Band structure of Charge Ordered Doped Antiferromagnets

0.73 MB·English

by  Mats Granath

Band structure of Charge Ordered Doped Antiferromagnets.
Mats Granath∗
Chalmers Technical University
G¨oteborg 41296
Sweden
(Dated: February 2, 2008)
4
0
We study the distribution of electronic spectral weight in a doped antiferromagnet with various
0
types of charge order and compare to angle resolved photoemission experiments on lightly doped
2
La2−xSrxCuO4 (LSCO)andelectrondopedNd2−xCexCuO4±δ. Calculationsonin-phasestripeand
n bubblephasesfortheelectrondopedsystemarebothingood agreementwithexperimentincluding
a in particular the existence of in-gap spectral weight. In addition we find that for in-phase stripes,
J in contrast to anti-phase stripes, the chemical potential is likely to move with doping. For the

6 hole doped system we find that “staircase” stripes which are globally diagonal but locally vertical
or horizontal can reproduce the photoemission data whereas pure diagonal stripes cannot. We

] also calculate the magnetic structure factors of such staircase stripes and find that as the stripe
n
separation isdecreased with increased dopingtheseevolve from diagonal toverticalseparated bya
o
coexistence region. The results suggest that thetransition from horizontal to diagonal stripes seen
c
inneutronscatteringonunderdopedLSCOmaybeacrossoverbetween aregime wherethetypical
-
r length of straight stripe segments is longer than the inter-stripe spacing to one where it is shorter
p
and that locally thestripes are always aligned with theCu-O bonds.
u
s
.
t I. INTRODUCTION pickedpotentialwhichemulatesthelocalenvironmentof
a
m electronsinastripeorderedantiferromagnet. Thepoten-
tialissimplyastaggeredfieldrepresentingthelocalanti-
- In several families of superconducting cuprates there
d ferromagnetismtogetherwithanti-phasedomainwallsof
is evidence for stripes which are regularly spaced quasi-
n suppressedfield strength representing stripes. The spac-
o one dimensional structures where the doped charge ings of the stripes are chosen in such a way that the
c assembles.1 At the same time there is a well developed
model by construction will reproduce the diffraction re-
[ theory of high-temperature superconductivity in a sys-
sponseofastripeorderedsystem. Thestripeplacements
tem of weakly coupled Hubbard ladders.2,3,4 It is thus
1 arestatic but may be chosenin anirregularfashionsim-
v quite natural to speculate that such a theory is in fact ulating quenched disorder or dynamic stripes which are
3 realized in the cuprates. One obvious objection to the
fluctuating slowly compared to time scales of the local
6 stripe scenario of superconductivity is the lack of con-
electron dynamics. Using this model for a disordered
0 vincing evidence for the existence of stripes in several
1 array of quarter filled “bond-aligned”12 stripes it was
materialsgiventhatthestrongestevidenceisfoundinthe
0 found that the low energy spectral weight forms a two-
relatively low-T La Sr CuO (LSCO) family.5 From
4 c 2−x x 4 dimensional Fermi surface with, in particular, spectral
a theory point of view the lack of direct evidence is not
0 weight in the “nodal regions”, near (π/2,π/2), which
t/ immediately discouragingbecause stripe order is bad for naively corresponds to propagation diagonally with re-
a superconductivity whereas more elusive dynamic charge
spect to the stripe direction. A more detailed study of
m stripe correlations are good.6,7 However, even in a sys-
this model was carried out in Ref. 13 where it was real-
tem such as LSCO where stripes are well established
- izedthatevenasinglestripewithone-dimensionalstates
d there are still issues about their fundamental implica-
localized transverse to the stripe captures the qualita-
n tions on the electron dynamics. In particular, it seems
o tive features of the low-energyspectral weight as seen in
the distribution of low-energy spectral weight in k-space
c ARPES in the under and optimally doped LSCO with
as probed by angle-resolved photoemission spectroscopy
: the full spectral weight of the stripe lying roughly on
v (ARPES) does not show any clear evidence of the sug-
i the diamond Fermi surface of a half-filled nearest neigh-
X gested quasi-one-dimensional nature of the electronic
bor tight-binding model. Here also issues of interactions
states, looking instead like the Fermi surface of a fully
r along a stripe was addressed and it was found that the
a two-dimensionalsystem.8OntheotherhandARPESalso
distinctnon-Fermiliquidpropertiesofaninteractingone-
provides clear evidence of exotic physics, perhaps elec-
dimensional electron gas may be displayed also in direc-
tron fractionalization9, with spectral functions that are
tions not aligned with the stripes.
typically very broad in energy and not consistent with a
Fermi liquid based quasi-particle description.10
InthepresentpaperweextendtheworkofRef. 11and
A simple model for studying the distribution of sin- 13to lookatabroaderrangeoforderedstructures. This
gle electron spectral weight in a charge ordered doped ismotivatedbyasearchforamorecomprehensiveunder-
antiferromagnet was introduced by Salkola et al.11 in standing of charge order in a doped antiferromagnetbut
whichallthecomplicatedphysicswhichisresponsiblefor alsobydirectorindirectobservationsofstructureswhich
the stripe formation and integrity is replaced by a hand are not consistent with bond-aligned anti-phase stripes.
