Possible origin of 60-K plateau in the YBa2Cu3O(6+y) phase diagram

0.47 MB·English

by  T. A. Zaleski

2 3 6+y
Possible origin of 60-K plateau in the YBa Cu O phase diagram
T. A. Zaleski
Department of Physi
s, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA and
Institute of Low Temperature and Stru
ture Resear
h,
6 Polish A
ademy of S
ien
es P.O. Box 1410, 50-950 Wro
ªaw, Poland
0
0 T. K. Kope¢
2
Institute of Low Temperature and Stru
ture Resear
h,

n Polish A
ademy of S
ien
es P.O. Box 1410, 50-950 Wro
ªaw, Poland
a
2 3 6+y
J We study a model of YBa Cu O to investigate the in(cid:29)uen
e of oxygen ordering and doping
2 imbalan
eonthe
riti
altemperatureTc(y)andtoelu
idateapossibleoriginofwell-knownfeature
1 of YBCO phase diagram: the 60-K plateau. Fo
using on (cid:16)phase only(cid:17) des
ription of the high-
temperature super
ondu
ting systemin terms of
olle
tive variables we utilize a three-dimensional

] semimi
ros
opi
XYmodelwithtwo-
omponentve
torsthatinvolvephasevariablesandadjustable
n parameters representing mi
ros
opi
phase sti(cid:27)nesses. The model
aptures
hara
teristi
energy
o
s
ales present in YBCO and allows for strong anisotropy within basal planes to simulate oxygen
c
ordering. Applying spheri
al
losure relation we have solved the phase XY model with the help of
- Tc
r transfermatrixmethodand
al
ulated for
hosensystemparameters. Furthermore,weinvestigate
p the in(cid:29)uen
e of oxygen ordering and doping imbalan
e on the shape of YBCO phase diagram. We
u (cid:28)nd it unlikely that oxygen ordering alone
an be responsible for the existen
e of 60-K plateau.
s
Relying on experimental data unveiling that oxygen doping of YBCO may introdu
e signi(cid:28)
ant
. 2
t
harge imbalan
e between CuO planes and other sites, we show that simultaneously the former
a
m areunderdoped,while thelatter (cid:21) strongly overdopedalmostin thewhole region of oxygendoping
in whi
h YBCO is super
ondu
ting. As a result, while oxygen
ontent is in
reased, this provides
- two
ounter a
ting fa
tors, whi
h possibly lead to rise of 60K plateau. Additionally, our result
an
d
provide an important
ontribution to understanding of experimental data supporting existen
e of
n
multi
omponentsuper
ondu
tivityin YBCO.
o
c
[ PACSnumbers: 74.20.-z,74.72.Bk,74.62.-

1
v I. INTRODUCTION seams that holes introdu
ed in su
h way tend to remain
7 2 3
inCuO layers. Amountofoxygenalsodeterminesele
-
5 y < 0.4
troni
state of the system. For the material
2 y > 0.4
1 The YB2Cu3O6+y
ompound (YBCO), as dis
ov- is insulating, while for be
omes super
ondu
t-
0 ered in 1987 by Wu and
o-workers, is the (cid:28)rst ma- ing. The
riti
al temperature is very sensitive to oxy-
6 terial that be
ame super
ondu
ting in boiling nitrogen gen doping and the temperature-oxygen amount phase
0 temperature.1 The material
ontains three
opper-oxide diagram
ontainstwo
hara
teristi
plateausat 60Kand
t/ layers in a unit
ell: two of them are separated by Yt- 90K. While the latter is now interpreted as an optimum
a triumatom,thethird(cid:21)basalplane,issurroundedbytwo doping with small overdoping region, the origin of the
m
Barium atoms (see, Fig. 1). The oxygen
an be intro- 60K plateau is still not fully
lear. It was argued that
nd- dfruo
med6inttoo7thpeebrafsoarlmpulalane(0an≤d iyts≤
o1n)t.e2ntFo
arnyb=e v0a,riaeldl iwtitmhiignhtthbeebeaxsapllapinlaende.b4y,5,6o,r7dOerningthoefotthheeroxhyagnedn, iattowmass
o Ob1 and Ob2 sites are empty and the system is tetrago- alsosuggestedthatthereasonmightbepurelyele
troni
:
c nal. As the oxygen
ontent in
reases, additional atoms super
ondu
tivity is weakened at the
arrier
on
entra-
v: o

upy Ob1 and Ob2 sites randomly up to a
riti
al dop- tion of 1/8 leading to the plateau.8 It is a goal of the
i ing,forwhi
htetragonal-orthorhombi
(T-O)phasetran- present paper to investigate the in(cid:29)uen
e of oxygen or-
X sition o

urs. For higher dopings oxygen in Ob1 and deringand dopingimbalan
eon the
riti
altemperature
r Ob2 sites be
omes partially ordered forming (depending of YBCO using a model that
an a

ommodate strong
a
on oxygen
ontent) one of three di(cid:27)erent orthorhombi
anisotropy and
hara
teristi
energy s
ales in order to
phases: ortho-I (with fragments of
opper-oxide
hains determine apossibleorigin of 60-Kplateau on the phase
in the basal plane); ortho-II (with
hains in every se
- diagram.
b1
ond O site); and ortho-III(with alternatingoneempty Binding of ele
trons into pairs is essential in form-
y = 1
and two (cid:28)lled
hains). Finally for (maximum ing the super
ondu
ting state, however its remarkable
b1
oxygen
ontent), all O sites be
ome o

