The role of Zhang-Rice singlet-like excitations in one-dimensional cuprates

4 Pages·0.12 MB·English

by  J. Richter

The role of Zhang-Rice singlet-like excitations in one-dimensional cuprates
J. Richter, C. Waidacher, and K. W. Becker
Institut fu¨r Theoretische Physik, Technische Universita¨t Dresden, D-01062 Dresden, Germany
We present the first calculation of the electron-energy loss spectrum of infinite one-dimensional
undopedCuO3 chainswithinamulti-bandHubbardmodel. Theresultsshowgood agreement with
experimentalspectraofSr2CuO3. Themainfeatureinthespectraisfoundtobeduetotheformation
0
of Zhang-Rice singlet-like excitations. The q-dependence of these excitations is a consequence of
0
the inner structure of the Zhang-Rice singlet. This makes the inclusion of the oxygen degrees of
0
freedom essential for the description of the relevant excitations. We observe that no enhanced
2
intersite Coulomb repulsion is necessary to explain theexperimental data.

n
a PACS numbers: 71.27.+a, 71.45.Gm, 71.10.Fd
J

7
2 Recently, charge excitations in the quasi one- where d† (p† ) create a hole with spin σ in the i-th Cu
iσ jσ
dimensional compound Sr2CuO3 have been investigated 3d orbital (j-th O 2p orbital), while nd (np ) are the
l] bothexperimentally1−3 andtheoretically.2−6 Sr2CuO3 is corresponding number operators. The ifiσrst ajσnd second
e composed of chains formed by CuO4 plaquettes which terminEq.(1)representtheatomicpartoftheHamilto-
-
r share the corner oxygens. The magnetic properties of nian, with the charge-transfer energy ∆, and the on-site
t
s thesechainshavebeensuccessfullydescribedusingaone- CoulombrepulsionU betweenCu3dholes. Thelasttwo
t. dimensional spin-12 Heisenberg antiferromagnet.7−9 terms in Eq. (1) aredthe hybridization of Cu 3d and O
a
Experimentally, the electron-energy loss spectrum 2p orbitals (hopping strength t ) and of O 2p orbitals
m pd
(EELS) of Sr2CuO33 shows several interesting features (hopping strength t ). The factors φij and φjj′ give
- (see Fig. 1): For small momentum transfer (q = 0.08 pp pd pp
d the correct sign for the hopping processes. Finally, hiji
˚A−1)paralleltothechaindirection,oneobservesabroad
n denotes the summation over nearest neighbor pairs.
o peakaround2.4eVenergyloss,andtworelativelysharp, ThelossfunctioninEELSexperimentsisdirectlypro-
c smaller maxima at 4.5 and 5.2 eV. With increasing mo-
portional to the dynamical density-density correlation
[ mentumtransfer,thelowest-energyfeatureshiftstowards function χ (ω,q),10 which depends on the energy loss
1 higher energy, reaching 3.2 eV at the zone boundary ω andmomρentumtransferq. χ (ω,q)iscalculatedfrom
v (q = 0.8 ˚A−1). Thereby its spectral width decreases. ρ
0 In addition, the peaks at 4.5 and 5.2 eV lose spectral 1 ∞
0 weight as the momentum transfer increases, while some χρ(ω,q)= i dt e−iωthΨ|[ρ−q(0),ρq(t)]|Ψi , (2)
4 less well-defined structures emerge around 6 eV. Z0
1
So far, these results have been compared only to cal- with
0
culations in an extended one-band Hubbard model.3,6
00 From this comparison, Neudert et al.3 concluded that in ρq = ndiσeiqri + npjσeiqrj , (3)
t/ Sr2CuO3 there is an unusually strong intersite Coulomb Xiσ Xjσ
a repulsionV: Intheone-bandmodelitisoftheorderof1
where|ΨiisthegroundstateofH,andρ istheFourier
m eV. It is argued that this large value of V allows for the q
transformed hole density. The ground state |Ψi is ap-
d- feoxrpmeraitmioennto.fOexnceitoofntihcesatiamtessowfthhiicshpaarpeerobissteorvsehdowinththaet proximated as follows:11 We start from a N´eel-ordered
n nointersiteCoulombrepulsionisnecessarytoexplainthe state |ΨNi with singly occupied Cu 3d orbitals (with al-
o ternating spin direction) andempty O 2p orbitals. Fluc-
basicfeatures ofthe experiment,if the O degreesoffree-
c tuations are added to |Ψ i using an exponential form
: dom are taken into account within the framework of a N
v
multi-band Hubbard model.
i
X We investigate the EELS spectrum of a one- |Ψi=exp λ F |Ψ i . (4)
α i,α N
r dimensional CuO3 chain system, using a multi-band iα !
a Hubbard Hamiltonian at half-filling. In the hole picture X
The fluctuation operators F describe various delocal-
this Hamiltonian reads i,α
ization processes of a hole initially located in the Cu 3d
H =∆ np +U nd nd orbital at site i, where a summation over equivalent fi-
jσ d i↑ i↓ nal sites takes place.11 The parameters λ in Eq. (4)
jσ i α
X X describe the strength of the delocalization processes and
+tpd φipjd(p†jσdiσ +h.c.) are determined self-consistently by solving the system of
<Xij>σ equations hΨ|LF0†,α|Ψi=0, where L is the Liouville op-
+tpp φjpjp′p†jσpj′σ , (1) erator,definedasLA=[H,A]foranyoperatorA. These
