The spin excitation spectrum in striped bilayer compounds

0.75 MB·English

by  Frank Krüger

The spin excitation spectrum in striped bilayer compounds
Frank Kru¨ger and Stefan Scheidl
Institut fu¨r Theoretische Physik, Universit¨at zu K¨oln, Zu¨lpicher Str. 77, D-50937 K¨oln, Germany
(Dated: February 2, 2008)
4 The spin dynamics of bilayer cuprate compounds are studied in a basic model. The magnetic
0 spectral properties are calculated in linear spin-wave theory for several stripe configurations which
0 differ by the relative location of the stripes in the layers. We focus on the bilayer splitting of
2 the magnon bands near the incommensurate low energy peaks as well as near the π resonance,
distinguishing between the odd and even channel. We find that a x-shaped dispersion near the
n
π resonance is generic for stripes. By comparison of our results to neutron scattering data for
a
J YBa2Cu3O6+x weconcludethatthestripemodelisconsistentwithcharacteristicfeaturesofbilayer
0 high-Tc compounds.
2
PACSnumbers: 75.10.Jm,74.72.-h,75.30.Ds,76.50.+g

]
n
o I. INTRODUCTION Furthermore,thedopingdependenceoftheresonancefre-
c quency was found in good agreement with experimental
r- Subsequent to predictions of stripe formation,1,2,3 observations.
p In this work we extend this model to bilayer systems
characteristic signatures of spin- and charge order
u in order to predict the corresponding features of the
have been found in a variety of high-T cuprate
s c
magnon band structure and the magnetic structure fac-
. superconductors, including La Sr CuO (LSCO)
at and YBa Cu O (YBCO). 2−Nxeuxtron 4scattering tor. Within each layer,holes are assumed to form unidi-
2 3 6+x
m experiments4,5 have provided evidence for spin order at rectionalsite-centeredstripes. Weconsiderseveralpossi-
bilities (parallel and perpendicular relative orientations)
- lowenergiesthrougha patternofincommensuratepeaks
d of the charge order in the antiferromagnetically coupled
around the antiferromagnetic wave vector. Although
n neighboring layers. The band structure and the T = 0
more difficult to detect, charge order has been observed
o
inLSCOco-dopedwithNd6 aswellasinYBCOwithout inelasticstructurefactorforevenandoddexcitationsare
c
[ codoping.7 calculated in linear spin-wave theory. Particular atten-
tion is paid to the band-splitting in the vicinity of the
Since LSCO and YBCO are paradigmatic for mono-
1 antiferromagneticwavevectorandtotheinfluenceofthe
layer and bilayer compounds, stripe-like “low”-energy
v
interlayer coupling on the π-resonance energy.
4 response is characteristic for both classes of materi-
The outline of this paper is as follows. In Sec. II
5 als. On the other hand, at “high” energies spin fluc-
3 tuations appeared to be qualitatively different since a the spin-only model for a bilayer system is introduced
1 commensurate π-resonance had been observed only in and motivated. Classical ground states and the result-
0 bilayer compounds, notably in YBa Cu O 8,9 and ing phase diagrams for competing types of magnetic or-
4 Bi Sr CaCu O ,10 whereas it seeme2d to3be6+axbsent in der are obtained. They are needed as starting point for
0 2 2 2 8+x
the linear spin-wave theory. A customized formulation
monolayer compounds. This apparent distinction be-
/
at tweenmono- andbilayercompounds lost its justification thereof is outlined in Sec. III. The results, namely the
spin-wavebandstructure,thezero-temperaturestructure
m only recently, when the π-resonance was discovered in
Tl Ba CuO 11 as the first monolayer compound. The factor for even and odd excitations, and the dependence
- 2 2 6+x
ofthebandsplittingattheantiferromagneticwave-vector
d fact that the π-resonant mode has not been detected in
n LSCOsofarcanpossiblybeascribedtoalargereffective on the strength of the interlayer coupling are presented
o in Sec. IV and compared to experiments in Sec. V.
strength of disorder, since the Sr-dopants are randomly
c
distributed whereasin the oxygendoped compounds the
:
v access oxygen orders in chains. Thus, one may believe
Xi that, in principle, mono- and bilayer compounds have II. MODEL
qualitativelysimilarfeaturesalsoathigherenergies. This
r
a universality of low- and high-energy features calls for an Stripes are a combined charge-and spin-density wave.
even more unifying framework.12 If the charge period pa is a multiple of the Cu spacing a
In a recent article13 we have analyzed an elementary withintegerp,lock-ineffectstendtosuppressphason-like
monolayermodelassumingthatchargesformaperfectly fluctuations of the density modulation. In a reductionist
orderedsite-centeredstripe arraywhich imposes a static real-space picture, one may think of the holes forming
spatial modulation of spin-exchange couplings. The re- parallel site-centered rivers of width a, which act as an-
sultingspindynamicswasstudiedusinglinearspin-wave tiphasedomainboundariesfortheantiferromagneticspin
theory. As a result, we found that the incommensu- domainsinbetween.6 Thisimplies thatthe periodofthe
rability and the π-resonance appear as complementary spin modulation is twice that of the charge modulation.
features of the band structure at different energy scales. Toimplementthatthechargestripesactlikeantiphase
2
boundarieswefollowourpreviouswork13 andchoosethe
simplest possible implementation of exchange couplings
within the layersstabilizing this magnetic structure: an-
tiferromagnetic exchange couplings J between neighbor-
ing spins within the domains and antiferromagneticcou-
plings λJ between closest spins across a stripe.