upied and all properties(cid:21)zero resistan
e and Meissner e(cid:27)e
t(cid:21)require
b2
O (cid:21) empty. The oxygen in the basal plane a
ts as a phase
oheren
e among the pairs as well. While the
2

harge reservoir introdu
ing holes into CuO plane
op- phase order is unimportant for determining the value
T
c
per atoms. Charge
on
entration
an be also
hanged of the transition temperature in
onventional BCS
by substituting Yttrium atomswithCal
ium, howeverit super
ondu
tors, in materials with low
arrier density
2
dimensional XY model. Furthermore, we solve it
in the spheri
al approximation with the help of the
transfer matrix method and obtain a dependen
e of
T =
c
the
riti
al temperature on model parameters:
T J , J , J′, J′ , η
c k ⊥ k ⊥ . Subsequently, in Se
tion III we
(cid:16) (cid:17) Tc
elaborate on in(cid:29)uen
e of oxygen ordering on . We
modelvaluesofin-planephasesti(cid:27)nessesasafun
tionof
oxygen amount and determine values of model param-
eters for whi
h the YBCO phase diagram
an be re-
produ
ed. In Se
tion IV, relying on experimental data
unveiling that oxygen doping of YBCO may introdu
e
2
signi(cid:28)
ant
harge imbalan
e between CuO planes and
other oxygen sites, we show that the former are under-
2 3 6+y
Figure 1: (Color online) Crystal stru
ture of YBa Cu O . doped, while the latter (cid:21) strongly overdoped almost in
Grey and open
ir
les are
opper and oxide atoms, respe
- thewholeregionofoxygendopingin whi
hYBCOissu-
tively. Additionally, oxygen atoms are divided into three per
ondu
ting. Finally, in Se
tion V we summarize the
p
groups: from fully oxidized
opper-oxide layers (O ), api
al
on
lusions to be drawn from our work.
ap b1 b2
(O ),fromoxygende(cid:28)
ientlayers(O ,O ). E(cid:27)e
tonoxy-
y = 0 b1 b2
gen dopingyi∈s a(l0s,o1)shownb1: a) b2 (cid:21) O and O sites are
empty; b) (cid:21) O and O sites o

upied in random
y y =1
II. MODEL
or ordered manner, depending on a
tual value of ;
)
b1
(cid:21)oxygenisorderedforming
hains: allO sitesareo

upied
b2
and O sites are empty. Sin
e, in underdoped high-temperature super
ondu
-
tors,twotemperatures
alesofshort-lengthpairing
orre-
lationsandlong-rangesuper
ondu
tingorderseemto be
c 13
su
h as high-T oxide super
ondu
tors, phase (cid:29)u
tua- well separated, we
onsider the situation, in whi
h lo-
tions may have aprofound in(cid:29)uen
e on low temperature
alsuper
ondu
tingpair
orrelationsareestablishedand
9
properties. Measurements of the frequen
y-dependent the relev0antϕde(grre)es<o2fπfreedom arerepresentedby phase
ℓ i

ondu
tivity, in the frequen
y range 100-600 GTHz, show fa
tors ≤ pla
ed in a latrti
e with nearest
c i
that phase
orrelations indeed persist above , where neighborintera
tions. Inournotation, numberslatti
e
ℓ ab
the phase dynami
s is governed by the bare mi
ros
opi
siteswithin -th plane. Thesystembe
omessupϕer
(orn)-
10 ℓ i
phase sti(cid:27)nesses. As a result, for underdoped
uprate du
tingon
eU(1) symmetrygroupgoverningthe
super
ondu
tors, the
onventional ordering of binding fa
toeriϕsℓi(sri)spontaneously broken and the non-zero value
andphasesti(cid:27)nessenergiesappearstobereversed. Thus, of appears signaling the long-range phase or-
theissuehowphase
orrelationsdevelopisa
entralprob- der(cid:10). The H(cid:11)amiltonian that we
onsider
onsists of four
T
c
lem of high- super
ondu
tivity. Furthermore, it was parts
dire
tly shownttht′aUt inJtegrating the ele
troni
degrees of H[ϕ]=H +H +H′ +H′
freedomoutin - - - Hubbardmodel,whi
hisbelieved k ⊥ k ⊥ (1)
to
orre
tly des
ribe strongly intera
ting systems, leads
11
to phase-only des
ription of super
ondu
tivity.
ontaining various mi
ros
opi
phase sti(cid:27)nesses repre-
In the present paper, we propose a semi-mi
ros
opi
senting
hara
teristi
energy s
ales present in YBCO
model of YBCO, whi
h is founded on mi
ros
opi
phase (see, Fig. 2) :
sti(cid:27)nesses that set the
hara
teristi
energy s
ales: in-
J J J >0
plane k and inter-plane ⊥
ouplings of CuO2 layers, 1. in-plane
oupling k within CuO2 layers:
J′ J′
in-plane k
oupling of basal planes and inter-plane ⊥ H = J cos[ϕ (r ) ϕ (r )]

oupling between neighboring basal and CuO2 planes. k − k { 3ℓ i − 3ℓ j
η ℓ i<j
XX
Additionally, the modeJl′
ontains a parameter , whi
h +cos[ϕ (r ) ϕ (r )] ;

