Origin and roles of a strong electron-phonon interaction in cuprate oxide superconductors

0.31 MB·English

by  Fusayoshi J. Ohkawa

Origin and roles of a strong electron-phonon interaction
in cuprate oxide superconductors
Fusayoshi J. Ohkawa
7 Department of Physics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan∗
0 (Received 21 July 2006; revised manuscript received 1 January 2007; accepted for publication in PRB)
0
Astrongelectron-phononinteractionarisesfromthemodulationofthesuperexchangeinteraction
2
bylatticevibrations. Itisresponsibleforthesofteningofthehalf-breathingmodesaround(±π/a,0)

n and (0,±π/a) in the two-dimensional Brillouin zone, with a being the lattice constant of CuO2
a planes, as is studied in Phys. Rev. B 70, 184514 (2004). Provided that antiferromagnetic spin
J fluctuations are developed around Q=(±3π/4a,±π/a) and (±π/a,±3π/4a), the electron-phonon

5 interaction can also cause the softening of Cu-O bond stretching modes around 2Q, or around
(±π/2a,0) and(0,±π/2a). Thesofteningaround2Qisaccompanied bythedevelopmentofcharge

] fluctuations corresponding to the so called 4a-period stripe or 4a×4a-period checker-board state.
n
However,anobservationthatthe4a-periodmodulatingpartorthe2Qpartofthedensityofstatesis
o
almostsymmetricwithrespecttothechemicalpotentialcontradictsascenariothatthestabilization
c
- of a single-2Q or double-2Q charge density wave following the complete softening of the 2Q bond
r stretching modes is responsible for the ordered stripe or checker-board state. It is proposed that
p
thestripeorchecker-boardstateissimplyasingle-Qordouble-Qspindensitywave,whosesecond-
u
harmonic effects can explain the observed almost symmetric 2Q part of the density of states. The
s
. strong electron-phonon interaction can play no or only a minor role in the occurrence of dγ-wave
t
a superconductivityin cuprateoxides.
m
PACSnumbers: 71.38.-k,74.20.-z,75.30.Et
-
d
n
o I. INTRODUCTION strong electron-phonon interaction.
c
The Hubbard model is one of the simplest effective
[
It is one of the most interesting and important is- Hamiltonians for strongly correlated electron liquids. In

1 sues in condensed-matter physics to elucidate the mech- Hubbard’s approximation,12,13 a band splits into two
v anism of high critical temperature (high-T ) supercon- subbands when the on-site repulsion U is so large that
8 c
ductivity occurring in cuprate oxides.1 The oxides are U &W, with W being the bandwidthof unrenormalized
8
0 highly anisotropic quasi-two-dimensional oxides, whose electrons. The subbands are called the upper Hubbard
1 main compositions are CuO2 planes. High-Tc super- band (UHB) and the lower Hubbard band (LHB), and a
0 conductivity occurs on the CuO planes. There are gap between UHB and LHB is called the Hubbard gap.
2
7 pieces of evidence that the electron-phonon interaction InGutzwiller’sapproximation,14,15,16 anarrowquasipar-
0 is strong on the CuO planes: the softening of the half- ticle band appears around the chemical potential. The
2
t/ breathing modes around ( π/a,0) and (0, π/a) in the band and quasiparticles are called the Gutzwiller band
a ± ±
two-dimensionalBrillouinzone(2DBZ),2,3,4,5,6withabe- and quasiparticles, respectively. It is plausible to spec-
m
ingthelatticeconstantoftheCuO planes,thesoftening ulate that the density of states has in fact a three-
2
- of Cu-O bond stretching modes around ( π/2a,0) and peak structure, with the Gutzwiller band between UHB
d
n (0, π/2a)in2DBZ,7,8 kinksinthedispers±ionrelationof and LHB. Both of the approximationsare single-site ap-
±
o quasiparticles,9,10 and so on. It may be arguedtherefore proximations (SSA). Another SSA theory confirms the
c that the electron-phonon interaction must play a major speculation,17 showingthatthe Gutzwillerbandappears
v: role in the occurrence of high-Tc superconductivity. On atthetopofLHBwhentheelectrondensityperunitcell
Xi the otherhand, observedisotope shifts ofTc aresmall,11 is less than one. The nature of the ground state of the
whichimpliesthatthestrongelectron-phononinteraction Hubbard model depends onthe nature of the Gutzwiller
r can play only a minor role in high-T superconductivity quasiparticles.
a c
itself. Theoriginandrolesofthestrongelectron-phonon
The SSA that considers all the single-site terms is re-
interaction should be clarified in order that the issue of
ducetodeterminingandsolvingself-consistentlytheAn-
high-Tc superconductivity might be solved. derson model,18,19,20 which is one of the simplest effec-
Parent cuprate oxides with no doping are Mott in- tive Hamiltonian for the Kondo problem. Hence, the
sulators. When holes or electrons are doped into the three-peak structure corresponds to the Kondo peak be-
the Mott insulators, high-T superconductivity appears. tween two subpeaks in the Anderson model, or in the
c
Cuprate oxide superconductors lie in the vicinity of the Kondo problem. The s-d model is also one of the sim-
Mott metal-insulator transition or crossover. It may be plest effective Hamiltonian for the Kondo problem. Ac-
argued therefore that strong electron correlations must cording to Yosida’s perturbation theory21 and Wilson’s
play a crucial role not only in the occurrence of high-T renormalization-group theory,22 the ground state of the
c
superconductivity but also in the origin and roles of the s-d model is a singlet or a normal Fermi liquid provided
2
thattheFermisurfaceofconductionelectronsispresent. T superconductivity occurs in an intermediate-coupling
c
Since thes-dmodelisderivedfromtheAndersonmodel, regime J /W∗ = 0.3-0.5 for superconductivity, which is
| |
the ground state of the Andersonmodel is also a normal realized in the strong-coupling regime for electron corre-
Fermi liquid. It is certain therefore that under the SSA lations defined by U/W &1.
