Table Of ContentFebruary 2, 2008 12:40 WSPC/INSTRUCTION FILE Pms
5
0
0
2
n
a
J INCOMMENSURATE SPIN DYNAMICS
8
IN UNDERDOPED CUPRATE PEROVSKITES
1
]
l
e
A.Sherman
-
r Institute of Physics, Universityof Tartu, Riia 142, 51014 Tartu, Estonia
t
s alexei@fi.tartu.ee
.
t
a
m M.Schreiber
- Institut fu¨rPhysik, Technische Universit¨at,D-09107 Chemnitz, Federal Republic of Germany
d
n
o ReceivedDayMonthYear
c RevisedDayMonthYear
[
Theincommensuratemagneticresponseobservedinnormal-statecuprateperovskitesis
1
interpretedbasedontheprojectionoperatorformalismandthet-JmodelofCu-Oplanes.
v
Inagreementwithexperimentthecalculated dispersionofmaximainthesusceptibility
8
hastheshapeoftwoparabolaswithupwardanddownwardbrancheswhichconvergeat
1
theantiferromagneticwavevector.Themaximaarelocatedatthemomenta(1,1±δ),
4 2 2
1 (12±δ,12)andat(12±δ,12±δ),(12±δ,21∓δ)inthelowerandupperparabolas,respectively.
0 The upper parabola reflects the dispersion of magnetic excitations of the localized Cu
5 spins,whilethelowerparabolaarisesduetoadipinthespin-excitationdampingatthe
0 antiferromagneticwavevector.Formoderatedopingthisdipstemsfromtheweaknessof
/ theinteractionbetweenthespinexcitationsandholesnearthehotspots.Thefrequency
t dependence of the susceptibility is shown to depend strongly on the hole bandwidth
a
m anddamping andvariesfromtheshape observedinYBa2Cu3O7−y tothat inherent in
La2−xSrxCuO4.
-
d
Keywords: Cupratesuperconductors; magneticproperties;t-J model.
n
o
c 1. Introduction
:
v
Oneofthemostinterestingfeaturesoftheinelasticneutronscatteringinlanthanum
i
X
cupratesisthatforholeconcentrationsx&0.04,lowtemperaturesandsmallenergy
r transfersthescatteringispeakedatincommensuratemomenta(1,1 δ),(1 δ,1)in
a 2 2± 2± 2
the reciprocallattice units 2π/awith the lattice perioda.1 For x.0.12the incom-
mensurabilityparameterδ isapproximatelyequaltox.2 Forlargerxtheparameter
saturatesnearthevalueδ 0.12.Theincommensurateresponsewasobservedboth
≈
belowandaboveT .3Recentlytheanalogouslow-frequencyincommensurabilitywas
c
observedalsoinYBa Cu O .4 Thisgivesgroundtosupposethatthe incommen-
2 3 7−y
surabilityisacommonfeatureofcuprateperovskiteswhichdoesnotdependonsub-
tle details ofthe energystructure.However,forlargerfrequencies the susceptibility
differs essentially in these two types of cuprates.In the underdoped YBa Cu O
2 3 7−y
and some other cuprates both below and above T a pronounced maximum is ob-
c
1
February 2, 2008 12:40 WSPC/INSTRUCTION FILE Pms
2 A. Sherman & M. Schreiber
served at frequencies ω = 25 40 meV.5 In the momentum space the magnetic
r
−
responseissharplypeakedatthe antiferromagneticwavevectorQ=(1,1)forthis
2 2
frequency. Contrastingly, no maximum at ω was observed in lanthanum cuprates.