In our previous work13 we have studied this model for
asinglelayerallowingfordiagonalandverticalstripeori-
entations. Here we focus on vertical stripes as observed
inthesuperconductingcupratesandrestrictouranalysis
to the representative case p = 4. This corresponds to a
doping of one hole per 8 Cu sites since the rivers have
a line charge of only half a hole per lattice constant. In
additiontothein-planecouplingsweconsideranantifer-
romagnetic exchange µJ between two layers(cf. Fig. 1).
The Hamiltonian of this bilayer model is given by
H = H +H , (1a)
α 1,2
αX=1,2
∗
1
H = J (r,r′)S (r)S (r′), (1b)
α α α α
2
Xr,r′
FIG. 1: Classical ground states for bilayer systems with par-
∗
allel (upper row), shifted parallel (middle row), and perpen-
H = µJ S (r)S (r), (1c)
1,2 1 2 dicular hole stripes (lower row) for a stripe spacing p = 4.
Xr The exchange couplings of the simple model are illustrated
in the lower row: AF couplings J > 0 for nearest neigh-
where r specifies the square-lattice position and α=1,2
bors within the domains (bold dashed), λJ between nearest
numbersthelayers. Theasterisksindicatethatthe sums
neighbors across a hole stripe(zig-zag) and couplings µJ be-
do not include positions of charge rivers. The in-plane
tween spins one above the other (dashed). Frustration of ex-
couplings Jα(r,r′) defined in the text above are illus- changecouplingmayleadtoacantingofspins(calculatedfor
trated in Fig. 1. They explicitly depend on the layer µ=0.09andλ=0.07inthemiddleandbottomrow,respec-
index if the charge distribution is different in both lay- tively). Possiblemagneticunitcellsareoutlinedbygraylines,
ers. identical gray levels of spins correspond to identical canting
For simplicity we neglect spin anisotropy, the weak angles.
3D coupling between bilayers, and more compli-
cated exchange processes such as cyclic exchange or
Dzyaloshinskii-Moriyainteractions,whichallmaybeim- shifted with respect to each other by half a stripe spac-
portantforquantitativepurposes. Obviously,thissimple ing. (Inoursimplemodel,whereholesareassumedtobe
spin-only model does not account for electronic correla- site centered, this configuration is only compatible with
tion effects, e.g. a spin gap at low energies due to the evenstripe spacingsp.) The gaininCoulombenergyhas
formation of Cooper pairs is not incorporated. Never- to be paid by a loss of exchange energy. For certain pa-
thelessweexpectthatourmodelprovidesaqualitatively rameters, a third configuration may be favorable, where
adequate description of the spin fluctuations well above the charge stripes of the two layers are perpendicular.
the gap energy. For the later analysis it is instructive to anticipate
The actual stripe configuration is determined by sev- thatfortheseconfigurationstheHamiltonianhasdiscrete
eral influences. Besides the magnetic exchange energy symmetries. We focus on symmetries involving an ex-
one also has to take into account the Coulomb energy, changeoflayers. Forparallelandshiftedparallelstripes,
and in principle also a further reduction of the four- this symmetry is just the reflection z z combined
→ −
fold symmetry of CuO2 planes in orthorhombic struc- with a translation (coordinates are chosen such that the
tures whichmayfavora certainalignmentofthe stripes. planes are parallel to the xy plane). For perpendicular
InYBa2Cu3O6+x the formationofCuOchainsalongthe stripes, one needs to add a rotation around the z axis.
b-direction may favor a parallel alignment of stripes.
We find that three different stripe configurations may
be realized physically (see Fig. 1). The exchange energy
A. Energetic estimates
favors parallel stripes lying exactly on top of each other.
This configuration is free of magnetic exchange frustra-
tion, each bond can be fully saturated. However, this To estimate the Coulomb energy for the three stripe
configurationisdisfavoredbythe Coulombenergywhich configurations, we assume a charge-density modula-
wouldfavoraconfigurationwherestripesareparallel but tion ρ(r) = ρ (r)δ(z) + ρ (r)δ(z d) with ρ (r) =
1 2 α
−
3
ρ(0)cos(k r) where the planes separated by d are per- neighborexchangeacrossastripeintherange0<λ<λ
α α c
pendicular to the z-direction. For simplicity, only the (λ 0.59 for shifted parallel and λ 0.35 for per-
c c
≈ ≈
firstharmonicofthe chargemodulationisretained. Par- pendicular stripes) the groundstate has a canted planar
allel stripes are described by k = k = ke and topology up to a value µ (λ) of the interlayer exchange
1 2 x c
ρ(0) = ρ(0) = ρ¯, shifted parallel stripes are realized for (cf. Fig. 2). For µ > µc(λ) spins lock into a collinear
1 2
k =k = ke and ρ(0) = ρ(0) = ρ¯, and perpendicular texture.
1 2 x 1 − 2
stripesfork =ke ,k =ke andρ(0) =ρ(0) =ρ¯. Fora
1 x 2 y 1 2
stripespacingpathecharge-modulationwavevectorsare
given by k = 2π/(pa), the amplitude by ρ¯ = e/(2pa2).