thegroundstateoftheHubbardmodelisanormalFermi Sincechargefluctuationsaresuppressedbystrongelec-
liquid or a metal.23 Even if the Hubbard gap opens, the troncorrelations,theconventionalelectron-phononinter-
Fermi surface of the Gutzwiller quasiparticles is present. action arising from charge-channel interactions must be
The SSA can also be formulated as the dynamical weak in cuprate oxide superconductors. On the other
mean-field theory24 and the dynamical coherent poten- hand, an electron-phonon interaction arising from spin-
tial approximation.25 In the SSA, local fluctuations are channel interactions can be strong. For example, an
rigorously considered but Weiss mean fields, which are electron-phononinteraction arising from the modulation
responsible for the appearance of the corresponding or- of a magnetic exchange interaction by phonons plays a
der parameter, are ignored. Hence, the SSA is rigorous significant role in the spin-Peierls effect. It has been
for infinite dimensions within the Hilbert subspace with shown in a previous paper36 that an electron-phonon
no orderparameter.26 In Kondo-latticetheory,18,19,20 an interaction arising from the modulation of the superex-
unperturbed state is the normal Fermi liquid, which is changeinteractionby phonons is strongin cuprate oxide
constructedinthe non-perturbativeSSAtheory,andnot superconductors. The electron-phonon interaction can
onlyeffectsofintersitefluctuationsbutalsoorderingdue explainthesofteningofthehalf-breathingmodesaround
to Weiss mean fields such as magnetism, superconduc- ( π/a,0) and (0, π/a) in 2DBZ. It has been predicted
tivity and so on are perturbatively considered. Kondo- t±hat the softening±must be small around ( π/a, π/a)
lattice theory can also be formulated as 1/d expansion in 2DBZ. An attractive mutual interaction±due to±such
theory, with d being the spatial dimensionality. an electron-phonon interaction is strong between quasi-
The d-p model, where Cu 3d and O 2p orbits are ex- particles on next-nearest-neighbor Cu ions, but is very
plicitlyconsidered,isoneofthesimplesteffectiveHamil- weakbetweenthoseonnearest-neighborCuions. There-
toniansfor cuprateoxidesuperconductors. Since the on- fore, the mutual interaction can play no significant role
site repulsion U plays a crucial role in the d-p model as in the binding of dγ-wave Cooper pairs. Observed small
it does in the Hubbard model, it is straightforward to isotope shifts of T can never contradict the presence of
c
extend the analysis for the Hubbard model to the d-p the strong electron-phonon interaction.
model. Observed quasiparticle states, which are often
The so called 4a-period stripes and 4a 4a-period
called mid-gap states, are simply the Gutzwiller quasi- ×
checker boards are observed in under-doped cuprate ox-
particle states, which can also be renormalized by inter- idesuperconductors,37,38,39,40,41,42 whosedopingconcen-
sitefluctuations. Whenobservedspecificheatcoefficients
trations are smaller than those of optimal-doped ones.
as large as γ 14 mJ/K2mol are used,27,28 the Fermi-
≃ The wave numbers of Cu-O bond stretching modes,
liquid relation gives29,30
( π/2a,0) and (0, π/2a) in 2DBZ, correspond to the
± ±
W∗ =0.3-0.4 eV, (1.1) period 4a of stripes and checker boards. The softening
of the stretching modes is accompanied by the develop-
for the effective bandwidth of the Gutzwiller quasipar- mentof4a-periodor4a 4a-periodfluctuationsincharge
×
ticles in optimal-doped cuprate oxide superconductors, channels, which are simply stripe or checker-board fluc-
where T is the highest as a function of doping concen- tuations. It may be argued therefore that a charge den-
c
trations. According to field theory, the superexchange sitywave(CDW) followingthe complete softeningofthe
interaction arises from the virtual exchange of a pair bond stretching modes is responsible for ordered stripes
excitation of electrons across the Hubbard gap.31 Since and checker boards.