<jj′>σ equations have to hold if |Ψi is the ground state of H.
X
1
Using Eqs. (2) and (4), we calculate the EELS spec-
trum by means of Mori-Zwanzig projection technique.12
(b)
For a set of operators Dµ, the so-called dynamical vari- q = 0.7 A−1
aXbγle"s,ztδhµeγf−ollXo×ηwihhnΨΨg||DDmµγ††aLtzrD−i1xηLe|ΨqDuiνa(cid:0)|thΨiΨoin|D=aη†phDpΨrγ|o|DΨxiµ†imD(cid:1)−aνt1|e#Ψly×ih,o(ld5s) Loss function (arbitrary units) qq == 00..35 AA−−11
(a)
wherez =ω+i0. InEq.(5)thesetofdynamicalvariables q = 0.1 A−1
was assumed to be sufficiently large so that self-energy 0
2.2 2.4 2.6 2.8 3 3.2 3.4
contributions can be neglected. The set {D } contains Energy loss ω (eV)
µ
the dynamical variable D0 = ρq. Therefore, by solving
Eq. (5), an approximation for Eq. (2) can be obtained.
Besides D0, the set includes Dα =ρqF0,α for all α. The
Fst0a,αteaErqe.t(h4e),flwuitchtuoauttiotnheospuemramtoartsiounsoedverineqtuhievagleronutnfid- tatFioIGn.a2t.2.4T−he3.1theeVorewtiictahlarebsuroltasdefonrintgheofd0o.m02ineaVnt. eTxchie-
momentum dependence of the spectrum is due to two dif-
nal sites. We use altogether 12 dynamical variables and
ferent effects. First, with increasing q the spectral weight
observe good convergence of the spectral function.
shifts from excitation (a) to (b). Second, the energies are
q-dependent. Botheffectscontributetoaboutonehalfofthe
full momentum dependence.
Thetheoreticalspectraconsistoftwoexcitations. The
q = 0.7 A−1 dominant excitation is at 2.45 eV for q = 0.1 ˚A−1, and
nits) shifts to 3.05 eV for q = 0.7 ˚A−1. Besides, a second
u
ary q = 0.5 A−1 excitation appears at 6.4 eV which has no dispersion.
arbitr The lowenergypeakstructure is showninmoredetail
on ( inFig.2,whereasmallerpeakbroadeninghasbeenused.
uncti q = 0.3 A−1 As will be explained below, mainly two different Zhang-
oss f Ricesinglet-likeexcitations15 leadtothispeakstructure.
L
The q-dependence of the spectrum is due to two effects.
q = 0.1 A−1 Firstly, one observes a shift of spectral weight with in-
creasingq between two excitations labelled with (a) and
0 (b) in Fig. 2. Secondly, with increasing q the energies of
1 2 3 4 5 6 7 1 2 3 4 5 6 7
Energy loss ω (eV) Energy loss ω (eV) the two peaks shift to higher values.
FIG. 1. Comparison of experimental data for Sr2CuO3 The shift of spectral weight can be attributed to dif-
(left),takenfromRef.3,andthepresenttheoreticalresultsfor
ferent delocalization properties of the two final states.
theone-dimensional multi-bandHubbardmodel(right). The
Theexcitedstate(a)inFig.2whichdominatesthespec-
theoreticallinespectrahavebeenconvolutedwithaGaussian
trum for small momentum transfer is rather extended,
function of width 0.1 eV. For details see the text.
see Fig. 3(a). This state has a rather small probability
for the hole at its original plaquette. With increasing q
In Fig. 1 the obtained results are compared to the ex-
thespectralweightshiftstoanotherexcitedstate,shown
perimental spectra from Ref.3. The parameters in the
in Fig. 3(b), with a higher probability for the hole on its
Hamiltonian are chosen as follows: U = 8.8 eV and
d original Cu-site. This means that the character of the
t = 0.65 eV are kept constant at typical values.13 The
pp excitation changes from an extended to a more localized
valuesof∆=4.3eVandt =1.5eVhavebeenadjusted
pd one, while still forming a Zhang-Rice singlet.
toobtainthecorrectpositionofthelowestenergyfeature
at2.5eVforq =0.01˚A−1,andat3.1eVforq =0.7˚A−1. This behaviorcanbe understoodby analyzingthe rel-
evantexpectationvalues in Eq.(5). Forsmallvalues of q
Thus, we effectively use only two free parameters. It is
the frequency term hΨ|D†LD |Ψi can be approximated
found that the value of ∆ dominates the excitation en- µ ν
ergy, which increases with increasing ∆. The dispersion by expanding eiqr ≈ 1 + iqr in Eq. (5). This gives
of the peak depends mainly on tpd with increasing dis- hΨ|F0†,µLF0,ν|Ψi × q(rµ−rν) which is proportional to
persion for increasing hopping parameter. As compared the fluctuation distance, thus favoring far-reaching exci-
to the standard value 1.3 eV,13 t = 1.5 eV is slightly tations. Thispicturechangesforlargevaluesofq,where
pd
enhanced,inagreementwithrecentresultsofbandstruc- stronger oscillations of the phase factor lead to a cance-
ture calculations.14 lationofextendedexcitations. The resultis a transferof
2
spectral weight from delocalized towards more localized TheimportantroleoftheZhang-Ricesingletformation
excitations with increasing q. hasbeenstudiedpreviouslyalsoinaneffectivemodelfor
excitons in the CuO2 plane.17 Like the one-band model,
this effective model neglects inner degrees of freedom of
(a) the Zhang-Rice singlet. If this model is reduced to the