2
In the very lightly doped, non-superconducting, phase tureofwhicharethe “staircasestripes”showninFig. 4.
of LSCO there are diffraction patterns consistent with In Fig. 6 is shown the low energy spectral weight which
diagonal stripes14,15 instead of the bond-aligned stripes is concentrated around the nodal region in a hole doped
seen at higher doping. The transition between bond- array of staircase stripes.
alignedanddiagonalstripescoincideswiththesupercon-
An interesting aspect of these staircase stripes is the
ductor to insulator transition at doping x 0.055, sug-
magnetic structure factors which depend on the ratio
≈
gesting a close link between the two properties. Within
between the length of the bond-aligned segments, the
the non-interacting model considered here we cannot di-
“step” length l, and the distance between neighboring
rectly addressthe connectionbetweensuperconductivity
stripesd. AsshowninFig. 8wefindthatthecorrespond-
and stripe orientation but by comparing the predicted
ing Bragg peaks can be classified in three main regimes.
spectral weight of diagonal stripes with that observed
For l d there are two peaks corresponding to diagonal
in ARPES16 we gain some insight into the nature of ≈
stripes along the xˆ+yˆdirection at (π δ ,π δ ),
diag diag
suchdiagonalstripes. Recently,there hasalsobeenindi- ± ∓
for l 2d the diagonal peaks coexist with four peaks at
rect evidence both from NMR17 and thermal conductiv- ≈
(π,π δ )and(π δ ,π)correspondingtotheresponse
col col
ity measurements18 of an inhomogeneous charge struc- ± ±
expectedfrombond-alignedstripesalongthexˆandyˆdi-
ture inthe electron-dopedcuprates (Pr,La) Ce CuO
2−x x 4 rections, while for l d there are only the four bond-
(PLCCO). However, in electron doped materials there ≫
aligned peaks but they are now shifted away slightly
has been no evidence for incommensurate magnetism19
from the square lattice axes. The qualitative features
effectivelyrulingoutthepossibilityofanti-phasestripes.
areverysimilartowhatisfoundfromneutronscattering
For this reason we have studied in-phase structures and in LSCO as a function of doping15, although the rela-
compared to ARPES results on the Nd Ce CuO
2−x x 4±δ tiveincommensurabilityδ /δ isnotquiteaccurately
diag col
(NCCO) family.20
reproduced in the coexistence regime. Nevertheless, this
Ourresultscanbesummarizedasfollows. Asexempli- suggestsascenarioinwhichthestripesintheorthorhom-
fiedinFig. 2wefindthatthespectralweightoflocalized bic phase of LSCO are always locally bond-aligned but
states is centered on the Brillouin zone (BZ) diagonals, in some sense globally diagonal with a crossover as the
givenbycos(k )+cos(k )=0,independentlyofthemag- stripe spacing is decreased with increased doping and
x y
nitude of second and third nearest neighbor hopping t′ not a first order transition as suggested by the neutron
and t′′ as well as the shape and form of the charge “im- scattering data. Corroboratingsuch a crossoverscenario
purity”. This shows why in general one may not expect is the fact that the ARPES spectra evolve smoothly
anydramaticsignatureofstripesorotherinhomogeneous through the diagonal to bond-aligned stripe transition
electronicstructureonthek-spacedistributionoflowen- and that the Fermi velocity in the nodal direction is
ergy spectral weight, as this is where the Fermi surface roughlyindependentofdoping,indicatingthatifthelow-
is located within a nearly half-filled tight-binding model energy spectral weight is stripe related the local char-
which is dominated by nearest neighbor hopping. How- acter of the stripes does not change dramatically with
ever, as shown in Fig. 7, 12, and 16, a defining feature doping.16Inaddition,theholemobilityatmoderatetem-
of the stripe states is that they have an energy which peratureschangesbyonlyafactorofthreefromverylight
is within the Mott gap of the undoped system. Such (x = 0.01) to optimal (x = 0.17) doping which is very
in-gap states appear to be a common feature of the evo- naturally understood within a stripe model in which the
lution of the band structures which doping as measured local character of the stripes is roughly independent of
in ARPESandwhichwe believe is a strongindicationof doping.22
an inhomogeneous charge distribution.21
In Fig. 12 we show the the band structure of a sys-
Wecomparethedispersionanddistributionofspectral tem with disordered in-phase stripes and in Fig. 11 the
weightofdiagonalandbond-alignedstripesandfindthat correspondingelectrondoped lowenergyspectralweight
pure diagonal stripes cannot reproduce the characteris- which is in qualitative agreementwith the ARPES data.
tic“Fermiarc”centeredaroundthenodalregionwhichis Themostinterestingpartoftheseresultsistheevolution
seen in ARPES on lightly doped LSCO. Instead we find ofspectralweightinthenodalregion,whereatlightdop-
that the spectral weight of a hole doped diagonal stripe ing (4%) there is in-gap spectral weight which at higher
is concentrated to the “anti-nodal” BZ regions around doping (10%) broadens as a consequence of the shorter
(π,0) with very little weight in the nodal region. In ad- inter-stripe distance and reaches the Fermi surface. The
dition, the band width of states on a diagonal stripe is evolutionof the Fermi surface with doping can be repro-
expectedtoberoughly2t′ .0.2eVwhichisinconsistent ducedinmean-fieldtheoryofHubbardmodelwithlonger
| |
with the ARPES data where the band width of the in- range hopping by allowing for a doping dependent inter-
gap states is of the order of 1eV (assuming half is seen). action U.23 The difference within a stripe model is that
However, it turns out that the spectral distribution and the low-energy states are dynamically one-dimensional,
band width of bond-aligned stripes is qualitatively con- being localized transverse to the stripes (Fig. 13) and
sistentwiththeARPESdata. Forthisreasonwesuggest the existence of in-gap spectral weight. In addition we
that the diagonal stripe phase consists of stripes which find that for in-phase stripes, in contrast to anti-phase
are globally diagonal but locally bond-aligned, a carica- stripes, the chemical potential is likely to move into the
3
upper Hubbard band with electron doping because the regions where V(x,y) = 0 will in general give rise to
in-phase stripe states lie close to the upper and lower localized “impurity” states within the gap.