ontrols anisotropy of k gradually turning o(cid:27) basal in- 3ℓ+1 i − 3ℓ+1 j } (2)
b η
plane
oupling along dire
tion while
hanges from 1
12 J > 0
to 0. Using our previous results, we model values of ⊥
2. inter-plane
oupling between neighboring
in-planephasesti(cid:27)nessesasafun
tionofoxygenamount 2
CuO layer:
to reprodu
e YBCO phase diagram and to elu
idate the
origin of the 60-K plateau. Our approa
h goes beyond H⊥ = J⊥ cos[ϕ3ℓ(ri) ϕ3ℓ+1(ri)];
− − (3)
the mean (cid:28)eld level and is able to
apture both the ef- ℓ i<j
XX
fe
ts of phase (cid:29)u
tuations and huge anisotropy on the
super
ondu
ting phase transition. J′ >0
The outline of the reminder of the paper is as fol- 3. in-plane
oupling k withinbasalplanes,whi
h
b
lows. In Se
tion II we
onstru
t an anisotropi
three-
an be gradually turned o(cid:27) along dire
tion by
3
β = 1/k T T
B
where with being the Ste(mrp)era=-
ℓ i
[tSure.(r )I,ntSrod(ur
)in]g two-dimensionalSv2e(
rto)r=s S2 (r )+
S2xℓ(ri) = 1yℓ i of theSu(nrit)le=ng[ctohsϕℓ (ri),sinxϕℓ (ri )],
yℓ i de(cid:28)ned by ℓ i ℓ i ℓ i
the Hamiltonian
an be expressed in a ve
tor form and
the partition fun
tion written as:
2π
Z = d2S (r )δ S2(r ) 1 e−βH[S],
 ℓ i ℓ i −  (7)
Z0 Yℓ,i (cid:2) (cid:3)
 S2(r) = 1
where the unit length
onstraint ( ℓ i ) is intro-
δ
du
edbythesetofDira
- fun
tions. Unfortunately,the
partition fun
tion in Eq. (6)
annot be
al
ulated ex-
a
tly. However, the model be
omes solvable, while the
rigidlength
onstraintin Eq. (7)isrepla
edbyaweaker
14
spheri
al
losure relation
1
δ S2(r ) 1 δ S2(r ) 1 .
ℓ i − → N ℓ i −  (8)
i,ℓ
(cid:2) (cid:3) X
 
Introdu
ing di(cid:27)erent mi
ros
opi
phase sti(cid:27)nesses for
2
Figure 2: (Color online) Stru
ture of the YBCO super
on- CuO cand basal planes breaks translational symmetry
du
tor. Basal plane is in the midJdl′e of the pi
ture with in- along axis,sin
e the3inter-planeand in-plane
ouplings
plane mi
ros
opi
phase sti(cid:27)ness k, whi
h
an be strongly vary with period of , when moving from one plane to
η η=1
anisotropi
when parameterissmall(orisotropi
for ). another. As a result, standard way of diagonalizing
2
BasJa′l plane
oupling with neighboring CuO planes is given the Hamiltonian usSing(rth)ree-dimensional Fourier trans-
by ⊥. In-planeandinter-planemi
ros
opi
phasesti(cid:27)nesses form of variables ( ℓ i in this
ase) fails, be
ause of
2 Jk J⊥
of CuO layers are and , respe
tively. the la
k of
omplete translational symmetry: To over-

omethis di(cid:30)
ulty, weimplement a
ombinationoftwo-
η [0,1] dimensional Fourier transform for in-plane ve
tor vari-
anisotropy parameter ∈ to simulate oxygen ables
Hk′ o=rde−rJink′g: {cos[ϕℓ(ri)−ϕℓ(ri+jaˆ)] Sℓ(ri)= N1k Xk Skℓe−ikri. (9)
ℓ i j=−1,1
XX X
+ηcos ϕ (r ) ϕ r +jˆb ; and transfer matrix method for one-dimensional de
o-
ℓ i ℓ i c
− (4) 15
rated stru
ture along -axis. This former operation di-
h (cid:16) (cid:17)io
4. inter-plane
oupling J⊥′ > 0 between adja
ent aregsopnea
ltizteoska,llletaevrminsgitnhethdeepHeanmdeilnt
oenioannlainyeErqin.d(e1x)ℓwuitnh-
2
hanged. Therefore,thepartitionfun
tion
anbewritten
CuO layer and basal plane:
in the form:
H′ = J′ cos[ϕ (r ) ϕ (r )]
⊥ − ⊥ { 3ℓ+1 i − 3ℓ+2 i +∞ dλ 1 +∞
Xℓ Xi<j Z = exp Nλ+ ln d2Skℓ
+cos[ϕ3ℓ+2(ri)−ϕ3ℓ+3(ri)]}. (5) Z−∞ 2πi  2 Z−∞ Yk,ℓ
The indi
es i,j gℓo=fr1o,m...,1Nto/3Nk beingNthe number of ×exp−N1  SkℓAℓNℓ⊥′ (k)S−kℓ′, (10)
sites in a plane, N = N⊥N, where ⊥ denotes the k kX,ℓ,ℓ′ 
number of layers and k ⊥ is the total number of  
sites. The partition fun
tion of the system reads: where AℓNℓ⊥′ (k) is an element of a square N⊥×N⊥ band
2π
Z = dϕ (r )e−βH[ϕ], matrix, appearicng as a result of non-trivial
oupling
ℓ i
(6) stru
ture along -dire
tion:
Z0 ℓ,i
Y
4
λ βJk(k) βJ⊥′ 0 0 0 0 0
 −βJ⊥′2 λ −βJ2k′(k) βJ⊥′ 0 ··· 0 0 0 
 −02 − βJ⊥′2 λ −βJ2k(k) βJ⊥ ··· 0 0 0 
 − 2 − 2 − 2 ··· 
 0 0 −βJ2⊥ λ− βJk2(k) ... ... ... ... 
AN⊥(k)=  ... ... ... ... λ− βJk2(k) −βJ2⊥ 0 0 . (11)
 0 0 0 ··· −βJ2⊥ λ− βJk2(k) −βJ2⊥′ 0 
 0 0 0 0 βJ⊥′ λ βJk′(k) βJ⊥′ 
 0 0 0 ··· 0 −02 −βJ⊥′2 λ −βJ2k(k) 
 ··· − 2 − 2 
 