r
Instead for low temperatures and frequencies of several millielectronvolts a broad
feature was detected.6 For even larger frequencies the magnetic response becomes
againincommensurateinbothtypesofcuprateswithpeakslocatedat(1 δ,1 δ),
2± 2±
(1 δ,1 δ).5,7,8,9Incontrasttothelow-frequencyincommensurabilityinwhichthe
2± 2∓
incommensurability parameter decreases with increasing frequency, the parameter
of the high-frequency incommensurability grows or remains practically unchanged
with frequency. Thus, the dispersion of maxima in the susceptibility resembles two
parabolaswith upward-and downward-directedbrancheswhich convergeat Q and
near the frequency ω .4,9
r
The nature of the magnetic incommensurability is the subject of active discus-
sion now. The most frequently used approaches for its explanation are based on
the picture of itinerant electrons with the susceptibility calculated in the random
phase approximation10,11 and on the stripe domain picture.9,12 In the former ap-
proach the low-frequency incommensurability is connected with the Fermi surface
nesting in the normal state or with the nesting in constant-energy contours in the
superconducting case. This imposes rather stringent requirements on the electron
energy spectrum, since the nesting has to persist in the range of hole concentra-
tions 0.04 . x . 0.18 where the incommensurability is observed and the nesting
momentum has to change in a specific manner with doping to ensure the known
dependence of the incommensurability parameter δ on x. It is unlikely that these
conditions arefulfilled inLa Sr CuO .13 Besides,the applicabilityofthe picture
2−x x 4
of itinerant electrons for underdoped cuprates casts doubts. As for the second no-
tion, it should be noted that in the elastic neutron scattering the charge-density
wave connected with stripes is observed only in crystals with the low-temperature
tetragonal or the low-temperature less-orthorhombic phases (La Ba CuO
2−x x 4
and La Nd Sr CuO ) and is not observed in the crystal La Sr CuO in
2−y−x y x 4 2−x x 4
the low-temperature orthorhombic phase.14 At the same time the magnetic incom-
mensurability is similar in these phases. It can be supposed that the magnetic
incommensurability is the cause rather than the effect of stripes which are formed
with an assistance of phonons.
In the present work the general formula for the magnetic susceptibility derived
in the projection operator formalism15 is used. For the description of spin excita-
tions in the doped antiferromagnet the t-J model of a Cu-O plane is employed. In
thisapproachthementionedpeculiaritiesofthemagneticpropertiesofcupratesare
reproducedincludingtheproperfrequencyandmomentumlocationofthesuscepti-
bilitymaxima.Theincommensurabilityforω >ω isconnectedwiththedispersion
r
of spin excitations.16,17 The incommensurability for lower frequencies is related to
the dip in the spin-excitation damping at Q. For small x the dip appears due to
the nesting of the hole pockets around ( 1, 1) forming the Fermi surface.18 For
±4 ±4
February 2, 2008 12:40 WSPC/INSTRUCTION FILE Pms
Incommensurate spin dynamics in underdoped cuprate perovskites 3
moderate doping this dip stems from the weakness of the interaction between the
spin excitations and holes near the hot spots – the intersection points of the Fermi
surface and the boundary of the magnetic Brillouin zone. Such a weak interaction
followsfromthefactthatduetoashort-rangeinteractionbetweenholesandspinsa
decayingsitespinexcitationcreatesafermionpairwithcomponentsresidingonthe
sameandneighborsites.Thespin-excitationdampingwasfoundtodependstrongly
on details of the hole dispersion, bandwidth and damping, so that the change in
thesecharacteristicsleadsto the conversionofwell-definedspinexcitationstoover-
dampedones.As thistakesplace,the frequency dependence ofthe susceptibility at
Q is transformed from a pronounced maximum5 at ω which is inherent in under-
r
dopedYBa Cu O toabroadlow-frequencyfeaturecharacteristicforlanthanum
2 3 7−y
cuprates.6 The increased spin-excitationdamping has no markedeffect on the low-
frequency incommensurability, however for ω > ω the incommensurate peaks are
r
shifted to Q andforma broadmaximum.Suchformofthe momentumdependence
of the susceptibility is also observed experimentally.19
2. Main formulas
The imaginary part of the magnetic susceptibility which determines the cross-
section of the magnetic scattering20 is calculated from the relations χ′′(kω) =
4µ2 sz sz , sz sz = ω((sz sz )) (sz,sz ). Here µ is the Bohr
− Bℑhh k| −kiiω hh k| −kiiω k| −k ω − k −k B
magneton, sz sz and ((sz sz )) are the Fourier transforms of the retarded
hh k| −kiiω k| −k ω
Green’s and Kubo’s relaxation functions,
∞
sz sz = iθ(t) [sz(t),sz ] , ((sz sz )) =θ(t) dt′ [sz(t′),sz ] ,
hh k| −kiit − h k −k i k| −k t Z h k −k i
t
sz =N−1/2 e−iknsz withthenumberofsitesN andthezcomponentofthespin
k n n
sz onthe latPticesiten,forarbitraryoperatorsAandB (A,B)=i ∞dt [A(t),B]
n 0 h i
where the angularbrackets denote the statistical averagingand A(tR)=eiHtAe−iHt
with the Hamiltonian H.