Calculatingthe Coulombcouplingenergypersquarelat-
tice site
1 a2 ρ (r)ρ (r′)
E = d3r d3r′ 1 2 , (2)
C 4πǫ0 A Z Z r r′
| − |
where A denotes the area of the planes, we find, in the
limit A , a vanishing Coulomb coupling for perpen-
→∞
dicular stripes, an energy cost
e2 d FIG.2: Classicalground-statephasediagramsforshiftedpar-
E = exp 2π (3)
C 32πǫ pa (cid:18)− pa(cid:19) allel and perpendicular stripes. For µ < µc(λ) the ground
0
statesshowacantedplanarspinpatternillustratedinFig.1.
for parallel stripes, and an energy gain of the same size Forµ>µc(λ)thetopology ofthegroundstateschangesinto
for shifted parallel stripes. For YBCO with a 3.85 ˚A, a collinear pattern where spins lying on top of each other
d 3.34˚A,J =125meV,S = 1 andforastrip≈espacing are strictly antiparallel and nearest neighbors across a stripe
p=≈4 we obtain ∆E 29 meV2. are parallel. For λ → λc (λc ≈ 0.59 for shifted parallel and
For antiferromagnCet≈ic YBCO the magnetic interlayer λc ≈0.35forperpendicularstripes)µcgoestoinfinity. Above
superexchange is reported to be µ 0.08.14 For paral- λc theground states are always planar.
≈
lel stripes, spins are not frustrated and, in a classical
picture, antiparallel in different layers, S (r) = S (r). To characterize these different phases, we start with
1 2
−
Thus, the exchange coupling roughly leads to an energy the planar one. As already indicated above, the frustra-
gain of order µJS2 3 meV, whereas the energy gain tion can lead to a canting of spins. The origin of the
≈
will be smaller for the other two configurations due to canting is easily understood. For µ = 0 the layers are
frustration. decoupled and the sublattice magnetization in both lay-
Thus, within our rough estimate, the Coulomb energy ers can have an arbitrary relative orientation. For small
appears to be up to one order of magnitude larger than interlayercouplingµthespinsstarttocantstartingfrom
theexchangeenergy,suchthatonemightexpectthepar- aconfigurationwherespinslyingontopofeachotherare
allelshiftedconfigurationtobetheonlyphysicalone. On perpendicular. Only in this case the interlayer couplings
the other hand the actual Coulomb energy may be sig- lead to an energy gain proportional to small canting an-
nificantly smaller than the result of our estimate since gleswhiletheintralayercouplingsleadtoanenergycosts
we have completely neglected screening. For almost un- of second order in the canting angles. Such canted pla-
doped YBCO a relative large value of ǫ 15 for the nar ground states are illustrated in Fig. 1. In Fig. 3 the
≈
staticdielectricconstantatT =4Kisreported.15 There- corresponding tilting angles are plotted for λ = 0.1 as a
fore the Coulomb energy might be of the same order of function of µ. The tilting angles increase monotonously
magnitude as the magnetic exchange energy. Due to the in a way that spins lying on top of each other become
crudeness of our estimate no stripe configuration can be increasingly antiparallel with increasing µ.
strictly ruled out. In the other phase, for µ > µ (λ), the interlayer
c
coupling µ dominates the coupling λ across the stripes
and the topology of the ground state changes into a
B. Classical ground states collinear configuration where the spins lying on top of
each other are strictly antiparallel and nearest neighbor
Duetofrustrationeffects,theground-statestructureis spins across a stripe are strictly parallel although they
nontrivial for shifted parallel and perpendicular stripes. are antiferromagnetically coupled. This configuration is
We now determine these ground states treating spins as stable against a canting of the spins because for small λ
classical. Thesegroundstateswillbeanecessaryprereq- the energy gain for λ-bonds and the energy costs for µ-
uisiteforthesubsequentspin-waveanalysis. Wecontinue bonds aswellasthe couplingswithin the domains would
to focus on the representative case p=4. be quadratic in the tilting angles. Since this ground
Depending on the values of the couplings λ and µ we state has lost the antiphase-boundary character of the
find two different types of ground states. For a nearest charge stripes it resembles a diluted antiferromagnet.
4
A. Holstein-Primakoff representation
The ground-stateanalysisofthe precedingsectionhas
madeclearthatspinwavesnowhavetobeintroducedas
excitation of a non-collinear ground state. However, our
numerical calculation of the classical ground states have
shown planar spin textures (here, a collinear texture is
considered as a special subcase of a planar texture).
In the following we consider a general planar ground
state which can be captured by a vector field S (r) =
α
cosφ (r),sinφ (r),0 , where the tilting angles of the
α α
{ }
spinsobeythetranslationalsymmetryφ (r)=φ (r+A)
α α
for an arbitrary magnetic lattice vector A = m A(1) +
1
m A(2). For the spin textures displayed in Fig. 1, corre-
2
sponding magnetic unit cells are given by A(1) = (4,1)
and A(2) =(0,2) for parallel stripes and for shifted par-
allel stripes, and by A(1) = (8,0) and A(2) = (0,8) for
perpendicular stripes.