 
λ=λ f(λ)
0
where: dle point of :
J (k) = 2J [cos(ak )+cos(bk )],
k k x y
∂f(λ)
J′ (k) = 2J′[cos(ak )+ηcos(bk )] =0
k k x y (12) ∂λ (cid:12)λ=λ0 (14)
(cid:12)
λ (cid:12)
and is a Lagrange multiplier introdu
ed by represent- (cid:12)
δ δ(x) = f(λ=λ )
in+g∞the Dira
- fun
tion in a spe
tral form and where 0 be
omes a free energy. In the
dλ/2πiexp( λx)
−∞ − . The problem redu
es then to spheri
almodel,theemergen
eofthe
riti
alpointissig-
Aℓℓ′ (k)
Revaluation of a determinant of the N⊥ matrix (for naledbydivergen
eoftheorderparametersus
eptibility:
te
hni
al details, we refer readersto Ref. [12℄).
G−1(k=0)=0,
The partition fun
tion in Eq. (10)
an be written as:
(15)
+∞ dλ
Z =Z−∞ 2πiexp[−Nβf(λ)]. (13) where G−1(k) ≡ hSk,ℓS−k,ℓi and h...i is the statisti
al
average. Eq. (15) determines the value of the Lagrange
N λ
In the thermodynami
limit → ∞ the dominant
on- multiplier . Consequently,thefreeenergyofthesystem
tribution to the integralin Eq. (13)
omesfrom the sad- reads:
f = λ 1 +πa +πb dkxdky ln 1 (βJ )2 [Λ(k)]2 Λ′(k)+2 βJ′ 2Λ(k)
β − 3β Z−πa Z−πb (2π)2/(ab) (cid:20)16(cid:18)n ⊥ − o (cid:16) ⊥(cid:17) (cid:19)
+ (βJ )2 [Λ(k)]2 (βJ )2[Λ′(k)]2 2 βJ′ 2 Λ(k)Λ′(k) 2 ,
sn ⊥ − o(cid:26) ⊥ −h (cid:0) ⊥(cid:1) − i (cid:27)# (16)
Λ(k)=βJ (k) 2λ Λ′(k)=βJ′ (k) 2λ
where fun
tions k − 0 and k − 0. The Lagrangemultiplier reads:
λ = β 4J +2J′ (1+η)+J + 8 J′ 2+ J +4J +2J′(1 η) 2 .
0 4 ( k k ⊥ r ⊥ ⊥ k k − ) (17)
(cid:0) (cid:1) h i
Finally, the Eq. (14) leads to expression for the
riti
al temperature:
2
2 +1 +1 2 J⊥′ +(J⊥)2−4α α+2α′
β = dξ dζρ(ξ)ρ(ζ) ,
c 3Z−1 Z−1 J′(cid:16)2 (cid:17)2αα′ 2 (J α′(cid:16))2 (2α)(cid:17)2 J2 (18)
s ⊥ − − ⊥ − ⊥
(cid:26)h(cid:0) (cid:1) i (cid:27)h i
5
α λ /β J (ξ+ζ) α′ λ /β J′(ξ+ηζ)
where ≡ 0 − k , ≡ 0 − k and 175
ρ(ε)
isa density ofstates ofthe
hain(one-dimensional)
150
latti
e given by: 1)
=
1 1 h
ρ(ε)= Θ(1 ε), 125 y,
π√1 ε2 −| | (19) (
Tc
− K]100
Θ(x)
[
and is the unit-step fun
tion.
Tc 75 h=0
1>h>0
50 0)
III. INFLUENCE OF OXYGEN ORDERING ON =
h
THE CRITICAL TEMPERATURE h=1 y,
(
25 Tc
The result in Eq. (18) provides us with a tool to an-
0
alyze the in(cid:29)uen
e of anisotropy and
hara
teristi
en-
0.2 0.4 0.6 0.8 1
ergys
alespresentinYBCOonthe
riti
altemperature. y
However,inordertodes
ribeYBCOphasediagram,itis
ne
essaryto
onne
t themodel parameterswith amount
of oxygen doping. In our previous paper we have pro- Figure 3: Constru
tion ofTYc(ByC,ηO)phase diagram wηi=thi1n pre-
posedaphenomenologi
aldependen
e ofin-planemi
ro- sented model: η
u=rve0s for for disordered ( ) and
2 fully ordered ( ) oxygen set limits for the a
tual phase
s
opi
phase sti(cid:27)ness of CuO plane on
harge (hole) y <0.5
diagram. For low dopings ( ) intera
tions in the basal

on
entration, whi
h appeared to su

essfully des
ribe Tc(y) η = 1
properties of super
ondu
ting homologousseries:12 planes are isotropi
and y=de1penden
e falls on