Using the projection operator technique15 the relaxation function ((sz sz ))
k| −k ω
can be calculated from the recursive relations
R (ω)=[ω E F R (ω)]−1, n=0,1,2,... (1)
n n n n+1
− −
where R (ω) is the Laplace transform of R (t) = (A ,A†)(A ,A†)−1, the time
n n nt n n n
dependence in A is determined by the relation
nt
n−1
d
i A = (1 P )[A ,H], A =A
nt k nt n,t=0 n
dt −
kY=0
with the projection operators P defined as P B = (B,A†)(A ,A†)−1A . The pa-
k k k k k k
rameters E and F in relations (1) and operators A in the functions R (t) are
n n n n
calculated recursively using the procedure17
[A ,H]=E A +A +F A , E =([A ,H],A†)(A ,A†)−1,
n n n n+1 n−1 n−1 n n n n n
February 2, 2008 12:40 WSPC/INSTRUCTION FILE Pms
4 A. Sherman & M. Schreiber
(2)
F =(A ,A†)(A ,A† )−1, F =0.
n−1 n n n−1 n−1 −1
As the starting operator for this procedure we set A = sz. In this case
0 k
((sz sz )) =(sz,sz )R (ω) where R (ω) is calculated from Eq. (1).
k| −k ω k −k 0 0
To describe the spin excitations of Cu-O planes which determine the magnetic
properties of cuprates20 the t-J model21 is used. The model was shown to describe
correctly the low-energy part of the spectrum of the realistic extended Hubbard
model.22,23 The Hamiltonian of the two-dimensional t-J model reads
1
H = t a† a + J s s , (3)
nm nσ mσ 2 nm n m
nXmσ Xnm
where a = nσ n0 is the hole annihilation operator, n and m label sites of
nσ
| ih |
the square lattice, σ = 1 is the spin projection, J and t are the exchange
nm nm
±
and hopping constants, respectively, nσ and n0 are site states corresponding to
| i | i
the absence and presence of a hole on the site. These states may be considered as
linear combinations of the products of the 3dx2−y2 copper and 2pσ oxygen orbitals
of the extended Hubbard model.23 The spin-1 operators can be written as sz =
2 n
1 σ nσ nσ and sσ = nσ n, σ .
2 σ | ih | n | ih − |
PWith Hamiltonian (3) and A =sz we find from Eq. (2)
0 k
E (sz,sz )=(is˙z,sz )= [sz,sz ] =0,
0 k −k k −k h k −k i
1
A =As+Ah = e−ikl J (δ δ )s+1s−1 (4)
1 1 1 2√N (cid:20) mn ln− lm n m
Xl Xnm
+ t (δ δ )σa† a ,
mn lm− ln nσ mσ(cid:21)
nXmσ
where is˙z = [sz,H]. To obtain a tractable form for the spin-excitation damping it
k k
is convenient to approximate the quantity (A ,A†) in the R (ω) by the sum
1t 1 1
(Ah(t),Ah†)+(As ,As†)
1 1 1t 1
where the first term describes the influence of holes on the spin excitations. Con-
tinuing calculations (2) with the second term of the sum we get
F =4JC (γ 1)(sz,sz )−1, E =0, (5)
0 1 k− k −k 1
where only the nearest neighbor interaction between spins was taken into account,
J = J δ , the four vectors a connect the nearest neighbor sites, C =
nm a n,m+a 1
hs+n1s−n+1aiPis the spin correlation on neighbor sites and γk = 21[cos(kx)+cos(ky)].