Hubbard bands (Fig. 14). TheobjectwhichwasstudiedindetailinRef13isthe
Finally, we present similar results from a calculation site-centered anti-phase stripe given by
on a “bubble” phase where the doped charge is con-
fined to small “0-dimensional” droplets instead of the 1, x>0
1-dimensional stripes. Bubbles would arise naturally in- Vstripe(x,y)= 0, x=0, (2)

steadofstripesinat J modelwithlong-rangeCoulomb 1, x<0
− −
repulsion in the limit t J because of the lower mag- 
≪ and displayed graphically in Fig. 1a. Here we also show
neticenergy. Thusintheelectrondopedmaterials,which
two other charge structures, diagonal stripes and bub-
appear to have “stronger” antiferromagnetism than the
bles, which we will consider in more detail subsequently.
holedopedmaterials,onemayspeculatethatbubblesare
In a real system we want to consider ordered or disor-
favoredoverstripes. Asfarasthedistributionofspectral
dered arrays structures such that at some finite doping
weight is concerned (Fig. 15 and 16) there is little qual-
the corresponding states are partly occupied.
itative difference between bubbles and stripes. For the
bubbles we find that the nodal spectral weight broadens
asthesizeofthebubblesincreasewithdopinginanalogy
with the increasing density of stripes.
II. THE MODEL
Wewillconsideratight-bindingmodelonasquarelat-
ticewithfirst,secondandthirdnearestneighborhopping FIG. 1: Graphic representation of the potential V(x,y) for
together with a static potential which represents stripes various charge structures. Here black corresponds to V =0,
or other charge structures. The Hamiltonian reads gray to V = 1, white to V = −1 and each square represents
a site (x,y) on the lattice. (a) is a bond-aligned anti-phase
H =− t (c†r,σcr′σ+H.C.) stripe, (b) is an anti-phase diagonal stripe and (c) a bubble.
hrXr′iσ
− t′ (c†r,σcr′σ+H.C.)
hrXr′i′σ A. Spectral weight of localized states
− t′′hrXr′i′′σ(c†r,σcr′σ+H.C.) Herewewillshowthatastatelocalizedonanimpurity
where the staggered field is zero has its spectral weight
+ m σ(−1)x+yV(x,y)c†x,y,σcx,y,σ , (1) centered within a region m of the cos(kx)+cos(ky) = 0
X
x,y,σ diamond,regardlessofthevaluesofthelongerrangehop-
ping t′ and t′′ and the geometry of the impurity. This
where c is the electron destruction operator at site
r,σ result is really a trivial consequence of the fact that the
r =(x,y) and with spin σ = . The hopping is given in
a standard fashion where rr±′ indicates nearest neigh- nearest neighbor hopping t connects the two sublattices
h i ofthe staggeredfield while t′ and t′′ does not, or equiva-
bors, rr′ ′ next-nearest neighbors, and rr′ ′′ next-next
h i h i lentlythesymmetryornotofthedispersionwithrespect
nearest neighbors. In what follows we will use energy
to a shift of the momenta by scattering vector (π,π) of
units such that t=1 and we take t′/t<0 and t′′/t>0.
the staggeredfield.
Unless stated otherwisewe will be using m=t=1. The
We define a potential for an arbitrary impurity
physical intuition for the field m is that it is the energy
cost of moving a hole from a stripe into the antiferro- V(x,y)= m( 1)x+y, x,y impurity
magneticbackgroundanditisthusexpectedtobeofthe − { }6∈
V(x,y)= 0, x,y impurity, (3)
orderofthespinexchangeJ. Wealsouseunitssuchthat { }∈
the lattice constant is equal to one and ~=1.
and write the tight-binding dispersions
The potential m( 1)x+yV(x,y) describes the collec-
−
tive field which defines the stripe order. We will use the ε = ε0 +ε1
simplestpossibleformandtakeV(x,y)=1,0or-1,where ~k ~k ~k
ε0 = 2t(cos(k )+cos(k )) (4)
V(x,y) =1 or -1 represent antiferromagnetic regions re- ~k − x y
latedbyaπ phaseshiftandV(x,y)=0arethelocations ε1 = 4t′cos(k )cos(k ) 2t′′(cos(2k )+cos(2k )),
ofthestripes. ThecaseV(x,y)=1forallxandy corre- ~k − x y − x y
sponds to the standardmean-fieldresult ofthe Hubbard With this we solve for the eigenstates in the bulk
modelathalf-filling24givinganupperandlowerHubbard
band separated by the Mott-Hubbard gap. Introducing ψ =a ei~k·~x+b ei(~k+~π)·~x, (5)
~k ~k ~k
4
with energy What we can contribute to this discussion within our
model is a comparison between the expected distribu-
E =ε1 (ε0)2+m2, (6) tion of spectral weight between bond-aligned and diago-
~k ~k±q ~k nalstripes. ItwasalreadyshowninRef. 11and13thata
disordered bond-aligned stripe array can well reproduce
and ratio of coefficients
thequalitativefeaturesofthenearoptimallydopedsam-
ples. It is natural to do a similar analysis for the lightly
b /a =( ε0 (ε0)2+m2)/m. (7)
~k ~k − ~k±q ~k doped samples x = 3−5% studied by ARPES in Ref.