Tostudy the quantumfluctuationaroundthe classical
ground state we rotate all spins by their planar angles
φ (r) according to
α
FIG.3: Upperrow: EnergyperlatticesiteinunitsofJS2 as
afunctionofµforλ=0.1. Forbothstripeconfigurationsthe Sαx(r) = S˜αx(r)cosφα(r)−S˜αy(r)sinφα(r), (4a)
energiesofthecantedplanarandthecollinearspinpatternare Sy(r) = S˜x(r)sinφ (r)+S˜y(r)cosφ (r), (4b)
plotted. The curves intersect at µ = µc where the topology α α α α α
of the ground states changes. Lower row: Relative values of Sz(r) = S˜z(r), (4c)
α α
thetiltinganglesofthespinsintheplanarconfigurationasa
function of the interlayercoupling µ for λ=0.1. suchthatS˜(r)hasaclassicalferromagneticgroundstate
S˜(r) = S 1,0,0 . In the transformed spin basis we in-
{ }
troduce Holstein-Primakoff(HP) bosons in the standard
way (using S˜± =S˜y iS˜z),
This would lead to a static magnetic response at the an-
±
tiferromagnetic wave vector in disagreementwith exper- S˜+(r) = 2S nˆ b , (5a)
imental observations. Therefore, these collinear phases α − r,α r,α
probably are not relevant for the magnetic properties of S˜α−(r) = bp†r,α 2S−nˆr,α, (5b)
the cuprate compounds. S˜x(r) = nˆp+S, (5c)
For small values of λ the phase boundary is approx- α − r,α
and obtain the spin-wave Hamiltonian
imately given by µ (λ) 2λ for both stripe configura-
c
≈
ctµiaconngtseode(cspf.tlaoFniiangrfi.fn2oi)rt.ya.IlnlAvtbahloeuveelsimλoicftttλhhee→ignrλtoecurlntahdyeesrctraciotteuicspalrlienvmgaalµuin.e H = S2 Xr,∗r′ αX,α′nfα,α′(r,r′)hb†rαbr′α′ +brαb†r′α′i
Comparing the classical magnetic ground-state ener-
gies for the two frustrated configurations, we find that – + gα,α′(r,r′) brαbr′α′ +b†rαb†r′α′ ,(6)
h io
in contrast to the Coulomb energy – the exchange cou-
pling favors perpendicular stripes over shifted parallel where the functions f and g are defined by
stripes. For this reason we retain perpendicular stripes
1
in our consideration. fα,α′(r,r′) = [Jα(r,r′)δα,α′ +µJδr,r′(1 δα,α′)]
2 −
[∆α,α′(r,r′)+1]
×
δr,r′δα,α′ Jα(r,r′)∆α,α′(r,r′)
III. SPIN-WAVE THEORY −
Xr′
µJδr,r′δα,α′ (1 δα,α′)∆α,α′(r,r′)
In this analytic part we derive general expressions for − −
Xα′
the magnon band structure and the spectral weight at
(7a)
zerotemperatureinaframeworkoflinearspin-wavethe-
1
ory (for a review in the contextof cuprates,see e.g. Ref. gα,α′(r,r′) = [Jα(r,r′)δα,α′ +µJδr,r′(1 δα,α′)]
2 −
16). Theseexpressionsareevaluatednumericallylateron
in Sec.IV for parallel,shifted paralleland perpendicular [∆α,α′(r,r′) 1] (7b)
× −
stripes and fixed stripe spacing p=4. ∆α,α′(r,r′) = cos[φα(r) φα′(r′)]. (7c)
−
5
TodiagonalizetheHamiltonian,weFouriertransformthe Here, 0 denotes the ground state (magnon vacuum)
| i
bosonic operators via b (r) = exp(ikr)b (k), where characterized by b (q)0 = 0 and we consider only
=(2π)−2 d2k and tαhe k intRekgrals run ovαer the Bril- single-magnon finalγsta|teis F = b†(q)0 with excita-
k | i γ | i
Rlouin zone oRf the square lattice with an area (2π/a)2. tion energy ωF := EF E0. k = (kx,ky) denotes
−
Following our calculations for the monolayer system13 the in-plane wave-vector, odd excitations correspond to
we decompose a square lattice vector r into a magnetic kz− = (2n +1)π/d [L− = (2n+ 1)c/(2d) in reciprocal
lattice vector A and a decoration vector a (r = A+a). lattice units], even ones to kz+ = 2nπ/d (L+ = nc/d),
The numberofvectorsais denotedby n(the areaofthe where d is the distance of the two layers within the or-
magnetic unit cell). In momentum space, the reciprocal thorhombic unit cell. For YBCO with d 3.34 ˚A and
magnetic basis Q(i), i = 1,2, spans the corresponding c 11.7 ˚A the corresponding values for ≈even and odd
magnetic Brillouin zone ( ). Wave vectors k can be mo≈des are L− 1.75,5.25 and L+ 0,3.5.