the Gutzwiller quasiparticles, which are responsible for One of the purposes of this paper is to show that the
metallicproperties,playsnosignificantroleinthevirtual strong electron-phonon interaction can also explain the
exchange process, the superexchange interaction is rele- softening of Cu-O bond stretching modes in cuprate ox-
vanteveninametallicphase,providedthattheHubbard ide superconductors. The other purpose is to examine
gap opens. Cooper-pairs can also be bound by a mag- critically the relevance of the CDW scenario,whether or
neticexchangeinteraction.32Sincethesuperexchangein- not the CDW is actually responsible for ordered stripes
teraction constant is as strong as33 J = (0.10-0.15)eV and checker boards. This paper is organized as fol-
−
between nearest-neighbor Cu ions on a CuO plane, ob- lows: Preliminary discussions are presented in Sec. II;
2
served high T can be easily reproduced. In actual, it the derivation of the electron-phonon interaction is re-
c
has already been proposed that the condensation of dγ- viewed in Sec. IIA and Kondo-lattice theory is reviewed
waveCooperpairsbetweenthe Gutzwiller quasiparticles in Sec. IIB. The softening of the bond stretching modes
due to the superexchange interaction is responsible for around( π/2a,0) and (0, π/2a) in 2DBZ is studied in
± ±
high-T superconductivity.34,35 Since the superexchange Sec. III. The relevance of the CDW scenario for stripes
c
interaction is strong only between nearest-neighbor Cu and checker boards is critically examined in Sec. IV. An
ions, it is definite that theoretical T of the dγ wave is argumentonthemechanismofhigh-T superconductivity
c c
much higher than those of other waves. In fact, high- is given in Sec. V. Conclusion is presented in Sec. VI.
3
II. PRELIMINARIES with R and R = (R +R )/2 being positions of the
i [ij] i j
ithCuand[ij]thOions,M andM massesofCuandO
d p
A. Electron-phonon interaction ions, b† and b creation and annihilation operators of
λq λq
aphononwithapolarizationλandawavevectorq,ω
λq
In cuprate oxide superconductors, the superexchange aphononenergy,ǫλq =(ǫλq,x,ǫλq,y,ǫλq,z)apolarization
interaction arises from the virtual exchange of a pair ex- vector, and N the number of unit cells. Here, only lon-
citation of 3d electrons between UHB and LHB that are gitudinal phonons are considered so that it is assumed
strongly hybridized subbands between Cu 3d and O 2p that ǫλq =(qx,qy,qz)/q for q within the first Brillouin
| |
orbits.31 When the broadening or finite bandwidths of zone. The q dependence of vd,λq and vp,λq is crucial.
UHBandLHBareignored,theexchangeinteractioncon- For example, vd,λq =0 and vp,λq =O(1) for modes that
stantbetweennearest-neighborCuionsonaCuO plane bring no change in adjacent Cu-Cu distances.
2
is given by Two types of electron-phonon interactions arise from
the modulations of the superexchange interaction J by
4V4 1 1 the vibrations of O and Cu ions. When they are con-
J = + , (2.1)
−(ǫ ǫ +U)2 ǫ ǫ +U U sidered, it is convenient to define a dual-spin operator.
d− p (cid:18) d− p (cid:19) First, a single-spin operator is defined by
with V being the hybridization matrix between nearest-
1 1
onfeiCghubo3rdOan2dpOan2dpCleuve3lsd,oarnbditsU, ǫtdheanodn-ǫspitethreepduelpstiohns S(q)= √N kαβ 2σαβd†(k+21q)αd(k−12q)β, (2.4)
X
between Cu 3d electrons.
with σ = (σ ,σ ,σ ) being the Pauli matrixes and d†
Doped holes reside mainly at O ions. The preferential x y z kσ
and d being creation and annihilation operators, re-
dopingsuggeststhatO2plevelsareshallowerthanCu3d kσ
spectively, of 3d electrons with wave number k. Then,
levels or that parent cuprate oxides with no hole doping
the dual-spin operator is defined by
must be charge-transfer insulators rather than Mott in-
sulators; charge-transfer insulators and Mott insulators 1
(q)= η (q′) S q′+1q S q′+1q , (2.5)
are characterized by ǫp > ǫd and ǫp < ǫd, respectively. PΓ 2 Γ 2 · − 2
Since the hybridization between Cu 3d and O 2p orbits Xq′ (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
must be strong, it may also be argued that Cu 3d lev- with
els are much deeper than O 2p levels, that is, ǫ ǫ
p d
≫ η (q)=cos(q a)+cos(q a), (2.6)
rather than ǫ >ǫ , to explain the observed preferential s x y
p d
doping. However, the suggested level scheme of ǫ > ǫ
p d and
or ǫ ǫ disagrees with the prediction of Mott insu-
p d