16. Experimentally it is found that the spectral weight
The localized state can be expanded in terms of the in the anti-nodal,(π,0), regionsof the BZ which is most
complete set of states ψ which are the solutions in the prominent for the under and optimally doped samples is
~k
bulk. Clearlyanystatewhichislocalizedattheimpurity gappedawayfromtheFermienergy,andinsteadthelow-
mustbesensitivetothestaggeredfield. Thismeansthat energy spectral weightconsists of disjoint arcs of “Fermi
it can only contain bulk solutions which are substantial surface”centerednearthe nodal,(π/2,π/2),regions. At
superpositions of~k and~k+~π, i.e. b /a 1. From the afirstglanceitisverytemptingtoidentifythesefeatures
~k ~k ∼
expression, Eq. 7, for b /a we find ε0 = 1(a/b b/a)m with that shown in Fig. 2b for a diagonal stripe. How-
~k ~k ~k 2 − ever, although it is probably not completely ruled out
which implies that~k is constrained to the volume ε0 .
| ~k| that this simple picture is correct a closer study of the
m.
diagonal stripe reveals a serious problem. The problem
is that for t′ < 0, a less then half-filled, i.e. hole doped,
diagonalstripewillhaveverylittlespectralweightinthe
nodal region, but only near the anti-nodal regions.
In orderto understandthis statement we canconsider
thedispersionofthe diagonalandbond-alignedstripeas
a function of the conserved momentum along the stripe.
We will restrict ourselves to the limit m and nu-
→ ∞
merically confirm the qualitative correctness for m = 1.
FIG.2: Spectralweightofvariouschargepotentials. (a)isa In this limit it is trivial to solve for the spectrum on the
stripe along they direction with m=2,t′=−.1,t′′=.1, (b) stripe because the problemreduces to a one dimensional
is a diagonal stripe along x+y direction with m = 0.5,t′ = tight-binding chain. For the bond-aligned stripe there is
−.1,t′′ =0 and (c) is a size 10×10 bubble with m=1,t′ = the nearest neighbor hopping t while t′′ acts as a next-
−.1,t′′=0.
nearest neighbor hopping, resulting in a dispersion
In Fig. 2 is shown the full spectral weight of the three εcol(kk)m→=∞ 2tcos(kk) 2t′′cos(2kk). (8)
− −
types of potentials displayed in Fig. 1, confirming our
analytic result. The broader distribution in k-kspace for Forthediagonalstripethenext-nearestneighborhopping
on the 2D lattice t′ acts as a nearest neighbor hopping
larger m is consistent with a shorter localization length
in real-space. Note alsothat for the diagonalstripe, Fig. on the chain, giving a dispersion
2b,thespectralweightisconcentratedtothosesegments
ε (k )m→=∞ 2t′cos(k ). (9)
of the diamond which have momenta orthogonal to the diag k k
−
stripe direction, a feature which will be important when
InFig. 3areshownnumericallycalculateddispersionsfor
analyzing diagonal stripes in the next section.
m=1 which agrees qualitatively with the large m limit.
Notethatforthebond-alignedstripethereisafoldingof
the BZdue tothe antiferromagneticscatteringalongthe
III. DIAGONAL STRIPES
stripe direction, which is absent for the diagonal stripe.
The conclusion we want to draw from these dispersions
One of the most interesting recent stripe related find- is that for the diagonal stripe at any finite hole doping
ings is the diagonal stripes seen by quasi-elastic neu- themomentaoffilledstateswhichmaycontributetothe
tron scattering in very underdoped insulating phase of low energy spectral weight are confined to k > π/2.
k
LSCO.14AsdiscussedinSec. Ithereisacrossoverregion From the distribution of spectral weight for a diagonal
where both diagonal and bond-aligned stripes appear to stripe, Fig. 2b, we find that this implies that there is
coexist and which coincides with the rapid drop of the very little spectral weight in the nodal regions. We can
superconductingtransitiontemperatureswithdecreasing easilyconvinceourselvesthatthisdistributionofspectral
doping,15pointingtoastrongconnectionbetweenstripes weightofadiagonalstripeisageneralconsequenceofthe
andsuperconductivity. Ontheotherhand,wealsonoted factthat a state with some momentumk will, as shown
k
that other properties, such as the nodal Fermi velocity in Sec.IIA, have its spectral weight concentrated to the
andtheholemobilitydoesnotshowanydramaticchange intersection of the line k +k = k with the cos(k )+
x y k x
across the bond-aligned to diagonal transition. cos(k ) = 0 diamond. This implies that for k = π the
y k
| |
5
weight will spread out over the whole intersection which stripes are not consistent with the ARPES data of lightly
isaline,whilefor k <πtheweightwillbeconcentrated doped LSCO.
k
| |
to the two points of intersection.
A. Staircase Stripes
Giventhe difficulties with matchingthe modelusing a
diagonalstripeconfigurationtotheARPESdataitisin-
steadtempting to look atbond-alignedstripes. Now, we
know from neutron scattering that bond-aligned stripes
are not seen in these very lightly doped materials, but
only diagonal stripes. This led us to investigating the
properties of stripes which are locally bond-aligned but
globally diagonal. A natural and most simple candidate
for such a construction is a “staircase” stripe. We can
defineastaircasestripe alongthe x+y directionbutlet-
ting it run alternately along the xˆ and yˆdirections with
some step length l. For an ordered array of such stair-
case stripes we also introduce a stripe distance d defined
accordingto Fig. 4. In the case of anti-phase stripes the
potential has the symmetries
V(x+l,y+l) = V(x,y),
V(x+d,y d) = V(x,y), (10)
− −
FIG.3: Dispersions of in-gap states for m=t=1, t′=−.1,
and t′′ = 0 for bond-aligned stripe (a) along y with kk = ky whichalsogivesthe primitive cellas indicatedinthe fig-
and diagonal stripe (b) along x+y with kk =kx+ky. Line ure. Thesamesymmetriesholdtrueforthefullpotential
marked “0%” indicates no doped holes (half-filling). of Eq. 1 which is simply multiplied by a factor ( 1)x+y
−
to account for the staggered field.