urve. On theotherhand, for , all addedoxygenatoms
Jk(δ)=( J0kfho1r−δ 0.1001.0(δ5−or0δ.15)20i.25fo.r 0.05<δ <0.025 lfiomrmit o
hfaηin=s a0n.dPtlhaetea
uritri
eagliotnemispaer
artousrηseovreera
bheeTtswctehene oththoeser
≤ ≥ two regimes in whi
h variable value of keeps
onA. J.
Leggett
(20)
To adapt it to the presentmodel it is ne
essaryto relate
2 Figure 4: A(cid:30)liation: Department of Physi
s, University of
oxygen amount to
harge
on
entration in CuO planes,
Illinois at Urbana-Champaign,Urbana,Illinois 61801, USAs-
sin
e YBCO phase diagram is presented as an oxygen
tant.
doping fun
tion of temperature. This relation was ex-
perimentally determined by Tallon, et at. and found
out to be roughly linear in the supeδr=
on0d.0u5
tingyr=egi0o.n4 In order to realize this s
enario within the presented
t(o
hδar=ge0.
1o7nf
oernytr=ati1o)n.16
hCaonngseesqfureonmtly,thein-pfloarnephase mJko,dJekl′,itJi⊥s,nJe⊥
′esasnadry(cid:28)tnod(cid:28)ax dvaeplueensdoenf
meoodfelηpoanradmoeptienrgs
y
sti(cid:27)ness from Eq. (20) expressedasafun
tion of oxygen
J (y)=J g(y) , whi
h would results in
onstant
riti
al temperature
k k
amount reads , where:
in the plateau region of theJphase dJia′gram. We assume
g(y)= 10−for0.5y152 (y0.−4.0.95)2 for 0.4≤y ≤1 (21) (teh2a1t),doapnidn,gfodrepsiemndpelin

itey,ofJ⊥k≡anJd⊥′ . kExapreergimiveenntablydEatqa.
(cid:26) ≤
for anisotropyJof /pJenet=ratλi2on/λd2epth i1n00YBCO provideus
with ratio of ⊥ k ab c ≃ .18 Furthermore,
Although,itisusuallystatedintheliterature,T-Ophase J J′
transitionand emergen
eofsuper
ondu
tivity in YBCO values of k and k
an be dedu
ed from the ratio of
17
arenotsimultaneousintermsofdoping. Thus,wesug- expressionsforthe
riti
al temperatures (see, Fig. 4) for
60K 93K
gest the following s
enario des
ribing the phase diagram plateau ( ) and optimum doping ( ):
of YBCO (see, Fig. 4): with in
reasing oxygen doping
y = 0 T J J′, y =0.5, η =1
(starting from ), the onset of super
ondu
tivity c k, k 60
y = 0.4 = .
is rea
hed ( ). The
riti
al temperature is ris- T (cid:16)J J′, y =0.95, η =0(cid:17) 93 (22)
ing up to a point, in whi
h T-O phase transition o

urs c k, k
y = 0.5
(cid:16) (cid:17)
( ). For higher dopings, oxygen
hains start to
η(δ)
form. That results in intera
tions in basal planes be- Finally, dependen
e of oxygen ordering on doping

oming more one-dimensional, thus phase (cid:29)u
tuations
an be foundJnu=m1e9r.i3
a2lly forJ
′a=l
u3l3a.t8e1dmveaVlueJs o=f mJo′d=el
rise signi(cid:28)
antly and enough to keep the
riti
al tem- parameters( k meV, k , ⊥ ⊥
0.58meV
perature
onstant. Further, while all the oxygen is or-
)andit
anbyapproximatedbytheexpression:
dered in
hains, the
riti
al temperature roughly follows
J 1
in-plane phase sti(cid:27)ness k dependen
e on doping rea
h- η(δ)= (2 2δ)15.2+(2 2δ)7.9+(2 2δ)3.6 .
93K y =0.95 3 − − −
ingits maximumvalue of for anddropping
h i
slightly later. (23)
6
100
120
J'/J=1.50
|| ||
80 100 J||'/J||=1.60
J'/J=1.75
|| ||
80 J'/J=1.90
] 60 || ||
K K] J'/J=1.95
[ || ||
[ 60
c
T c
40 T
40
Beyer, Ref. 19
Cava, Ref. 20
20 Segawa, Ref. 8 20
Theory (a)
Theory (b)
0
0
0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
y y
Figure 5: (Color online) Comparison of experimental phase F′igure6: Comparisonofphase diagrams forvariousratios of
J /J
diagramsofYBCOwithresultsofthepresentedmodelfora) k k. Plots are normalized to have the
riti
al temperature
60K
linear and b) non-linear (more exa
t) approximation of data
of theplateau region equal to .
of Tallon, et al. (see, Ref 16).
tio of
riti
al temperatures for systems with anisotropi
η =0 η =1
TheresultingphasediagramispresentedinFig. 5(thi
k ( )andisotropi
( )intera
tionsin basalplanes
solid line) along with experimental results, for
ompar- reads:
ison. It is
learly seen that the pro
edure reprodu
es T (η =0)
c

