.05 20.4 -0.07 -0.86
-0.05 -0.05 20.6 0.5 -2.01
(c)
30
10 direct comparisonswith experiments. Such a theory will
require much larger values of V0 and V2 to give a gap of
30 meV, thereby placing the coupling constants in the
0.0 0.2 0.4 0.6 0.8
intermediate coupling regime.
x = 1-n
The ratio 2∆ (k ,T = 0)/K T as a function of
d F,max B d
FIG. 2. Gaps (measured in meV) for (a) V0 = −250, doping is non-universal and larger than the value ob-
V2 =150meV;(b)V0 =50,V2=150meV;and(c)V0 =250, tained for a parabolic band (4.14). There is a plateau,
V2 =50 meV. which ranges from x = 0.05 to 0.4, where the ratio is
almost constant and equal to 4.35, while it grows for
smaller or larger doping, even reaching a value of 7 for
the gaps of the t1 − t2 model have peaks at the same x=0.65.
doping concentrations. We therefore conclude that their
The presentmodel provides a universalvalue x0 =0.2
existence and position is due purely to filling effects and
for optimal doping, defined as the hole concentration
nottothe peculiarities ofthe bandstructure. The arrow
which yields the maximum critical temperature. We
in Fig. 2(c) marks the boundary of stability x between
c should caution however that experimentally, ∆ (T = 0)
d
thedandthesstates,thefreeenergyofthedstatebeing ∗
andthetemperatureT atwhichthepseudogapbeginsto
lower for x<x .
c open up are monotonically decreasing functions of dop-
Therearecompellingreasonstobelievethattherenor-
inganddonotpeakatx0,whiletheexperimentalTd has
malized on-site interaction of this low energy model a maximum at about x0 = 0.15.26 RPA theories22,23 of
should still be strongly repulsive, so we focus our study
s-wavesuperconductors,whichpermittostudytheinter-
on the negative V0 quadrantnow. A first point to notice mediate and strong coupling regimes find that electrons
is that the d-wave superconducting solution disappears ∗
begin to pair up at a temperature T which is higher
forvaluesofthe on-siteeffectiverepulsionofthe orderof
than the critical temperature T where pairs condense.
c
two times V2. This fact will set some constraints on the We study now how orthorrombicity effects can induce
possible parent Hamiltonian of this model if it survives
mixing of d and s states as might happen in YBCO, a
the inclusion of fluctuations over the Mean Field (MF)
cuprate where the a and b crystal axes become substan-
solution. Moreover,we estimate that the strength of the tially inequivalent.3 In order to do so, we redefine the
effectivenearestneighborattractionneededtoobtainthe
hopping integrals t =t (1±α) for i= 1,3,4 so that
i,x,y i
experimentalvalueofthegapforoptimallydopedBi2212
the probability amplitude for electrons to leap between
(30 meV) is of about 140 meV, while V0 can range from twositesdependsonthespatialdirectionofthehop. We
-320 to 0 meV. Such values, compared to a band width
alsosupposethattheorthorrombicityleadstoadifferent
of1.1eV,placetheresultingmodelwithintheweakcou-
nearest neighbors interaction in the x and y directions,
pling regime. This result leads to question the quanti-
V2,x,y =V2(1±β). We find that a finite β doesn’t mod-
tative accuracy of the BCS gap and of the whole MF-
ify much the d-wave gap, but indeed induces a finite s
RPA22 scheme at intermediate couplings even at T = 0.
component(see tableI).Orthorrombicityinthe hopping
What actually happens is that suchtheories providefine
integrals produces appreciable changes in ∆ , but leads
d
estimates for the gap and other physical magnitudes for
tosmallermodificationsinthes-stategaps. Noticefinally
large values of the coupling constants while for interme-
how a sign change in α and/or β inverts the sign of the
diatecouplingsandlowdimensions,theresultsaremuch
s components. This effect, which was already deduced
poorer. AcomparisonofMFtheoryandBetheansatzre- phenomenologically using Ginsburg-Landau theory,27,28
sults for the 1d Hubbard model at half filling and T =0
implies that the s-wave component of the gap has oppo-
can be found in Ref. 24, where it is shown that the size
site sign in different twin domains of twinned samples of
ofthegapisgrosslyoverestimatedbytheMFsolutionat
YBCO. This is also the explanationgivenfor the Fraun-
intermediate coupling.25 We think that the gap should
hofer pattern observed in a recent c-axis Josephson tun-
be estimated beyond MF-RPA theory in order to make
3
ACKNOWLEDGMENTS
5
The authors gratefully acknowledge financial support
4 (a) (b)
fromthe Spanish Direcci´on Generalde Ensen˜anzaSupe-
) 3 rior, Project No. PB96-0080-C02.