These results for the distribution of spectral weight of
aholedopeddiagonalstripemaybecontrastedwiththat
of a bond-aligned stripe. Here we find from 3a comple-
mentedwith Fig. 2a thatfor finite hole-dopingthe spec-
tral weight may be spread out over the whole diamond.
The details, of course, depending on the doping, the pa-
rameters used and on the density of stripes. The latter
being particularly important in that for a short inter-
stripe distance the stripe states will overlap and form
bands of momenta transverse to stripes (see Fig. 14).
One might, inaneffective modelsuchas this, attempt
tofittheexperimentaldatabytakingt′ >0whichwould
allow for spectral weight concentrated in the nodal re-
gion. However the ARPES data for the lightly doped
samples indicate that most of the additional weight in-
troduced with doping is in fact in the anti-nodal region
onlythatitisgappedawayfromtheFermisurface. Ifwe
taket′ >0 we wouldvacate the anti-nodalstates andwe
would not be able to reproduce this qualitative feature.
Relatedtothisthereisamorequantitativeproblemfora
diagonalstripeascontrastedwiththeARPESdatawhich
shouldbe moregeneralthanourmodel, namely thatthe FIG.4: Graphicalrepresentationofastaircasestripedefined
band width of a purely diagonalstripe is expected to be by thestep length l and thestripe distance d. The rectangle
proportional to t′. Values of t′ in the literature is less indicates a primitive cell. The potential V(x,y) is given by
than 0.1 eV, implying a band width W < .2 eV, V=0fordarkergray,V=1forlightgray,andV=-1forwhite.
diagonal
whereas the band width from the in-gap states seen in
ARPES can be estimated at W .1 eV which looks We will return to magnetic structure factors of such
in-gap
more consistent with the band width W 2t of bond- staircase stripes below, but it is easy to see that as long
col
∼
alignedstripes. Tosummarize,wefindthatpurediagonal as l d the main magnetic diffraction peaks of such
≤
6
a staircase stripe are equivalent to an array of purely
diagonal stripes with the inter stripe distance 2d along
the x and y direction.
We turn now to the distribution of spectral weight
of staircase stripes. As an example we look at at a
system with l = 8 and d = 8, were d is chosen such
that the magnetic structure factor has main peaks at
(π δ/√2,π δ/√2) with δ/√2 = 1/32 which corre-
± ∓
sponds roughly to the δ 1/25 seen in Neutron scat-
≈
tering at 4% doping. We diagonalize this system numer-
ically to find the single particle eigenstates ψα(~k) with FIG. 5: Ordered array of staircase stripes with l = 8 and
energies Eα in terms of which we calculate the single d = 8, using t = m = 1, t′ = −.1 and t′′ = 0. Momentum
particle spectral function space (a) and real space (b) spectral weight integrated over
anenergywindowof0.2aroundtheFermienergyat4% hole
A(~k,ω)= ψ (~k)2δ(E ω) (11) doping. Thefullsystemsizeis256×256whilein(b)isshown
α α
| | −
X a 100×100 section.
α
where δ(E ω) is the Kronecker delta, and the local
α
−
density of states
R(~r,ω)= ψ (~r)2δ(E ω) (12)
α α
| | −
X
α
withψ (~r)=1/L L ei~k·~rψ (~k)beingtheeigenfunc-
α x y ~k α
tions in real space. InPFig. 5 is shown spectral weight
distribution in k-space and real-space when integrated
over an energy window ∆ω = .2 around the Fermi en-
ergy at 4% doping, i.e. calculating
EF+∆ω/2
I(~k)= A(~k,ω)dω, (13)
Z
EF−∆ω/2
and FIG.6: SameasFig. 5butsymmetrizedwithrespecttothe
stripeorientation and inthefirst quadrantoftheBZ.Shown
EF+∆ω/2
is also theintensity map which has a linear scale.
R(~r)= R(~r,ω)dω. (14)
Z
EF−∆ω/2
We find that the low-energy spectral weight is concen- function, A(~k,ω), over a broader energy window along
tratednearthenodalregion. Itishighlyanisotropicwith
the high-symmetry directions, as shown in Fig. 7. The
most of the spectral weight parallel to the overall stripe
qualitative features of the spectral weight deriving from
direction along x + y in contrast to the pure diagonal
the staircase stripes can be directly linked to the prop-
stripe shown in Fig. 2b. We have tried making the step
erties of ordinary bond-aligned stripes. The high energy
lengthl shorter,thisgivesresultsclosertothepurediag-
(away from E ) spectral weight is concentrated around
F
onal stripe with most of the spectral weight in the anti-
theanti-nodal,(π,0),regionwhereasthelowenergyspec-
nodal region. In Fig. 6 the results aresymmetrized with
tral weight close to the Fermi energy is focused to the
respect to the stripe direction, so that (π,π) = (π, π)
− nodal region around (π/2,π/2). This is what we would
etc. Herethedisjointfeaturesmergeintoasinglepieceof
find from a lightly hole doped bond-aligned stripe with
“Fermi arc” in each quadrant of the BZ, similar to what
thedispersioninFig. 3a,giventhe factthatthe spectral
isseeninARPES.NotethatinFig. 5bitisnotthestripe
weight is concentrated to the intersection of k = k or
k x
potential V(x,y) which is plotted, but the amplitude in
k with the BZ diamond.