lators,≫ǫd−ǫp ≃ 1 eV, by band calculations.43,44,45 The ηd(q)=cos(qxa)−cos(qya). (2.7)
preferential doping does not necessarily mean that the
It is assumed in this paper that the x and y axes are
parent cuprate oxides are charge-transferinsulators, but
within CuO planes and the z axis is perpendicular to
it simply means that the local charge susceptibility of 2
CuO planes. The electron-phonon interactions are sim-
3d electrons is much smaller than that of 2p electrons, 2
ply given by
which implies that the effective on-site repulsion U be-
tween 3d electrons is very strong. It is assumed in this = iC ~vp,λq b† +b
dpiacpteerdtbhyatbaVnd≃ca1l.c6uleaVtioannsd.43ǫ,d44−,45ǫpSi≃nc1eetVhe, aosn-issitperUe- Hp pXq 2NMpωλq (cid:16) λ−q λq(cid:17)
should be so large that the Hubbard gap might open, ×η¯s(qp) ηΓ 12q PΓ(q), (2.8)
it is assumed that U 5 eV. Then, Eq. (2.1) gives Γ=s,d
≃ X (cid:0) (cid:1)
J 0.27 eV. This is about twice as large as the ex-
per≃im−ental J = (0.10–0.15) eV.33 This discrepancy is and
− ~v
resolvedwhennonzerobandwidthsofUHB andLHBare = iC d,λq b† +b
considered.31 Hd d 2NM ω λ−q λq
Displacements of the ith Cu ion and the [ij]th O ion, Xq d λq (cid:16) (cid:17)
which lies between the nearest-neighbor ith and jth Cu pη¯Γ(q) Γ(q), (2.9)
× P
ions, are given by Γ=s,d
X
with C and C being real constants, which are given in
~v p d
ui = 2NMd,λqω eiq·Riǫλq b†λ−q+bλq , (2.2) the previous paper,36 and
Xλq d λq (cid:16) (cid:17) η¯ (q)=2 e sin qxa +e sin qya , (2.10)
p s λx 2 λy 2
and
h (cid:16) (cid:17) (cid:16) (cid:17)i
and
~v
u[ij] =Xλq 2NMp,λpqωλqeiq·R[ij]ǫλq(cid:16)b†λ−q+bλq(cid:17), (2.3) η¯d(q)=2heλxsin(cid:16)qx2a(cid:17)−eλysin(cid:16)qy2a(cid:17)i. (2.11)
p
4
B. Kondo-lattice theory Thesingle-siteself-energyΣ˜ (iε )isgivenbythatofthe
σ n
Anderson model. It is expanded as
OneofthesimplesteffectiveHamiltoniansfortheelec-
Σ˜ (ε+i0) = Σ˜ + 1 φ˜ ε
tron part of cuprate oxide superconductors is the d-p σ 0 − γ
modelonasquarelattice. Sincetheanisotropyislarge,it + 1 (cid:0)φ˜ 1σ(cid:1)gµ H +O(ε2), (2.17)
s B
is convenient to consider phenomenologically quasi-two- − 2
dimensional features. The d-p model is approximately (cid:0) (cid:1)
at T = 0 K in the presence of an infinitesimally small
mapped to the t-J model or the t-J-infinite-U model:46
Zeeman energy gµ H, with g the g factor and µ the
B B
Ht-J = tijd†iσdjσ − 21J (Si·Sj) φB˜soharremaallgrneeatlo;nφ˜.sTh2eφ˜eγxpan1sifoonrncoeffi1c.ieWnthseΣn˜0E,qφ˜.γ(,2.a1n7d)
Xijσ Xhiji is used and the m≃ulti-site≫self-energ≃y is ignored, the dis-
+U d† d d† d , (2.12) persion relation of the Gutzwiller quasiparticles is given
∞ i↑ i↑ i↓ i↓
by
i
X
mwiatthiotnheshsouumldmbaetimonadoeveorvehrijnieianrdeisctantienigghtbhoartstahnedsum- ξσ(k)= φ˜1 Σ˜0+E(k)−µ − 12σW˜sgµBH, (2.18)
γ
h i
S = 1σαβd† d . (2.13) with
i 2 iα iβ
Xαβ W˜s =φ˜s/φ˜γ, (2.19)
The carrier density per unit cell is defined by
being the so called Wilson ratio for the Kondo problem.
1 The irreducible polarization function in spin channels
n= N d†iσdiσ . (2.14) is alsodividedinto single-siteandmulti-site polarization
i functions:
X(cid:10) (cid:11)
It should be noted that the infinitely large on-site repul-
π (iω ,q)=π˜ (iω )+∆π (iω ,q). (2.20)
sion U is introduced to exclude double occupancy so s l s l s l
∞
that n can never be larger than unity, or 0 n 1.
The single-site polarization function π˜ (iω ) is given by
≤ ≤ s l
The electron and hole pictures should be taken for the
that of the Anderson model. The spin susceptibilities of
so called hole-doped (n <1) and electron-doped (n >1)
the Anderson and t-J models are given, respectively, by
cuprate oxide superconductors, respectively, so that n is
the electron density for hole-doped ones and is the hole 2π˜ (iω )
s l
χ˜ (iω )= , (2.21)
density for electron-doped ones. The doping concentra- s l 1 U π˜ (iω )
∞ s l
tion is defined by δ =1 n, and the optimal concentra- −
−
tion, where superconducting T is the highest as a func- and
c
tion of δ, is δ 0.15. Then, δ . 0.15 and δ & 0.15 are
called under-d≃oped and over-doped concentrations, re- χ (iω ,q)= 2πs(iωl,q) , (2.22)
spectively. When transfer integrals between nearest and s l 1− 14J(q)+U∞ πs(iωl,q)
next-nearest neighbors, which are denoted by t and t′,
with (cid:2) (cid:3)
are only considered, the dispersion relation of electrons
or holes is given by
J(q)=2J[cos(q a)+cos(q a)]. (2.23)
x y
E(k) = 2t[cos(k a)+cos(k a)]