hara
teristi
features of YBCO phase diagram: 60-K =0.949.
plateau, 93K maximum
riti
al temperature with small Tc(η =1) (24)
overdoped region. However, trying to (cid:28)t experimental J = J′ = J J′ = 0.1J J = J
data better, we have performed more exa
t (non-linear) Similarly, for || k , ⊥ and || ,
approximation of data from Ref. [16℄ that Jleka(dy)to dop- J|′| = J⊥′ = 0.5J and J⊥ = 0.1J the ratio is 0.983. The
ing dependen
e of in-plane phase sti(cid:27)ness . Cor-
in(cid:29)uen
eofbasalplaneanisotropyistoosmalltonoti
e-
responding phase diagram is also presented in Fig. 5 as
thi
k dashed line. It does not provide any new quality ably
hanJg′ethe
riti
altemperatureuntilJin-planephase
sti(cid:27)ness k is in
reasedto be higherthan k. This, how-
in
reasing the width of 60-K plateau only very slightly,
J (y)
k ever, does notseemto be reasonablefrom physi
alpoint
thus it is reasonableto use simpler form of depen-
den
e, as in Eq. (21). It
an be also noti
ed that the of view.JE′/vJolution of the phase diagram with
hanging
ratio of k k is presented in Fig. 6. When the ratio is
model predi
ts 60-K plateau to be a little bit narrower
and moved toward lower doping region than experimen- being de
reased, the plateau region is disappearing. To
tal results show. It
an be argued that sin
e it is very summarize, we (cid:28)nd it rather unlikely that oxygen order-
hardto
ontroltheamountofoxygeninYBCOpre
isely, ing into
hains alone
an explain the existen
e of 60-K
somemeasurements,espe
iallybasedonmultigrainpow- plateau.
der samples,
an be ina

urate. However newer results
are based on un-twinned single
rystals and seem to be
reliable,alsobe
auseofgood
onsisten
eamongdi(cid:27)erent IV. INFLUENCE OF DOPING IMBALANCE ON
resear
hgroups. THE CRITICAL TEMPERATURE
Unfortunately, the most serious (cid:29)aw in the presented
s
enario are the a
tual values of model parameters re- YBCO
an be doped not only by adding the oxy-
quired to reprodu
e experimental phase diagram. It genatoms,butalsobysubstitutingthree-valentYttrium
is ne
essaJr′y for basal plane in-plane mi
ros
opi
phase atoms with two-valent Cal
iums. This introdu
es holes
sti(cid:27)ness k to be almost two times bigger than CuO2 dire
tly to CuO2 planes leaving api
al and
hain sites
planes
oupling Jk. For more reasonable values Jk > untou
hed:3eventhoughthe
harge
on
entrationwithin
J′, J , J′
k ⊥ ⊥, the in(cid:29)uen
e of anisotropy in basal planes
opper-oxideplanesishighenough,la
kofdopingofapi-
J = J′ = J
alsitesresultsininterplane
ouplingbeingsmallandthe
is almost negligible. For example for || k ,
J′ = 0.1J J = 0.01J J = 19.3meV onset of super
ondu
tivity is not rea
hed. This suggests
⊥ and ⊥ with , the ra- that
harge
on
entration within api
al sites along with
7
1.0
h(y)=y
h(y)=y3/2
) 0.8
s
t
i
n
u
0.6
.
b
r
a
( 0.4
'^
J

,
'J|| 0.2
0.0
0.2 0.4 0.6 0.8 1.0
y
2 3 7
Figure 7: (Color online) YBa Cu O stru
ture with
opper-
′
J
oxide
hainsfully formed. Figure 8: (Color online) Cal
ulated dependen
e of k and
′
J
⊥ on oxygen amount for various fa
tors due to number of
oxygen va
an
ies in
opper-oxide
hains. Dependen
es are
2
basal plane sites and CuO layers
an be signi(cid:28)
antly normalized to have maximumvalue equal to 1. Dashed lines
di(cid:27)erent. Similarly, as Yttrium substitution
an lead to areguidesforeyeextendingdependen
esbeyondregionwhere
introdu
tion of holes ex
lusively into CuO2 planes, one
al
ulation was possible.

an imagine that in
reasing of oxygen amounts
hanges

harge
on
entration primarily in the
hain regions and
only some of the
harges are transported to the
opper-
oxide layers. In fa
t, this s
enario has been
on(cid:28)rmed
3
bymeansofsite-spe
i(cid:28)
X-rayabsorptionspe
tros
opy.
Be
auseoverdopingofsomeregionsof YBCO
ould pro-
the qualitative results are the same and in reasonable
vide a fa
tor de
reasing the
riti
al temperature with
agreement with experimental data.
in
reasing oxygen doping and thus leading to the 60-K
plateau, we want to investigate su
h a possibility within
presented model.
Ourpro
edureisasfollows: weassumethatanisotropy
First, we want to emphasize the high signi(cid:28)
an
e of ratio between in-plane and inter-plane
oupling among
J /J = 0.0136 η
api
al site doapingc. Analyzing a proje
tion of the YBCO CuO2 planes i0s equal to ⊥ k ,12 parame-
stru
ture on − plane (presented in Fig. 7), it
an be taer is equal to (oxygen inJba′salplaJn′es is ordered along
noti
ed that asetof
hainsdi(cid:27)ersfrom full
opper-oxide dire
tion) and values of k and ⊥ are equal and of
layer only by a la
k of oxygens between CuO2 planes. J J′
the order of k (however k is modi(cid:28)ed by
hosen fa
tor
Thus oJn′e
an expe
t that the mi
ros
opi
phase sti(cid:27)- h(y) = yγ Jk J⊥ η
nesses J⊥′ relatedtoapi
alsitesandtheonewithinbasal ). Providing values of , and into Eq.
planes kareoftheorderofCuO2in-planephasesti(cid:27)ness (18) alongwith experimentally obtained
riti
altemper-
Jk atures based on resistivity measurements from Ref. [8℄,
. This suggests it is reasonable to investigate the role
wedetermine doping dependen
e of basaland api
al mi-
of doping of api
al and basal sites on the phase diagram

ros
opi
phase sti(cid:27)nesses. The results are presented in
ovafluYeBsCoOf .Jk′ToanidnvJe⊥s′titghaatet wthoiusld, wreesuwltanint Ttoc(dye)tedrempeinne- Fig. 8. Using expression in Eq. (20) it is also possible
to
al
ulate doping dependen
e of
orresponding
harge
den
e observed experimentally. For
larity, we assume