y
realspaceofthelowenergystates. Notsurprisinglythese
follow the potential quite closely, but there is some leak-
age of spectral weight into the antiferromagnetic regions
1. Magnetic Structure factors of Staircase Stripes
which appearsto smooththe kinks and make the stripes
more diagonal. An indication that the staircase stripes
arejust a caricaturewith the realstripes probablybeing In the previous section we found a qualitative agree-
smoother,butneverthelesslocallyclosertobond-aligned mentofthespectraldistributionofstaircasestripeswith
than to diagonal. that seen experimentally in lightly doped LSCO. Here
Wecangainabetterunderstandingoftheseresultsfor wehadto restrictourselvesto staircasestripes withstep
the low-energy spectral weight by studying the spectral length l less than the stripe distance d as defined in Fig.
7
because the difference is roughly the amplitude on the
stripes which is zero for the potential but slightly differ-
ent from zero for the actual spin density because of the
occupied stripe states.
Using the symmetry properties of a staircase stripe,
Eq. 10, we find that V(k ,k ) can only have non-zero
x y
components for
2π
k +k = N, N Integer (16)
x y
l ∈
2π
k k = N′, N′ odd Integer. (17)
x y
− 2d ∈
In Fig. 8 is shown the structure factors of three dif-
ferent stripe realizations, l = d, l = 2d and l = 5d. We
findthreeverydistinctdiffractionpatterns,werethefirst
correspondsto diagonalstripes, the second looks like di-
agonalstripes coexisting with both verticaland horizon-
tal bond-aligned stripes, and the last looks like vertical
and horizontal stripes but with an orientation which de-
viates slightly from bond-aligned. We also show diffrac-
tion patterns of samplings over disordered stripe config-
FIG. 7: Band structure of system with t=m=1, t′ =−.1 urations were we find that in general only the primary
andt′′ =0for(a) nostripesand(b)with staircase stripesas peaks survivewith secondary,lowerintensity peaks, get-
in Fig. 5. UHB and LHB indicate upper and lower Hubbard ting washed out, although this does not happen for the
bands respectively and EF is the Fermi energy at 4% hole second configuration with coexisting bond-aligned and
doping. Thespectralweightisindicatedbytheintensity,but
diagonal peaks. We have also studied stripe arrays with
onanon-linearscalewhichexaggerateslow-intensityfeatures.
largerunit cells where l is notan integer factor ofd. For
(b) is symmetrized with respect to thestripe direction.
disordered realizations it appears that these roughly fall
into one of the three characteristic regimes, with only a
quite narrow window of l 2d showing both diagonal
4inorderto haveamagnetic structurefactorwhichcor- ≈
and bond-aligned peaks of appreciable amplitude.
responds to diagonal stripes. A natural extension is to
These results for the diffraction pattern of staircase
study also stairs with l > d. Certainly, in limit l d
≫ stripes appear qualitatively very similar to what is seen
we expect very little influence from the kinks and the
in neutron diffraction experiments on LSCO in the low-
system will become equivalent to that of bond-aligned
temperature orthorhombic (LTO) phase.15 In the non-
stripes, both for the spectral distribution and structure
superconducting phase at very low doping (x . 6%) is
factors. Wefoundthatthedistributionofspectralweight
found peaks consistent with diagonalstripes, whereas at
couldbe rationalizedintermsofbond-alignedstripes al-
higher doping (x 10 13%) are found a pattern con-
readyforthesystemwithl=dsothatforl>dwewould ≈ −
sistent with stripes that are close to bond-aligned but
notexpectanyqualitativedifferencebetweenthespectral
shifted by a few degrees from the tetragonal axes. More
distributionofstaircasestripeswiththatofbond-aligned
stripes studied in the earlier work.11,13 What is more in- recentlywasfound thatveryclose to the insulator to su-
perconducting transition (x 6%) both patterns coex-
terestingistostudythemagneticstructurefactor,which ≈
ist. This may suggest that the stripes in the LTO phase
is sensitive to the global properties of the system.
of LSCO are always of the staircase type and that the
Physically,therelevantentityisthemagneticstructure
transition between bond-aligned and diagonal stripes is a
factor which is the amplitude Sz(~q)2 of of the Fourier
| | crossoverfromaregimewherel>2dtoonewherel <2d.
transformedspindensitywhichatzerotemperaturereads
Particularly interesting would be to have more detailed
experimental data from neutron scattering on the un-
Sz(~q)=1/L2 eiq~·~rσ ψ (~r)2Θ(E E ). (15) der doped superconducting regime x 10%. From the
X | ασ | F − α staircase stripe scenario we would exp≈ect a larger angle
~r,α,σ
of deviation from the tetragonal axes than for the sam-
However, for a large system, such as for disordered pleswithhigherdoping,possiblytogetherwithsecondary
stripes, this is difficult to calculate because of the need peaks in the diagonaldirection oronthe tetragonalaxes
to diagonalize the system. Much simpler to find is the as shown in Fig. 9.
amplitude squaredof the Fourier transformof the stripy We should note that the correspondence between ex-
potential ( 1)x+yV(x,y). We have checked for several periment and our results for staircase stripes is not per-
−
ordered stripe arrays that close to half-filling the two fect. In particular there appears to be a discrepancy
calculations give very similar results. This is expected with the relation between the “incommensurabilty” δ,
8
FIG.9: Magneticstructurefactorsofdisorderedrealizations
of staircase stripes with l = 20 and (a) d = 5 and (b) d = 6
which could roughly correspond to 10% and 8% doping re-
spectively.