uniquely decomposed intoBkZ= Q+q with q and Expressing t≈he spin operators b≈y the final bosonic
Q=m1Q(1)+m2Q(2). Within the Brillouinz∈onBeZof the operators bγ(q) it is straightforward to calculate the
square lattice there are n vectors Q which we denote by structure factor. Using a pseudo-Dirac notation and
Q . Usingthesedecompositionswerewritethespin-wave denoting the 2n-dimensional cartesian basis by ν,α
ν | ii
Hamiltonian as (ν = 1,...N,α = 1,2) and the orthonormal eigenba-
sis ofM−1/2KM−1/2 by γ , the structure factor canbe
H = 21Z Fνα,ν′α′(q)[b†α,q+Qνbα′,q+Qν′ rewritten in a compact fo|rmii,
qXν,ν′αX,α′
in(q+Q ,ω) = S ±(q+Q )δ(ω ω (q)),(13a)
+bα,−q−Qνb†α′,−q−Qν′] S± ν Xγ Sγ ν − γ
+12ZqXν,ν′αX,α′Gνα,ν′α′(q)[b†α,q+Qνb†α′,−q−Qν′ Sγ±(q+Qν) = 12X=CX,S,ωγ−1Khhν,±|XM−1/2|γii
+bα,−q−Qνbα′,q+Qν′], (8) 1 γ M−1/2Xν, , (13b)
where ×ωγhh | | ±ii
S
Fνα,ν′α′(q) = n fα,α′(a+A,a′) where we have defined |ν,±ii = (1/√2)[|ν,1ii±|ν,2ii]
XA Xa,a′ and introduced the matrices S and C according to
cos[qA+q(a a′)+Qνa Qν′a(′]9) ∗
× − − 1
is essentially the Fourier transform of f, sνα,ν′α′ = nδαα′ sinφα(a)ei(Qν−Qν′)a, (14a)
Xa
S ∗
nfα,α′(Qν +q,Qν′ +q′)=δ(q+q′)Fνα,ν′α′(q). (10) cνα,ν′α′ = 1δαα′ cosφα(a)ei(Qν−Qν′)a. (14b)
n
Xa
Analogous expressions relate G to g. The Hamiltonian
(8) has exactly the same structure as in the monolayer
case [compare Eq. 8 in Ref. 13] and can be diagonal- IV. RESULTS
izedbyaBogoliubovtransformationinananalogousway.
The final diagonal form is given by
We now evaluate the the magnon dispersion and the
2n inelastic structure factor for even and odd excitations
1
= ω (q) b†(q)b (q)+ , (11) numerically. From a comparison of our findings for the
H Z γ (cid:26) γ γ 2(cid:27)
γX=1 q monolayer system to neutron scattering data for the
cuprate compounds we found13 the coupling λJ across
wherethe squaredenergiesω2 areeigenvaluesoftheher-
γ astripetobeaboutoneorderofmagnitudesmallerthan
mitian matrix M−1/2KM−1/2. Thereby M−1 = F G thenearestneighborcouplingJ withinthe domains. For
−
denotes the inverse mass matrix and K = F + G the the coupling µJ between the layers a value µ 0.08 is
coupling matrix. reported14forantiferromagneticYBCOinthea≈bsenceof
stripes. Therefore in the stripe system the couplings λ
and µ can be assumed to be of the same order. In the
B. Structure Factor followingwe keepthe value ofλ fixedand discuss the ef-
fects of increasing µ starting from the case of decoupled
We now proceed to calculate the inelastic zero- layers (µ = 0) where the band structure of the mono-
temperaturestructurefactorforevenandoddexcitations layer system13 should be recovered. In this parameter
regimetheclassicalgroundstatesforshiftedparalleland
S±in(k,ω) := |hF|S1j(k)±S2j(k)|0i|2 perpendicularchargestripesshowthe cantedplanartex-
XF j=Xx,y,z tureandtheantiphasedomainboundarycharacterofthe
δ(ω ω ). (12) charge stripe is weakened by the interlayer coupling but
F
× −
6
still pronounced. Finally we shortly present the excita-
tionspectraforshiftedparallelandperpendicularstripes
for parameters belonging to the collinear ground state
regime.
In the case of decoupled layers (µ = 0) the results of
the monolayer system are trivially recovered. Since the
twolayersareuncorrelated,thestructurefactordoesnot
depend on the L component of the wave vector. For
parallel stripes (with or without a relative shift of the
stripes) where the charge modulation is unidirectional
with Qch = Qch = (1/4,0) we just obtain an additional
1 2
twofolddegeneracyof eachofthe three bands due to the
equivalence of the two layers. Therefore the degenera-
tion of the bands is fourfold since in the monolayer case
each band is twofold degenerated due to the equivalence
of the two sublattices.13 The lowest,acousticalband has
zeros at the magnetic superstructure vectors which are
located at (j/4,0) and (j/4 + 1/8,1/2), j = 0,...,3,
withintheBrillouinzoneofthesquarelattice(wechoose
0 H,K < 1). The spectral weight is concentrated
≤
near the lowest harmonic incommensurate wave vectors
Q=(1/2 1/8,1/2). Withincreasingenergytheincom-
± FIG. 4: Band structure and spectral weight along the
mensurabilitydecreasesandthe branchesofthe acoustic (H,0.5,L±)and(0.5,H,L±)directionsforparallelstripesly-
magnonband close at the antiferromagneticwave vector
ing on top of each other and couplings λ = 0.15 and µ = 0,
(1/2,1/2)and an energyω which we associate with the
π 0.08. The last row shows the band structure of a twinned
π-resonance. Along the (H,1/2) direction the acoustic sample(seetext). L+ correspondstoeven,L− tooddexcita-
bandisgappedtotheoverlyingopticalmagnonband(see tions. Darkerandlargerpointscorrespond toalarger weight
upperleftpanelsinFigs.4and5). Alongthe orthogonal of the inelastic structurefactor.