To calculate the quantity (sz,sz ) let us notice that in accord with procedure
k −k
(2) the interruption of calculations at this stage actually means that (A ,A†) in
2 2
the parameter F is set to zero. Here A = i2s¨z F sz. The substitution of this
1 2 k − 0 k
expression into (A ,A†) = 0 gives an equation for (sz,sz ). Using the decoupling
2 2 k −k
in calculating i2s¨z we get17
k
(sz,sz )−1 =4αJ(∆+1+γ ), (6)
k −k k
February 2, 2008 12:40 WSPC/INSTRUCTION FILE Pms
Incommensurate spin dynamics in underdoped cuprate perovskites 5
whereα 1isthedecouplingparameter.24Themeaningoftheparameter∆,which
∼
can be expressed in terms of spin correlations,will be discussed later.
Using the decoupling in (Ah(t),Ah†) we find from the above formulas
1 1
4µ2ω R(kω)
χ′′(kω)= B ℑ , (7)
−[ω2 ωf R(kω) ω2]2+[ωf R(kω)]2
− kℜ − k kℑ
where
f−1 =4J C (1 γ ), ω2 =16J2αC (1 γ )(∆+1+γ ),
k | 1| − k k | 1| − k k
8πω2 ∞
ℑR(kω)= Nk gk2k′Z dω′A(k′ω′) (8)
Xk′ −∞
n (ω+ω′) n (ω′)
A(k+k′,ω+ω′) F − F ,
× ω
the interaction constant gkk′ = tk′ −tk+k′ with tk = neik(n−m)tnm, nF(ω) =
[exp(ω/T)+1]−1,T isthetemperatureandA(kω)isthehPolespectralfunction.Since
the incoherentpartof the spectralfunction is unlikely to leadto sharpstructure in
χ′′, only the coherent part of A(kω) is taken into account in this work,
η/π
A(kω)= . (9)
(ω ε +µ)2+η2
k
−
Hereµisthechemicalpotential,ηistheartificialbroadening,andε istheholedis-
k
persion.TherealpartofR(kω)canbe calculatedfromthe imaginarypart R(kω)
ℑ
and the Kramers-Kronigrelation.
Noticethattheinteractionconstantgkk′ isdeterminedbytheFouriertransform
of the hole hopping constant t . If the hopping to the nearest and next nearest
nm
sites is taken into account the constant acquires the form
gkk′ =t(γk′ −γk+k′)+t′(γk′′ −γk′+k′), (10)
where γ′ = cos(k )cos(k ). This constant vanishes for k=Q when the vector k′
k x y
is locatedatthe boundary ofthe magneticBrillouinzone.In otherwords,fermions
near hot spots interact weakly with spin excitations. This is connected with the
short-range character of the interaction described by constant (10) – the decaying
spinexcitationonthesitencreatesthefermionpaironthesameandneighborsites
which is reflected in the above form of the interaction constant.
As the quantity ωf R(kω) influences the frequency of spin excitations only
k
ℜ
nearQ, it is convenientto incorporateit inω . This modifies the parameter∆>0
k
which,asseenfromEqs.(7)and(8),describesagapinthespin-excitationspectrum
at the antiferromagnetic wave vector Q. The most exact way to determine this
parameter is to use the constraint of zero site magnetization
1
sz = (1 x) s−1s+1 =0, (11)
h ni 2 − −h n n i
which has to be fulfilled in the short-range antiferromagnetic ordering. It can be
shown that ∆ ξ−2 where ξ is the correlation length of the short-range order.25
∝
February 2, 2008 12:40 WSPC/INSTRUCTION FILE Pms
6 A. Sherman & M. Schreiber
120
80
V)
e
m
ω (k
40
0
0.4 0.5 0.6
k (r.l.u.)