The experimentalhallmark ofanti-phasestripes is the
incommensurate magnetismdetected by neutrondiffrac-
tion,wheretheweightisshiftedawayfromtheantiferro-
magnetic ordering vector Q~ =(π,π). A system with or-
deredin-phasestripesontheotherhandwillhaveamain
diffractionpeak atthe antiferromagneticorderingvector
FIG. 8: Magnetic structure factors of the three different withsatellitesatpositionsshifted byintegermultiples of
regimesofstaircasestripesdiscussedintext. Theleftcolumn 2π/d, where d is the distance between stripes. However,
shows the unit cells which in (a) is 80×80 with step length
if the stripes are not static and ordered, but fluctuating
l = 20 and stripe distance d = 20, in (d) is 40×40 with
ordisordered,thesatellitepeakswilleasilybewashedout
l=20,d=10andin(g)is80×80withl=20,d=4. Black
and the weight absorbedinto the AF peak. In Fig. 10 is
are the stripes where the potential is zero, gray and white
shownthe amplitude squaredofthe Fouriertransformof
arethetwoπ-shifteddomainsoftheantiferromagnetic order,
the stripe potential ( 1)x+yV(x,y) along the transverse
and the rectangles are primitive cells. The middle column,
−
(b),(e)and(h),showsthecorrespondingstructurefactorsin direction for ordered and disordered anti-phase and in-
reciprocal lattice units centered around (1/2,1/2) = (π,π). phase stripes and the for the charge density represented
The right column, (c), (f) and (i), give the structure factors by V(x,y). For the charge order there is no difference
| |
ofasumovertensize80×80disorderedconfigurationsofthe between anti-phase and in-phase stripes. Nevertheless,
corresponding ordered states to the left, where the length of charge order is more difficult to detect even for systems
the legs and the distance between stripes is allowed to vary withstaticorder. Thereasonforthisisthatneutronsdo
with a flat random distribution without allowing stripes to
not couple directly to the charge order and that the su-
comecloserthanonesiteseparation. Thewidthofthepeaks
perlattice peaks relatedto chargestripe order arise from
of the right column is simply related to the fraction of the
modulations of the uniform charge density, Q~ = (0,0),
BZviewed,giventhatinallcalculationsthesamecorrelation
length is used. which dominates the structure factor. As shown in Fig.
10 these peaks are also readily destroyed by disorder.
(In fact, the charge structure factor as estimated here is
definedas the shiftofa peakfrom(π,π), ofthe diagonal equivalenttothespinstructurefactorofin-phasestripes
and bond-aligned components. For the staircase stripes but shifted by π.) Direct signatures of in-phase stripes
we find δ = δ /√2, whereas experimentally is seen fromneutron orx-rayscattering will thus be muchmore
diag col
δ δ . The former relation follows directly from difficult to find, most likely requiring a stripe ordered
diag col
≈
the symmetry of the stripy potential, Eq. 17, andconse- material.
quently is not sensitive to small changes of the potential Nevertheless, given the indirect indications for the ex-
such as for instance smoothing of the kinks. istence of in-phase stripes in electron doped cuprates
it may be interesting to study the implications of such
structures to the distribution of spectral weight. In Fig.
IV. IN-PHASE STRIPES 11 is shown the integrated spectral weight close to the
Fermi energy for disordered in-phase stripes with inter-
Wenowturntoastudyofthespectralweightdistribu- stripe distances 9-15 with mean 12 and 1-7 with mean 4
tionofin-phasestripes. AsdiscussedinSec. Ithereisin- for a system with t = m = 1, t′ = .2 and t′′ = .1. The
directevidenceforaninhomogeneouschargedistribution parameterst′ andt′′ arechosensuc−hastoroughlyrepro-
in the electron doped PLCCO17,18. In fact, the possibil- duce the band structure of the undoped system seen in
ity of in-phase stripes had already been suggested from ARPES on the electron doped NCCO20 with the small-
theory for hole doped stripes at lower hole densities.25 est spectral gap at (π/2,π/2) as shown in Fig. 12a. The
9
FIG. 10: Structure factors (in arbitrary units) of spin and
charge density for anti-phase and in-phase stripes as a func-
tion ofmomentumtransversetothestripesinr.l.u.. Theleft
column shows ordered arrays with stripe distance d =4 and
the right disordered with a flat distribution d = 2−6. (a)
and (b) shows the spin order for anti-phase stripes, (c) and
(d) the spin order for in-phase stripes, and (e) and (f) the
chargeorderwhichisindependentofthetypeofstripe. Inall FIG.12: Bandstructureofsystemwitht=m=1,t′ =−0.2,
figures is used a finite correlation length of 40 sites and the and t′′ = 0.1 and in-phase stripes as in Fig. 12. (a) is no
disordered systems are averaged over 1000 samples. stripes, (b) mean period 12 stripes and (c) mean period 4
stripes.
weightstartstolookmorelikeafullFermisurfaceclosed
around (π,π) but with weight missing at “hot-spots”
where the putative Fermi surface cuts the BZ diago-
nal (diamond). Similar results have been reproduced by
mean-field theory of the t t′ t′′ U Hubbard model
− − −
by allowing for a doping dependent U.23 This is in fact
themodelweconsiderbutwithoutthestripesandwitha
magnitude of the staggeredfield m whichdepends onU.