V
e 2
m
( 1 x=0.02 x=0.02
∆ 0 x=0.20 x=0.20

og −1 x=0.30 x=0.30
l x=0.50 x=0.40
−2
x=0.60 x=0.50
−3 1H.Ding et al.,Phys. Rev.B 54, R9678 (1996).
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0 2
Properties of High Temperature Superconductors V,edited
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One may compare the results obtained within this 10H.Ding et al.,Nature 382, 51 (1996).
tight-binding model with the predictions of a much sim- 11A. V. Puchkov, P. Fournier, T. Timusk, and N. N.
pler isotropic Hamiltonian, with a linearized dispersion Kolesnikov, Phys.Rev.Lett. 77, 1853 (1996).
relation and pairing interaction of the form 12T. Watanabe, T. Fujii, and A. Matsuda, Phys. Rev. Lett.
ξk = h¯22mk2 −µ≃¯hvF(k−kF) 137D9.,J2.1S1c3al(a1p9i9n7o),.Physics Reports 250, 330 (1995).
V(~k±k~′)≃−VL−VLcos(2θ)cos(2θ′) (5) 14A. V. Chubukov, D. Pines, and B. P. Stojkovic, J. Phys.
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In this case, the gap function ∆L = ∆L + ∆Lcos(2θ) 15X.G.WenandP.A.Lee,Phys.Rev.Lett.76,503(1996).
s d
obeys the following saddle point equations30 16J. Engelbrecht, A. Nazarenko, M. Randeria, and E.
Dagotto, condmat/9705166 (unpublished).
∆L =gLVL 2π dθ EDdε FsL,d(θ)∆L(θ) , (6) 17A.V.BalatskyandM.I.Salkola,Phys.Rev.Lett.76,2386
s,d 0,2Z0 2π Z0 ε2+|∆L(θ)|2 (1996).
p 18P. M. A. Cook, R. Raimondi, and C. J. Lambert, Phys.
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dimensional density of states. Now, expanding Eq. (4) 19W.C.Wu,B.W.Statt,Y.-W.Hsueh,andJ.P.Carbotte,
uptoorder(kξ)4 andcomparingwithEq.(6),we obtain Phys.Rev.B 56, R2952 (1997).
the following relations: gLV0L = gV0, gLV2L = γ2gV2, 20M. R. Norman, M. Randeria, H. Ding, and J. C. Cam-
∆L = ∆ and ∆L = γ∆ , where g is an effective tight- puzano, Phys.Rev.B 52, 615 (1995).
bisndingdsensityodfstatesadndγ =−(k ξ)2/2+(k ξ)4/24. 21H.Ding et al.,Phys. Rev.Lett. 78, 2628 (1997).
F F
These two parameters, together with the cut-off E , can 22J. R.Engelbrecht, M. Randeria, and C. A. R.S´ade Melo,
d
rbeespuescetdivteolyfi.tA∆sss,dhowwitnhin∆LsF,diga.lo3n,gwteheinVd0eeadnfidnVd2lianxeeasr, 23PVh.yMs..RLeovk.teBv5a5n,d1S5.15G3.(S1h9a9r7a)p.ov, condmat/9706285 (un-
behavior for log∆ as a function of 1/V, from which we published).
24J. Ferrer, Phys. Rev.B 51, 8310 (1995).
extractvalues of k ξ/π ranging from1.21 to 1.27,and g
from 1.1 to 1.6 eVF−1. 25ThemainroleofRPAistoincorporatethecollectivemodes
to the MF solution; RPA corrections therefore diverge in
In conclusion, we have proposed and studied an effec-
1devenatT =0becauseoftheMermin-Wagnertheorem.
tive tight-binding model for the superconducting state
26H.Ding et al.,condmat/9712100 (unpublished).
of the cuprates, guided mostly by results of ARPES ex- 27M. Sigrist et al.,Phys. Rev.B 53, 2835 (1996).
periments. In particular, we have delineated its weak- 28M.E.ZhitomirskyandM.B.Walker,Phys.Rev.Lett.79,
coupling phase diagrams for different hole doping con-
1734 (1997).
centrations close to the experimental optimum doping. 29K.A.Kouznetsovet al.,Phys.Rev.Lett.79,3050(1997).
We have been able to establish the region of parameters 30K. A. Musaelian, J. Betouras, A. V. Chubukov, and R.
whichis likelyto be relevantforthe cupratesinabsolute Joynt,Phys. Rev.B 53, 3598 (1996).
energy units (meV). We have finally mapped the tight-
bindingmodelontoamuchsimplerlinearizedmodeland
obtained similar phase diagrams.
4