direction (1/2,K), one optical band has vanishing spec-
tralweightandonlytwobandsarevisible(seemiddle-left
panels in Figs. 4 and 5). tosplitwithdifferentdistributionsofthespectralweights
In twinned samples with stripe domains oriented or- in the odd and even channel (cf. Figs. 4, 5, and 6). For
thogonal to each other, a scan along the (H,1/2) di- parallel and shifted parallel stripes the Hamiltonian is
rection results in the superposition of the signals ob- invariant under the reflection z z combined with a
→ −
tained from scans in directions (H,1/2) and (1/2,H) of translation. This implies that the magnonstates – mod-
a single-domain sample. For domains of equal size, one ulo a phase factor which does not enter the structure
thus obtains an apparent symmetry (H,K) (K,H) factor – have a well defined parity with respect to an
↔
and a fourfold pattern of the static incommensurate exchange of both layers. As a consequence, nondegener-
wave vectors located at Q = (1/2 1/8,1/2) and Q = ate bands are visible only either in the even or the odd
±
(1/2,1/2 1/8) also for (shifted) parallel stripes. In channel.
±
Figs.4and5,thepanelsinthethirdrowarejustobtained Nevertheless the excitation spectra of the two parallel
by superimposing the panels of the first and second row. stripe configurations deviate significantly, e.g. the even
Since the acoustic band of the monolayer system has a excitations are gapped for parallel stripes whereas for
saddlepointattheantiferromagneticwavevector,there- shifted parallel stripes the intensity of even excitations
sulting band structure is x-shaped in the vicinity of the is only reduced at low energies (cf. middle columns of
π-resonance energy. Figs. 4, 5). For stripes on top of each other each band –
Theconfigurationofholestripeslyingperpendicularto which is fourfold degenerate at µ = 0 – splits up into
each other corresponds to charge modulation wave vec- twofold degenerate bands which have identical parity.
tors Qch = (1/4,0) and Qch = (0,1/4). For decoupled For shifted stripes each band splits up into three bands.
1 2
layers, the resulting band structure contains the bands One of them is twofold degenerate and both subbands
of the monolayer system and the same bands rotated by are of opposite parity. Therefore this degenerated band
90 degrees leading to the symmetry ω(H,K)=ω(K,H) is visible in both channels (cf. Fig. 5).
and therefore to a fourfold pattern of the static incom- For perpendicular stripes the symmetry is more com-
mensurate wave vectors located at Q=(1/2 1/8,1/2) plicated. The Hamiltonian is invariant under a reflec-
±
andQ=(1/2,1/2 1/8). Thus,for µ=0, the structure tion z z in combination with a 90◦-rotation along
factor is identical f±or perpendicular stripes and twinned the z a→xis.−Since this rotation mixes different wave vec-
parallel stripes (left lower panel in Figs. 4 and 5). tors, almost all eigenstates do not have a well defined
With increasing interlayer coupling µ the bands start parity and will be partially visible in the odd and even
7
FIG. 6: Band structure and spectral weight for even (left
FIG. 5: Band structure and spectral weight along the panel) and odd (right panel) excitations along (H,0.5,L±)
(H,0.5,L±) and (0.5,H,L±) directions for shifted parallel direction for perpendicular stripes with couplings λ = 0.15
stripes and couplings λ = 0.15 and µ = 0,0.08. The last across the stripes and interlayer couplings µ = 0.04 (upper
row shows theresulting band structureof a twinned sample. row) and µ=0.08 (lower row).
parallel shifted parallel perpendicular
channel. The exception are modes at particular wave
k=(H,0.5)
vectors such as the antiferromagnetic wave vector which
aremappedontothemselves(modulo areciprocallattice w p+ w p+ w p+
vector). Only there the excitations can be classified due w w p- ww pp-0 ww pp-0
to their symmetry. Like for the shifted parallel stripes
the excitations are not gapped in the even channel (cf.
Fig. 6). 0.5 0.5 0.5
H
We now focus on the band splitting and the distribu-
tion of the spectral weights of even and odd excitations
FIG. 7: Schematic illustration of the band splitting in the
at the antiferromagnetic wave-vector (1/2,1/2). With vicinityoftheantiferromagnetic wavevector(1/2,1/2) along
increasing interlayer coupling µ, the resonance energy the (H,1/2) direction. In the cases of parallel stripes the
ω splits up into two different energies ω− and ω+ for band structures for twinned samples are shown. Even and
π π π
centered-parallel stripes and into three energies ω−, ω0 odd bands are gathered together.