Here, we keep the parameters fixed but vary the stripe
FIG. 11: Spectral weight close to EF (window .2) of dis-
ordered in-phase stripes with t = m = 1, t′ = −0.2, and spacing as a function of doping. There are however very
t′′ = 0.1. Mean stripe distance in (a) is 12 with EF at 4% distinct differences between the implications of the two
doping and in (b) mean distance is 4 with EF at 10% dop- scenarios. In the stripe model there are mid gap states,
ing. The results are symmetrized with respect to the stripe notpresentforthe modelwithauniformstaggeredfield.
direction. The system size is 320×320 . This is particularly visible for the lightly doped system,
Fig. 12b, where along the Γ = (0,0) to (π,π) direc-
tion there is spectral weight between E and the lower
F
density ofstripes is assumedto be roughlyasinthe hole Hubbard band. Precisely such a feature is seen in the
doped materials with stripe spacing d given by the dop- ARPES data for the 4% doped sample. Secondly, the
ing n according to n 1/2d, which for electron doping low-energy states of the stripe model are dynamically
≈
corresponds to 3/4 filled stripes. one-dimensional, i.e. they are localized transverse to
ClearlytheseresultsreproducetheexperimentalFermi a stripe but have a well defined momentum along the
surface quite well, at light doping there are patches of stripe, as shown in Fig. 13. Thus, one of the main
spectral weight around (π,0) while at higher doping the conclusions is that the distribution of low-energy spec-
10
tral weight, the “Fermi surface”, may be indistinguish-
able from a homogeneous two-dimensional system even
though the states are dynamically one-dimensional.
FIG. 13: Example of a single state ψ of system in Fig. 12c,
withenergyE ≈0andmomentumalongstripedirectionky =
π/2,showinglocalizationinthetransversestripedirection. In
(a) isshown thereal spaceamplitude |ψ(x,y)|2 asafunction
oftransversedirectionxforarbitraryyandin(b)thek-space
amplitude |ψ(kx)|2. We can estimate a localization length
ξ≈2 from thereal space peak amplitude.
Let us now look at how we can understand this evolu-
tionofspectralweightwithdopingandcorrespondingin-
creasingstripe density fromthe properties ofa single in-
phase stripe. Fig. 14 shows the spectra of systems with
sparse, mean distance 12, and dense, mean distance 4,
stripes. Forthesparsestripes,Fig. 14a,thestripestates FIG. 14: Spectrum for |E|<2 of in-phase stripe systems as
are clearly visible as the isolated in-gap bands. Because inFig. 12withmeanstripeperiod12in(a)and4in(b)asa
there is no broadeningofthe stripe statesthe stripesare functionofmomentumkk=ky paralleltothestripedirection
clearly sufficiently far apart as to be effectively indepen- y. The black curves indicate the bounds of the upper and
dent. We find that in-phase stripes at moderate m have lower Hubbard bands for the corresponding system without
abandstructurewhichisdistinctlydifferentfromthatof stripes.
anti-phase stripes as shown in Fig. 3a, with the former
stayingclosetotheupperandlowerHubbardbands. (We
have checked that in the limit m , where in-phase A. Bubbles
→ ∞
and anti-phase stripes are equivalent, the dispersion Eq.
8 is correctly reproduced also for in-phase stripes.) As Because of the lack of direct diffraction evidence for
the stripe density is increasedthe stripe states will over- stripes one may be free to speculate on other forms of
lap and form bands transverse to the stripes, Fig. 14b, charge order in these materials. The microscopic mo-
with states fromthe lowerbranchcrossingthe Fermi en- tivation for stripe formation is the tendency of the an-
ergy. Because of disorder, these states may nevertheless tiferromagnet to expel extra charge which will disrupt
be strongly localized as shown in Fig. 13. thelocalantiferromagnetism. Theformationofstripesis
An important point about in-phase stripes which is thenacompromisebetweentheminimizingthemagnetic
demonstrated here is that the chemical potential may energy by concentrating the holes, kinetic energy by al-
move as a function of doping and is not necessarily lowing holes to delocalize along the stripes, and possibly
pinned within the gap. This is particularly evident in thechargingenergyduetolongrangeCoulombrepulsion
comparing the 0% and 4% samples, Fig. 12a and b, by not allowing a macroscopic charge inhomogeneity.26
where in the former the chemical potential is in the gap However, if the magnetism is relatively stronger than
( 0.6 < µ < 0.2) whereas in the latter it has moved the kinetic energycontributionthe systemmay preferto
− −
up to cut the upper Hubbard band (µ 0). The motion keepthe extrachargeinzero-dimensional“bubbles”(see
≈
of the chemical potential is a necessary consequence of Fig. 1c) in order to minimize the disruption of the lo-
the dispersion of in-phase stripes as shown in Fig. 14a, calmagneticorder.27 Roughlyspeaking,foracompletely
with the stripe states “hugging” the upper and lower filledstripewithtwoelectronspersiteonaone-sitewide
Hubbard bands. This is in sharp contrast to anti-phase stripetherearethreebadbondswherethespinexchange
stripes(Fig. 7and3),wherethe“stripy”spectralweight is destroyed per stripe site, while for a bubble there are
is “mid-gap” implying that the chemical potential may only two bad bonds per site. (The sites on the perime-
stay fixed with doping. Motion of the chemical potential ter of the bubble have three bad bonds). In addition,
with doping is thus not necessarily an indication of the because of charging energy, the bubbles would have to
absence of stripes.20,28 be limited to a microscopic size. Clearly, the putative