Fig.1. Thedispersionofspinexcitationscalculatedina20×20latticeforx=0.06andT =17K
(filledsquares).17ThesolidlineisthefitofEq.(12)tothesedata.Opensquaresarethedispersion
ofthepeakintheoddsusceptibilityinYBa2Cu3O6.5 (x≈0.075,Ref.26)atT =5K.5
Thus, in this case the frequency of spin excitations at Q is nonzero, in contrast to
the classical antiferromagnetic magnons. As follows from Eq. (8), the dispersion of
spin excitations has a local minimum at Q and can be approximated as
ω =[ω2 +c2(k Q)2]1/2 (12)
k Q −
near this momentum. In Fig. 1 the calculated dispersion of spin excitations17
near Q is compared with the dispersion of the maximum in the susceptibility in
YBa Cu O .5 This is a bilayer crystal and the symmetry allows one to divide
2 3 6.5
the susceptibility into odd and even parts. For the antiferromagnetic intrabilayer
coupling the dispersionof the maximumin the odd partcanbe comparedwith our
calculations carried out for a single layer. This comparison demonstrates that the
observeddispersion of the susceptibility maxima aboveω , which we identify with
Q
the resonance frequency ω , is closely related to the dispersion of spin excitations.
r
Previouscalculations25 showthatthe variationofthetemperatureinthe range
from 0 to approximately 100 K leads only to some broadening of maxima in the
susceptibility. Therefore to simplify calculations and use larger lattices, which is
necessary to resolve the low-frequency incommensurability, let us set T = 0. In
calculating R(kω) the integration over frequencies in Eq. (8) is the most time-
ℑ
consuming operation. For T =0 and ω 0 this integral reduces to
≥
0
dω′A(k′ω′)A(k+k′,ω+ω′)
Z
−ω
and is easily integrated for the spectral function (9). The same result is obtained
for ω < 0, since R(kω) is an even function of frequency. Notice that for η ω
ℑ ≪
the states with energies
ω <εk′ µ<0 and 0<εk+k′ µ<ω (13)
− − −
February 2, 2008 12:40 WSPC/INSTRUCTION FILE Pms
Incommensurate spin dynamics in underdoped cuprate perovskites 7
make the main contribution to this integral.
In the following, we use the values of C , ∆ and α calculated self-consistent-
1
ly in the t-J model on a 20 20 lattice for the range of hole concentrations 0
× ≤
x . 0.16.25 The calculations were carried out for the parameters t = 0.5 eV and
J =0.1eV correspondingto hole-dopedcuprates.27 In Eq.(9), for ε we apply the
k
hole dispersion
ε = 0.0879+0.5547γ 0.1327γ′ 0.0132γ
k − k− k− 2k
+0.09245[cos(2k )cos(k )+cos(k )cos(2k )] 0.0265γ′ (14)
x y x y − 2k
proposed from the analysis of photoemission data in Bi Sr CaCu O .28 Here the
2 2 2 8
coefficients are in electronvolts. Results which are analogous to those discussed in
the next section can also be obtained with other model dispersions suggested for
cuprates.10,11,28 Resultsdo notchangequalitativelyeither withthe variationofthe
parametert′ inEq.(10)in the rangefrom0to 0.4t(notice that parameterst and
−
t′ of the hole hopping part of Hamiltonian (3) are only indirectly connected with
the coefficients in Eq. (14), since to a great extent the hole dispersion is shaped by
the interaction between holes and spin excitations29).