x y In Eqs. (2.21) and (2.22), the conventionalfactor 1g2µ2
+4t′cos(kxa)cos(kya). (2.15) is not included. A physical picture for Kondo la4tticeBs
is that local spin fluctuations at different sites interact
Accordingtobandcalculations,43,44,45 itfollowsthatt=
with each other by an intersite exchange interaction. In
(0.3-0.5) eV and t′ 0.3t for electrons in hole-doped
Kondo-latticetheory,therefore,anintersiteexchangein-
− ≃−
cuprateoxidesuperconductorsandt= (0.3-0.5)eVand
teraction I (iω ,q) is defined by
t′ +0.3t for holes in electron-doped o−nes. s l
≃
Every physical quantity is divided into single-site and χ˜ (iω )
χ (iω ,q)= s l . (2.24)
multi-site terms. Calculating the single-site term is re- s l 1 1I (iω ,q)χ˜ (iω )
ducedtodeterminingandsolvingselfconsistentlytheAn- − 4 s l s l
derson model, as is discussed in the Introduction. When It follows that
itisassumedthatthere isnoorderparameter,forexam-
ple, the self-energy of electrons is divided into single-site I (iω ,q)=J(q)+2U2 ∆π (iω ,q). (2.25)
s l ∞ s l
and multi-site self-energies:
The derivation of Eq. (2.25) from Eqs. (2.21) and
Σ (iε ,k)=Σ˜ (iε )+∆Σ (iε ,k). (2.16) (2.22) is rigorous because ignored terms, which are
σ n σ n σ n
5
O[1/U χ˜ (iω )],vanishforinfinitelylargeU . Theterm III. SOFTENING OF PHONONS DUE TO
∞ s l ∞
of 2U2 ∆π (iω ,q) is composed of two terms:47 ANTIFERROMAGNETIC SPIN FLUCTUATIONS
∞ s l
2U2 ∆π (iω ,q)=J (iω ,q) 4Λ(iω ,q). (2.26)
∞ s l Q l − l AneffectiveHamiltoniantobeeventuallyexaminedin
The firstterm JQ(iωl,q) is an exchangeinteractionaris- this paper is
ing from the virtual exchange of a pair excitation of
the Gutzwiller quasiparticles. According to the Ward = t-J + ph+ p+ d, (3.1)
relation,48 the static component of the single-site irre- H H H H H
ducible vertex function in spin channels is given by with
λ˜ =φ˜ [1 U π˜ (0)]. (2.27) 1
s s − ∞ s Hph = ωλq b†λqbλq+ 2 , (3.2)
Then, it follows that λq (cid:18) (cid:19)
X
U λ˜ =2φ˜ /χ˜ (0). (2.28)
∞ s s s and and defined by Eqs. (2.8) and (2.9), respec-
p d
H H
Whenthevertexcorrectionλ˜sgivenbyEq.(2.28)isused, tively. The t-J model Ht-J is defined on a square lat-
it follows that tice in Sec. IIB but it should be defined on a quasi-two-
dimensionallatticehere. Althoughnointerlayercoupling
4W˜ 2 1
J (iω ,q)= s P(iω ,q) P(iω ,q) , is included in t-J, the nature of quasi-two-dimensional
Q l χ˜2s(0)" l − N q l # AF spin fluctuHations is phenomenologically considered,
X (2.29) whichplaysone ofthe mostcrucialrolesinthe softening
with of Cu-O bond stretching modes, as is examined below.
In the absence of and , the Green function for
P(iω ,q)= 1 f[ξσ(k)]−f[ξσ(k+q)], (2.30) phonons is given by Hp Hd
l N ξ (k+q) ξ (k)+iω
σ σ l
kσ −
X 2ω
with Dλ(0)(iωl,q)= (iω )2λqω2 . (3.3)
1 l − λq
f(ε)= . (2.31)
eε/kBT +1 When the self-energy for phonons is denoted by
In Eq. (2.29), the single-site term is subtracted because Σ(λph)(iωl,q), which can be perturbatively calculated in
it is considered in the SSA. The strength of the ex- termsof p and d,therenormalizedGreenfunctionfor
H H
change interaction is proportional to the width of the phonons is given by
Gutzwillerband. Sincethechemicalpotentialliesaround
the center of the Gutzwiller band or the nesting of the Dλ(iωl,q) = Dλ(0)(iωl,q)
Fermi surface is sharp, the exchange interaction is an- +D(0)(iω ,q)Σ(ph)(iω ,q)D (iω ,q)
tiferromagnetic (AF) in cuprate oxide superconductors. λ l λ l λ l
2ω
The second term 4Λ(iωl,q) corresponds to the mode- = λq . (3.4)
mode coupling ter−m in the self-consistent renormaliza- (iω )2 ω2 2ω Σ(ph)(iω ,q)
l − λq− λq λ l
tiontheoryofspinfluctuations,49 whichisrelevantinthe
weak-couplingregimeforelectroncorrelationsdefinedby The renormalized energy of phonons, which is denoted
U/W .1. by ω∗ , is given by
λq
When Eq.(2.28) is used, the mutual interactionmedi-
ated by intersite spin fluctuations, which works between (ω∗ )2 ω2 2ω Σ(ph)(ω∗ +i0,q)=0. (3.5)
λq − λq− λq λ λq
the Gutzwiller quasiparticles, is given by
The renormalization of phonon energies is given by
(U λ˜ )2[χ (iω ,q) χ˜ (iω )]=φ˜2I∗(iω ,q), (2.32)
∞ s s l − s l s s l
∆ω = ω∗ ω
with λq λq− λq
I∗(iω ,q)= Is(iωl,q) . (2.33) = Σ(λph)(ωλq+i0,q)
s l 1− 41Is(iωl,q)χ˜s(iωl) − Σ(λph)(ωλq+i0,q) 2/2ωλq+··· , (3.6)
In Eq. (2.32), the single-site term is subtracted because
it is considered in the SSA. The exchange interaction unless Σ(ph)(ω (cid:2)+i0,q) is larger(cid:3)than ω .