on
entration (see, Fig. 9). As it is apparent, almost
that oxygen is fully ordered in
hains to study the e(cid:27)e
t
in the whole region of oxygen doping in whi
h YBCO is
of
harge imbalan
e alone. It is also ne
essary to model
super
ondu
ting,basalplanesalongwithapi
al sites are
yan<in1(cid:29)uen
eofthenumberofva
an
ieJs′inbasalplanesfor overdoped while simultaneously CuO2 planes are under-
on the in-plane phase sti(cid:27)ness k. This relation is
doped. While oxygen
ontent is in
reased, this provides
unfortunately notobvious and it is hardto (cid:28)nd any hint
two
ountera
tingfa
tors,whi
h may lJea′d to rJis′eof 6y0K
aboutitspe
hi(cid:28)(
y)fo=rmy. ThusJ,′westartw(yi)th=aysγimJ′p(ley)linear plateau. Although spe
i(cid:28)
relation of k and ⊥ on is
dependen
e ,so: keffective k with h(y) = yγ
γ =1 γ =3/2 dependent on assumed fa
tor , the results are
21
. Later, we
hoose di(cid:27)erent valJu′e of J′ and qualitatively similar and in reasonable agreement with
noti
ethatalthoughspe
i(cid:28)
valuesof k and ⊥
hange, experimental data (see stars, in Fig. 9).
8
Basal plane & apical sites
0.25 parameter. The model fully implements
ompli
ated en-
ergy s
ales present in YBCO also allowing for strong
d anisotropy within basal planes in order to simulate oxy-
e h(y)=y
p genordering. Applyingspheri
al
losurerelationwehave
0.20 overdo h(y)=y3/2 msoalvteridxtmheethpohdasaendXY
alm
uoldateeldwTitchfotrhe
hhoseelpn soyfstteramnspfear-
rameters. Furthermore, by making a physi
ally justi(cid:28)ed
d 0.15
assumption regarding the doping dependen
e of the mi-

ros
opi
phase sti(cid:27)nesses we are able to re
reate the
0.10 underdoped C u O(fro2pmlaRenf. e1s6) Wdopfihaeaogsxrdeayemgtdeei,rnamig.oreiarn.dmee6rt0oih-nfKagtYap
BnlhaCdatrOedaa
ouatp,enirn
diagsntiini
mbveefbesaatrlietagu
narr
teeeeastoteohndfetibthiysne(cid:29)sepuh(cid:27)heaenap

steees.
0.05 of oxygen ordering. However, the spe
i(cid:28)
values of the
model parameters for whi
h this result is obtained seem
to be a little bit hard to justify. Furthermore, relying
0.2 0.4 0.6 0.8 1.0
on experimental data unveiling that oxygen doping of
y
YBCO may introdu
e signi(cid:28)
ant
harge imbalan
e be-
2
tweenCuO planesandotheroxygensites,weshowthat
the former are underdoped, while the latter (cid:21) strongly
Figure 9: (Color online) Charge
on
entration within basal
2 overdopedalmostinthewholeregionofoxygendopingin
planes, api
al sites and in CuO planes. Stars denoteexperi-
mental values of hole doping in basal planes and api
al sites whi
h YBCO is super
ondu
ting. In
reasing of the oxy-
(from Ref. [3℄). Dashed lines are guides for eye extending gen
ontent provides then a natural me
hanism of two
dependen
esbeyond region where
al
ulation was possible.
ounter a
ting fa
tors that leads to emergen
e of 60-K
plateau. Additionally, our result
an provide an impor-
tant
ontribution to solve a
ontroversy of the symme-
V. SUMMARY AND CONCLUSIONS try of YBCO order parameter, for whi
h various experi-
d s
ments give
ontradi
toryanswers suggesting or -wave
Wehave
onsideredamodelofYBa2Cu3O6+y tostudy (although the fa
t that YBCO is orthorhombi
sshould
tthheei
nr(cid:29)itui
eanl
eteomfopxeyragteunroerTdcer(iyn)gaannddtdoopeilnug
iidmabtealaanp
oesosin- adl-swoavleea)d.22toFuorrtdheerrmpaorraem, emteerasbuerienmgeantms ioxftuthree o
fompalnedx
bleoriginofwell-knownfeatureofYBCOphasediagram:
ondu
tivityofhighqualityYBCO
rystalsshowathird
the 60-K plateau. Motivated by the experimental evi- peak in the normal
ondu
tivity at608K0K along with en-
23
den
e that the ordering of the phase degrees of freedom han
ed pair
dondu
tivity below ∼ . Authors show
is responsible for the emergen
e of the super
ondu
ting thatasingle -waveorderparameterisinsu(cid:30)
ienttode-
statewithlong-rangeorder,wefo
usonthe(cid:16)phaseonly(cid:17) s
ribethe dataand su

essfully
onsidertwo-
omponent
des
riptionofthehigh-temperaturesuper
ondu
tingsys- model of super
ondu
tivity in YBCO. They
laim that
tem. Inourapproa
h, the vanishingofthe super
ondu
- it would be tempting to assign the two super
ondu
ting
tivitywithunderdoping
anbeunderstoodbytheredu
-
omponents with the asso
iated
ondensates residing on
2
tionofthein-planephasesti(cid:27)nessesand
anbelinkedto CuO planes and
hains, respe
tively, however they do
the manifestation of Mott physi
s (no double o

upan
y not see any justi(cid:28)
ation of su
h situation. Thus, our re-
due to the large Coulomb on-site repulsion) that leads sult
an providea nastural answerforplausibility of su
h
to the loss of long-rangeyp=ha0se
oheren
e whyile=m0oving as
enario,in whi
hd -wave
omponent
omes from over2-
toward half-(cid:28)lled limit ( ). For doping , the doped
hains and -wave one (cid:21) from underdoped CuO
(cid:28)xed ele
tron number implies large (cid:29)u
tuations in the planes.