3. Magnetic susceptibility
Themomentumdependenceofχ′′(kω)calculatedwiththeaboveequationsforthree
energy transfers are shown in Fig. 2. The contour plots of the susceptibility for the
same parameters are demonstrated in Fig. 3. As seen from these figures, there
are three frequency regions with different shapes of the momentum dependence of
χ′′(kω).Thefirstregionisthevicinityofthefrequencyω ofthegapinthedisper-
Q
sionofspinexcitationsatthe antiferromagneticwavevectorQ.Forthe parameters
ofFig.2ω 37meV.Inthisregionthesusceptibilityispeakedatthewavevector
Q
≈
Q.Forsmallerandlargerfrequenciesthemagneticresponseisincommensurate.The
dispersion of maxima in χ′′(kω) for scans along the edge and the diagonal of the
Brillouinzoneandtheirfullwidthsathalfmaximum(FWHM)areshowninFig.4.
Analogous dispersion was obtained in Ref. 28 in the itinerant-carrier approach for
the superconducting state.
The momentum dependencies of the susceptibility which are similar to those
shown in Fig. 2 and 3 were observed in yttrium and lanthanum cuprates.3,4,9 The
dispersion of the peaks in χ′′(kω) which is similar to that shown in Fig. 4 was
derivedfromexperimentaldatainYBa Cu O andLa Ba CuO inRefs.4,9.
2 3 7−y 2−x x 4
AsseenfromFig.2,forfrequenciesω <ω the susceptibilityispeakedatthe wave
Q
vectors k=(1,1 δ) and (1 δ,1), while for ω >ω the maxima are located at
2 2 ± 2 ± 2 Q
(1 δ,1 δ),(1 δ,1 δ)forthe parametersused.Thisresultisalsoinagreement
2± 2± 2± 2∓
withexperimentalobservations.4,8,9 Notice,however,thatforω >ω the positions
Q
of maxima in the momentum space may vary with parameters.
To understand the above results one should notice that Eq. (7) contains the
resonancedenominator which will dominate in the momentum dependence for ω
≥
February 2, 2008 12:40 WSPC/INSTRUCTION FILE Pms
8 A. Sherman & M. Schreiber
1.0
(a)
0.5
0.0
units) 1.0 (b)
b.
ar
’’ ( 0.5
0.0
1.0
(c)
0.5
0.0
0.3 0.4 0.5 0.6 0.7
(r.l.u.)
Fig. 2. The momentum dependence of χ′′(kω) for T = 0, x ≈ 0.12, µ = −40 meV, t′ = −0.2t
and ω =70meV, η =30 meV (a), ω=35 meV, η=15meV (b), ω =2 meV, η=1.5 meV (c).
Calculationswerecarriedoutina1200×1200lattice.Thesolidlinescorrespondtothescansalong
theedgeoftheBrillouinzone, k=(κ,1);thedashedlinesareforthezonediagonal,k=(κ,κ).
2
ω if the spin excitations are not overdamped. Parameters of Fig. 2 correspond to
Q
this case.For ω ω the equationω =ω determines the positions of the maxima
Q k
≥
in χ′′(kω) which are somewhat shifted by the momentum dependence of the spin-
excitation damping f R(kω). Using Eq. (12) we find that the maxima in χ′′(kω)
k
ℑ
are positioned near a circle centered at Q with the radius c−1(ω2 ω2)1/2.16,17
− Q
In the region ω < ω the nature of the incommensurability is completely dif-
Q
ferent. It is most easily seen in the limit of small frequencies when Eq. (7) reduces
to
R(kω)
χ′′(kω) 4µ2ωℑ . (15)
≈− B ω4
k
As seeninFig.1,ω−4 isadecreasingfunctionofthedifference k Qwhichactsin
k −
favorofacommensuratepeak.However,if R(kω)inthenumeratorofEq.(15)has
ℑ
a pronounced dip at Q the commensurate peak splits into several incommensurate
maxima. For hole concentrations x.0.06, when the Fermi surface consists of four
ellipsescenteredat( 1, 1),21,29,30 R(kω)hasadipatQduetothenestingofthe
±4 ±4 ℑ
ellipses with this wavevector.18 For largerx the mechanismof the dip formationis
thefollowing.AsfollowsfromEq.(13),fork=Qandsmallfrequenciesωholestates
which make the main contribution to the spin-excitation damping (8) are located
February 2, 2008 12:40 WSPC/INSTRUCTION FILE Pms
Incommensurate spin dynamics in underdoped cuprate perovskites 9
0.7 0.7
(a) (b)
0.6 0.4 0.6
0.2
0.7
0.1
ky0.5 ky0.5 0.4
0.9 0.6
0.8
0.3
0.4 0.4
0.03
0.3 0.3
0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7
kx kx
0.7
(c)
0.4
0.6
0.9
0.7
ky 0.9 0.6 0.9
0.5 0.7
0.9
0.4
0.5 0.6
0.3
0.3 0.4 0.5 0.6 0.7
kx
Fig.3. Thecontourplotsofχ′′(kω).Parametersinparts(a),(b)and(c)arethesameasinthe
respectivepartsofFig.2.