| λ λq | | λq|
I (iω ,q)isenhancedintoI∗(iω ,q)byspinfluctuations. As well as AF spin fluctuations are developed around
s l s l
The mutual interaction mediated by spin fluctuations is Q due to I (iω ,q), dγ-wave superconducting (SC) and
s l
essentiallythe same as that due to the exchangeinterac- chargebondorder(CBO)fluctuationsarealsodeveloped
tion I (iω ,q) or I∗(iω ,q). due to I (iω ,q) or I∗(iω ,q). Although charge fluctua-
s l s l s l s l
In Kondo-lattice theory, a unperturbed state is con- tions are never much developed, charge-channel fluctu-
structed in the non-perturbative SSA theory and multi- ations can also contribute to the softening of phonons,
site or intersite effects are perturbatively considered in as well as AF, SC, and CBO fluctuations, provided that
terms of I (iω ,q) or I∗(iω ,q). vertexcorrectionsforthe dualspin operatorin spin, SC,
s l s l
6
and CBO channels are properly treated. According to with
the previous paper,36 the softening of the half-breathing
modes is mainly caused by the charge-channel fluctua- Y (q)=η¯ (q) C v η 1q +C v Mp , (3.9)
tions. Since the charge-channel fluctuations are signifi- Γ s p p,λq Γ 2 d d,λq M
" r d#
cant in the metallic phase, the softening is large in the (cid:0) (cid:1)
metallic phase but is small in the insulating phase. and
Since phonons can couple with two lines or two chan-
ntheelsdoufaslp-sipninfluocpteuraattioorn,satsoitsheshloowwnestinorEqfisr.st(2o.r8d)eranind XΓΓ′(iωl,q)= kNBT ηΓ(p)ηΓ′(p)χs iωl′,p+12q
(2.9), AF spin fluctuations around ( 3π/4a, π/a) and ωXl′p (cid:0) (cid:1)
( π/a, 3π/4a) in 2DBZ can play a±signific±ant role in ×χs −iωl′ −iωl,−p+21q . (3.10)
± ±
the softening of Cu-O bond stretching modes around
In Eq. (3.10), two χ(cid:0)’s appear because o(cid:1)f the dual-
( π/2a,0) and (0, π/2a) in 2DBZ. According to Mer- s
m±in and Wagner,5±0 if the N´eel temperature T were spin operator. It should be noted that 2Q’s are
N equivalent to ( π/2a,0) and (0, π/2a): 2Q G =
nonzero in two dimensions integrated effects of two- ± ± −
( π/2a,0) and (0, π/2a), with G = ( 2π/a,0) and
dimensional critical AF spin fluctuations would be di- ± ± ±
(0, 2π/a) being reciprocal lattice vectors in 2DBZ.
vergent at TN, which leads to a conclusion that TN ±
Then, Cu-O bond stretching modes around ( π/2a,0)
must be zero in two dimensions. Their argument im- ±
and (0, π/2a) in 2DBZ can be soft provided that
plies that quasi-two-dimensional critical AF spin fluctu- ±
AF fluctuations around Q = ( 3π/4a, π/a) and
ations can play a crucial role in the softening, at least, ± ±
( π/a, 3π/4a)in 2DBZ are developed.
inanAFcriticalregionofcuprateoxidesuperconductors ± ±
SinceCu-Obondstretchingmodesaround2Qarecon-
provided that the anisotropy of critical AF spin fluctua-
sidered, the vibrations of Cu ions are ignored, that is, it
tions is large. In order to examine how cruciala role the
is assumed that
anisotropyplaysinthesoftening,itismoreconvenientto
use a phenomenological expression for the spin suscepti-
C v M /M =0, (3.11)
bility, which includes explicitly the anisotropy factor for d d,λq p d
| |
AF spin fluctuations, than to calculate microscopically q
thespinsusceptibilityforquasi-two-dimensionalsystems. and
The superexchange interaction J(q), which is given by
C v =c eV/˚A, (3.12)
Eq. (2.23), has broad peaks at ( π/a, π/a) in 2DBZ p p,λq p
| |
± ±
and the exchange interaction J (0,q), which is given by
Q where c is a dimensionless constant and it is likely36
Eq. (2.29), has sharp peaks at nesting wave numbers of p
the Fermi surface. Therefore,it is assumedin this paper
c =O(1). (3.13)
thatI (0,q)ismaximalatQ=( 3π/4a, π/a,Q )and p
s z
± ±
( π/a, 3π/4a,Q ) or that χ (0,q) is maximal at Q,
± ± z s Since the contribution from small p is large in the sum-
and that the spin susceptibility (2.24) is approximately
mation over p in Eq. (3.10), only the contribution from
but well described by
the Γ=s channel is considered. Then, it follows that
χ (0,Q)κ2
χs(iωl,Q+q)= κ2+(q a)2s+δ2(q c)2+ |ωl| , (3.7) Σ(λph)(ωλq+i0,q)=−AqΞ(ωλq+i0,q), (3.14)
k z Γ
AF with
around each of Q’s, with q = (q ,q ) being the com-
ponent parallel to CuO2 plaknes, qzxthye component per- A = ~2 3 Γ χ (0,Q)κ2 2 C v 2, (3.15)
pendicular to CuO2 planes, c the lattice constant along q 2Mpωλq42 AF s | p p,λq|
the z axis, and Γ the energy scale of AF spin fluctua- (cid:2) (cid:3)
AF
tions. The anisotropy factor δ is introduced to consider and
quasi-two-dimensional AF spin fluctuations. The corre-
X (ω +i0,q)
lation length within the x-y plane is a/κ and that along Ξ(ωλq+i0,q)=η¯s2(q)ηs2 12q Γss [χλq(Q)κ2]2 . (3.16)
the z axis is δc/κ. A cut-off q = π/3a is introduced AF s
c
(cid:0) (cid:1)
in such a way that χ (iω ,Q+q) = 0 for q > q or
q > q . The anisotsropylof the lattice con|stxa|nts pclays It should be noted that Ξ(iωl,q) is defined as a dimen-
| y| c sionless quantity. The effective transfer integralbetween
no role when δ and q are defined in these ways.