π π
and ω+ for the other stripe configurations as schemat-
π
ically illustrated in Fig. 7. It is common to all stripe
configurations that ω− has a finite spectral weight only and µ>µ (λ) where the ground-statesare collinear and
π c
in the odd channel, whereas ω+ has a finite weight only thechargestripeslosetheiranti-phasedomainboundary
π
in the even channel. For shifted parallel and perpendic- character. We implicitly assume that µ is not too large,
ular stripes, in both channels a finite intensity is found otherwise spins on top of each other dimerize and lose
at the intermediate energy ω0. This intensity is however their magnetization. In this regime the magnetic fluctu-
π
smaller than at ω±. ations are drastically changed. For both stripe orienta-
π
The splitting of the resonance energy for shifted par- tions, the odd channel now has a static signal at the an-
allel and perpendicular stripes looks quite similar. ω− tiferromagneticwavevector,whereasintheevenchannel
π
and ω+ are almost equidistant to the intermediate en- the spectral weight is concentrated at incommensurate
π
ergy ω0 which increases only slightly with µ (cf. Fig. 8). positions (1/2 1/4,1/2) (cf. Fig. 9). For perpendicu-
π ±
For small couplings the splitting is quadratic in µ. For lar stripes we also find small intensity at this positions
centered-parallel stripes the splitting looks different, ω+ in the odd channel. The incommensurability is doubled
π
increases almost linearly with µ whereas ω− is almost compared to the regime of canted planar ground-states
π
independent of the interlayer coupling. reflecting that the charge stripes do not act like anti-
Finally, we calculate the band structures for shifted phase domain boundaries in the regime of strongly cou-
parallel and perpendicular stripes for couplings λ < λ pled layers. In the even channel the intensity at the an-
c
8
parallel shifted parallel perpendicular pothesis that the stripe picture is a suitable approachto
1.6(cid:13)
describe spin fluctuations. Furthermore, we hope that a
1.2(cid:13)
/ (JS)0.8(cid:13) k=(0.5,0.5,L+) claotmiopnasrwisoilnl hoeflpfuttouriedeenxtpifeyritmheenrteaallidzaedtastwriitpheocounrficgaulcrua--
p 1.6(cid:13) tion.
w
1.2(cid:13) Since a spin gap with an energy ω – e.g. due
0.8(cid:13) k=(0.5,0.5,L-) to Cooper-pair formation – is not incograpporated in our
0(cid:13) 0.04(cid:13) 0.08(cid:13) 0.12(cid:13)0(cid:13) 0.04(cid:13) 0.08(cid:13) 0.12(cid:13)0(cid:13) 0.04(cid:13) 0.08(cid:13) 0.12(cid:13) model, the results apply only to energies above ω
m gap
wherethemagnondispersionisnotmaskedbythesuper-
FIG. 8: Splitting of the resonance energy as a function of conducting condensate. In particular in the underdoped
the interlayer coupling µ for λ = 0.15. In the odd channel regime where ωgap decreases with the doping level, the
(L = L−) the spectral weight is concentrated at ωπ− and no calculated spectral features become visible over an in-
intensity is found at ωπ+, in the even channel (L = L+) no creasing energy range. Our calculations are restricted
excitations at ωπ− are observable and the spectral weight is to zero temperature. Therefore, a comparison can also
concentrated at ωπ+. For shifted and perpendicular stripes in be made only to experiments performedattemperatures
both channels a small intensity is found at the intermediate
well below the superconducting transition temperature.
energy ωπ0. Experiments17,18,19 in (partially) detwinned YBCO
provideevidenceforunidirectionalorder,i.e.,thatafour-
fold pattern of incommensurate peaks near the antifer-
tiferromagnetic wave-vector is peaked at an energy ω
π
romagnetic wave vector k = (1/2,1/2) results only
which increases with the interlayercoupling µ and is ap- AF
from the twinning. The stripes seem to be parallel and
proximately the same for both stripe configurations.
oriented along the direction of the oxygen chains in the
adjacent planes. This immediately speaks against the
scenario of perpendicular stripes for which detwinning
would not affect the fourfold symmetry.
We briefly recall some neutron scattering measure-
ments on YBa Cu O which provide insight into the
2 3 6+x
incommensurabilityandtheπresonanceoverawidedop-
ingandtemperaturerange. Itwascontroversialforquite
some time whether both phenomena would exist above
T untilinunderdopedmaterialstheincommensurability
c
was found also above T .20 Likewise, the appearance of
c
the magnetic resonance was found above T , occurring
c
together with the pseudogap at a temperature T∗ > T
c
determined from transport and nuclear resonance.21 Al-
thoughthe π-resonancepersists as a welldefined feature
also in the normal state above T , its intensity can be
c
reduced significantly at T .19 For near optimally doped
c
compounds,theresonanceisnotdetectableinthenormal
phase22 since T∗ almost coincides with T . Dai et al.23
c
concluded that the resonance exists above T for x 0.8
c
≤
andthatincommensuratespinfluctuationsappearinthe
normal state for x 0.6. Arai et al.24 also observed
≤
incommensurate fluctuations in the normal state for a
FIG. 9: Band structure in the collinear regime µ > µc(λ) sample with an oxygen concentration of x = 0.7. Thus,
along the direction (H,1/2,L±) for shifted parallel and per- superconductivity is not a prerequisite for incommensu-
pendicular stripes with λ=0.07 and µ=0.50. rability and π resonancein bilayercompounds as well as
in monolayer compounds.