onjugatephase variable, whi
hnaturallytranslatesinto
theredu
tionofthemi
ros
opi
in-planephasesti(cid:27)nesses
and destru
tion of the super
ondu
ting long-rangeorder
y A
knowledgments
in this limit. In the opposite region of large , the onset
Tof∗a pair-breaking e(cid:27)e
t (at the pseudogap temperature
)
an deplete the mi
ros
opi
phase sti(cid:27)nesses, thus T. A. Z. would like to thank The Foundation for Pol-
redu
ing the
riti
al temperature. ish S
ien
e for supporting his stay at the University of
Tothis end, wehaveutilized a three-dimensionalsemi Illinois within Foreign Postdo
Fellowships program. T.
mi
ros
opi
XY model with two-
omponentve
torsthat A. Z. would also like to thank Prof. Leggett for hospi-
involvephase variablesand adjustableparametersrepre- tality during the stay, fruitful dis
ussionsand invaluable
senting mi
ros
opi
phase sti(cid:27)nesses and an anisotropy
omments to the manus
ript.
9
1
M.K.Wu,J.R.Ashburn,C.J.Torng,P.H.Hor,R.L.Meng, E. Stanley, ibid. 176, 718 (1968); G. S. Joy
e, Phys. Rev.
L. Gao, Z.J. Huang, Y.Q. Wang, and C.W. Chu, Phys. 146, 349 (1966); G. S. Joy
e, in Phase Transitions and
Rev. Lett. 58, 908 (1987). Criti
al Phenomena, edited byC. Domband M. S. Green
2
J.M.S. Skakle,Mater. S
i. Eng. R23, 1 (1998). (A
ademi
,New York, 1972), Vol. 2, p. 375.
3 15
M.Merz,N.Nü
ker,P.S
hweiss,S.S
huppler,C.T.Chen, S.Fishman,T.A.L.Ziman,Phys.Rev.B26,1258(1982).
16
V. Chakarian, J. Freeland, Y. U. Idzerda, M. Kläser, G. J.L. Tallon,C.Bernhard,H.Shaked,R.L.Hitterman,and
Müller-Vogt, and Th. Wolf, Phys. Rev. Lett. 80, 5192 J.D. Jorgensen, Phys. Rev. B 51, R12911 (1995).
17
(1998). L.Li, S.Cao, F.Liu, W.Li, C.Chi, C.Jing, andJ. Zhan,
4
A. Ourmazd, J.C.H. Spen
e, Nature (London) 329, 425 Physi
a C 418, 43 (2005).
18
(1987). C. Panagopoulos, J.R. Cooper, T. Xiang, G.B. Pea
o
k,
5
C. Chaillout, M.A. Alario-Fran
o, J.J. Capponi, J. I. Gameson, P.P. Edwards, W. S
hmidbauer, and J.W.
Chenavas, J.L. Hodeau, and M. Marezio, Phys. Rev. B Hodby,Physi
a C 282-287, 145 (1997).
19
36, 7118 (1987). R.Beyers,B.T.Ahn,G.Gorman,V.Y.Lee,S.S.P.Parkin,
6
H.F. Poulsen, N.H. Andersen, J.V. Andersen, H. Bohrt, M.L. Ramirez, K.P. Ro
he, J.E. Vazquez, T.M. Gür, and
and O.G. Mouritsen, Nature (London) 349, 594 (1991). R.A.Huggins, Nature (London) 340, 619 (1989).
7 20
Y.Yan,M.G.Blan
hin,C.Pi
ard,P.Gerdanian,J.Mater. R.J.Cava,B.Batlogg, C.H.Chen,E.A.Rietman,S.M.Za-
Chem. 3, 603 (1993). hurak, and D. Werder,Nature (London) 329, 423 (1987).
8 21
K. Segawa and Y. Ando, J. Low Temp. Phys. 131, 822 In order to (cid:28)nd exa
t′in(cid:29)uen
e of number of oxygen va-
J
(2003).
an
ies in
hains on k it would be ne
essary to
al
ulate
9
V.J.Emery,andS.A.Kivelson,Nature(London)374,434 doping-dependent anisotropi
penetration depth and (cid:28)ne
(1995). tuneit to experimentalresults.
10 J. Corson, R. Mallozzi, J. Orenstein, J. N. E
kstein, and 22 R.Combes
otandX.Leyronas,Phys.Rev.Lett.75,3732
I. Bozovi
, Nature (London) 398, 221 (1999). (1995), and referen
es therein.
11 T.K. Kope¢, Phys.Rev. B 70, 054518 (2004). 23 H. Srikanth, B.A. Willemsen, T. Ja
obs, S. Sridhar, A.
12
T.A. Zaleski and T.K. Kope¢, Phys. Rev. B 71, 014519 Erb,E.Walker,andR.Flükiger,Phys.Rev.B55,R14733
(2005). (1997).
13
V. J. Emery,S. A. Kivelson, Nature 374, 434 (1995).
14
T. H. Berlin and M. Ka
, Phys. Rev. 86, 821 (1952); H.