c nearest neighbors for the Gutzwiller quasiparticles is
When AF spin fluctuations are only considered, the
self-energy for phonons is given by36 t∗ =t/φ˜ . (3.17)
γ
~2 3
Σ(λph)(iωl,q) = −2M ω 42 YΓ(q)YΓ′(q) According to Eq. (1.1), a plausible number for t∗ is
p λq ΓΓ′
X
XΓΓ′(iωl,q), (3.8) t∗ W∗/8 40-50 meV. (3.18)
× | |≃ ≃
7
FIG. 1: (a) ReˆΞ(ωλq+i0,2Q0)˜ as a function of κ2, with Q0 =(−3π/4a,π/a), and (b) ReˆΞ(ωλq+i0,q)˜ as a function of
q (−2π/a≤q ≤−π/a)for q =2π/a. Fortheanisotropy factor, (i) δ=1,(ii) δ=10−1/2, (iii) δ=10−1, and(iv)δ=10−3.
x x y
In each figure, solid, dotted,dashed, and dashed chain lines are for ωλq/ΓAF = 0.2, 0.4, 0.8, and 1.6, respectively.
According to a microscopic calculation for the spin Sincethesofteningissmallwhenκ2 islargeorAFspin
susceptibility, it follows that Γ /t∗ = O(1) and fluctuations are not developed, it must be small in over-
AF
| |
χ (0,Q)κ2 t∗ =O(1). Itisassumed,forthesakeofsim- dopedcuprateoxidesuperconductors,whosedopingcon-
s
| |
plicity, that the energy of Cu-O bond stretching modes centrations are larger than those of optimal-doped ones.
is constant and is as large as When AF spin fluctuations are developed similarly or
differentlybetween( 3π/4a, π/a)and( π/a, 3π/4a)
± ± ± ±
ωλq =50 meV. (3.19) because of the anisotropy of the Fermi surface within
2DBZ, the softening must also occurs similarly or differ-
Then, Aq defined by Eq. (3.15) is approximately given entlybetweenthexandyaxesorbetween( π/2a,0)and
±
by (0, π/2a). These two predictions are consistent with
obs±ervations.7,8
A 10 c2ΓAF χ (0,Q)κ2 t∗ 2 meV
q ≃ × p t∗ s | |
| |
10c2 meV. (cid:2) (cid:3) (3.20) IV. STRIPES AND CHECKER BOARDS
≃ p
In this paper, T = 0 K is assumed in the ωl′ sum of Since the 4a-period and 4a 4a-period correspond to
Eq. (3.10). ×
2Q, with Q = ( 3π/4a, π/a) and ( π/a, 3π/4a), a
The softening around one of 2Q’s or 2Q , with Q = ± ± ± ±
0 0 plausible scenario for the stripes and checker boards is
( 3π/4a,π/a) in 2DBZ is considered; 2Q is equivalent
0 that the complete softening is followed by the stabiliza-
−
to (π/2a,0). Figure 1 shows the dependence of Ξ(ωλq+ tion of CDW with 2Q. In general,the 2Q component of
i0,q) on κ, δ, Γ , and q; Fig. 1(a) and Fig. 1(b) show
AF the density of states, ρ2Q(ε), as a function of ε is com-
Re[Ξ(ω +i0,2Q )]asa functionofκ2 andRe[Ξ(ω +
λq 0 λq posedofsymmetricandasymmetriccomponentswithre-
i0,q)] as a function of q , respectively,for severalsets of
x specttothechemicalpotentialorε=0. Theasymmetric
δ and ωλq/ΓAF. According to Fig. 1(b), Ξ(ωλq +i0,q) component is large when CDW with 2Q is stabilized as
has a maximum, that is, Re Σ(λph)(ωλq + i0,q) has a a fundamental 2Q effect. According to an experiment,37
minimum around 2Q as a function of q. According to the symmetric component is larger than the asymmetric
0
(cid:2) (cid:3)
Eq. (3.20), Fig. 1(a), and Fig. 1(b), it is likely that the one, which contradicts the scenario of CDW even if the
softening at 2Q is as large as (10-20)meV for κ2 1 softening of the 2Q modes is large and the 2Q fluctua-
0
− ≪
andδ 1. Itshouldbenotedthatthesofteningcanonly tionsarewelldeveloped. Ontheotherhand,thesymmet-
≪
be large provided that κ2 1 and δ 1, as is implied ric component is large when the 2Q modulation is due
by Mermin and Wagner’s a≪rgument;50≪integrated effects to a simple second-harmonic effect of an ordered SDW
on the softening are never divergent even in the limit of with Q.51 The second-harmonic effect of the SDW can
κ 0 and δ 0. explain the observed almost symmetric ρ (ε). When
→ → 2Q
It is definite that κ2 1 in the critical region of anti- stripesandchecker-boardsarereallystaticorders,stripes
≪
ferromagneticorderingorspindensity wave(SDW), and must be due to single-Q SDW and checker-boards must
it is certain that the anisotropy is as large as δ < 10−3 be due to double-Q SDW. It is predicted that magne-
in cuprate oxide superconductors. Then, the second- tizations of the two waves must be orthogonal to each
harmonic effect of AF spin fluctuations can explain the otherin double-Q SDW.52,53 It is interestingto examine
observed softening7,8 as large as (10-20) meV around whether the prediction actually holds in cuprate oxide
−
2Q or 2Q. superconductors.