ForunderdopedYBCOwithvariousoxygenconcentra-
tions, the experimentally observed spin dynamics data
V. DISCUSSION (see Table I) look qualitatively very similar. There is
a systematic increase of the incommensurability and of
In this section we compare our results to neu- the π resonance frequency with doping, which is consis-
tron scattering data for the bilayer high-T compound tent with our model. We have shown this recently for a
c
YBa Cu O . We wish to stress that – because of the monolayer model.13 The bilayer stripe model shares this
2 3 6+x
simplifications assumed in our model – it is not our goal feature and therefore we focus in this paper exclusively
to obtain a quantitative agreement. Rather we wish to on specific bilayer features.
draw a qualitative comparisonin order to fortify the hy- Experimentally, constant energy scans slightly above
9
x 0.35 0.45 0.5 0.5 0.6 0.7 0.7 0.7 ate frequency ω0 (cf. Fig. 7). Experimentally,19,27,28 a
π
Tc (K) 39 48 52 59 63 67 67 74 strong oscillatory dependence of the scattering intensity
δ (r.l.u.) 1/16 0.08 0.10 1/8 0.1 onLshowsthattheresonancefrequenciesintheoddand
evenchannelarewellseparated. Energyscansatthe an-
p 8 6.25 5 4 5
ωπ− (meV) 23 30.5 31.5 33 34 36 33 37 tcihfearnrnoemlaagnndeωti+cwinavtheeveevcteonrcshhaonwneple,ankospaetaωkπ−atinthteheinotedrd-
ωπ+ (meV) 41 50 mediate energπyω0 which shouldbe visible inboth chan-
π
Ref. 7 23 23 19 23 24 25 23 nels is resolved.25 This again favors unshifted parallel
stripes,which(incontrasttoshiftedparallelandperpen-
dicular stripes) have no shared resonance frequency ω0.
TABLE I: Spin dynamics data for YBa2Cu3O6+x for various π
Although we restricted our comparison to experiments
oxygenconcentrationsxcharacterizedbythecriticaltemper-
ature Tc, incommensurability δ, corresponding stripe period onunderdopedsamples,overdopedcompoundsalsoshow
p,theresonanceenergyωπ− observedintheoddchannel,and two distinct resonance modes of opposite symmetry,29
ωπ+. which could be identified with ωπ− and ωπ+.
From a comparisonof the band splitting ∆ω =ω+
ω− to experimental values (cf. Tab. I) we canπestimπat−e
the gap in the odd channel along (H,1/2,L−) show π
the strength of the interlayer coupling µ. For λ = 0.15
a broad intensity peak at k , before incommensurate
AF we find µ 0.02 0.06 almost independent of the stripe
scattering sets in and the data can be compared to our ≈ −
configuration. Thisvalueisreasonablesincetheeffective
model. The intensity shows magnetic peaks at a dis-
couplingµinthestripesystemshouldbeslightlyreduced
tance δk(ω) away from k . The incommensurability δ
AF comparedto the undoped casewhere a value ofµ 0.08
is determined by extrapolating δk(ω) to ω = 0 and it is is reported.14 ≈
connected to the stripe spacing p through δ = 1/(2p).
Above ω the response is found to become incommen-
The incommensurate peaks are best defined if the stripe π
surate again with increasing separation δk(ω). The mo-
spacingisnearlyamultipleofthelatticespacing(integer
mentum width is larger and the intensity is weaker than
p) since the stripes are stabilized by the lattice.7
below ω . Overall, the dispersion is “x-shaped”. As
Thethreestripeconfigurationsexaminedforourmodel π
pointed out in Sec. IV such a shape appears basically
arenotequivalentintheirlow-energybehavior. For(un-
foreverynon-unidirectionalstripeconfiguration,forpar-
shifted)parallelstripes(seeFig.4),anincommensurabil-
allel stripes in twinned crystals as well as in perpendic-
ityisvisibleatlowenergiesonlyintheoddchannelsince
ular stripes. The x-shape has been observed explicitly
the evenchannelhasa relativelylargegapnotrelatedto
in Refs. 7,24,25,26. It would be interesting to verify in
superconductivity. In contrast, for shifted parallel and
detwinnedsamplesthattherelativeintensitiesoftheup-
perpendicularstripesthe evenchannelshowsincommen-
per and lower branches of the x-shape are related to the
surate response down to the superconducting gap. Ex-
population ratio of the twin domains.
perimental evidence21,25,26 for a large gap in the even
In conclusion, we have calculated the bilayer effects in
channel(wellabovetheresonanceenergyinoddchannel)
the magnetic excitation spectrum in striped states. As a
therefore favorsthe configurationwith unshifted parallel
generic feature of the stripe model we find x-shaped dis-
stripes.
persionrelationin the vicinity ofthe π resonance,which
Withincreasingenergy,theseparationδk(ω)ofthein-
is consistent with experimental data. We have obtained
commensuratepeaksdecreasesandthe branchescloseat
a bilayer splitting of single-layer bands into two or three
k atcertainenergiesω . Dependingonthestripecon-
AF π
bilayerbands. Fromthethreestripeconfigurationsstud-
figuration,there are two or three suchenergies, compare
ied, the unshifted parallel case overallis most consistent
Fig. 7. According to our model, an energy scan of the
with neutron scattering data, although it seems to be
odd channel at k would show a first resonance at the
AF
intersection with the lowest magnon band at ω− which energetically unfavorable at first sight.
π
we identify with the resonance frequency.8,27 For shifted
parallel and perpendicular stripes, a second line at ω0
π
contributes to the odd channel. It has significantly less
Acknowledgments
weightandisseparatedfromthefirstonebyonlyasmall
energysplitting (oftheorderofafewmeV) whichwould
be hard to be resolved experimentally. We gratefullyacknowledgehelpful discussionswithM.
In a similar way, the even channel has a resonance at Braden, Y. Sidis, and J.M. Tranquada. This project
an energy ω+ > ω−, and for shifted parallel an perpen- was supported financially by Deutsche Forschungsge-
π π
dicular stripes also a weaker resonance at an intermedi- meinschaft (SFB608).
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