near the hot spots (see Fig. 5). For these wave vectors the interaction constant
gQk′, Eq. (10), is small which leads to the smallness of R(Qω). With the wave
ℑ
vector moving away from Q momenta of states contributing to the spin-excitation
damping recede from the hot spots, the interaction constant grows, and with it
the spin-excitation damping. Thus, the damping has a dip at Q which leads to the
low-frequency incommensurability shown in Fig. 2c.
Letuscomparethediscussedmechanismsofthelow-andhigh-frequencyincom-
mensurabilitywiththosebasedonthepictureofitinerantelectronsandtherandom
phaseapproximation.Inthislatterapproachincommensurabilityarisesduetomax-
imainthenoninteractingsusceptibilityχ describedbythefermionbubbles.10,11,28
0
For low frequencies such a maximum appears if the Fermi surface has nesting. As
mentioned, this mechanism imposes rather stringent requirements on the electron
energyspectrum, because to reproduce knownexperimentalresults the nesting has
to persist in the wide range of hole concentrations and the nesting momentum has
tochangeinaspecificmannerwithdoping.InRef.11thenotionwasproposedthat
the nesting for constant-energy contours can appear in the superconducting state.
February 2, 2008 12:40 WSPC/INSTRUCTION FILE Pms
10 A. Sherman & M. Schreiber
150
100
V)
e
m
ω (
50
0
0.3 0.4 0.5 0.6 0.7
κ (r.l.u.)
Fig.4. Thedispersionofmaximainχ′′(kω)forscansalongtheedge[k=(κ,1),solidlines]and
2
the diagonal [k=(κ,κ), dashed lines]ofthe Brillouinzone. Thedispersionalong thediagonal is
shownonlyinthefrequencyrangeinwhichthesemaximaaremoreintensivethanthosealongthe
edge.ParametersarethesameasinFig.2.HorizontalbarsareFWHMformaximaalongtheedge
oftheBrillouinzone.
3
2
1
ky0 Q
-1
-2
-3
-3 -2 -1 0 1 2 3
k
x
Fig.5. TheFermisurfacefordispersion(14)and µ=−40meV(solidlines).Dashedlinesshow
the boundary of the magnetic Brillouinzone, gray circles are the hot spots, the dotted arrow is
theantiferromagneticwavevector.
The application of this idea also requires fine tuning of parameters.28 Besides, this
mechanismcannotexplaintheincommensurabilityaboveT whichisobservedboth
c
in lanthanum and yttrium cuprates.3,4,9 In the approach discussed in this paper
requirements on the Fermi surface are substantially relaxed: the Fermi surface has
tointersectwiththeboundaryofthemagneticBrillouinzone,i.e.theFermisurface
has to contain hot spots where the interaction constant gkk′ is small which leads
to the dip in R at Q. According to the available photoemission data13,30 Fermi
ℑ
surfaces of this type, which resemble that shown in Fig. 5, are indeed observed in
underdoped cuprates. Apart from Eq. (14) we used some other model dispersions
present in the literature10,11,28 and obtained results which are qualitatively similar