0
8
The appearance or stabilization of SDW is a transi- electrons in spin channels across the Hubbard gap be-
tion rather than a crossover. However, no specific heat cause the energy scales of spin fluctuations or spin ex-
anomalyhasbeenreportedsofarexceptfortheanomaly citations are totally different from each other in the two
due to superconductivity. The absence of any specific physicalprocesses. Themainpartoftheattractiveinter-
heat anomaly implies that, even if SDW is stabilized, action in cuprate oxide superconductors must be the su-
SDW is never a homogeneous phase but is an inhomo- perexchange interaction rather than the interaction me-
geneous phase, which is composed of many domains. If diated by low-energy AF spin fluctuations. Since the
the transition temperatures of SDW can be different in superexchange interaction is as strong as J = (0.10-
−
different domains, no significant specific anomaly can be 0.15) eV,33 observed high T can be easily reproduced,
c
observed. ItisplausiblethatSDWisandisorder-induced as is discussed in the Introduction.
SDW.47
On the other hand, it is proposed42 that a stripe or
a checker-boardat rather high temperatures must be an VI. CONCLUSION
exoticorderedstate,thatis,afluctuatingstateinaquan-
tum disordered phase. It should be examined whether
it is actually such an exotic state. Another possibil- In cuprate oxides superconductors, the electron-
ity is that it is a rather normal low-energy fluctuating phonon interaction arising from the modulation of the
state, whose energy scale is as small as that of the soft superexchange interaction by lattice vibrations is strong
phonons. The other possibility is the disorder-induced enough to cause the softening of not only the half-
SDW,47 which can behave as a fluctuating state because breathing modes around ( π/a,0) and (0, π/a) in
± ±
it is inhomogeneous. the two-dimensional Brillouin zone but also Cu-O bond
stretchingmodes around( π/2a,0)and(0, π/2a). Al-
± ±
though the softening of the bond stretching modes is
V. ATTRACTIVE INTERACTION responsible for stripe and checker-board fluctuations in
charge channels, the stabilization of a charge density
Although the electron-phonon interaction plays no or wave state following the complete softening of the bond
only a minor role in the formation of dγ-wave Cooper stretching modes can never be any relevant scenario
pairs, as is discussed in the Introduction, isotope shifts for ordered stripe and checker-board states. The or-
of T can arise from the depression of superconductivity dered stripe or checker-board state must be simply a
c
by the 2Q fluctuations, whose development depends on single-Qordouble-Qspindensity wavestate,whose Q’s
the mass of O ions. are ( 3π/4a, π/a) and ( π/a, 3π/4a). The strong
± ± ± ±
In Kondo-lattice theory, cuprate oxide superconduc- electron-phonon interaction can play no or only a minor
torscanberelevantlytreatedasoneofthetypicalKondo role in the binding of dγ-wave Cooper pairs in cuprate
lattices. AccordingtoEq.(2.32),whichisoneofthemost oxide superconductors, because the attractive interac-
crucial results of Kondo-lattice theory, two mechanisms tion arising from the virtual exchange of a phonon is
ofattractiveinteractions,thespin-fluctuationmechanism neverstrongbetweenquasi-particlesonnearest-neighbor
and the exchange-interaction mechanism, are essentially Cu ions on a CuO plane. However, isotope shifts of T
2 c
the same as each other. However, the attractive inter- canarisefromthedepressionofsuperconductivitybythe
action mediated by low-energy spin fluctuations such as stripe or checker-board fluctuations. Since the superex-
those observed by low-energy neutron inelastic scatter- change interaction is as strong as J = (0.10-0.15) eV
−
ings or those described by the phenomenological spin betweennearest-neighborCuions,thesuperexchangein-
susceptibility (2.24) is physically different from the at- teraction must be mainly responsible for the binding of
tractiveinteractionduetothesuperexchangeinteraction, the dγ-waveCooper pairs in cuprate oxide superconduc-
which arises from the exchange of a pair excitation of tors.
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