CuO3 chain, q-dependent energies are only possible for
a non-vanishing O on-site Coulomb repulsionU 6=0. In
p
contrast to these results, we find q-dependent energies
for U =0. As described above this effect cannot be ex-
p
plained in a model which neglects the inner structure of
(b) the singlet.
Thus, our results show that both an inclusion of the
O-sites and a complete description of the excitation is
necessary to obtain the full dispersion. The O-sites are
essentialfor the correct description of the different char-
acters of the singlet excitations, which leads to the shift
ofspectralweightfromoneexcitationtoanother. Onthe
(c)
other hand, taking account of the inner degrees of free-
domoftheZhang-Ricesingletleadstothe q-dependence
of the energies.
Theresultsoftheprojectiontechniquedonotcorrectly
describe the experimentally observed width of the peak
for small momentum transfer. A possible explanation
FIG. 3. Hole delocalization properties of different excited
is that not all excitations are included in the projec-
states. Larger(smaller)dotssymbolizealarger(smaller)den-
sity of the hole originally located at the central plaquette. tionspace. Theabovediscussionsuggeststhatthewidth
States(a)and(b)areZhang-Ricesinglet-likeexcitationswith should be due to the presence of additional delocalized
different delocalization properties. Excitation (a) has the excitations. Processeswhichareneglectedinthe present
largestspectralweightforsmallq,whereasexcitation(b)has calculationinvolveless importantmultiple excitations of
dominantspectralweightforlargeq,seealsoFig.2. Part(c) holes beyond their original plaquette.
showsthelocalexcitationat6.4eV,wheretheholesurrounds Finally, although they are not the focus of this paper,
thecentral Cu site. we discuss some high-energy features. The excitation at
6.4 eV in the theoretical spectra is due to a local pro-
The q-dependence of the energies, on the other hand, cess on the plaquette itself. Here, the hole is excited
isaconsequenceoftheinnerstructureofthe Zhang-Rice to the surrounding O sites, without leaving its original
singlet-like excitations. In both excitations (a) and (b) plaquette, see Fig. 3(c). The energy of this structure
a hole hops onto the Cu site of its nearest neighbor pla- doesnotshiftasafunctionofmomentumtransfer. Once
quette,seeFig.3. Duetothe CoulombrepulsionU ,the again, a transfer of spectral weight towards this local-
d
holewhichhadoriginallyoccupiedthis Cusite ispushed ized excitation with increasing values of q is observed.
away onto the surrounding O sites. Depending on the The plaquette excitation has a highly local character.
direction of this delocalization, this process leads to a Therefore, its spectral weight increases as a function of
q-dependence of the excitation energy. q compared to the more delocalized Zhang-Rice singlet
Next, we wantto stress that the claimin3 for the one- excitations. For small q the spectral weight of the pla-
bandHubbardmodelthatonlytheinclusionofthenext- quette peak is about 6 times smaller than that of the
neighbor repulsion leads to the possibility of the forma- Zhang-Rice peak. As q increases,this ratio increasesto
tion of an excitonic state is not consistent with our re- about one half. One should note that the experimental
sults. In the one-band model such a repulsion leads to a spectra show no obvious features above 6 eV. However,
binding energy of empty and doubly occupied sites due since many different orbitals may contribute in this en-
to the reduction of neighboring interactions. This bind- ergyrange,wecannotexpectarealisticdescriptionusing
ingenergyisproportionaltoV. However,ascanbe seen amodelthatcontainsonlyCu3dandO2porbitals. This
from exact diagonalization calculations in the one-band applies also to the experimentalstructure around4.5 eV
model,16 the intersite repulsion mainly leads to an en- for small momentum transfer, which is not described by
ergy shift of the EELS spectra. Thus, the parameter V thepresentmodel. We assumethatthis featureisdueto
in the one-bandmodel servesonly to adjust the energet- excitations which involveSr orbitals,as has been argued
ical position of the spectra, and is not necessary in more before.3
realisticmodels. Inthe multi-bandmodel,the formation In comparison with earlier works on Cu2p3/2 X-ray
of an exciton is only driven by the energetically favored photoemission spectroscopy using the same theoretical
18
formation of a Zhang-Rice singlet, and no further inclu- approach, we find that the characterof the excitations
sion of next-neighbor repulsion is necessary. in both experiments is very similar. Zhang-Rice singlet
3
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